- No tags were found...

Algebraic development of many-body perturbation theory in ...

INSTITUTE OF THEORETICAL PHYSICS AND ASTRONOMYOF VILNIUS UNIVERSITYRytis Juršėnas**Algebraic** **development** **of** **many**-**body** **perturbation** **theory****in** theoretical atomic spectroscopyDoctoral DissertationPhysical Sciences, Physics (02P)Mathematical and general theoretical physics, classical mechanics, quantum mechanics,relativity, gravitation, statistical physics, thermodynamics (P190)Vilnius, 2010

The thesis was prepared at Institute **of** Theoretical Physics and Astronomy **of** Vilnius University**in** 2006-2010.Scientific supervisor:Dr. G**in**taras Merkelis (Institute **of** Theoretical Physics and Astronomy **of** Vilnius University,02P: Physical Sciences, Physics; P190: mathematical and general theoretical physics, classicalmechanics, quantum mechanics, relativity, gravitation, statistical physics, thermodynamics)

VILNIAUS UNIVERSITETOTEORINĖS FIZIKOS IR ASTRONOMIJOS INSTITUTASRytis JuršėnasAlgebr**in**is daugiadalelės trikdžių teorijos plėtojimasteor**in**ėje atomo spektroskopijojeDaktaro disertacijaFiz**in**iai mokslai, fizika (02P)Matemat**in**ė ir bendroji teor**in**ė fizika, klasik**in**ė mechanika, kvant**in**ė mechanika,reliatyvizmas, gravitacija, statist**in**ė fizika, termod**in**amika (P190)Vilnius, 2010

Disertacija rengta Vilniaus universiteto Teor**in**ės fizikos ir astronomijos **in**stitute 2006-2010metais.Moksl**in**is vadovas:Dr. G**in**taras Merkelis (Vilniaus universiteto Teor**in**ės fizikos ir astronomijos **in**stitutas, 02P:fiz**in**iai mokslai, fizika; P190: matemat**in**ė ir bendroji teor**in**ė fizika, klasik**in**ė mechanika, kvant**in**ėmechanika, reliatyvizmas, gravitacija, statist**in**ė fizika, termod**in**amika)

5Contents1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1 The ma**in** goals **of** present work . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 The ma**in** tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 The scientific novelty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Statements to be defended . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 List **of** publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 List **of** abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Partition**in**g **of** function space and basis transformation properties . . . . . . . . 132.1 The **in**tegrals **of** motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Coord**in**ate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 Spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 RCGC technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 System **of** variable particle number . . . . . . . . . . . . . . . . . . . . . . . . 222.3.1 Orthogonal subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 Effective operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4 Conclud**in**g remarks and discussion . . . . . . . . . . . . . . . . . . . . . . . 313 Irreducible tensor operator techniques **in** atomic spectroscopy . . . . . . . . . . 333.1 Restriction **of** tensor space **of** complex antisymmetric tensors . . . . . . . . . . 333.1.1 Classification **of** angular reduction schemes . . . . . . . . . . . . . . . 333.1.2 Correspondence **of** reduction schemes . . . . . . . . . . . . . . . . . . 363.1.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1.4 Equivalent permutations . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.1 A two-particle operator . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.2 A three-particle operator . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Summary and conclud**in**g remarks . . . . . . . . . . . . . . . . . . . . . . . . 574 Applications to the third-order MBPT . . . . . . . . . . . . . . . . . . . . . . 594.1 The treatment **of** terms **of** the second-order wave operator . . . . . . . . . . . . 604.2 The treatment **of** terms **of** the third-order effective Hamiltonian . . . . . . . . . 644.3 Conclud**in**g remarks and discussion . . . . . . . . . . . . . . . . . . . . . . . 675 Prime results and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 70A Basis coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72BThe classification **of** three-particle operators act**in**g on l = 2, 3, 4, 5, 6 electronshells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.1 2–shell case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.2 3–shell case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.3 4–shell case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80B.4 5–shell case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81B.5 6–shell case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82B.6 Identification **of** operators associated to classes . . . . . . . . . . . . . . . . . 83C SU(2)–**in**variant part **of** the second-order wave operator . . . . . . . . . . . . . 85C.1 One-**body** part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85C.2 Two-**body** part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85C.3 Three-**body** part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86C.4 Four-**body** part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88D Symbolic computations with NCoperators . . . . . . . . . . . . . . . . . . . . 89D.1 SQR and AMT blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90D.2 RSPT block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92D.3 UEP block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6List **of** Tables1 The values for parameters characteristic to the SU(2)–irreducible matrix representationparametrised by the coord**in**ates **of** S 2 × S 2 . . . . . . . . . . . . . . 152 Numerical values **of** reduced matrix element **of** SO(3)–irreducible tensor operatorS k for several **in**tegers l, l ′ . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Reduction schemes **of** Ô2−5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Reduction schemes **of** Ô6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 The schemes associated to A 0 , A 1 , A 2 . . . . . . . . . . . . . . . . . . . . . . 356 A connection between irreducible tensor operators on H λ and H q . . . . . . . . 467 The parameters for three-particle matrix elements: l = 2 . . . . . . . . . . . . 568 The parameters for three-particle matrix elements: l = 3 . . . . . . . . . . . . 569 The parameters for three-particle matrix elements: l = 4 . . . . . . . . . . . . 5610 The parameters for three-particle matrix elements: l = 5 . . . . . . . . . . . . 5611 Possible values **of** m, n, ξ necessary to build the (m + n − ξ)–**body** terms **of** Ĥ (3) 5912 The multipliers for one-particle effective matrix elements **of** ̂Ω (2) . . . . . . . . 6113 The multipliers for two-particle effective matrix elements **of** ̂Ω (2) . . . . . . . . 6114 The multipliers for three- and four-particle effective matrix elements **of** ̂Ω (2) . . 6115 The expansion coefficients for one-**body** terms **of** the third-order contribution tothe effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6516 The expansion coefficients for two-**body** terms **of** the third-order contributionto the effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 6517 The expansion coefficients for two-**body** terms **of** the third-order contributionto the effective Hamiltonian (cont**in**ued) . . . . . . . . . . . . . . . . . . . . . 6618 The amount **of** one-**body** terms **of** Ĥ (3) . . . . . . . . . . . . . . . . . . . . . 6719 The amount **of** two-**body** terms **of** Ĥ (3) . . . . . . . . . . . . . . . . . . . . . 6820 The class X 2 (0, 0): d 2 = 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7821 The class X 2 (+1, −1): d 2 = 15 . . . . . . . . . . . . . . . . . . . . . . . . . 7822 The class X 2 (+2, −2): d 2 = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 7823 The class X 3 (0, 0, 0): d 3 = 21 . . . . . . . . . . . . . . . . . . . . . . . . . . 7824 The class X 3 (+2, −1, −1): d 3 = 24 . . . . . . . . . . . . . . . . . . . . . . . 7925 The class X 3 (+3, −2, −1): d 3 = 3 . . . . . . . . . . . . . . . . . . . . . . . . 7926 The class X 3 (+1, −1, 0): d 3 = 45 . . . . . . . . . . . . . . . . . . . . . . . . 7927 The class X 3 (+2, −2, 0): d 3 = 9 . . . . . . . . . . . . . . . . . . . . . . . . . 7928 The class X 4 (+1, +1, −1, −1): d 4 = 72 . . . . . . . . . . . . . . . . . . . . . 8029 The class X 4 (+2, −2, +1, −1): d 4 = 9 . . . . . . . . . . . . . . . . . . . . . 8030 The class X 4 (+3, −1, −1, −1): d 4 = 6 . . . . . . . . . . . . . . . . . . . . . 8031 The class X 4 (+1, −1, 0, 0): d 4 = 36 . . . . . . . . . . . . . . . . . . . . . . . 8032 The class X 4 (+2, −1, −1, 0): d 4 = 18 . . . . . . . . . . . . . . . . . . . . . . 8133 The class X 5 (+2, +1, −1, −1, −1): d 5 = 18 . . . . . . . . . . . . . . . . . . 8134 The class X 5 (+1, +1, −1, −1, 0): d 5 = 36 . . . . . . . . . . . . . . . . . . . 8235 The class X 6 (+1, +1, +1, −1, −1, −1): d 6 = 36 . . . . . . . . . . . . . . . . 8236 The classes for 3–shell case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8337 The classes for 4–shell case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8338 The classes for 5–shell case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8439 The classes for 6–shell case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8440 Phase factors Z α ′ β ′ ¯µ ′¯ν ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89List **of** Figures1 A computation **of** recoupl**in**g coefficient ( j 1 j 2 (j 12 )j 3 j ∣ ∣j 2 j 3 (j 23 )j 1 j ) with NCoperators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902 The usage **of** Def**in**ition[] **in** NCoperators . . . . . . . . . . . . . . . . . 913 Manipulations with the antisymmetric Fock space operators **in** NCoperators . . 914 Wigner–Eckart theorem **in** NCoperators . . . . . . . . . . . . . . . . . . . . . 91

75 The generation **of** ĥ(3) 11;1 terms with NCoperators . . . . . . . . . . . . . . . . . 926 The generation **of** ω (2)11;1 terms: part 1 . . . . . . . . . . . . . . . . . . . . . . . 947 The generation **of** ω (2)11;1 terms: part 2 . . . . . . . . . . . . . . . . . . . . . . . 948 Example **of** an application **of** the UEP block . . . . . . . . . . . . . . . . . . . 959 The diagrammatic visualisation **of** :{ ̂P } 2 : terms . . . . . . . . . . . . . 96̂R̂V 2̂Ω(1) 1

1 Introduction 81 IntroductionIn atomic spectroscopy, a powerful mathematical tool for theoretical study **of** electron correlationeffects as well as the atomic parity violation, ultra-cold collisions and much greater**of** **many**-electron problems, is based on irreducible tensor formalisms and, consequently, onsymmetry pr**in**ciples, simplify**in**g various expressions, thus considerably reduc**in**g amount **of**further on theoretical calculations **of** desirable physical quantities. The mathematical formulation**of** irreducible tensor operators ga**in**ed appropriate grade **in** modern physics due to welldevelopedtheories **of** group representations. Namely, **in** the manner **of** representations, thegroup operations are accomplished on vector spaces over real, complex, etc. fields, def**in****in**gvector-valued functions and their behaviour under various transformations. There**in** the fundamentalconnection between measurable physical quantities and abstract operators on Hilbertspace—particularly those transform**in**g under irreducible representations—is realised throughthe bil**in**ear functionals which ascerta**in**, **in** general, the mapp**in**g from the Kronecker product **of**vector spaces **in**to some given unitary or Euclidean vector space. In physical applications, thesebil**in**ear forms, usually called the matrix elements on given basis, are selected to be self-adjo**in**t.The physical processes and various spectroscopic magnitudes, such as, for example, electrontransition probability, energy width **of** level or lifetime **of** state **of** level, electron **in**teractionsand **many** more, are uniformly estimated by the correspond**in**g operator matrix elementson the basis **of** eigenfunctions **of** the Hamiltonian which characterises the studied process. Tothis day, the most widely used method to construct the basis functions is based on the atomicshell model suggested by N. Bohr [1] and later adapted to the nuclear shell model that was firstproposed by M. G. Mayer and J. H. D. Jensen [2–4]. In this model, electron states **in** atom arecharacterised by the nonnegative **in**tegers which **in** their turn form the set **of** quantum numbersdescrib**in**g the Hamiltonian **of** a local or stationary system. For the most part, the nonnegative**in**tegers that describe the dynamics **of** such system simply mark **of**f the irreducible group representationsif the group operators commute with a Hamiltonian. Particularly, **in** the atomiccentral-field approximation, the Hamiltonian is **in**variant under reflection and rotation **in** R 3 ,thus the eigenstate **of** such Hamiltonian is characterised by the parity Π **of** configuration andby the SO(3)–irreducible representation L, also known as the angular momentum. With**in** theframework **of** the last approximation, the atomic Hamiltonian may be constructed by mak**in**git the SU(2)–**in**variant, s**in**ce SU(2) is a double cover**in**g group **of** SO(3). Then the eigenstateis characterised additionally by the SU(2)–irreducible representation J, also known as the totalangular momentum. The **theory** **of** angular momentum was first **of**fered by E. U. Condon, G.H. Shortley [5] and later much more extended by E. Wigner, G. Racah [6–9] and A. P. Jucyset. al. [10–12]. Although the methods to reduce the Kronecker products **of** the irreduciblerepresentations which label, particularly, the irreducible tensor operators, are extensively developedby **many** researchers until now [13–17], still there are a lot **of** predicaments to choosea convenient reduction scheme which ought to dim**in**ish the time resources for a large scale**of** theoretical calculations. The problems to prepare the effective techniques **of** reduction aredom**in**ant especially **in** the studies **of** open-shell atoms, when deal**in**g with the physical as wellas the effective none scalar irreducible tensor operators and their matrix elements on the basis**of** complex configuration functions.A total eigenfunction **of** atomic stationary Hamiltonian is built up beyond the central-fieldapproach and it is, by the orig**in**, the major object **of** the **many**-**body** theories. Unfortunately, theexact eigenstates can not be found, thus the f**in**al results that characterise the dynamics **of** suchcomplex system are not yet possible. From the mathematical po**in**t **of** view, the eigenstates **of**Hamiltonian form some l**in**ear space. If the spectrum **of** Hamiltonian is described by discretelevels, then the eigenstates characterised by the nonnegative **in**tegers form a separable Hilbertspace; otherwise, the l**in**ear space is, **in** general, non-separable. Through ignorance **of** the structure**of** exact eigenstates, there are formed the l**in**ear comb**in**ations **of** the basis functions that areusually far from the exact picture. The basis functions are selected to be the eigenstates **of** thecentral-field Hamiltonian. This yields various versions **of** the multi-configuration Hartree–Fock(MCHF) approach based on the variation **of** the energy functional with respect to s**in**gle-electronwave functions. In this approach, a huge number **of** admixed configurations together with highorder **of** energy matrices need to be taken **in**to account [18–20]. By the mathematical formu-

1 Introduction 9lation, the multi-configuration function is considered as the superposition **of** the configurationstate functions (CSF), thus the present approximation is conf**in**ed to be **in** operation on a s**in**gle**many**-particle Hilbert space. Contrary to this model, there exists another extremely differentmethod to build the exact eigenfunction **of** **many**-particle system. Of special **in**terest is theatomic **many**-**body** **perturbation** **theory** (MBPT) that accounts for a variable number **of** **many**particleHilbert spaces simultaneously. Therefore, the latter approximation is **in** operation on aFock space.The MBPT—due to its versatility—is widespread until nowadays not only **in** atomic physics.Namely, the ma**in** idea **of** present approximation has been appropriated, as it became usual, fromthe **theory** **of** nucleus followed by the works **of** K. A. Brueckner and J. Goldstone [21–23], andafterwards adapted to atomic physics and quantum chemistry [24–33]. In most general case, theeigenfunction **of** **many**-particle Hamiltonian is generated by the exponential ansatz which actson the unit vector **of** the entire Hilbert space, also known as the genu**in**e vacuum. In **perturbation****theory** (PT), the exponential ansatz is frequently def**in**ed to obey the form **of** the so-called waveoperator act**in**g on a reference function or else the physical vacuum. Such formulation is followedby the Fock space **theory**, studied **in** detail by W. Kutzelnigg [31] and by the particle-holepicture, recently exploited by **many** authors **in** modern MBPT [34–38]. For the closed-shellatoms, the reference function simply denotes a s**in**gle Slater determ**in**ant which represents aneigenfunction **of** the central-field Hamiltonian. Attempts to construct the reference function forthe open-shell atoms lead to a much more complicated task. By traditional procedure, the entireHilbert space is partitioned **in**to two subspaces, where the first one is spanned by the multiconfigurationstate functions associated with the eigenvalues **of** the central-field Hamiltonian,and the second one is formed from the functions which are absent **in** the first subspace. Thereason for such partition**in**g is that for open-shell atoms the energy levels are degenerate andthe full set **of** reference functions is not always determ**in**ed **in**itially. Therefore the number **of**selected functions denotes the dimension **of** a subspace usually called the model space [39].A universal algorithm to form the model space **in** open-shell MBPT is yet impossible and theproblem under consideration still **in**sists on further studies.A significant advantage **of** the MBPT is that the exact eigenvalues **of** atomic Hamiltonianare obta**in**ed even without know**in**g the exact eigenfunctions. The number **of** solutions for energiesis made dependent on a dimension **of** model space. To solve this task, the eigenvalueequation is addressed to f**in**d**in**g the operator which acts on chosen model space. The form **of**the latter operator, called the effective Hamiltonian [34], is closely related to the form **of** waveoperator. Usually, there are dist**in**guished Hilbert-space and Fock-space approaches, **in** orderto specify this operator. In the Hilbert-space approach, the model space is chosen to **in**cludea fixed number **of** electrons. Then the wave operator is determ**in**ed by eigenvalue equation **of**atomic Hamiltonian for a s**in**gle **many**-particle Hilbert space only. In this sense, it is similarto the multi-configuration approach. In the Fock-space approach, the wave operator is representedequally on all **many**-particle Hilbert spaces which are formed by the functions with avariable number **of** valence electrons **of** open-shell atom. By the mathematical formulation, theFock-space approximation is based on the occupation-number representation. Consequently,this treatment suggests the possibility **of** simultaneously tak**in**g **in**to account for the effects thatare conditioned by the variable number **of** particles. Start**in**g from this po**in**t **of** view, severalvariations to construct the effective **in**teraction operator on a model space are separated result**in**g**in** different versions **of** the PT. Nevertheless, a general idea embodied **in** all **perturbation** theoriesstates that the Hamiltonian **of** **many**-particle system splits up **in**to the unperturbed Hamiltonianand the **perturbation** which characterises the **in**homogeneity **of** system. In atomic physics, theunperturbed Hamiltonian stands for a usual central-field Hamiltonian. The major difficultiesarise due to the **perturbation**. In various versions **of** PT, the techniques to account for the **perturbation**differ. The Rayleigh–Schröd**in**ger (RS) and coupled-cluster (CC) theories comb**in**edwith the second quantisation representation (SQR) are the most common approaches used **in**theoretical atomic physics. The **perturbation** series is built by us**in**g the Wick’s theorem [40]which makes it possible to evaluate the products **of** the Fock space operators. The number **of**these products grows rapidly as the order **of** **perturbation** **in**creases. For this reason, the RSPTis applied to a f**in**ite-order **perturbation**, when construct**in**g the **many**-electron wave function**of** a fixed order m, start**in**g from m = 0 step by step [36, 37, 41]. In CC **theory**, the **in**itially

1 Introduction 10given exponential ansatz is represented, **in** general, as an **in**f**in**ite sum **of** Taylor series. The totaleigenfunction is then expressed by the sum **of** all-order n–particle (n = 0, 1, 2, . . .) functions,denot**in**g zero-, s**in**gle-, double-, etc. excitations [35,42]. However, **in** practical applications, thesum **of** terms is also f**in**ite. To this day, the progressive attempts to evaluate the terms **of** atomicPT by us**in**g the computer algebra systems have been reported by a number **of** authors [43–45].One more problem ord**in**ary to open-shell MBPT is to handle the generated terms **of** PT. Inaddition to a large number **of** terms from given scale, each term has to be separately workedup for a convenient usage, **in** order to calculate the energy corrections efficiently. This is doneby us**in**g the angular momentum **theory** (AMT) comb**in**ed with the tensor formalism. In atomicspectroscopy, the theoretical foundation **of** tensor operator technique has been built by Judd et.al. [46–49] and later extended by Rudzikas et. al. [50–55]. In the occupation-number representation,the terms **of** PT are reduced to the effective n–particle operators. In most cases, authorsaccount for the zeroth, s**in**gle and double (n = 0, 1, 2) particle-hole excitations. First **of** all, thisis due to their biggest part **of** contribution to the correlation energy. Secondly, it is determ**in**edby the complexity **of** structure **of** the irreducible tensor operators that act on more than fouropen shells (n > 2). For example, **in** his study **of** the wave functions for atomic beryllium [56],Bunge calculated that the contribution **of** double excitations to the correlation energy for theBe atom represented about 95%, while the triple excitations made approximately 1% **of** thecontribution. On the other hand, **in** modern physics, the high-level accuracy measured bellow0.1% is desirable especially **in** the studies **of** atomic parity violation [57] or when account**in**g forradiative corrections **of** hyperf**in**e splitt**in**gs **in** alkali metals or highly charged ions [58]. Suchlevel **of** accuracy is obta**in**ed when the triple excitations are **in**volved **in** the series **of** the PT, asdemonstrated by Porsev et. al. [42]. This **in**dicates that the mathematical techniques applied tothe reduction **of** the tensor products **of** the Fock space operators still are urgent and **in**evitable.1.1 The ma**in** goals **of** present work1. To work out the versatile disposition methods and forms pert**in**ent to the tensor products **of** theirreducible tensor operators which represent either physical or effective **in**teractions considered**in** the atomic open-shell **many**-**body** **perturbation** **theory**.2. To create a symbolic computer algebra package that handles complex algebraic manipulationsused **in** modern theoretical atomic spectroscopy.3. Mak**in**g use **of** the symbolic computer algebra and mathematical techniques, to explore theterms **of** atomic open-shell **many**-**body** **perturbation** **theory** **in** a Fock-space approach, pay**in**gspecial attention to the construction **of** a model space and the **development** **of** angular reduction**of** terms that fit a fixed-order **perturbation**. Meanwhile, to elaborate the reduction schemesuitable for an arbitrary order **perturbation** or a coupled-cluster expansion.1.2 The ma**in** tasks1. To f**in**d regularities responsible for the behaviour **of** operators on various subspaces **of** theentire Fock space. To study the properties and consequential causes made dependent on thecondition that a set **of** eigenvalues **of** Hamiltonian on the **in**f**in**ite-dimensional Hilbert spaceconta**in**s a subset **of** eigenvalues **of** Hamiltonian projected onto the f**in**ite-dimensional subspace.2. To classify the totally antisymmetric tensors determ**in**ed by the Fock space operator str**in**g**of** any length. To determ**in**e the transfer attributes **of** irreducible tensor operators associated todist**in**ct angular reduction schemes.3. To generate the terms **of** the second-order wave operator and the third-order effective Hamiltonianon the constructed f**in**ite-dimensional subspace by us**in**g a produced symbolic computeralgebra package. To develop the approach **of** **many**-particle effective matrix elements so thatthe projection-**in**dependent parts could be easy to vary rema**in****in**g steady the tensor structure **of**expansion terms.

1 Introduction 111.3 The scientific novelty1. Opposed to the usual Slater-type orbitals, the SU(2)–irreducible matrix representations havebeen demonstrated to be a convenient basis for the calculation **of** matrix elements **of** atomicquantities. A prom**in**ent part **of** such type **of** computations is **in** debt to the newly found SO(3)–irreducible tensor operators.2. Bear**in**g **in** m**in**d that the theoretical **in**terpretation concerned with the model space **in** openshell**many**-**body** **perturbation** **theory** is poorly def**in**ed so far, attempts to give rise to moreclarity have been **in**itiated. The key result which causes to dim**in**ish the number **of** expansionterms is that only a fixed number **of** types **of** the Hilbert space operators with respect to thes**in**gle-electron states attach the non-zero effective operators on the constructed model space.3. The algorithm to classify the operators observed **in** the applications **of** effective operatorapproach to the atomic open-shell **many**-**body** **perturbation** **theory** has been produced. Theclassification **of** three-particle effective operators that act on 2, 3, 4, 5, 6 electron shells **of** atomhas been performed expressly. As a result, the calculation **of** matrix elements **of** three-particleoperators associated to any angular reduction scheme becomes easily performed.4. The angular reduction **of** terms **of** the third-order effective Hamiltonian on the constructedmodel space has been performed **in** extremely different way than it has been done so far. Tocompare with a usual diagrammatic formulation **of** atomic **perturbation** **theory**, the pr**in**cipaladvantages **of** such technique are: (i) the ability to vary the amplitudes **of** electron excitationsuitable for special cases **of** **in**terest – the tensor structure is free from the change; (ii) the abilityto enclose a number **of** Goldstone diagrams by the sole tensor structure. As a result, the problem**of** evaluation **of** each separate diagram is elim**in**ated.1.4 Statements to be defended1. The **in**tegrals over S 2 **of** the SO(3)–irreducible matrix representation parametrised by thecoord**in**ates **of** S 2 × S 2 constitute a set **of** components **of** SO(3)–irreducible tensor operator.2. There exists a f**in**ite-dimensional subspace **of** **in**f**in**ite-dimensional **many**-electron Hilbertspace such that the non-zero terms **of** effective atomic Hamiltonian on the subspace are generatedby a maximum **of** eight types **of** the n–**body** parts **of** wave operator with respect to thes**in**gle-electron states for all nonnegative **in**tegers n.3. The method developed by mak**in**g use **of** the S l –irreducible representations, the tuples andthe commutative diagrams **of** maps associat**in**g dist**in**ct angular reduction schemes makes itpossible to classify the angular reduction schemes **of** antisymmetric tensors **of** any length **in**an easy to use form that stipulates an efficiency **of** calculation **of** matrix elements **of** complexirreducible tensor operators.4. The restriction **of** **many**-electron Hilbert space the wave operator acts on to its SU(2)–irreducible subspaces guarantees the ability to enclose a number **of** Goldstone diagrams bythe sole tensor structure so that the amplitudes **of** electron excitation are easy to vary depend**in**gon the specific cases **of** **in**terest, but the tensor structure **of** expansion terms is fixed.

1 Introduction 121.5 List **of** publications1. R. Juršėnas and G. Merkelis, Coupl**in**g schemes for two-particle operator used **in** atomiccalculations, Lithuanian J. Phys. 47, no. 3, 255 (2007)2. R. Juršėnas, G. Merkelis, Coupled tensorial form for atomic relativistic two-particle operatorgiven **in** second quantization representation, Cent. Eur. J. Phys. 8, no. 3, 480(2010)3. R. Juršėnas and G. Merkelis, Coupled tensorial forms **of** the second-order effective Hamiltonianfor open-subshell atoms **in** jj-coupl**in**g, At. Data Nucl. Data Tables (2010),doi:10.1016/j.adt.2010.08.0014. R. Juršėnas and G. Merkelis, Application **of** symbolic programm**in**g for atomic **many**-**body****theory**, Materials Physics and Mechanics 9, no. 1, 42 (2010)5. R. Juršėnas, G. Merkelis, The transformation **of** irreducible tensor operators under sphericalfunctions, Int. J. Theor. Phys. 49, no. 9, 2230 (2010)6. R. Juršėnas, G. Merkelis, Irreducible tensor form **of** three-particle operator for open-shellatoms, Cent. Eur. J. Phys. (2010), doi: 10.2478/s11534-010-0082-07. R. Juršėnas and G. Merkelis, Development **of** algebraic techniques for the atomic openshellMBPT (3), to appear **in** J. Math. Phys. (2010)1.6 List **of** abstracts1. R. Juršėnas, G. Merkelis, Coupl**in**g schemes for two-particle operator used **in** atomiccalculations, 37th Lithuanian National Physics Conference, Vilnius, 2007, Abstracts, p.2192. R. Juršėnas, Coupled tensorial forms **of** atomic two-particle operator, 40th EGAS Conference,Graz, 2008, Abstracts, p. 453. R. Juršėnas, G. Merkelis, Coupled tensorial forms **of** the second-order effective Hamiltonianfor open-subshell atoms **in** jj-coupl**in**g, 38th Lithuanian National Physics Conference,Vilnius, 2009, p. 2294. R. Juršėnas, G. Merkelis, Symbolic programm**in**g applications for atomic **many**-**body** **theory**,13th International Workshop on New Approaches to High Tech: Nano Design, Technology,Computer Simulations, Vilnius, 2009, Abstracts, p. 225. R. Juršėnas, G. Merkelis, The MBPT study **of** electron correlation effects **in** open-shellatoms us**in**g symbolic programm**in**g language Mathematica, 41st EGAS Conference, Gdansk,2009, Abstracts, p. 1026. R. Juršėnas and G. Merkelis, **Algebraic** exploration **of** the third-order MBPT, Conferenceon Computational Physics, Trondheim, 2010, Abstracts, p. 2137. R. Juršėnas, G. Merkelis, The transformation **of** irreducible tensor operators under thespherical functions, ECAMP10, Salamanca, 2010, Abstracts, p. 878. R. Juršėnas, G. Merkelis, The generation and analysis **of** expansion terms **in** the atomicstationary **perturbation** **theory**, ICAMDATA 7, Vilnius, 2010, Abstracts, p. 86

2 Partition**in**g **of** function space and basis transformation properties 132 Partition**in**g **of** function space and basis transformation propertiesTwo mathematical notions exploited **in** atomic physics are discussed: the first and second quantisationrepresentations. In the first representation, the basis transformation properties—fixedto a convenient choice—are developed. In the second representation, **many**-electron systemswith variable particle number are studied—concentrat**in**g on partition**in**g techniques **of** functionspace—to improve the efficiency **of** effective operator approach.The key results are the composed SO(3)–irreducible tensor operators, the technique—basedon coord**in**ate transformations—to calculate the **in**tegrals **of** **many**-electron angular parts, theFock space formulation **of** the generalised Bloch equation, the theorem that determ**in**es nonzeroeffective operators on the bounded subspace **of** **in**f**in**ite-dimensional **many**-electron Hilbertspace.This section is organised as follows. In Sec. 2.1, the widespread methods to construct atotal wave function **of** atomic **many**-**body** system are briefly presented. Sec. 2.2 studies theSU(2)–irreducible matrix representations and their parametrisations. The **in**spiration for theparametrisation **of** matrix representation **in** a specified form came from the properties characteristicto irreducible tensor operators, usually studied **in** theoretical atomic spectroscopy. Sec.2.2.2 demonstrates the application **of** method based on the properties **of** founded new set **of**irreducible tensor operators. In Sec. 2.3, the second quantised formalism applied to the atomicsystems is developed. The advantages **of** perturbative methods to compare with the variationalapproach are revealed. The effective operator approach, as a direct consequence **of** the so-calledpartition**in**g technique (Sec. 2.3.1), is developed **in** Sec. 2.3.2.2.1 The **in**tegrals **of** motionThe quantum mechanical **in**terpretation **of** atomic **many**-**body** system is found to be closelyrelated to the construction **of** Hamiltonian H. In a time-**in**dependent approach, this Hamiltoniancorresponds to the total energy E **of** system. To f**in**d E, the eigenvalue equation **of** H must besolved. The eigenfunction Ψ **of** H depends on the symmetries that are hidden **in** the **many**-**body**system Hamiltonian. The group-theoretic formulation **of** the problem is to f**in**d the group Gsuch that the operators ĝ ∈ G commute with H. That is, if [H, ĝ] = 0, then Ψ is characterisedby the irreducible representations (or «irrep» for short) **of** G. A well-known example is theBohr or central-field Hamiltonian ˜H 0 = T + U C **of** the N–electron atom, where T representsthe k**in**etic energy **of** electrons and U C denotes the Coulomb (electron-nucleus) potential. ThisHamiltonian is **in**variant under the rotation group G = SO(3) with ĝ ∈ {̂L 1 , ̂L 2 , ̂L 3 } be**in**g the**in**f**in**itesimal operator or else the angular momentum operator. Consequently, the eigenfunctionsΨ are characterised by the irreducible representations L = 0, 1, . . . **of** SO(3) and by the **in**dicesM = −L, −L + 1, . . . , L − 1, L that mark **of**f the eigenstates **of** ̂L i (i = 1, 2, 3). In this case,we write Ψ ≡ Ψ(ΓLM|x 1 , x 2 , . . . , x N ) ≡ Ψ(ΓLM), where x ξ ≡ r ξ̂x ξ denotes the radial r ξand spherical ̂x ξ ≡ θ ξ ϕ ξ coord**in**ates **of** the ξth electron. The quantity Γ denotes the rest **of**numbers that append the other, if any, symmetry properties **in**volved **in** ˜H 0 . Particularly, theBohr Hamiltonian ˜H 0 is also **in**variant under the reflection characterised by the parity Π. Thus˜H 0 implicates the symmetry group O(3).The **in**f**in**itesimal operators ̂L i form the B 1 Lie algebra (**in** Cartan’s classification) which isisomorphic to the A 1 algebra formed by the generators Ĵi. Besides, it is known [59, Sec. 5.5.2,p. 99] that the angular momentum operator ̂L i is the sum **of** two **in**dependent A 1 Lie algebrasformed by Ĵ ± i . This implies that ˜H 0 is also **in**variant under SU(2) operations and thus Ψ maybe characterised by the SU(2)–irreducible representation J = 1/2, 3/2, . . . which particularlycharacterises the sp**in**- 1 /2 particles (electrons).Regardless **of** well-def**in**ed symmetries appropriated by ˜H 0 , the central-field approximationdoes not account for the **in**teractions between electrons. In order to do so, the total HamiltonianH **of** N–electron system is written as followsH = H 0 + V, H 0 = ˜H 0 + U. (2.1)The Hamiltonian H 0 perta**in**s to the symmetry properties **of** ˜H 0 s**in**ce U represents the central-

2 Partition**in**g **of** function space and basis transformation properties 14field potential which has the mean**in**g **of** the average Coulomb **in**teraction **of** electron with theother electrons **in** atom [34, Sec. 5.4, p. 114]. All electron **in**teractions along with externalfields, if such exist, are drawn **in** the **perturbation** V . It seems to be obvious that the structure **of**H is much more complicated than the structure **of** H 0 .Hav**in**g def**in**ed the Hamiltonian H and assum**in**g that the eigenfunctions {Ψ i } ∞ i=1 ≡ Xform the **in**f**in**ite-dimensional Hilbert space H with the scalar product 〈, ·, 〉 H : X × X −→ R,the energy levels E i are expressed by 〈Ψ i · HΨ i 〉 H ≡ 〈Ψ i |H|Ψ i 〉. S**in**ce the functions Ψ i areunknown, usually they are expressed by the l**in**ear comb**in**ation **of** some **in**itially known basisΦ i . With such def**in**ition, the relation readsΨ(Γ i Π i Λ i M i ) = ∑ e Γic eΓi Π i Λ iΦ(˜Γ i Π i Λ i M i ), c eΓi Π i Λ i∈ R. (2.2)Here Λ i M i ≡ L i S i M Li M Si or Λ i M i ≡ J i M Ji depend on the symmetry group assum**in**g thatthe functions Φ i represent the eigenfunctions **of** H 0 . The real numbers c eΓi Π i Λ iare found bydiagonaliz**in**g the matrix **of** H on the basis Φ i , where the entries **of** matrix areH Γi Γ ′ i = δ Π i Π ′ i δ Λ i Λ ′ i δ M i M ′ i 〈Γ iΠ i Λ i M i |H|Γ ′ iΠ ′ iΛ ′ iM ′ i〉. (2.3)If, particularly, H is the scalar operator, then the H Γi Γ ′ i do not depend on M i.The solutions Φ i **of** central-field equation, the configuration state functions (CSF), are foundby mak**in**g the antisymmetric products or Slater determ**in**ants [60, p. 1300] **of** s**in**gle-electronfunctions R(n ξ κ ξ |r ξ )φ(λ ξ m ξ |̂x ξ ), characterised by the numbers λ ξ ≡ l ξ 1/2 or λ ξ ≡ j ξ , whereξ = 1, 2, . . . , N, l ξ = 0, 1, . . . and j ξ = 1/2, 3/2, . . .. The quantity κ ξ depends on the symmetrygroup. For G = SO(3), κ ξ ≡ l ξ ; for G = SU(2), κ ξ ≡ l ξ j ξ , where l ξ = 2j ξ ± 1. The functionsφ(λ ξ m ξ |̂x ξ ) are transformed by the irreducible unitary matrix representations **of** G = SU(2).These representations denote the matrix D λ ξ (g) **of** dimension dim Dλ ξ(g) = 2λξ + 1, whereg ∈ G. That is,D λ ξ † (g)φ(λ ξ m ξ |̂x ξ ) = ∑ em ξD λ ξm ξ em ξ(g)φ(λ ξ ˜m ξ |̂x ξ ). (2.4)The explicit form **of** element D λ ξm ξ em ξ(g) depends on the parameters g and basis. The parametrisation**of** D λ ξm ξ em ξ(Ω) by the Euler angles Ω = (Φ, Θ, Ψ) was first carried out by Wigner [61]D λ ξm ξ em ξ(Ω) = exp (m ξ Φ + ˜m ξ Ψ)P λ ξm ξ em ξ(cos Θ). (2.5)The properties **of** P λ ξm ξ em ξ(z) were comprehensively studied by Vilenk**in** [62, p. 121]. Particularly,for arbitrary k ∈ Z + or k ∈ Q + = {r+1/2, r ∈ Z + } and q, q ′ = −k, −k+1, . . . , k−1, k,Pqq k ′(z) can be represented by( 1 − zPqq k ′(z) a(k, q, q ′ )=(−1)q−q′ 1 + z×m**in** (k−q ′ ,k+q)∑p=max (0,q−q ′ )b p (k, q, q ′ )) q−q ′2( 1 − z1 + z) k( 1 + z2) p, (2.6)a(k, q, q ′ ) def= i √ q′ −q(k + q)!(k − q)!(k + q ′ )!(k − q ′ )!, (2.7)b p (k, q, q ′ ) def=(−1) pp!(p + q ′ − q)!(k + q − p)!(k − q ′ − p)! . (2.8)Eq. (2.4) associated to any f**in**ite unitary group G has a significant mean**in**g **in** representation**theory** as well as **in** theoretical atomic spectroscopy. First **of** all, it allows one to f**in**d aconvenient basis for which the matrix elements **in** Eq. (2.3) appropriate the simplest form [63].

2 Partition**in**g **of** function space and basis transformation properties 15Secondly, for a particular G = SU(2), the Kronecker product **of** irreducible matrix representationsis reduced **in** accordance with the rule D λ ξ (g) × Dλ ζ(g) = ⊕λ D λ (g) which makes itpossible to form the basis **of** the typeΦ(λ ξ λ ζ λm|̂x ξ , ̂x ζ ) = ∑ m ξ m ζφ(λ ξ m ξ |̂x ξ )φ(λ ζ m ζ |̂x ζ )〈λ ξ m ξ λ ζ m ζ |λm〉 (2.9)The coefficient 〈λ ξ m ξ λ ζ m ζ |λm〉 that transforms one basis **in**to another is called the Clebsch–Gordan coefficient (CGC) **of** SU(2), also denoted as [ λ ξ λ ζ]λm ξ m ζ m [10, 12, 64]. The basis constructedby us**in**g Eq. (2.9) is convenient to calculate matrix elements if apply**in**g the Wigner–Eckart theorem. In order to do so, the Hamiltonian H on H is represented by the sum **of**irreducible tensor operators H Λ that act on the subspaces H Λ **of** H. For example, the HamiltonianH 0 is obta**in**ed from H if H is conf**in**ed to operate on H Λ = H 0 which is a scalar spacespanned by the functions Φ i . On the other hand, Bhatia et. al. [65, Eq. (47)] demonstrated thatthe angular part Φ(λm|̂x 1 , ̂x 2 ) **of** two-electron wave function was found to be represented **in**terms **of** Dm λ em(Ω). Conversely, such basis is **in**convenient for the application **of** Wigner–Eckarttheorem. Moreover, to calculate matrix elements **of** irreducible tensor operator H Λ , the **in**tegral**of** type∫∫d̂x 1 d̂x 2 Dm λ em (Ω)HΛ Dm λ′′ em ′(Ω) (2.10)S 2must be calculated. The **in**tegration becomes complicated s**in**ce Ω depends on ̂x 1 , ̂x 2 . To performthe latter **in**tegration, the function Ω(̂x 1 , ̂x 2 ) should be found. Afterwards it has to besubstituted **in** Eq. (2.5).The last simple example suggests that the parametrisation **of** the irreducible matrix representationD λ (g) by the spherical coord**in**ates **of** S 2 × S 2 makes sense. Here and elsewhere S 2denotes a 2–dimensional sphere.2.2 Coord**in**ate transformations2.2.1 Spherical functionsSuppose given a map Ω: S 2 × S 2 −→ SO(3) represented on R 3 , a 3–dimensional vectorspace, by ̂r 2 = D(3, 2)̂r 1 , where ̂r i = r i /r i = (s**in** θ i cos ϕ i s**in** θ i s**in** ϕ i cos θ i ) T . The 3 × 3rotation matrix, D(3, 2) ∈ SO(3), is parametrised by the Euler angles Φ, Θ, Ψ [63, p. 84, Eqs.(7.24)-(7.25)]. In Ref. [66], it was proved that the map Ω is realised on S 2 ⊂ R 3 ifΦ = ϕ 2 + α π 2 , Θ = β(θ 1 − γθ 2 ) + 2πn, Ψ = −ϕ 1 + δ π 2 + 2πn′ (2.11)Tab. 1: The values for parameters characteristic to the SU(2)–irreducible matrix representationparametrised by the coord**in**ates **of** S 2 × S 2The maps α β γ δ n The maps α β γ δ nΩ ± 1 Ω + 1 Ω + 11 + + + − 0 Ω ± 2 Ω + 2 + + − + 0Ω + 12 − + + +Ω − 1 Ω − 11 + − + − Ω − 2 − − − − 1Ω − 12 − − + +with n, n ′ ∈ Z + . The parameters α, β, γ, δ, n are presented **in** Tab. 1. Then the sphericalfunction Dqq k ′(Ω) is parametrised as follows(n, n ′ ; α, β, γ, δ|̂x 1 , ̂x 2 ) k qq ′ =iαq+δq′ (−1) 2(nk+n′ q ′) β q′ −q a(k, q, q ′ )e i(qϕ 2−q ′ ϕ 1 ) {cos [ 1 2 (θ 1 − γθ 2 )]} 2k× ∑ pb p (k, q, q ′ ){tan [ 1 2 (θ 1 − γθ 2 )]} 2p+q′ −q . (2.12)

2 Partition**in**g **of** function space and basis transformation properties 162.2.1 Remark. The exploitation **of** Euler’s formula for the parameter (θ 1 − γθ 2 )/2 **in**dicates thefollow**in**g alternative **of** parametrisation(n, n ′ ; α, β, γ, δ|̂x 1 , ̂x 2 ) k qq ′ = 1 4 k i(α−1)q+(δ+1)q′ (−1) 2(nk+n′ q ′) β q′ −q a(k, q, q ′ )h(q, q ′ )× ∑ rs(−1) r W k rs(q − q ′ ) exp {i[(r + s − k)θ 1 − q ′ ϕ 1 ]}× exp {−i[(r + s − k)γθ 2 − qϕ 2 ]}, (2.13)( )( )q − qWrs(q k − q ′ ) def= [1 + π(q, q ′ ′ 2k − q + q′θ(q − q ′ ))]s r (k − q)!(k + q ′ )!(q − q ′ )![−k + q − k − q′ 1× 6 F (q − 2 q′ 1+ 1) (q − 2 q′ + 2)5q − q ′ 1(q − 2 q′ 1− 2k) (q − 2 q′ − 2k + 1)1(q − 2 q′ 1+ r − 2k) (q − ]2 q′ + r − 2k + 1)1(q − 2 q′ 1− s + 1) (q − ; 1 . (2.14)2 q′ − s + 2)The function h(q, q ′ ) = 1 if q ≠ q ′ , otherwise h(q, q) = 1/2. The quantity θ(q − q ′ ) denotesthe Heaviside step function. The permutation operator π(q, q ′ ) reverses q and q ′ that are on theright hand side **of** π(q, q ′ ).Pro**of**. The Euler’s formula for x ∈ R reads e ix = cos x + i s**in** x. Deduce( ) α α∑ ( )( ) β i α 1β∑( βs**in** α x = (−1) s e i(2s−α)x , cos β x =e2s2 r)i(2r−β)x ,s=0r=0where the b**in**omial formula for (e ix ± e −ix ) y has been used. In this case (see Eq. (2.12)),x = 1 2 (θ 1 − γθ 2 ), α = 2p + q ′ − q, β = 2k − 2p − q ′ + q.These values are substituted **in** Eq. (2.12). The exponents with the parameters p vanish s**in**ceα ∝ +2p and β ∝ −2p. In addition, the summation over p can be proceeded for the construction∑( )( )2p + q(−1) p b p (k, q, q ′ ′ − q 2k − 2p − q ′ + q).srpBy pass**in**g to Eq. (2.8), we get for q ≥ q ′ , the b**in**omial series( )( )q − q′ 2k − q + q′1s r (k − q)!(k + q ′ )!(q − q ′ )![−k + q − k − q′ 1× 6 F (q − 2 q′ 1+ 1) (q − 2 q′ + 2)5q − q ′ 1(q − 2 q′ 1− 2k) (q − 2 q′ − 2k + 1)1(q − 2 q′ 1+ r − 2k) (q − ]2 q′ + r − 2k + 1)1(q − 2 q′ 1− s + 1) (q − ; 1 .2 q′ − s + 2)If q ≤ q ′ , the last expression rema**in**s irrelevant if replac**in**g q with q ′ . Thus for any q, q ′ , it isconvenient to use the Heaviside step functions θ(q − q ′ ) and θ(q ′ − q). This proves Remark2.2.1.If follows from Eqs. (2.12)-(2.13) and Tab. 1 that the four spherical functions on S 2 × S 2serve for the irreducible matrix representation Dqq k ′(Ω). Each **of** the function corresponds toDqq k ′(Ω) **in** dist**in**ct areas **of** S2 . The spherical functions are considered as follows{+ ξqq k ′(̂x L1, ̂x 2 ) :2 (Ω + 11) def= {ϕ 2 ∈ [0, π]; θ 2 ∈ [0, θ 1 ], n ′ = 1, 2}L 2 (Ω − 12) def= {ϕ 2 ∈ [π, 2π]; θ 2 ∈ [θ 1 , π], n ′ = 0, 1} , (2.15a)

2 Partition**in**g **of** function space and basis transformation properties 17− ξ k qq ′(̂x 1, ̂x 2 ) :{L 2 (Ω − 11) def= {ϕ 2 ∈ [0, π]; θ 2 ∈ [θ 1 , π], n ′ = 1, 2}L 2 (Ω + 12) def= {ϕ 2 ∈ [π, 2π]; θ 2 ∈ [0, θ 1 ], n ′ = 0, 1} , (2.15b)+ ζqq k ′(̂x 1, ̂x 2 ) : L 2 (Ω + 2 ) def= {ϕ 2 ∈ [0, 3π/2]; θ 2 ∈ M + θ 1, n ′ = 0, 1}, (2.15c)− ζqq k ′(̂x 1, ̂x 2 ) : L 2 (Ω − 2 ) def= {ϕ 2 ∈ [π/2, 2π]; θ 2 ∈ M − θ 1, n ′ = 1, 2}. (2.15d)The compacts M ± θ 1⊆ [0, π] are def**in**ed by{ { (0, π], θ1 = 0,π, θ1 = 0,M + defθ 1= 0, θ 1 = π, M − defθ 1= [0, π] , θ 1 = π, (2.16)(0, π − θ 1 ], θ 1 ∈ (0, π),[π − θ 1 , π] , θ 1 ∈ (0, π).The spherical functions are related to each other by the phase factors: − ξqq k = + ′ (−1)q−q′ ξ qq ′,− ζqq k = + ′ (−1)2q′ ζqq k ′. If± τqq k ∈ ′ {ηk qq ′, ± ζqq k ′}, ηk qq ∈ ′ {+ ξqq k ′, − ξqq k ′}, (2.17)then the functions ± τqq k satisfy′∑+ τqq k − τ k ′ −q−q = δ ′′ q ′ q ′′, ±τqq k = (−1) q−q′ ± τ k ′ −q−q ′. (2.18)qThe products **of** spherical functions are reducible by us**in**g the rule ± τ k 1× ± τ k 2= ⊕ k ± τ k ,which is obvious s**in**ce the ± τ k (̂x 1 , ̂x 2 ) represent the D k (Ω) **in** different areas L 2 (Ω) ⊂ S 2 .Example. Assume that ̂x 1 = ( π /6, π /4) and ̂x 2 = ( π /3, π). Possible rotations are realised on S 2by the angles Ω − 11 = ( 3π /2, π /6, 5π /4) and Ω + 2 = ( 3π /2, π /2, π /4). In accordance with Eq. (2.15), fork = 5 /2, q = − 1 /2, q ′ = 3 /2, the functions are− ξ 5 /2−1/2 3/2 (π /6, π /4, π /3, π) = D 5 /2−1/2 3/2 (3π /2, π /6, 5π /4) = 1 /32 (−1) 1 /8(13 − 3 √ 3),+ ζ 5 /2−1/2 3/2 (π /6, π /4, π /3, π) = D 5 /2−1/2 3/2 (3π /2, π /2, π /4) = (−1) 5 /8 1/4.Obta**in**ed spherical functions ± τ k qq ′(̂x 1, ̂x 2 ) are suitable for their direct realisation throughEq. (2.4) which also makes it possible to carry out the **in**tegration **in** Eq. (2.10) easily enough.2.2.2 RCGC techniqueIt is natural to make use **of** the spherical functions ± τ λ µν(̂x 1 , ̂x 2 ) **in** selection **of** a convenientbasis, as described **in** Eq. (2.4). The argument becomes more motivated recall**in**g that theirreducible tensor operators T λ —be**in**g **of** special **in**terest **in** atomic physics—also transformunder the irreducible matrix representations D λ (Ω). In general, the 2λ + 1 components T λ µ **of**T λ on H λ transform under the unitary matrix representation D λ (g) as follows [64, p. 70, Eq.(3.57)]T λ†ν T λ µ T λ ν = ∑ ρD λ µρ(g)T λ ρ . (2.19)It is assumed that each **in**variant subspace H λ is spanned by the orthonormal basis φ(λµ|̂x µ ),for which Eq. (2.4) holds true. Alike the case **of** the basis **in** Eq. (2.9), for G = SU(2), Eq.(2.19) allows each tensor product T λ 1× T λ 2to be reduced by[T λ 1× T λ 2] λ µ = ∑ µ 1 µ 2T λ 1µ 1T λ 2µ 2〈λ 1 µ 1 λ 2 µ 2 |λµ〉, (2.20)where the irreducible tensor operator [T λ 1× T λ 2] λ transforms under D λ (Ω).Most **of** the physical operators T λ —basically studied **in** atomic spectroscopy—are expressed**in** terms **of** D λ and their various comb**in**ations. These are, for example, the normalised spherical

