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INSTITUTE OF THEORETICAL PHYSICS AND ASTRONOMYOF VILNIUS UNIVERSITYRytis JuršėnasAlgebraic development of many-body perturbation theoryin 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 Universityin 2006-2010.Scientific supervisor:Dr. Gintaras 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 JuršėnasAlgebrinis daugiadalelės trikdžių teorijos plėtojimasteorinėje atomo spektroskopijojeDaktaro disertacijaFiziniai mokslai, fizika (02P)Matematinė ir bendroji teorinė fizika, klasikinė mechanika, kvantinė mechanika,reliatyvizmas, gravitacija, statistinė fizika, termodinamika (P190)Vilnius, 2010


Disertacija rengta Vilniaus universiteto Teorinės fizikos ir astronomijos institute 2006-2010metais.Mokslinis vadovas:Dr. Gintaras Merkelis (Vilniaus universiteto Teorinės fizikos ir astronomijos institutas, 02P:fiziniai mokslai, fizika; P190: matematinė ir bendroji teorinė fizika, klasikinė mechanika, kvantinėmechanika, reliatyvizmas, gravitacija, statistinė fizika, termodinamika)


5Contents1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1 The main goals of present work . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 The main tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 The scientific novelty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Statements to be defended . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 List of abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Partitioning of function space and basis transformation properties . . . . . . . . 132.1 The integrals of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Coordinate 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 Concluding 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 concluding 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 Concluding remarks and discussion . . . . . . . . . . . . . . . . . . . . . . . 675 Prime results and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 70A Basis coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72BThe classification of three-particle operators acting 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)–invariant 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 coordinates of S 2 × S 2 . . . . . . . . . . . . . . 152 Numerical values of reduced matrix element of SO(3)–irreducible tensor operatorS k for several integers 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 (continued) . . . . . . . . . . . . . . . . . . . . . 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 recoupling 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 Definition[] 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 greaterof many-electron problems, is based on irreducible tensor formalisms and, consequently, onsymmetry principles, simplifying various expressions, thus considerably reducing amount offurther on theoretical calculations of desirable physical quantities. The mathematical formulationof irreducible tensor operators gained 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, definingvector-valued functions and their behaviour under various transformations. Therein the fundamentalconnection between measurable physical quantities and abstract operators on Hilbertspace—particularly those transforming under irreducible representations—is realised throughthe bilinear functionals which ascertain, in general, the mapping from the Kronecker product ofvector spaces into some given unitary or Euclidean vector space. In physical applications, thesebilinear forms, usually called the matrix elements on given basis, are selected to be self-adjoint.The physical processes and various spectroscopic magnitudes, such as, for example, electrontransition probability, energy width of level or lifetime of state of level, electron interactionsand many more, are uniformly estimated by the corresponding 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 integers which in their turn form the set of quantum numbersdescribing the Hamiltonian of a local or stationary system. For the most part, the nonnegativeintegers that describe the dynamics of such system simply mark off the irreducible group representationsif the group operators commute with a Hamiltonian. Particularly, in the atomiccentral-field approximation, the Hamiltonian is invariant 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. Within theframework of the last approximation, the atomic Hamiltonian may be constructed by makingit the SU(2)–invariant, since SU(2) is a double covering 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 offered 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 diminish the time resources for a large scaleof theoretical calculations. The problems to prepare the effective techniques of reduction aredominant especially in the studies of open-shell atoms, when dealing with the physical as wellas the effective none scalar irreducible tensor operators and their matrix elements on the basisof complex configuration functions.A total eigenfunction of atomic stationary Hamiltonian is built up beyond the central-fieldapproach and it is, by the origin, the major object of the many-body theories. Unfortunately, theexact eigenstates can not be found, thus the final results that characterise the dynamics of suchcomplex system are not yet possible. From the mathematical point of view, the eigenstates ofHamiltonian form some linear space. If the spectrum of Hamiltonian is described by discretelevels, then the eigenstates characterised by the nonnegative integers form a separable Hilbertspace; otherwise, the linear space is, in general, non-separable. Through ignorance of the structureof exact eigenstates, there are formed the linear combinations 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 single-electronwave functions. In this approach, a huge number of admixed configurations together with highorder of energy matrices need to be taken into 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 confined to be in operation on a singlemany-particle Hilbert space. Contrary to this model, there exists another extremely differentmethod to build the exact eigenfunction of many-particle system. Of special interest is theatomic many-body perturbation theory (MBPT) that accounts for a variable number of manyparticleHilbert 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 main 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 genuine vacuum. In perturbationtheory (PT), the exponential ansatz is frequently defined to obey the form of the so-called waveoperator acting 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 single Slater determinant 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 into 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 partitioning is that for open-shell atoms the energy levels are degenerate andthe full set of reference functions is not always determined initially. Therefore the number ofselected 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 insists on further studies.A significant advantage of the MBPT is that the exact eigenvalues of atomic Hamiltonianare obtained even without knowing 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 finding the operator which acts on chosen model space. The form ofthe latter operator, called the effective Hamiltonian [34], is closely related to the form of waveoperator. Usually, there are distinguished Hilbert-space and Fock-space approaches, in orderto specify this operator. In the Hilbert-space approach, the model space is chosen to includea fixed number of electrons. Then the wave operator is determined by eigenvalue equation ofatomic Hamiltonian for a single 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 taking into account for the effects thatare conditioned by the variable number of particles. Starting from this point of view, severalvariations to construct the effective interaction operator on a model space are separated resultingin different versions of the PT. Nevertheless, a general idea embodied in all perturbation theoriesstates that the Hamiltonian of many-particle system splits up into the unperturbed Hamiltonianand the perturbation which characterises the inhomogeneity 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 perturbationdiffer. The Rayleigh–Schrödinger (RS) and coupled-cluster (CC) theories combinedwith the second quantisation representation (SQR) are the most common approaches used intheoretical atomic physics. The perturbation series is built by using the Wick’s theorem [40]which makes it possible to evaluate the products of the Fock space operators. The number ofthese products grows rapidly as the order of perturbation increases. For this reason, the RSPTis applied to a finite-order perturbation, when constructing the many-electron wave functionof a fixed order m, starting from m = 0 step by step [36, 37, 41]. In CC theory, the initially


