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

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)