2 Partition**in**g **of** function space and basis transformation properties 18harmonics Cq k (̂x) = i k Dq0(¯Ω), k ¯Ω = (Φ, Θ, 0) with Φ = ϕ + π/2, Θ = θ; the spherical harmonicsYq k (̂x) = √ (2k + 1)/4πCq k (̂x). The sp**in** operator S 1 , the angular momentum operator L 1are also expressed **in** terms **of** the irreducible matrix representation D 1 , as it was demonstrated**in** Refs. [52, 67]. The announced particular cases **of** T λ represent the operators that dependon the coord**in**ate system. Be**in**g more tight, these spherical tensor operators are **in** turn thefunctions **of** ̂x ≡ θϕ. With this **in** m**in**d, we may writeT λ µ (̂x 2 ) = ∑ ρ± τ λ µρ(̂x 1 , ̂x 2 )T λ ρ (̂x 1 ). (2.21)Eq. (2.21) also applies for the basis φ(λµ|̂x 2 ). Then the follow**in**g result is immediateΦ(λ 1 λ 2 λµ|̂x 1 , ̂x 2 ) = ∑ ( )˜λ λ1 λ 2 λ; ̂xν µ 1 , ̂x 2¯Φ(λ 1 λ 2˜λν|̂x1 ), (2.22)λν e( )˜λ λ1 λ 2 λdef; ̂xν µ 1 , ̂x 2 = ∑ ( ) [ ]λ1 λ 2˜λµ 1 µ 2 ν ; ̂x λ1 λ1, ̂x 2 λ2 , (2.23)µ 1 µ 2 µµ 1 µ 2( )λ1 λ 2˜λµ 1 µ 2 ν ; ̂x def1, ̂x 2 = ∑ ± τ λ 2µ 2 eµ 2(̂x 1 , ̂x 2 )eµ 2[λ1 λ 2˜λµ 1 ˜µ 2 ν], (2.24)where the basis Φ(λ 1 λ 2 λµ|̂x 1 , ̂x 2 ) is def**in**ed **in** Eq. (2.9) and ¯Φ(λ 1 λ 2˜λν|̂x1 ) ≡ Φ(λ 1 λ 2˜λν|̂x1 , ̂x 1 )is the transformed basis. The quantities ( λ 1 λ 2 λ eµ 1 µ 2 ν ; ̂x ) (1, ̂x 2 and eλ)λ1 λ 2 λν µ; ̂x 1 , ̂x 2 are called rotatedClebsch–Gordan coefficients (RCGC) **of** the first and second type [66, Sec. 6], respectively.Particularly,( ) [λ1 λ 2˜λµ 1 µ 2 ν ; ̂x 1, ̂x 1 = λ1 λ 2˜λµ 1 µ 2 νThe RCGCs are reducible. For example,],( ˜λ λ1 λ 2 λν µ( ) ( )λ1 λ 2 λµ 1 µ 2 µ ; ̂x 1, ̂x ˜λ1 ˜λ2 ˜λ2ν 1 ν 2 ν ; ̂x 1, ̂x 2 = (−1)e λ 2 +1[˜λ 2 ] 1/2( ) [× λ2 Λ 2˜λ2 λ1 λ; ̂x−ρ 2 M 2 ˜ρ 1 , ̂x 2 λ22 µ 1 ρ 2 µThe abbreviation [x] 1/2 ≡ √ 2x + 1.] [˜λ1 ˜λ2 ˜λν 1 ˜ρ 2 ν; ̂x 1 , ̂x 1)= δ λ e λδ µν . (2.25)∑(−1) Λ 2[Λ 2 ] ∑ 1/2 (−1) eρ 2Λ 2 ρ 2 eρ 2] [ ]λ2 ˜λ2 Λ 2. (2.26)µ 2 ν 2 M 2The specific feature **of** technique based on the coord**in**ate transformations (or simplyRCGC technique) is the ability to transform the coord**in**ate-dependence **of** thebasis Φ(ΓΠΛM|̂x 1 , ̂x 2 , . . . , ̂x N ) **in** H Λ preserv**in**g its **in**ner structure. The transformedbasis ¯Φ(˜Γ˜Π˜Λ˜M|̂x ξ ) **in** H eΛ implicates the tensor structure **of** the **in**itial basisΦ(ΓΠΛM|̂x 1 , ̂x 2 , . . . , ̂x N ), but the coord**in**ate-dependence is represented by thefunction **of** arbitrary variable ̂x ξ , where ξ acquires any value from 1, 2, . . . , N.A particular case **of** the two-electron basis function **in** Eq. (2.22) along with the properties**of** RCGCs (see Eqs. (2.25)-(2.26)) **in**itiates a possibility to change the calculation **of** multiple**in**tegrals with the calculation **of** a s**in**gle one. This argument also fits the **in**tegrals **of** the type**in** Eq. (2.10). In a two-electron case, a simple evaluation **in**dicates that the **in**tegration **of**the operator [T k 1(̂x 1 ) × T k 2(̂x 2 )] k (see Eq. (2.20)) on the basis Φ(λ 1 λ 2 λµ|̂x 1 , ̂x 2 ) (see Eq.(2.9)) is transformed **in**to a s**in**gle **in**tegral **of** the transformed operator [T e k 1(̂x 1 ) × T e k 2(̂x 1 )] e kon the transformed basis ¯Φ(˜λ 1˜λ2˜λ˜µ|̂x1 ). The obta**in**ed s**in**gle **in**tegral becomes even simplerif T k acquires some special values. For **in**stance, if T k ≡ C k , then the transformed operatorequals to i e k 1 + e k 2 − ek C ek (̂x 1 )〈˜k 1 0˜k 2 0|˜k0〉. Moreover, if the basis φ(λ 1 µ 1 |̂x 1 ) is written **in** terms **of**D λ 1λµ 1 0(¯Ω 1 ) [67, Eq. (38)], then the transformed basis reads D e eµ0(¯Ω 1 )〈˜λ 1 0˜λ 2 0|˜λ0〉. In addition,due to transformations, the three RCGCs II arise (see Eqs. (2.22)-(2.23)). Their product is

2 Partition**in**g **of** function space and basis transformation properties 19reduced **in** accordance with Eqs. (2.23), (2.26). The obta**in**ed RCGC II ( ) Λ e Λ1 Λ 2 Λ; ̂xeɛ ɛ 1 , ̂x 2 is**in**tegrated over ̂x 2 , and the resultant function also depends on ̂x 1 only. It goes without say**in**gthat the procedure **of** **in**tegration is suitable for the N–electron case. However, to improve theapplicability **of** this algorithm, the **in**tegrals **of** the RCGCs or, what is equivalent, the **in**tegrals**of** spherical functions ± τqq k ′(̂x 1, ̂x 2 ) should be calculated.In Ref. [66, Sec. 5], it was proved that the **in**tegral **of** ± τqq k ′(̂x 1, ̂x 2 ) over ̂x 2 ∈ S 2 is the sum**of** **in**tegrals determ**in**ed **in** the areas L 2 (Ω). That is,( ∫ )Sqq k ′(̂x 1) = + d̂xL 2 (Ω + 11∫L + 2 ξ k) 2 (Ω − 12 ) qq ′(̂x 1, ̂x 2 )( ∫ )+ + d̂xL 2 (Ω − 11∫L − 2 ξ k) 2 (Ω + 12 ) qq ′(̂x 1, ̂x 2 ), (2.27)where the measure d̂x 2 = dϕ 2 dθ 2 s**in** θ 2 ; a normalisation ∫ S 2 d̂x 2 = 4π. A direct **in**tegrationleads toSqq k ′(̂x 1) =λ q ′(ϕ 1 )i q−q′ −1 (−1)q − 1( (−1)q ′ + 1 ) a(k, q, q ′ )e −iq′ ϕ 1q× ∑ pb p (k, q, q ′ ) ( pI k qq ′(θ 1; 0, θ 1 ) + (−1) q−q′ pI k qq ′(θ 1; θ 1 , π) ) . (2.28)The functions λ q ′(ϕ 1 ) and p Iqq k ′(θ 1; a, b) are def**in**ed by⎧⎨ (−1) q′ , ϕ 1 ∈ [0, π/2]λ q ′(ϕ 1 ) def= (−1)⎩2q′ , ϕ 1 ∈ (π/2, 3π/2](−1) 3q′ , ϕ 1 ∈ (3π/2, 2π], (2.29)pIqq k ′(θ 1; a, b) def= 2 { }2I1(a, p b) cos θ 1 + (I2(a, p b) − I0(a, p b)) s**in** θ 1 , (2.30)Is p (a, b) def= Is p (tan [ 1(θ 2 1 − b)]) − Is p (tan [ 1(θ 2 1 − a)]), s = 0, 1, 2, (2.31)Is p (z) def=z 2p+q′ −q+s+12p + q ′ − q + s + 1( 2p + q ′ − q + s + 1× 2 F 1 , k + 2; 2p + )q′ − q + s + 3; −z 2 . (2.32)22It appears from Eqs. (2.28)-(2.32) that the function Sqq k ′(̂x) is represented by the sum **of**Gauss’s hypergeometric functions. If, particularly, l ∈ Z + , then Sµm(̂x) l = δ µ0 S0m(̂x). l Indeed,it follows from Eq. (2.28) that for l ∈ Z + ,S l 0m(̂x) = 2πl! √ (l + m)!(l − m)!e −imϕ2.2.2 Def**in**ition. The functionsm**in** (l,l−m)∑p=max (0,−m)(−1) p pI l 0m(θ; 0, π)p!(p + m)!(l − p)!(l − m − p)! . (2.33)Sm(̂x) l def= S0m(̂x) l (2.34)with l ∈ Z + defform the set L l such that the card**in**ality #L l = 2l + 1, and the **in**dices satisfym = −l, −l + 1, . . . , l − 1, l.2.2.3 Proposition. For l ∈ Z + , the #L l functions S l m(̂x) constitute a set L l **of** components **of**the SO(3)–irreducible tensor operator S l (̂x).Pro**of**. To prove the proposition, it suffices to demonstrate that S l m(̂x) transforms under D l (Ω)(see Eq. (2.19)) or, equivalently, under η l (̂x 1 , ̂x 2 ) (see Eqs. (2.17)-(2.21)), where the squarematrix η l (̂x 1 , ̂x 2 ) ∈ { + ξ l (̂x 1 , ̂x 2 ), − ξ l (̂x 1 , ̂x 2 )}.

2 Partition**in**g **of** function space and basis transformation properties 20To beg**in** with, **in**tegrate both sides **of** Eq. (2.21) over ̂x 2 on S 2 . For positive **in**teger λ ≡ l,the result reads∫S 2 d̂x 2 T l µ(̂x 2 ) = δ µ0l∑Sρ(̂x l 1 )Tρ(̂x l 1 ), (2.35)The left hand side **of** Eq. (2.35) does not depend on ̂x 1 and it is a function **of** l. This impliesthat the relationl∑Sρ(̂x l 1 )Tρ(̂x l 1 ) =ρ=−lρ=−ll∑Sρ(̂x l 2 )Tρ(̂x l 2 )ρ=−lis valid for any ̂x 1 , ̂x 2 ∈ S 2 . Apply Eq. (2.21) for T l ρ(̂x 2 ) once aga**in**. Thenl∑Sρ(̂x l 1 )Tρ(̂x l 1 ) =ρ=−ll∑ρ,ν=−lS l ρ(̂x 2 )η l ρν(̂x 1 , ̂x 2 )T l ν(̂x 1 ).F**in**ally, replace ρ with ν on the right hand side and list the common terms next to T l ρ(̂x 1 ) fromboth sides **of** expression. After replac**in**g ̂x 1 with ̂x 2 , the result readsS l ρ(̂x 2 ) =l∑ηνρ(̂x l 2 , ̂x 1 )Sν(̂x l 1 ), ρ = −l, −l + 1, . . . , l − 1, l. (2.36)ν=−lThis proves the proposition.Accord**in**g to Proposition 2.2.3, the tensor products **of** SO(3)–irreducible tensor operatorsS l (̂x) are reduced by us**in**g Eq. (2.20).Conjecture. Unlike the case **of** SO(3), the transformation properties **of** functions Sqq k ′(̂x) withrational numbers k ∈ Q + are not so clear. In this case, Eq. (2.36) is not valid. This is becausea direct **in**tegration, as **in** Eq. (2.35), over ̂x 2 on S 2 can not be performed correctly due to thespecific properties **of** Sqq k ′(̂x) for k ∈ Q+ . That is, for l ∈ Z + ,∫Sm(̂x l 1 ) def= d̂x 2 η0m(̂x l 1 , ̂x 2 )S 2for any ̂x 1 ∈ S 2 . To compare with, see Eq. (2.27). This means, Eq. (2.21) applies forη l µρ(̂x 1 , ̂x 2 ) ∈ { + ξ l µρ(̂x 1 , ̂x 2 ), − ξ l µρ(̂x 1 , ̂x 2 )}, for all **in**tegers µ, ρ and for all possible ̂x 1 , ̂x 2 .Conversely, for k ∈ Q + , the latter expression does not fit, as q, q ′ are the rational numbers.However, the numerical analysis enforces to make a prediction that particularlyS k qq ′(̂x 2; ξ) = −k∑Q=−k+ ξQq k ′(̂x 2, ̂x 1 )SqQ(̂x k 1 ; ξ), Sqq k ′(̂x 1; ξ) def=∫S 2 d̂x 2 + ξ k qq ′(̂x 1, ̂x 2 ).Know**in**g the connection between SO(3) and SU(2), it turns out that there must exist the transformationproperties for S k qq ′ with k ∈ Q+ , similar to Eq. (2.36) and to this day, the SU(2) caseis an open question yet.Hav**in**g def**in**ed the functions Sqq k ′(̂x), the calculation **of** **in**tegrals **in** Eq. (2.10) requires littleeffort. Besides, this is a simpler case than the **in**tegration on the basis Φ(ΓΠΛM|̂x 1 , ̂x 2 , . . . , ̂x N )s**in**ce the product **of** Dm λ em(Ω) and Dλ′m ′ em(Ω) is reduced to a s**in**gle spherical function D Λ ′ MM f(Ω)which is replaced by ± τ Λ MM f(̂x 1, ̂x 2 ). If H Λ represents the angular part (C k (̂x 1 ) · C k (̂x 2 )) **of** theCoulomb **in**teraction operator 1/r 12 , then Λ = 0 and a double **in**tegral is transformed to a s**in**gleone as follows

2 Partition**in**g **of** function space and basis transformation properties 21∫∫d̂x 1 d̂x 2 Dm l em (Ω)(Ck (̂x 1 ) · C k (̂x 2 ))Dm l′ ′ em ′(Ω) = 4π[k]−1/2 (−1) k+m′ + emS 2× ∑ [K] −1/2 [k‖S K ‖k] ∑ [ ] [ ] []l l′L l l′L L k K−m m ′ m ′ − m − ˜m ˜m ′ ˜m ′ − ˜m m ′ − m m − m ′ 0KL× ∑ [] []k k K L k Km ′ − m Q + m − m ′ Q ˜m ′ − ˜m ˜m − ˜m ′ . (2.37)− Q −QQ=evenTab. 2: Numerical values **of** reduced matrix element **of** SO(3)–irreducible tensor operator S k for several**in**tegers l, l ′l l ′ k (4π) −1 [l‖S k ‖l ′ ] l l ′ k (4π) −1 [l‖S k ‖l ′ ] l l ′ k (4π) −1 [l‖S k ‖l ′ ]√20 0 0 1 2 4 21 5 4 − 15·7 27√√ 51 1 0 3 1 3 4 − 2 2 4 4 − 2 55·27 27 7·11√√√1 21 21 2·71 1 2 2 2 2 3 5 25 5 3 7 3 3·5√1 31 3 2 3 3 0 7 4 6 2 √ 35 5 5·11Reduced matrix elements [l‖S k ‖l ′ ] are found from the Wigner–Eckart theorem. That is,[l‖S k ‖l ′ ] = [l]1/2[l ′ ] 1/2 [l′ ‖S k ‖l] = ∑qmm ′ 〈lm|S k q |l ′ m ′ 〉〈l ′ m ′ kq|lm〉, (2.38)where the matrix element is calculated on the basis **of** spherical harmonics. Some **of** the values**of** [l‖S k ‖l ′ ] are listed **in** Tab. 2.In general, each N-**in**tegral∫ ∫d̂x 1 d̂x 2 . . . d̂x N Φ∫S † (Γ bra Π bra Λ bra M bra |̂x 1 , ̂x 2 , . . . , ̂x N )TQ K (̂x 1 , ̂x 2 , . . . , ̂x N )2 S 2 S 2× Φ(Γ ket Π ket Λ ket M ket |̂x 1 , ̂x 2 , . . . , ̂x N )is replaced by the sum **of** s**in**gle **in**tegrals∫d̂x ¯Φ † (˜Γ bra ˜Πbra˜Λbra ˜M bra |̂x) ¯T K eeQ (̂x)S Λ 2M 2 M(̂x)S Λ ′ 3S 2 2 M 3 M(̂x) . . . S Λ3′ NM N M(̂x)N ′× ¯Φ(˜Γ ket ˜Πket˜Λket ˜M ket |̂x).Instead **of** that the N − 1 functions S are produced. At least for Λ i ∈ Z + ∀i = 2, 3, . . . , N, thelast **in**tegral can acquire the follow**in**g matrix representation (see Proposition 2.2.3)〈˜Γ bra Π bra˜Λbra ˜M bra |[. . . [[ ¯T eK × S Λ 2] E 2× S Λ 3] E 3× . . . × S Λ N] E N¯M N|˜Γ ket Π ket˜Λket ˜M ket 〉, (2.39)recall**in**g that the parity is **in**variant under coord**in**ate transformation. That is, Π bra,ket = ˜Π bra,ket .Eq. (2.39) represents thus the s**in**gle-particle matrix element on the basis **of** transformed functions.The calculation **of** spherical tensor operator matrix element on the basis functionsΦ(ΓΠΛM|̂x 1 , ̂x 2 , . . . , ̂x N ) assigns to calculate the N-**in**tegral over the sphericalcoord**in**ates ̂x ξ ∀ξ = 1, 2, . . . , N. If the basis is represented **in** terms **of** Slater determ**in**ants,then a usual technique based on the Wigner–Eckart theorem is convenient.That is, the N–electron matrix element is taken to be the product **of** s**in**gle-electron

2 Partition**in**g **of** function space and basis transformation properties 22matrix elements. In the **in**dividual cases, the calculation may be performed **in** adifferent way. If the RCGC technique is exploited, the N-**in**tegral is reduced tothe sum **of** s**in**gle **in**tegrals. The technique based on the coord**in**ate transformationsis even more applicable for the basis expressed by the spherical functions D Λ (Ω).The example **of** helium-like atoms confirms this clearly.2.3 System **of** variable particle numberUnlike the case **of** Slater determ**in**ants it is more convenient to form the basis Φ(ΓΠΛM) fromthe s**in**gle-electron quantum states |λm〉—known as the vectors **of** Hilbert space—that are createdby the 2λ + 1 components **of** irreducible tensor operator a λ —known as the Fock spaceoperator—act**in**g on the genu**in**e vacuum |0〉. In this case, the quantum mechanical **many****body**system is characterised by the number **of** particles rather than their coord**in**ates. Thecharacteristic operator is the particle number operator ̂N = −[λ] 1/2 W 0 (λ˜λ), where the irreducibletensor operator W Λ (λ 1˜λ2 ) = [a λ 1× ã λ 2] Λ . The transposed annihilation operatorã λ m = (−1) λ−m a λ†−m, where a λ†−m annihilates the state |λ − m〉. It is assumed that the irreducibletensor operator W Λ (λ 1˜λ2 ) acts on the irreducible tensor space H Λ if W Λ (λ 1˜λ2 ) transforms underthe G–irreducible matrix representation D Λ (g). On the other hand, H Λ may be reducibleH L × H S (Λ ≡ LS for LS-coupl**in**g). Then W LS (l 1˜l2 ) transforms under both D L (g), D S (g)irreducible matrix representations **in**dependently. If, however, H Λ is irreducible, then Λ ≡ J(jj-coupl**in**g).Judd [46] demonstrated that the application **of** a second quantised representation **of** atomic**many**-**body** system appears to be especially comfortable for the group-theoretic classification**of** the states **of** equivalent electrons **of** atom. The key feature is that the products **of** a λ λ†m and a e emform the Lie algebra A Nl −1, where N l = max N = 4l + 2 is a maximal number **of** electrons**in** the shell l N . This implies that the branch**in**g rules for the states **of** l N are to be obta**in**ed.For LS-coupl**in**g, the typical reduction scheme reads U(N l ) → Sp(N l ) → SO L (3) × SU S (2).Particularly, the multiplicities **of** Sp(N l )–irreducible representations determ**in**e the so-called seniorityquantum number v, first **in**troduced by Racah [9, Sec. 6-2]. In this case, it is convenientto form the tensor operators W κλ (λ 1 λ 2 ) = [a 1 2 λ 1× a 1 2 λ 2] κλ on H q ≡ H Q × H Λ , where H Q denotesthe quasisp**in** space. The quasisp**in** quantum number Q relates to v by Q = ([λ] − 2v)/4.Various useful properties **of** operators on H q were studied by Rudzikas et. al. [50]. However,start**in**g from f 3 electrons, the last scheme is **in**sufficient and thus additional characteristicnumbers are necessary. The complete classification **of** terms **of** d N and f N configurations wastabulated by Wybourne et. al. [68, 69].In order to write the Hamiltonian Ĥ that describes a system with variable particle number,it is sufficient to express Ĥ by its matrix elements as followsĤ = Ĥ0 + ̂V , Ĥ 0 = ∑ f∑Ô 1 (αα)ε α1 , ̂V = ̂V n , ̂Vn = F n [v], (2.40)α 1F n [v] def= ∑I n(α ¯β)n=0Ô n (α ¯β)v n (α ¯β), (2.41)defÔ n (α ¯β) = :a α1 a α2 . . . a αn−1 a αn a †¯βn a †¯βn−1 . . . a †¯β2 a †¯β1 :, Ô 0 (α ¯β) = 1, (2.42)defv n (α ¯β) = v α1 α 2 ...α n−1 α n ¯β1 ¯β2 ... ¯β n−1 ¯βn = 〈α 1 α 2 . . . α n−1 α n |h(n)| ¯β 1 ¯β2 . . . ¯β n−1 ¯βn 〉, (2.43)where I n (α ¯β) = {α 1 , α 2 , . . . , α n−1 , α n , ¯β 1 , ¯β 2 , . . . , ¯β n−1 , ¯β n } is the set **of** numbers α i and ¯β j∀i, j = 1, 2, . . . , n that characterise the states |x i 〉 = a xi |0〉, where x i = α i , ¯β i and a xi ≡ a λx im xi .By default, it is assumed that each operator a xi is additionally characterised by the pr**in**cipalquantum number n xi . The notation : : denotes the normal order (or normal form). The operatorsh(n) with the eigenvalues ε xi represent the Hamiltonians that are particular for the s**in**gleparticles. Their sum forms the total Hamiltonian H. For the atomic case, see Eq. (2.1). Thenumber f depends on the concrete **many**-**body** system. For the atoms and ions, f = 2, as all**in**teraction operators h(n) used **in** atomic spectroscopy are obta**in**ed from the Feynman diagram

2 Partition**in**g **of** function space and basis transformation properties 23121◦′◦ 2′which requires effort to demonstrate that the path **in**tegral is determ**in**ed **in** terms **of** the **in**teraction( 1 − (α 1 · α 2 ) ) exp [ ]i|ε 1 − ε 1 ′|r 12 /r12 , where α i = ( 0 σ i)σ i 0 are the Dirac matrices andσ i denote the 2 × 2 Pauli matrices. Thus the matrix element **of** the latter **in**teraction operatoron the basis **of** Dirac 4–sp**in**ors results to the sum **of** Coulomb **in**teraction 1/r 12 and the Breit**in**teraction which particularly is expressed **in** relativistic and nonrelativistic forms [52, Secs.1.3, 2.2].It can be readily checked that the matrix elements **of** N–electron Hamiltonians H and Ĥ onthe correspond**in**g basis Φ(ΓΠΛM) or else |ΓΠΛM〉 are equal. However, Ĥ has the eigenstatesfor all N, while H only for a specified N. Accord**in**g to Kutzelnigg [31, 70], Ĥ is called theFock space Hamiltonian.2.3.1 Orthogonal subspacesThe task to f**in**d the set X ≡ {|Ψ i 〉} ∞ i=1 **of** eigenfunctions **of** Ĥ (see Eq. (2.2)) is found tobe partially solvable by us**in**g the partition**in**g technique, first **in**troduced by Feshbach [71].Later, it was demonstrated by L**in**dgren et. al. [34] that the present approximation leads tothe effective operator approach. In their used formalism, on the other hand, L**in**dgren andthe authors beh**in**d [36, 38] commonly regarded the matrix representation **of** tensor operators.This, however, is a more comfortable representation for practical applications, though it is lessuniversal. The significant opportunities **of** the irreducible tensor operator techniques **in** the**many**-**body** **perturbation** **theory** (MBPT) were demonstrated **in** Refs. [54, 55, 72, 73].To f**in**d a subset Y ≡ {|Ψ j 〉} d j=1 ⊂ X **of** functions |Ψ j 〉 (with d < ∞), the follow**in**g spacepartition**in**g procedure is performed.2.3.1 Def**in**ition. A subset Ỹ ≡ {|Φ k〉} d k=1 ⊂ ˜X ≡ {|Φ p 〉} ∞ p=1 satisfies:(a) the configuration parity Π k ≡ Π eY ∀k = 1, 2, . . . , d is a constant for all N k –electron configurationstate functions |Φ k 〉 ≡ |Φ e Yk 〉 ≡ |Γ k Π eY Λ k M k 〉 ∈ Ỹ ;(b) the eigenstates |Φ e Yk 〉 **of** Ĥ0 conta**in** the configurations **of** two types:(1) fully occupied l N l ktktconfigurations that particularly determ**in**e either core (c) or valenceY(v) orbitals; the core orbitals are present **in** all |Φ e k 〉 for all **in**tegers t < u c k and for allk, where u c k is the number **of** closed shells **in** |Φ Y e k 〉; the valence orbitals are present **in**Ysome **of** the functions |Φ e k 〉;(2) partially occupied l N kzkzconfigurations that determ**in**e valence (v) orbitals for all **in**tegersz ≤ u o k , where uo k is the number **of** open shells **in** |Φ Y e k 〉;(c) the subsetways.Ỹ is complete by means **of** the allocation **of** valence electrons **in** all possibleSeveral mean**in**gful conclusions immediately follow from the def**in**ition **of** Ỹ .1. The number N k **of** electrons **in** |Φ e Yk 〉 equals to⎛⎞u c ku∑c ku∑∑o kN k = Nk c + Nk, o Nk c = N lkt = 2 ⎝u c k + 2 l kt⎠ , Nk o = N kz , (2.44)t=1where Nk c and N k o denote the electron occupation numbers **in** closed and open shells.t=1z=1

2 Partition**in**g **of** function space and basis transformation properties 242. The subset Ỹ is partitioned **in**to several subsets Ỹn, each **of** them def**in**ed byA⋃Ỹ = Ỹ n ,n=1def YỸ n = {|Φ e kn〉} dnk n=d n−1 +1 , d 0 = 0, d A = d. (2.45)The subsets Ỹn Yare assumed to conta**in** the N n –electron basis functions |Φ e kn〉, where theidentities N n ≡ N dn−1 +1 = N dn−1 +2 = . . . = N dn and N 1 ≠ N 2 ≠ . . . ≠ N A hold true.This implies that Ĥ0 Yhas the eigenstates ∀ |Φ e kn〉 ∈ Ỹn, ∀n = 1, 2, . . . , A, while H 0 has theYeigenstates |Φ e kn〉 for which δ NNn ≠ 0. Thus only one specified subset Ỹn fits the eigenvalueequation **of** H 0 . It is found to be the subset Ỹen with N en = N.3. Items (a), (c) **in** Def**in**ition 2.3.1 stipulate that the subset ˜Z ≡ ˜X\Ỹ = {|Θ l〉} ∞ l=1 formedfrom the functions |Θ l 〉 ≡ |Φ d+l 〉 represents the orthogonal complement **of** Ỹ . That is,Ỹ ∩ ˜Z = ∅. The s**in**gle-electron orbitals that form the configurations **in** |Θ l 〉 will be calledexcited (e) or virtual orbitals. These orbitals are absent **in** Ỹ .The conclusions **in** items 1, 3 agree with those **in**ferred by L**in**dgren [34, Sec. 9.4, p. 199],who used to exploit the traditional Hilbert space approach. On the other hand, item 2 extendsthis approach to the systems **of** variable particle number.Hav**in**g def**in**ed the subsets Ỹ , ˜Z ⊂ ˜X **of** vectors **of** **many**-**body** Hilbert spaces, it is sufficientto **in**troduce the subspaces as follows.2.3.2 Def**in**ition. The functions |Φ e Ykn〉 ∈ Ỹn form the N n –electron subspacedefP n = { Y|Φ e Ykn〉 : 〈Φ e Ykn|Φ e k ′ n〉 Hn = δ Γkn Γ k ′ δ Λkn Λ n k ′ δ Mkn M n k ′ ≡ δ knk ′ ,n n}∀k n , k n ′ = d n−1 + 1, d n−1 + 2, . . . , d n (2.46)**of** dimension dim P n = d n − d n−1 , where H n denotes the **in**f**in**ite-dimensional N n –electronHilbert space, spanned by all N n –electron functions |Φ pn 〉 from ˜X n ⊂ ˜X.2.3.3 Corollary. In accordance with item 2, if n = ñ, then H en = H denotes the **in**f**in**itedimensionalN–electron Hilbert space, while P en = P denotes the N–electron subspace **of** Hwith dim P = d en − d en−1 ≡ D.def2.3.4 Corollary. Accord**in**g to item 3, the orthogonal complement Q n = H n ⊖ P n **of** P n isspanned by the N n –electron functions |Θ ln 〉 ∈ ˜Z n ⊂ ˜Z. That is,〈Θ ln |Φ e Ykn〉 Hn = 0, ∀l = 1, 2, . . . , ∞, ∀k = 1, 2, . . . , d, ∀n = 1, 2, . . . , A. (2.47)If particularly n = ñ, then Q en = Q denotes the orthogonal complement **of** P.Y2.3.5 Def**in**ition. The functions |Φ e k 〉 ∈ Ỹ form the subspace{W def Y= |Φ e Yk 〉 : 〈Φ e Yk |Φ e k ′〉 F =}∀k, k ′ = 1, 2, . . . , d =where F denotes the Fock space.A∑Y〈Φ e Ykn|Φ e k ′ n〉 Hn = δ Γk Γ k ′δ Λk Λ k ′δ Mk M k ′ ≡ δ kk ′,n=1A⊕n=1P n ⊂ F def=A⊕H n ⊂ F, (2.48)2.3.6 Corollary. The orthogonal complement U def= F ⊖ W **of** W is spanned by the functions|Θ l 〉 ∈ ˜Z.Hav**in**g def**in**ed the **many**-electron Hilbert spaces, the follow**in**g proposition is straightforward.n=1

2 Partition**in**g **of** function space and basis transformation properties 252.3.7 Proposition. The form ̂1 n : H n −→ H n , expressed byis a unit operator on H n .∞∑̂1 n = |Φ pn 〉〈Φ pn |, |Φ pn 〉 ∈ ˜X n , (2.49)p n=1Pro**of**. For the basis |Φ pn 〉, it is evident that (see Def**in**ition 2.3.2)̂1 n |Φ pn 〉 = ∑ |Φ p ′ n〉〈Φ p ′ n|Φ pn 〉 Hn = |Φ pn 〉.p ′ nFor the l**in**ear comb**in**ations |Ψ M 〉 ≡ ∑ Mp n=1 c p n|Φ pn 〉 with c pnobta**in**able. That is, ̂1 n |Ψ M 〉 = ∑ Mp n=1 c p n̂1 n |Φ pn 〉 = |Ψ M 〉.2.3.8 Corollary. The form ̂1: F −→ F, expressed by∈ K = R, C this is also readilyis a unit operator on F.A∑̂1 = ̂1 n , (2.50)n=1Pro**of**. The pro**of** directly follows from Def**in**ition 2.3.5 and Proposition 2.3.7, recall**in**g that thefunctions |Φ p 〉 ∈ ˜X **of** F determ**in**e any function from the sets ˜X 1 , ˜X 2 , . . . , ˜X A .Select another basis Y ≡ {|Ψ j 〉} d j=1 **of** W ⊂ F which is partitioned **in**to the subsets Y n **of**defP n ⊂ H n , def**in**ed by Y n = {|Ψ jn 〉} dnj n=d n−1 +1 with d 0 = 0, d A = d (see Eq. (2.45)). As usually(see Eq. (2.2)), it is assumed that the functions |Ψ j 〉 ∈ Y designate the eigenstates **of** Ĥ on F,while the functions |Ψ jen 〉 designate the eigenstates **of** H on H. Then it is easy to verify that forany **in**teger n ≤ A,̂1 n |Ψ jn 〉 = |Ψ jn 〉 = |Φ P j n〉 + ̂Q n |Ψ jn 〉,̂P ndef=d n∑k n=d n−1 +1Y|Φ e Ykn〉〈Φ e defkn|, ̂Qn =|Φ P j n〉 def= ̂P n |Ψ jn 〉, (2.51)∞∑|Θ ln 〉〈Θ ln |, ̂Pn + ̂Q n = ̂1 n . (2.52)If the operator ̂Ω: P n −→ H n is def**in**ed by ̂Ω(n) ̂P n = ̂1 n , then ̂Q n = ̂Ω(n) ̂P n − ̂P n andl n=1|Ψ jn 〉 = ̂Ω(n)|Φ P j n〉. (2.53)̂Ω(n) acts on H n and it is called the wave operator [34, Sec. 9.4.2, p. 202, Eq. (9.66)]. Thisimplies that the eigenfunctions |Ψ jen 〉 **of** H are generated by the wave operator ̂Ω(ñ) ≡ ̂Ω on theN–electron Hilbert space H. The functions |Φ P j n〉 are called the model functions **of** P n . It wasL**in**dgren [28, 34] who first proved that the wave operator ̂Ω satisfies the so-called generalisedBloch equation[̂Ω, H 0 ] ̂P = V ̂Ω ̂P − ̂Ω ̂P V ̂Ω ̂P (2.54)which is obta**in**ed from the eigenvalue equations **of** H 0 and H tak**in**g **in**to consideration that[H 0 , ̂P ] = 0, where ̂P ≡ ̂P en (it is also considered that ̂Q ≡ ̂Q en ).For the systems with variable particle number, the action **of** ̂Ω(n) must be extended. Thiswill be done by **in**troduc**in**g the Fock space operator (see Eq. (2.41))Ŝ def= ̂1 +∞∑Ŝ n ,n=1defŜ n = F n [ω], (2.55)

2 Partition**in**g **of** function space and basis transformation properties 27Ĥ = ̂P Ĥ ̂P + Ŵ ,Ŵ def=∞∑̂P (̂V 1 + ̂V 2 )̂Ω n ̂P , (2.61)where ̂Ω n is **of** the form presented **in** Eq. (2.59). Eqs. (2.59), (2.61) po**in**t to at least two types**of** Hilbert space operators: ̂P Ô n (α ¯β) ̂P and ̂QÔn(α ¯β) ̂P (see Eq. (2.42)). To determ**in**e theirbehaviour for a given set I n (α ¯β) **of** s**in**gle-electron orbitals, redef**in**e items (b)(1)-(2), (c), 3 **in**Sec. 2.3.1 **in** a more strict mannern=1(A) a c ̂P = 0, (C) av ̂P ≠ 0,(B) a † ̂P ē = 0, (D) a †¯v ̂P ≠ 0.(2.62)As already po**in**ted out, items (A)-(B) agree with Ref. [34, Sec. 13.1.2, p. 288, Eq. (13.3)].Items (C)-(D) em**body** a mathematical formulation **of** item (c) **in** Def**in**ition 2.3.1 and are **of**special significance s**in**ce they def**in**e the so-called complete model space.The normal orders **of** products **of** creation and annihilation operators **in** Ôn(α ¯β) for thespecified types (v, e, c) **of** α, β are thesev¯v⊼eēc¯c(2.63)where ⊼ (up) and (down) denote the direction **of** electron propagation. For the states createdby a α , write |α〉. For the states annihilated by a †¯β, write | ¯β〉. Hereafter, the over bar designatesannihilated states, but both α and β determ**in**e the type **of** orbital: v, e or c. Accord**in**g to Eq.(2.63), permitted propagations for α and β electrons are to be upwards for α, β = e, v anddownwards for α, β = c. In algebraic form **of** Eq. (2.63), write :a α a †¯β: = a α a †¯β for α, β = v, eand :a †¯βa α : = a †¯βa α for α, β = c.2.3.9 Lemma. If Ôn(α ¯β) is a Fock space operator and ̂P , ̂Q are the orthogonal projections on**in**f**in**ite-dimensional N–electron Hilbert space H, then for any **in**teger n ≤ N, the follow**in**gassertions are straightforward:i) ̂P Ô n (α ¯β) ̂P ≠ 0 iff α, β = v;ii) ̂QÔ n (α ¯β) ̂P ≠ 0 iff α = v, e and β = v, c;iii) ̂QÔ n (v¯v) ̂P = 0 iff ∑ ni=1 (l v i+ l¯vi ) ∈ 2Z + .Observ**in**g that self-adjo**in**t operator Ô† n(α ¯β) = Ôn( ¯βα), the follow**in**g statements are trueif Lemma 2.3.9 is valid.2.3.10 Corollary. The operator ̂P Ôn(α ¯β) ̂Q ≠ 0 iff α = v, c and β = v, e.2.3.11 Corollary. The operator ̂P Ôn(v¯v) ̂Q = 0 iff ∑ ni=1 (l v i+ l¯vi ) ∈ 2Z + .Corollaries 2.3.10-2.3.11 are to be proved simply replac**in**g α with β **in** Lemma 2.3.9.Pro**of** **of** Lemma 2.3.9. To prove the lemma, start with item i) which is easy to confirm by pass**in**gto Eq. (2.62). To prove item ii), write:1. ̂Qav ̂P = av ̂P − ̂P av ̂P ≠ 0 due to items (C)-(D), i).†2. ̂Qa ̂P ¯v = a †¯v ̂P − ̂P a †¯v ̂P ≠ 0 due to items (C)-(D), i).

2 Partition**in**g **of** function space and basis transformation properties 283. ̂Qae ̂P = ae ̂P − ̂P ae ̂P = ae ̂P = ae − a e ̂Q ≠ 0 due to item i) and Corollary 2.3.4.†4. ̂Qa ̂P ē = a † ̂P ē − ̂P a † ̂P ē = 0 due to items (B), i).5. ̂Qac ̂P = ac ̂P − ̂P ac ̂P = 0 due to items (A), i).†6. ̂Qa ̂P ¯c = a †¯c ̂P − ̂P a †¯c ̂P = a †¯c ̂P ≠ 0 due to item i).YTo prove item iii), write |Φ e ken〉 **in** an explicit form as follows (see Def**in**ition 2.3.1)|l N k en 1k en 1 lN k en 2k en 2. . . l N k en r−1k en r−1 lN k en rk en r lN k en r+1k en r+1. . . k en u kenlN k en u kenΓ ken Π eY Λ ken M ken 〉,where u ken = u c k en+ u o k en. Start from n = 0. This is a trivial case, as Ô0(α ¯β) = 1 and ̂Q ̂P = 0.Suppose n = 1. Then̂Qa v1 a †¯v 1̂P =d en∑k en =d en−1 +1∞∑|Θ len 〉〈Θ len |Φ ′ Yk en〉 H 〈Φ e ken|. (2.64)l en =1The N–electron function |Φ ′ k en〉 ≡ a v1 a †¯v Y1|Φ e ken〉 **in** an explicit form reads{0, Nken s = 4lδ(l¯v1 , l ken r)δ(l v1 , lken s + 2,ken s)1, otherwise× |l N k en 1k en 1 lN k en 2k en 2. . . l N k en r−1k en r−1 lN k en r−1k en rl N k en r+1k en r+1. . . lN k en s−1k en s−1 lN k en s+1k en sl N k en s+1k en s+1. . . k en u kenlN k en u kenΓ ′ k enΠ ′ k enΛ ′ k enM k ′ en〉.In Eq. (2.64), the sum runs over all k en . Consequently, there exists at least one function |Φ e Yken〉from the complete set Ỹen such that l¯v1 = l ken r and l v1 = l ken s with N ken s < 4l ken s + 2. Thenthe parity **of** obta**in**ed non-zero function |Φ ′ k en〉 equals to Π ′ k en= (−1) lv 1 +l¯v e1 ΠY . In addition, ifl v1 + l¯v1 is even, then Π ′ k en= Π eY and thus |Φ ′ Yk en〉 is equal to |Φ e k(see item (c) **of** Def**in**itionen〉 ′2.3.1) up to multiplier, where ken ′ acquires any values from d Yen−1 + 1 to d en . But 〈Θ len |Φ e k en〉 ′ H = 0due to Corollary 2.3.4.For n > 1, the consideration is consequential and easy to prove. In this case, the parity **of**|Φ ′ k en〉 ≡ Ôn(v¯v)|Φ Y e ken〉 equals to Π ′ k en= (−1) ϑn Π eY , where ϑ n = ∑ ni=1 (l v i+ l¯vi ) assum**in**g thatl ken r i= l vi and l ken s i= l¯vi for all i = 1, 2, . . . , n and for all r i , s i **in** the doma**in** **of** **in**tegers[1, u ken ].Item iii) **of** Lemma 2.3.9 may be thought **of** as an additional parity selection rule whoseapplication to the effective operator approach is **of** special mean**in**g. The ma**in** purpose **of**the rule is to reject the terms **of** ̂Ω n (see Eq. (2.59)) with zero-valued energy denom**in**ators.These energy denom**in**ators are obta**in**ed from the generalised Bloch equation. Indeed, for thecommutator **of** Eq. (2.58), writêQ[̂Ω, Ĥ0] ̂P = ∑ |Θ len 〉〈Θ len |[̂Ω, Ĥ0]|Φ Y e Yken〉〈Φ e ken| = ∑ ∑Y|Θ len 〉〈Θ len |̂Ω n |Φ e Yken〉〈Φ e Yken|(E e ken−E ken ),k en l en n k en l enYwhere E e kenand E ken denote the eigenvalues **of** Ĥ0 Yfor the eigenfunctions |Φ e ken〉 and |Θ len 〉, respectively.The right hand side **of** Eq. (2.58) may be expressed by∑|Θ len 〉〈Θ len | ̂ Y V |Φ e Yken〉〈Φ e ken|,k en l enwhere the effective operator ̂ V denotes the sum **of** Fn [v eff ] (see Eq. (2.41)) with v eff be**in**gsome effective **in**teraction. Then̂Ω n = ̂QF n [v eff ] ̂P .YE e ken− E ken

2 Partition**in**g **of** function space and basis transformation properties 29YBut each eigenvalue E e ken, E ken is the sum **of** s**in**gle-electron energies ε i which are found bysolv**in**g the s**in**gle-electron eigenvalue equations for the s**in**gle-electron eigenstates that correspond**in**glyform |Φ e kenY〉 and |Θ len 〉. Hence,̂Ω n = ∑I n(α ¯β)̂QÔn(α ¯β) ̂P ω n (α ¯β),defω n (α ¯β)(α ¯β)D n (α ¯β) ,= veff nD defn(α ¯β) =n∑(ε ¯βi −ε αi ), (2.65)where coefficients ω n (α ¯β) also characterise the Fock space operator Ŝ (see Eq. (2.55)). Itimmediately follows that for α = β = v, the energy denom**in**ator D n (v¯v) = 0. Therefore itemiii) **of** Lemma 2.3.9 omits this result for even **in**tegers ∑ ni=1 (l v i+ l¯vi ).The D–dimensional subspace P **of** **in**f**in**ite-dimensional N–electron separable Hilbertspace H is formed **of** the set Ỹen **of** the same parity Π eY configuration state functions|Φ e kenY〉 by allocat**in**g the valence electrons **in** all possible ways (complete modelspace). In addition, to avoid the divergence **of** PT terms, the parity selection rule isassumed to be valid. The subspace P will be called the model space.The effective operator Ĥ **in** Eq. (2.61) is usually called the effective Hamiltonian or theeffective **in**teraction operator. This is because the eigenvalue equation **of** Hamiltonian Ĥ for|Ψ jen 〉 is found to be partially solved on the model space P by solv**in**g the eigenvalue equation**of** Ĥ for the model functions |Φ P j en〉 (see Eq. (2.60)).The second quantised effective operators ̂P Ĥ ̂P and ̂Ω are written **in** normal order (see Eqs.(2.40)-(2.42), (2.59), (2.61)), while the operator Ŵ is not. To «normalise» Ŵ , the Wick’stheorem [40, Eq. (8)] is applied. Then Ŵ = :Ŵ: + ∑ ξ :{Ŵ } ξ:, where the last term denotes thesum **of** normal-ordered terms with all possible ξ–pair contractions between the m–**body** part **of****perturbation** ̂V (for m = 1, 2) and the n–**body** part **of** wave operator ̂Ω (for n ∈ Z + ). In thiscase, 1 ≤ ξ ≤ m**in** (2m, 2n). In accordance with Lemma 2.3.9, the result :Ŵ:= 0 is immediate.Thus the operator Ŵ **in** normal order readsŴ =∞∑2∑n=1 m=1m**in** (2m,2n)∑ξ=1i=1:{ ̂P ̂V m̂Ωn ̂P }ξ : . (2.66)2.3.12 Theorem. The non-zero terms **of** effective Hamiltonian Ĥ on the model space P aregenerated by a maximum **of** eight types **of** the n–**body** parts **of** wave operator ̂Ω with respect tothe s**in**gle-electron states **of** the set I n (α ¯β) for all n ∈ Z + .Pro**of**. The pro**of** is implemented mak**in**g use **of** Lemma 2.3.9. The terms **of** wave operator ̂Ωthat generates Ĥ are drawn **in** Ŵ . Therefore it suffices to prove the theorem for the effectiveoperator Ŵ .Expand the sums **in** Eq. (2.66) as followsŴ =2∑ξ,m=1:{ ̂P ̂V m̂Ω1 ̂P }ξ : +∞∑ ( 2∑:{ ̂P ̂V 1̂Ωn ̂P }ξ : +n=2ξ=14∑ξ=1:{ ̂P ̂V)2̂Ωn ̂P }ξ : .It turns out that for n = 1, the follow**in**g three sets I 1 (α ¯β) (see items ii), iii) **of** Lemma 2.3.9)are valid **in** Eq. (2.65):I (1)1 ≡ {e, ¯v}, I (2)1 ≡ {v, ¯c}, I (3)1 ≡ {e, ¯c}, (2.67)avoid**in**g for simplicity the subscripts 1 **in** α 1 , ¯β 1 .For n ≥ 2, the T –**body** terms **of** Ŵ are derived. Here T = n−2, n−1, n, n+1. Particularly,the (n − 2)–**body** terms are derived by mak**in**g the four-pair contractions **in** { ̂P ̂V 2̂Ωn ̂P }4 . In

2 Partition**in**g **of** function space and basis transformation properties 30accordance with items ii), iii) **of** Lemma 2.3.9, it immediately follows that the n–**body** part **of**wave operator ̂Ω must **in**clude at least n−2 creation and n−2 annihilation operators, designat**in**gthe valence orbitals (item i) **of** Lemma 2.3.9). Possible sets I n (α ¯β) **of** s**in**gle-electron states thatprovide non-zero T –**body** terms are these:I n (1) ≡{v 1 , v 2 , . . . , v i−1 , e, v i+1 , . . . , v n , ¯v 1 , ¯v 2 , . . . , ¯v n }, (2.68a)I n (2) ≡{v 1 , v 2 , . . . , v n , ¯v 1 , ¯v 2 , . . . , ¯v i−1 , ¯c, ¯v i+1 , . . . , ¯v n }, (2.68b)I n (3) ≡{v 1 , v 2 , . . . , v i−1 , e, v i+1 , . . . , v n , ¯v 1 , ¯v 2 , . . . , ¯v j−1 , ¯c, ¯v j+1 , . . . , ¯v n }, (2.68c)I n (4) ≡{v 1 , v 2 , . . . , v i−1 , e, v i+1 , . . . , v j−1 , e ′ , v j+1 , . . . , v n , ¯v 1 , ¯v 2 , . . . , ¯v n }, (2.68d)I n (5) ≡{v 1 , v 2 , . . . , v n , ¯v 1 , ¯v 2 , . . . , ¯v i−1 , ¯c, ¯v i+1 , . . . , ¯v j−1 , ¯c ′ , ¯v j+1 , . . . , ¯v n }, (2.68e)I n (6) ≡{v 1 , v 2 , . . . , v i−1 , e, v i+1 , . . . , v n , ¯v 1 , ¯v 2 , . . . , ¯v j−1 , ¯c, ¯v j+1 , . . . , ¯v k−1 ,¯c ′ , ¯v k+1 , . . . , ¯v n },(2.68f)I (7)n ≡{v 1 , v 2 , . . . , v i−1 , e, v i+1 , . . . , v j−1 , e ′ , v j+1 , . . . , v n , ¯v 1 , ¯v 2 , . . . , ¯v k−1 ,¯c, ¯v k+1 , . . . , ¯v n },I n (8) ≡{v 1 , v 2 , . . . , v i−1 , e, v i+1 , . . . , v j−1 , e ′ , v j+1 , . . . , v n , ¯v 1 , ¯v 2 , . . . , ¯v k−1 ,¯c, ¯v k+1 , . . . , ¯v l−1 , ¯c ′ , ¯v l+1 , . . . , ¯v n }.This proves the theorem.(2.68g)(2.68h)Theorem 2.3.12 determ**in**es that for each fixed Fock space operator Ŝn, there are maximumeight types **of** the Hilbert space wave operators ̂Ω n . For example, if the wave function |Ψ jen 〉dist**in**guishes between s**in**gle-, two-, three-, four-particle effects (n = 1, 2, 3, 4), then Theorem2.3.12 allows to write ̂Ω n as followŝΩ 1 = ∑ I (1)1̂Ω 2 = ∑ ′I (4,5,8)2a e a †¯vω e¯v + ∑ I (2)1a v a †¯cω v¯c + ∑ I (3)1a α a α ′a †¯β′a †¯βω αα ′ ¯β ¯β′ + ∑ I (3)2+ ∑ ′a α a α ′a †¯va †¯cω αα′¯c¯v,I (2,7)2a e a †¯cω e¯c ,̂Ω 3 = ∑ ′a α a α ′a µ a †¯v a †¯v a †¯vω ′′ ′ αα ′ µ¯v¯v ′¯v ′′ + ∑ ′I (1,4)3+ ∑ ′I (3,6,7,8)3a α a α ′a µ a †¯ca †¯β′a †¯vω αα ′ µ¯v ¯β ′¯c,(2.69c)a e a v a †¯ca †¯vω ev¯v¯c + ∑ ′a e a v a †¯β′a †¯βω ev ¯β ¯β′I (2,5)3̂Ω 4 = ∑ ′a e a α ′a v a v ′′a †¯v a †¯v a †¯v a †¯vω ′′′ ′′ ′ eα ′ vv ′′¯v¯v ′¯v ′′¯v ′′′ + ∑ ′I (1,4)4+ ∑ ′I (3,6,7,8)4a e a α ′a v a v ′′a †¯ca †¯β′′a †¯v a †¯vω ′ eα ′ vv ′′¯v¯v ′ ¯β ′′¯c,(2.69d)I (1,6)2a v a v ′a v ′′a †¯ca †¯β′a †¯vω vv ′ v ′′¯v ¯β ′¯cI (2,5)4(2.69a)(2.69b)a v a v ′a v ′′a v ′′′a †¯ca †¯β′′a †¯v a †¯vω ′ vv ′ v ′′ v ′′′¯v¯v ′ ¯β ′′¯cwhere the s**in**gle-electron states α i , ¯β j are replaced with α, α ′ , µ, ¯β, e, ¯v, etc. Here and elsewhere,it is assumed that the Greek letters denote all three types **of** s**in**gle-electron states. Thesums with primes denote the follow**in**g operations