1 Introduction 10given exponential ansatz is represented, in general, as an infinite sum of Taylor series. The totaleigenfunction is then expressed by the sum of all-order n–particle (n = 0, 1, 2, . . .) functions,denoting zero-, single-, double-, etc. excitations [35,42]. However, in practical applications, thesum of terms is also finite. To this day, the progressive attempts to evaluate the terms of atomicPT by using the computer algebra systems have been reported by a number of authors [43–45].One more problem ordinary 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 using the angular momentum theory (AMT) combined 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, single 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 determinedby 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 accounting forradiative corrections of hyperfine splittings in alkali metals or highly charged ions [58]. Suchlevel of accuracy is obtained when the triple excitations are involved in the series of the PT, asdemonstrated by Porsev et. al. [42]. This indicates that the mathematical techniques applied tothe reduction of the tensor products of the Fock space operators still are urgent and inevitable.1.1 The main goals of present work1. To work out the versatile disposition methods and forms pertinent to the tensor products of theirreducible tensor operators which represent either physical or effective interactions consideredin 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. Making 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, payingspecial attention to the construction of a model space and the development of angular reductionof 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 main tasks1. To find 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 infinite-dimensional Hilbert spacecontains a subset of eigenvalues of Hamiltonian projected onto the finite-dimensional subspace.2. To classify the totally antisymmetric tensors determined by the Fock space operator stringof any length. To determine the transfer attributes of irreducible tensor operators associated todistinct angular reduction schemes.3. To generate the terms of the second-order wave operator and the third-order effective Hamiltonianon the constructed finite-dimensional subspace by using a produced symbolic computeralgebra package. To develop the approach of many-particle effective matrix elements so thatthe projection-independent parts could be easy to vary remaining steady the tensor structure ofexpansion 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 prominent part of such type of computations is in debt to the newly found SO(3)–irreducible tensor operators.2. Bearing in mind that the theoretical interpretation concerned with the model space in openshellmany-body perturbation theory is poorly defined so far, attempts to give rise to moreclarity have been initiated. The key result which causes to diminish the number of expansionterms is that only a fixed number of types of the Hilbert space operators with respect to thesingle-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 principaladvantages of such technique are: (i) the ability to vary the amplitudes of electron excitationsuitable for special cases of interest – 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 problemof evaluation of each separate diagram is eliminated.1.4 Statements to be defended1. The integrals over S 2 of the SO(3)–irreducible matrix representation parametrised by thecoordinates of S 2 × S 2 constitute a set of components of SO(3)–irreducible tensor operator.2. There exists a finite-dimensional subspace of infinite-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 thesingle-electron states for all nonnegative integers n.3. The method developed by making use of the S l –irreducible representations, the tuples andthe commutative diagrams of maps associating distinct angular reduction schemes makes itpossible to classify the angular reduction schemes of antisymmetric tensors of any length inan 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 dependingon the specific cases of interest, but the tensor structure of expansion terms is fixed.


1 Introduction 121.5 List of publications1. R. Juršėnas and G. Merkelis, Coupling 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-coupling, At. Data Nucl. Data Tables (2010),doi:10.1016/j.adt.2010.08.0014. R. Juršėnas and G. Merkelis, Application of symbolic programming for atomic many-bodytheory, 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, Coupling 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-coupling, 38th Lithuanian National Physics Conference,Vilnius, 2009, p. 2294. R. Juršėnas, G. Merkelis, Symbolic programming 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 using symbolic programming 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 Partitioning of function space and basis transformation properties 132 Partitioning 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—concentrating on partitioning 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 coordinate transformations—to calculate the integrals of many-electron angular parts, theFock space formulation of the generalised Bloch equation, the theorem that determines nonzeroeffective operators on the bounded subspace of infinite-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 inspiration 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 ofirreducible 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-calledpartitioning technique (Sec. 2.3.1), is developed in Sec. 2.3.2.2.1 The integrals of motionThe quantum mechanical interpretation of atomic many-body system is found to be closelyrelated to the construction of Hamiltonian H. In a time-independent approach, this Hamiltoniancorresponds to the total energy E of system. To find E, the eigenvalue equation of H must besolved. The eigenfunction Ψ of H depends on the symmetries that are hidden in the many-bodysystem Hamiltonian. The group-theoretic formulation of the problem is to find 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 kinetic energy of electrons and U C denotes the Coulomb (electron-nucleus) potential. ThisHamiltonian is invariant under the rotation group G = SO(3) with ĝ ∈ {̂L 1 , ̂L 2 , ̂L 3 } being theinfinitesimal 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 indicesM = −L, −L + 1, . . . , L − 1, L that mark off 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 coordinates of the ξth electron. The quantity Γ denotes the rest ofnumbers that append the other, if any, symmetry properties involved in ˜H 0 . Particularly, theBohr Hamiltonian ˜H 0 is also invariant under the reflection characterised by the parity Π. Thus˜H 0 implicates the symmetry group O(3).The infinitesimal 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 independent A 1 Lie algebrasformed by Ĵ ± i . This implies that ˜H 0 is also invariant under SU(2) operations and thus Ψ maybe characterised by the SU(2)–irreducible representation J = 1/2, 3/2, . . . which particularlycharacterises the spin- 1 /2 particles (electrons).Regardless of well-defined symmetries appropriated by ˜H 0 , the central-field approximationdoes not account for the interactions 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 pertains to the symmetry properties of ˜H 0 since U represents the central-


2 Partitioning of function space and basis transformation properties 14field potential which has the meaning of the average Coulomb interaction of electron with theother electrons in atom [34, Sec. 5.4, p. 114]. All electron interactions along with externalfields, if such exist, are drawn in the perturbation V . It seems to be obvious that the structure ofH is much more complicated than the structure of H 0 .Having defined the Hamiltonian H and assuming that the eigenfunctions {Ψ i } ∞ i=1 ≡ Xform the infinite-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 〉. Since the functions Ψ i areunknown, usually they are expressed by the linear combination of some initially known basisΦ i . With such definition, 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 assuming thatthe functions Φ i represent the eigenfunctions of H 0 . The real numbers c eΓi Π i Λ iare found bydiagonalizing 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 making the antisymmetric products or Slater determinants [60, p. 1300] of single-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 parametrisationof 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 Vilenkin [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×min (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 finite unitary group G has a significant meaning in representationtheory as well as in theoretical atomic spectroscopy. First of all, it allows one to find aconvenient basis for which the matrix elements in Eq. (2.3) appropriate the simplest form [63].


2 Partitioning 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 into 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 using Eq. (2.9) is convenient to calculate matrix elements if applying the Wigner–Eckart theorem. In order to do so, the Hamiltonian H on H is represented by the sum ofirreducible tensor operators H Λ that act on the subspaces H Λ of H. For example, the HamiltonianH 0 is obtained from H if H is confined 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 interms of Dm λ em(Ω). Conversely, such basis is inconvenient for the application of Wigner–Eckarttheorem. Moreover, to calculate matrix elements of irreducible tensor operator H Λ , the integralof type∫∫d̂x 1 d̂x 2 Dm λ em (Ω)HΛ Dm λ′′ em ′(Ω) (2.10)S 2must be calculated. The integration becomes complicated since Ω depends on ̂x 1 , ̂x 2 . To performthe latter integration, 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 coordinates of S 2 × S 2 makes sense. Here and elsewhere S 2denotes a 2–dimensional sphere.2.2 Coordinate 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 = (sin θ i cos ϕ i sin θ i sin 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 coordinates 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 Partitioning 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 indicates thefollowing 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 ′ ).Proof. The Euler’s formula for x ∈ R reads e ix = cos x + i sin x. Deduce( ) α α∑ ( )( ) β i α 1β∑( βsin α x = (−1) s e i(2s−α)x , cos β x =e2s2 r)i(2r−β)x ,s=0r=0where the binomial 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 sinceα ∝ +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 passing to Eq. (2.8), we get for q ≥ q ′ , the binomial 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 remains irrelevant if replacing 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 distinct 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 Partitioning 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 defined 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 using the rule ± τ k 1× ± τ k 2= ⊕ k ± τ k ,which is obvious since 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.Obtained 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 integration 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 recalling that theirreducible tensor operators T λ —being of special interest in atomic physics—also transformunder the irreducible matrix representations D λ (Ω). In general, the 2λ + 1 components T λ µ ofT 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 invariant 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 expressedin terms of D λ and their various combinations. These are, for example, the normalised spherical