2 Partition**in**g **of** function space and basis transformation properties 31∑ ′I (4,5,8)2∑ ′I x(a,b)∑ ′I x(a,b)∑ ′I (1,4)3∑ ′I (3,6,7,8)3∑ ′≡ ∑ I (4)2≡ ∑ I (a)x≡ ∑ I (a)x≡ ∑ I (1)3≡ ∑ I (3)3δ αe δ βv + ∑ I (5)2δ βv + ∑ I (b)xδ αv + ∑ I (b)xδ αv δ µe + ∑ I (4)3δ αv δ βv δ µe + ∑ I (6)3δ αv δ βc + ∑ I (8)2δ αe δ βc ,(2.70a)δ βc , for x = 2, a = 1, b = 6 and x = 3, 4, a = 2, b = 5, (2.70b)δ αe , for x = 2, a = 2, b = 7 and x = 4, a = 1, b = 4, (2.70c)δ αe δ µv ,δ αv δ βc δ µe + ∑ I (7)3δ αe δ βv δ µv + ∑ I (8)3δ αe δ βc δ µv ,(2.70d)(2.70e)≡ ∑ δ αv δ βv + ∑ δ αv δ βc + ∑ δ αe δ βv + ∑ δ αe δ βc .(2.70f)I (3,6,7,8)4I (3)4I (6)4I (7)4I (8)4For the sake **of** brevity, **in** Eqs. (2.70a)-(2.70f) and wherever possible elsewhere, only the types**of** s**in**gle-particle orbitals will be designated, but not their values. For **in**stance, **in** Eq. (2.69b),the term ∑ ′aI (1,6) e a v a †¯β′a †¯βω ev ¯β ¯β′ conta**in**s two two-particle effective matrix elements ω ev ¯β ¯β′ with2β = v and β = c (see Eq. (2.70b)). This implies that two s**in**gle-particle orbitals ¯β and ¯β ′ are**of** whether valence (v) or core (c) type. That is, the values **of** orbitals are correspond**in**gly ¯v, ¯v ′for β = v and ¯c, ¯c ′ for β = c.2.4 Conclud**in**g remarks and discussionFor the first time, the method to calculate matrix elements—based on coord**in**ate transformationsor else the RCGC technique—has been developed (Sec. 2.2). The technique solves twoma**in** tasks. First **of** all, it reduces the number **of** multiple **in**tegrals **of** **many**-electron angularparts up to a s**in**gle one. Obta**in**ed s**in**gle **in**tegral is calculated by **in**troduc**in**g the reduced matrixelement **of** founded functions that particularly form the set **of** SO(3)–irreducible tensor operators(Sec. 2.2.2, Def**in**ition 2.2.2, Proposition 2.2.3). Secondly, the technique makes it possibleto calculate the matrix elements **of** irreducible tensor operators on the SU(2)–irreducible matrixrepresentations efficiently. As a result, the method developed here comes down to two dist**in**ctl**in**eages, one that can be traced to the basis **of** separable Hilbert space (Eq. (2.2)), and the otherto irreducible matrix representations (Eq. (2.5)) [52, 65, 67].Sec. 2.3 extends the formulation **of** **many**-electron system studied **in** Sec. 2.1 and Sec. 2.2 tothe system with variable particle number. The non-variational or else the perturbative approachto f**in**d a f**in**ite set **of** solutions **of** eigenvalue equation **of** atomic Hamiltonian has been developed.Based on Feshbach’s partition**in**g technique [71], the generalised Bloch equation [34] has beenrewritten **in** a Fock space approach (Sec. 2.3.1) [39, 70]. As a consequence, proposed algebraictechnique based on the effective operator approach (Sec. 2.3.2) leads to mean**in**gful results(Theorem 2.3.12) that directly govern the solutions **of** generalised Bloch equation as well asthe number **of** terms **of** effective Hamiltonian (Eq. (2.61)). These results become even morevaluable for the higher-order **perturbation**s.The foremost consequence followed by the suggested Fock space partition**in**g procedure isthat not all computed terms **of** wave operator (see Eq. (2.59)) attach non-zero contributions tothe terms **of** effective Hamiltonian. This fact, undoubtedly, significantly decreases the amount**of** computations necessary to f**in**d, subsequently, a fixed number **of** energy levels **of** atom.To what has been found **in** the present section so far, a no less important task is to realisea developed foundation for the effective operators **in** a second quantised representation. Twomajor problems take place to solve the task **of** perturbative expansion for atomic systems. Thefirst one is to solve the generalised Bloch equation for the wave operator on **many**-electronHilbert space (see Eqs. (2.58)-(2.59)). The wave operator is listed by the **in**f**in**ite sum **of** n–**body** terms. Consequently, for practical computations, it has to be **in**terrupted to a f**in**ite number

2 Partition**in**g **of** function space and basis transformation properties 32**of** terms by exclud**in**g the rem**in**der term. The procedure, obviously, leads to approximate values**of** operator and the wave function as well. Meanwhile, there are also good news. As alreadypo**in**ted out, Theorem 2.3.12 allows us to reduce significantly the number **of** n–**body** parts **of**wave operator. This even more simplifies computations for the approximate wave operator.Blundell et. al. [35] demonstrated that their all-order calculations **of** energies **in** cesium withomitted triple and higher-order excitations (n = 1, 2) differ from the experimental data at around1%. In their study for the beryllium-like ions, Safronova et. al. [36] used to exploit the secondorderMBPT, clearly demonstrat**in**g that their calculations differ from exist**in**g experimentaldata for the nuclear charges rang**in**g from 4 to 100 at the level **of** 50cm −1 for triplet states and500cm −1 for s**in**glet states. These results obta**in**ed by us**in**g perturbative methods with relativelylow-order excitations are **in** a smooth agreement with experimental data and thus they stronglymotivate for further **development** **of** the approach under consideration.There are two ways to solve the second quantised generalised Bloch equation for the waveoperator, one that is to express it by the sum **of** n–**body** terms, as **in** Eq. (2.59), and the other isto express it **in** terms **of** the kth–order wave operator, where k = 1, 2, . . .. The first approach iscalled the coupled-cluster (CC) approximation and the second approach is called the Rayleigh–Schröd**in**ger **perturbation** **theory** (RSPT). For the RSPT, the wave operator on N–particle Hilbertspace is found to be **of** the form̂Ω = ̂1 en +∞∑̂Ω (k) ,k=1̂Ω(k)ndef̂Ω(k) =2k∑n=1̂Ω (k)n . (2.71)To f**in**d the kth–order n–**body** terms **of** wave operator, it is sufficient to express them byEq. (2.65) replac**in**g the effective matrix elements ω n (α ¯β) with the kth–order effective matrixelements ω n(k) (α ¯β). For example, if n = 1, 2, 3, 4, then is given by Eq. (2.69) if ω’s arereplaced with the correspond**in**g ω (k) ’s.It turns out that it is a fair **of** choice which form **of** the solutions **of** wave operator to beselected, as the mathematical formulation **of** tensor structure **of** effective operators rema**in**sirrelevant. That is, the difference is **in** the ω elements only. To f**in**d the irreducible tensorform **of** effective operators such that both – CC and RSPT – approaches could be applicablesimply replac**in**g the effective matrix elements, is the second major problem **of** perturbativeformulation, studied here.To this day, the authors who work **in** the MBPT usually consider the diagrammatic representation**of** expansion terms, followed by Goldstone [23] and later by L**in**dgren et. al. [34].However, recent works **in** the higher-order **perturbation** **theory** [36, 42, 74, 75] demonstrate, **in**pr**in**cipal, the **in**efficiency **of** such representation, though it is beautiful to behold and efficient**in** **many** **in**dividual cases. The motivation to develop an algebraic technique suitable for bothMBPT approaches is encouraged by the features **of** nowadays symbolic programm**in**g tools aswell. The algebraic method that solves the tasks studied **in** the present section will be developed**in** thereafter sections.̂Ω(k)n

3 Irreducible tensor operator techniques **in** atomic spectroscopy 333 Irreducible tensor operator techniques **in** atomic spectroscopyIn the present section, the angular reduction schemes **of** totally antisymmetric tensors Ôn (Sec.2.3, Eq. (2.42)) are studied. The properties **of** irreducible tensor operators on different tensorspaces and on their tensor products are studied as well. The methods developed here are applicableto both physical operators considered **in** atomic spectroscopy and effective operatorsparticular to the atomic MBPT. Special attention is paid to the cases n = 1, 2, 3, as the contributions**of** s**in**gle, double and triple excitations are most valuable for energy corrections observed**in** MBPT.The prime issue is the developed technique to reduce a tensor space H q 1× H q 2× . . . × H q ldenoted by H l to its irreducible subspaces H q . Irreducible tensor operators on H q are classifiedby the angular reduction schemes that **in** turn are obta**in**ed mak**in**g use **of**: the l–numbers forl ∈ Z + , the irreducible representations [λ] **of** symmetric group S l , the l 2 –tuples, the permutationrepresentations ̂π **of** S l . To provide suggested technique efficiently, the method **of** commutativediagrams is orig**in**ated. The advantage **of** developed reduction technique to compare with ausual diagrammatic method used **in** the angular momentum **theory** is clarified. Namely, it isa convenience to apply it to the systems considered by a large number **of** momenta or otherSU(2)–irreducible representations.Sec. 3.1 conta**in**s a description **of** a method to classify angular reduction schemes **of** Ôn andestablishes the connection between them. In Sec. 3.2, the application to the atomic systems isdemonstrated: the irreducible tensor operators are considered when they act on l ≤ 6 electronshells.3.1 Restriction **of** tensor space **of** complex antisymmetric tensors3.1.1 Classification **of** angular reduction schemesTo classify reduction schemes, the decomposition **of** Ôn **in**to irreducible tensor series **of** operatorson H q is the most convenient choice. In this case, all creation a αi ≡ a λ iµ iand annihilationa †¯βj ≡ a λ j†µ joperators are represented by a α kβ k. Also, it makes sense to consider the notation **of**operator str**in**g as followsdefÔ l = a α 1β 1a α 2β 2. . . a α lβ l, α k ≡ 1 /2 λ k , β k ≡ ±1 /2 µ k , (3.1)where the irreducible tensor operators a α k satisfy the follow**in**g anticommutation rule{aα k}β k, a α lβ l = (−1)α k −β k +1 δ(α k , α l )δ(β k , −β l ). (3.2)Then for l ∈ 2Z + , the l–length str**in**g Ôl represents Ôl/2 on H l if ∑ lk=1 β k = ∑ lk=1 µ k. Thatis, the sum **of** quasisp**in** basis **in**dices is zero and thus the matrix representation **of** Ôl is diagonalwith respect to the particle number. Throughout this text, the last condition is always satisfiedfor l ∈ 2Z + , unless explicitly stated otherwise.A classification **of** reduction schemes **of** Ôl with l = 2, 3, . . . , 5 has been first establishedby Jucys et. al. [10, Sec. 5-21]. However, only three types **of** schemes have been considered(A 0 , A 1 , A 2 ), concentrat**in**g ma**in**ly on the permutation properties **of** angular momentum. Incontrast, a general algorithm to classify reduction schemes suitable for Ôl with l ∈ Z + will bedemonstrated here. The schemes will be explicitly listed for l = 2, 3, . . . , 6.To solve the task for l–length str**in**g Ôl, it is convenient to **in**troduce the l–number thatconta**in**s numerals 1 and 2 only.3.1.1 Def**in**ition. The l–number is a number conta**in****in**g l 2 numerals whose values are 1 and 2only.It follows from Def**in**ition 3.1.1 thatwhere h 1 and h 2 denote the multiplicities **of** 1 and 2 **in** l.l 2 = h 1 + h 2 = l − h 2 , (3.3)

3 Irreducible tensor operator techniques **in** atomic spectroscopy 34Example. For example, there are two 3–numbers: 12 and 21; h 1 = h 2 = 1, l 2 = 2. On the otherhand, number 3 is partitioned by several different ways which, **in** general, are easily obta**in**edfrom the tables **of** irreducible representations (recall «irreps») [λ] **of** symmetric group S l (see,for example, [76, Appendix 1, p. 261]). In this case, [λ] = [3], [21], [1 3 ]. Pick out irreps withthe numbers that are present **in** 3–number. This is [λ] = [21]. Consequently, the 3–numbers 12and 21 are to be convenient to characterise them by the S 3 –irreducible representation [21]; thusthey belong to the same conjugacy class (α) = (1 1 2 2 ) **of** S 3 .In general, the l–numbers are characterised by the S l –irreducible representations **of** the type[2 h 21 h 1]. Particularly, if h 1 = 0 and h 2 = 1, the symmetric S 2 –irreducible representation [2] isomitted. Instead **of** that, the antisymmetric representation [1 2 ] is chosen.The example considered above demonstrates that the characterisation **of** l–numbers by theS l –irreducible representations is **in**sufficient s**in**ce both 3–numbers belong to the same class **of**S 3 . Extra characteristics are necessary for a unique labell**in**g **of** l–numbers. The numbers 12and 21 differ by the order**in**g **of** numerals 1 and 2. Consequently, it is convenient to exploitthe properties **of** l 2 –tuple which is an ordered list **of** l 2 numerals 1, 2. Then 3–numbers areadditionally characterised by the two dist**in**ct 2–tuples [[12]] and [[21]].3.1.2 Proposition. For a fixed l ∈ Z + , the l–numbers form a set E l with card**in**ality#E l = l 2!h 1 !h 2 ! , l 2 = h 1 + h 2 , (3.4)where h 1 and h 2 are the multiplicities **of** numbers 1 and 2 **in** the S l –irreducible representation[λ] ≡ [2 h 21 h 1].Pro**of**. The pro**of** is a simple comb**in**atorial task to f**in**d all possible permutations **of** numbers 1,2 whose multiplicities are h 1 , h 2 , where h 1 + h 2 = l 2 . If h 2 = 1, there are l 2 ways to permute 2**in** l. If h 2 = 2, there are l 2 (l 2 − 1)/2 ways to permute both 2 numerals **in** l, etc. For arbitraryh 2 , there are ( l 2h 2)ways to permute h2 numerals 2 **in** l.Still, one more property **of** l–numbers ought to be determ**in**ed. It is a connection betweentwo l–numbers with different l. First **of** all, pick out the κth l–number from the set E l , whereκ = 1, 2, . . . , #E l . This l–number is uniquely characterised by the S l –irreducible representation[λ] and by the l 2 (κ)–tuple. Secondly, make a change 2 → 1 ′ , where a prime over numeral1 is written to dist**in**guish 1 ′ from 1. By do**in**g this, we restricted S l to its subgroup S l2 . That is,2 → 1 ′ leads to S l → S l ′, where l ′ = l 2 = l − h 2 (see Eq. (3.3)). Obta**in**ed number conta**in**sh 1 numerals 1 and h 2 numerals 1 ′ . The l ′ –number must be made **of** it. This is done **in** the sameway as for l–number. F**in**d all S l ′–irreducible representations [λ ′ ] ≡ [2 h′ 2 1h ′ 1 ]. Then each κ ′ thl ′ –number from the set E l ′ is uniquely characterised by the S l ′–irreducible representation [λ ′ ]and by the l ′ 2(κ ′ )–tuple **of** length l ′ 2 = l ′ − h ′ 2. F**in**ally, pick out those l ′ 2(κ ′ )–tuples which aretransformed **in**to the l 2 (κ)–tuple if revers**in**g 1 ′ to 2 back aga**in**.Example. Let us go through an example **in** detail. Assume that the irrep [λ] = [21 2 ]. By Proposition3.1.2, h 1 = 2, h 2 = 1, l = 4, l 2 = 3, #E 4 = 3. This means E 4 conta**in**s three 4–numberscharacterised by the S 4 –irreducible representation [21 2 ]. That is, E 4 = {112, 121, 211}. Pickout the 1st (κ = 1) 4–number 112 which is additionally labelled by the 3(1)–tuple [[112]]. Now,make a change 2 → 1 ′ which leads to the restriction S 4 → S 3 . Obta**in**ed number is 111 ′ . TheS 3 –irreducible representation **of** the type [2 h′ 2 1h ′ 1 ] is [λ ′ ] = [21]. Consequently, h ′ 1 = h ′ 2 = 1,l ′ 2 = 2 and #E 3 = 2. The set E 3 = {12, 21}, where the 3–numbers 12 (κ ′ = 1) and 21 (κ ′ = 2)are additionally labelled by the 2(1)–tuple [[12]] and by the 2(2)–tuple [[21]], respectively. Thenumbers 2 **in** 2(κ ′ )–tuples are obta**in**ed from 111 ′ by mak**in**g the sum 2 = 1 + 1 ′ for κ ′ = 1and the sum 2 = 1 + 1 for κ ′ = 2. The second sum is excluded s**in**ce **in** this case—revers**in**g1 ′ to 2 back aga**in**—we get a tuple [[22]] which is absent **in** E 4 . The tuple [[22]] is characteristicfor the 4–number that is labelled by the S 4 –irreducible representation [2 2 ]. Conversely, the sum2 = 1 + 1 ′ fits the irrep [21 2 ]. Indeed, if 1 ′ → 2, then [[12]] → [[111 ′ ]] → [[112]]. We write[[112]] ⋉ [[12]] by us**in**g the semijo**in** notation ⋉.

3 Irreducible tensor operator techniques **in** atomic spectroscopy 35The algorithm considered above is easy to apply to the l–length str**in**g Ôl for the classification**of** angular reduction schemes if replac**in**g 1 with α k and 2 with α kk ′, where the irreps α kk ′are **in** the Kronecker product α k × α k ′. Then each tuple t [λ]κ characteristic for the κth scheme(κth l–number) obeys the mean**in**g **of** reduction scheme **of** Ôl.Example. The tuple [[12]] stipulates the scheme t [21]1 = (α 1 , α 2 α 3 (α 23 )α), where, **in** general,α 12...l ≡ α.Tab. 3: Reduction schemes **of** Ô2−5(α) Tuples Scheme(21 ) [[2]] T [12 ]1(1 1 2 1) [[12]] T [21]1[[21]] T [21]2(22 ) [[22]] T [22 ]1(1 1 3 1) [[211]] ⋉ T [21]2 T [212 ]1[[121]] ⋉ T [21]2,1 T [212 ]2,3[[112]] ⋉ T [21]1 T [212 ]4(2 1 3 1) [[221]] ⋉ T [21]2,1 T [22 1]1,2[[122]] ⋉ T [21]1,2 T [22 1]3,4[[212]] ⋉ T [21]2,1 T [22 1]5,6(1 1 4 1) [[2111]] ⋉ T [212 ]1 T [213 ]1[[1211]] ⋉ T [212 ]1,3,2 T [213 ]2,3,4[[1121]] ⋉ T [212 ]4,2,3 T [213 ]5,6,7[[1112]] ⋉ T [212 ]4 T [213 ]8Tab. 4: Reduction schemes **of** Ô6(α) Tuples Scheme( ) 32[[222]] ⋉ T [21]2,1 T [23 ]1,2(2 1 4 1) [[2211]] ⋉ T [212 ]1,2,3 T [22 1 2 ]1,2,3[[1221]] ⋉ T [22 ]1 T [22 1 2 ]4[[1221]] ⋉ T [212 ]1−4 T [22 1 2 ]5−8[[1122]] ⋉ T [212 ]2,3,4 T [22 1 2 ]9,10,11[[2121]] ⋉ T [22 ]1 T [22 1 2 ]12[[2121]] ⋉ T [212 ]1−4 T [22 1 2 ]13−16[[2112]] ⋉ T [22 ]1 T [22 1 2 ]17[[2112]] ⋉ T [212 ]1,4 T [22 1 2 ]18,19[[1212]] ⋉ T [22 ]1 T [22 1 2 ]20[[1212]] ⋉ T [212 ]1−4 T [22 1 2 ]21−24(1 1 5 1) [[21111]] ⋉ T [213 ]1 T [214 ]1[[12111]] ⋉ T [213 ]1−4 T [214 ]2−5[[11211]] ⋉ T [213 ]2−7 T [214 ]6−11[[11121]] ⋉ T [213 ]5−8 T [214 ]12−15[[11112]] ⋉ T [213 ]8 T [214 ]16Tab. 5: The schemes associated to A 0 , A 1 , A 2A p a \ l 2 3 4 5 6A 0 T [12 ]1 T [21]2 T [212 ]1 T [213 ]1 T [214 ]1A 1 T [12 ]1 T [21]2 T [22 ]1 T [22 1]2 T [22 1 2 ]3A 2 – – T [22 ]1 T [22 1]1 T [23 ]1a [10, Sec. 5-21, Eq. (21.12)]Algorithm. The procedure to classify reduction schemes **of** Ôl is applicable to any l if all otherschemes **of** Ôl ′ with l′ = 2, 3, . . . , l − 1 are determ**in**ed. To perform a task, the follow**in**g stepsshould be accomplished.I. For a given l, make the set E l **of** l–numbers (Def**in**ition 3.1.1, Proposition 3.1.2) such thateach l–number is characterised by the S l –irreducible representation [λ] ≡ [2 h 21 h 1] and bythe l 2 (κ)–tuple t [λ]κ **of** length l 2 = l − h 2 , where κ = 1, 2, . . . , #E l (Eq. (3.4)).II. Make a restriction 2 → 1 ′ so that S l → S l2 .

3 Irreducible tensor operator techniques **in** atomic spectroscopy 36a) If l 2 ≤ 3, then pick out the l ′ 2(κ ′ )–tuples t [λ′ ]κwhich characterise l ′2 –numbers labelledby the S l2 –irreducible representations [λ ′ ] ≡ [2 h′ 2 1h ′ 1 ] and which are transformed **in**tothe l 2 (κ)–tuple when revers**in**g 1 ′ to 2 back aga**in**. Write a correspondence by t [λ]κ ⋉t [λ′ ]κ. ′b) If l 2 > 3, repeat the procedure **of** restriction for S l2 to its subgroup S l ′2. If l ′ 2 ≤ 3, thenperform item a), otherwise make another restriction S l ′2→ S l ′′, etc. At each step **of**2restriction, pick out the tuples that transform to the **in**itial one by mak**in**g the change1 ′ → 2. Write t κ [λ] ⋉ t [λ′ ]κ⋉ . . . ⋉ t [λ′′ ]′ κ, where the last tuple is **of** length l ′′′′ 2 ≤ 3.III. Mark **of**f determ**in**ed l 2 (κ)–tuple t [λ]κT [λ]eκexpressly by T [λ] defeκ= t [λ]κ ⋉ t [λ′ ]κ⋉ . . . ⋉ t [λ′′ ]′ κ, where′′denotes reduction scheme **of** Ôl when the changes 1 → α k and 2 → α kk ′ are made.As a result, the semijo**in** notation obeys its orig**in**al mean**in**g: the reduction scheme T [λ]eκ**of** Ôl consists **of** only those reduced Kronecker products which are determ**in**ed **in** thereduction scheme T [λ′ ]κ**of** ′ Ôl ′. The numeration **of** **in**dex ˜κ is arbitrary. Particularly,= t [λ]κ if l ≤ 3.T [λ]eκAll angular reduction schemes **of** Ôl for l = 1, 2, . . . , 6 are listed **in** Tabs. 3-4, wherenotations T [λ]a,b,...= t κ[λ] ⋉ T [λ]′[λ]a ′ ,b ′ ,...and Ta−b = t[λ] κ ⋉ T [λ]′a ′ −bsignify that T [λ]′a = t [λ]κ ⋉ T [λ]′a, ′T [λ]b= t [λ]κ ⋉ T [λ]′b, T [λ]′ a+1 = t [λ]κ ⋉ T [λ]′a ′ +1 , etc. For l = 2, 4, 6, the operators Ô1, Ô 2 , Ô 3 arereduced accord**in**g to the schemes: (i) T [12 ]1 for n = 1; (ii) T [22 ]κ , T [212 ]κfor n = 2; (iii) T [23 ]′ κ ,T [22 1 2 ]κ ′ , T [214 ]κ ′′ for n = 3.that are related to A p schemes with p = 0, 1, 2 are listed **in** Tab.5, where A 2 exists if l ≥ 4. It is clear that Tabs. 3-4 significantly extend the classificationpresented **in** Ref. [10].Reduction schemes T [λ]κ3.1.2 Correspondence **of** reduction schemesDist**in**ct schemes T κ[λ] associated to Ôn have been widely studied **in** several works so far. InRefs. [14, 15], the authors considered schemes T [12 ]1 , T [21]1,2 , T [22 ]1 which are also widespread**in** Refs. [54, 55, 77]. This is, however, a case n = 1, 2. The schemes T [23 ]1,2 , T [22 1 2 ]12 , T [22 1 2 ]17associated to Ô3 have been studied **in** Refs. [49, 54].Any scheme T κ [λ] **of** Ôl is l**in**ked to another one T [λ′ ]κuniquely. To f**in**d coefficients thattransform one scheme **in**to another, is the ma**in** task **of** the ′present section. The transformationcoefficients for schemes with l = 3, 4, 5 can be found **in** [10, Sec. 5]. Therefore the transformationproperties **of** schemes associated to Ô3 will be studied only.It can be verified by pass**in**g to Tab. 4 that there are **in** total 42 · 42 = 1, 764 such transformationcoefficients for n = 3. It can be also easily verified that it suffices to determ**in**e 42coefficients that relate one separate scheme with all the rest, **in**clud**in**g itself. These coefficientswill be called the basis coefficients.Assume that **in** any scheme, irreps α k (see Eq. (3.1)) are listed **in** the order α 1 , α 2 , . . . , α 6 ;irreps **in** the Kronecker product α i × α j will be labelled by α ij . Expand the operator Ô6 on H 6by the sum **of** irreducible tensor operators Ôα β ([λ]κ) on Hq , where each Ôα β ([λ]κ) is associatedto reduction scheme T κ[λ] . ThenÔ 6 = ∑ ∑Ôβ α ([λ]κ) I([λ]κ), (3.5)αβ α ζ∈Γξwhere the **in**dices ζ ∈ Γ ξ = {ζ 1 , ζ 2 , ζ 3 , ζ 4 } that label **in**termediate irreps depend on a specifiedscheme. The coefficients I([λ]κ) are found from the expressionÔβ α ([λ]κ) = ∑ Ô 6 I([λ]κ), I def= {1, 2, . . . , 6}. (3.6)β i∈I

3 Irreducible tensor operator techniques **in** atomic spectroscopy 37For example, if [λ] = [2 2 1 2 ], κ = 12, then Ôα β ([22 1 2 ]12) is associated to the scheme (see Tab.4) T [22 1 2 ]12 = [[2121]] ⋉ [[22]] = ( α 1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ) and thusÔβ α ([2 2 1 2 ]12) =[[[a α 1× a α 2] α 12× a α 3] α 123× [[a α 4× a α 5] α 45× a α 6] α 456] α β, (3.7)I([2 2 1 2 ]12) = ∑〈α 1 β 1 α 2 β 2 |α 12 β 12 〉〈α 12 β 12 α 3 β 3 |α 123 β 123 〉〈α 4 β 4 α 5 β 5 |α 45 β 45 〉β 12 β 123β 45 β 456× 〈α 45 β 45 α 6 β 6 |α 456 β 456 〉〈α 123 β 123 α 456 β 456 |αβ〉. (3.8)3.1.3 Def**in**ition. The coefficients ɛ ξ , ξ ∈ {1, 2, . . . , 42} such thatɛ ξ ≡ ( ∣T [22 1 2 ] ∣T [λ] ) def∑=defI ξ =⎧⎨⎩are called the basis coefficients.12ξβ i∈II([2 2 1 2 ]12) I ξ , (3.9)I([2 3 ]ξ), ξ = 1, 2,I([2 2 1 2 ] ξ − 2), ξ = 3, 4, . . . , 26,I([21 4 ] ξ − 26), ξ = 27, 28, . . . , 42(3.10)It is easy to verify that there exists a map τ ξ : T [22 1 2 ]12 −→ T κ[λ] such thatÔβ α ([λ]κ) =∑Ôβ α ([2 2 1 2 ]12)ɛ ξ . (3.11)α η∈Υ\ΥξThe sum is over all irreps α η whose **in**dices η are absent **in** T κ[λ] . Υ ξ is a set **of** **in**dices **of**irreps **in** T κ [λ] . Particularly, Υ 14 ≡ Υ. Eq. (3.9) **in**dicates that there exists an **in**verse mapτ −1ξ: T [λ′ ]′ κ−→ T [22 1 2 ]′ 12 . Then the composition τ ξ ◦τ −1ξ: T [λ′ ]′ κ ′tensor operators Ôα β ([λ]κ) and Ôα β ([λ′ ]κ ′ ) byÔ α β ([λ]κ) =If ξ = ξ ′ , then τ ξ ◦ τ −1ξ∑α η∈Υξξ ′(identity with respect to operations on irreducible representations**in** given scheme T [λ]κÔ α β ([λ ′ ]κ ′ )ɛ ξ ɛ ξ ′, Υ ξξ ′= id T[λ]κ) and thus−→ T [λ]κrelates two irreducibledef= ( (Υ\Υ ξ ) ⋃ (Υ ξ ′\Υ) ) \Υ ξ . (3.12)∑Eqs. (3.12)-(3.13) stipulate the follow**in**g corollary.α η∈Υ\Υξɛ 2 ξ = 1. (3.13)3.1.4 Corollary. The entriesE ξξ ′ = E ξ ′ ξdef= ∑ɛ ξ ɛ ξ ′, M ξξ ′def= (Υ\Υ ξ ) ⋃ (Υ\Υ ξ ′) (3.14)α η∈Mξξ ′form a 42 × 42 transformation matrix E such thatÔβ α ([λ]κ) =∑E ξξ ′α η∈Υξ ′ \Υ ξÔβ α ([λ ′ ]κ ′ ). (3.15)It is by no means obvious that E ξ 14 = E 14 ξ = ɛ ξ .Explicit expressions **of** ɛ ξ are to be found exploit**in**g the angular momentum technique basedon whether diagrammatic representation followed by Refs. [10, 12] or algebraic manipulations.Here, the algebraic approach actualised by a symbolic programm**in**g package NCoperators [78](see also Appendix D) is a preferable one. Obta**in**ed expressions are listed **in** Appendix A.

3 Irreducible tensor operator techniques **in** atomic spectroscopy 38Def**in**ition 3.1.3 **in**dicates that the coefficients ɛ ξ are basis-**in**dependent. Indeed, Eqs. (A.1a)-(A.42a) assert that the basis coefficients are represented by 3nj–symbols. Specifically { }a b ed c fdenotes 6j–symbol. Expressions **of** 12j–symbols **of** the second k**in**d (see Eqs. (A.12a), (A.34a),(A.37a), (A.40a)) are determ**in**ed accord**in**g to Eq. (19.3) **in** Ref. [10, Sec. 4-19, p. 102]. Thedef**in**ition **of** 15j–symbol **of** the second k**in**d (see Eq. (A.33a)) is found **in** Ref. [12, Sec. 4-34,p. 212, Eq. (34.1)]. The triangular deltas ∆(a, b, c) are encountered due to the orthogonalitycondition **of** CGCs.Operators Ôα ([1 2 ]1) associated to the scheme T [12 ]1 are found to be the build**in**g blocks(recall numerals 2 **in** l–numbers) along with a α k (numerals 1 **in** l–numbers) **in** all other operatorsÔ α ([λ]κ). Therefore their properties stipulate the carriage **of** any operator Ôα ([λ]κ). Theprimary and most precious characteristics **of** Ôα ([1 2 ]1) are received from Eq. (3.2). For thesake **of** clarity, it is useful to alter Ôα ([1 2 ]1) by a usual notation W α kl (λk λ l ).Reduce the Kronecker products α k × α l , α l × α k **in** Eq. (3.2). It requires little effort to f**in**dthatW α klβ kl(λ k λ l ) + (−1) α k+α l +α klW α klβ kl(λ l λ k ) = −[α k ] 1/2 δ(λ k , λ l )δ(α kl , 0). (3.16)The follow**in**g results are up front.W α kl(λ k λ l ) =(−1) α k+α l +α kl +1 W α kl(λ l λ k ), λ k ≠ λ l , (3.17a)W 0 (λ k λ k ) = − [λ k ] 1/2 / √ 2,(3.17b)W α kk(λ k λ k ) =0, α kk ∈ 2Z + , (3.17c)W α kk(λ k λ k ) ≠0, α kk ∈ 2Z + + 1. (3.17d)If W α kl (λk λ l ) acts on a tensor space H Q kl × HL kl × HS kl, then Eq. (3.17) agrees with Eqs.(8.16), (12.39) **in** Ref. [50]. It is worth to give heed to operators W 0λ kk and W1λ kk. The firstone is a scalar on H Q kk , while the second is not. In agreement with Eq. (3.17d), W0λ kk is nonzeroif λ kk is odd; W 1λ kk is non-zero if λkk is even. On the other hand, the irreducible tensoroperator W λ kk (λk λ k ) on H Λ kk is non-zero if only λkk is even. This is easy to verify from theanticommutation rule {a λ kµ k, a λ lµ l} = 0 that holds for creation operators. But W λ kk is also a scalaron H Q kk . This immediately implies that the antisymmetric two-electron state characterised bya configuration lk 2 (**in** LS-coupl**in**g) is generated by W 1λ kk or Wλ kk depend**in**g on the tensorspace under consideration. It is rather not embed by a tensor operator W 0λ kk .3.1.3 PermutationsThe classification **of** angular reduction schemes T κ[λ] **of** Ôl is still **in**sufficient. To accomplishthe study to its f**in**al stage, the basis coefficients (Appendix A) are to be supplemented by thequantities that relate not only reduction schemes but also reduction schemes with somehowpermuted irreps with**in** them.The permutations **in** schemes are realised through the permutation (reducible) representationŝπ or else the permutation operators **of** S l [79, Sec. 1.3, p. 9] such that( )1 2 · · · l̂π α i = α π(i) ∀i = 1, 2, . . . , l, π =∈ Sπ(1) π(2) · · · π(l) l . (3.18)Here and elsewhere, the **in**dices **of** irreps α i enumerate operators **in** a l–length str**in**g Ôl; thevalues **of** α i and α j with i ≠ j can be identical, though (see Sec. 3.1.4).For l ≤ 5, the permutation properties related to the angular reduction schemes A 0 , A 1 , A 2have been considered **in** Ref. [10]. This is, aga**in**, a motive to concentrate on the case l = 6only.The symmetry group S 6 conta**in**s 6! elements π, each **of** which is a composition **of** 2–cyclesor else transpositions (ij). For example, π = ( 1 2 3 4 5 63 1 2 4 5 6)= (132) is a product **of** two 2–cycles:(13)(23), (12)(13), (23)(12). It is easy to verify that there are l(l − 1)/2 = 15 non-trivial2–cycles that generate elements π **of** S 6 :

3 Irreducible tensor operator techniques **in** atomic spectroscopy 39(12) (23) (34) (45) (56)(13) (24) (35) (46)(14) (25) (36)(15) (26)(16)(3.19)To cognise the correspondence **of** the permutation operator ̂π to the associate element π = (ij),the notation ̂π ij is preferred. For example (see Eq. (3.17a)),̂π ij W α ij(λ i λ j ) = W α ij(λ j λ i ) = ϖ ij W α ijdef(λ i λ j ), ϖ ij = (−1) α i+α j +α ij +1 . (3.20)Note, Eq. (3.18) **in**dicates that ̂π ij α ij = α ji . But |α i − α j | ≤ α ij ≤ α i + α j and thus α ij = α ji .In general, if there is a map p ij : T [22 1 2 ]12 −→ (̂π ij T ) [22 1 2 ]12 such that̂π ij Ôβ α ([2 2 1 2 ]12) =∑ε ij Ôβ α ([2 2 1 2 ]12), (3.21)then (see Eq. (3.11))̂π ij Ô α β ([λ]κ) =∑α η∈Υ\bπij Υα η∈Aijξξ ′ ε ij ɛ ξ ′ɛ (ij)(ξ)Ôβ α ([λ ′ ]κ ′ def), ɛ (ij)(ξ) = ̂π ij ɛ ξ , (3.22)A ijξξ ′ def= ( (Υ\̂π ij Υ) ⋃ (̂π ij (Υ\Υ ξ )) ⋃ (Υ ξ ′\Υ) ) \̂π ij Υ ξ , (3.23)where it is assumed that irreps α i are ordered (α 1 , α 2 , . . . , α 6 ) **in** Ôα β ([λ′ ]κ ′ ), while the ith andjth irreps are permuted **in** ̂π ij Ôβ α ([λ]κ). Eq. (3.22) relates two irreducible tensor operators associatedto dist**in**ct angular reduction schemes (T κ [λ] and (̂π ij T ) [λ′ ]κ) with the different order**in**g **of**irreps with**in** them. The relation is realised through basis coefficients ′and the recoupl**in**g coefficientsdef**in**ed **in** Eq. (3.21). Eq. (3.22) is easy to generalise for any permutation (ij)(kl) . . . (pq)by repeat**in**g the procedure several times.Eq. (3.22) is more preferable if rewritten as follows.3.1.5 Def**in**ition. The map τ ξ ◦ p ij ◦ τ −1ξ: T [λ′ ]′ κ−→ (̂π ′ij T ) [λ]κ is realised through the entriesE ijξ ′ ξ = ̂π ijE ijξξ ′ def= ∑α η∈Nξ ′ ξε ij ɛ ξ ′ɛ (ij)(ξ) ,N ξ ′ ξdef= (Υ\̂π ij Υ) ⋃ (Υ\Υ ξ ′) ⋃ (̂π ij (Υ\Υ ξ )) (3.24)**of** 15 42 × 42 transformation matrices E ij so that∑̂π ij Ôβ α ([λ]κ) = E ijξ ′ ξα η∈Υξ ′ \bπ ij Υ ξ Ôα β ([λ ′ ]κ ′ ). (3.25)3.1.6 Proposition. The transformation coefficients E ξ ′ ξ and E ijξ ′∑ξE ξ ′ ξ =α η∈Rξ ′′E ijξ ′ ξ ′′ E ijξ ′′ ξ ,are related byRdefξ ′′ = ( ) ⋃((̂π ij Υ ξ ′′)\Υ ξ (̂πij Υ ξ ′′)\Υ ξ ′), ∀i, j ∈ I. (3.26)Pro**of**. The most appropriate tool to prove the proposition is the exploitation **of** map products.First **of** all, note that p ij ◦ p ij = id [2 T 2 1 2 ] : T [22 1 2 ]12 −→ T [22 1 2 ]12 is an identity **in** T [22 1 2 ]12 . It is12obvious thatwhich can be rewritten byτ ξ ′′ ◦ p ij ◦ τ −1ξ◦ τ ξ ◦ τ −1ξ ′ = τ ξ ′′ ◦ p ij ◦ τ ξ ′τ ξ ′ ◦ τ −1ξ= (τ ξ ′ ◦ p ij ◦ τ −1ξ ′′ ) ◦ (τ ξ ′′ ◦ p ij ◦ τ −1ξ).

3 Irreducible tensor operator techniques **in** atomic spectroscopy 40Accord**in**g to Corollary 3.1.4 and Def**in**ition 3.1.5, the three maps τ ξ ′ ◦ τ −1ξ, τ ξ ′ ◦ p ij ◦ τ −1ξ ′′and τ ξ ′′ ◦ p ij ◦ τ −1ξare realised through the transformation coefficients E ξξ ′, E ijξ ′′ ξand E ij′ ξξ, ′′respectively. Thus E ξξ ′ = ∑ E ijξξE ij′′ ξ ′′ ξ. Replac**in**g ξ with ξ ′ and recall**in**g that E ′ ξξ ′ = E ξ ′ ξ, Eq.(3.26) is satisfied. The **in**dices **of** summation are easily derived subtract**in**g the sets **of** **in**dices**of** irreps that are present on both sides **of** expression.3.1.7 Corollary. If the map p ij : T [22 1 2 ]12 −→ (̂π ij T ) [22 1 2 ]12 is realised through the coefficient ε ij ,then∑ε 2 ij = 1. (3.27)α η∈Υ\bπij ΥPro**of**. Eq. (3.27) directly follows from the fact that p ij ◦ p ij = id [2 T 2 1 2 ]. The **in**dices **of**12summation are easily found recall**in**g that (see Def**in**ition 3.1.3)1 = ( ∣T [22 1 2 ] ∣T [22 1 2 ]) ∑=1212α η∈Υ\bπij Υ( ∣[2 T 2 1 2 ] ∣(̂π ij T ) [22 1 2 ])( ∣12 (̂πij T ) [22 1 2 ]1212∣T [22 1 2 ]12).In agreement with Eq. (3.21), it is useful to generalise Eq. (3.20) by **in**troduc**in**g a mapp ′ ij : T κ[λ] −→ (̂π ij T ) [λ]κ ,X [22 1 2 ]12̂π ij Ô([λ]κ) = ϖ ij Ô([λ]κ) (3.28)which is valid for a specified [λ], κ, (ij). Then for any operator ̂X([λ]κ) associated to thescheme X κ[λ] **in** Tab. 4, the paths τ ξ , τ −1ξ, p ij , p ′ ij satisfy the follow**in**g equivalentsτ ξ X κ [λ] ≡ ∑(3.29a)X [22 1 2 ]12X [λ]κτ −1ξX [22 1 2 ]12≡α η∈Υ\Υξɛ ξ ̂Xα β ([2 2 1 2 ]12),∑α η∈Υξ \Υp ij (̂π ij X) [22 1 2 ]12≡ ∑α η∈Υ\bπij Υɛ ξ ̂Xα β ([2 2 1 2 ]12),ε ij ̂Xα β ([2 2 1 2 ]12),(3.29b)(3.29c)X [λ]κp ′ ij(̂π ij X) [λ]κ ≡ ϖ ij ̂Xα β ([λ]κ). (3.29d)Note, X does not necessary represent T . It also fits the quantities ̂π ij...l T , ̂π iĵπ kl . . . T , etc.which particularly denote schemes with permuted irreps with**in** them. The same argument foldsfor operators ̂X.The f**in**al task to accomplish the classification **of** angular reduction schemes **of** irreducibletensor operators Ôα ([λ]κ) for l = 6, is to f**in**d the transformation coefficients E ijξ ′ ξ(see Def**in**ition3.1.5), as all other coefficients Eξ π ′ ξ aris**in**g duo to permutations **of** ̂π = ̂π ij . . . ̂π kl̂π pq arefound by apply**in**g Eq. (3.25) for several times. To f**in**d E ijξ ′ ξ , recoupl**in**g coefficients ε ij are tobe established. By Eq. (3.19), there are 15 recoupl**in**g coefficients, each **of** which is caused bythe transposition (ij). The realisation **of** an assignment by us**in**g the traditional method, that is,to recouple Kronecker products **of** irreps (particularly, angular momenta) **of** given scheme, is**in**efficient due to the complexity **of** schemes. Therefore another method suitable for any l if thebasis coefficients are known (see items I-III **in** Sec. 3.1.1) will be developed.Algorithm (Method **of** commutative diagrams). The idea is simple: f**in**d a commutative diagram

3 Irreducible tensor operator techniques **in** atomic spectroscopy 41(̂π ij T ) [22 1 2 ]p ijφ 2s12 E [λ φ2s] 2s−1κ 2sD [λ φ2s−1] 2s−2κ 2s−1 · · ·φ s+1(̂π ij C) [λ s+1]κ s+1p ′ ij(3.30)T [22 1 2 ]12φ 1 A [λ 2]κ 2φ 2 B [λ 3]κ 3φ 3 · · ·φ s C [λ s+1]κ s+1such that for a m**in**imal number ν s = 2s + 1 **of** paths, the composition equals toφ 2s ◦ φ 2s−1 ◦ . . . ◦ φ s+1 ◦ p ′ ij ◦ φ s ◦ φ s−1 ◦ . . . ◦ φ 1 = p ij . (3.31)In Eq. (3.30), A, B, . . . , E denote angular reduction schemes; the maps φ a depend on specifiedschemes and they mark **of**f any suitable map τ ξ , τ −1ξor p ′ kl ((kl) ≠ (ij)). Eq. (3.30) **in**dicates t**of****in**d a specified operator Ĉα ([λ s+1 ]κ s+1 ) associated to the scheme C [λ s+1]κ s+1 such that Eq. (3.28)holds true. Once the operator Ĉα ([λ s+1 ]κ s+1 ) is found, the paths from C to E are easy toperform. F**in**ally, obta**in**ed commutative diagram is rewritten **in** algebraic form by us**in**g Eq.(3.29).The simplest to f**in**d are recoupl**in**g coefficients ε ij with j = i + 1 (see the first row **in** Eq.(3.19)). For all **of** them, except for ε 12 = ϖ 12 , ε 45 = ϖ 45 , the commutative diagram readsτ −1ξ(̂π i i+1 T ) [22 1 2 ]12(̂π i i+1 T ) [λ]κp i i+1p ′ i i+1(3.32)T [22 1 2 ]12τ ξT [λ]κand thus ν s = ν 1 = 3, p i i+1 = τ −1ξ◦ p ′ i i+1 ◦ τ ξ . This is because ̂π i i+1 permutes any twoirreps adjacent to one another with**in** a given scheme. The values **of** [λ], κ and ξ depend oni ≤ 5. The last step is to write an algebraic expression **of** given diagram. Start from the pathτ ξ : T [22 1 2 ]12 −→ T κ[λ] . By Eq. (3.29a), write ∑ α η∈Υ\Υξɛ ξ Ôβ α([22 1 2 ]12). For the next pathp ′ i i+1 : T [λ]κ−→ (̂π i i+1 T ) [λ]κ , write (see Eq. (3.29d)) ∑ α η∈Υ\Υξϖ i i+1 ɛ ξ Ô α β ([22 1 2 ]12). F**in**ally,for τ −1ξ: (̂π i i+1 T ) [λ]κ −→ (̂π i i+1 T ) [22 1 2 ]12 , write∑̂π i i+1 Ôβ α ([2 2 1 2 ]12) =α η∈Fξ \bπ i i+1 Υϖ i i+1 ɛ ξ ɛ (i i+1)(ξ) Ô α β ([2 2 1 2 ]12), (3.33)defwhere F ξ = (Υ\Υ ξ ) ⋃ (̂π i i+1 (Υ ξ \Υ)). The converse mapp**in**g p i i+1 : T [22 1 2 ]12 −→ (̂π i i+1 T ) [22 1 2 ]12is realised by Eq. (3.21). Thus both right hand sides **of** Eqs. (3.21), (3.33) are equal. This impliesε i i+1 =∑ϖ i i+1 ɛ ξ ɛ (i i+1)(ξ) . (3.34)α η∈Fξ \ΥFor i = 2, ξ = 6; for i = 3, ξ ∈ {1, 2, . . . , 5}; for i = 5, ξ = 19.The procedure to f**in**d commutative diagrams for the rest **of** ε ij coefficients with **in**dicesj = i + 2, i + 3, i + 4, i + 5 is analogous, except that the diagrams are more complicated. Forexample, the coefficient ε 24 is found from the diagramτ −16(̂π 24 T ) [22 1 2 ]12 (̂π 24 T ) [22 1 2 ]4p 24p ′ 34(̂π 234 T ) [22 1 2 ]4τ 6(̂π 234 T ) [22 1 2 ]12τ −1ξ(̂π 234 T ) [λ]κp ′ 24T [22 1 2 ]12τ 6 T [22 1 2 ]4p ′ 23(̂π 23 T ) [22 1 2 ]4τ −16(̂π 23 T ) [22 1 2 ]12τ ξ (̂π 23 T ) [λ]κ

3 Irreducible tensor operator techniques **in** atomic spectroscopy 42which is equivalent for several schemes with [λ] = [2 3 ], κ = 1, 2 and [λ] = [2 2 1 2 ], κ = 1, 2, 3.ThenT [22 1 2 ] τ 6 12 T [22 1 2 ]≡∑ɛ46 Ôβ α ([2 2 1 2 ]12), Υ\Υ 6 = {12},α η∈Υ\Υ6T [22 1 2 ]4p ′ 23(̂π 23 T ) [22 1 2 ]4≡ ∑ α 12ϖ 23 ɛ 6 Ô α β ([2 2 1 2 ]12),(̂π 23 T ) [22 1 2 ]4(̂π 23 T ) [22 1 2 ]12(̂π 23 T ) [λ]κ(̂π 234 T ) [λ]κτ −16(̂π 23 T ) [22 1 2 ]12τ ξp ′ 24τ −1ξ≡ ∑ α 12 (̂π 23 T ) κ [λ] ≡ ∑(̂π 234 T ) [λ]κ ≡ ∑(̂π 234 T ) [22 1 2 ]12∑α η∈bπ23 (Υ 6 \Υ)∑α 12 α 23 α η∈bπ23 (Υ\Υ ξ )∑α 12 α 23 α η∈bπ23 (Υ\Υ ξ )≡∑α 12 α 23∑ϖ 23 ɛ 6 ɛ (23)(6) Ô α β ([2 2 1 2 ]12), Υ 6 \Υ = {23},ϖ 23 ɛ 6 ɛ (23)(6) ɛ (23)(ξ) Ô α β ([2 2 1 2 ]12),ϖ 23 ϖ 24 ɛ 6 ɛ (23)(6) ɛ (23)(ξ) Ô α β ([2 2 1 2 ]12),α η∈Aξϖ 23 ϖ 24 ɛ 6 ɛ (23)(6) ɛ (23)(ξ) ɛ (234)(ξ) Ô α β ([2 2 1 2 ]12),(̂π 234 T ) [22 1 2 ]12τ 6 (̂π 234 T ) [22 1 2 ]4≡= ( (̂π 23 (Υ\Υ ξ )) ⋃ (̂π 234 (Υ ξ \Υ)) ) ,∑ ∑ ∑ϖ 23 ϖ 24 ɛ 6 ɛ (23)(6) ɛ (23)(ξ) ɛ (234)(ξ)A ξdefα 12 α 23 α η∈Aξ α η ′ ∈bπ234 (Υ\Υ 6 )(̂π 234 T ) [22 1 2 ]4(̂π 24 T ) [22 1 2 ]4p ′ 34(̂π 24 T ) [22 1 2 ]4τ −16(̂π 24 T ) [22 1 2 ]12≡× ɛ (234)(6) Ô α β ([2 2 1 2 ]12),∑α 12 α 23 α 13∑α η∈Aξϖ 23 ϖ 24 ϖ 34 ɛ 6 ɛ (23)(6) ɛ (23)(ξ) ɛ (234)(ξ)× ɛ (234)(6) Ôβ α ([2 2 1 2 ]12),≡∑ ∑ ∑α 12 α 23 α 13 α η∈Aξ α η ′ ∈bπ24 (Υ 6 \Υ)ϖ 23 ϖ 24 ϖ 34 ɛ 6 ɛ (23)(6) ɛ (23)(ξ)Thus× ɛ (234)(ξ) ɛ (234)(6) ɛ (24)(6) Ôβ α ([2 2 1 2 ]12).ε 24 =∑ ∑f 24 ɛ 6 ɛ (23)(6) ɛ (23)(ξ) ɛ (234)(ξ) ɛ (234)(6) ɛ (24)(6) , (3.35)α 13 α 23 α 34 α η∈Aξ \Υwhere the multiplier f ij denotes the follow**in**g productdeff ij =j−i∏k=1j−i−1∏l=1ϖ i i+k ϖ i+l j . (3.36)Other coefficients ε i i+2 are found from the commutative diagrams with ν s = ν 2 = 5 so thatp ′ tu ◦ τ −1ξ◦ p ′ i i+2 ◦ τ ξ ◦ p ′ rs = p i i+2 . A composition is realised for the numbers r, s, t, u such that̂π i i+2 = ̂π tûπ i i+2̂π rs . It is to be whether (rs) = (i i + 1), (tu) = (i + 1 i + 2) or (tu) = (i i + 1),(rs) = (i + 1 i + 2). Then