2 Partitioning 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 spin operator S 1 , the angular momentum operator L 1are also expressed in terms of the irreducible matrix representation D 1 , as it was demonstratedin Refs. [52, 67]. The announced particular cases of T λ represent the operators that dependon the coordinate system. Being more tight, these spherical tensor operators are in turn thefunctions of ̂x ≡ θϕ. With this in mind, 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 following 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 defined 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 coordinate transformations (or simplyRCGC technique) is the ability to transform the coordinate-dependence of thebasis Φ(ΓΠΛM|̂x 1 , ̂x 2 , . . . , ̂x N ) in H Λ preserving its inner structure. The transformedbasis ¯Φ(˜Γ˜Π˜Λ˜M|̂x ξ ) in H eΛ implicates the tensor structure of the initial basisΦ(ΓΠΛM|̂x 1 , ̂x 2 , . . . , ̂x N ), but the coordinate-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 propertiesof RCGCs (see Eqs. (2.25)-(2.26)) initiates a possibility to change the calculation of multipleintegrals with the calculation of a single one. This argument also fits the integrals of the typein Eq. (2.10). In a two-electron case, a simple evaluation indicates that the integration ofthe 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 into a single integral 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 obtained single integral becomes even simplerif T k acquires some special values. For instance, 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 ofD 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 Partitioning of function space and basis transformation properties 19reduced in accordance with Eqs. (2.23), (2.26). The obtained RCGC II ( ) Λ e Λ1 Λ 2 Λ; ̂xeɛ ɛ 1 , ̂x 2 isintegrated over ̂x 2 , and the resultant function also depends on ̂x 1 only. It goes without sayingthat the procedure of integration is suitable for the N–electron case. However, to improve theapplicability of this algorithm, the integrals of the RCGCs or, what is equivalent, the integralsof spherical functions ± τqq k ′(̂x 1, ̂x 2 ) should be calculated.In Ref. [66, Sec. 5], it was proved that the integral of ± τqq k ′(̂x 1, ̂x 2 ) over ̂x 2 ∈ S 2 is the sumof integrals determined 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 sin θ 2 ; a normalisation ∫ S 2 d̂x 2 = 4π. A direct integrationleads 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 defined 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)) sin 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 ofGauss’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 Definition. The functionsmin (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 cardinality #L l = 2l + 1, and the indices 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 ofthe SO(3)–irreducible tensor operator S l (̂x).Proof. 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 Partitioning of function space and basis transformation properties 20To begin with, integrate both sides of Eq. (2.21) over ̂x 2 on S 2 . For positive integer 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 again. 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 ).Finally, 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 replacing ̂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.According to Proposition 2.2.3, the tensor products of SO(3)–irreducible tensor operatorsS l (̂x) are reduced by using 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 integration, 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 integers 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 ).Knowing 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.Having defined the functions Sqq k ′(̂x), the calculation of integrals in Eq. (2.10) requires littleeffort. Besides, this is a simpler case than the integration on the basis Φ(ΓΠΛM|̂x 1 , ̂x 2 , . . . , ̂x N )since the product of Dm λ em(Ω) and Dλ′m ′ em(Ω) is reduced to a single 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 interaction operator 1/r 12 , then Λ = 0 and a double integral is transformed to a singleone as follows


2 Partitioning 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 severalintegers 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 valuesof [l‖S k ‖l ′ ] are listed in Tab. 2.In general, each N-integral∫ ∫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 single integrals∫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 integral can acquire the following 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)recalling that the parity is invariant under coordinate transformation. That is, Π bra,ket = ˜Π bra,ket .Eq. (2.39) represents thus the single-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-integral over the sphericalcoordinates ̂x ξ ∀ξ = 1, 2, . . . , N. If the basis is represented in terms of Slater determinants,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 single-electron


2 Partitioning of function space and basis transformation properties 22matrix elements. In the individual cases, the calculation may be performed in adifferent way. If the RCGC technique is exploited, the N-integral is reduced tothe sum of single integrals. The technique based on the coordinate 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 determinants it is more convenient to form the basis Φ(ΓΠΛM) fromthe single-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—acting on the genuine vacuum |0〉. In this case, the quantum mechanical manybodysystem is characterised by the number of particles rather than their coordinates. 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-coupling). Then W LS (l 1˜l2 ) transforms under both D L (g), D S (g)irreducible matrix representations independently. If, however, H Λ is irreducible, then Λ ≡ J(jj-coupling).Judd [46] demonstrated that the application of a second quantised representation of atomicmany-body system appears to be especially comfortable for the group-theoretic classificationof 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 electronsin the shell l N . This implies that the branching rules for the states of l N are to be obtained.For LS-coupling, 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 determine the so-called seniorityquantum number v, first introduced 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 quasispin space. The quasispin 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,starting from f 3 electrons, the last scheme is insufficient 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 principalquantum 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 singleparticles. 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 allinteraction operators h(n) used in atomic spectroscopy are obtained from the Feynman diagram


2 Partitioning of function space and basis transformation properties 23121◦′◦ 2′which requires effort to demonstrate that the path integral is determined in terms of the interaction( 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 interaction operatoron the basis of Dirac 4–spinors results to the sum of Coulomb interaction 1/r 12 and the Breitinteraction 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 corresponding basis Φ(ΓΠΛM) or else |ΓΠΛM〉 are equal. However, Ĥ has the eigenstatesfor all N, while H only for a specified N. According to Kutzelnigg [31, 70], Ĥ is called theFock space Hamiltonian.2.3.1 Orthogonal subspacesThe task to find the set X ≡ {|Ψ i 〉} ∞ i=1 of eigenfunctions of Ĥ (see Eq. (2.2)) is found tobe partially solvable by using the partitioning technique, first introduced by Feshbach [71].Later, it was demonstrated by Lindgren et. al. [34] that the present approximation leads tothe effective operator approach. In their used formalism, on the other hand, Lindgren andthe authors behind [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 themany-body perturbation theory (MBPT) were demonstrated in Refs. [54, 55, 72, 73].To find a subset Y ≡ {|Ψ j 〉} d j=1 ⊂ X of functions |Ψ j 〉 (with d < ∞), the following spacepartitioning procedure is performed.2.3.1 Definition. 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 contain the configurations of two types:(1) fully occupied l N l ktktconfigurations that particularly determine either core (c) or valenceY(v) orbitals; the core orbitals are present in all |Φ e k 〉 for all integers 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 inYsome of the functions |Φ e k 〉;(2) partially occupied l N kzkzconfigurations that determine valence (v) orbitals for all integersz ≤ 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 meaningful conclusions immediately follow from the definition 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 Partitioning of function space and basis transformation properties 242. The subset Ỹ is partitioned into several subsets Ỹn, each of them defined 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 contain 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 Definition 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 single-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 inferred by Lindgren [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.Having defined the subsets Ỹ , ˜Z ⊂ ˜X of vectors of many-body Hilbert spaces, it is sufficientto introduce the subspaces as follows.2.3.2 Definition. 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 infinite-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 infinitedimensionalN–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. According 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 Definition. 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.Having defined the many-electron Hilbert spaces, the following proposition is straightforward.n=1


2 Partitioning 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=1Proof. For the basis |Φ pn 〉, it is evident that (see Definition 2.3.2)̂1 n |Φ pn 〉 = ∑ |Φ p ′ n〉〈Φ p ′ n|Φ pn 〉 Hn = |Φ pn 〉.p ′ nFor the linear combinations |Ψ M 〉 ≡ ∑ Mp n=1 c p n|Φ pn 〉 with c pnobtainable. 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=1Proof. The proof directly follows from Definition 2.3.5 and Proposition 2.3.7, recalling that thefunctions |Φ p 〉 ∈ ˜X of F determine 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 into the subsets Y n ofdefP n ⊂ H n , defined 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 integer 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 defined 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 wasLindgren [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 obtained from the eigenvalue equations of H 0 and H taking into 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 introducing the Fock space operator (see Eq. (2.41))Ŝ def= ̂1 +∞∑Ŝ n ,n=1defŜ n = F n [ω], (2.55)


2 Partitioning 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) point to at least two typesof Hilbert space operators: ̂P Ô n (α ¯β) ̂P and ̂QÔn(α ¯β) ̂P (see Eq. (2.42)). To determine theirbehaviour for a given set I n (α ¯β) of single-electron orbitals, redefine items (b)(1)-(2), (c), 3 inSec. 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 pointed out, items (A)-(B) agree with Ref. [34, Sec. 13.1.2, p. 288, Eq. (13.3)].Items (C)-(D) embody a mathematical formulation of item (c) in Definition 2.3.1 and are ofspecial significance since they define 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 determine the type of orbital: v, e or c. According 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 oninfinite-dimensional N–electron Hilbert space H, then for any integer n ≤ N, the followingassertions 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 + .Observing that self-adjoint operator Ô† n(α ¯β) = Ôn( ¯βα), the following 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 replacing α with β in Lemma 2.3.9.Proof of Lemma 2.3.9. To prove the lemma, start with item i) which is easy to confirm by passingto 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 Partitioning 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 Definition 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 obtained 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 Definitionen〉 ′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 ) assuming 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 domain of integers[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 meaning. The main purpose ofthe rule is to reject the terms of ̂Ω n (see Eq. (2.59)) with zero-valued energy denominators.These energy denominators are obtained 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 beingsome effective interaction. Then̂Ω n = ̂QF n [v eff ] ̂P .YE e ken− E ken


2 Partitioning of function space and basis transformation properties 29YBut each eigenvalue E e ken, E ken is the sum of single-electron energies ε i which are found bysolving the single-electron eigenvalue equations for the single-electron eigenstates that correspondinglyform |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 denominator D n (v¯v) = 0. Therefore itemiii) of Lemma 2.3.9 omits this result for even integers ∑ ni=1 (l v i+ l¯vi ).The D–dimensional subspace P of infinite-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 allocating 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 interaction 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 solving the eigenvalue equationof 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 ofperturbation ̂V (for m = 1, 2) and the n–body part of wave operator ̂Ω (for n ∈ Z + ). In thiscase, 1 ≤ ξ ≤ min (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=1min (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 single-electron states of the set I n (α ¯β) for all n ∈ Z + .Proof. The proof is implemented making 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 following 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)avoiding 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 making the four-pair contractions in { ̂P ̂V 2̂Ωn ̂P }4 . In


2 Partitioning 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 ofwave operator ̂Ω must include at least n−2 creation and n−2 annihilation operators, designatingthe valence orbitals (item i) of Lemma 2.3.9). Possible sets I n (α ¯β) of single-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 determines 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 〉distinguishes between single-, 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 single-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 single-electron states. Thesums with primes denote the following operations


2 Partitioning 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 typesof single-particle orbitals will be designated, but not their values. For instance, in Eq. (2.69b),the term ∑ ′aI (1,6) e a v a †¯β′a †¯βω ev ¯β ¯β′ contains two two-particle effective matrix elements ω ev ¯β ¯β′ with2β = v and β = c (see Eq. (2.70b)). This implies that two single-particle orbitals ¯β and ¯β ′ areof whether valence (v) or core (c) type. That is, the values of orbitals are correspondingly ¯v, ¯v ′for β = v and ¯c, ¯c ′ for β = c.2.4 Concluding remarks and discussionFor the first time, the method to calculate matrix elements—based on coordinate transformationsor else the RCGC technique—has been developed (Sec. 2.2). The technique solves twomain tasks. First of all, it reduces the number of multiple integrals of many-electron angularparts up to a single one. Obtained single integral is calculated by introducing the reduced matrixelement of founded functions that particularly form the set of SO(3)–irreducible tensor operators(Sec. 2.2.2, Definition 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 distinctlineages, 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 find a finite set of solutions of eigenvalue equation of atomic Hamiltonian has been developed.Based on Feshbach’s partitioning 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 meaningful 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 perturbations.The foremost consequence followed by the suggested Fock space partitioning 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 amountof computations necessary to find, 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 infinite sum of n–body terms. Consequently, for practical computations, it has to be interrupted to a finite number


2 Partitioning of function space and basis transformation properties 32of terms by excluding the reminder term. The procedure, obviously, leads to approximate valuesof operator and the wave function as well. Meanwhile, there are also good news. As alreadypointed out, Theorem 2.3.12 allows us to reduce significantly the number of n–body parts ofwave 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 demonstrating that their calculations differ from existing experimentaldata for the nuclear charges ranging from 4 to 100 at the level of 50cm −1 for triplet states and500cm −1 for singlet states. These results obtained by using 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–Schrödinger 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 find the kth–order n–body terms of wave operator, it is sufficient to express them byEq. (2.65) replacing 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 corresponding 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 remainsirrelevant. That is, the difference is in the ω elements only. To find the irreducible tensorform of effective operators such that both – CC and RSPT – approaches could be applicablesimply replacing 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 representationof expansion terms, followed by Goldstone [23] and later by Lindgren et. al. [34].However, recent works in the higher-order perturbation theory [36, 42, 74, 75] demonstrate, inprincipal, the inefficiency of such representation, though it is beautiful to behold and efficientin many individual cases. The motivation to develop an algebraic technique suitable for bothMBPT approaches is encouraged by the features of nowadays symbolic programming tools aswell. The algebraic method that solves the tasks studied in the present section will be developedin 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 contributionsof single, double and triple excitations are most valuable for energy corrections observedin 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 obtained making 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 originated. 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 contains 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 into 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 ofoperator string 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 following 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 string Ôl represents Ôl/2 on H l if ∑ lk=1 β k = ∑ lk=1 µ k. Thatis, the sum of quasispin basis indices 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 ), concentrating mainly 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 string Ôl, it is convenient to introduce the l–number thatcontains numerals 1 and 2 only.3.1.1 Definition. The l–number is a number containing l 2 numerals whose values are 1 and 2only.It follows from Definition 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 obtainedfrom 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 insufficient since both 3–numbers belong to the same class ofS 3 . Extra characteristics are necessary for a unique labelling of l–numbers. The numbers 12and 21 differ by the ordering 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 distinct 2–tuples [[12]] and [[21]].3.1.2 Proposition. For a fixed l ∈ Z + , the l–numbers form a set E l with cardinality#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].Proof. The proof is a simple combinatorial task to find 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 2in 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 determined. 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 distinguish 1 ′ from 1. By doing 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)). Obtained number containsh 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. Find 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. Finally, pick out those l ′ 2(κ ′ )–tuples which aretransformed into the l 2 (κ)–tuple if reversing 1 ′ to 2 back again.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 contains 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 . Obtained 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 obtained from 111 ′ by making the sum 2 = 1 + 1 ′ for κ ′ = 1and the sum 2 = 1 + 1 for κ ′ = 2. The second sum is excluded since in this case—reversing1 ′ to 2 back again—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 using the semijoin notation ⋉.