3 Irreducible tensor operator techniques **in** atomic spectroscopy 43ε i i+2 =∑α η∈Bξ \Υf i i+2 ɛ (i i+1)(ξ) ɛ (i i+1 i+2)(ξ) , i = 1, 4, (3.37)B ξdef= (̂π i i+1 (Υ\Υ ξ ) ) ⋃(̂πi i+1 i+2 (Υ ξ \Υ) ) . (3.38)ε 35 =∑f 35 ɛ (45)(ξ) ɛ (354)(ξ) , (3.39)α η∈B ′ξ\ΥB ξ′ def= (̂π i+1 i+2 (Υ\Υ ξ ) ) ⋃(̂πi i+2 i+1 (Υ ξ \Υ) ) , i = 3. (3.40)For i = 1, ξ = 6; for i = 3, ξ ∈ {1, 2, . . . , 5}; for i = 4, ξ = 19.The coefficients ε i i+3 are characterised by the commutative diagrams with ν s = ν 5 = 11 sothatp 14 =p ′ 24 ◦ τ6 −1 ◦ p ′ 34 ◦ τ 6 ◦ τ −1ξ◦ p ′ 14 ◦ τ ξ ◦ τ6 −1 ◦ p ′ 13 ◦ τ 6 ◦ p ′ 12, ξ ∈ {1, 2, . . . , 5}, (3.41a)p 25 =p ′ 24 ◦ τ6 −1 ◦ p ′ 35 ◦ τ 6 ◦ τ −1ξp 36 =τ19 −1 ◦ p ′ 35 ◦ τ 19 ◦ p ′ 34 ◦ τ −1ξFrom these diagrams it follows thatε 14 =∑ ∑f 14 ɛ (12)(6) ɛ (123)(6) ɛ (123)(ξ) ɛ (1234)(ξ) ɛ (1234)(6) ɛ (124)(6) ,ε 25 =α 13 α 23 α 34 α η∈Cξ \Υ∑∑α 13 α 23 α 35 α η∈C ′ \Υ ξ◦ p ′ 25 ◦ τ ξ ◦ τ −16 ◦ p ′ 23 ◦ τ 6 ◦ p ′ 45, ξ ∈ {1, 2, . . . , 5}, (3.41b)◦ p ′ 36 ◦ τ ξ ◦ p ′ 46 ◦ τ −119 ◦ p ′ 56 ◦ τ 19 , ξ ∈ {1, 2}. (3.41c)f 25 ɛ (45)(6) ɛ (23)(45)(6) ɛ (23)(45)(ξ) ɛ (2354)(ξ) ɛ (2354)(6) ɛ (254)(6) ,(3.42a)(3.42b)ε 36 =∑ ∑f 36 ɛ 19 ɛ (56)(19) ɛ (465)(ξ) ɛ (3654)(ξ) ɛ (365)(19) ɛ (36)(19) ,(3.42c)α 34 α 35 α 56 α η∈C ′′ξ \Υdefwhere C ξ = (̂π 123 (Υ\Υ ξ ) ) ⋃(̂π 1234 (Υ ξ \Υ) ) , C ξ′ def= (̂π 23̂π 45 (Υ\Υ ξ ) ) ⋃(̂π 2354 (Υ ξ \Υ) ) andC ξ′′ def= (̂π 465 (Υ\Υ ξ ) ) ⋃(̂π 3654 (Υ ξ \Υ) ) .The advantage **of** method **of** commutative diagrams as a language **of** comb**in**atorial computationappears to be evident especially **in** the cases j = i + 4, i + 5. The coefficients ε 15 , ε 26 , ε 16can be found from the commutative diagrams with ν s = ν 6 = 13 for j = i+4 and ν s = ν 7 = 15for j = i + 5. That is,p 15 =p ′ 25 ◦ p ′ 14 ◦ τ6 −1 ◦ p ′ 35 ◦ τ 6 ◦ τ −1ξ◦ p ′ 15 ◦ τ ξ ◦ τ6 −1 ◦ p ′ 45 ◦ p ′ 13 ◦ τ 6 ◦ p ′ 12, (3.43a)p 26 =τ −122 ◦ p ′ 25 ◦ p ′ 36 ◦ τ 22 ◦ p ′ 24 ◦ τ −1ξ◦ p ′ 26 ◦ τ ξ ◦ p ′ 46 ◦ τ −122 ◦ p ′ 56 ◦ p ′ 23 ◦ τ 22 , (3.43b)p 16 =p ′ 26 ◦ τ22 −1 ◦ p ′ 15 ◦ p ′ 36 ◦ τ 22 ◦ p ′ 14 ◦ τ −1ξ◦ p ′ 16 ◦ τ ξ ◦ p ′ 46 ◦ τ22 −1 ◦ p ′ 56 ◦ p ′ 13◦ τ 22 ◦ p ′ 12,where ξ ∈ {1, 2, . . . , 5}. The correspond**in**g coefficients areε 15 =∑ ∑f 15 ɛ (12)(6) ɛ (123)(45)(6) ɛ (123)(45)(ξ) ɛ (12354)(ξ) ɛ (12354)(6) ɛ (1254)(6) ,ε 26 =ε 16 =α 13 α 23 α 35 α η∈Dξ \Υ∑∑α 13 α 23 α 24 αα 25 α 36 α η∈D ′ \Υ56 ξ∑∑α 13 α 14 α 15 αα 23 α 36 α η∈Eξ \Υ56f 26 ɛ 22 ɛ (23)(56)(22) ɛ (23)(465)(ξ) ɛ (23654)(ξ) ɛ (2365)(22) ɛ (26)(22) ,f 16 ɛ (12)(22) ɛ (123)(56)(22) ɛ (123)(465)(ξ) ɛ (123654)(ξ) ɛ (12365)(22) ɛ (126)(22)(3.43c)(3.44a)(3.44b)(3.44c)defwith D ξ = (̂π 123̂π 45 (Υ\Υ ξ ) ⋃(̂π 12354 (Υ ξ \Υ) ) , D ξ′ def= (̂π 23̂π 465 (Υ\Υ ξ ) ) ⋃(̂π 23654 (Υ ξ \Υ) )defand E ξ = (̂π 123̂π 465 (Υ\Υ ξ ) ) ⋃(̂π 123654 (Υ ξ \Υ) ) .In [80, Tab. 3], the coefficients ε 26 and ε 16 have been found to be obta**in**ed from the expressions

3 Irreducible tensor operator techniques **in** atomic spectroscopy 44̂π 26 Ô α β ([2 2 1 2 ]12) = ∑α 13 α 36ϖ 36 ɛ (26)(6) ɛ (236)(6)̂π 23̂π 36 Ô α β ([2 2 1 2 ]12),̂π 26 Ô α β ([2 2 1 2 ]12) =ϖ 16̂π 12̂π 16 Ô α β ([2 2 1 2 ]12).In the language **of** commutative diagrams, the coefficients ε 26 and ε 16 are found from the diagramswith ν s = ν 8 = 17 and ν s = ν 9 = 19 compositions, respectively. These numbers po**in**tto an ocular demonstration **of** the advantage **of** diagrammatic method presented here.To conclude, it is worth to mention that most **of** commutative diagrams studied **in** this sectioncan be drawn **in** several different ways with the same number ν s . For **in**stance,p 25 = τ6 −1 ◦ p ′ 35 ◦ τ 6 ◦ p ′ 24 ◦ τ −1ξ◦ p ′ 25 ◦ τ ξ ◦ τ6 −1 ◦ p ′ 45 ◦ p ′ 23 ◦ τ 6is also a composition **of** ν s = ν 5 = 11 maps, thus it is equivalent to the diagram characterisedby the composition presented **in** Eq. (3.41b). If, for **in**stance, φ i ◦ φ j = φ j ◦ φ i , then it ought tobe possible to construct the classes C i that conta**in** equivalent elements φ i = φ −1j ◦ φ i ◦ φ j withall possible j **in**clud**in**g j = i. Such classification is still to be clarified.3.1.4 Equivalent permutationsIn the present section, the follow**in**g statement will be proved.3.1.8 Theorem. Let ̂π be a permutation representation **of** S l . If the irreducible tensor operatorÔ α ([λ]κ) on H q conta**in**s a set **of** equal irreducible representations α s , s = 1, 2, . . . , t < l, thenthere exists a permutation representation ̂π m**in** which represents a cycle **of** the smallest possiblelength or a product **of** cycles **of** the smallest possible length such that the correspondenceE π ξ ′ ξ = E π m**in**ξ ′ ξ(3.45)is satisfied, though the map τ ξ ◦ p πm**in** ◦ τ −1ξ: T [λ′ ]′ κ−→ (̂π ′m**in** T ) [λ]κ does not exist.An **in**citement to f**in**d the smallest possible length ̂π m**in** associated to ̂π is caused by aprospect to reduce the number **of** **in**termediate irreps that appear due to transformations: theless number **of** transpositions, the less number **of** **in**dices **of** summation.To avoid further mislead**in**g, it is more suitable to label irreps α s with s ≤ t by ς x , wherex = i, j, k, l, p, q, . . .. That is, α s designates the irrep **of** the sth operator a αs ; ς x denotes itsvalue, though.Suppose there is operator Ôα [21]([21]2) associated to the scheme T 2 (see Tab. 3) with irrepsα 1 = α 2 = ς i , α 3 = ς j , α = ς, β = ι. In accordance with Eq. (3.17d), α 12 = ς ii is odd. Thenϖ 12 = (−1) α 1+α 2 +α 12 +1 = (−1) 2ς i+ς ii +1 = 1 s**in**ce (−1) 2ς i= (−1) 1+2λ i= 1 (see Eq. (3.1)):(−1) 2λ i= (−1) 1+2l i= −1 (l i = 0, 1, . . .) for LS-coupl**in**g and (−1) 2λ i= (−1) 2j i= −1(j i = 1 /2, 3 /2, . . .) for jj-coupl**in**g.Consider two operators: ̂π 132 Ôι ς ([21]2), ̂π 13 Ôι ς ([21]2). The permutation representation ̂π 132is realised by the permutation (132) = ( 1 2 33 1 2); the permutation representation ̂π13 acts on irrepsα s , s = 1, 2, 3 by transpos**in**g them with (13) = ( 1 2 33 2 1). Consequently, the anticommutationrule **in** Eq. (3.2) enables us to writêπ 132 [W ς ii(λ i λ i ) × a ς j] ς ι = [W ς ij(λ j λ i ) × a ς i] ς ι ≡ [W α 13(λ 3 λ 1 ) × a α 2] α β= ∑C 132 [W ς ii(λ i λ i ) × a ς j] ς ι ≡∑C 132 [W α 12(λ 1 λ 2 ) × a α 3] α β,ς ii =oddα 12 =odd(3.46a)defC 132 = ∑ 〈ς j ι j ς i ι i1 |ς ij ι ij 〉〈ς ij ι ij ς i ι i2 |ςι〉〈ς i ι i1 ς i ι i2 |ς ii ι ii 〉〈ς ii ι ii ς j ι j |ςι〉. (3.46b)ι i1 ι i2ι j ι ij ι iiThe summation over **in**dices causes the appearance **of** 6j–symbols. However, it is unimportant**in** the present case. The transposition (13) permutes operators with the same value ς i . Thus

3 Irreducible tensor operator techniques **in** atomic spectroscopy 45̂π 13 [W ς ii(λ i λ i ) × a ς j] ς ι = [W ς ij(λ j λ i ) × a ς i] ς ι ≡ [W α 23(λ 3 λ 2 ) × a α 1] α β= ∑C 13 [W ς ii(λ i λ i ) × a ς j] ς ι + δ(ς j , ς)(−1) ς i+ς ij −ς+1 [ς ij ] 1/2 [ς] −1/2 a ς ι≡ς ii =odd∑α 12 =oddC 13 [W α 12(λ 1 λ 2 ) × a α 3] α β + δ(α 3 , α)(−1) α 1+α 13 −α+1 [α 13 ] 1/2 [α] −1/2 a α β,(3.47a)defC 13 = − ∑ 〈ς j ι j ς i ι i2 |ς ij ι ij 〉〈ς ij ι ij ς i ι i1 |ςι〉〈ς i ι i1 ς i ι i2 |ς ii ι ii 〉〈ς ii ι ii ς j ι j |ςι〉. (3.47b)ι i1 ι i2ι j ι ij ι iiIn Eq. (3.47b), the third CGC equals to 〈ς i ι i1 ς i ι i2 |ς ii ι ii 〉 = −ϖ 12 〈ς i ι i2 ς i ι i1 |ς ii ι ii 〉. But ϖ 12 = 1.Replace ς i ι i1 with ς i ι i2 **in** Eq. (3.46b). ThenC 132 = C 13 . (3.48)Eq. (3.48) has a significant mean**in**g that makes sense for any operator Ôα ([λ]κ). A generalisationis straightforward due to the structure **of** coefficients C 13 , C 132 : it is a product **of** CGCs.Consequently, it is a manner **of** quantity **of** transpositions that permute the same value irrepsonly, but for both even and odd permutations, the coefficients co**in**cide with each other becausefor any equal irreps α s , α s ′, the phase multiplier ϖ ss ′ = 1. Thus for the map τ ξ ◦ p 13 ◦ p 23 ◦ τ −1ξ, ′Eξ 132 ′ ξ = E ξ 13 ′ ξ . Indeed, (132) = (13)(23) and∑̂π 132 Ôβ α ([2 2 1 2 ]12) =̂π 13 ε 23 Ôβ α ([2 2 1 2 ]12) = ∑ ε (13)(23) ε 13 Ôβ α ([2 2 1 2 ]12)α 12 α 12= ∑ α 12ε 132 Ô α β ([2 2 1 2 ]12),where ε (13)(23) ≡ ̂π 13 ε 23 , ξ = ξ ′ = 14. Write ε 132 = ε 13 ε (13)(23) . But ε 23 = ∑ α 23ϖ 23 ɛ 6 ɛ (23)(6)(see Eq. (3.34)). Thus ε (13)(23) = ∑ α 12ɛ (13)(6) ɛ (132)(6) , as ̂π 13 ϖ 23 = ϖ 12 = 1. Now refer to Eq.(A.6a) **in** Appendix A. This implies ɛ (13)(6) = ɛ (132)(6) = ɛ 6 . Then ε (13)(23) = ∑ α 12ɛ 2 6 = 1 (seeEq. (3.13)), recall**in**g that α 1 = α 2 and thus α 13 = α 23 . The result is ε 132 = ε 13 .The map τ ξ ◦ p 13 ◦ p 23 ◦ τ −1ξ: T [λ′ ]′ κ ′Eq. (3.24))E 132ξ ′ ξ = ∑−→ (̂π 132 T ) [λ]κα η∈Nξ ′ ξε 132 ɛ ξ ′ɛ (132)(ξ) .is realised through the coefficient (seeIt has been already po**in**ted out that ε 132 = ε 13 . The coefficient ɛ (132)(ξ) = ɛ (13)(ξ) : for ̂π 13 ,α 1 ↔ α 3 ; for ̂π 132 , α 1 → α 3 , α 2 → α 1 , α 3 → α 2 , therefore (α 1 = α 2 ) α 1 → α 3 , α 3 → α 1 ⇒α 1 ↔ α 3 . ThenEξ 132 ′ ξ = ∑ε 13 ɛ ξ ′ɛ (13)(ξ) = Eξ 13ξ. ′α η∈Nξ ′ ξIt turns out that although a one-to-one correspondence τ ξ ◦ p 13 ◦ τ −1ξ: T [λ′ ]′ κ−→ (̂π ′13 T ) [λ]κdoes not exist (the permutation **of** equal representations causes additional terms due to non-zero−→ (̂π 132 T ) [λ]κKronecker deltas), but the realisation **of** a bijective map τ ξ ◦p 13 ◦p 23 ◦τ −1ξ: T [λ′ ]′ κ ′is actualised by the coefficient Eξ 13 ′ ξ that relates operators ̂π 13Ôα ([λ]κ) and Ôα ([λ ′ ]κ ′ ) if onlyα 1 ≠ α 2 , that is, if only the map τ ξ ◦ p 13 ◦ τ −1ξis valid.′A generalisation is pla**in** due to the peculiarity **of** method based on commutative diagrams(Sec. 3.1.3). That is, each permutation operator ̂π realises a permutation π **of** irreps α 1 , α 2 , . . .by mak**in**g the transformations so that only the irreps α s and α s ′ with**in** W α ss ′ (λ s λ s ′) are permuted(the maps τ ξ , τ −1ξ). If the irreducible tensor operator Ôα ([λ]κ) associated to the schemeconta**in**s a set **of** several irreps α s that are equal, then a m**in**imal card**in**ality **of** such set isT [λ]κ

3 Irreducible tensor operator techniques **in** atomic spectroscopy 473.2.1 Remark. Despite **of** different representations **in** dist**in**ct tensor spaces, it is by no meansobvious that there exists a one-to-one correspondence **of** basis associated to irreducible tensoroperators so that the matrix representations **of** Ôλ and Ôα co**in**cide. Consequently, the choice**of** considered tensor space is arbitrary and it depends on a specified subject matter.3.2.1 A two-particle operatorIn this section, the tensor properties **of** a two-particle operator—either physical or effective—are studied. The systematic study is essentially required due to the subsequent computationrequirements needed to produce high-precision atomic structure data by generat**in**g large sets **of**matrix elements for a two-particle operator. The operator is **of** the form (refer to Eq. (2.69b))∑Ĝ[ω] def= X ω a α a β a †¯νa †¯µω αβ ¯µ¯ν . (3.52)I 2Much the same as **in** Sec. 2.3, the Greek letters α, β, . . . designate the s**in**gle-electron statescharacterised by the sets **of** numbers {n α , λ α , m α }, {n β , λ β , m β }, . . .. As usually, I 2 denotesthe set **of** s**in**gle-electron states. The multiplier X ω is closely related to the structure **of** a twoparticlematrix element ω αβ ¯µ¯ν . If ω ≡ g, that is, if Ĝ[g] ≡ Ĝ denotes a two-particle operatorrepresent**in**g some physical (Coulomb, Breit) **in**teraction, then X ω = 1 /2 anddefg αβ ¯µ¯ν = 〈αβ|g 12 |¯µ¯ν〉. (3.53)A two-particle matrix representation g αβ ¯µ¯ν is sometimes useful to represent by [70, Eq. (38)]g αβ ¯µ¯ν = 1 /2 ˜g αβ ¯µ¯ν ,def˜g αβ ¯µ¯ν = g αβ ¯µ¯ν − g αβ¯ν ¯µ . (3.54)Then for Ĝ[˜g] ≡ ĜA , X ω = 1 /4. The superscript A over Ĝ **in**dicates the presence **of** antisymmetrictwo-particle matrix representation ˜g αβ ¯µ¯ν . For the symmetric and self-adjo**in**t **in**teractionoperator g 12 = g 21 = g 12, † the symmetry properties **of** its matrix representations areg αβ ¯µ¯ν =g βα¯ν ¯µ = g¯ν ¯µβα = g¯µ¯ναβ ,(3.55a)˜g αβ ¯µ¯ν =˜g βα¯ν ¯µ = ˜g¯µ¯ναβ = ˜g¯ν ¯µβα= − ˜g αβ¯ν ¯µ = −˜g βα¯µ¯ν = −˜g¯µ¯νβα = −˜g¯ν ¯µαβ . (3.55b)To f**in**d the irreducible tensor form **of** Ĝ[ω], a usual decomposition (see, for example, Eq.(3.5)) is performed.Ĝ[ω] = ∑ I 2∑Λ∑+ΛM=−ΛĜ Λ M. (3.56)Each irreducible tensor operator ĜΛ acts on H Λ .3.2.2 Remark. In the present section, operators act**in**g on the irreducible tensor space H Λ willbe considered. Their correspondence to operators on H q is associated by us**in**g the data **in** Tab.6 and by apply**in**g Eq. (3.50).For the irreducible tensor operators ĜΛ associated to the angular reduction scheme T [22 ]1(z-scheme), it is easy to deduce thatĜ Λ def= − ∑ [W Λ 1(λ α λ β ) × W Λ 2(˜λ¯µ˜λ¯ν )] Λ Ω αβ ¯µ¯ν (Λ 1 Λ 2 Λ), (3.57)Λ 1 Λ 2Ω αβ ¯µ¯ν (Λ 1 Λ 2 Λ) def= ∑(−1) λ¯µ+λ¯ν+m¯µ+m¯ν Em αm βm¯µm¯ν( )λα λ β λ¯µ λ¯νm α m β; Λ−m¯µ −m¯ν 1 Λ 2 ΛM ω αβ ¯µ¯ν . (3.58)A decomposition **in** Eq. (3.56) is convenient to f**in**d matrix elements **of** Ĝ[ω] on the basis|Γ i Π i Λ i M i 〉 which is a l**in**ear comb**in**ation **of** CSFs (see Eq. (2.2)). That is, a matrix representation**of** Ĝ[ω] on exact wave functions is the sum **of** matrix representations **of** ĜΛ on the

3 Irreducible tensor operator techniques **in** atomic spectroscopy 48eigenfunctions **of** atomic central-field Hamiltonian. Meanwhile, to f**in**d the matrix element **of**Ĝ Λ means to f**in**d the matrix element **of** ÔΛ ([2 2 ]1) given on the right hand side **of** Eq. (3.57).These particular matrix representations **of** ÔΛ ([2 2 ]1) have been comprehensively studied **in**Refs. [46, 50]. Consequently, to accomplish the formation **of** ĜΛ , the basis-**in**dependent quantitiesΩ αβ ¯µ¯ν (Λ 1 Λ 2 Λ) are to be found. These quantities are related to the orig**in** **of** **in**teractionoperator (not necessarily physical) g 12 , as demonstrated **in** Eq. (3.58), where, **in** general,E(λ 1 λ 2 λ 3 λ 4µ 1 µ 2 µ 3 µ 4; λ 12 λ 34 λµ)def= 〈λ 1 µ 1 λ 2 µ 2 |λ 12 µ 12 〉〈λ 3 µ 3 λ 4 µ 4 |λ 34 µ 34 〉× 〈λ 12 µ 12 λ 34 µ 34 |λµ〉. (3.59)For some permutation representations ̂π **of** S 4 , the coefficient E satisfies()(λ̂πE 1 λ 2 λ 3 λ 4λµ 1 µ 2 µ 3 µ 4; λ 12 λ 34 λµ = Z π(1)π(2)π(3)π(4) (λ 12 λ 34 λ)E 1 λ 2 λ 3 λ 4µ 1 µ 2 µ 3 µ 4; λ 12 λ 34 λµ π). (3.60)If π = 1 4 , (14)(23), then Z 1234 = Z 4321 = 1, µ 14 = µ, µ (14)(23) = −µ. If π = (12), (1423),then Z 2134 = Z 4312 = a(λ 1 λ 2 λ 12 ), µ (12) = µ, µ (1423) = −µ and a(λ 1 λ 2 λ 12 ) def= (−1) λ 1+λ 2 +λ 12.If π = (34), (1324), then Z 1243 = Z 3421 = a(λ 3 λ 4 λ 34 ), µ (34) = µ and µ (1324) = −µ. F**in**ally,if π = (12)(34), (13)(24), then Z 2143 = Z 3412 = a(λ 1 λ 2 λ 12 )a(λ 3 λ 4 λ 34 ), µ (12)(34) = µ andµ (13)(24) = −µ.For the physical **in**teractions g 12 , the quantity Ω αβ ¯µ¯ν (Λ 1 Λ 2 Λ) equals—up to a multiplier—tothe reduced matrix element **of** g Λ , where g Λ is def**in**ed much the same as ĜΛ for Ĝ[ω] (see Eq.(3.56)). Indeed, if g αβ ¯µ¯ν is calculated on the basis **of** s**in**gle-electron eigenstates |α〉, . . . , |¯ν〉,thenwhereΩ αβ ¯µ¯ν (Λ 1 Λ 2 Λ) = a(λ¯µ , λ¯ν , Λ 2 )z(Λλ α λ β λ¯ν λ¯µ Λ 1 Λ 2 ), (3.61)z(Λλ α λ β λ¯ν λ¯µ Λ 1 Λ 2 ) def= 1 /2 [Λ 1 ] 1/2 [Λ] −1/2 [n α λ α n β λ β Λ 1 ‖g Λ ‖n¯µ λ¯µ n¯ν λ¯ν Λ 2 ] (3.62)or on the contraryg αβ ¯µ¯ν = 2 ∑( )(−1) Λ 2+m¯µ+m¯ν λα λ β λ¯µ λ¯νm α m β; Λ−m¯µ −m¯ν 1 Λ 2 ΛM z(Λλ α λ β λ¯ν λ¯µ Λ 1 Λ 2 ). (3.63)Λ 1 Λ 2 ΛThe coefficient z that represents the basis-**in**dependent part **of** ˜g αβ ¯µ¯ν equals to˜z(Λλ α λ β λ¯ν λ¯µ Λ 1 Λ 2 ) def= z(Λλ α λ β λ¯ν λ¯µ Λ 1 Λ 2 ) − a(λ¯µ λ¯ν Λ 2 )z(Λλ α λ β λ¯µ λ¯ν Λ 1 Λ 2 ). (3.64)Explicit expressions **of** z depend on a specified **in**teraction operator g 12 .Example (Coulomb **in**teraction). Assume that a s**in**gle-electron eigenstate |α〉 is a 4–sp**in**or suchthat [82, p. 106, Eq. (19.2)]( )f(nα l|α〉 ≡ |n α l α j α m α 〉 =α j α |r)|l α j α m α 〉(−1) ϑα g(n α l αj ′ α |r)|l αj ′ (3.65)α m α 〉defand ϑ α = l α − j α + 1 /2, l α′ def= 2j α − l α , l α = j α ± 1 /2. The 2–sp**in**ors |l α j α m α 〉 are constructedas follows [52, Sec. 2.3, p. 13, Eq. (2.16)]|l α j α m α 〉 def=+1/2∑µ=−1/2( )δ(µ, + 1/2)〈l α m α − µ 1 /2 µ|j α m α 〉Ym lαα−µ(̂x)δ(µ, − 1 . (3.66)/2)Then the matrix representation z **of** 1/r 12 on the basis **in** Eq. (3.65) equals toz(0l α j α l β j β l¯ν j¯ν l¯µ j¯µ J 1 J 2 ) = 1 /2 δ(J 1 , J 2 )(−1) jα−j¯ν+(l¯µ+l¯ν−lα−l β)/2× [j α , j β , j¯µ , j¯ν , J 1 ] 1/2 ∑ k[k] −1 {jα j¯µ kj¯ν j β J 1}〈j α1/2 j¯µ − 1 /2|k0〉〈j β1/2 j¯ν − 1 /2|k0〉

3 Irreducible tensor operator techniques **in** atomic spectroscopy 49× ( R k (αβ, ¯µ¯ν) + R k (αβ ′ , ¯µ¯ν ′ ) + R k (α ′ β, ¯µ ′¯ν) + R k (α ′ β ′ , ¯µ ′¯ν ′ ) ) . (3.67)The radial **in**tegrals R k are def**in**ed **in** Ref. [52, Sec. 19.3, p. 232, Eq. (19.72)]. The primes overs**in**gle-electron states designate the correspond**in**g small component g **of** relativistic radial wavefunction.Example (Coulomb **in**teraction #2). Bhatia et. al. [65, Sec. 5, Eq. (55)] constructed a totaltwo-electron wave function so that for S = 0, 1,|Π 12 LSM〉 = ∑ ′′κ[f+κ ( 2S+1 L|r 1 , r 2 )D L +Mκ (Ω)+f − κ ( 2S+1 L|r 1 , r 2 )D L −Mκ (Ω)] , Π 12 = (−1) l 1+l 2.The sum with double prime runs over κ = 0, 2, . . . , k ≤ L, k ∈ 2Z + if parity Π 12 = 1 and overκ = 1, 3, . . . , k ≤ L, k ∈ 2Z + + 1 if parity Π 12 = −1. The radial functions satisfyf κ + ( 1 L|r 2 , r 1 ) = (−1) L+κ f κ + ( 1 L|r 1 , r 2 ), fκ − ( 1 L|r 2 , r 1 ) = (−1) L+κ+1 fκ − ( 1 L|r 1 , r 2 ),f κ + ( 3 L|r 2 , r 1 ) = (−1) L+κ+1 f κ + ( 3 L|r 1 , r 2 ), fκ − ( 3 L|r 2 , r 1 ) = (−1) L+κ fκ − ( 3 L|r 1 , r 2 ).The radial equations for f κ± are listed **in** Eq. (70) **of** Ref. [65], where it is also demonstratedthat they are the same as that **of** Breit [83]. Ω ≡ (Φ, Θ, Ψ) denotes the usual Euler angles thatconnect the coord**in**ates ̂x 1 , ̂x 2 **of** both electrons. The spherical functions D L ±Mκ(Ω) are def**in**edbyD L + defMκ(Ω) = DL Mκ (Ω) + (−1)κ DM−κ L √ [ √ (Ω) ] ,2 1 + δ(κ, 0)( 2 − 1)D L − defMκ(Ω) = DL Mκ (Ω) − (−1)κ DM−κ L √ (Ω) .2 iTo f**in**d the matrix representation z **of** 1/r 12 , it is more convenient to **in**troduce the radial function[g µ ( 2S+1 L|r 1 , r 2 ) def= 1 / √ 2 f+µ ( 2S+1 L|r 1 , r 2 ) − ifµ − ( 2S+1 L|r 1 , r 2 ) ] , for µ ≥ 0,[g µ ( 2S+1 L|r 1 , r 2 ) def= (−1)µ / √ 2 f+|µ| ( 2S+1 L|r 1 , r 2 ) + if −|µ| ( 2S+1 L|r 1 , r 2 ) ] , for µ < 0which leads to a compact form [65, Sec. 5, Eq. (47)]|Π 12 LSM〉 =L∑µ=−Lg µ ( 2S+1 L|r 1 , r 2 )D L Mµ(Ω). (3.68)Eq. (3.68) is convenient to apply the RCGC technique suggested **in** Sec. 2.2.2, whereas thespherical functions DMµ L (Ω) are the eigenfunctions **of** Laplacian on SU(2) [62, Sec. III-4.8, p.14, Eq. (5)]def∆ Ω = ∂2∂Θ + cot Θ ∂2 ∂Θ + 1 ( )∂2s**in** 2 Θ ∂Φ − 2 cot Θ∂22 ∂Φ∂Ψ + ∂2,∂Ψ 2much the same as the spherical harmonics Y L M (̂x) are the eigenfunctions **of** Laplacian ∆ bx [11,Sec. I-1, p. 13, Eq. (1.4)] on SO(3). By us**in**g Eq. (2.37), it is easy to deduce thatz(00l α l β l¯ν l¯µ L 1 S 1 L 2 S 2 ) = 2πδ(Π αβ , Π¯µ¯ν )δ(L 1 , L 2 )δ(S 1 , S 2 )[L 1 ] −1/2× ∑k∑∑L 1(−1) k [k] −1 [k‖S k ‖k] 〈k0kQ|kQ〉 Fκ( k 2S1+1 L 1 ), (3.69)k∈2Z + Q=−kκ=−L 1where the radial **in**tegral∫∫Fκ( k 2S+1 L) def= dr 1 dr 2 r1r 2 2 r k ∣k+1 κ ( 2S+1 L|r 1 , r 2 ) ∣ 2 .Reduced matrix elements **of** the SO(3)–irreducible tensor operators (Proposition 2.2.3) S k arefound from Eq. (2.38). For the detailed study **of** radial functions g κ , see Ref. [65, Sec. VII, p.1057].

3 Irreducible tensor operator techniques **in** atomic spectroscopy 50Example (Magnetic **in**teraction). One **of** the components **of** Breit operator is the so-called magnetic**in**teraction [− ( α 1 1 · α 1 2)]/r12 , where each Dirac matrix α 1 i is referred to the ith electron.The irreducible tensor form **of** magnetic **in**teraction can be found **in** Ref. [84, Sec. 2-5, p. 67,Eq. (5.86a)]. Then the matrix representation z on the 4–sp**in**ors (see Eq. (3.65)) obeys the formz(0l α j α l β j β l¯ν j¯ν l¯µ j¯µ J 1 J 2 ) = 1 /2 δ(J 1 , J 2 )(−1) jα−j¯ν+(l¯µ+l¯ν−lα−l β)/2 [j α , j β , j¯µ , j¯ν , J 1 ] 1/2× ∑ { }(−1) K+k [k] −1 jα j¯µ K〈jj¯ν j β J α1/2 j¯µ − 1 /2|K0〉〈j β1/2 j¯ν − 1 /2|K0〉1kK× R k (αβ¯ν ¯µ)MF kkK (αβ¯ν ¯µ), (3.70)R k (αβ¯ν ¯µ) def= ( R k (α ′ β ′ , ¯µ¯ν) R k (α ′ β, ¯µ¯ν ′ ) R k (αβ ′ , ¯µ ′¯ν) R k (αβ, ¯µ ′¯ν ′ ) ) ,⎛⎞⎛⎞1 1 1 1A(k 1 , K)A(k 2 , K)(3.71)M def ⎜−1 1 −1 1⎟= ⎝−1 −1 1 1⎠ , F k1 k 2 K(αβ¯ν ¯µ) def ⎜ A(k=1 , K)B β¯ν (k 2 , K) ⎟⎝B α¯µ (k 1 , K)A(k 2 , K)⎠ , (3.72)1 −1 −1 1B α¯µ (k 1 , K)B β¯ν (k 2 , K)A(k, K) def= 1 /2[1 − (−1) K+k ]〈10K0|k0〉, (3.73)B ρσ (k, K) def= (−1) K+k+lσ+jσ− 1 [j σ ] + (−1) jρ+jσ+K [j ρ ]2 √ 〈11K − 1|k0〉 (3.74)2K(K + 1)with ρσ ≡ α¯µ and ρσ ≡ β¯ν.Example (Retard**in**g **in**teraction). The second (and the last) component **of** relativistic Breit operatoris the so-called **in**teraction **of** retardation [− ( α 1 1 · ∇ 1 1)(α12 · ∇ 1 2)]r12 /2 whose irreducibletensor form is given **in** Ref. [84, Sec. 2-5, p. 67, Eq. (5.87a)]. The matrix representationz(0l α j α l β j β l¯ν j¯ν l¯µ j¯µ J 1 J 2 ) = 1 /2 δ(J 1 , J 2 )(−1) jα−j¯ν+(l¯µ+l¯ν−lα−l β)/2 [j α , j β , j¯µ , j¯ν , J 1 ] 1/2× ∑ ({ }[k] −1 k jα j¯µ k − 1〈j2k − 1 j¯ν j β J α1/2 j¯µ − 1 /2|k − 1 0〉〈j β1/2 j¯ν − 1 /2|k − 1 0〉1k√ { }k + 1 jα j¯µ k + 1× R k (αβ¯ν ¯µ)MF kkk−1 (αβ¯ν ¯µ) +〈j2k + 3 j¯ν j β J α1/2 j¯µ − 1 /2|k + 1 0〉1(√× 〈j β1/2 j¯ν − 1 /2|k + 1 0〉 [k] −1√ k + 2k + 1R k (αβ¯ν ¯µ)MF kkk+1 (αβ¯ν ¯µ) +2× [ P (12)k(αβ¯ν ¯µ)MF kk+2 k+1 (αβ¯ν ¯µ) + P (21)k(αβ¯ν ¯µ)MF k+2 k k+1 (αβ¯ν ¯µ) ])) . (3.75)Matrices R k (αβ¯ν ¯µ), M and F k1 k 2 K are def**in**ed **in** Eqs. (3.71)-(3.72). The matrix P (ij)k(αβ¯ν ¯µ)is considered byP (ij)k(αβ¯ν ¯µ) def= ( P (ij)k(α ′ β ′ , ¯µ¯ν) P (ij)k(α ′ β, ¯µ¯ν ′ ) P (ij)k(αβ ′ , ¯µ ′¯ν) P (ij)where the radial **in**tegral P (ij)kk(αβ, ¯µ ′¯ν ′ ) ) , (3.76)(αβ, ¯µ¯ν) is considered **in** the same way as R k (αβ, ¯µ¯ν) by replac**in**gr k+1 with r k+2i /r k+3j − ri k /r k+1j .3.2.3 Remark. From the above studied examples it appears that the physical two-particle operatorsg Λ = g 0 observed **in** atomic physics are scalars. This significantly simplifies further oncalculations. On the other hand, it is not necessary true for the effective two-particle operators.The irreducible tensor operators ĜΛ associated to the angular reduction scheme (̂π 243 T ) [22 ]1(b-scheme) are derived from the irreducible tensor operators ĜΛ associated to T [22 ]1 by us**in**gEq. (3.49) which clearly establishes the sum **of** two irreducible tensor operators. Consequently,the scheme (̂π 243 T ) [22 ]1 is less convenient from the po**in**t **of** view **of** tensor structure derivedfrom Eq. (3.52). On the other hand, whilst on the subject **of** a particular operator, the presentreduction scheme is more preferable to compare with T [22 ]1 , as it provides **in**formation relatedto the **in**ner structure **of** g Λ expressly. This is easily seen from the equation

3 Irreducible tensor operator techniques **in** atomic spectroscopy 51g αβ ¯µ¯ν =2 ∑Λ ′ 1 Λ′ 2 Λ (−1) λ¯µ+λ¯ν+m¯µ+m¯ν E( )λα λ¯µ λ β λ¯νm α −m¯µ m β; Λ ′ −m¯ν 1Λ ′ 2ΛM× b(Λλ α λ β λ¯ν λ¯µ Λ ′ 1Λ ′ 2), (3.77)where b coefficient is such thatz(Λλ α λ β λ¯ν λ¯µ Λ 1 Λ 2 ) =(−1) λ¯µ+λ¯ν+Λ 2[Λ 1 , Λ 2 ] ∑ { }λ¯µ λ¯ν Λ 21/2 [Λ ′ 1, Λ ′ 2] 1/2 Λ ′ 1 Λ ′ 2 ΛΛ ′ λ1 Λ′ 2α λ β Λ 1× b(Λλ α λ β λ¯ν λ¯µ Λ ′ 1Λ ′ 2). (3.78)In Eq. (3.77), the structure **of** E **in**dicates that the irreps Λ ′ 1, Λ ′ 2—despite **of** a resultant irrepΛ—designate **in**termediate irreps **of** g Λ that is found by apply**in**g a usual technique **of** separation**of** variables (coord**in**ates). That is,g Λ = ∑ Λ ′ 1 Λ′ 2g(r 1 , r 2 )[g Λ′ 1 (̂x1 ) × g Λ′ 2 (̂x2 )] Λ , (3.79)where g(r 1 , r 2 ) is a radial function. Conversely, the coefficient E **in** Eq. (3.63) **in**volves theirreps Λ 1 , Λ 2 that are supplemental, as it is demonstrated **in** Eq. (3.78). For this reason, theangular reduction scheme T [22 ]1 supplies more versatility to compare with (̂π 243 T ) [22 ]1 , but atthe same time it leads to additional summations. The latter fact, however, will be argued aga**in**stwhen study**in**g the two-particle effective matrix elements ω αβ ¯µ¯ν **in** more detail (Sec. 4).Another circumstance **of** a widespread application **of** b-scheme is that for LS-coupl**in**g, theirreducible tensor operators ̂π 243 Ô λ ([2 2 ]1) are simply the tensor products **of** generators W λα¯µ ,W λ β¯ν **of** accord**in**gly U(Nlα ), U(N lβ ) if l α = l¯µ and l β = l¯ν (recall that N l = 4l + 2). But thisis a case **of** equivalent electrons **of** atom. The Lie algebra **of** the present generators is expresslydef**in**ed and studied **in** Ref. [50, Sec. 6, p. 46, Eq. (6.20)]. The b coefficients **in** H q space werefirst orig**in**ated by Kaniauskas et. al. [85, Eq. (2.11)]. Later, the representation **of** operators **in** b-scheme found **many** applications **in** theoretical atomic spectroscopy [14,15,54,55,81,87] as wellas **in** MBPT [36, 37, 86, 88]. In contrast, a more versatile z-scheme that was first systematicallydeveloped **in** Refs. [77, 81] found a natural application to the effective operator approach **in**MBPT [72, 73]. The present technique is considered **in** Sec. 4.The efficiency **of** calculation **of** matrix elements—as a pr**in**cipal motive—strongly dependson the preparation **of** expressions **of** the irreducible tensor operators ĜΛ . In atomic physics, atypical task is to calculate the matrix element **of** ĜΛ that acts on l = 1, 2, 3, 4 electron shells **of**atom. Consequently, the values **of** irreps λ α , λ β , λ¯µ , λ¯ν **in** ÔΛ ([2 2 ]1) depend on l. Particularly,if l = 1, then λ α = λ β = λ¯µ = λ¯ν ; if l = 4, then λ α ≠ λ β ≠ λ¯µ ≠ λ¯ν . Possible preparations**of** ĜΛ that acts on l = 1, 2, 3, 4 electron shells are displayed **in** Ref. [14, Tab. 1], whereboth schemes are mixed depend**in**g on a specified case, and **in** Ref. [77, Tabs. 1-3], whereboth schemes are strictly separated. It is to be noted the essential difference between methodsused **in** these two works: it is a manner **of** distribution **of** irreps and **of** the choice **of** angularreduction scheme. The follow**in**g example **of** application **of** b-scheme clearly demonstrates that.Assume that l = 4. In Ref. [77, Tab. 3, case:ϱ = 1], the irreducible tensor operator ÔΛ ([21 2 ]1)associated to the angular reduction scheme T [212 ]1 (Tab. 3) is considered, while **in** Ref. [14, Fig.A1, A 7 ], the irreducible tensor operator ÔΛ ([2 2 ]1) associated to T [22 ]1 is preferred. As a result,**in** the first approach, the angular coefficient conta**in**s a product **of** two 6j–symbols; **in** the secondapproach [14, Eq. (52)], the angular coefficient conta**in**s a 9j–symbol, though.3.2.2 A three-particle operatorUnlike the case **of** a two-particle operator, the study **of** a three-particle operator is much morecomplicated and not so well fulfilled. First attempts to clarify the tensor structure **of** scalarthree-particle operators belong to Judd [47, 89] who specified on the classification **of** terms **of**f N configuration. Later, Leavitt et. al. [49, 90, 91] comprehensively considered effective threeparticleoperators act**in**g with**in** the d, f shells. Recently, a few works [42,92,93] devoted to the

3 Irreducible tensor operator techniques **in** atomic spectroscopy 52effective operator approach that accounts for triple excitations are found. None **of** them providea systematic classification **of** operators act**in**g on l = 2, 3, . . . , 6 electron shells.A three-particle operator is **of** the form (refer to Eq. (2.69c))̂L def= ∑ I 3a α a β a ζ a †¯ηa †¯νa †¯µ ω αβζ ¯µ¯ν ¯η . (3.80)The effective three-particle matrix element ω αβζ ¯µ¯ν ¯η depends on the case under consideration.However, formally the Wigner–Eckart theorem can be applied. The result readŝL = ∑ I 3∑Λ∑+ΛM=−Λ̂L Λ , (3.81)where the irreducible tensor operator ̂L Λ acts on H Λ and it is associated to the angular reductionscheme T [22 1 2 ]12 (Tab. 4) so that̂L Λ def= (−1) ∑ [ λα+λ β+λ ζ +λ¯µ+λ¯ν+λ¯η+1 [WE 1(λ α λ β ) × a λ ζ] Λ 1× [W E 2(˜λ¯µ˜λ¯ν ) × ã λ¯η ] ] Λ Λ 2E 1 Λ 1E 2 Λ 2× (−1) Λ 1+Λ 2[E 1 , E 2 , Λ 1 , Λ 2 ] ∑ ∑1/2 (−1) M ′ [Λ ′ 1, Λ ′ 2, Λ 3 , Λ ′ ] 1/2 Ω αβζ ¯µ¯ν ¯η (Λ ′ 1Λ ′ 2Λ 3 Λ ′ )Λ ′ 1 Λ′ M 2 3 M ′Λ 3 Λ ′{ } { } { }λ× 〈Λ 3 M 3 Λ ′ − M ′ λα λ|ΛM〉 β E 1 λ¯µ λ¯ν E α λ¯µ Λ 32λ ζ Λ 1 Λ ′ 1 λ¯η Λ 2 Λ ′ Λ ′ 1 Λ ′ 2 Λ ′ . (3.82)2Λ 1 Λ 2 ΛThe projection-**in**dependent Ω αβζ ¯µ¯ν ¯η (Λ ′ 1Λ ′ 2Λ 3 Λ ′ ) is def**in**ed much the same as the analogousquantity **in** a two-particle case (see Eq. (3.58)). That is, Ω αβζ ¯µ¯ν ¯η (Λ ′ 1Λ ′ 2Λ 3 Λ ′ ) satisfies the equation∑ ∑ω αβζ ¯µ¯ν ¯η =(−1) λα+λ β+λ ζ +λ¯µ+λ¯ν+λ¯η(−1) M ′ 〈λ α m α λ¯µ − m¯µ |Λ 3 M 3 〉Λ ′ 1 Λ′ Λ 2 3 Λ( ′ λβ λ× E ζ λ¯ν λ¯ηm β m ζ; Λ ′ −m¯ν −m¯η 1Λ ′ 2Λ ′ − M)Ω ′ αβζ ¯µ¯ν ¯η (Λ ′ 1Λ ′ 2Λ 3 Λ ′ ). (3.83)By a decomposition **in** Eq. (3.81), the chore to f**in**d matrix elements **of** the three-particle operator̂L is dis**in**tegrated **in**to two dist**in**ct tasks: the computation **of** matrix elements **of** the irreducibletensor operators ÔΛ ([2 2 1 2 ]12) and the establishment **of** projection-**in**dependent quantitiesΩ αβζ ¯µ¯ν ¯η that particularly play the role **of** the effective reduced matrix elements. The lattertask depends on a concrete case to be studied. For the expansion terms **in** the third-order approximation**of** MBPT, the quantities Ω αβζ ¯µ¯ν ¯η are found **in** Sec. 4 (also, see Appendix C).Conversely, the first task—determ**in**ation **of** matrix elements—is **in**dependent **of** the dynamics**of** system under consideration and thus it can be solved **in** general.In Sec. 3.1, the classification **of** angular reduction schemes **of** operator str**in**g Ôl, **in**clud**in**gl = 6, has been accomplished. The case l = 6 fits Ô3 that represents a product **of** three creationand three annihilation operators (see Eq. (2.42)). A decomposition **of** Ô6 is actualised by Eq.(3.5). F**in**ally, the connection between irreducible tensor operators Ôα ([λ]κ) is associated bythe entries **of** transformation matrices (Corollary 3.1.4, Def**in**ition 3.1.5) E and E ij . These resultsare sufficient to f**in**d expressions **of** matrix elements **of** the sole irreducible tensor operatorÔ α ([λ]κ) associated to a specified angular reduction scheme T κ[λ] : all other matrix elements **of**the irreducible tensor operators Ôα ([λ ′ ]κ ′ ) with λ ≠ λ ′ , κ ≠ κ ′ are found **in**stantly through thetransformation coefficients E ξξ ′ and E ijξ ′ ξ . In particular cases, E ξ 14 = ɛ ξ and E ij14 14 = ε ij take onthe values **of** the basis (Appendix A) and recoupl**in**g (Sec. 3.1.3) coefficients. Thus the mostnatural choice is the scheme T [22 1 2 ]12 .

3 Irreducible tensor operator techniques **in** atomic spectroscopy 53From now on, the operators Ôα ([λ]κ) that act on the irreducible tensor space H q will beconsidered. The notations identified **in** Sec. 3.1.4 will be used. In the f**in**al result, these operatorsare associated to ÔΛ ([λ]κ) act**in**g on H Λ by us**in**g Eq. (3.51) for [λ] = [2 2 1 2 ], κ = 12. That is,[λ N Γ¯Λ‖ÔΛ ([2 2 1 2 ]12)‖λ N Γ ′ ¯Λ′ ] = 1 2[λΓQ¯Λ||| Ô 0Λ ([2 2 1 2 ]12)|||λΓ ′ Q¯Λ ′]+ 1 2 〈Q′ M Q 20|QM Q 〉 [ λΓQ¯Λ|||Ô2Λ ([2 2 1 2 ]12)|||λΓ ′ Q ′ ¯Λ′ ]+ 32 √ 5 〈Q′ M Q 10|QM Q 〉 [ λΓQ¯Λ|||Ô1Λ ([2 2 1 2 ]12)|||λΓ ′ Q ′ ¯Λ′ ]+ 12 √ 5 〈Q′ M Q 30|QM Q 〉 [ λΓQ¯Λ|||Ô3Λ ([2 2 1 2 ]12)|||λΓ ′ Q ] ′ ¯Λ′ , (3.84)where it is assumed that λ ≡ l 1 /2 for LS-coupl**in**g (Λ ≡ LS) and λ ≡ j for jj-coupl**in**g(Λ ≡ J). The quasisp**in** number Q = 1 /2 (λ − v + 1 /2); the basis **in**dex M Q = 1 /2 (N − λ − 1 /2).Eq. (3.84) **in**dicates that it is a l = 1–shell case, where the irreps with**in** Ôς ([2 2 1 2 ]12) (recallthat α ≡ ς ≡ κΛ) satisfy α 1 = α 2 = . . . = α 6 = ς x ≡ 1 /2 λ, where as usually, x = i, j, k, l, p, q.Eq. (3.84) is noth**in**g else but the mathematical realisation **of** Remark 3.2.1: a one-toonecorrespondence for the basis function |λ N ΓΛM〉 **of** H Λ is the function |λΓQΛM Q M〉 **of**H Q × H Λ . It is understood that a particular s**in**gle-shell case can be extended to l = 2, 3, 4, 5, 6.However, **in** all these cases a correspondence **in** Eq. (3.51) holds true.To f**in**d the (reduced) matrix element **of** Ôς ([2 2 1 2 ]12) that acts on l = 2, 3, . . . , 6 electronshells, an efficient preparation **of** Ôς ([2 2 1 2 ]12) needs to be established **in** the same manner as ithas been done for the two-particle case. To solve the present task, a classification **of** operatorsthat act on different number **of** electron shells is required.Assum**in**g that the numbers x ∈ {i, j, k, l, p, q} designate the **in**dices **of** irreps ς x with**in**Ô ς ([2 2 1 2 ]12) and at the same time they mark the xth electron shell, make the follow**in**g convenientnotations. (In contrast, the **in**dex i = 1, 2, . . . , 6 **of** irrep α i labels the ith operator a α iwith**in** Ôς ([2 2 1 2 ]12).)3.2.4 Def**in**ition. The set **of** operatorsÔ ς ([2 2 1 2 ]12) ≡ ̂T ς (λ i λ j λ k λ l λ p λ q ) ≡ [ [W ς ij(λ i λ j ) × a ς k] ς ijk× [W ς lp(λ l λ p ) × a ςq ]lpq] ς ς(3.85)associated to the angular reduction scheme (i, j, k, l, p, q ∈ I = {1, 2, 3, 4, 5, 6}){T [22 1 2 ]〈x〉, if i ≤ j ≤ k ≤ l ≤ p ≤ q,12 ≡ 〈ijklpq〉 def=(3.86)〈x π 〉, otherwise,forms the parent class X l (∆ 1 , ∆ 2 , . . . , ∆ l ), l ≤ 6 **of** dimension d l if the subtraction **of** multiplicities**of** the same value numbers from the set s = {i, j, k} and from the set s ′ = {l, p, q}equals to ∆ x , wherel∑∆ x = 0. (3.87)3.2.5 Def**in**ition. The class Xl ∗(∆ 1, ∆ 2 , . . . , ∆ l ) **of** dimension d ∗ l such thatx=1is a dual class.X ∗ l (∆ 1 , ∆ 2 , . . . , ∆ l ) = X l (−∆ 1 , −∆ 2 , . . . , −∆ l ), d ∗ l = d l (3.88)In other words, the parent class X l (∆ 1 , ∆ 2 , . . . , ∆ l ) def**in**es the set **of** operators Ôς ([2 2 1 2 ]12)associated to the same angular reduction scheme T [22 1 2 ]12 but with dist**in**ct labell**in**g **of** irreps sothat Eq. (3.87) is valid. The dimension d l **of** class equals to the number **of** such operators.Example. Suppose there are two operators such that α 1 = α 2 = . . . = α 5 = ς 1 , α 6 = ς 2for the first one and α 1 = α 2 = α 3 = α 4 = α 6 = ς 1 , α 5 = ς 2 for the second one. Bothoperators associated to the schemes 〈111112〉 and 〈111121〉 act on two electron shells. Checkthe conditions **in** Eq. (3.87). For the first operator, write s = {i = 1, j = 1, k = 1} and s ′ =

3 Irreducible tensor operator techniques **in** atomic spectroscopy 54{l = 1, p = 1, q = 2}; the multiplicities **of** 1 and 2 **in** s are 3 and 0, respectively. Analogously,the multiplicities **of** 1 and 2 **in** s ′ are 2 and 1. Write ∆ 1 = 3 − 2 = +1, ∆ 2 = 0 − 1 = −1.For the second operator, write s = {i = 1, j = 1, k = 1}, s ′ = {l = 1, p = 2, q = 1} and∆ 1 = 3 − 2 = +1, ∆ 2 = 0 − 1 = −1. Then ∆ 1 + ∆ 2 = +1 + (−1) = 0. Consequently,both operators belong to the same parent class X 2 (+1, −1). By Def**in**ition 3.2.5, the dual classis X 2 (−1, +1), and the operators that correspond to the previous ones are associated to theschemes 〈222221〉, 〈222212〉.The choice which class is parent and which one is dual is optional, as it follows from theirdef**in**itions.3.2.6 Corollary. The mapp π : 〈x〉 −→ 〈x π 〉, ( ) ( )i ′ j ′ k ′ l ′ p ′ q ′1 2 3 4 5 6 ↦→ i ′ j ′ k ′ l ′ p ′ q ′π(1) π(2) π(3) π(4) π(5) π(6)is realised by≡ ( )i j k l p q1 2 3 4 5 6(3.89)̂π ̂T ς (λ i ′λ j ′λ k ′λ l ′λ p ′λ q ′) = ̂T ς (λ i λ j λ k λ l λ p λ q ). (3.90)It follows from Corollary 3.2.6 that for π = (ij), Eq. (3.90) co**in**cides with Eq. (3.21). Ingeneral, the permutation π must be expanded **in**to the product **of** 2–cycles (transpositions) **in**order to apply Eq. (3.21) for several times. However, a general solution to count the number **of**ways **in** which a given permutation π can be factored **in**to a given number **of** transpositions (ij)is absent so far. In algebraic comb**in**atorics, authors usually refer to the so-called Hurwitz’s formulafor the number **of** m**in**imal transitive factorisations. In a particular case, this formula yieldsthat the number **of** factorisations **of** a full cycle **in** S l **in**to l − 1 transpositions is l l−2 [94,95]. InRef. [94], it was also demonstrated that there exists an elegant connection between the primitivefactorisation and Jucys–Murphy elements [96–98]. Due to the absence **of** a general factorisationformula, each permutation π must be expanded **in** a unique way whenever it appears.3.2.7 Theorem. If the operators ̂T ς (λ i ′λ j ′λ k ′λ l ′λ p ′λ q ′) and ̂T ς (λ m ′λ n ′λ r ′λ s ′λ t ′λ u ′) are associatedto the schemes 〈x〉 and 〈y〉, and ̂T ς (λ i λ j λ k λ l λ p λ q ), ̂T ς (λ m λ n λ r λ s λ t λ u ) are associated tothe schemes 〈x π 〉, 〈y π ′〉 so that for some ̂Π, ̂Πλ i = λ m , ̂Πλ j = λ n , ̂Πλ k = λ r , ̂Πλ l = λ s ,̂Πλ p = λ t , ̂Πλ q = λ u , then for the given two maps p π : 〈x〉 −→ 〈x π 〉, p π ′ : 〈y〉 −→ 〈y π ′〉, thereexists a map p˜π such that the follow**in**g diagram is commutative〈y〉p π ′ 〈y π ′〉(3.91)p πp˜π 〈y π 〉and the permutation representation ̂˜π **of** S 6 is found from̂˜π ̂T ς (λ m ′λ n ′λ r ′λ s ′λ t ′λ u ′) = ̂T ς (λ Π(i ′ )λ Π(j ′ )λ Π(k ′ )λ Π(l ′ )λ Π(p ′ )λ Π(q ′ )). (3.92)Pro**of**. To prove the theorem, it suffices to demonstrate that the commutator [̂π, ̂Π −1̂˜π] = 0,assum**in**g that ̂Π −1 ̂Π = 16 . Indeed, if the latter commutator equals to zero, then, by pass**in**g toEq. (3.92) and the def**in**ition **of** ̂Π, it is true that ̂Π̂π ̂Π −1̂˜π = ̂π ′ = ̂˜π̂π and thus p π ′ = p eπ ◦ p π .F**in**d [̂π, ̂Π −1̂˜π] ̂T ς (λ m ′λ n ′λ r ′λ s ′λ t ′λ u ′). For ̂π ̂Π −1̂˜π, writêπ ̂Π −1̂˜π ̂T ς (λ m ′λ n ′λ r ′λ s ′λ t ′λ u ′) = ̂π ̂Π −1 ̂T ς (λ Π(i ′ )λ Π(j ′ )λ Π(k ′ )λ Π(l ′ )λ Π(p ′ )λ Π(q ′ ))= ̂π ̂T ς (λ i ′λ j ′λ k ′λ l ′λ p ′λ q ′) = ̂T ς (λ i λ j λ k λ l λ p λ q ).For ̂Π −1̂˜π̂π, writêΠ −1̂˜π̂π ̂T ς (λ m ′λ n ′λ r ′λ s ′λ t ′λ u ′) = ̂T ς (λ i λ j λ k λ l λ p λ q )

3 Irreducible tensor operator techniques **in** atomic spectroscopy 55if [̂π, ̂Π −1̂˜π] = 0. Then̂˜π̂π ̂T ς (λ m ′λ n ′λ r ′λ s ′λ t ′λ u ′) = ̂Π ̂T ς (λ i λ j λ k λ l λ p λ q ) = ̂T ς (λ m λ n λ r λ s λ t λ u )= ̂π ′ ̂T ς (λ m ′λ n ′λ r ′λ s ′λ t ′λ u ′)and ̂π ′ = ̂˜π̂π.3.2.8 Corollary. If the operator ̂T ς (λ i λ j λ k λ l λ p λ q ) associated to the scheme 〈x π 〉 belongs tothe class X l (∆ 1 , ∆ 2 , . . . , ∆ l ) and it is found by realis**in**g the map p π : 〈x〉 −→ 〈x π 〉 for̂T ς (λ i ′λ j ′λ k ′λ l ′λ p ′λ q ′) with i ′ ≤ j ′ ≤ k ′ ≤ l ′ ≤ p ′ ≤ q ′ , then any other irreducible tensoroperator ̂T ς (λ m λ n λ r λ s λ t λ u ) = ̂Π ̂T ς (λ i λ j λ k λ l λ p λ q ) associated to the scheme 〈y π ′〉 **of** classX l (∆ ′ 1, ∆ ′ 2, . . . , ∆ ′ l ) is found by realis**in**g the map p π ′ = p eπ ◦p π for ̂T ς (λ m ′λ n ′λ r ′λ s ′λ t ′λ u ′) withm ′ ≤ n ′ ≤ r ′ ≤ s ′ ≤ t ′ ≤ u ′ , where ̂˜π is found from Eq. (3.92). The class X l (∆ ′ 1, ∆ ′ 2, . . . , ∆ ′ l )is called the derived class **of** dimension d l .Pro**of**. By Def**in**ition 3.2.4, the pro**of** is similar to the one applied to Theorem 3.2.7.3.2.9 Corollary. The dual class X ∗ l (∆ 1, ∆ 2 , . . . , ∆ l ) is a particular case **of** the derived classX l (∆ ′ 1, ∆ ′ 2, . . . , ∆ ′ l ) if ∆′ 1 = −∆ 1 , ∆ ′ 2 = −∆ 2 , . . . , ∆ ′ l = −∆ l; each derived class has its dualone which is **of** the same dimension.Corollary 3.2.9 **in**dicates that for a given parent class, all three types **of** classes—parent,dual, derived—are **of** the identical dimension.Theorem 3.2.7 allows us to reduce the number **of** classes that need to be studied to classifythe irreducible tensor operators which act on l = 2, 3, . . . , 6 electron shells. The classification**of** all the rest operators that belong to the correspond**in**g derived class is performed immediatelyby us**in**g Eq. (3.91), once the permutation representation ̂˜π is found. In addition, it ought to beby no means obvious that Theorem 3.1.8 also fits Eq. (3.90), thus the permutations that arisedur**in**g the transformations are «simplified», as it can be recognised from the tables **in** AppendixB.In Tabs. 20-35 (Appendix B), the dual classes are not listed. The reason for not do**in**gso is simple. If the operator ̂L expanded by us**in**g Eq. (3.81) belongs to the parent classX l (∆ 1 , ∆ 2 , . . . , ∆ l ), then its adjo**in**t operator ̂L † belongs to the dual class Xl ∗(∆ 1, ∆ 2 , . . . , ∆ l )which is noth**in**g else but X l (−∆ 1 , −∆ 2 , . . . , −∆ l ) (Def**in**ition 3.2.5). Consequently, the matrixelement 〈Ψ i |̂L † |Ψ j 〉 on the basis given **in** Eq. (2.2) simply equals to the matrix element〈Ψ j |̂L|Ψ i 〉 if recall**in**g that the matrix element on the **in**f**in**ite-dimensional N–electron Hilbertspace H is def**in**ed by the map X × X −→ R, where X denotes the set **of** basis Ψ i (Sec. 2.1).The similar argument holds for the dual classes Xl ∗(∆′ 1, ∆ ′ 2, . . . , ∆ ′ l ) associated to the derivedclasses X l (∆ ′ 1, ∆ ′ 2, . . . , ∆ ′ l ). Thus, the only task is to identify a given irreducible tensor operatorT ς (λ i λ j λ k λ l λ p λ q ) by us**in**g Def**in**ition 3.2.4. This is easily done utilis**in**g Tabs. 36-39 **in**Appendix B. For a particular two-shell case, the identification **of** operators is trivial: there areonly parent and derived classes; the derived classes co**in**cide with the dual ones.Example (Operator identification). Consider operator ̂T ς (λ 5 λ 3 λ 2 λ 4 λ 1 λ 4 ) associated to the angularreduction scheme 〈532414〉. This is a 5–shell case. F**in**d the associated class. By Def**in**ition3.2.4, write s = {i = 5, j = 3, k = 2}, s ′ = {l = 4, p = 1, q = 4}: ∆ 1 = 0 − 1 = −1,∆ 2 = 1−0 = 1, ∆ 3 = 1−0 = 1, ∆ 4 = 0−2 = −2, ∆ 5 = 1−0 = 1. The condition ∑ 5x=1 ∆ x =0 is satisfied, thus the operator belongs to the class X 5 (−1, +1, +1, −2, +1). Identify the class**in** Tab. 38 (Appendix B). The present class is the dual class **of** X 5 (+1, −1, −1, +2, −1) whichis the derived class associated to the parent class X 5 (+2, +1, −1, −1, −1). This implies that thematrix element **of** ̂L ∝ ∑ a 5 a 3 a 2 a † 4a † 1a † 4 (see Eq. (3.80)) is equal to the matrix element **of** ̂L ′ ∝∑a4 a 1 a 4 a † 2a † 3a † 5, where ̂L † = ̂L ′ . Consequently, the operator ̂T ς (λ 4 λ 1 λ 4 λ 2 λ 3 λ 5 ) **of** the derivedclass X 5 (+1, −1, −1, +2, −1) must be studied. Refer to Tab. 33. S**in**ce 〈y π ′〉 = 〈414235〉 and〈y〉 = 〈123445〉, π ′ = (142)(35), where (142) = (12)(14). By Eq. (3.21),

3 Irreducible tensor operator techniques **in** atomic spectroscopy 56̂π ′ ̂T ς (λ 1 λ 2 λ 3 λ 4 λ 4 λ 5 ) = ̂T ς (λ 4 λ 1 λ 4 λ 2 λ 3 λ 5 ) = ∑ε π ′̂T ς (λ 1 λ 2 λ 3 λ 4 λ 4 λ 5 ),ς 12 ς 123ς 44 ς 445ε π ′ =∑ε (142)(35) ε (12)(14) ε 12 .ς 24 ς 134 ς 245Other 17 operators that belong to the derived class X 5 (+1, −1, −1, +2, −1) are obta**in**ed mak**in**guse **of** Eq. (3.91) for ˜π = (153)(24).Tab. 7: The parameters for three-particle matrix elements: l = 2X 2 (∆ 1, ∆ 2) 〈x〉 ξ w 1 w 2X 2 (0, 0) 〈111122〉 20 1111 22〈112222〉 5 2222 −〈111112〉 27 1111 11111X 2 (+1, −1) 〈111222〉 14 − −〈122222〉 30 2222 22222Tab. 8: The parameters for three-particle matrix elements: l = 3X 3 (∆ 1, ∆ 2, ∆ 3) 〈x〉 ξ w 1 w 2 w 3X 3 (0, 0, 0) 〈112233〉 1 22 33 1122〈111123〉 27 1111 11112 −X 3 (+2, −1, −1) 〈112223〉 4 222 11222 −〈112333〉 14 − − −X 3 (+3, −2, −1) 〈111223〉 15 11122 − −〈123333〉 5 3333 − −X 3 (+1, −1, 0) 〈111233〉 20 1112 33 −〈122233〉 24 22 33 222Derived class〈y〉X 3 (−1, +2, −1) 〈122223〉 31 222 2222 12222〈122333〉 6 22 − −Tab. 9: The parameters for three-particle matrix elements: l = 4X 4 (∆ 1, ∆ 2, ∆ 3, ∆ 4) 〈x〉 ξ w 1 w 2 w 3X 4 (+1, +1, −1, −1) 〈111234〉 27 1112 11123 −〈122234〉 29 222 1222 12223〈123334〉 4 333 12333 −〈123444〉 14 − − −X 4 (+2, −2, +1, −1) 〈112234〉 3 22 1122 11223Derived classes〈y〉X 4 (+2, +1, −2, −1) 〈112334〉 15 11233 − −X 4 (+2, +1, −1, −2) 〈112344〉 20 44 1123 −X 4 (+1, +2, −2, −1) 〈122334〉 7 12233 − −X 4 (+1, +2, −1, −2) 〈122344〉 23 44 1223 −X 4 (+1, −1, +2, −2) 〈123344〉 1 33 44 1233Tab. 10: The parameters for three-particle matrix elements: l = 5X 5 (∆ 1, ∆ 2, ∆ 3, ∆ 4, ∆ 5) 〈x〉 ξ w 1 w 2 w 3X 5 (+2, +1, −1, −1, −1) 〈112345〉 27 1123 11234 −X 5 (+1, +1, −1, −1, 0) 〈123455〉 20 55 1234 −Derived classes〈y〉X 5 (+1, +2, −1, −1, −1) 〈122345〉 28 22 1223 12234X 5 (−1, −1, +2, +1, −1) 〈123345〉 3 33 1233 12334X 5 (−1, +1, −1, +2, −1) 〈123445〉 15 12344 − −To conclude, a general formula **of** reduced matrix element **of** ̂T ς (λ i λ j λ k λ l λ p λ q ) is displayedassum**in**g that l > 1 and i ≤ j ≤ k ≤ l ≤ p ≤ q.

3 Irreducible tensor operator techniques **in** atomic spectroscopy 57[(λ1 + λ 2 + . . . + λ l )Γ 1 q 1 Γ 2 q 2 q 12 . . . Γ l q l q||| ̂T ς (λ i λ j λ k λ l λ p λ q )|||(λ 1 + λ 2 + . . . + λ l )¯Γ 1¯q 1¯Γ2¯q 2¯q 12 . . . ¯Γ l¯q l¯q ] = (−1) Φ l∑ɛ ξς w∈Lξ x=1l∏[q 12...x , q x+1 , ¯q 12...x+1 , ς px+1] 1/2× [ ] {¯q }12...x ¯q x+1 ¯q 12...x+1λ x Γ x q x |||Û ςpxx (λ x λ x . . . λ x )|||λ x¯Γx¯q x ς px ς px+1 x+1ς px+1 , (3.93)q 12...x q x+1 q 12...x+1where q ≡ QΛ, q 12...l = q, ¯q 12...l = ¯q, ς pl = ς. In Eq. (3.93), the N x –length numbersp x = 11 . . . 122 . . . 2xx . . . x, p xx = xx . . . x, where N x is the multiplicity **of** equal irreps ς xwith**in** ̂T ς (λ i λ j λ k λ l λ p λ q ). The phase multiplierdefΦ l =l∑ ((N x + N x + ∆ x )x=1l∑ ∑x−1)N y + N x (N z + ∆ z ) . (3.94)y>xN x denotes the number **of** electrons **in** the shell characterised by the numbers λ x Γ x Q x Λ x M Qx M x ;M Qx = 1 /2 (N x − λ x − 1 /2); ∆ x is recognised from the class that conta**in**s a given irreducibletensor operator. The **in**dices **of** summation w ∈ L ξ = {w 1 , w 2 , w 3 } depend on a specified operator.Possible values are listed **in** Tabs. 7-10. Particularly for l = 6, ξ = 27, w 1 = 1234,w 2 = 12345, w 3 is absent.The operator Û ςpxx (λ x λ x . . . λ x ) is associated to the angular reduction schemes: T [12 ]1 forN x = 2; T [21]2 for N x = 3; T [212 ]1 for N x = 4; T [213 ]1 for N x = 5; T [22 1 2 ]12 for N x = 6. Thereduced matrix elements **of** Û Λ on H Λ are known [54, Eq. (25)]. Thus the connection with Û ςis actualised mak**in**g use **of** Tab. 6 and Eq. (3.51) which fits the case N x = 6 (see also Eq.(3.84)).3.3 Summary and conclud**in**g remarksTheoretical atomic physics deals with various irreducible tensor operators that allow us to accountfor the contribution **of** atomic as well as effective **in**teractions **in** a simplified form: theirreducible tensor operators attach the symmetry properties **of** atom. In mathematical formulation,the contributions are evaluated by calculat**in**g matrix elements – the real scalar products onthe **many**-electron Hilbert spaces. In practice, the calculation **of** matrix elements on the basis**of** **many**-electron wave functions is a very complicated task. The reason for this is a complextensor structure **of** **many**-electron operators. The one-electron and two-electron operators thatare most common **in** atomic physics are exam**in**ed very well. In contrast, the study **of** triple oreven higher excitations is already troublesome and an uncerta**in** one so far. Due to a complexity,it has become standard **in** **many** cases **of** MBPT to account for the contributions **of** s**in**gleor double excitations only. But the approximation stipulates the physical problems that can besolved theoretically. On the other hand, the practice requires a more valued precision. Thebest example is the atomic parity violation [42, 57], the study **of** which is a very popular taskamong the atomists nowadays: the contribution **of** at least triple excitations becomes **in**evitable.Therefore **in** most cases, Sec. 3 is concentrated on the present problem.Sec. 3.1.1 provides an opportunity to classify the angular reduction schemes **of** operatorstr**in**g Ôl (see Eq. (3.1)) for any **in**teger l. The classification is performed by us**in**g: the l–numbers (Def**in**ition 3.1.1), the S l –irreducible representations [λ], the l 2 –tuples. The idea—**in**its most general form—is simple: every angular reduction scheme is characterised by the irrep[λ], thus mak**in**g the restriction (items I-III **in** Sec. 3.1.1) S l → S l ′ → . . . → S l ′′ until l ′′ ≤ 3,a given complex structure associated to [λ] is transformed to the scheme associated to either[21] or [1 2 ] (Tabs. 3-4). The path **of** such a restriction is therefore a unique angular reductionscheme.Sec. 3.1.2 concentrates on the case l = 6 which characterises the three-particle operators.The correspondence **of** angular reduction schemes (42 **in** total) is actualised by us**in**g the entriesE ξξ ′, ξ, ξ ′ = 1, 2, . . . , 42 **of** a 42 × 42 transformation matrix E (Corollary 3.1.4). These entriesare constructed from the so-called basis coefficients ɛ ξ that relate operators associated to thez=1

3 Irreducible tensor operator techniques **in** atomic spectroscopy 58scheme T [22 1 2 ]12 with all the rest operators that are constructed by us**in**g different schemes T κ[λ](refer to Eq. (3.11)). The basis coefficients are displayed **in** Appendix A.In Sec. 3.1.3, the permutation properties **of** the irreps with**in** a given operator Ôα ([λ]κ)associated to the angular reduction scheme T κ[λ] are studied for l = 6. The symmetry groupS 6 conta**in**s 6! permutations at all. However, each permutation or a product **of** permutations isa product **of** transpositions whose total number is 15 (see Eq. (3.19)). Thus the task that dealswith operators with different order**in**g **of** irreps is conf**in**ed to the task to f**in**d the transformationcoefficients E ijξ ′ ξ (Def**in**ition 3.1.5) formed from the so-called recoupl**in**g coefficients ε ij (see Eq.(3.21)). These coefficients relate operators constructed by us**in**g the order**in**g (α 1 , α 2 , . . . , α 6 )with the operators constructed by us**in**g the order**in**g (α π(1) , α π(2) , . . . , α π(6) ), where π ∈ S 6 isa transposition. The expressions **of** ε ij are found by mak**in**g use **of** the so-called commutativediagrams (see Eq. (3.30)). The orig**in**ated method based on Eq. (3.29) allows us to f**in**d therecoupl**in**g coefficients by mak**in**g the least number **of** transformations that are required by thepermutation representation ̂π. Consequently, it reduces the number **of** **in**termediate irreps.Sec. 3.1.4 is a very important addition to the method presented **in** Sec. 3.1.3. The ma**in** ideais listed **in** Theorem 3.1.8 which is, aga**in**, a statement that allows us to reduce the number **of****in**dices **of** summation. Meanwhile, the present theorem is one **of** the key features that stipulatethe further on foundation **of** the simplification **of** classification **of** the three-particle operatorsact**in**g on **many**-electron shells (Appendix B).In Sec. 3.2, the irreducible tensor operators associated to some special angular reductionschemes are considered. For the two-particle case (Sec. 3.2.1), the most common schemes areT [22 ]1 and (̂π 243 T ) [22 ]1 which are also known [77] as the z-scheme and b-scheme, respectively.The b-scheme first orig**in**ated by Kaniauskas [85] is the most typical one. However, **in** Sec. 4,the advantage **of** a less common z-scheme will be demonstrated.F**in**ally, Sec. 3.2.2 demonstrates the method to classify the three-particle operators that acton l = 2, 3, 4, 5, 6 electron shells. The algorithm is based on Theorem 3.2.7; the realisation – onTheorem 3.1.8. The operators are grouped **in**to the three types **of** classes: parent, dual, derived(Def**in**ition 3.2.4, Def**in**ition 3.2.5, Corollary 3.2.8). The identification **of** operators that belongto a specified class is performed mak**in**g use **of** the tables **in** Appendix B. The present identificationmakes it possible to calculate the matrix elements **of** three-particle operators efficiently: thetables **of** classes display the relationship between the operators that belong to the parent classand the operators that belong to the derived classes. Thus it suffices to f**in**d the expression **of**matrix element (see Eq. (3.93)) for the sole operator – the matrix elements **of** other operatorsare found **in**stantly.In Sec. 2.3, a general expression **of** the Fock space operator has been presented (refer to Eq.(2.41)). In Sec. 2.3.1, its conf**in**ement on the **many**-electron Hilbert space has been obta**in**ed(to compare with, see Eqs. (2.55), (2.59)) lead**in**g to Lemma 2.3.9 and Theorem 2.3.12. As aresult, Eq. (2.69) has been derived for the n–**body** parts (n = 1, 2, 3, 4) **of** wave operator that isnecessary to compute the terms **of** effective Hamiltonian **in** Eq. (2.61). Based on the methods,developed **in** the present section, it is now possible to form the irreducible tensor operators**of** both the wave operator and the effective Hamiltonian **in** a systematic way. However, stillone more problem rema**in**s unsolved. This is the n–**body** effective matrix element ω n (see Eq.(2.65)). As already po**in**ted out **in** Sec. 3.2, these elements are related to a specified systemunder consideration. In the next section, the terms **of** the third-order MBPT (k = 2 **in** Eq.(2.71)) will be considered and the n–**body** (n = 1, 2, 3, 4) effective matrix elements **of** thekth–order (k = 1, 2, 3) will be displayed expressly.

4 Applications to the third-order MBPT 594 Applications to the third-order MBPTThe ma**in** goal **of** the present section is a symbolic preparation **of** terms **of** the third-order effectiveHamiltonian Ĥ (3) so that the expressions attach an applicability to the coupled-cluster(CC) approach as well.To br**in**g the purpose to a successful end, two tasks are to be solved: the generation **of**expansion terms and their appropriation.The generation **of** terms is related to the selection **of** model space **in** open-shell MBPT. Themodel space P has been selected **in** Sec. 2.3 (also, see Def**in**ition 2.3.2 and Corollary 2.3.3).To generate the terms **of** Ĥ (3) , the n–**body** parts **of** ̂Ω (1) and ̂Ω (2) are required (see Eqs. (2.61),(2.71)). These are found from the generalised Bloch equation (refer to Eqs. (2.58), (2.59))̂Ω(k)n[̂Ω (1) , Ĥ0] ̂P = ̂Q(̂V 1 + ̂V 2 ) ̂P , ̂Ω(1) =2∑n=1[̂Ω (2) , Ĥ0] ̂P = ̂Q(̂V 1 + ̂V 2 )̂Ω (1) ̂P , ̂Ω(2) =̂Ω (1)n , (4.1)4∑n=1̂Ω (2)n . (4.2)The solutions for are **of** the form given by Eq. (2.69) replac**in**g ω with ω (k) . Then thethird-order contribution to Ĥ reads (see Eq. (2.66))Ĥ (3) =∑2∑4∑I m+n−ξ m=1 n=1m**in**(2m,2n)∑ξ=1ĥ (3)mn;ξ ,ĥ(3)defmn;ξ= :{ ̂P ̂V ̂P m̂Ω(2) n } ξ :, (4.3)where ĥ(3) mn;ξ, the (m + n − ξ)–**body** operator (Tab. 11), must be expanded **in**to the sum **of**irreducible tensor operators. The conf**in**ement **of** the space the operator acts on is the secondtask to be solved. A common technique **of** expansion **of** N–electron model space P **in**to theorthogonal sum **of** its SU(2)–irreducible subspaces P Λ leads toĥ (3)mn;ξ = ∑ Λ∑+ΛM=−Λ∑ΓÔM([λ]κ) Λ h (3)mn;ξ(ΓΛ). (4.4)Tab. 11: Possible values **of** m, n, ξ necessary to build the (m + n − ξ)–**body** terms **of** Ĥ (3)m + n − ξ m n ξ m + n − ξ m n ξ m + n − ξ m n ξ0 1 1 2 2 2 1 1 3 1 3 12 2 4 1 2 1 2 3 21 1 1 1 2 2 2 1 4 22 1 2 1 3 2 2 4 31 2 2 2 3 3 4 2 3 12 2 3 2 4 4 1 4 12 3 4 3 2 2 1 2 4 25 2 4 1By Lemma 2.3.9, the irreducible tensor operator ÔΛ ([λ]κ) associated to the angular reductionscheme T κ[λ] conta**in**s creation and transposed annihilation operators labell**in**g the valence statesonly. As usually, Γ denotes additional numbers that are necessary to characterise studied operators.h (3)mn;ξ(ΓΛ) denotes the projection-**in**dependent coefficient; thus it is the SU(2)–**in**variant.At this step, it should be po**in**ted out the two pr**in**cipal differences to compare with the traditionalversion **of** MBPT, though they are easily seen from Eqs. (2.69), (4.2), (4.4). These differences,however, comprise their own benefits that are the key features **of** the method presented here.1. The generalised Bloch equation for ̂Ω (k) **in**dicates that the solutions are proportional to thekth power **of** **perturbation** ̂V . For **in**stance, ̂Ω (1) ∝ ̂V (see Eq. (4.1)), ̂Ω (2) ∝ ̂V ̂Ω (1) ∝ ̂V 2

4 Applications to the third-order MBPT 60(see Eq. (4.2)), etc. This is also true for the terms **of** effective Hamiltonian (see Eq. (4.3)):Ĥ (3) ∝ ̂V ̂Ω (2) ∝ ̂V 3 , etc. Each m–**body** part **of** **perturbation** ̂V (refer to Eqs. (2.40)-(2.43)) conta**in**s the m–**body** matrix element v m (α ¯β). Consequently, ̂Ω (k) conta**in**s the product**of** k such elements. Moreover, every s**in**gle term **in** ̂Ω (k) —obta**in**ed by us**in**g the Wick’stheorem—is evaluated separately. A typical example is the classical work **of** Ho et. al. [37].(The latter method **of** evaluation did not change until nowadays.) It should be obvious thatsuch **in**terpretation becomes tedious when a huge number **of** diagrams is generated. This factmakes sense especially for the higher-order PT. In the present case, on the other hand, thesolutions for ̂Ω (k) are given by Eq. (2.69) with a s**in**gle element ω (k) . Thus ω (k) representsthe product **of** k matrix elements **of** ̂V m with energy denom**in**ators **in**cluded. In other words,is the kth–order n–**body** effective matrix element that characterises the n–**body** part **of**the kth–order wave operator. In addition to the convenience **of** otherwise marked product**of** matrix elements v m , there is also an essential peculiarity: by Theorem 2.3.12, the terms**of** ̂Ω (k) are comb**in**ed **in** groups (refer to Eq. (2.69)) related to the different types (core, valence,excited) **of** s**in**gle-electron orbitals. That is, a number **of** Golstone diagrams drawn**in** ω (k) are characterised by the sole tensor structure and thus the problem **of** evaluation **of**each separate diagram is elim**in**ated. Meanwhile, **in** CC approach, the n–particle effects areembodied **in** the so-called amplitudes ρ n (see, for example, Ref. [42]) **of** excitation that are,ω (k)n**in** pr**in**cipal, the n–**body** effective matrix elements. Therefore, if replac**in**g ω n(k) with ρ n , thetensor structure **of** terms rema**in**s steady and thus such formulation is applicable to at leasttwo approaches **of** PT.2. In traditional MBPT, the two-particle matrix elements v 2 (α ¯β) or else g αβ ¯µ¯ν (see Eq. (3.53))are expressed **in** terms **of** b-coefficient (Sec. 3.2.1). In this case, as it will be confirmed later,a z-scheme is preferred.4.1 The treatment **of** terms **of** the second-order wave operatorThe generation **of** expansion terms **of** ̂Ω (2) is clearly a computational task and it is the most timeconsum**in**g process. The nowadays s**of**tware programs are capable to solve the problems **of** thepresent type. To generate the terms, the symbolic package NCoperators written on Mathematicais used. The features **of** the package are studied **in** a more detail **in** Appendix D, while **in** thepresent section, the study **of** already generated terms is argued.Despite **of** a large number **of** generated terms **of** ̂Ω (2) , fortunately, there are only a few **of**fundamental constructions that need to be considered: all the rest terms are obta**in**ed vary**in**gthe known expressions.In total, there are 13 such constructions (Tabs. 12-14). In tables, g αβ ¯µ¯ν and ˜g αβ ¯µ¯ν , the matrix representations**of** a two-particle **in**teraction operator g 12 , are def**in**ed by Eqs. (3.53)-(3.54). Theirexplicit expressions are found from Eqs. (3.63)-(3.64). For the typical **in**teraction operators g 12observed **in** atomic physics, the expressions **of** z coefficients are listed **in** the examples **of** Sec.3.2.1. Here, Remark 3.2.3 is taken **in**to account. The s**in**gle-particle matrix representation v α ¯β**of** the self-adjo**in**t **in**teraction operator v i reads (refer to Eq. (2.43))defv α ¯β = v ¯βα = 〈α|v i | ¯β〉 = (−1) λ ¯β +m ¯β f(τi λ α λ ¯β)〈λ α m α λ ¯β − m ¯β|τ i m i 〉, (4.5)f(τ i λ α λ ¯β) def= − [λ α] 1/2 [nα λ[τ i ] 1/2 α ‖v τ i]‖n ¯βλ ¯β . (4.6)In Eqs. (4.5)-(4.6), v τ idenotes the SO(3)–irreducible tensor operator that acts on the subspace**of** the space the operator v i acts on. It is assumed that the s**in**gle-particle Slater **in**tegral(s) is**in**volved **in** f(τ i λ α λ ¯β). The number **of** such **in**tegrals depends on the basis |n α λ α m α 〉. Thenumbers f(τ i λ α λ ¯β) are complex, **in** general, and they satisfyf(τ i λ ¯βλ α ) = o(τ i λ α λ ¯β)f(τ i λ α λ ¯β), (4.7)

4 Applications to the third-order MBPT 61Tab. 12: The multipliers for one-particle effective matrix elements **of**Construction∑µ v αµω (1)µ ¯β∑ζ ¯µ v ζ ¯µ˜ω (1)¯µαζ ¯β(1)∑ζ ¯µ ˜g¯µαζ ¯βωζ ¯µ∑ρηζ ˜g αζρη ˜ω (1)ρηζ ¯βSU(2)–**in**variant: ExpressionS α ¯β(τ 1 τ 2 τ):(−1) λ ¯β −λα−τ [τ 1 , τ 2 ] 1/2 ∑ µ×〈τ 1 m 1 τ 2 m 2 |τm〉˜S α ¯β(τ 1 ):λα+λ2(−1)¯β∑ζ ¯µ (−1)λ f(τ ζ+λ¯µ 1λ ζ λ¯µ)ε ζ ¯β−ε α¯µ×˜z(0λ¯µ λ α λ ¯βλ ζ uu)˜S ′ α ¯β(τ 2 ):̂R ( ¯βζ→¯µα¯µ→ζ) ˜Sα ¯β(τ 2 )˜S α ¯β:̂Ω(2)f(τ 1λ αλ µ)f(τ 2λ µλ {¯β) τ1 τ 2 τ }ε ¯β−ε µλ ¯β λ α λ µ4δ λαλ [λ ¯β α ] −1/2 ∑ u∑ρηζ (−1)λρ−λη−u 1ε ¯βζ −ε ρη×˜z(0λ α λ ζ λ η λ ρ uu)˜z(0λ ρ λ η λ α λ ζ uu)∑ { }u [u]1/2 τ 1 λ ζ λ¯µu λ α λ ¯βTab. 13: The multipliers for two-particle effective matrix elements **of**Constructionv α¯µ ω (1)β¯νSU(2)–**in**variant: ExpressionD αβ ¯µ¯ν (udτ):[τ 1 , τ 2 , u, d] 1/2 f(τ1λαλ¯µ)f(τ2λ βλ¯ν)ε¯ν−ε β〈τ 1 m 1 τ 2 m 2 |τm〉{ }λα λ β u× dλ¯µ λ¯ντ 1 τ 2 τ∑ζ v αζω (1)ζβ ¯µ¯νD αβ ¯µ¯ν (Uuτ 1 ):2(−1) λα−λ β+λ¯µ+λ¯ν+τ 1[U] 1/2 ∑ }ζ×{τ1 λ α λ ζλ β u U∑ζ g ζβ ¯µ¯νω (1)αζD ′ αβ ¯µ¯ν (Uuτ 2):∑ζρ g αβζρω (1)ρζ ¯µ¯ν∑ζρ g αζ ¯µρω (1)ρβ¯νζ̂R ( ¯µ¯ν→ζβζ→α)Dαβ ¯µ¯ν (Uuτ 2 )D αβ ¯µ¯ν (uu):4(−1) λ¯µ+λ¯ν+u [u] −1/2 ∑ ζρ∆ αβ ¯µ¯ν (UU):4(−1) U−λ¯µ [U] 1/2 ∑ ud×{λα λ β Uλ ζ d λ¯νu λ ρ λ¯µ}̂Ω(2)f(τ 1λ αλ ζ )ε¯µ¯ν−ε βζz(0λ ζ λ β λ¯ν λ¯µ uu)z(0λ αλ β λ ρλ ζ uu)z(0λ ρλ ζ λ¯νλ¯µuu)ε¯µ¯ν−ε ζρ∑ζρ (−1)λ ζ+d [u,d] 1/2ε¯νζ −ε βρz(0λ α λ ζ λ ρ λ¯µ uu)z(0λ ρ λ β λ ζ λ¯ν dd)Tab. 14: The multipliers for three- and four-particle effective matrix elements **of**ConstructionSU(2)–**in**variant: Expressionv α¯µ ω (1)βζ ¯ν ¯ηT αβζ ¯µ¯ν ¯η (uτ 1 ):2(−1) λ¯ν+λ¯η+u f(τ 1λ αλ¯µ)ε¯ν ¯η−ε βζg βζ ¯ν ¯η ω (1)α¯µ T ′ αβζ ¯µ¯ν ¯η (uτ 2):∑ρ g αβ ¯µρω (1)ρζ ¯ν ¯ηg αβ ¯µ¯ν ω (1)ζρ¯η¯σ̂R (¯ν ¯η→¯µβζ→α)Tαβζ ¯µ¯ν ¯η (uτ 2 )(−1) M T αβζ ¯µ¯ν ¯η (DdU):z(0λ β λ ζ λ¯η λ¯ν uu)(−1) λρ4(−1) λ¯η−λ¯ν+λζ+U [D] 1/2 ∑ uρ ε¯ν ¯η−ε ζρ[u] 1/2 z(0λ α λ β λ ρ λ¯µ uu)×z(0λ ρ λ ζ λ¯η λ¯ν dd) { } { }D d U λ α λ β uλ ρ λ β λ ζλ ρ λ¯µ UQ αβζρ¯µ¯ν ¯η¯σ (ud):4ε¯η¯σ−ε ζρ(−1) u+d+λ¯µ+λ¯ν+λ¯η+λ¯σ z(0λ α λ β λ¯ν λ¯µ uu)z(0λ ζ λ ρ λ¯σ λ¯η dd)̂Ω(2)

4 Applications to the third-order MBPT 62v τ i†m i∑+τ io(τ i λ α λ ¯β) def= (−1) λα−λ ¯β [τi ] −1m i =−τ i(−1) m i−e(τ i m i ) , (4.8)which is easy to derive consider**in**g the properties **of** the self-adjo**in**t operator v i : v α ¯β = v ¯βα and= (−1) e(τ im i ) v τ i−m i. The phase multiplier e(τ i m i ) is optional. Usually, two possibilities areexploited, one that is given by e(τ i m i ) = m i [8, Eq. (25)], and the other by e(τ i m i ) = τ i − m i[11, Eq. (2.3)]. It is to be noticed that throughout the present text, the second possibility ispreferred. In this case, o(τ i λ α λ ¯β) = (−1) λα−λ ¯β +τ i.The **in**dex i dist**in**guishes the **in**teraction operators v i that befit to a second quantised form̂V 1 : for ̂V 1 **in** Eq. (4.3), i = 0; for ̂V 1 **in** Eq. (4.2), i = 1; for ̂V 1 **in** Eq. (4.1), i = 2.In Tabs. 12-14, the abbreviation ε xy...z = ε x + ε y + . . . + ε z , where ε x is the s**in**gle-electronenergy (refer to Eq. (2.65)). The operator ̂R replaces orbitals **in** energy denom**in**ators. Thus,for example, ̂R (¯ν ¯η→¯µβζ→α)(ε¯ν ¯η − ε βζ ) −1 = (ε¯µ − ε α ) −1 .The first-order effective matrix elements ω (1) , ˜ω (1) are def**in**ed by (see Eq. (2.65))ω (1)α ¯βdef= v α ¯β , ω (1) defαβ ¯µ¯ν= g αβ ¯µ¯ν, ˜ω (1) defαβ ¯µ¯ν= ω (1)αβ ¯µ¯νε ¯β − ε α ε¯µ¯ν − ε − ω(1) αβ¯ν ¯µ . (4.9)αβRecall**in**g the properties **of** v α ¯β, g αβ ¯µ¯ν (also, see Eq. (3.55)), it is easy to deduce that the matrixrepresentations ω (1) , ˜ω (1) satisfy ω (1) = −ω(1)α ¯β ¯βα andω (1)αβ ¯µ¯ν = ω(1) βα¯ν ¯µ = −ω(1) ¯ν ¯µβα = −ω(1) ¯µ¯ναβ ,˜ω (1)αβ ¯µ¯ν = ˜ω(1) βα¯ν ¯µ = ˜ω(1) ¯µ¯νβα = ˜ω(1) ¯ν ¯µαβ(4.10a)= −˜ω (1)αβ¯ν ¯µ = −˜ω(1) βα¯µ¯ν = −˜ω(1) ¯ν ¯µβα = −˜ω(1) ¯µ¯ναβ .(4.10b)Now, it becomes clear why it suffices to consider only 13 constructions from the large number**of** terms.To dist**in**guish the SU(2)–**in**variants by the summation parameters, the follow**in**g specialnotations are used.µ c v eS α ¯β(τ 1 τ 2 τ) S α ¯β(τ 1 τ 2 τ) Ṡ α ¯β(τ 1 τ 2 τ) ¨Sα ¯β(τ 1 τ 2 τ)ζ c c¯µ v e˜S α ¯β(τ 1 )˜Ṡ α ¯β(τ 1 )˜¨S α ¯β(τ 1 )˜S ′ α ¯β(τ 2 )˜Ṡ ′ α ¯β(τ 2 )˜¨S ′ α ¯β(τ 2 )ζ c v eD αβ ¯µ¯ν (Uuτ 1 ) D αβ ¯µ¯ν (Uuτ 1 ) Ḋ αβ ¯µ¯ν (Uuτ 1 ) ¨Dαβ ¯µ¯ν (Uuτ 1 )D αβ ′ ¯µ¯ν (Uuτ 2) D αβ ′ ¯µ¯ν (Uuτ 2) Ḋ αβ ′ ¯µ¯ν (Uuτ 2) ¨D′ αβ ¯µ¯ν (Uuτ 2 )ζ c v e e vρ c v e v e...Dαβ ¯µ¯ν(uu)D αβ ¯µ¯ν (uu) D αβ ¯µ¯ν (uu) Ḋ αβ ¯µ¯ν (uu) ¨Dαβ ¯µ¯ν (uu)ζ c cρ v e∆ αβ ¯µ¯ν (UU) ˙∆αβ ¯µ¯ν (UU) ¨∆αβ ¯µ¯ν (UU)ρ c v eT αβζ ¯µ¯ν ¯η (DdU) T αβζ ¯µ¯ν ¯η (DdU) ˙ T αβζ ¯µ¯ν ¯η (DdU)...Dαβ ¯µ¯ν(uu)¨Tαβζ ¯µ¯ν ¯η (DdU)(4.11)(4.12)(4.13)(4.14)(4.15)(4.16)

4 Applications to the third-order MBPT 63Particularly, for ˜S α ¯β, ζ = c, η = v, ρ = e. To designate that the direct and exchanged parts **of** atwo-particle matrix element are **in**volved, the additional tildes are exploited. Thus ˜D αβ ¯µ¯ν (Uuτ 1 )implies that the two-particle matrix representation reads ˜ω (1)ζβ ¯µ¯ν . Conversely, ˜D ′ αβ ¯µ¯ν(Uuτ 2 ) impliesthat the two-particle matrix representation reads ˜g ζβ ¯µ¯ν . If written D αβ ¯µ¯ν (ũu) (∆ αβ ¯µ¯ν (ŨU)),then the considered matrix representation is ˜g αβζρ (˜g αζ ¯µρ ); if written D αβ ¯µ¯ν (uũ) (∆ αβ ¯µ¯ν (UŨ)),then the correspond**in**g matrix representation is ˜ω (1)ρζ ¯µ¯ν (˜ω ρβ¯νζ); if written ˜D αβ ¯µ¯ν (uu) ( ˜∆ αβ ¯µ¯ν (UU)),then the matrix representations g and ω (1) are both with tildes. The similar arguments hold andfor the rest **of** SU(2)–**in**variants. Also, it should be noted that for each two-particle matrixrepresentation with tilde, the correspond**in**g z coefficient is given by Eq. (3.64).Example. Consider the sum **of** three Goldstone diagrams:vv¯cτ· 1 τ v 1 ·1 v 1 ·e¯c¯cv ′cvτ· 2 τ v 2 · 2 v 2 ·where the double arrow dist**in**guishes the valence electron states. S**in**ce τ 1 ≠ 0, τ 2 ≠ 0, thesediagrams **in**volve the **in**teraction **of** atom with some external field (e.g. electric). By Lemma2.3.9, given diagrams denote the one-**body** terms **of** ̂Ω (2) . Their algebraic equivalent readsv¯cev ′τ 1 τ 2 ∑ ∑a v a †¯cv ve ω (1)e¯c+ ∑ ∑a v a †¯cv vv ′ω (1)v ′¯c + ∑ ∑a v a †¯cε¯c − ε vv¯cε¯c − ε vv¯c cv 1v 2v c¯c ω (1)vcε v − ε¯c.Now, arrange a given expression **in** terms **of** SU(2)–**in**variants. By us**in**g Tab. 12 and Eq.(4.11), it follows that the first two terms are easy to obta**in** with α = v, ¯β = ¯c and µ = e, v ′ .The correspond**in**g SU(2)–**in**variants are ¨S v¯c (τ 1 τ 2 τ) (for µ = e) and Ṡv¯c(τ 1 τ 2 τ) (for µ = v ′ ).The last term (diagram) must be written **in** a standard form, as it does not satisfy a givenconstruction for S α ¯β(τ 1 τ 2 τ). Make use **of** the symmetry properties **of** v α ¯β and ω (1) . Thenα ¯β∑ ∑a v a †¯cv¯ccv c¯c ω (1)vcε v − ε¯c= ∑ v¯c∑a v a †¯cv¯cc ω cv(1)ε¯c − ε vand thus the correspond**in**g SU(2)–**in**variant is S¯cv (τ 1 τ 2 τ) with α = ¯c, ¯β = v, µ = c. F**in**ally,the given sum **of** diagrams yields∑(ε¯c −ε v ) ∑ (−1 Wm(λ τ v˜λ¯c ) ( Ṡ v¯c (τ 1 τ 2 τ)+ ¨S v¯c (τ 1 τ 2 τ) ) )+(−1) λv−λ¯c+m W−m(λ τ v˜λ¯c )S¯cv (τ 1 τ 2 τ) .v¯cτmAlso, it is **in**terest**in**g to notice that particularly for e(τ i m i ) = τ i − m i (see Eqs. (4.7)-(4.8)),S¯cv (τ 1 τ 2 τ) = (−1) λv−λ¯c+τ 1+τ 2 ̂R(¯c→vc→c)Sv¯c (τ 1 τ 2 τ).Thus the third diagram represents some complex conjugate—the reflection about a horizontalaxis—**of** a diagram that is characterised by the SU(2)–**in**variant S v¯c (τ 1 τ 2 τ).The example **in**dicates that the Goldstone diagrams are represented by the two types **of**irreducible tensor operators: ÔΛ +M and ÔΛ −M . The analysis **of** all generated terms **of** ̂Ω (2) leadsto the follow**in**g conclusions. The terms (diagrams) associated to the tensor structure ÔΛ −Mare: (i) the folded diagrams; (ii) some diagrams, obta**in**ed by contract**in**g the core orbitals **in**Wick’s series; (iii) the diagrams, acceded to the reflection about a horizontal axis **of** a number**of** diagrams associated to the tensor structure ÔΛ +M .c

4 Applications to the third-order MBPT 64To dist**in**guish the considered two types **of** terms, the superscripts ± over the matrix representationsare used so that for n = 1, 2, 3, 4, the SU(2)–**in**variants Ω (2)± **of** each n–**body** part **of**̂Ω (2) are given by∑ω (2)±α ¯β = (−1) t± 1t + 1ω (2)±αβ ¯µ¯ν = (−1)t± 2Λdef= λ ¯β + m ¯β, t − 1∑EΛ 1 Λ 2 Λ〈λ α m α λ ¯β − m ¯β|Λ ± M〉Ω (2)±α¯β(Λ), (4.17a)def= λ α + m α , (4.17b))Ω (2)±αβ ¯µ¯ν (Λ 1Λ 2 Λ), (4.18a)(λα λ β λ¯µ λ¯νm α m β −m¯µ −m¯ν; Λ 1 Λ 2 Λ ± Mt + def2 = λ¯µ + λ¯ν + m¯µ + m¯ν , t − def2 = λ α + λ β + m α + m β , (4.18b)ω (2)±αβζ ¯µ¯ν ¯η = ∑(−1)t± 3(−1) M 〈λ α m α λ¯µ − m¯µ |Λ 3 ± M 3 〉Ω (2)±αβζ ¯µ¯ν ¯η (Λ 1Λ 2 Λ 3 Λ)∑Λ 1 Λ 2 Λ 3 Λ(λβ λ× E ζ λ¯ν λ¯ηm β m ζ; Λ−m¯ν −m¯η 1 Λ 2 Λ ∓ Mt + 3def= λ¯µ + λ¯ν + λ¯η + m¯µ + m¯ν + m¯η , t − 3ω (2)αβζρ¯µ¯ν ¯η¯σ = ∑(−1)t 4Λ 1 Λ 2Λ 3 Λ 4 Λ(λα λ× Eβ λ¯µ λ¯νm α m β; Λ−m¯µ −m¯ν 1 Λ 2 ΛM), (4.19a)def= λ α + λ β + λ ζ + m α + m β + m ζ , (4.19b)(−1) M Ω (2)αβζρ¯µ¯ν ¯η¯σ (Λ 1Λ 2 Λ 3 Λ 4 Λ))E( )λζ λ ρ λ¯η λ¯σm ζ m ρ; Λ−m¯η −m¯σ 3 Λ 4 Λ − M , (4.20a)deft 4 = λ¯µ + λ¯ν + λ¯η + λ¯σ + m¯µ + m¯ν + m¯η + m¯σ (4.20b)and ω (2) = ω (2)+ + ω (2)− . The coefficients Ω (2)± are displayed **in** Appendix C.4.1.1 Remark. In Tab. 12-14, the SU(2)–**in**variants are also obta**in**ed by given Eqs. (4.17)-(4.20)for the effective matrix elements with the plus sign, the basis **in**dices +M, +M 3 and elim**in**atedphase multiplier (−1) M .Eqs. (4.17)-(4.20) direct attention to the convenience **of** z-scheme to compare with b-scheme, though both **of** them are equivalent. The attraction is due to the coefficients E andtheir symmetry properties **in** Eq. (3.60). Conversely, for b-scheme, the associated coefficientreads E ′ ≡ ̂π 23 E, and thus it does not satisfy the same properties as that **of** E. For **in**stance, therelationship between ̂π 24 E ′ and E ′ is realised by( )λα λ¯ν λEβ λ¯µm α −m¯ν m β; Λ−m¯µ 1 Λ 2 ΛM = ∑ (−1) Λ2+1 a(λ¯µ λ¯ν Λ 2 )[Λ 1 , Λ 2 , Λ 1 , Λ 2 ] 1/2Λ 1 Λ 2⎧ ⎫⎨λ α λ¯ν Λ 1 ⎬ ( )× λ¯µ λ β Λ 2⎩Λ 1 Λ 2 Λ⎭ E λα λ¯µ λ β λ¯νm α −m¯µ m β; Λ−m¯ν 1 Λ 2 ΛMwhich implies the appearance **of** additional summation.4.2 The treatment **of** terms **of** the third-order effective HamiltonianBy Lemma 2.3.9, this part **of** computation requires significantly less time to compare with thehandl**in**g **of** ̂Ω (2) . Moreover, the non-zero terms ĥ(3) mn;ξare derived **in** accordance with Theorem2.3.12 which makes it possible to reject a large amount **of** terms **of** ̂Ω (2) attach**in**g the zero-valuedcontributions. As already po**in**ted out, the terms **of** ̂Ω (2) that provide non-zero contributions to, are listed **in** Appendix C.ĥ (3)mn;ξIn the present section, the s**in**gle-particle and two-particle operators **of** Ĥ (3) will be considered.Then (refer to Eq. (4.4)) ÔΛ ([λ]κ) ≡ W Λ (λ v˜λ¯v ) is associated to the angular reductionscheme T [12 ]1 for m+n−ξ = 1, and ÔΛ ([λ]κ) ≡ −[W Λ 1(λ v λ v ′)×W Λ 2(˜λ¯v˜λ¯v ′)] Λ is associatedto the scheme T [22 ]1 for m + n − ξ = 2. Possible values **of** m, n, ξ are listed **in** Tab. 11.