3 Irreducible tensor operator techniques in atomic spectroscopy 35The algorithm considered above is easy to apply to the l–length string Ôl for the classificationof angular reduction schemes if replacing 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 meaning 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 determined. To perform a task, the following stepsshould be accomplished.I. For a given l, make the set E l of l–numbers (Definition 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 intothe l 2 (κ)–tuple when reversing 1 ′ to 2 back again. 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 of2restriction, pick out the tuples that transform to the initial one by making 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 off determined 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 semijoin notation obeys its original meaning: the reduction scheme T [λ]eκof Ôl consists of only those reduced Kronecker products which are determined in thereduction scheme T [λ′ ]κof ′ Ôl ′. The numeration of index ˜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 according 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 schemesDistinct 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 widespreadin 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 linked to another one T [λ′ ]κuniquely. To find coefficients thattransform one scheme into another, is the main 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 passing 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 determine 42coefficients that relate one separate scheme with all the rest, including 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 indices ζ ∈ Γ ξ = {ζ 1 , ζ 2 , ζ 3 , ζ 4 } that label intermediate 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 Definition. 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 indices η are absent in T κ[λ] . Υ ξ is a set of indices ofirreps in T κ [λ] . Particularly, Υ 14 ≡ Υ. Eq. (3.9) indicates that there exists an inverse 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 representationsin 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 following 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 exploiting the angular momentum technique basedon whether diagrammatic representation followed by Refs. [10, 12] or algebraic manipulations.Here, the algebraic approach actualised by a symbolic programming package NCoperators [78](see also Appendix D) is a preferable one. Obtained expressions are listed in Appendix A.


3 Irreducible tensor operator techniques in atomic spectroscopy 38Definition 3.1.3 indicates that the coefficients ɛ ξ are basis-independent. 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 kind (see Eqs. (A.12a), (A.34a),(A.37a), (A.40a)) are determined according to Eq. (19.3) in Ref. [10, Sec. 4-19, p. 102]. Thedefinition of 15j–symbol of the second kind (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 building 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 findthatW 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 following 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-coupling) is generated by W 1λ kk or Wλ kk depending 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 insufficient. To accomplishthe study to its final 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 within 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 indices of irreps α i enumerate operators in a l–length string 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, again, a motive to concentrate on the case l = 6only.The symmetry group S 6 contains 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) indicates 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 distinct angular reduction schemes (T κ [λ] and (̂π ij T ) [λ′ ]κ) with the different ordering ofirreps within them. The relation is realised through basis coefficients ′and the recoupling coefficientsdefined in Eq. (3.21). Eq. (3.22) is easy to generalise for any permutation (ij)(kl) . . . (pq)by repeating the procedure several times.Eq. (3.22) is more preferable if rewritten as follows.3.1.5 Definition. 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)Proof. 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 40According to Corollary 3.1.4 and Definition 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. Replacing ξ with ξ ′ and recalling that E ′ ξξ ′ = E ξ ′ ξ, Eq.(3.26) is satisfied. The indices of summation are easily derived subtracting the sets of indicesof 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 ΥProof. Eq. (3.27) directly follows from the fact that p ij ◦ p ij = id [2 T 2 1 2 ]. The indices of12summation are easily found recalling that (see Definition 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 introducing 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 following 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 within them. The same argument foldsfor operators ̂X.The final task to accomplish the classification of angular reduction schemes of irreducibletensor operators Ôα ([λ]κ) for l = 6, is to find the transformation coefficients E ijξ ′ ξ(see Definition3.1.5), as all other coefficients Eξ π ′ ξ arising duo to permutations of ̂π = ̂π ij . . . ̂π kl̂π pq arefound by applying Eq. (3.25) for several times. To find E ijξ ′ ξ , recoupling coefficients ε ij are tobe established. By Eq. (3.19), there are 15 recoupling coefficients, each of which is caused bythe transposition (ij). The realisation of an assignment by using the traditional method, that is,to recouple Kronecker products of irreps (particularly, angular momenta) of given scheme, isinefficient 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: find 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 minimal 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 off any suitable map τ ξ , τ −1ξor p ′ kl ((kl) ≠ (ij)). Eq. (3.30) indicates tofind 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. Finally, obtained commutative diagram is rewritten in algebraic form by using Eq.(3.29).The simplest to find are recoupling 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 within 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). Finally,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 mapping 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 find commutative diagrams for the rest of ε ij coefficients with indicesj = 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 following 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 combinatorial 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 corresponding 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 obtained 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 pointto 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 instance,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 instance, φ i ◦ φ j = φ j ◦ φ i , then it ought tobe possible to construct the classes C i that contain equivalent elements φ i = φ −1j ◦ φ i ◦ φ j withall possible j including j = i. Such classification is still to be clarified.3.1.4 Equivalent permutationsIn the present section, the following 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 contains a set of equal irreducible representations α s , s = 1, 2, . . . , t < l, thenthere exists a permutation representation ̂π min 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 minξ ′ ξ(3.45)is satisfied, though the map τ ξ ◦ p πmin ◦ τ −1ξ: T [λ′ ]′ κ−→ (̂π ′min T ) [λ]κ does not exist.An incitement to find the smallest possible length ̂π min associated to ̂π is caused by aprospect to reduce the number of intermediate irreps that appear due to transformations: theless number of transpositions, the less number of indices of summation.To avoid further misleading, 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 since (−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-coupling and (−1) 2λ i= (−1) 2j i= −1(j i = 1 /2, 3 /2, . . .) for jj-coupling.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 transposing 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 indices causes the appearance of 6j–symbols. However, it is unimportantin 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 meaning 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 coincide 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)), recalling 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 pointed 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 plain 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 making the transformations so that only the irreps α s and α s ′ within 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 schemecontains a set of several irreps α s that are equal, then a minimal cardinality of such set isT [λ]κ


3 Irreducible tensor operator techniques in atomic spectroscopy 473.2.1 Remark. Despite of different representations in distinct 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 coincide. Consequently, the choiceof 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 generating large sets ofmatrix 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 single-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 single-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 operatorrepresenting some physical (Coulomb, Breit) interaction, 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 indicates the presence of antisymmetrictwo-particle matrix representation ˜g αβ ¯µ¯ν . For the symmetric and self-adjoint interactionoperator 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 find 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 acting on the irreducible tensor space H Λ willbe considered. Their correspondence to operators on H q is associated by using the data in Tab.6 and by applying 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 find matrix elements of Ĝ[ω] on the basis|Γ i Π i Λ i M i 〉 which is a linear combination of CSFs (see Eq. (2.2)). That is, a matrix representationof 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 find the matrix element ofĜ Λ means to find 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 inRefs. [46, 50]. Consequently, to accomplish the formation of ĜΛ , the basis-independent quantitiesΩ αβ ¯µ¯ν (Λ 1 Λ 2 Λ) are to be found. These quantities are related to the origin of interactionoperator (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. Finally,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 interactions 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 defined much the same as ĜΛ for Ĝ[ω] (see Eq.(3.56)). Indeed, if g αβ ¯µ¯ν is calculated on the basis of single-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-independent 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 interaction operator g 12 .Example (Coulomb interaction). Assume that a single-electron eigenstate |α〉 is a 4–spinor 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–spinors |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 integrals R k are defined in Ref. [52, Sec. 19.3, p. 232, Eq. (19.72)]. The primes oversingle-electron states designate the corresponding small component g of relativistic radial wavefunction.Example (Coulomb interaction #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 coordinates ̂x 1 , ̂x 2 of both electrons. The spherical functions D L ±Mκ(Ω) are definedbyD 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 find the matrix representation z of 1/r 12 , it is more convenient to introduce 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 ( )∂2sin 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 using 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 integral∫∫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 interaction). One of the components of Breit operator is the so-called magneticinteraction [− ( 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 interaction can be found in Ref. [84, Sec. 2-5, p. 67,Eq. (5.86a)]. Then the matrix representation z on the 4–spinors (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 (Retarding interaction). The second (and the last) component of relativistic Breit operatoris the so-called interaction 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 defined 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 integral P (ij)kk(αβ, ¯µ ′¯ν ′ ) ) , (3.76)(αβ, ¯µ¯ν) is considered in the same way as R k (αβ, ¯µ¯ν) by replacingr 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 usingEq. (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 point 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 information relatedto the inner 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 indicates that the irreps Λ ′ 1, Λ ′ 2—despite of a resultant irrepΛ—designate intermediate irreps of g Λ that is found by applying a usual technique of separationof variables (coordinates). 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) involves 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 againstwhen studying 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-coupling, 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 accordingly 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 expresslydefined and studied in Ref. [50, Sec. 6, p. 46, Eq. (6.20)]. The b coefficients in H q space werefirst originated 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 inMBPT [72, 73]. The present technique is considered in Sec. 4.The efficiency of calculation of matrix elements—as a principal 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 ofatom. 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 preparationsof ĜΛ that acts on l = 1, 2, 3, 4 electron shells are displayed in Ref. [14, Tab. 1], whereboth schemes are mixed depending 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 following 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 contains a product of two 6j–symbols; in the secondapproach [14, Eq. (52)], the angular coefficient contains 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 off N configuration. Later, Leavitt et. al. [49, 90, 91] comprehensively considered effective threeparticleoperators acting within 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 acting 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-independent Ω αβζ ¯µ¯ν ¯η (Λ ′ 1Λ ′ 2Λ 3 Λ ′ ) is defined 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 find matrix elements of the three-particle operator̂L is disintegrated into two distinct tasks: the computation of matrix elements of the irreducibletensor operators ÔΛ ([2 2 1 2 ]12) and the establishment of projection-independent 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 approximationof MBPT, the quantities Ω αβζ ¯µ¯ν ¯η are found in Sec. 4 (also, see Appendix C).Conversely, the first task—determination of matrix elements—is independent of the dynamicsof system under consideration and thus it can be solved in general.In Sec. 3.1, the classification of angular reduction schemes of operator string Ôl, includingl = 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). Finally, the connection between irreducible tensor operators Ôα ([λ]κ) is associated bythe entries of transformation matrices (Corollary 3.1.4, Definition 3.1.5) E and E ij . These resultsare sufficient to find 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 ofthe irreducible tensor operators Ôα ([λ ′ ]κ ′ ) with λ ≠ λ ′ , κ ≠ κ ′ are found instantly 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 recoupling (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 final result, these operatorsare associated to ÔΛ ([λ]κ) acting on H Λ by using 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-coupling (Λ ≡ LS) and λ ≡ j for jj-coupling(Λ ≡ J). The quasispin number Q = 1 /2 (λ − v + 1 /2); the basis index M Q = 1 /2 (N − λ − 1 /2).Eq. (3.84) indicates that it is a l = 1–shell case, where the irreps within 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 nothing 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〉 ofH Q × H Λ . It is understood that a particular single-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 find 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.Assuming that the numbers x ∈ {i, j, k, l, p, q} designate the indices of irreps ς x withinÔ ς ([2 2 1 2 ]12) and at the same time they mark the xth electron shell, make the following convenientnotations. (In contrast, the index i = 1, 2, . . . , 6 of irrep α i labels the ith operator a α iwithin Ôς ([2 2 1 2 ]12).)3.2.4 Definition. 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 multiplicitiesof 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 Definition. 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 ) defines the set of operators Ôς ([2 2 1 2 ]12)associated to the same angular reduction scheme T [22 1 2 ]12 but with distinct labelling 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 Definition 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 theirdefinitions.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) coincides with Eq. (3.21). Ingeneral, the permutation π must be expanded into the product of 2–cycles (transpositions) inorder to apply Eq. (3.21) for several times. However, a general solution to count the number ofways in which a given permutation π can be factored into a given number of transpositions (ij)is absent so far. In algebraic combinatorics, authors usually refer to the so-called Hurwitz’s formulafor the number of minimal transitive factorisations. In a particular case, this formula yieldsthat the number of factorisations of a full cycle in S l into 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 following 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)Proof. To prove the theorem, it suffices to demonstrate that the commutator [̂π, ̂Π −1̂˜π] = 0,assuming that ̂Π −1 ̂Π = 16 . Indeed, if the latter commutator equals to zero, then, by passing toEq. (3.92) and the definition of ̂Π, it is true that ̂Π̂π ̂Π −1̂˜π = ̂π ′ = ̂˜π̂π and thus p π ′ = p eπ ◦ p π .Find [̂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 realising 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 realising 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 .Proof. By Definition 3.2.4, the proof 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 indicates 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 classificationof all the rest operators that belong to the corresponding derived class is performed immediatelyby using 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 ariseduring 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 doingso is simple. If the operator ̂L expanded by using Eq. (3.81) belongs to the parent classX l (∆ 1 , ∆ 2 , . . . , ∆ l ), then its adjoint operator ̂L † belongs to the dual class Xl ∗(∆ 1, ∆ 2 , . . . , ∆ l )which is nothing else but X l (−∆ 1 , −∆ 2 , . . . , −∆ l ) (Definition 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 recalling that the matrix element on the infinite-dimensional N–electron Hilbertspace H is defined 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 using Definition 3.2.4. This is easily done utilising Tabs. 36-39 inAppendix B. For a particular two-shell case, the identification of operators is trivial: there areonly parent and derived classes; the derived classes coincide 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. Find the associated class. By Definition3.2.4, 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 classin 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. Since 〈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 obtained makinguse 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 displayedassuming 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 xwithin ̂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 contains a given irreducibletensor operator. The indices 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 making use of Tab. 6 and Eq. (3.51) which fits the case N x = 6 (see also Eq.