4 Applications to the third-order MBPT 65Tab. 15: The expansion coefficients for one-**body** terms **of** the third-order contribution to the effectiveHamiltonian(mnξ) h (3)+mn;ξ (Λ)(111)(−1) λv−λ¯v [τ 0 ] 1/2 ∑ (Λ [Λ]1/2 〈τ 0 m 0 Λ M|ΛM〉−(−1) Λ ∑ { })c f(τ 0λ c λ¯v )Ω (2)+vc (Λ)τ 0 Λ Λλ v λ¯v λ c(212) 2(−1) λv−λ¯v ∑ α=v,e(−1) Λ ∑ e f(τ 0λ v λ e )Ω (2)+e¯v{ }(Λ)τ 0 Λ Λλ¯v λ v λ e∑c ∑ { }(−1)λ α ′ −λc u ˜z(0λ cλ v λ¯v λ α ′uu)Ω (2)+Λα ′ c (Λ)[u]1/2 λα ′ λ cu λ v λ¯v(122)(223)(234)(−1) Λ [τ 0 ] 1/2 ∑ Λ 1Λ 2Λ (−1)Λ [Λ 1 , Λ 2 , Λ] 1/2 〈τ 0 m 0 Λ M|ΛM〉 ∑ ∑c(v ′(−1)λc−λ v ′ f(τ 0 λ c λ v ′){×˜Ω (2)+v ′ vc¯v (Λ λv}′ λ v Λ 11Λ 2 Λ)− ∑ } )e a(λ eλ¯v Λ 2 )f(τ 0 λ c λ e )Ω (2)+ev¯vc(Λ 1 Λ 2 Λ)λ c λ¯v Λ 2τ 0 Λ Λ2 ∑ Λ 1Λ 2[Λ 1 ] 1/2 ∑ cc ′ (a(λ v λ¯v Λ) ∑ v ′ ˜z(0λ cλ c ′λ v ′λ¯v Λ 2 Λ 2 )˜Ω (2)+vv ′ cc ′(Λ 1Λ 2 Λ){λe λ v Λ 1λ c λ¯v Λ 2τ 0 Λ Λ{Λ1 Λ 2 Λλ¯v λ v λ v ′−a(Λ 1 Λ 2 Λ) ∑ e ˜z(0λ cλ c ′λ¯v λ e Λ 2 Λ 2 )Ω (2)+evcc ′(Λ 1Λ 2 Λ) { Λ 1 Λ 2} ) Λλ¯v λ v λ e+ 2 ∑ Λ 1Λ 2(−1) Λ1 [Λ 2 ] 1/2{ Λ1 Λ 2 Λ× ∑ } (c λ¯v λ v λ ca(λ v λ¯v Λ) ∑ µ=v,e (−1)λ µ ′ +λ µ ′′ ˜z(0λ v λ c λ µ ′′λ µ ′Λ 1 Λ 1 )Ω (2)+µ ′ µ ′′ c¯v (Λ 1Λ 2 Λ)−a(Λ 1 Λ 2 Λ) ∑ )ev ′(−1)λe+λ v ′ ˜z(0λ c λ v λ v ′λ e Λ 1 Λ 1 )Ω (2)+ev ′¯vc (Λ 1Λ 2 Λ)2 ∑ ∑(cc∑µ=v,e′ Λ 2a(λ c λ c ′Λ 2 ) ˜z(0λ c λ c ′λ v ′′λ µ ′Λ 2 Λ 2 )Ω (2)+vv ′′ µ ′¯vc ′ c (Λ 2Λ 2 Λ0)+ ∑ Λ (−1)λ¯v+Λ3+M 1Λ 3Λ[Λ 1 , Λ 3 , Λ] 1/2 〈Λ 3 M 3 Λ M|ΛM〉 ( (−1) λ v ′′×˜z(0λ c λ c ′λ µ ′λ v ′′Λ 2 Λ 2 )Ω (2)+v ′′ vµ ′¯vc ′ c (Λ 1Λ 2 Λ 3 Λ)×˜z(0λ c λ c ′λ µ ′′λ µ ′Λ 2 Λ 2 )Ω (2)+µ ′′ µ ′ v¯vc ′ c (Λ 1Λ 2 Λ 3 Λ){Λ1 Λ 2 Λλ v ′′ λ v λ µ ′{Λ1 Λ 2 Λλ µ ′′ λ v λ µ ′}} { }Λ 3 Λ Λλ v λ¯v λ v ′′} {Λ3 Λ Λ+ a(Λ 1 Λ 2 λ v )} ) )λ v λ¯v λ µ ′′Tab. 16: The expansion coefficients for two-**body** terms **of** the third-order contribution to the effectiveHamiltonian(mnξ) h (3)+mn;ξ (Λ 1Λ 2 Λ) a−[τ 0 ] 1/2 ∑ { }Λ [Λ 1Λ 1, Λ]((−1) 1/2 Λ1 a(λ v λ v ′τ 0 )a(Λ 1 Λ 2 Λ)[Λ 1 ] 1/2 〈τ 0 m 0 Λ M|ΛM〉τ0 Λ 1 Λ 1(121)(211){τ0 λ v λ eΛ 2 Λ Λ× ∑ }e f(τ 0λ v λ e )Ω (2)+ev ′¯v¯v ′(Λ 1Λ 2 Λ)λ v ′ Λ 1 Λ 1+ (−1) m0 a(Λ 1 Λ 2 Λ)[Λ 2 ] 1/2 〈τ 0 − m 0 Λ M|ΛM〉{ }×τ0 Λ 2 Λ 1∑ { })Λ 1 Λ Λ c f(τ 0λ c λ¯v ′)Ω (2)+vv ′ c¯v (Λ τ0 λ¯v ′ λ1Λ 1 Λ)cλ¯v Λ 1 Λ 2((−1) λ¯v ′ +Λ [Λ 2 ] 1/2 ∑ { }e ˜z(0λ (−1)λe v λ v ′λ e λ¯v Λ 1 Λ 1 )Ω (2)+e¯v (Λ) Λ1 Λ 2 Λ′ λ¯v ′ λ e− (−1) λ¯v+Λ1 [Λλ¯v 1 ] 1/2× ∑ { })c ˜z(0λ cλ v λ¯v ′λ¯v Λ 2 Λ 2 )Ω (2)+v ′ c (Λ) Λ1 Λ 2 Λλ c λ v ′ λ v(222)(132)a(λ¯v λ¯v ′Λ 2 )[Λ 2 ] −1/2 ∑ cc ˜z(0λ cλ ′ c ′λ¯v ′λ¯v Λ 2 Λ 2 )Ω (2)+vv ′ cc ′(Λ 1Λ 2 Λ) + [Λ 1 ] −1/2× ∑ e∑µ=v,e ˜z(0λ vλ v ′λ µ ′′λ e Λ 1 Λ 1 )Ω (2)+eµ ′′¯v¯v ′(Λ 1Λ 2 Λ) + 2(−1) Λ1+Λ2 [Λ 1 , Λ 2 ] 1/2 ∑ Λ [Λ 1Λ 2u 1] 1/2×[Λ 2 , u] 1/2 ∑ (c(−1) ∑ { }Λ1+Λ2e ˜z(0λ vλ c λ e λ¯v ′uu)Ω (2)+ev ′¯vc (Λ λ¯v ′ λ e Λ 1 Λ 21Λ 2 Λ) u λ v ′ Λ λ¯vλ c λ v Λ 1 Λ 2+(−1) λ ∑ { }v ′ +λ¯v v ′′(−1)λc+λ v ′′ ˜z(0λ v λ c λ v ′′λ¯v ′uu)˜Ω (2)+v ′ v ′′ c¯v (Λ λ¯v ′ λ v ′′ Λ 1 Λ 2)1Λ 2 Λ) u λ v ′ Λ λ¯vλ c λ v Λ 1 Λ 2(−1) τ0+M [τ 0 , Λ 2 , Λ] 1/2 ∑ ∑ ∑( c µ=v,e Λ 2Λ 3Λϑ (−1)Λ [Λ 2 , Λ 3 , ϑ] 1/2 〈Λ 3 M 3 ϑϱ|Λ M〉×〈τ 0 m 0 Λ − M|ϑϱ〉 (−1) λc−λ µ ′′ +Λ [Λ 1 ] 1/2 ∑ Λ 1[Λ 1 ] 1/2 f(τ 0 λ c λ µ ′′)} (Ω(2)+×vv ′ µ ′′¯v¯v ′ c (Λ 1Λ 2 Λ 3 Λ) − δ µv a(λ v ′λ v ′′Λ 1 ){λ¯v Λ 2 Λ 3 Λ λ v Λ 1λ¯v ′ ϑ λ v ′Λ 2 λ c Λ τ 0 Λ 1 λ µ ′′×Ω (2)+vv ′′ v ′¯v¯v ′ c (Λ 1Λ 2 Λ 3 Λ) ) + (−1) λ¯v ′ +λ v{ } ′′ +Λ2+Λ3+ϑ δ µv f(τ 0 λ c λ v ′′)Ω (2)+v ′′ vv ′¯v¯v ′ c (Λ 1Λ 2 Λ 3 Λ)λc Λ 2 Λ Λ 3)× λ¯v ′ Λ 1 ϑ λ v ′′λ¯v Λ 2 Λ τ 0a The 12j–symbol **of** the first k**in**d is given **in** Ref. [12, Sec. 4-33, p. 207, Eq. (33.17)]

4 Applications to the third-order MBPT 66Tab. 17: The expansion coefficients for two-**body** terms **of** the third-order contribution to the effectiveHamiltonian (cont**in**ued)(mnξ) h (3)+mn;ξ (Λ 1Λ 2 Λ)(233)(244)2[Λ 2 ] 1/2 ∑ c (−1)λc ∑ Λ 1Λ 2Λ 3Λ (−1)M [Λ 3 , Λ] 1/2 ((−1) Λ 〈Λ 3 M 3 Λ − M|ΛM〉 [∑ c ′ ([Λ1 , Λ 1 ] 1/2×(−1) λ ∑ c ′ +Λ1 µ=v,e ˜z(0λ cλ c ′λ¯v ′λ µ ′′Λ 2 Λ 2 ){δ µv Ω (2)+vµ ′′ v ′¯vc ′ c (Λ 1Λ 2 Λ 3 Λ) − a(λ v ′λ µ ′′Λ 1 ){ } { }×Ω (2)+vv ′ µ ′′¯vc ′ c (Λ λ v λ1Λ 2 Λ 3 Λ)}Λ1 Λ 2 Λv ′ Λ 1λ¯v λ¯v ′ Λλ¯v ′ λ v ′ λ 2µ− δ′′Λ1Λ 1(−1) λ c ′ +Λ1Λ 3 Λ Λ× ∑ { } { }v ˜z(0λ cλ ′′ c ′λ¯v ′λ v ′′Λ 2 Λ 2 )Ω (2)+v ′′ vv ′¯vc ′ c (Λ Λ1Λ 2 Λ 3 Λ) 2 Λ 2 Λ 3 Λ1 Λ 2 Λ )λ v ′′ λ¯v λ¯v ′ Λ 3 Λ Λ 2− (−1)λ v ′ +Λ 1×[Λ 1 , Λ 2 ] 1/2 ∑ ∑v ′′ µ=v,e (−1)λ v ′′ +λ¯µ ′′ +Λ1+Λ2 ˜z(0λ v λ c λ¯µ ′′λ v ′′Λ 1 Λ 1 )Ω (2)+v ′ v ′′ ¯µ ′′¯v¯v ′ c (Λ 1Λ 2 Λ 3 Λ){ } { }λ v ′ λ v Λ 1Λ× 1 Λ 2 Λ]λ¯v λ¯v ′ Λ 2λ¯v ′ λ v λ c+ (−1)λ¯v+Λ 2+Λ 3+Λ [Λ 1 , Λ 1 , Λ 2 ] 1/2 〈Λ 3 M 3 Λ M|ΛM〉 ∑ u [u]1/2Λ 3 Λ Λ}{ u λ¯v λ c λ¯v ′ λ v Λ 2× [ (−1) ∑ Λ1 µ=v,e ˜z(0λ vλ c λ¯µ ′′λ µ ′′uu)Ω (2)+µ ′′ ¯µ ′′ v ′¯v¯v ′ c (Λ 1Λ 2 Λ 3 Λ)+(−1) λ ∑ (∑v ′ v ′′ e (−1)λe ˜z(0λ v λ c λ e λ v ′′uu)Ω (2)+v ′′ v ′ e¯v¯v ′ c (Λ 1Λ 2 Λ 3 Λ){ u λ¯v λ c λ¯v ′ λ v Λ 2}× λ v ′′ Λ 2 Λ 1 − (−1) Λ1+u ∑¯v ′′(−1)λ¯v ′′ ˜z(0λ v λ c λ¯v ′′λ v ′′uu)λ e Λ 3 Λ 1 Λ λ v ′{Λu λ¯v λ c λ¯v ′ λ v Λ 2×Ω (2)+¯v ′′ v ′ v ′′¯v¯v ′ c (Λ 1Λ 2 Λ 3 Λ)λ¯v ′′ Λ 2 Λ 1λ v ′′ Λ 3 Λ 1 Λ λ v ′ Λ} )] )λ µ ′′ Λ 2 Λ 1λ¯µ ′′ Λ 3 Λ 1 Λ λ v ′ Λ2δ Λ1Λ 2δ Λ0 (−1) λ¯v+λ¯v ′ ∑ cc ′ Λ 2( ∑v ′′ (∑¯v ′′ a(λ v ′′λ¯v ′′Λ 1)˜z(0λ c λ c ′λ¯v ′′λ v ′′Λ 2 Λ 2 )×Ω (2)vv ′ v ′′¯v ′′¯v ′¯vc ′ c (Λ 1Λ 1 Λ 2 Λ 2 0) + ∑ eΛ [Λ][a(λ eλ v ′′Λ 2 )[Λ 1 , Λ 2 ] −1/2˜z(0λ c λ c ′λ v ′′λ e Λ 2 Λ 2 )×Ω (2)ev ′′ vv ′¯v ′¯vc ′ c (Λ 2Λ 1 Λ 1 Λ 2 Λ) − ∑ { } { }Λ 3Λ 4[Λ 2 , Λ 3 ] 1/2 (−1) Λ4 Λa(λ e λ v ′Λ 1 ) 1 Λ 2 Λ Λ 3 Λ 4 Λλ e λ v ′ λ v λ e λ v ′ λ v ′′×˜z(0λ c λ c ′λ v ′′λ e Λ 4 Λ 4 ){Ω (2)evv ′ v ′′¯v ′¯vc ′ c (Λ 2Λ 1 Λ 3 Λ 4 Λ) − a(λ v ′λ v ′′Λ 3 )Ω (2)evv ′′ v ′¯v ′¯vc ′ c (Λ 2Λ 1 Λ 3 Λ 4 Λ)}] )+ ∑ µ=v,e∑Λ [Λ][Λ 1, Λ 2 ] −1/2˜z(0λ c λ c ′λ¯µ ′′λ µ ′′Λ 2 Λ 2 )Ω (2)µ ′′ ¯µ ′′ vv ′¯v ′¯vc ′ c (Λ 2Λ 1 Λ 1 Λ 2 Λ)+ ∑ v∑Λ ′′¯v a(Λ ′′ 3Λ 4Λ 1Λ 3 λ¯v ′′)[Λ][Λ 2 , Λ 3 ] 1/2˜z(0λ { } { }Λ c λ c ′λ¯v ′′λ v ′′Λ 4 Λ 4 ) 1 Λ 2 Λ Λ 3 Λ 4 Λλ¯v ′′ λ v ′ λ v λ¯v ′′ λ v ′ λ v ′′×[Ω (2)¯v ′′ vv ′′ v ′¯v ′¯vc ′ c (Λ 2Λ 1 Λ 3 Λ 4 Λ) + a(λ v λ v ′Λ 2 )a(λ v ′′λ¯v ′′Λ 3 )Ω (2)v¯v ′′ v ′ v ′′¯v ′¯vc ′ c (Λ 2Λ 1 Λ 3 Λ 4 Λ)−a(Λ 3 Λ 4 λ v ′){Ω (2)¯v ′′ vv ′ v ′′¯v ′¯vc ′ c (Λ 2Λ 1 Λ 3 Λ 4 Λ) + (−1) λ v)′′ +Λ3 a(λ v λ v ′Λ 2 )×Ω (2)v¯v ′′ v ′′ v ′¯v ′¯vc ′ c (Λ 2Λ 1 Λ 3 Λ 4 Λ)}]The expansion coefficients h (3)mn;ξ associated to the irreducible tensor operator ÔΛ are givenby the sum **of** h (3)+mn;ξand h(3)−mn;ξ. The sign **of** h(3)±mn;ξ is related to the sign **of** ̂Ω (2)± . Particularly, thecoefficients h (3)+mn;ξare found **in** Tabs. 15-17. The coefficients h(3)−mn;ξare derived from h(3)+mn;ξ bymak**in**g the follow**in**g alterations:(a) Ω (2)+α ¯β(Λ) → (−1)λα+λ ¯β +M+1 Ω (2)− (Λ)(b) Ω (2)+αβ ¯µ¯ν (Λ 1Λ 2 Λ) → (−1) λα+λ β+λ¯µ+λ¯ν+M Ω (2)−αβ ¯µ¯ν (Λ 1Λ 2 Λ)α ¯β(c) Ω (2)+αβζ ¯µ¯ν ¯η (Λ 1Λ 2 Λ 3 Λ) → (−1) λα+λ β+λ ζ +λ¯µ+λ¯ν+λ¯η+M+M 3 +1 Ω (2)−αβζ ¯µ¯ν ¯η (Λ 1Λ 2 Λ 3 Λ)In addition, there holds one more rule: (d) each basis **in**dex drawn **in** h (3)+mn;ξis replaced by theopposite sign **in**dex except for m 0 . In tables, the quantities ˜Ω (2)± satisfy˜Ω (2)±αβ ¯µ¯ν (Λ 1Λ 2 Λ) = Ω (2)±αβ ¯µ¯ν (Λ 1Λ 2 Λ) − a(λ α λ β Λ 1 )Ω (2)±βα¯µ¯ν (Λ 1Λ 2 Λ). (4.21)It is now easy to br**in**g to a decision the applicability **of** terms **of** Ĥ (3) given by Eq. (4.4).1. The third-order contributions to the effective Hamiltonian Ĥ (3) are written **in** an operatorform provid**in**g an opportunity to construct their matrix representations efficiently

4 Applications to the third-order MBPT 67– the task is conf**in**ed to the calculation **of** matrix elements **of** irreducible tensor operatorsthat are written apart from the projection-**in**dependent parts. These particular angularcoefficients conta**in** Ω (2)± multiplied by the 3nj–symbols.2. The determ**in**ation **of** SU(2)–**in**variants **of** ̂Ω (2) makes it possible to settle down a number**of** terms (Goldstone diagrams) associated to the sole tensor structure. As a result, thelatter approach permits to ascerta**in** the contribution **of** n–particle (n = 1, 2, 3, 4) effects**in** CC approximation. This is done by simply replac**in**g Ω (2)± with Ω n found from Eqs.(4.17)-(4.20). In this case, replace the correspond**in**g effective matrix element ω (2) withω, where ω denotes the valence s**in**gles, doubles, triples and quadruples amplitude.3. The terms **of** Ĥ (3) account for the **in**teraction **of** atom with external field. This is drawn**in** the matrix representations **of** v i that represents the electric, magnetic, hyperf**in**e, etc.**in**teraction operator.4. The terms **of** Ĥ (3) are applicable to the study **of** both nonrelativistic and relativistic approximations.These effects are embodied **in** z coefficients.4.3 Conclud**in**g remarks and discussionIn Sec. 4, an algebraic technique to evaluate the terms **of** MBPT has been suggested. Themethod relies on Lemma 2.3.9 and Theorem 2.3.12. The advantage **of** method reveals itselfespecially **in** the higher-order MBPT, when a huge number **of** terms is generated.The restriction **of** space H the operator ̂Ω (2) acts on to its SU(2)–irreducible subspaces H Λmakes it possible to f**in**d the 13 non-equivalent SU(2)–**in**variants, as demonstrated **in** Tabs. 12-14. Due to the symmetry properties **of** matrix representations v α ¯β, g αβ ¯µ¯ν , these **in**variants areenough to evaluate all generated terms **of** ̂Ω (2) . As a result, the quantities Ω (2)± are determ**in**ed(Appendix C). Consequently, the restriction **of** space P the effective Hamiltonian Ĥ (3) acts onto its subspaces P Λ (see Eq. (4.4)) permits to express the expansion coefficients h (3)mn;ξ**in** terms**of** Ω (2)± (Tabs. 15-17). The key feature is that the coefficients Ω (2)± can be replaced while on aparticular case **of** **in**terest, but the tensor structure **of** operators on P Λ rema**in**s steady.Tab. 18: The amount **of** one-**body** terms **of** Ĥ (3)(mnξ) d + ¯d + d − ¯d −(111) 13 0 3 0(122) 37 0 18 0(212) 14 2 2 0(223) 67 34 29 2(234) 57 36 18 18Total: 188 72 70 20In total, there are computed 188+70 = 258 direct one-**body** terms **of** Ĥ (3) and 72+20 = 92direct one-**body** terms **of** Ĥ (3) **in**clud**in**g the two-particle **in**teractions g 0 only (**in** Tab. 18, thesign ± over d or ¯d takes possession **of** the sign **of** ĥ(3)± mn;ξ). To compare with, Blundell et. al. [41,Sec. II, Eq. (8)] calculated 84 diagrams contribut**in**g to the third-order mono-valent removal energy.Their studied energies E (3)A –E(3) H , E(3) I, E (3)Jand E (3)K , E(3) Lconform to the matrix elements**of** terms drawn **in** accord**in**gly ĥ(3) 22;3, ĥ(3) 23;4 and ĥ(3) 21;2 if g 0 represents the Coulomb **in**teraction. For**in**stance, E (3)A = ∑ ee ′ c ˜g vce ′ e ω (2)ee ′ c¯v which conforms to ˜z(0λ vλ c λ µ ′′λ µ ′Λ 1 Λ 1 )Ω (2)µ ′ µ ′′ c¯v (Λ 1Λ 2 Λ)perta**in**ed to h (3)22;3(Λ) with µ = e (Tab. 15).Tab. 19 lists the amount **of** two-**body** terms **of** Ĥ (3) . There are computed 217 + 82 = 299direct two-**body** terms and 125 + 42 = 167 direct two-**body** terms **in**clud**in**g the two-particle**in**teractions g 0 only. For **in**stance, Ho et. al. [37] calculated 218 two-**body** diagrams **of** the

4 Applications to the third-order MBPT 68Tab. 19: The amount **of** two-**body** terms **of** Ĥ (3)(mnξ) d + ¯d + d − ¯d −(121) 20 0 10 0(211) 13 2 3 0(222) 64 32 31 4(132) 20 16 10 10(233) 75 50 28 28(244) 25 25 − −Total: 217 125 82 42third-order **perturbation**. Analogous disposition to account for the two-particle **in**teractions g 0only can be found and **in** other foremost works [36, 41].At this step, it should be made an explanatory **of** such a comparison with other works. Thepr**in**cipal difference **of** formulation **of** MBPT between the present and the other works is thedifference between algebraic and diagrammatic realisation – all the rest **of** dist**in**ctive featuresfollow the present one. Particularly, one **of** the typical features is the so-called factorisationtheorem [37, Eq. (10)] which makes it possible to comb**in**e the diagrams with the same energydenom**in**ators. Consequently, the comparison **of** the amount **of** computed terms with the amount**of** computed diagrams is a very conditional one.As it follows from Tab. 11, Ĥ (3) also **in**cludes the terms ĥ(3) mn;ξwith m + n − ξ = 0, 3, 4, 5.The zero-**body** terms (the scalars on P 0 ) are easy to derive. The result readsĥ (3)+11;2 = ∑ cĥ (3)−11;2 = ∑ cĥ (3)+22;4 =2 ∑ ∑( ∑cc ′ Λ 1µ=v,e∑(−1) λµ−λc+m 0f(τ 0 λ c λ µ )Ω (2)+µc (τ 0 ), (4.22a)µ=v,e∑µ=v,ef(τ 0 λ c λ µ )Ω (2)−µc (τ 0 ), (4.22b)(−1) λµ+λ µ ′ Ω (2)+µµ ′ cc(Λ ′ 1 Λ 1 0)˜z(0λ c λ c ′λ µ ′λ µ Λ 1 Λ 1 )+ ∑ )a(λ e λ v Λ 1 )Ω (2)+evcc(Λ ′ 1 Λ 1 0)˜z(0λ c λ c ′λ v λ e Λ 1 Λ 1 ) , (4.23a)v,eĥ (3)−22;4 =2 ∑ ∑ ( ∑a(λ c λ c ′Λ 1 ) Ω (2)−µµ ′ cc(Λ ′ 1 Λ 1 0)˜z(0λ c λ c ′λ µ ′λ µ Λ 1 Λ 1 )cc ′ Λ 1 µ=v,e+ ∑ )Ω (2)−evcc(Λ ′ 1 Λ 1 0)˜z(0λ c λ c ′λ v λ e Λ 1 Λ 1 ) , (4.23b)v,eand ĥ(3)− 11;2 vanishes if the s**in**gle-particle **in**teractions v i are neglected (see Eq. (C.1b) **in** AppendixC).The study **of** excitations with m + n − ξ > 2 is usually much more complicated. However,once the SU(2)–**in**variant coefficients Ω (2) are derived, the expansion coefficients h (3)mn;ξcan beobta**in**ed **in** the same manner. In Ref. [42, Eqs. (19)-(20)], the triple excitations for m = n = 2,ξ = 1 have been considered. In this case,h (3)+22;1 (E 1 Λ 1 E 2 Λ 2 Λ) = (−1) λ v ′′+λ¯v ′+λ¯v ′′+Λ2+Λ [Λ 1 , Λ 2 ] 1/2∑ (∑Λ 1a(λ v ′λ¯v Λ 2 )[E 1 , E 2 , Λ 1 ] 1/2Λ 2× ∑ { } { } { } { }˜z(0λ c λ v λ¯v ′λ¯v uu)Ω (2)+λv ′ v ′′ c¯v(Λ ′′ 1 Λ 2 Λ) v Λ 2 Λ 2 Λ 2 λ v Λ λv 2 λ v ′ E 1 λ¯v ′′ λ¯v ′ E 2λ¯v ′′ u λ c Λ 1 Λ Λ 1 λ v ′′ Λ 1 Λ 1 λ¯v Λ 2 ucu∑{ } {+ (−1) E 1+E 2˜z(0λ v λ v ′λ e λ¯v E 1 E 1 )Ω (2)+ev ′′¯v ′¯v Λ(Λ ′′ 1 E 2 Λ) 1 Λ 1 λ¯v Λ1 Λ 2 Λλ e E 1 λ v ′′ E 2 Λ 1}), (4.24)λ¯veand the correspond**in**g irreducible tensor operator **in** Eq. (4.4) is associated to the angular re-

4 Applications to the third-order MBPT 69duction scheme T [22 1 2 ]12 so thatÔ Λ M([2 2 1 2 ]12) ≡ [[W E 1(λ v λ v ′) × a λ v ′′ ] Λ 1× [W E 2(˜λ¯v ′′˜λ¯v ′) × ã λ¯v ] Λ 2] Λ M,where ĥ(3) 22;1 conta**in**s 30 direct three-**body** terms **of** Ĥ (3) . The study **of** other expansion coefficientssuitable for higher-order excitations is still **in** progress.

5 Prime results and conclusions 705 Prime results and conclusions1. The RCGC technique based on the constituted SO(3)–irreducible tensor operators has beenorig**in**ated. The key feature **of** proposed technique is the ability to reduce the N–electron angular**in**tegrals **in**to the sum **of** s**in**gle **in**tegrals. The method is especially convenient for thecalculation **of** matrix elements **of** **in**teraction operators on the basis **of** SU(2)–irreducible matrixrepresentations. As a result, the proposed technique makes it possible to turn to practical accountthe SU(2)–irreducible matrix representations as a convenient basis rather than the usualSlater-type orbitals.2. The effective operator approach has been developed. Based on the Feshbach’s space partition**in**gtechnique, the f**in**ite-dimensional **many**-electron model space has been constructed. Asa result, it has been determ**in**ed that only a fixed number **of** types **of** the Hilbert space operatorswith respect to the s**in**gle-electron states attach the non-zero effective operators on the givenmodel space. The result has a consequential mean**in**g **in** applications **of** atomic **many**-**body****perturbation** **theory**, as it permits to reduce the number **of** expansion terms significantly.3. A systematic way **of** **in**quiry **of** totally antisymmetric tensors has been brought to a moreadvanced state. Based on the S l –irreducible representations and the conception **of** tuples, themethod to classify the angular reduction schemes **of** operator str**in**g **of** any length l has been**in**itiated. Based on the commutative diagrams that realise the mapp**in**gs from a given angularreduction scheme to the required one, the permutation properties **of** antisymmetric tensors havebeen considered systematically. Special attention is paid to the case l = 6 which characterisesthe three-particle operators observed **in** the applications **of** effective operator approach to theatomic **perturbation** **theory**. As a result, the foundations developed for the irreducible tensoroperators associated to dist**in**ct angular reduction schemes appear to be well-suited with respectto facility to compute the matrix representations **of** given operators.4. The classification **of** three-particle operators that act on 2, 3, 4, 5, 6 electron shells **of** atomhas been performed. The irreducible tensor operators associated to their own angular reductionschemes are identified by the classes. The classes are characterised by the number **of** electronshells the operator acts on and by the number **of** electrons **in** a given shell. The proposedclassification is convenient to calculate the matrix elements **of** any three-particle operator. Thatis, the way **of** classification permits to establish the connection between the operators that belongto dist**in**ct classes, and thus it suffices to f**in**d the matrix representation for the sole operator – thematrix representations for the other operators are found **in**stantly by us**in**g the transformationcoefficients.5. The third-order **many**-**body** **perturbation** **theory** has been considered. To simplify the generation**of** expansion terms followed by the generalised Bloch equation, the symbolic programm**in**gpackage NCoperators written on Mathematica has been produced. The angular reduction**of** generated terms has been performed mak**in**g use **of** NCoperators too. The specifictechnique **of** reduction has been developed. The algorithm is based on the composed SU(2)–**in**variants which—owed to the symmetry properties **of** matrix representations **of** atomic **in**teractionoperators—take **in**to consideration all generated terms **of** the second-order wave operator.Therefore the irreducible tensor form **of** terms **of** the third-order effective Hamiltonian is applicableto other effective operator approaches used **in** MBPT: it is simply a manner **of** replacement**of** the excitation amplitudes suitable for some special cases **of** ones **in**terests. Obta**in**edsymbolic preparation **of** terms **of** the third-order MBPT is convenient to implement it **in** thecomputer codes for the calculations **of** characteristic quantitites **of** atoms with several valenceelectrons as well.

5 Prime results and conclusions 71AcknowledgmentI would like to express my gratitude to the scientific supervisor Dr. G**in**taras Alfredas Merkeliswhose critical remarks and at the same time tolerance left the work on the present thesis to beaccomplished to my discretion without loss **of** enthusiasm.

A Basis coefficients 72ABasis coefficients(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 α 2 (α 12 ), α 3 α 4 (α 34 )(α 1234 ), α 5 α 6 (α 56 )α )= (−1) ℘ 1∆ (α 1 , α 2 , α 12 ) [α 34 , α 45 , α 56 , α 123 , α 456 , α 1234 ] 1 2{ } { } { }α6 α×5 α 56 α12 α 3 α 123 α123 α 4 α 1234, (A.1a)α 4 α 456 α 45 α 4 α 1234 α 34 α 56 α α 456℘ 1 = α 5 + α 6 − α 56 + α − α 12 + α 123 − α 1234 − α 3 − α 4 − α 456 .(A.1b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 α 2 (α 12 ), α 3 α 4 (α 34 ), α 5 α 6 (α 56 )(α 3456 )α )= (−1) ℘ 2∆ (α 1 , α 2 , α 12 ) [α 34 , α 45 , α 56 , α 123 , α 456 , α 3456 ] 1 2{ } { } { }α6 α×5 α 56 α12 α 3 α 123 α3 α 4 α 34, (A.2a)α 4 α 456 α 45 α 456 α α 3456 α 56 α 3456 α 456℘ 2 = α 5 + α 6 − α 56 + α + α 12 + 2α 3 + α 3456 .(A.2b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 α 2 (α 12 ), α 3 α 4 (α 34 )(α 1234 )α 5 (α 12345 )α 6 α )= (−1) ℘ 3∆ (α 1 , α 2 , α 12 ) [α 34 , α 45 , α 123 , α 456 , α 1234 , α 12345 ] 1 2{ } { } { }α12 α×3 α 123 α4 α 5 α 45 α45 α 6 α 456, (A.3a)α 4 α 1234 α 34 α 12345 α 123 α 1234 α α 123 α 12345℘ 3 = −α 45 + α 12345 − α − α 5 + α 6 − α 12 − α 1234 − α 3 .(A.3b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 α 2 (α 12 ), α 3 α 4 (α 34 )α 5 (α 345 )(α 12345 )α 6 α )= (−1) ℘ 4∆ (α 1 , α 2 , α 12 ) [α 34 , α 45 , α 123 , α 345 , α 456 , α 12345 ] 1 2{ } { } { }α12 α×3 α 123 α4 α 5 α 45 α45 α 6 α 456, (A.4a)α 45 α 12345 α 345 α 345 α 3 α 34 α α 123 α 12345℘ 4 = −α 4 − α 5 + α 345 − α − α 12 − α 123 − α 12345 − α 6 .(A.4b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 α 2 (α 12 ), α 3 α 4 (α 34 )α 5 (α 345 )α 6 (α 3456 )α )= (−1) ℘ 5∆ (α 1 , α 2 , α 12 ) [α 34 , α 45 , α 123 , α 345 , α 456 , α 3456 ] 1 2{ } { } { }α12 α×3 α 123 α4 α 5 α 45 α45 α 6 α 456, (A.5a)α 456 α α 3456 α 345 α 3 α 34 α 3456 α 3 α 345℘ 5 = −α 3 − α 4 + α 5 + α 6 + α 45 − α 12 + α 345 − α 456 − α 3456 − α.(A.5b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 , α 2 α 3 (α 23 )(α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α )= (−1) ℘ 6∆ (α 4 , α 5 , α 45 ) ∆ (α 45 , α 6 , α 456 ) ∆ (α 123 , α 456 , α) [α 12 , α 23 ] 1 2{ }α1 α×2 α 12, (A.6a)α 3 α 123 α 23℘ 6 = α 1 + α 2 + α 3 + α 123 .(A.6b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 α 3 (α 23 )(α 123 ), α 4 α 5 (α 45 )(α 12345 )α 6 α )= (−1) ℘ 7∆ (α 4 , α 5 , α 45 ) [α 12 , α 23 , α 456 , α 12345 ] 1 2{ } { }α1 α×2 α 12 α45 α 6 α 456, (A.7a)α 3 α 123 α 23 α α 123 α 12345℘ 7 = α 1 + α 2 + α 3 − α 6 − α 45 − α.(A.7b)

A Basis coefficients 73(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 α 3 (α 23 ), α 4 α 5 (α 45 )(α 2345 )(α 12345 )α 6 α )= (−1) ℘ 8∆ (α 4 , α 5 , α 45 ) [α 12 , α 23 , α 123 , α 456 , α 2345 , α 12345 ] 1 2{ } { } { }α1 α×2 α 12 α1 α 23 α 123 α45 α 6 α 456, (A.8a)α 3 α 123 α 23 α 45 α 12345 α 2345 α α 123 α 12345℘ 8 = α 2 + α 3 − α 12345 + 2α 1 + α 23 + α + α 6 .(A.8b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 α 3 (α 23 ), α 4 α 5 (α 45 )(α 2345 )α 6 (α 23456 )α )= (−1) ℘ 9∆ (α 4 , α 5 , α 45 ) [α 12 , α 23 , α 123 , α 456 , α 2345 , α 23456 ] 1 2{ } { } { }α1 α×2 α 12 α1 α 23 α 123 α45 α 6 α 456, (A.9a)α 3 α 123 α 23 α 456 α α 23456 α 23456 α 23 α 2345℘ 9 = α 2 + α 3 + α 123 + α 23456 + α 6 + α 45 − α 456 − α.(A.9b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 α 3 (α 23 ), α 4 α 5 (α 45 )α 6 (α 456 )(α 23456 )α )= (−1) ℘ 10∆ (α 4 , α 5 , α 45 ) ∆ (α 45 , α 6 , α 456 ) [α 12 , α 23 , α 123 , α 23456 ] 1 2{ } { }α1 α×2 α 12 α1 α 23 α 123, (A.10a)α 3 α 123 α 23 α 456 α α 23456℘ 10 = α − α 123 + α 2 − α 23 + α 3 + α 456 .(A.10b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 , α 2 , α 3 α 4 (α 34 )(α 234 )(α 1234 ), α 5 α 6 (α 56 )α )= (−1) ℘ 11[α 12 , α 34 , α 45 , α 56 , α 123 , α 456 , α 234 , α 1234 ] 1 2{ } { } { } { }α1 α×2 α 12 α12 α 3 α 123 α4 α 5 α 45 α4 α 56 α 456, (A.11a)α 34 α 1234 α 234 α 4 α 1234 α 34 α 6 α 456 α 56 α α 123 α 1234℘ 11 = −α 1 + α 2 + α 3 + α 12 + α 34 + α 5 + α 6 − α 56 + α 456 − α 4 − α 123 − α. (A.11b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 , α 2 , α 3 α 4 (α 34 )(α 234 ), α 5 α 6 (α 56 )(α 23456 )α )= (−1) ℘ 12[α 12 , α 34 , α 45 , α 56 , α 123 , α 234 , α 456 , α 23456 ] 1 2{ } [ ]αα4 α×5 α 2 α 23456 α 3 α 45645αα 6 α 456 α 12 α 34 α α 56 , (A.12a)56α 1 α 123 α 234 α 4℘ 12 = α 5 + α 6 − α 123 − α 4 − α 234 + α 23456 + α 12 − α 34 .(A.12b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 , α 2 , α 3 α 4 (α 34 ), α 5 α 6 (α 56 )(α 3456 )(α 23456 )α )= (−1) ℘ 13[α 12 , α 34 , α 45 , α 56 , α 123 , α 456 , α 3456 , α 23456 ] 1 2{ } { } { } { }α1 α×2 α 12 α12 α 3 α 123 α4 α 5 α 45 α4 α 56 α 456, (A.13a)α 3456 α α 23456 α 456 α α 3456 α 6 α 456 α 56 α 3456 α 3 α 34℘ 13 = α 5 + α 6 − α 56 − α 1 − α 2 + α 12 + 2α 3 .(A.13b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α )= ∆ (α 1 , α 2 , α 12 ) ∆ (α 12 , α 3 , α 123 ) ∆ (α 4 , α 5 , α 45 ) ∆ (α 45 , α 6 , α 456 )× ∆ (α 123 , α 456 , α) . (A.14)

A Basis coefficients 74(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )(α 12345 )α 6 α )= (−1) ℘ 15∆ (α 1 , α 2 , α 12 ) ∆ (α 12 , α 3 , α 123 ) ∆ (α 4 , α 5 , α 45 ) [α 456 , α 12345 ] 1 2{ }α45 α×6 α 456, (A.15a)α α 123 α 12345℘ 15 = α + α 123 + α 6 + α 45 .(A.15b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 α 2 (α 12 ), α 3 , α 4 α 5 (α 45 )(α 345 )(α 12345 )α 6 α )= (−1) ℘ 16∆ (α 1 , α 2 , α 12 ) ∆ (α 4 , α 5 , α 45 ) [α 123 , α 345 , α 456 , α 12345 ] 1 2{ } { }α12 α×3 α 123 α45 α 6 α 456, (A.16a)α 45 α 12345 α 345 α α 123 α 12345℘ 16 = α − α 12 + α 123 − α 12345 − α 3 + α 6 .(A.16b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 α 2 (α 12 ), α 3 , α 4 α 5 (α 45 )(α 345 )α 6 (α 3456 )α )= (−1) ℘ 17∆ (α 1 , α 2 , α 12 ) ∆ (α 4 , α 5 , α 45 ) [α 123 , α 345 , α 456 , α 3456 ] 1 2{ } { }α12 α×3 α 123 α45 α 6 α 456, (A.17a)α 456 α α 3456 α 3456 α 3 α 345℘ 17 = α + α 12 − α 3456 + α 45 − α 456 + α 6 .(A.17b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 α 2 (α 12 ), α 3 , α 4 α 5 (α 45 )α 6 (α 456 )(α 3456 )α )= (−1) ℘ 18∆ (α 1 , α 2 , α 12 ) ∆ (α 4 , α 5 , α 45 ) ∆ (α 45 , α 6 , α 456 ) [α 123 , α 3456 ] 1 2{ }α12 α×3 α 123, (A.18a)α 456 α α 3456℘ 18 = α + α 12 + α 3 + α 456 .(A.18b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 α 2 (α 12 )α 3 (α 123 ), α 4 , α 5 α 6 (α 56 )(α 456 )α )= (−1) ℘ 19∆ (α 1 , α 2 , α 12 ) ∆ (α 12 , α 3 , α 123 ) ∆ (α 123 , α 456 , α) [α 45 , α 56 ] 1 2{ }α4 α×5 α 45, (A.19a)α 6 α 456 α 56℘ 19 = α 4 + α 5 + α 6 + α 456 .(A.19b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 α 2 (α 12 )α 3 (α 123 )α 4 (α 1234 ), α 5 α 6 (α 56 )α )= (−1) ℘ 20∆ (α 1 , α 2 , α 12 ) ∆ (α 12 , α 3 , α 123 ) [α 45 , α 56 , α 456 , α 1234 ] 1 2{ } { }α4 α×5 α 45 α4 α 56 α 456, (A.20a)α 6 α 456 α 56 α α 123 α 1234℘ 20 = α + α 123 − α 456 + α 5 + α 6 − α 56 .(A.20b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 α 2 (α 12 ), α 3 , α 4 , α 5 α 6 (α 56 )(α 456 )(α 3456 )α )= (−1) ℘ 21∆ (α 1 , α 2 , α 12 ) [α 45 , α 56 , α 123 , α 3456 ] 1 2{ } { }α4 α×5 α 45 α12 α 3 α 123, (A.21a)α 6 α 456 α 56 α 456 α α 3456℘ 21 = −α 3 + α 4 + α 5 + α 6 − α 12 − α.(A.21b)

A Basis coefficients 75(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 α 3 (α 23 )(α 123 ), α 4 , α 5 α 6 (α 56 )(α 456 )α )= (−1) ℘ 22∆ (α 123 , α 456 , α) [α 12 , α 23 , α 45 , α 56 ] 1 2{ } { }α1 α×2 α 12 α4 α 5 α 45, (A.22a)α 3 α 123 α 23 α 6 α 456 α 56℘ 22 = α 1 + α 2 + α 3 + α 4 + α 5 + α 6 + α 123 + α 456 .(A.22b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 α 3 (α 23 )(α 123 )α 4 (α 1234 ), α 5 α 6 (α 56 )α )= (−1) ℘ 23[α 12 , α 23 , α 45 , α 56 , α 456 , α 1234 ] 1 2{ } { } { }α1 α×2 α 12 α4 α 5 α 45 α4 α 56 α 456, (A.23a)α 3 α 123 α 23 α 6 α 456 α 56 α α 123 α 1234℘ 23 = α 1 + α 2 + α 3 + α 5 + α 6 − α 56 − α + α 456 .(A.23b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 α 3 (α 23 )α 4 (α 234 )(α 1234 ), α 5 α 6 (α 56 )α )= (−1) ℘ 24[α 12 , α 23 , α 45 , α 56 , α 123 , α 234 , α 456 , α 1234 ] 1 2{ } { } { } { }α1 α×2 α 12 α1 α 23 α 123 α4 α 5 α 45 α4 α 56 α 456, (A.24a)α 3 α 123 α 23 α 4 α 1234 α 234 α 6 α 456 α 56 α α 123 α 1234℘ 24 = α 2 + α 3 − α 4 + α 5 + α 6 − α 23 − α 56 + α 456 − α − α 1234 .(A.24b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 , α 2 α 3 (α 23 )α 4 (α 234 ), α 5 α 6 (α 56 )(α 23456 )α )= (−1) ℘ 25[α 12 , α 23 , α 45 , α 56 , α 123 , α 234 , α 456 , α 23456 ] 1 2{ } { } { } { }α1 α×2 α 12 α1 α 23 α 123 α4 α 5 α 45 α4 α 56 α 456, (A.25a)α 3 α 123 α 23 α 456 α α 23456 α 6 α 456 α 56 α 23456 α 23 α 234℘ 25 = α 2 + α 3 − α 123 + α 5 + α 6 − α 56 + α + α 23456 .(A.25b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 , α 2 α 3 (α 23 ), α 4 , α 5 α 6 (α 56 )(α 456 )(α 23456 )α )= (−1) ℘ 26[α 12 , α 23 , α 45 , α 56 , α 123 , α 23456 ] 1 2{ } { } { }α1 α×2 α 12 α1 α 23 α 123 α4 α 5 α 45, (A.26a)α 3 α 123 α 23 α 456 α α 23456 α 6 α 456 α 56℘ 26 = α 2 + α 3 + α 4 + α 5 + α 6 − α 23 + α 123 − α.(A.26b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 α 2 (α 12 )α 3 (α 123 )α 4 (α 1234 )α 5 (α 12345 )α 6 α )= (−1) ℘ 27∆ (α 1 , α 2 , α 12 ) ∆ (α 12 , α 3 , α 123 ) [α 45 , α 456 , α 1234 , α 12345 ] 1 2{ } { }α4 α×5 α 45 α45 α 6 α 456, (A.27a)α 12345 α 123 α 1234 α α 123 α 12345℘ 27 = −α + α 12345 + α 4 + α 5 − α 6 − α 45 .(A.27b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 α 3 (α 23 )(α 123 )α 4 (α 1234 )α 5 (α 12345 )α 6 α )= (−1) ℘ 28[α 12 , α 23 , α 45 , α 456 , α 1234 , α 12345 ] 1 2{ } { } { }α1 α×2 α 12 α4 α 5 α 45 α45 α 6 α 456, (A.28a)α 3 α 123 α 23 α 12345 α 123 α 1234 α α 123 α 12345℘ 28 = α 1 + α 2 + α 3 + α 4 + α 5 + α 6 + α 45 − α 123 + α + α 12345 .(A.28b)

A Basis coefficients 76(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 α 3 (α 23 )α 4 (α 234 )(α 1234 )α 5 (α 12345 )α 6 α ){ }= (−1) ℘ 29[α 12 , α 23 , α 45 , α 123 , α 234 , α 456 , α 1234 , α 12345 ] 1 α1 α22 α 12α 3 α 123 α 23{ } { } { }α1 α×23 α 123 α4 α 5 α 45 α45 α 6 α 456, (A.29a)α 4 α 1234 α 234 α 12345 α 123 α 1234 α α 123 α 12345℘ 29 = α 2 + α 3 − α 23 + α 123 + α 1234 + α 12345 + α − α 5 + α 6 − α 45 .(A.29b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 α 3 (α 23 )α 4 (α 234 )α 5 (α 2345 )α 6 (α 23456 )α ){ }= (−1) ℘ 30[α 12 , α 23 , α 45 , α 123 , α 234 , α 456 , α 2345 , α 23456 ] 1 α1 α22 α 12α 3 α 123 α 23{ } { } { }α1 α×23 α 123 α4 α 5 α 45 α45 α 6 α 456, (A.30a)α 456 α α 23456 α 2345 α 23 α 234 α 23456 α 23 α 2345℘ 30 = α 2 + α 3 + α 4 + α 5 + α 6 − α 23 − α 45 − α 123 − α 2345 + α + α 23456 + α 456 . (A.30b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 α 3 (α 23 )α 4 (α 234 )α 5 (α 2345 )(α 12345 )α 6 α ){ }= (−1) ℘ 31[α 12 , α 23 , α 45 , α 123 , α 234 , α 456 , α 2345 , α 12345 ] 1 α1 α22 α 12α 3 α 123 α 23{ } { } { }α1 α×23 α 123 α4 α 5 α 45 α45 α 6 α 456, (A.31a)α 45 α 12345 α 2345 α 2345 α 23 α 234 α α 123 α 12345℘ 31 = α 2 + α 3 − α 4 − α 5 + α 6 + α 2345 + α + α 12345 .(A.31b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 , α 2 , α 3 α 4 (α 34 )(α 234 )(α 1234 )α 5 (α 12345 )α 6 α ){ }= (−1) ℘ 32[α 12 , α 34 , α 45 , α 123 , α 234 , α 456 , α 1234 , α 12345 ] 1 α1 α22 α 12α 34 α 1234 α 234{ } { } { }α12 α×3 α 123 α4 α 5 α 45 α45 α 6 α 456, (A.32a)α 4 α 1234 α 34 α 12345 α 123 α 1234 α α 123 α 12345℘ 32 = α 1 + α 2 + α 34 + α 6 + α 45 + α + α 5 + α 12345 + α 3 + α 12 .(A.32b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 , α 2 , α 3 α 4 (α 34 )(α 234 )α 5 (α 2345 )α 6 (α 23456 )α )= (−1) ℘ 33[α 12 , α 34 , α 45 , α 123 , α 234 , α 456 , α 2345 , α 23456 ] 1 2[ ]α1 α 2 α 234 α 2345 α 23456× α 12 α 34 α 5 α 6 α , (A.33a)α 123 α 3 α 4 α 45 α 456℘ 33 = α 1 + α 2 − α 12 + α 34 − α 3 − α 4 .(A.33b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 , α 3 α 4 (α 34 )(α 234 )α 5 (α 2345 )(α 12345 )α 6 α )= (−1) ℘ 34[α 12 , α 34 , α 45 , α 123 , α 234 , α 456 , α 2345 , α 12345 ] 1 2{ } [ ]αα45 α×6 α 2 α 2345 α 3 α 45456αα α 123 α 12 α 34 α 12345 α 5 , (A.34a)12345α 1 α 123 α 234 α 4℘ 34 = −α 234 + α 2345 + 2α 4 + α + α 6 + α 12 − α 34 .(A.34b)

A Basis coefficients 77(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 , α 3 α 4 (α 34 )α 5 (α 345 )α 6 (α 3456 )(α 23456 )α )= (−1) ℘ 35[α 12 , α 34 , α 45 , α 123 , α 345 , α 456 , α 3456 , α 23456 ] 1 2{ } { } { } { }α1 α×2 α 12 α12 α 3 α 123 α4 α 5 α 45 α45 α 6 α 456, (A.35a)α 3456 α α 23456 α 456 α α 3456 α 345 α 3 α 34 α 3456 α 3 α 345℘ 35 = α 345 − α 3 + α 4 + α 5 + α 45 + α 6 + α 456 − α 1 − α 2 + α 12 .(A.35b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 , α 3 α 4 (α 34 )α 5 (α 345 )(α 2345 )(α 12345 )α 6 α ){ }= (−1) ℘ 36[α 12 , α 34 , α 45 , α 123 , α 345 , α 456 , α 2345 , α 12345 ] 1 α1 α22 α 12α 345 α 12345 α 2345{ } { } { }α12 α×3 α 123 α4 α 5 α 45 α45 α 6 α 456, (A.36a)α 45 α 12345 α 345 α 345 α 3 α 34 α α 123 α 12345℘ 36 = α 1 + α 2 − α 4 − α 5 + α + α 12 − α 123 + α 6 .(A.36b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 , α 3 α 4 (α 34 )α 5 (α 345 )(α 2345 )α 6 (α 23456 )α )= (−1) ℘ 37[α 12 , α 34 , α 45 , α 123 , α 345 , α 456 , α 2345 , α 23456 ] 1 2{ } [ ]αα4 α×5 α 2 α 23456 α 3 α 45645αα 345 α 3 α 12 α 345 α α 6 , (A.37a)34α 1 α 123 α 2345 α 45℘ 37 = α 4 + α 5 + α 2345 − α 123 + α 3 − α 23456 + α 456 + α 12 .(A.37b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 , α 2 , α 3 , α 4 α 5 (α 45 )(α 345 )α 6 (α 3456 )(α 23456 )α )= (−1) ℘ 38∆ (α 4 , α 5 , α 45 ) [α 12 , α 123 , α 345 , α 456 , α 3456 , α 23456 ] 1 2{ } { } { }α1 α×2 α 12 α12 α 3 α 123 α45 α 6 α 456, (A.38a)α 3456 α α 23456 α 456 α α 3456 α 3456 α 3 α 345℘ 38 = α 6 + α 45 − α 456 + 2α − α 1 − α 2 − α 12 .(A.38b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ α1 , α 2 , α 3 , α 4 α 5 (α 45 )(α 345 )(α 2345 )(α 12345 )α 6 α )= (−1) ℘ 39∆ (α 4 , α 5 , α 45 ) [α 12 , α 123 , α 345 , α 456 , α 2345 , α 12345 ] 1 2{ } { } { }α1 α×2 α 12 α12 α 3 α 123 α45 α 6 α 456, (A.39a)α 345 α 12345 α 2345 α 45 α 12345 α 345 α α 123 α 12345℘ 39 = α 1 + α 2 + α 345 + α − α 12 + α 123 − α 3 + α 6 .(A.39b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 , α 3 , α 4 α 5 (α 45 )(α 345 )(α 2345 )α 6 (α 23456 )α )= (−1) ℘ 40∆ (α 4 , α 5 , α 45 ) [α 12 , α 123 , α 345 , α 456 , α 2345 , α 23456 ] 1 2[ ]α1 α 2345 α 123 α 45× α 12 α α 345 α 6 , (A.40a)α 2 α 3 α 23456 α 456℘ 40 = α 123 + α 2345 + 2α 2 + α 23456 − α 456 − α 12 − α 345 .(A.40b)(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 , α 3 , α 4 α 5 (α 45 )α 6 (α 456 )(α 3456 )(α 23456 )α )= (−1) ℘ 41∆ (α 4 , α 5 , α 45 ) ∆ (α 45 , α 6 , α 456 ) [α 12 , α 123 , α 3456 , α 23456 ] 1 2{ } { }α1 α×2 α 12 α12 α 3 α 123, (A.41a)α 3456 α α 23456 α 456 α α 3456℘ 41 = α 12 − α 1 − α 2 + α 3 + α 456 − α 3456 .(A.41b)

B The classification **of** three-particle operators act**in**g on l = 2, 3, 4, 5, 6 electron shells 78(α1 α 2 (α 12 )α 3 (α 123 ), α 4 α 5 (α 45 )α 6 (α 456 )α ∣ ∣α 1 , α 2 , α 3 , α 4 , α 5 α 6 (α 56 )(α 456 )(α 3456 )(α 23456 )α )= (−1) ℘ 42[α 12 , α 45 , α 56 , α 123 , α 3456 , α 23456 ] 1 2{ } { } { }α1 α×2 α 12 α12 α 3 α 123 α4 α 5 α 45, (A.42a)α 3456 α α 23456 α 456 α α 3456 α 6 α 456 α 56℘ 42 = −α 4 − α 5 − α 6 + α 12 − α 1 − α 2 + α 3 − α 3456 .(A.42b)B The classification **of** three-particle operators act**in**g on l = 2, 3, 4, 5, 6electron shellsB.1 2–shell caseTab. 20: The class X 2 (0, 0): d 2 = 12〈x π〉 π 〈x〉〈112112〉 (35) 〈111122〉〈121112〉 (25)〈211112〉 (15)〈121121〉 (26)〈112121〉 (36)〈112211〉 (35) (46)〈122122〉 (24) 〈112222〉〈122212〉 (25)〈122221〉 (26)〈212122〉 (14)〈212212〉 (15)〈221122〉 (13) (24)Tab. 21: The class X 2 (+1, −1): d 2 = 15〈x π〉 π 〈x〉〈111112〉 1 6 〈111112〉〈111121〉 (56)〈111211〉 (46)〈211122〉 (14) 〈111222〉〈121122〉 (24)〈112122〉 (34)〈211212〉 (15)〈121212〉 (25) 〈111222〉〈112212〉 (35)〈211221〉 (16)〈121221〉 (26)〈112221〉 (36)〈122222〉 1 6 〈122222〉〈212222〉 (12)〈221222〉 (13)Tab. 22: The class X 2 (+2, −2): d 2 = 6〈x π〉 π 〈x〉〈111122〉 1 6 〈111122〉〈111212〉 (45)〈111221〉 (46)〈112222〉 1 6 〈112222〉〈121222〉 (23)〈211222〉 (13)The class X 2 (+3, −3) conta**in**s the sole operator associated to the scheme 〈x π 〉 = 〈111222〉with π = 1 6 .B.2 3–shell caseTab. 23: The class X 3 (0, 0, 0): d 3 = 21〈x π〉 π 〈x〉 〈x π〉 π 〈x〉〈123 {123}〉 (24) (35) ϑ 〈112233〉 〈213213〉 (135) 〈112233〉〈132123〉 (254) 〈213312〉 (135) (46)〈132132〉 (264) 〈231123〉 (13) (254)〈132213〉 (25) 〈231132〉 (13) (264)〈132231〉 (26) 〈231213〉 (13) (25)〈132312〉 (25) (46) 〈312123〉 (154)〈213123〉 (14) (35) 〈312213〉 (15)〈213132〉 (14) (36) 〈321123〉 (154) (23)

B The classification **of** three-particle operators act**in**g on l = 2, 3, 4, 5, 6 electron shells 79Tab. 24: The class X 3 (+2, −1, −1): d 3 = 24〈x π〉 π 〈x〉 〈x π〉 π 〈x〉〈111 {123}〉 ϑ 〈111123〉 〈113233〉 (34) 〈112333〉〈112223〉 1 6 〈112223〉 〈113323〉 (35)〈112232〉 (56) 〈113332〉 (36)〈112322〉 (46) 〈131233〉 (243)〈121223〉 (23) 〈131323〉 (253)〈121232〉 (23) (56) 〈131332〉 (263)〈121322〉 (23) (46) 〈311233〉 (143)〈211223〉 (13) 〈311323〉 (153)〈211232〉 (13) (56) 〈311332〉 (163)〈211322〉 (13) (46)X 3(−1, +2, −1): X 3(−1, −1, +2):〈x〉 ˜π 〈y〉 〈x〉 ˜π 〈y〉〈111123〉 (15) 〈122223〉 〈111123〉 (16) (25) 〈123333〉〈112223〉 (14) (25) 〈111223〉 〈112223〉 (16) (25) 〈122233〉〈112333〉 (13) 〈122333〉 〈112333〉 (16) (25) 〈111233〉Tab. 25: The class X 3 (+3, −2, −1): d 3 = 3〈x π〉 π 〈x〉〈111223〉 1 6 〈111223〉〈111232〉 (56)〈111322〉 − (46)Derived classes ˜π 〈y〉X 3 (−2, +3, −1) (15) (24) 〈112223〉X 3 (−1, −2, +3) (16) (24) (35) 〈122333〉X 3 (+3, −1, −2) (46) 〈111233〉X 3 (−1, +3, −2) (146) 〈122233〉X 3 (−2, −1, +3) (14) (25) (36) 〈112333〉Tab. 26: The class X 3 (+1, −1, 0): d 3 = 45〈x π〉 π 〈x〉 〈x π〉 π 〈x〉〈133233〉 (24) 〈123333〉 〈313332〉 (162) 〈123333〉〈133323〉 (25) 〈331233〉 (13) (24)〈133332〉 (26) 〈331323〉 (13) (25)〈313233〉 (142) 〈331332〉 (13) (26)〈313323〉 (152)〈113 {123}〉 (354) ϑ 〈111233〉 〈{123} 223〉 (35) η 〈122233〉〈131 {123}〉 (254) ϑ 〈{123} 232〉 (36) η〈311 {123}〉 (154) ϑ 〈{123} 322〉 (35) (46) ηX 3 (+1, 0, −1):X 3 (0, +1, −1):〈x〉 ˜π 〈y〉 〈x〉 ˜π 〈y〉〈123333〉 (162) 〈122223〉 〈123333〉 (15) (26) 〈111123〉〈111233〉 (46) 〈111223〉 〈111233〉 (15) (246) 〈112223〉〈122233〉 (25) (36) 〈122333〉 〈122233〉 (135) (26) 〈112333〉Tab. 27: The class X 3 (+2, −2, 0): d 3 = 9〈x π〉 π 〈x〉 〈x π〉 π 〈x〉〈113223〉 (35) 〈112233〉 〈131322〉 (253) (46) 〈112233〉〈113232〉 (36) 〈311223〉 (153)〈113322〉 (35) (46) 〈311232〉 (163)〈131223〉 (253) 〈311322〉 (153) (46)〈131232〉 (263)Derived class ˜π 〈y〉 Derived class ˜π 〈y〉X 3 (+2, 0, −2) (35) (46) 〈112233〉 X 3 (0, +2, −2) (135) (246) 〈112233〉

B The classification **of** three-particle operators act**in**g on l = 2, 3, 4, 5, 6 electron shells 80B.3 4–shell caseTab. 28: The class X 4 (+1, +1, −1, −1): d 4 = 72〈x π〉 π 〈x〉 〈x π〉 π 〈x〉〈112 {134}〉 (34) ϑ 〈111234〉 〈122 {234}〉 ϑ 〈122234〉〈121 {134}〉 (24) ϑ 〈212 {234}〉 (12) ϑ〈211 {134}〉 (14) ϑ 〈221 {234}〉 (13) ϑ〈{123} 334〉 η 〈123334〉 〈{124} 344〉 (34) η 〈123444〉〈{123} 343〉 (56) η 〈{124} 434〉 (35) η〈{123} 433〉 (46) η 〈{124} 443〉 (36) ηX 4 (+1, −1, +1, −1): X 4 (+1, −1, −1, +1):〈x〉 ˜π 〈y〉 〈x〉 ˜π 〈y〉〈111234〉 (45) 〈111234〉 〈111234〉 (46) 〈111234〉〈122234〉 (25) 〈123334〉 〈122234〉 (26) (35) 〈123444〉〈123334〉 (25) 〈122234〉 〈123334〉 (26) 〈123334〉〈123444〉 (23) 〈123444〉 〈123444〉 (26) (35) 〈122234〉Tab. 29: The class X 4 (+2, −2, +1, −1): d 4 = 9〈x π〉 π 〈x〉 〈x π〉 π 〈x〉〈113224〉 (35) 〈112234〉 〈131422〉 (253) (46) 〈112234〉〈113242〉 (356) 〈311224〉 (1532)〈113422〉 (35) (46) 〈311242〉 (1563)〈131224〉 (253) 〈311422〉 (153) (46)〈131242〉 (2563)Derived classes ˜π 〈y〉 Derived classes ˜π 〈y〉X 4 (+2, −2, −1, +1) (56) 〈112234〉 X 4 (+1, +2, −2, −1) (135) 〈122334〉X 4 (+2, +1, −2, −1) (35) 〈112334〉 X 4 (+1, +2, −1, −2) (135) (46) 〈122344〉X 4 (+2, +1, −1, −2) (35) (46) 〈112344〉 X 4 (+1, −2, +2, −1) (15) (24) 〈122334〉X 4 (+2, −1, −2, +1) (356) 〈112334〉 X 4 (+1, −2, −1, +2) (15) (264) 〈122344〉X 4 (+2, −1, +1, −2) (36) (45) 〈112344〉 X 4 (+1, −1, +2, −2) (135) (246) 〈123344〉X 4 (+1, −1, −2, +2) (15) (26) 〈123344〉Tab. 30: The class X 4 (+3, −1, −1, −1): d 4 = 6〈x π〉 π 〈x〉〈111 {234}〉 ϑ 〈111234〉Derived classes ˜π 〈y〉X 4 (−1, +3, −1, −1) (14) 〈122234〉X 4 (−1, −1, +3, −1) (15) (24) 〈123334〉X 4 (−1, −1, −1, +3) (16) (24) (35) 〈123444〉Tab. 31: The class X 4 (+1, −1, 0, 0): d 4 = 36〈x π〉 π 〈x〉〈{134} {234}〉 − (24) (35) ηϑ 〈123344〉Derived classes ˜π 〈y〉X 4 (+1, 0, −1, 0) (24) 〈122344〉X 4 (+1, 0, 0, −1) (26) (35) 〈122334〉X 4 (0, +1, −1, 0) (13) (24) 〈112344〉X 4 (0, +1, 0, −1) (135) (26) 〈112334〉X 4 (0, 0, +1, −1) (15) (26) 〈112234〉

B The classification **of** three-particle operators act**in**g on l = 2, 3, 4, 5, 6 electron shells 81Tab. 32: The class X 4 (+2, −1, −1, 0): d 4 = 18〈x π〉 π 〈x〉〈114 {234}〉 (354) ϑ 〈112344〉〈141 {234}〉 (2543) ϑ〈411 {234}〉 (1543) ϑDerived classes ˜π 〈y〉X 4 (−1, +2, −1, 0) (13) 〈122344〉X 4 (+2, −1, 0, −1) (46) 〈112334〉X 4 (+2, 0, −1, −1) (36) (45) 〈112234〉X 4 (0, +2, −1, −1) (145) (236) 〈112234〉X 4 (0, −1, +2, −1) (15) (2436) 〈112334〉X 4 (−1, +2, 0, −1) (1364) 〈122334〉X 4 (−1, −1, +2, 0) (13) (24) 〈123344〉X 4 (−1, −1, 0, +2) (164) (253) 〈123344〉X 4 (−1, 0, +2, −1) (14) (25) (36) 〈122334〉X 4 (−1, 0, −1, +2) (1634) (25) 〈122344〉X 4 (0, −1, −1, +2) (15) (26) 〈112344〉B.4 5–shell caseTab. 33: The class X 5 (+2, +1, −1, −1, −1): d 5 = 18〈x π〉 π 〈x〉〈112 {345}〉 ϑ 〈112345〉〈121 {345}〉 (23) ϑ〈211 {345}〉 (13) ϑDerived classes ˜π 〈y〉X 5 (+2, −1, +1, −1, −1) (34) 〈112345〉X 5 (+2, −1, −1, +1, −1) (35)X 5 (+2, −1, −1, −1, +1) (36)X 5 (+1, +2, −1, −1, −1) (13) 〈122345〉X 5 (−1, +2, +1, −1, −1) (134)X 5 (−1, +2, −1, +1, −1) (135)X 5 (−1, +2, −1, −1, +1) (136)X 5 (−1, −1, +2, +1, −1) (135) (24) 〈123345〉X 5 (−1, +1, +2, −1, −1) (14) (23)X 5 (+1, −1, +2, −1, −1) (13) (24)X 5 (−1, −1, +2, −1, +1) (136) (24)X 5 (−1, +1, −1, +2, −1) (14) (253) 〈123445〉X 5 (+1, −1, −1, +2, −1) (153) (24)X 5 (−1, −1, +1, +2, −1) (15) (24)X 5 (−1, −1, −1, +2, +1) (1436) (25)X 5 (+1, −1, −1, −1, +2) (1543) (26) 〈123455〉X 5 (−1, −1, +1, −1, +2) (16) (254)X 5 (−1, +1, −1, −1, +2) (154) (263)X 5 (−1, −1, −1, +1, +2) (15) (2634)

B The classification **of** three-particle operators act**in**g on l = 2, 3, 4, 5, 6 electron shells 82Tab. 34: The class X 5 (+1, +1, −1, −1, 0): d 5 = 36〈x π〉 π 〈x〉〈{125} {345}〉 (354) ηϑ 〈123455〉Derived classes ˜π 〈y〉X 5 (+1, −1, +1, −1, 0) (23) 〈123455〉X 5 (+1, −1, −1, +1, 0) (24)X 5 (+1, +1, −1, 0, −1) (46) 〈123445〉X 5 (+1, −1, +1, 0, −1) (23) (46)X 5 (+1, −1, −1, 0, +1) (264)X 5 (+1, +1, 0, −1, −1) (35) (46) 〈123345〉X 5 (+1, −1, 0, +1, −1) (254) (36)X 5 (+1, −1, 0, −1, +1) (263) (45)X 5 (+1, 0, +1, −1, −1) (245) (36) 〈122345〉X 5 (+1, 0, −1, +1, −1) (25) (346)X 5 (+1, 0, −1, −1, +1) (26) (345)X 5 (0, +1, +1, −1, −1) (146) (235) 〈112345〉X 5 (0, +1, −1, +1, −1) (15) (2346)X 5 (0, +1, −1, −1, +1) (16) (2345)B.5 6–shell caseTab. 35: The class X 6 (+1, +1, +1, −1, −1, −1): d 6 = 36〈x π〉 π 〈x〉〈{123} {456}〉 ηϑ 〈123456〉Derived classes ˜π 〈y〉X 6 (+1, +1, −1, +1, −1, −1) (34) 〈123456〉X 6 (+1, +1, −1, −1, +1, −1) (35)X 6 (+1, +1, −1, −1, −1, +1) (36)X 6 (+1, −1, +1, +1, −1, −1) (24)X 6 (+1, −1, +1, −1, +1, −1) (254)X 6 (+1, −1, +1, −1, −1, +1) (264)X 6 (+1, −1, −1, +1, +1, −1) (24) (35)X 6 (+1, −1, −1, +1, −1, +1) (24) (36)X 6 (+1, −1, −1, −1, +1, +1) (25) (36)In tables, the notation 〈ijk{lpq}〉 considers the set{〈ijklpq〉, 〈ijklqp〉, 〈ijkplq〉, 〈ijkpql〉, 〈ijkqlp〉, 〈ijkqpl〉},where each **of** the scheme **in** a given set is prescribed by the correspond**in**g permutation ϑ fromthe setϑ ∈ {1 6 , (56), (45), (456), (465), (46)}.Similarly, the notation 〈{ijk}lpq〉 considers the set{〈ijklpq〉, 〈ikjlpq〉, 〈jiklpq〉, 〈jkilpq〉, 〈kijlpq〉, 〈kjilpq〉},and each **of** the scheme **in** the set is prescribed by the correspond**in**g permutation η from the setη ∈ {1 6 , (23), (12), (123), (132), (13)}.F**in**ally, the notation 〈{ijk}{lpq}〉 is a union **of** both sets.

B The classification **of** three-particle operators act**in**g on l = 2, 3, 4, 5, 6 electron shells 83B.6 Identification **of** operators associated to classesTab. 36: The classes for 3–shell caseX 3 (∆ 1 , ∆ 2 , ∆ 3 ) X3 ∗ (∆ 1, ∆ 2 , ∆ 3 ) X 3`∆′ 1 , ∆ ′ 2 , ´∆′ 3 X3∗ `∆′ 1 , ∆ ′ 2 , 3´∆′X 3 (0, 0, 0) X 3 (0, 0, 0) X 3 (0, 0, 0) X 3 (0, 0, 0)X 3 (+2, −1, −1) X 3 (−2, +1, +1) X 3 (−1, +2, −1) X 3 (+1, −2, +1)X 3 (−1, −1, +2) X 3 (+1, +1, −2)X 3 (+3, −2, −1) X 3 (−3, +2, +1) X 3 (−2, +3, −1) X 3 (+2, −3, +1)X 3 (−1, −2, +3) X 3 (+1, +2, −3)X 3 (+3, −1, −2) X 3 (−3, +1, +2)X 3 (−1, +3, −2) X 3 (+1, −3, +2)X 3 (−2, −1, +3) X 3 (+2, +1, −3)X 3 (+1, −1, 0) X 3 (−1, +1, 0) X 3 (+1, 0, −1) X 3 (−1, 0, +1)X 3 (0, +1, −1) X 3 (0, −1, +1)X 3 (+2, −2, 0) X 3 (−2, +2, 0) X 3 (+2, 0, −2) X 3 (−2, 0, +2)X 3 (0, +2, −2) X 3 (0, −2, +2)Tab. 37: The classes for 4–shell caseX 4 (∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 ) X4 ∗ (∆ 1, ∆ 2 , ∆ 3 , ∆ 4 ) X 4`∆′ 1 , ∆ ′ 2 , ∆′ 3 , ´∆′ 4 X4∗ `∆′ 1 , ∆ ′ 2 , ∆′ 3 , 4´∆′X 4 (+1, +1, −1, −1) X 4 (−1, −1, +1, +1) X 4 (+1, −1, +1, −1) X 4 (−1, +1, −1, +1)X 4 (+1, −1, −1, +1) X 4 (−1, +1, +1, −1)X 4 (+2, −2, +1, −1) X 4 (−2, +2, −1, +1) X 4 (+2, −2, −1, +1) X 4 (−2, +2, +1, −1)X 4 (+2, +1, −2, −1) X 4 (−2, −1, +2, +1)X 4 (+2, +1, −1, −2) X 4 (−2, −1, +1, +2)X 4 (+2, −1, −2, +1) X 4 (−2, +1, +2, −1)X 4 (+2, −1, +1, −2) X 4 (−2, +1, −1, +2)X 4 (+1, +2, −2, −1) X 4 (−1, −2, +2, +1)X 4 (+1, +2, −1, −2) X 4 (−1, −2, +1, +2)X 4 (+1, −2, +2, −1) X 4 (−1, +2, −2, +1)X 4 (+1, −2, −1, +2) X 4 (−1, +2, +1, −2)X 4 (+1, −1, +2, −2) X 4 (−1, +1, −2, +2)X 4 (+1, −1, −2, +2) X 4 (−1, +1, +2, −2)X 4 (+3, −1, −1, −1) X 4 (−3, +1, +1, +1) X 4 (−1, +3, −1, −1) X 4 (+1, −3, +1, +1)X 4 (−1, −1, +3, −1) X 4 (+1, +1, −3, +1)X 4 (−1, −1, −1, +3) X 4 (+1, +1, +1, −3)X 4 (+2, −1, −1, 0) X 4 (−2, +1, +1, 0) X 4 (−1, +2, −1, 0) X 4 (+1, −2, +1, 0)X 4 (+2, −1, 0, −1) X 4 (−2, +1, 0, +1)X 4 (+2, 0, −1, −1) X 4 (−2, 0, +1, +1)X 4 (0, +2, −1, −1) X 4 (0, −2, +1, +1)X 4 (0, −1, +2, −1) X 4 (0, +1, −2, +1)X 4 (−1, +2, 0, −1) X 4 (+1, −2, 0, +1)X 4 (−1, −1, 0, +2) X 4 (+1, +1, 0, −2)X 4 (−1, 0, +2, −1) X 4 (+1, 0, −2, +1)X 4 (−1, 0, −1, +2) X 4 (+1, 0, +1, −2)X 4 (0, −1, −1, +2) X 4 (0, +1, +1, −2)X 4 (+1, −1, 0, 0) X 4 (−1, +1, 0, 0) X 4 (+1, 0, −1, 0) X 4 (−1, 0, +1, 0)X 4 (+1, 0, 0, −1) X 4 (−1, 0, 0, +1)X 4 (0, +1, −1, 0) X 4 (0, −1, +1, 0)X 4 (0, +1, 0, −1) X 4 (0, −1, 0, +1)X 4 (0, 0, +1, −1) X 4 (0, 0, −1, +1)

B The classification **of** three-particle operators act**in**g on l = 2, 3, 4, 5, 6 electron shells 84Tab. 38: The classes for 5–shell caseX 5 (∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 , ∆ 5 ) X5 ∗ (∆ 1, ∆ 2 , ∆ 3 , ∆ 4 , ∆ 5 ) X 5`∆′ 1 , ∆ ′ 2 , ∆′ 3 , ∆′ 4 , ´∆′ 5 X5∗ `∆′ 1 , ∆ ′ 2 , ∆′ 3 , ∆′ 4 , 5´∆′X 5 (+2, +1, −1, −1, −1) X 5 (−2, −1, +1, +1, +1) X 5 (+2, −1, +1, −1, −1) X 5 (−2, +1, −1, +1, +1)X 5 (+2, −1, −1, +1, −1) X 5 (−2, +1, +1, −1, +1)X 5 (+2, −1, −1, −1, +1) X 5 (−2, +1, +1, +1, −1)X 5 (+1, +2, −1, −1, −1) X 5 (−1, −2, +1, +1, +1)X 5 (−1, +2, +1, −1, −1) X 5 (+1, −2, −1, +1, +1)X 5 (−1, +2, −1, +1, −1) X 5 (+1, −2, +1, −1, +1)X 5 (−1, +2, −1, −1, +1) X 5 (+1, −2, +1, +1, −1)X 5 (−1, −1, +2, +1, −1) X 5 (+1, +1, −2, −1, +1)X 5 (−1, +1, +2, −1, −1) X 5 (+1, −1, −2, +1, +1)X 5 (+1, −1, +2, −1, −1) X 5 (−1, +1, −2, +1, +1)X 5 (−1, −1, +2, −1, +1) X 5 (+1, +1, −2, +1, −1)X 5 (−1, +1, −1, +2, −1) X 5 (+1, −1, +1, −2, +1)X 5 (+1, −1, −1, +2, −1) X 5 (−1, +1, +1, −2, +1)X 5 (−1, −1, +1, +2, −1) X 5 (+1, +1, −1, −2, +1)X 5 (−1, −1, −1, +2, +1) X 5 (+1, +1, +1, −2, −1)X 5 (+1, −1, −1, −1, +2) X 5 (−1, +1, +1, +1, −2)X 5 (−1, −1, +1, −1, +2) X 5 (+1, +1, −1, +1, −2)X 5 (−1, +1, −1, −1, +2) X 5 (+1, −1, +1, +1, −2)X 5 (−1, −1, −1, +1, +2) X 5 (+1, +1, +1, −1, −2)X 5 (+1, +1, −1, −1, 0) X 5 (−1, −1, +1, +1, 0) X 5 (+1, −1, +1, −1, 0) X 5 (−1, +1, −1, +1, 0)X 5 (+1, −1, −1, +1, 0) X 5 (−1, +1, +1, −1, 0)X 5 (+1, +1, −1, 0, −1) X 5 (−1, −1, +1, 0, +1)X 5 (+1, −1, +1, 0, −1) X 5 (−1, +1, −1, 0, +1)X 5 (+1, −1, −1, 0, +1) X 5 (−1, +1, +1, 0, −1)X 5 (+1, +1, 0, −1, −1) X 5 (−1, −1, 0, +1, +1)X 5 (+1, −1, 0, +1, −1) X 5 (−1, +1, 0, −1, +1)X 5 (+1, −1, 0, −1, +1) X 5 (−1, +1, 0, +1, −1)X 5 (+1, 0, +1, −1, −1) X 5 (−1, 0, −1, +1, +1)X 5 (+1, 0, −1, +1, −1) X 5 (−1, 0, +1, −1, +1)X 5 (+1, 0, −1, −1, +1) X 5 (−1, 0, +1, +1, −1)X 5 (0, +1, +1, −1, −1) X 5 (0, −1, −1, +1, +1)X 5 (0, +1, −1, +1, −1) X 5 (0, −1, +1, −1, +1)X 5 (0, +1, −1, −1, +1) X 5 (0, −1, +1, +1, −1)Tab. 39: The classes for 6–shell caseX 6 (∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 , ∆ 5 , ∆ 6 ) X6 ∗ (∆ 1, ∆ 2 , ∆ 3 , ∆ 4 , ∆ 5 , ∆ 6 ) X 6`∆′ 1 , ∆ ′ 2 , ∆′ 3 , ∆′ 4 , ∆′ 5 , ´∆′ 6 X6∗ `∆′ 1 , ∆ ′ 2 , ∆′ 3 , ∆′ 4 , ∆′ 5 , 6´∆′X 6 (+1, +1, +1, −1, −1, −1) X 6 (−1, −1, −1, +1, +1, +1) X 6 (+1, +1, −1, +1, −1, −1) X 6 (−1, −1, +1, −1, +1, +1)X 6 (+1, +1, −1, −1, +1, −1) X 6 (−1, −1, +1, +1, −1, +1)X 6 (+1, +1, −1, −1, −1, +1) X 6 (−1, −1, +1, +1, +1, −1)X 6 (+1, −1, +1, +1, −1, −1) X 6 (−1, +1, −1, −1, +1, +1)X 6 (+1, −1, +1, −1, +1, −1) X 6 (−1, +1, −1, +1, −1, +1)X 6 (+1, −1, +1, −1, −1, +1) X 6 (−1, +1, −1, +1, +1, −1)X 6 (+1, −1, −1, +1, +1, −1) X 6 (−1, +1, +1, −1, −1, +1)X 6 (+1, −1, −1, +1, −1, +1) X 6 (−1, +1, +1, −1, +1, −1)X 6 (+1, −1, −1, −1, +1, +1) X 6 (−1, +1, +1, +1, −1, −1)

C SU(2)–**in**variant part **of** the second-order wave operator 85CSU(2)–**in**variant part **of** the second-order wave operatorC.1 One-**body** partΩ (2)+µc (Λ)(ε c − ε µ ) = δ Λτ [ ¨S µc (τ 1 τ 2 τ) + Ṡµc(τ 1 τ 2 τ)] + δ Λτ1 [˜¨Sµc (τ 1 ) + ˜Ṡ µc (τ 1 )]+δ Λτ2 [ ˜¨S′ µc (τ 2 ) + S ˜˙ ′ µc(τ 2 )] + δ Λ0 ˜Sµc ,(C.1a)Ω (2)−µc (Λ)(ε c − ε µ ) = δ Λτ S cµ (τ 1 τ 2 τ). (C.1b)Ω (2)+ev (Λ)(ε v − ε e ) = δ Λτ ¨Sev (τ 1 τ 2 τ) + δ Λτ1 [˜¨Sev (τ 1 ) + ˜Ṡ ev (τ 1 )]+δ Λτ2 [ ˜¨S′ ev (τ 2 ) + ˜Ṡ ′ ev(τ 2 )] + δ Λ0 ˜Sev ,(C.2a)C.2 Two-**body** partΩ (2)−ev (Λ)(ε v − ε e ) = δ Λτ [Ṡve(τ 1 τ 2 τ) + ¨S ve (τ 1 τ 2 τ)]. (C.2b)Ω (2)+µµ ′ cc(Λ ′ 1 Λ 2 Λ)(ε cc ′ − ε µµ ′)= δ Λ1 uδ Λ2 dδ Λτ D µµ ′ cc ′(udτ) + δ Λ 1 Uδ Λ2 uδ Λτ1 [X µµ ′ cc ′(Uuτ 1) + 1Ỹ 2 µµ ′ cc ′(Uuτ 1)]+ 1δ 2 Λ 1 Uδ Λ2 uδ Λτ2 Z µ ′ µcc ′(Uuτ 2)˜D ′ µ ′ µcc ′(Uuτ 2) − δ Λ1 Λ 2δ Λ1 uδ Λ0 [ 1 ¨D4 µµ ′ cc ′(ũu)+ 1Ḋ 4 µµ ′ cc ′(ũu) + ...12Z µµ ′ cc ′(ũu) + ˜˙∆µµ ′ cc ′(uu) + ˜¨∆µµ ′ cc ′(uu)],(C.3a)Ω (2)−µµ ′ cc ′ (Λ 1 Λ 2 Λ)(ε cc ′ − ε µµ ′)= − 1 2 δ Λ 1 Uδ Λ2 uδ Λτ1 Z c ′ cµµ ′(uUτ 1) ˜D c ′ cµµ ′(uUτ 1) + 1 2 δ Λ 1 Uδ Λ2 uδ Λτ2 Z c ′ cµµ ′(uUτ 2)×[˜¨D′ c ′ cµµ ′(uUτ 2) + ˜Ḋ ′ c ′ cµµ ′(uUτ 2)] + 1 4 δ Λ 1 Λ 2δ Λ1 uδ Λ0 Z cc ′ µµ ′(u)D cc ′ µµ ′(uũ).(C.3b)Ω (2)+evcc ′ (Λ 1 Λ 2 Λ)(ε cc ′ − ε ev )= −δ Λ1 uδ Λ2 dδ Λτ Z vecc ′(udτ)D vecc ′(udτ) + δ Λ1 Uδ Λ2 uδ Λτ1 [ ¨D evcc ′(Uuτ 1 )+Z vec ′ c(Uuτ 1 )Ḋvec ′ c(Uuτ 1 ) + 1 2˜Ḋ evcc ′(Uuτ 1 ) − 1 2 Z vecc ′(Uuτ 1) ˜¨Dvecc ′(Uuτ 1 )]+ 1 2 δ Λ 1 Uδ Λ2 uδ Λτ2 [−˜D ′ evcc ′(Uuτ 2) + Z vecc ′(Uuτ 2 )˜D ′ vecc ′(Uuτ 2)] − δ Λ1 Λ 2δ Λ1 uδ Λ0×[ ˜¨∆evcc ′(uu) + ˜˙∆evcc ′(uu) − Z vecc ′(u){ ˜¨∆vecc ′(uu) + ˜˙∆vecc ′(uu)}+Z evc ′ c(u) ...Devc ′ c(uũ)],Ω (2)−evcc(Λ ′ 1 Λ 2 Λ)(ε cc ′ − ε ev )= −δ Λ1 Uδ Λ2 uδ Λτ1 Z c ′ cev(uUτ 1 ) ˜D c ′ cev(uUτ 1 ) + δ Λ1 Uδ Λ2 uδ Λτ2 Z c ′ cev(uUτ 2 )×[˜¨D′ c ′ cev(uUτ 2 ) + ˜Ḋ ′ c ′ cev(uUτ 2 )].(C.4a)(C.4b)Ω (2)+µµ ′ c¯v (Λ 1Λ 2 Λ)(ε c¯v − ε µµ ′)= δ Λ1 uδ Λ2 dδ Λτ δ µe [D ee ′ c¯v(udτ) − Z ee′¯vc(udτ)D ee′¯vc(udτ)] + δ Λ1 Uδ Λ2 uδ Λτ1×[ ˜¨Dµµ ′ c¯v(Uuτ 1 ) − Z µµ′¯vc(Uuτ 1 ) ˜Ḋ µµ′¯vc(Uuτ 1 )] + δ Λ1 Uδ Λ2 uδ Λτ2 Z µ ′ µc¯v(Uuτ 2 )×˜D ′ µ ′ µc¯v(Uuτ 2 ) + δ Λ1 Λ 2δ Λ1 uδ Λ0 [Z µµ′¯vc(u){ ˜¨∆µµ′¯vc(uu) + ˜˙∆µµ′¯vc(uu)}− ˜¨∆µµ ′ c¯v(uu) − ˜˙∆µµ ′ c¯v(uu) −˜...12Dµµ ′ c¯v(uu)],Ω (2)−µµ ′ c¯v (Λ 1Λ 2 Λ)(ε c¯v − ε µµ ′)= −δ Λ1 Uδ Λ2 uδ Λτ1 [Z¯vcµµ ′(uUτ 1 ){Ḋ¯vcµµ ′(uUτ 1) + 1 2˜D¯vcµµ ′(uUτ 1 )} + Z c¯vµ ′ µ(uUτ 1 )×D c¯vµ ′ µ(uUτ 1 )] − δ Λ1 Uδ Λ2 uδ Λτ2 [Z c¯vµµ ′(uUτ 2 ){˜¨D′ c¯vµµ ′(uUτ 2 ) + ˜Ḋ ′ c¯vµµ ′(uUτ 2)}−Z¯vcµµ ′(uUτ 2 )˜¨D′¯vcµµ ′(uUτ 2 )].(C.5a)(C.5b)

C SU(2)–**in**variant part **of** the second-order wave operator 86Ω (2)+ev¯vc(Λ 1 Λ 2 Λ)(ε c¯v − ε ev )= δ Λ1 uδ Λ2 dδ Λτ [D ev¯vc (udτ) + Z vec¯v (udτ)D vec¯v (udτ)] + δ Λ1 Uδ Λ2 uδ Λτ1 [ ˜¨Dev¯vc (Uuτ 1 )+ ˜Ḋ ev¯vc (Uuτ 1 ) − Z ve¯vc (Uuτ 1 ){ ˜¨Dve¯vc (Uuτ 1 ) + ˜Ḋ ve¯vc (Uuτ 1 )}] + δ Λ1 Uδ Λ2 uδ Λτ2×[−˜D ′ ev¯vc(Uuτ 2 ) + Z ve¯vc (Uuτ 2 )˜D ′ ve¯vc(Uuτ 2 )] − δ Λ1 Λ 2δ Λ1 uδ Λ0 [ ˜¨∆ev¯vc (uu)+ ˜˙∆ev¯vc (uu) + ˜...Dev¯vc(uu) − Z ve¯vc (u){ ˜¨∆ve¯vc (uu) + ˜˙∆ve¯vc (uu)} − Z evc¯v (u)×{ ˜¨∆evc¯v (uu) + ˜˙∆evc¯v (uu)} + Z vec¯v (u){ ˜¨∆vec¯v (uu) + ˜˙∆vec¯v (uu)}],Ω (2)−ev¯vc(Λ 1 Λ 2 Λ)(ε c¯v − ε ev )= δ Λ1 Uδ Λ2 uδ Λτ1 [Z¯vcev (uUτ 1 ){ ˜D¯vcev (uUτ 1 ) − ˜Ḋ¯vcev (uUτ 1 )} − Z c¯vev (uUτ 1 )× ˜D c¯vev (uUτ 1 )] + δ Λ1 Uδ Λ2 uδ Λτ2 [−Z¯vcev (uUτ 2 )˜¨D′¯vcev (uUτ 2 ) + Z c¯vev (uUτ 2 )×{˜¨D′ c¯vev (uUτ 2 ) − ˜Ḋ ′ c¯vev(uUτ 2 )}].(C.6a)(C.6b)Ω (2)+ev¯v¯v ′ (Λ 1 Λ 2 Λ)(ε¯v¯v ′ − ε ev )= δ Λ1 Uδ Λ2 uδ Λτ1 [ ¨D ev¯v¯v ′(Uuτ 1 ) + Z ve¯v¯v ′(Uuτ 1 ){Ḋve¯v¯v ′(Uuτ 1) − 1 2˜¨D ve¯v¯v ′(Uuτ 1 )}]+ 1 2 δ Λ 1 Uδ Λ2 uδ Λτ2 [Z ve¯v¯v ′(Uuτ 2 )˜D ′ ve¯v¯v ′(Uuτ 2) − ˜D ′ ev¯v¯v ′(Uuτ 2) − ˜Ḋ ′ ev¯v¯v ′(Uuτ 2)]+δ Λ1 Λ 2δ Λ1 uδ Λ0 [− ˜¨∆ev¯v¯v ′(uu) − ˜˙∆ev¯v¯v ′(uu) + Z ve¯v¯v ′(u){ ˜¨∆ve¯v¯v ′(uu) + ˜˙∆ve¯v¯v ′(uu)}],Ω (2)−ev¯v¯v(Λ ′ 1 Λ 2 Λ)(ε¯v¯v ′ − ε ev )= −δ Λ1 Uδ Λ2 uδ Λτ1 Z¯v′¯vev(uUτ 1 )[ ˜D¯v′¯vev(uUτ 1 ) + ˜Ḋ¯v′¯vev(uUτ 1 )] + δ Λ1 Uδ Λ2 uδ Λτ2×Z¯v′¯vev(uUτ 2 ) ˜¨D¯v′¯vev(uUτ 2 ) − 1 2 δ Λ 1 Λ 2δ Λ1 uδ Λ0 Z¯v¯v ′ ev(u)Ḋ¯v¯v ′ ev(uũ).(C.7a)(C.7b)Ω (2)+ee ′¯v¯v (Λ ′ 1 Λ 2 Λ)(ε¯v¯v ′ − ε ee ′)= δ Λ1 uδ Λ2 dδ Λτ D ′(udτ) ee′¯v¯v + δ Λ 1 Uδ Λ2 uδ Λτ1 [˜¨D 1 ′(Uuτ 2 ee′¯v¯v 1) + Ḋee ′¯v¯v ′(Uuτ 1)]+ 1δ 2 Λ 1 Uδ Λ2 uδ Λτ2 Z e ′ e¯v¯v ′(Uuτ 2)˜D ′ e ′ e¯v¯v ′(Uuτ 2) − δ Λ1 Λ 2δ Λ1 uδ Λ0 [ 1{ 1 ¨D ′(ũu) (C.8a)2 2 ee′¯v¯v...−Z ee′¯v ′¯v(u) Dee ′¯v ′¯v(ũu)} + ˜¨∆ee′¯v¯v ′(uu) + ˜˙∆ee′¯v¯v ′(uu)],Ω (2)−ee ′¯v¯v (Λ ′ 1 Λ 2 Λ)(ε¯v¯v ′ − ε ee ′)= −δ Λ1 Uδ Λ2 uδ Λτ1 Z¯v′¯vee ′(uUτ 1)[D¯v′¯vee ′(uUτ 1) + ˜Ḋ¯v′¯vee ′(uUτ 1)] + 1δ 2 Λ 1 Uδ Λ2 uδ Λτ2×Z¯v′¯vee ′(uUτ 2)˜¨D′¯v′¯vee ′(uUτ 2) + 1δ 4 Λ 1 Λ 2δ Λ1 uδ Λ0 Z¯v¯v ′ ee ′(u)[D¯v¯v ′ ee ′(uũ)(C.8b)−Ḋ¯v¯v ′ ee ′(uũ)].In Eq. (C.3a), the quantities X µµ ′ cc ′, Y µµ ′ cc ′, Z µµ ′ cc ′ differ for dist**in**ct one-electron orbitalsµ = v, e. If µ = v, then X vv ′ cc ′ ≡ ¨D vv ′ cc ′, Y vv ′ cc ′ ≡ Ḋvv ′ cc ′ and Z vv ′ cc ′ ≡ −Z vv ′ c ′ c(u)D vv ′ c ′ c; ifµ = e, then X ee ′ cc ′ ≡ Ḋee ′ cc ′, Y ee ′ cc ′ ≡ ¨D ee ′ cc ′, Z ee ′ cc ′ ≡ D ee ′ cc ′.C.3 Three-**body** partΩ (2)+vv ′ v ′′¯vc ′ c (Λ 1Λ 2 Λ 3 Λ)(ε¯vc ′ c − ε vv ′ v ′′)= 1δ 2 ΛΛ 3δ MM3 [ ¨T vv ′ v ′′¯vc ′ c(˜Λ 1 Λ 2 Λ) + 1 T˜˙2 vv ′ v ′′¯vc ′ c(Λ 1 Λ 2 Λ) − ∑ J Λ 2 Λ vc¯vc ′(Λ Λ 2ΛΛ 2 Λ 1 )×{˜¨T vv ′ v ′′ c ′¯vc(Λ 1 Λ 2 Λ) + T ˜˙ vv ′ v ′′ c ′¯vc(Λ 1 Λ 2 Λ)}] + (−1) λv+λ c ′ Y vc¯vc ′(Λ 1 Λ 2 ΛΛ 3 τ 1 )∑×T vv ′ v ′′ c ′ c¯v(Λ 1 τ 1 ) + 1 2 u a(λ v ′λ¯vu)I vv ′ v ′′¯vc ′ c(Λ 1 Λ 2 Λ 3 τ 1 Λu) ˜T ′ v ′′ vv ′ c¯vc ′(uτ 2),Ω (2)−vv ′ v ′′¯vc ′ c (Λ 1Λ 2 Λ 3 Λ)(ε¯vc ′ c − ε vv ′ v ′′)= 1 2 δ ΛΛ 3δ MM3 a(λ v ′λ v ′′Λ 1 )[a(λ c λ c ′Λ 2 ) ˜T¯vc ′ cvv ′ v ′′(Λ 2Λ 1 Λ) + ∑ Λ 2 Λ a(λ cλ c ′Λ 2 )×J vc′¯vc(Λ Λ 2 ΛΛ 2 Λ 1 )T cc′¯vvv ′ v ′′(˜Λ 2 Λ 1 Λ)].(C.9a)(C.9b)