(3.84)).3.3 Summary and concluding remarksTheoretical atomic physics deals with various irreducible tensor operators that allow us to accountfor the contribution of atomic as well as effective interactions in a simplified form: theirreducible tensor operators attach the symmetry properties of atom. In mathematical formulation,the contributions are evaluated by calculating matrix elements – the real scalar products onthe many-electron Hilbert spaces. In practice, the calculation of matrix elements on the basisof 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 examined very well. In contrast, the study of triple oreven higher excitations is already troublesome and an uncertain one so far. Due to a complexity,it has become standard in many cases of MBPT to account for the contributions of singleor 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 inevitable.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 operatorstring Ôl (see Eq. (3.1)) for any integer l. The classification is performed by using: the l–numbers (Definition 3.1.1), the S l –irreducible representations [λ], the l 2 –tuples. The idea—inits most general form—is simple: every angular reduction scheme is characterised by the irrep[λ], thus making 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 using 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 using 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 within a given operator Ôα ([λ]κ)associated to the angular reduction scheme T κ[λ] are studied for l = 6. The symmetry groupS 6 contains 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 ordering of irreps is confined to the task to find the transformationcoefficients E ijξ ′ ξ (Definition 3.1.5) formed from the so-called recoupling coefficients ε ij (see Eq.(3.21)). These coefficients relate operators constructed by using the ordering (α 1 , α 2 , . . . , α 6 )with the operators constructed by using the ordering (α π(1) , α π(2) , . . . , α π(6) ), where π ∈ S 6 isa transposition. The expressions of ε ij are found by making use of the so-called commutativediagrams (see Eq. (3.30)). The originated method based on Eq. (3.29) allows us to find therecoupling coefficients by making the least number of transformations that are required by thepermutation representation ̂π. Consequently, it reduces the number of intermediate irreps.Sec. 3.1.4 is a very important addition to the method presented in Sec. 3.1.3. The main ideais listed in Theorem 3.1.8 which is, again, a statement that allows us to reduce the number ofindices 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 operatorsacting 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 originated by Kaniauskas [85] is the most typical one. However, in Sec. 4,the advantage of a less common z-scheme will be demonstrated.Finally, 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 into the three types of classes: parent, dual, derived(Definition 3.2.4, Definition 3.2.5, Corollary 3.2.8). The identification of operators that belongto a specified class is performed making 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 find the expression ofmatrix element (see Eq. (3.93)) for the sole operator – the matrix elements of other operatorsare found instantly.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 confinement on the many-electron Hilbert space has been obtained(to compare with, see Eqs. (2.55), (2.59)) leading 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 operatorsof both the wave operator and the effective Hamiltonian in a systematic way. However, stillone more problem remains unsolved. This is the n–body effective matrix element ω n (see Eq.(2.65)). As already pointed 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 main 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 bring the purpose to a successful end, two tasks are to be solved: the generation ofexpansion 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 Definition 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) replacing 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=1min(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 into the sum ofirreducible tensor operators. The confinement of the space the operator acts on is the secondtask to be solved. A common technique of expansion of N–electron model space P into 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] contains creation and transposed annihilation operators labelling the valence statesonly. As usually, Γ denotes additional numbers that are necessary to characterise studied operators.h (3)mn;ξ(ΓΛ) denotes the projection-independent coefficient; thus it is the SU(2)–invariant.At this step, it should be pointed out the two principal 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) indicates that the solutions are proportional to thekth power of perturbation ̂V . For instance, ̂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)) contains the m–body matrix element v m (α ¯β). Consequently, ̂Ω (k) contains the productof k such elements. Moreover, every single term in ̂Ω (k) —obtained by using 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 interpretation 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 single element ω (k) . Thus ω (k) representsthe product of k matrix elements of ̂V m with energy denominators included. In other words,is the kth–order n–body effective matrix element that characterises the n–body part ofthe kth–order wave operator. In addition to the convenience of otherwise marked productof matrix elements v m , there is also an essential peculiarity: by Theorem 2.3.12, the termsof ̂Ω (k) are combined in groups (refer to Eq. (2.69)) related to the different types (core, valence,excited) of single-electron orbitals. That is, a number of Golstone diagrams drawnin ω (k) are characterised by the sole tensor structure and thus the problem of evaluation ofeach separate diagram is eliminated. 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)nin principal, the n–body effective matrix elements. Therefore, if replacing ω n(k) with ρ n , thetensor structure of terms remains 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 timeconsuming process. The nowadays software 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 offundamental constructions that need to be considered: all the rest terms are obtained varyingthe 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 representationsof a two-particle interaction operator g 12 , are defined by Eqs. (3.53)-(3.54). Theirexplicit expressions are found from Eqs. (3.63)-(3.64). For the typical interaction 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 into account. The single-particle matrix representation v α ¯βof the self-adjoint interaction 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 subspaceof the space the operator v i acts on. It is assumed that the single-particle Slater integral(s) isinvolved in f(τ i λ α λ ¯β). The number of such integrals 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 ofConstruction∑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)–invariant: 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 ρηטz(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 ofConstructionv α¯µ ω (1)β¯νSU(2)–invariant: 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 ofConstructionSU(2)–invariant: 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 considering the properties of the self-adjoint 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 index i distinguishes the interaction 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 single-electronenergy (refer to Eq. (2.65)). The operator ̂R replaces orbitals in energy denominators. 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 defined 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)αβRecalling 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 numberof terms.To distinguish the SU(2)–invariants by the summation parameters, the following 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 involved, 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 corresponding 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)–invariants. Also, it should be noted that for each two-particle matrixrepresentation with tilde, the corresponding 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 distinguishes the valence electron states. Since τ 1 ≠ 0, τ 2 ≠ 0, thesediagrams involve the interaction 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)–invariants. By using Tab. 12 and Eq.(4.11), it follows that the first two terms are easy to obtain with α = v, ¯β = ¯c and µ = e, v ′ .The corresponding SU(2)–invariants 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 corresponding SU(2)–invariant is S¯cv (τ 1 τ 2 τ) with α = ¯c, ¯β = v, µ = c. Finally,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 interesting 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)–invariant S v¯c (τ 1 τ 2 τ).The example indicates that the Goldstone diagrams are represented by the two types ofirreducible tensor operators: ÔΛ +M and ÔΛ −M . The analysis of all generated terms of ̂Ω (2) leadsto the following conclusions. The terms (diagrams) associated to the tensor structure ÔΛ −Mare: (i) the folded diagrams; (ii) some diagrams, obtained by contracting the core orbitals inWick’s series; (iii) the diagrams, acceded to the reflection about a horizontal axis of a numberof diagrams associated to the tensor structure ÔΛ +M .c