C SU(2)–**in**variant part **of** the second-order wave operator 87Ω (2)+vv ′ e¯vc ′ c (Λ 1Λ 2 Λ 3 Λ)(ε¯vc ′ c − ε vv ′ e)= δ ΛΛ3 δ MM3 [ 1{−a(λ 2 cλ c ′Λ 2 ) T ˙ vv ′ e¯vcc ′(˜Λ 1 Λ 2 Λ) +˜¨T 1 2 vv ′ e¯vc ′ c(Λ 1 Λ 2 Λ)}−a(λ e λ v ′Λ 1 ){ ¨T vev′¯vc ′ c(˜Λ 1 Λ 2 Λ) + 1 T˜˙2 vev′¯vc ′ c(Λ 1 Λ 2 Λ)} − ∑ J Λ 2 Λ vc¯vc ′(Λ Λ 2ΛΛ 2 Λ 1 )×{ 1(˜˙T2 vv ′ ec ′¯vc(Λ 1 Λ 2 Λ) + ˜¨T vv ′ ec ′¯vc(Λ 1 Λ 2 Λ)) + a(λ e λ v ′Λ 1 )a(λ c λ¯v Λ 2 )×(˜˙ T vev ′ c ′ c¯v(Λ 1 Λ 2 Λ) + ˜¨T vev ′ c ′ c¯v(Λ 1 Λ 2 Λ))}] + (−1) λv+λ c ′ Y vc¯vc ′(Λ 1 Λ 2 ΛΛ 3 τ 1 )× ˜T vv ′ ec ′ c¯v(Λ 1 τ 1 ) + 1a(λ 2 eλ¯v Λ 2 )Y vv ′ ′ e¯v (τ 1Λ 1 ΛΛ 3 Λ 2 )T evv′¯vc ′ c(Λ 2 τ 1 )+ ∑ u [a(λ cλ v ′Λ 2 )I vv ′ e¯vc ′ c(Λ 1 Λ 2 Λ 3 τ 1 Λu)T evv ′ c ′ c¯v(uτ 1 ) + 1a(λ 2 v ′λ¯vu)×I vv ′ e¯vc ′ c(Λ 1 Λ 2 Λ 3 τ 2 Λu) ˜T ′ evv ′ c¯vc ′(uτ 2) + (−1) λv−λ¯v a(λ e λ v ′Λ 1 )×I vev′¯vc ′ c(Λ 1 Λ 2 Λ 3 τ 2 Λu) ˜T ′ v ′ evc¯vc ′(uτ 2)],Ω (2)−vv ′ e¯vc ′ c (Λ 1Λ 2 Λ 3 Λ)(ε¯vc ′ c − ε vv ′ e)= δ ΛΛ3 δ MM3 a(λ e λ v ′Λ 1 )[a(λ c λ c ′Λ 2 ) ˜T¯vc ′ cvv ′ e(Λ 2 Λ 1 Λ)+ 1 2∑Λ 2 Λ {J vc¯vc ′(Λ Λ 2ΛΛ 2 Λ 1 ) ˜T c ′ c¯vvv ′ e(Λ 2 Λ 1 Λ) + (−1) λv−λe× ∑ Λ 1I ′ vv ′ e¯vc ′ c (Λ 1Λ 2 Λ 1 Λ 2 ΛΛ)(T c ′ c¯vevv ′(˜Λ 2 Λ 1 Λ) − a(λ c λ¯v Λ 2 )× ˜T c′¯vcevv ′(Λ 2Λ 1 Λ))}].(C.10a)(C.10b)Ω (2)+ee ′ v¯vc ′ c (Λ 1Λ 2 Λ 3 Λ)(ε¯vc ′ c − ε ee ′ v)= δ ΛΛ3 δ MM3 [ 1{ ¨T 2 ee ′ v¯vc ′ c(˜Λ 1 Λ 2 Λ) + 1 T˜˙2 ee ′ v¯vc ′ c(Λ 1 Λ 2 Λ)} + a(λ v λ e ′Λ 1 ){a(λ c λ c ′Λ 2 )× T ˙ ′(˜Λ eve′¯vcc 1 Λ 2 Λ) −˜¨T 1 2 eve′¯vc ′ c(Λ 1 Λ 2 Λ)} + ∑ J Λ 2 Λ ec¯vc ′(Λ Λ 2ΛΛ 2 Λ 1 )×{ 1(a(λ 2 cλ¯v Λ 2 )˜¨T ee ′ vc ′ c¯v(Λ 1 Λ 2 Λ) − T ˜˙ ee ′ vc ′¯vc(Λ 1 Λ 2 Λ)) + a(λ v λ e ′Λ 1 )×(˜¨T eve ′ c ′¯vc(Λ 1 Λ 2 Λ) + T ˜˙ eve ′ c ′¯vc(Λ 1 Λ 2 Λ))}] + δ Λ1 Λ 2δ Λ3 τ 1δ Λ0 T ee ′ v¯vc ′ c(Λ 1 τ 1 )+(−1) λe+λ c ′ Y ec¯vc ′(Λ 1 Λ 2 ΛΛ 3 τ 1 ) ˜T ee ′ vc ′ c¯v(Λ 1 τ 1 ) + ∑ u [a(λ e ′λ c ′Λ 2)×I ee ′ v¯vcc ′(Λ 1Λ 2 Λ 3 τ 1 Λu)T vee ′ c ′ c¯v(uτ 1 ) + a(λ e ′λ¯v u){ 1I 2 ee ′ v¯vc ′ c(Λ 1 Λ 2 Λ 3 τ 2 Λu)× ˜T ′ vee ′ c¯vc ′(uτ 2) + (−1) λe−λv I eve′¯vc ′ c(Λ 1 Λ 2 Λ 3 τ 2 Λu) ˜T ′ e ′ vec¯vc ′(uτ 2)}],Ω (2)−ee ′ v¯vc ′ c (Λ 1Λ 2 Λ 3 Λ)(ε¯vc ′ c − ε ee ′ v)= δ ΛΛ3 δ MM3 a(λ v λ e ′Λ 1 )[a(λ c λ c ′Λ 2 ) ˜T¯vc ′ cee ′ v(Λ 2 Λ 1 Λ)+ 1 2∑Λ 2 Λ {J ec¯vc ′(Λ Λ 2ΛΛ 2 Λ 1 ) ˜T c ′ c¯vee ′ v(Λ 2 Λ 1 Λ) + (−1) λe−λv× ∑ Λ 1I ′ ee ′ v¯vc ′ c (Λ 1Λ 2 Λ 1 Λ 2 ΛΛ)(T c ′ c¯vvee ′(˜Λ 2 Λ 1 Λ) − a(λ c λ¯v Λ 2 ) ˜T c′¯vcvee ′(Λ 2Λ 1 Λ))}].(C.11a)(C.11b)Ω (2)+vv ′ v ′′¯v¯v ′ c (Λ 1Λ 2 Λ)(ε¯v¯v ′ c − ε vv ′ v ′′)= 1δ 2 ΛΛ 3δ MM3 [˜¨T vv ′ v ′′¯v¯v ′ c(Λ 1 Λ 2 Λ) + T ˜˙ vv ′ v ′′¯v¯v ′ c(Λ 1 Λ 2 Λ) + a(λ c λ¯v ′Λ 2 )× ∑ J Λ 2 Λ v¯v ′¯vc(Λ Λ 2 ΛΛ 2 Λ 1 ) ¨T vv ′ v ′′ c¯v¯v ′(˜Λ 1 Λ 2 Λ)],Ω (2)−vv ′ v ′′¯v¯v ′ c (Λ 1Λ 2 Λ)(ε¯v¯v ′ c − ε vv ′ v ′′)= δ ΛΛ3 δ MM3 a(λ v ′λ v ′′Λ 2 )[ 1 4 a(λ cλ¯v ′Λ 1 ) ˜T¯v¯v ′ cvv ′ v ′′(Λ 2Λ 1 Λ) − T¯vc¯v ′ vv ′ v ′′(˜Λ 2 Λ 1 Λ)].(C.12a)(C.12b)

C SU(2)–**in**variant part **of** the second-order wave operator 88Ω (2)+vv ′ e¯v¯v ′ c (Λ 1Λ 2 Λ 3 Λ)(ε¯v¯v ′ c − ε vv ′ e)= δ ΛΛ3 δ MM3 [ 1{˜¨T 2 vv ′ e¯v¯v ′ c(Λ 1 Λ 2 Λ) + T ˜˙ vv ′ e¯v¯v ′ c(Λ 1 Λ 2 Λ)} − a(λ e λ v ′Λ 1 )×{˜¨T vev′¯v¯v ′ c(Λ 1 Λ 2 Λ) + T ˜˙ vev′¯v¯v ′ c(Λ 1 Λ 2 Λ)} + a(λ c λ¯v ′Λ 2 ) ∑ J Λ 2 Λ v¯v ′¯vc(Λ Λ 2 ΛΛ 2 Λ 1 )×{˜¨T 1 4 vv ′ ec¯v¯v ′(Λ 1Λ 2 Λ) − 1 2 a(λ¯vλ¯v ′Λ 2 ) T ˙ vv ′ ec¯v ′¯v(˜Λ 1 Λ 2 Λ) − a(λ e λ v ′Λ 1 )× ¨T vev ′ c¯v¯v ′(˜Λ 1 Λ 2 Λ)}] + (−1) λe−λ¯v [(−1) λc+λ¯v ′ Y vv ′ ′ e¯v (τ 1Λ 1 ΛΛ 3 Λ 2 )T ′(Λ evv′¯vc¯v 2τ 1 )+a(λ v λ v ′Λ 2 )Y v¯v′¯vc(Λ 1 Λ 2 ΛΛ 3 τ 1 )T vev ′ c¯v¯v ′(Λ ∑1τ 1 )] + 1 2 u [(−1)λe−λ¯v a(λ v λ v ′Λ 1 )×I vev′¯v¯v ′ c(Λ 1 Λ 2 Λ 3 τ 2 Λu) ˜T ′ v ′ evc¯v¯v ′(uτ 2) + a(λ v ′λ¯v ′Λ 2 )I vv ′ e¯vc¯v ′(Λ 1Λ 2 Λ 3 τ 2 Λu)× ˜T ′ evv ′¯v ′ c¯v(uτ 2 )],Ω (2)−vv ′ e¯v¯v ′ c (Λ 1Λ 2 Λ 3 Λ)(ε¯v¯v ′ c − ε vv ′ e)= δ ΛΛ3 δ MM3 [ ˜T¯vc¯v ′ vev ′(Λ 2Λ 1 Λ) − 1a(λ 2 cλ¯v ′Λ 2 ) ˜T¯v¯v ′ cvev ′(Λ 2Λ 1 Λ)+ ∑ Λ 1 Λ 2 Λ a(λ vλ v ′Λ 2 ){a(λ¯v λ¯v ′Λ 1 )T c¯v¯v ′ evv ′(˜Λ 2 Λ 1 Λ)I ′ vv ′ e¯vc¯v ′(Λ 1Λ 2 Λ 1 Λ 2 ΛΛ)− 1 4 a(λ cλ¯v Λ 1 ) ˜T¯v′¯vcevv ′(Λ 2Λ 1 Λ)I ′ vv ′ e¯v¯v ′ c (Λ 1Λ 2 Λ 1 Λ 2 ΛΛ)}].(C.13a)(C.13b)Ω (2)+ee ′ v¯v¯v ′ c (Λ 1Λ 2 Λ 3 Λ)(ε¯v¯v ′ c − ε ee ′ v)= δ ΛΛ3 δ MM3 [ 1{˜¨T 2 ee ′ v¯v¯v ′ c(Λ 1 Λ 2 Λ) + T ˜˙ ee ′ v¯v¯v ′ c(Λ 1 Λ 2 Λ)} − a(λ e ′λ v Λ 1 )×{˜¨T eve′¯v¯v ′ c(Λ 1 Λ 2 Λ) + T ˜˙ eve′¯v¯v ′ c(Λ 1 Λ 2 Λ)} + a(λ c λ¯v ′Λ 2 ) ∑ J Λ 2 Λ e¯v ′¯vc(Λ Λ 2 ΛΛ 2 Λ 1 )×{ 1 ¨T2 ee ′ vc¯v¯v ′(˜Λ 1 Λ 2 Λ) − a(λ e ′λ v Λ 1 )(˜¨T 1 2 eve ′ c¯v¯v ′(Λ 1Λ 2 Λ) + T ˜˙ eve ′ c¯v¯v ′(Λ 1Λ 2 Λ))}]+δ Λ1 Λ 2δ Λ3 τ 1δ Λ0 ˜Tee ′ v¯v¯v ′ c(Λ 1 τ 1 ) + a(λ e λ¯v Λ 1 )(−1) Λ 2Y e¯v′¯vc(Λ 1 Λ 2 ΛΛ 3 τ 1 )×T ee ′ vc¯v¯v ′(Λ 1τ 1 ) + ∑ u [ 1a(λ 2 e ′λ¯vu){I ee ′ v¯v¯v ′ c(Λ 1 Λ 2 Λ 3 τ 1 Λu)T vee ′ c¯v¯v ′(uτ 1)+I ee ′ v¯v¯v ′ c(Λ 1 Λ 2 Λ 3 τ 2 Λu)T vee ′ ′ c¯v¯v ′(uτ 2)} + a(λ e ′λ v Λ 1 ){a(λ v λ¯v u)a(λ c λ¯v ′Λ 2 )×I ′(Λ eve′¯vc¯v 1Λ 2 Λ 3 τ 2 Λu) ˜T ′ e ′ ev¯v ′¯vc(uτ 2 ) + 1(−1)λe−λ¯v I2 eve′¯v¯v ′ c(Λ 1 Λ 2 Λ 3 τ 2 Λu)× ˜T ′ e ′ vec¯v¯v ′(uτ 2)}],Ω (2)−ee ′ v¯v¯v ′ c (Λ 1Λ 2 Λ 3 Λ)(ε¯v¯v ′ c − ε ee ′ v)= −δ ΛΛ3 δ MM3 [a(λ e ′λ v Λ 1 ){ ˜T¯vc¯v ′ ee ′ v(Λ 2 Λ 1 Λ) − 1a(λ 2 cλ¯v ′Λ 2 ) ˜T¯v¯v ′ cee ′ v(Λ 2 Λ 1 Λ)}+ ∑ Λ 1 Λ 2 Λ a(λ eλ e ′Λ 1 )a(λ c λ¯v Λ 2 ){(−1) Λ 2I ′ ee ′ v¯vc¯v ′(Λ 1Λ 2 Λ 1 Λ 2 ΛΛ)T c¯v¯v ′ vee ′(˜Λ 2 Λ 1 Λ)+ 1 4 I′ ee ′ v¯v¯v ′ c (Λ 1Λ 2 Λ 1 Λ 2 ΛΛ) ˜T¯v′¯vcvee ′(Λ 2Λ 1 Λ)}].C.4 Four-**body** partΩ (2)vv ′ v ′′ v ′′′¯v¯v ′ cc ′ (Λ 1 Λ 2 Λ 3 Λ 4 Λ)(ε¯v¯v ′ cc ′ − ε vv ′ v ′′ v ′′′)= 1 2 (−1)Λ 3a(λ c λ¯v Λ 1 )F c ′ c¯v¯v ′(Λ 1Λ 2 Λ 3 Λ 4 Λ)Q vv ′ v ′′ v ′′′¯vcc ′¯v ′(˜Λ 1 Λ 3 ).(C.14a)(C.14b)(C.15)Ω (2)evv ′ v ′′¯v¯v ′ cc(Λ ′ 1 Λ 2 Λ 3 Λ 4 Λ)(ε¯v¯v ′ cc ′ − ε evv ′ v ′′)= 1δ 4 Λ 1 Λ 2δ Λ3 Λ 4δ Λ0 Q evv ′ v ′′¯v¯v ′ cc ′(˜Λ 1 Λ 3 ) + (−1) Λ 3a(λ e λ v Λ 1 )F c ′ c¯v¯v ′(Λ 1Λ 2 Λ 3 Λ 4 Λ)×Q vev ′ v ′′ c¯vc ′¯v ′(˜Λ 1 Λ 3 ) + 1 2 (−1)Λ 3a(λ e λ v Λ 4 )F evv ′ v ′′(Λ 4Λ 3 Λ 2 Λ 1 Λ)Q vv ′ ev ′′ cc ′¯v¯v ′(Λ (C.16)4Λ 2 )∑+ 1 2 ud G evv ′ v ′′¯v¯v ′ cc ′(udΛ 1Λ 3 Λ 2 Λ 4 Λ) ˜Q vv ′ ev ′′¯vcc ′¯v ′(ud).Ω (2)ee ′ vv ′¯v¯v ′ cc(Λ ′ 1 Λ 2 Λ 3 Λ 4 Λ)(ε¯v¯v ′ cc ′ − ε ee ′ vv ′)= 1δ 4 Λ 1 Λ 2δ Λ3 Λ 4δ Λ0 {Q ee ′ vv ′¯v¯v ′ cc ′(˜Λ 1 Λ 3 ) + Q vv ′ ee ′ cc ′¯v¯v ′(˜Λ 3 Λ 1 )} − 1 2 (−1)Λ 3×{F c ′ c¯v¯v ′(Λ 1Λ 2 Λ 3 Λ 4 Λ)Q ee ′ vv ′ c¯vc ′¯v ′(˜Λ 1 Λ 3 ) + (−1) Λ 1+Λ 4F e ′ evv ′(Λ 4Λ 3 Λ 2 Λ 1 Λ)×Q eve ′ v ′ cc ′¯v¯v ′(˜Λ 4 Λ 2 ) + a(λ¯v λ¯v ′Λ 2 )a(λ c λ c ′Λ 4 )F ′¯v(Λ cc′¯v 1 Λ 2 Λ 3 Λ 4 Λ)×Q vv ′ ee ′ c¯vc ′¯v ′(˜Λ 3 Λ 1 ) + a(λ v λ v ′Λ 3 )a(λ e λ e ′Λ 4 )F ee ′ v ′ v(Λ 4 Λ 3 Λ 2 Λ 1 Λ)×Q eve ′ v ′¯v¯v ′ cc ′(˜Λ 2 Λ 4 )} + a(λ e λ e ′Λ 1 ) ∑ ud a(λ c ′λ¯v ′d) ˜Q eve ′ v ′¯vc¯v ′ c ′(ud)×G e ′ evv ′¯v¯v ′ cc ′(udΛ 1Λ 3 Λ 2 Λ 4 Λ).(C.17)

D Symbolic computations with NCoperators 89Tab. 40: Phase factors Z α ′ β ′ ¯µ ′¯ν ′λ α ′ λ β ′ λ¯µ ′ λ¯ν ′ Z α ′ β ′ ¯µ ′¯ν ′(Λ 1Λ 2 Λ)λ α λ β λ¯µ λ¯ν 1λ α λ β λ¯ν λ¯µ a(λ¯µ λ¯ν Λ 2 )λ β λ α λ¯µ λ¯ν a(λ α λ β Λ 1 )λ β λ α λ¯ν λ¯µ a(λ α λ β Λ 1 ) a(λ¯µ λ¯ν Λ 2 )λ¯µ λ¯ν λ α λ β a(λ α λ β Λ 2 ) a(λ¯µ λ¯ν Λ 1 )λ¯µ λ¯ν λ β λ α a(λ¯µ λ¯ν Λ 1 )λ¯ν λ¯µ λ α λ β a(λ α λ β Λ 2 )λ¯ν λ¯µ λ β λ α 1The coefficients J, Y , Y ′ , I, I ′ , F , G are def**in**ed by the follow**in**g formulas{ }J αβ ¯µ¯ν (Λ 1 Λ 2 Λ 1 Λ 2 Λ) def= (−1) Λ 2+Λ 2[Λ 1 ][Λ 2 , Λ 2 ] 1/2 λα λ¯ν Λ 1λ¯µ λ β Λ 2 , (C.18a)Λ 1 Λ 2 ΛY αβ ¯µ¯ν (Λ 1 Λ 2 Λ 1 Λ 2 Λ) def= (−1) Λ 1+Λ 2 +Λ 1 +Λ 2[Λ 1 , Λ 2 , Λ] 1/2{ } {Λ× 〈ΛMΛ 1 M 1 |Λ 2 M 2 〉 1 Λ 2 Λ 1 Λ 1 Λ 2 Λλ¯ν λ¯µ λ β, (C.18b)λ α λ¯ν λ¯µ}Y αβ ′ ¯µ¯ν(Λ 1 Λ 2 Λ 1 Λ 2 Λ) def= (−1) Λ 1+Λ 2[Λ 1 , Λ 1 , Λ 2 ] 1/2{ } { }Λ× 〈Λ 1 M 1 Λ 1 M 1 |Λ 2 M 2 〉 1 Λ 2 Λ Λ 1 Λ 2 Λ 1λ β λ α λ¯µ λ¯ν λ¯µ λ α, (C.18c)I αβζ ¯µ¯ν ¯η (Λ 1 Λ 2 Λ 1 Λ 2 ΛΛ) def= (−1) Λ+Λ 1[Λ 1 , Λ 2 , Λ 2 , Λ, Λ] 1/2{ } { }λ λα λ× 〈Λ 2 M 2 ΛM|Λ 1 M 1 〉 β Λ β λ ζ Λ 1λ¯ν λ¯η Λ 2 , (C.18d)λ¯ν λ¯µ Λ 1Λ 1 Λ 2 ΛI αβζ ′ ¯µ¯ν ¯η(Λ 1 Λ 2 Λ 1 Λ 2 ΛΛ) def= (−1) λα+λ ζ+λ¯µ+λ¯η+Λ 1 +Λ 1 +Λ 2 +Λ+Λ[ Λ λβ]× [Λ][Λ 1 , Λ 2 , Λ 1 , Λ 2 ] 1/2 λ¯η Λλ¯µ Λ 2 Λ 1 λ ζ , (C.18e)λ α Λ 2 Λ 1 λ¯ν{ } {F αβζρ (Λ 1 Λ 2 Λ 1 Λ 2 Λ) def= (−1) Λ 2[Λ 2 , Λ 2 ] 1/2 Λ1 Λ 2 Λ Λ 1 Λ 2 Λλ ρ λ β λ ζ, (C.18f)λ β λ ρ λ α}G αβζρ¯µ¯ν ¯η¯σ (Λ 1 Λ 2 Λ 1 Λ 2 Λ 1 Λ 2 Λ) def= (−1) Λ a(λ α λ β Λ 1 )a(λ¯ν λ¯σ Λ 2 ){× [Λ 1 , Λ 2 , Λ 1 , Λ 2 , Λ 1 , Λ 2 ] 1/2 λ β λ¯µ Λ 1 Λ 1 λ α λ¯νΛ 1 Λ Λ 2 , (C.18g)λ ζ λ¯η Λ 2 Λ 2 λ ρ λ¯σ}where the last quantity—the 15j–symbol **of** the third k**in**d—is def**in**ed accord**in**g to Ref. [10,Sec. 4-20, p. 207, Eq. (20.3)]. In Eqs. (C.3)-(C.8), the expressions for Z α ′ β ′ ¯µ ′¯ν ′(Λ 1Λ 2 Λ), where{α ′ , β ′ , ¯µ ′ , ¯ν ′ } denotes somehow permuted orbitals {α, β, ¯µ, ¯ν} **of** the coefficient Ω (2)αβ ¯µ¯ν (Λ 1Λ 2 Λ),are displayed **in** Tab. 40. Particularly, the abbreviation Z α ′ β ′ ¯µ ′¯ν ′(Λ 1Λ 1 0) ≡ Z α ′ β ′ ¯µ ′¯ν ′(Λ 1) isused.DSymbolic computations with NCoperatorsThe «Non-Commutative operators» package runs under Mathematica [99–101], a computationals**of**tware program. To take all advantage **of** NCoperators, especially when manipulat**in**gwith the antisymmetric tensors, a free ware package NCAlgebra developed by Helton, Stankuset. al. [102] must be compiled. The NCoperators package has been written for Unix OS, but it iseasy to adapt it for W**in**dows OS as well if some **of** the parameters **of** compilation are changed.The NCoperators is a composition **of** a large number **of** packages with extension *.m thatcan be divided **in**to four ma**in** blocks:

D Symbolic computations with NCoperators 901. Second Quantisation Representation (SQR)2. Angular Momentum Theory (AMT)3. Rayleigh–Schröd**in**ger Perturbation Theory (RSPT)4. Unstructured External Programm**in**g (UEP)The packages store Mathematica codes that are loaded **in**to a Mathematica session by us**in**gthe function «“NCoperators.m” or Get[“NCoperators.m”]. The NCoperatorsis designed so that each block can be brought **in**to action separately. This is easily done withGet[“PackageName.m”]. All these functions (Get[]) are located **in** a s**in**gle packageNCoperators.m.D.1 SQR and AMT blocksThe SQR and AMT blocks are closely related and therefore they will be considered together.The present blocks conta**in** a huge number **of** functions, each **of** them be**in**g useful for a specialcase **of** **in**terest. In the present overview, only a few **of** them will be discussed.In theoretical atomic physics, a typical task is to f**in**d the angular coefficients that relate twodist**in**ct ( momenta coupl**in**g schemes. Consider, for example, the momenta recoupl**in**g coefficientj1 j 2 (j 12 )j 3 j ∣ ∣j 2 j 3 (j 23 )j 1 j ) . The expression is known, and it is{ }(−1) 2j+j 2+j 3 −j 23[j 12 , j 23 ] 1/2 j1 j 2 j 12.j 3 j j 23Mak**in**g use **of** NCoperators, the algorithm to obta**in** the latter formula is displayed **in** Fig. 1.Fig. 1: A computation **of** recoupl**in**g coefficient ( j 1 j 2 (j 12 )j 3 j ∣ ∣ j2 j 3 (j 23 )j 1 j ) with NCoperatorsOut[25] conta**in**s two Clebsch–Gordan coefficients labelled by CGcoeff[] – the standardMathematica function ClebschGordan[] is **in**sufficient for symbolic manipulations.The function Recoupl**in**g[] **in**itiates the momenta recoupl**in**g followed by Refs. [10–12].SymbolSum[] plays a role **of** a standard Mathematica function Sum[] adapted to symboliccomputations. The function Make[] **in**itiates the momenta recoupl**in**g with**in** a given Clebsch–Gordan coefficient. The simplification **of** considered sum is performed by SumSimplify.Out[31] is pr**in**ted **in** a standard Unix output form specified by the font family «cmmi10». Obta**in**edformula is easy to convert to a L A TEX language by mak**in**g use **of** Mathematica function

D Symbolic computations with NCoperators 91Export[“DirectoryName/FileName.tex”,%31]. A brief description **of** all functionsis lited by us**in**g Def**in**ition[function] or simply ?function. The example ispresented **in** Fig. 2.Fig. 2: The usage **of** Def**in**ition[] **in** NCoperatorsIn Fig. 2, the function TAntiCommutator1[] along with TAntiCommutator2[],TAntiCommutator3[] and TAntiCommutator4[] **in**itiates the (four) anticommutationrules for irreducible tensor operators a λα im αi , ã λα im αi (RSPT block). To compare with, the anticommutationproperties **of** the Fock space operators a αi , a † α iare realised through the NCoperatorsfunctions AntiCommutator1[], AntiCommutator2[], AntiCommutator3[] andAntiCommutator4[]. The examples are listed **in** Fig. 3. Particularly, In[61] computesthe one-**body** terms {Ô2(αβ)Ô1(µν)} 2 which are found **in** the Wick’s series for ĥ(3) mn;ξwith the**in**dices m = 2, n = 1, ξ = 2 (refer to Eqs. (2.42), (4.3)); NonCommutOpMulti[n,α,β] isnoth**in**g else but Ôn(αβ). In addition, the function NonCommutOpMulti[n,α,β,m,µ,ν]corresponds to Ôn(αβ)Ôm(µν).Fig. 3: Manipulations with the antisymmetric Fock space operators **in** NCoperatorsFig. 4: Wigner–Eckart theorem **in** NCoperatorsThere are **many** more functions **in** NCoperators.Many tasks **in** atomic physics arerelated to the Wigner–Eckart theorem. Thistheorem is also found **in** the present package.The example **of** usage is demonstrated**in** Fig. 4, where the projection-**in**dependentquantity [ ]j α ‖v (k) ‖j β denotes a usual reducedmatrix element **of** the irreducible tensor operatorv (k) . Note, throughout the present text,the notation v k is preferred.

D Symbolic computations with NCoperators 92D.2 RSPT blockThe present block **in**volves functions suitable for the symbolic manipulations observed **in** thestationary Rayleigh–Schröd**in**ger **perturbation** **theory**. In pr**in**cipal, the RSPT block applies thepreviously summarised blocks with some specific functions particular with the MBPT. Withoutgo**in**g **in**to too much details concerned with the structure **of** programmed codes, consider anexample related to the third-order MBPT (Sec. 4). Select the one-**body** term ĥ(3) 11;1 : P −→ P.By Eq. (4.3),The operators ̂V 1 : F −→ F,equal toĥ (3)11;1 =:{ ̂P ̂V 1̂Ω(2) 1̂P } 1 : .̂Ω(2)1 : P −→ H given by Eqs. (2.40), (2.69a) are deduced to bêV 1 = ∑ αβa α a †¯βv α ¯β,̂Ω(2) 1 = ∑ ev(2)a e a†¯vω e¯v + ∑ vc(2)a v a†¯cω v¯c + ∑ ec(2)a e a†¯cω e¯c ,where the s**in**gle-particle matrix element v α ¯β is def**in**ed **in** Eq. (4.5). Recall that only the types**of** s**in**gle-electron orbitals are written below the sums, but not their values.Fig. 5: The generation **of** ĥ(3) 11;1 terms with NCoperatorsIn Fig. 5, the generation **of** ĥ(3) 11;1 terms is demonstrated. Many **of** the functions **in** Fig.5 are easy to detect by their names: NormalOrder[] denotes ::; OneContraction[]denotes the one-pair contractions (ξ = 1) between ̂V (2)1 and ̂Ω 1 ; KronDelta[] is obviouslythe Kronecker delta function. Other supplementary functions are to be used for various technicalsimplifications. In Out[4], 〈a|v eff2 |b〉 ≡ ω (2)ab (ε b − ε a ) (recall the irrep τ 2 ) and 〈a|v 1 |b〉 ≡ v ab(recall the irrep τ 1 ).In NCoperators, the three types—core (c), valence (v), excited (e)—**of** s**in**gle-electron orbitalsare designated bya c : cc1, cc2, etc.; a †¯c : ca1, ca2, etc.a v : vc1, vc2, etc.; a †¯v : va1, va2, etc.a e : ec1, ec2, etc.; a † ē : ea1, ea2, etc.In a standard output (such as Out[4]), the notations **of** orbitals are simplified to

D Symbolic computations with NCoperators 93c : a 1 , b 1 , c 1 , d 1 , e 1 , f 1 ¯c : a 2 , b 2 , c 2 , d 2 , e 2 , f 2v : m 1 , n 1 , p 1 , q 1 , k 1 , l 1 ¯v : m 2 , n 2 , p 2 , q 2 , k 2 , l 2e : r 1 , s 1 , t 1 , u 1 , w 1 , x 1 ē : r 2 , s 2 , t 2 , u 2 , w 2 , x 2Therefore, **in** accordance with Fig. 5, the generated terms ĥ(3) 11;1 areĥ (3)11;1 = ∑ eva v a †¯vv ve ω (2)e¯v− ∑ cva v a †¯vv c¯v ω (2)vc .(D.1)The next step is to restrict the model space P to its irreducible subspaces P Λ , that is, to expandĥ (3)11;1 **in**to the sum **of** irreducible tensor operators W Λ (λ v˜λ¯v ). Refer to Eqs. (4.4), (4.17a).Consider the operators ĥ(3)+ 11;1 associated to ω (2)+α ¯β . In this case, it is easily done by us**in**g theexpression for SU(2)–**in**variant **in** the first row **of** Tab. 12: simply replace f(τ 2 λ µ λ ¯β)/(ε ¯β − ε µ )with Ω (2)+ (Λ) (recall Remark 4.1.1). But the present **in**variant has been obta**in**ed also mak**in**gµ ¯βuse **of** NCoperators. Thus, for the sake **of** clarity, the full procedure **of** angular reduction willbe demonstrated. This is, however, easy to perform.In Eq. (D.1), the first term conta**in**s the product **of** type ∑ µ v αµω (2)+ , while the second one –µ ¯βthe product **of** type ∑ µ v (2)+µ ¯βω αµ . These products determ**in**e the s**in**gle-particle effective matrixelements. Then∑e∑cv ve ω (2)+e¯v = ∑ λ em e(−1) λe+me f(τ 1 λ v λ e )〈λ v m v λ e − m e |τ 1 m 1 〉× (−1) λ¯v+m¯v ∑ΛΩ (2)+e¯v (Λ)〈λ e m e λ¯v − m¯v |ΛM〉,v c¯v ω (2)+vc = ∑ λ cm c(−1) λ¯v+m¯v f(τ 1 λ c λ¯v )〈λ c m c λ¯v − m¯v |τ 1 m 1 〉× (−1) λc+mc ∑ΛΩ (2)+vc (Λ)〈λ v m v λ c − m c |ΛM〉.F**in**ally, recall**in**g that a v a †¯v = (−1) ∑ λ¯v+m¯v Λ W M Λ (λ v˜λ¯v )〈λ v m v λ¯v − m¯v |ΛM〉, the expression**in** the first row **of** Tab. 15 (the case m = n = ξ = 1) is obta**in**ed by us**in**g the result **in** Fig. 1.A much more complicated task is to f**in**d Ω (2)+ (Λ). To generate the s**in**gle-particle effectiveα ¯βmatrix elements ω (2) , make use **of** the generalised Bloch equation **in** Eq. (4.2). It follows thatα ¯βω (2) is the sum **of** ω(2)α ¯β ij;i+j−1 (α ¯β) ∀i, j = 1, 2, where the coefficients ω (2)ij;i+j−1 (α ¯β) are obta**in**edfrom the terms:{ ̂R̂V îΩ(1) ĵP −̂R̂Ω(1)ĵP ̂V i ̂P }i+j−1 : .The operator ̂R—also known as the resolvent [34]—is def**in**ed here by the action on somefunctional F (x † f 1, x † f 2, . . . , x † f a, x i1 , x i2 , . . . , x ia ) on H so that̂RF (x † f 1, x † f 2, . . . , x † f a, x ¯f1 , x ¯f2 , . . . , x ¯fa ) = [ D a (f ¯f) ] −1 ̂QF (x†f 1, x † f 2, . . . , x † f a, x ¯f1 , x ¯f2 , . . . , x ¯fa ).The energy denom**in**ator D a (f ¯f) is found from Eq. (2.65); the vectors x ¯fk (x † f k) ∀k = 1, 2, . . . , a**of** the s**in**gle-particle Hilbert space are identified to be the s**in**gle-particle eigenstates | ¯f k 〉 (〈f k |).The example how to generate the ω (2)11;1 terms is demonstrated **in** Figs. 6-7: **in** Fig. 6, thêR̂V 1̂Ω(1) 1̂R̂Ω(1)1terms { ̂P } 1 are generated, while **in** Fig. 7, the terms {result, ω (2)11;1 conta**in**s 9 expansion termsω 11;1(α (2) ¯β)(ε ¯β − ε α ) = ∑ v αµ ω (1)µ ¯β − ∑v ν ¯βω αν (1) ,µ(αβ)ν(αβ)̂P ̂V 1 ̂P }1 are derived. As a

D Symbolic computations with NCoperators 94Fig. 6: The generation **of** ω (2)11;1 terms: part 1Fig. 7: The generation **of** ω (2)11;1 terms: part 2

D Symbolic computations with NCoperators 95where µ(vc) = µ(ec) = v, e; µ(ev) = e; ν(vc) = ν(ec) = c; ν(ev) = v, c. Analogouscomputations must be performed for the rest **of** coefficients ω 12;2, (2) ω 21;2, (2) ω 22;3, (2) and thuswhereω (2) = ω(2)α ¯β 11;1(α ¯β) + ω 12;2(α (2) ¯β) + ω 21;2(α (2) ¯β) + ω 22;3(α (2) ¯β),ω (2)12;2(α ¯β)(ε ¯β − ε α ) = ∑ cω (2)21;2(α ¯β)(ε ¯β − ε α ) = ∑ c∑µ=v,e∑µ=v,ev cµ˜ω (1)µαc ¯β ,ṽ cαµ ¯βω (1)µc ,ω 22;3(α (2) ¯β)(ε ¯β − ε α ) = ∑ ṽ cαve˜ω (1)evc ¯β .c,v,eThe expressions are identified **in** Tab. 12, where the correspond**in**g SU(2)–**in**variants are selected(recall the dist**in**ct notations for dist**in**ct **in**termediate irreps def**in**ed by Eqs. (4.11)-(4.16)). Then split the solution for ω (2)α ¯β**in**to the sum **of** ω(2)+α ¯βand ω (2)−α ¯β. Make use **of** Eq.(4.17a). F**in**ally, the solutions for Ω (2)±α ¯β are displayed **in** Eqs. (C.1)-(C.2) **in** Appendix C.The algorithm **of** generation and angular reduction **of** ĥ(3) 11;1 terms is suited to the rest **of**operators ĥ(3) mn;ξelements associated t**of**or all **of** them:. It is by no means obvious that the procedure to f**in**d out the effective matrix̂Ω(2)nwith n ≥ 2 is much more complicated, though the idea holds truen = 2 : ω (2)αβ ¯µ¯ν = ∑ 2i,j=1 ω(2) ij;i+j−2 (αβ ¯µ¯ν), ̂Ω(2) 2 = :{n = 3 : ω (2)αβζ ¯µ¯ν ¯η = ∑ 2i,j=1 ω(2) ij;i+j−3 (αβζ ¯µ¯ν ¯η), ̂Ω(2) 3 = :{n = 4 : ω (2)αβζρ¯µ¯ν ¯η¯σ = ω(2) 22;0(αβζρ¯µ¯ν ¯η¯σ), ̂Ω(2) 4 = :assum**in**g that for ξ = 2, 3, the condition i + j − ξ ≥ 0 is valid.D.3 UEP block̂R̂V îΩ(1) ĵR̂V îΩ(1) ĵR̂V 2̂Ω(1) 2̂P −̂P −̂P: − :̂R̂Ω(1)ĵR̂Ω(1)ĵR̂Ω(1)2̂P ̂V i ̂P }i+j−2 :,̂P ̂V i ̂P }i+j−3 :,̂P ̂V 2 ̂P:,The present block is the most technical one, as it is closely related to the operat**in**g system used.The UEP block has been designed for the Unix systems and it can be skipped for those who arekeen on other systems. Nevertheless, the block actualises **many** useful features, though it is stillpermanently augmented. The ma**in** idea is to convert obta**in**ed symbolic preparation **in**to the Ccode for a more rapid calculation **of** the quantities under consideration.External programm**in**g is a way **of** communication between Mathematica **in**terface and someother (external) programs (such as C). Differently from the structured programm**in**g actualisedby us**in**g MathL**in**k, the unstructured programm**in**g does not require any other external «tunnels»except for the user’s own term**in**al.To make the UEP block operate, theMathematica header file mdefs.h must be located**in** the directory, where all system headerfiles are situated. In most Unix systems,the directory is usr/**in**clude/. The headeris supplemented with the C-based functionsthat are necessary for the atomic calculations:ClebschGordanC[], SixJSymbolC[],N**in**eJSymbolC[], etc. The MathL**in**k librarieslibML32i3.a, libML32i.so are substituted**in** /usr/lib or /usr/local/lib.The example **of** the usage **of** C-basedFig. 8: Example **of** an application **of** the UEP block functions with**in** Mathematica **in**terface isdemonstrated **in** Fig. 8, where the function

D Symbolic computations with NCoperators 96Fig. 9: The diagrammatic visualisation **of** :{ ̂R ̂V 2 ̂Ω(1) 1̂P } 2 : terms

D Symbolic computations with NCoperators 97Tim**in**g[] clearly demonstrates the ma**in** benefit **of** such type **of** programm**in**g that stipulates,therefore, the further on improvements **in** the same direction.Perspectives In addition to the briefly studied possibilities **of** the application **of** NCoperators,one more option can be found **in** the present package. It is a diagrammatic visualisation **of**expansion terms. The example is demonstrated **in** Fig. 9. A newbie equipment **of** NCoperatorsis actualised by us**in**g the Mathematica functions Graphics[], Show[]. Likewise **in** thecase **of** UEP block, the present feature is not fully operational at this time.The diagrammatic visualisation has the only one advantage to compare with the algebraicformulation **of** atomic MBPT. It is an easy to behold visualisation **of** complex algebraic expressions.It is not a mere mnemonic device, though. A number **of** rules to handle the diagramsdesigned for atomic and nuclear spectroscopy are formulated. However, as demonstrated **in** Sec.4, such visualisation is **in**efficient due to a huge number **of** expansion terms. Consequently, thereare two alternatives: whether to write programs that are capable to visualise expansion termsdiagrammatically and afterwards to convert them **in**to a usual algebraic form or to take aim atthe algebraic approach such as developed **in** Sec. 4. The NCoperators package has been designedto be **of** versatile disposition as much as it is possible. Therefore, the direction that mustbe chosen strongly depends on the future demands.

D Symbolic computations with NCoperators 98References[1] N. Bohr, Philosophical Magaz**in**e 26, 1 (1913)[2] Maria G. Mayer, Phys. Rev. A 74, no. 3, 235 (1948)[3] Maria G. Mayer, Phys. Rev. A 75, no. 13, 1969 (1949)[4] O. Haxel, J. Hans D. Jensen, Hans E. Suess, Phys. Rev. A 75, no. 11, 1766 (1949)[5] E. U. Condon, G. H. Shortley, The Theory **of** Atomic Spectra, Cambridge, CambridgeUniv. Press (1935)[6] E. P. Wigner, “ On the matrices which reduce the Kronecker products **of** representations**of** simply reducible groups”, **in** Quantum Theory **of** Angular Momentum edited by L. C.Biedenharn and J. D. Louck, New York, Academic (1965)[7] G. Racah, Phys. Rev. 61, 186 (1942)[8] G. Racah, Phys. Rev. 62, 438 (1942)[9] G. Racah, Phys. Rev. 63, 367 (1943)[10] A. Jucys, Y. Lev**in**son and V. Vanagas, Mathematical Apparatus **of** the Theory **of** AngularMomentum, Vilnius, 3rd ed. (Russ. Orig**in**al, Gospolitnauchizdat, 1960)[11] A. P. Jucys, A. J. Savukynas, Mathematical Foundations **of** the Atomic Theory, Vilnius(1973) (**in** Russian)[12] A. P. Jucys and A. A. Bandzaitis, Theory **of** Angular Momentum **in** Quantum Mechanics,Mokslas publishers, Vilnius (1977) (**in** Russian)[13] D. J. Newman and J. Wallis, J. Phys. A: Math. Gen. 9, no. 12, 2021 (1976)[14] G. Gaigalas, Z. Rudzikas and Ch. F. Fischer, J. Phys. B: At. Mol. Opt. Phys. 30, 3747(1997)[15] G. Gaigalas, S. Fritzsche, I. P. Grant, Comput. Phys. Comm. 139, no. 3, 263 (2001)[16] G. Gaigalas and S. Fritzsche, Comput. Phys. Comm. 148, no. 3, 349 (2002)[17] S. Fritzsche, Comput. Phys. Comm. 180, no. 10, 2021 (2009)[18] Ch. F. Fischer, J. Phys. B: At. Mol. Phys. 3, no. 6, 779 (1970)[19] M. R. Godefroid, Ch. F. Fischer and P. Jönsson, Phys. Scr. 1996, T65, 70 (1996)[20] Ch. F. Fischer, A. Ynnerman, G. Gaigalas, Phys. Rev. A 51, no. 6, 4611 (1995)[21] K. A. Brueckner, Phys. Rev. 97, no. 5, 1344 (1955)[22] K. A. Brueckner, Phys. Rev. 100, no. 1, 36 (1955)[23] J. Goldstone, J. Proc. R. Soc. Lond. A 239, 267 (1957)[24] H. P. Kelly, Phys. Rev. 131, 684 (1963)[25] H. P. Kelly, Phys. Rev. 134, A1450 (1964)[26] H. P. Kelly, Phys. Rev. 144, 39 (1966)[27] D. Mukherjee, R. K. Moitra and A. Mukhopadhyay, Mol. Phys. 30, 1861 (1975)

D Symbolic computations with NCoperators 99[28] I. L**in**dgren, Int. J. Quant. Chem. S12, 33 (1978)[29] J. Morrison and S. Salomonson, Phys. Scr. 21, 343 (1980)[30] V. Kvasnička, Chem. Phys. Lett. 79, no. 1, 89 (1981)[31] W. Kutzelnigg, Chem. Phys. Lett. 83, 156 (1981)[32] I. L**in**dgren, Phys. Rev. A 31, no. 3, 1273 (1985)[33] D. Mukherjee, Chem. Phys. Lett. 125, no. 3, 207 (1986)[34] I. L**in**dgren, J. Morrison, Atomic Many-Body Theory, Spr**in**ger Series **in** ChemicalPhysics, Vol. 13 (1982)[35] S. A. Blundell, W. R. Johnson and J. Sapirste**in**, Phys. Rev. A 43, no. 7, 3407 (1991)[36] M. S. Safronova, W. R. Johnson and U. I. Safronova, Phys. Rev. A 53, no. 53, 4036 (1996)[37] H. C. Ho and W. R. Johnson et. al., Phys. Rev. A 74, 022510, 1 (2006)[38] A. Derevianko, J. Phys. B: At. Mol. Opt. Phys. 43, 074001, 1 (2010)[39] I. L**in**dgren and D. Mukherjee, Phys. Rep. 151, no. 2, 93 (1987)[40] G. C. Wick, Phys. Rev. 80, no. 2, 268 (1950)[41] S. A. Blundell, W. R. Johnson and J. Sapirste**in**, Phys. Rev. A 42, no. 7, 3751 (1990)[42] S. G. Porsev, A. Derevianko, Phys. Rev. A 73, 012501 (2006)[43] V. Dzuba, Comput. Phys. Comm. 180, no. 3, 392 (2009)[44] http://wolfweb.unr.edu/homepage/andrei/tap.html[45] Z. Csepes and J. Pipek, J. Comput. Phys. 77, no. 1, 1 (1988)[46] B. R. Judd, Operator Techniques **in** Atomic Spectroscopy, McGraw-Hill, New York (1963)[47] B. R. Judd, Second Quantization and Atomic Spectroscopy, Baltimore, MD: Johns Hopk**in**s(1967)[48] B. R. Judd and R. C. Leavitt, J. Phys. B: At. Mol. Phys. 15, 1457 (1982)[49] R. C. Leavitt, J. Phys. B: At. Mol. Phys. 21, 2363 (1987)[50] Z. Rudzikas, J. Kaniauskas, Quasisp**in** and Isosp**in** **in** the Theory **of** Atom, Vilnius, MokslasPublishers (1984) (**in** Russian)[51] J. M. Kaniauskas, V. Č. Šimonis and Z. B. Rudzikas, J. Phys. B: At. Mol. Phys. 20, 3267(1987)[52] Z. Rudzikas, Theoretical Atomic Spectroscopy, Cambridge, Cambridge Univ. Press (1997)[53] G. Gaigalas, J. Kaniauskas et. al., Phys. Scr. 49, 135 (1994)[54] G. Merkelis, Phys. Scr. 61, 662 (2000)[55] G. Merkelis, Phys. Scr. 63, 289 (2001)[56] C. F. Bunge, At. Data Nucl. Data Tables 18, 293 (1976)[57] W. J. Marciano, J. L. Rosner, Phys. Rev. Lett. 65, 2963 (1990)

D Symbolic computations with NCoperators 100[58] J. Sapirste**in**, K. T. Cheng, Phys. Rev. A 67, 022512 (2003)[59] P. Ramond, Group Theory: A Physicist’s Survey, Cambridge, Cambridge Univ. Press(2010)[60] J. C. Slater, Phys. Rev. 34, no. 10, 1293 (1929)[61] E. P. Wigner, Group Theory and its Applications to the Quantum Mechanics **of** AtomicSpectra, New York, Academic Press (1959)[62] N. J. Vilenk**in**, Special Functions and the Theory **of** Group Representations, 2nd ed.,Moscow, Nauka (1991) (**in** Russian)[63] V. V. Vanagas, **Algebraic** Foundation **of** the Microscopic Nuclear Theory, Moscow, Nauka(1988) (**in** Russian)[64] A. U. Klimyk, Matrix Elements and Clebsch-Gordan Coefficients **of** Representations **of**Groups, Kiev, Naukova Dumka (1979) (**in** Russian)[65] A. K. Bhatia, A. Temk**in**, Rev. Mod. Phys. 36, 1050 (1964)[66] R. Juršėnas and G. Merkelis, Int. J. Theor. Phys. 49, no. 9, 2230 (2010)[67] Z. B. Rudzikas, J. M. Kaniauskas, Int. J. Quant. Chem. 10, 837 (1976)[68] P. R. Smith and B. G. Wybourne, J. Math. Phys. 9, 1040 (1968)[69] B. G. Wybourne, Spectroscopic Properties **of** Rare Earths, New York, John Wiley & Sons,Inc. (1965)[70] W. Kutzelnigg, Int. J. Quant. Chem. 109, 3858 (2009)[71] H. Feshbach, Ann. Phys. (N.Y.) 5, 357 (1958)[72] R. Juršėnas and G. Merkelis, At. Data Nucl. Data Tables (2010),doi:10.1016/j.adt.2010.08.001[73] R. Juršėnas and G. Merkelis, to appear **in** J. Math. Phys. (2010)[74] A. Derevianko, S. G. Porsev, Phys. Rev. A 71, 032509 (2005)[75] A. Derevianko, S. G. Porsev and K. Beloy, Phys. Rev. A 78, 010503(R) (2008)[76] V. Vanagas, **Algebraic** Methods **in** Nuclear Theory, Vilnius (1971) (**in** Russian)[77] R. Juršėnas, G. Merkelis, Cent. Eur. J. Phys. 8, no. 3, 480 (2010)[78] R. Juršėnas and G. Merkelis, MPM e-journal 9, no. 1, 42 (2010)[79] W. Fulton, J. Harris, Representation Theory: A First Course, New York, Spr**in**ger-VerlagInc. (1991)[80] R. Juršėnas, G. Merkelis, Cent. Eur. J. Phys. (2010),doi:10.2478/s11534-010-0082-0[81] R. Juršėnas and G. Merkelis, Lithuanian J. Phys. 47, no. 3, 255 (2007)[82] M. A. Braun, A. D. Gurchumelia, U. I. Safronova, Relativistic Atom Theory, Moscow,Nauka (1984) (**in** Russian)[83] G. Breit, Phys. Rev. 35, 569 (1930)

D Symbolic computations with NCoperators 101[84] A. A. Nikit**in**, Z. B. Rudzikas, Foundations **of** the Theory **of** the Spectra **of** Atoms andIons, Moscow, Nauka (1983) (**in** Russian)[85] J. M. Kaniauskas, V. V. Špakauskas, Z. B. Rudzikas, Lietuvos Fizikos R**in**k**in**ys 19, no. 1,21 (1979) (**in** Russian)[86] G. V. Merkelis, G. A. Gaigalas, Z. B. Rudzikas, Lietuvos Fizikos R**in**k**in**ys 25, no. 6, 14(1985) (**in** Russian)[87] G. Merkelis, Nucl. Instr. Meth. Phys. Research B 235, 184 (2005)[88] R. Pal et. al., Phys. Rev. A 74, 042515 (2007)[89] B. R. Judd, Phys. Rev. 141, 4 (1966)[90] H. Dothe et. al. J. Phys. B: At. Mol. Phys. 18, 1061 (1985)[91] P. H. M. Uyl**in**gs, J. Phys. B: At. Mol. Phys. 17, 2375 (1984)[92] J. Noga and R. J. Barllet, Chem. Phys. Lett. 134, no. 2, 126 (1987)[93] L. Meissner, P. Mal**in**owski and J. Gryniakow, J. Phys. B: At. Mol. Phys. 37, 2387 (2004)[94] Sh. Matsumoto and J. Novak, DMTCS proc., 403 (2010)[95] T. Graber and R. Vakil, Compositio Math. 135, no. 1, 25 (2003)[96] A. A. Jucys, Lietuvos Fizikos R**in**k**in**ys 11, no. 1, 5 (1971) (**in** Russian)[97] A. A. Jucys, Rep. Math. Phys. 5, 107 (1974)[98] G. E. Murphy, J. Algebra 69, 287 (1981)[99] S. Wolfram, The Mathematica Book, Wolfram Media, Cambridge Univ. Press, 4th ed.(1999)[100] http://www.wolfram.com/[101] A. Dargys, A. Acus, Fizika su kompiuteriu, Ciklonas, Vilnius (2003) (**in** Lithuanian)[102] http://www.math.ucsd.edu/~ncalg/