4 Applications to the third-order MBPT 64To distinguish the considered two types of terms, the superscripts ± over the matrix representationsare used so that for n = 1, 2, 3, 4, the SU(2)–invariants Ω (2)± of each n–body part of̂Ω (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)–invariants are also obtained by given Eqs. (4.17)-(4.20)for the effective matrix elements with the plus sign, the basis indices +M, +M 3 and eliminatedphase 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 instance, 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 thehandling 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) attaching the zero-valuedcontributions. As already pointed 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 single-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 ′′טz(0λ c λ c ′λ µ ′λ v ′′Λ 2 Λ 2 )Ω (2)+v ′′ vµ ′¯vc ′ c (Λ 1Λ 2 Λ 3 Λ)טz(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 kind 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 (continued)(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 ′′טz(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 bymaking the following 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 index drawn in h (3)+mn;ξis replaced by theopposite sign index 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 bring 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 providing an opportunity to construct their matrix representations efficiently


4 Applications to the third-order MBPT 67– the task is confined to the calculation of matrix elements of irreducible tensor operatorsthat are written apart from the projection-independent parts. These particular angularcoefficients contain Ω (2)± multiplied by the 3nj–symbols.2. The determination of SU(2)–invariants of ̂Ω (2) makes it possible to settle down a numberof terms (Goldstone diagrams) associated to the sole tensor structure. As a result, thelatter approach permits to ascertain the contribution of n–particle (n = 1, 2, 3, 4) effectsin CC approximation. This is done by simply replacing Ω (2)± with Ω n found from Eqs.(4.17)-(4.20). In this case, replace the corresponding effective matrix element ω (2) withω, where ω denotes the valence singles, doubles, triples and quadruples amplitude.3. The terms of Ĥ (3) account for the interaction of atom with external field. This is drawnin the matrix representations of v i that represents the electric, magnetic, hyperfine, etc.interaction 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 Concluding 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 find the 13 non-equivalent SU(2)–invariants, as demonstrated in Tabs. 12-14. Due to the symmetry properties of matrix representations v α ¯β, g αβ ¯µ¯ν , these invariants areenough to evaluate all generated terms of ̂Ω (2) . As a result, the quantities Ω (2)± are determined(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;ξin termsof Ω (2)± (Tabs. 15-17). The key feature is that the coefficients Ω (2)± can be replaced while on aparticular case of interest, but the tensor structure of operators on P Λ remains 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) including the two-particle interactions 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 contributing 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 elementsof terms drawn in accordingly ĥ(3) 22;3, ĥ(3) 23;4 and ĥ(3) 21;2 if g 0 represents the Coulomb interaction. Forinstance, 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)pertained 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 including the two-particleinteractions g 0 only. For instance, 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 interactions 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. Theprincipal difference of formulation of MBPT between the present and the other works is thedifference between algebraic and diagrammatic realisation – all the rest of distinctive featuresfollow the present one. Particularly, one of the typical features is the so-called factorisationtheorem [37, Eq. (10)] which makes it possible to combine the diagrams with the same energydenominators. Consequently, the comparison of the amount of computed terms with the amountof computed diagrams is a very conditional one.As it follows from Tab. 11, Ĥ (3) also includes 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 single-particle interactions 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)–invariant coefficients Ω (2) are derived, the expansion coefficients h (3)mn;ξcan beobtained 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 corresponding 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 contains 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 beenoriginated. The key feature of proposed technique is the ability to reduce the N–electron angularintegrals into the sum of single integrals. The method is especially convenient for thecalculation of matrix elements of interaction 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 partitioningtechnique, the finite-dimensional many-electron model space has been constructed. Asa result, it has been determined that only a fixed number of types of the Hilbert space operatorswith respect to the single-electron states attach the non-zero effective operators on the givenmodel space. The result has a consequential meaning in applications of atomic many-bodyperturbation theory, as it permits to reduce the number of expansion terms significantly.3. A systematic way of inquiry 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 string of any length l has beeninitiated. Based on the commutative diagrams that realise the mappings 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 distinct 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 distinct classes, and thus it suffices to find the matrix representation for the sole operator – thematrix representations for the other operators are found instantly by using the transformationcoefficients.5. The third-order many-body perturbation theory has been considered. To simplify the generationof expansion terms followed by the generalised Bloch equation, the symbolic programmingpackage NCoperators written on Mathematica has been produced. The angular reductionof generated terms has been performed making use of NCoperators too. The specifictechnique of reduction has been developed. The algorithm is based on the composed SU(2)–invariants which—owed to the symmetry properties of matrix representations of atomic interactionoperators—take into 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 replacementof the excitation amplitudes suitable for some special cases of ones interests. Obtainedsymbolic 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. Gintaras 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 acting 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 acting 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) contains 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 acting 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 acting 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 acting 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 acting 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 corresponding 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 corresponding permutation η from the setη ∈ {1 6 , (23), (12), (123), (132), (13)}.Finally, the notation 〈{ijk}{lpq}〉 is a union of both sets.


B The classification of three-particle operators acting 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 acting 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)–invariant part of the second-order wave operator 85CSU(2)–invariant 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 )טD ′ 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)–invariant 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 distinct 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)–invariant 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)–invariant 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 defined by the following 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 kind—is defined according 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 computationalsoftware program. To take all advantage of NCoperators, especially when manipulatingwith 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 Windows 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 into four main blocks:


D Symbolic computations with NCoperators 901. Second Quantisation Representation (SQR)2. Angular Momentum Theory (AMT)3. Rayleigh–Schrödinger Perturbation Theory (RSPT)4. Unstructured External Programming (UEP)The packages store Mathematica codes that are loaded into a Mathematica session by usingthe function «“NCoperators.m” or Get[“NCoperators.m”]. The NCoperatorsis designed so that each block can be brought into action separately. This is easily done withGet[“PackageName.m”]. All these functions (Get[]) are located in a single 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 contain a huge number of functions, each of them being useful for a specialcase of interest. In the present overview, only a few of them will be discussed.In theoretical atomic physics, a typical task is to find the angular coefficients that relate twodistinct ( momenta coupling schemes. Consider, for example, the momenta recoupling 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 23Making use of NCoperators, the algorithm to obtain the latter formula is displayed in Fig. 1.Fig. 1: A computation of recoupling coefficient ( j 1 j 2 (j 12 )j 3 j ∣ ∣ j2 j 3 (j 23 )j 1 j ) with NCoperatorsOut[25] contains two Clebsch–Gordan coefficients labelled by CGcoeff[] – the standardMathematica function ClebschGordan[] is insufficient for symbolic manipulations.The function Recoupling[] initiates the momenta recoupling followed by Refs. [10–12].SymbolSum[] plays a role of a standard Mathematica function Sum[] adapted to symboliccomputations. The function Make[] initiates the momenta recoupling within a given Clebsch–Gordan coefficient. The simplification of considered sum is performed by SumSimplify.Out[31] is printed in a standard Unix output form specified by the font family «cmmi10». Obtainedformula is easy to convert to a L A TEX language by making use of Mathematica function


D Symbolic computations with NCoperators 91Export[“DirectoryName/FileName.tex”,%31]. A brief description of all functionsis lited by using Definition[function] or simply ?function. The example ispresented in Fig. 2.Fig. 2: The usage of Definition[] in NCoperatorsIn Fig. 2, the function TAntiCommutator1[] along with TAntiCommutator2[],TAntiCommutator3[] and TAntiCommutator4[] initiates 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 theindices m = 2, n = 1, ξ = 2 (refer to Eqs. (2.42), (4.3)); NonCommutOpMulti[n,α,β] isnothing 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 demonstratedin Fig. 4, where the projection-independentquantity [ ]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 involves functions suitable for the symbolic manipulations observed in thestationary Rayleigh–Schrödinger perturbation theory. In principal, the RSPT block applies thepreviously summarised blocks with some specific functions particular with the MBPT. Withoutgoing into 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 single-particle matrix element v α ¯β is defined in Eq. (4.5). Recall that only the typesof single-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 single-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 into 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 using theexpression for SU(2)–invariant 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 invariant has been obtained also makingµ ¯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 contains 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 determine the single-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〉.Finally, recalling 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 expressionin the first row of Tab. 15 (the case m = n = ξ = 1) is obtained by using the result in Fig. 1.A much more complicated task is to find Ω (2)+ (Λ). To generate the single-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 obtainedfrom 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 defined 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 denominator D a (f ¯f) is found from Eq. (2.65); the vectors x ¯fk (x † f k) ∀k = 1, 2, . . . , aof the single-particle Hilbert space are identified to be the single-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 contains 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 corresponding SU(2)–invariants are selected(recall the distinct notations for distinct intermediate irreps defined by Eqs. (4.11)-(4.16)). Then split the solution for ω (2)α ¯βinto the sum of ω(2)+α ¯βand ω (2)−α ¯β. Make use of Eq.(4.17a). Finally, 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 ofoperators ĥ(3) mn;ξelements associated tofor all of them:. It is by no means obvious that the procedure to find 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 = :assuming 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 operating 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 main idea is to convert obtained symbolic preparation into the Ccode for a more rapid calculation of the quantities under consideration.External programming is a way of communication between Mathematica interface and someother (external) programs (such as C). Differently from the structured programming actualisedby using MathLink, the unstructured programming does not require any other external «tunnels»except for the user’s own terminal.To make the UEP block operate, theMathematica header file mdefs.h must be locatedin the directory, where all system headerfiles are situated. In most Unix systems,the directory is usr/include/. The headeris supplemented with the C-based functionsthat are necessary for the atomic calculations:ClebschGordanC[], SixJSymbolC[],NineJSymbolC[], etc. The MathLink librarieslibML32i3.a, libML32i.so are substitutedin /usr/lib or /usr/local/lib.The example of the usage of C-basedFig. 8: Example of an application of the UEP block functions within Mathematica interface 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 97Timing[] clearly demonstrates the main benefit of such type of programming 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 ofexpansion terms. The example is demonstrated in Fig. 9. A newbie equipment of NCoperatorsis actualised by using 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 inefficient 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 into 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.


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