One-body operators

In this appendix, explicit functionals of the density representing the different relativistic corrections are found by performing expectation values taking Slater determinants of local plane waves as in the standard Fermi gas model. For a one-body operator we use ... 

Making use of the property of a semi-definite one-body operator, say, L ... 

The superscript 2 is added to emphasize that these interactions are two-body. That is, these operators are orthogonal to all scalar and one-body operators with respect to the trace scalar product. 

In analytical investigations it is often desirable to leave the particle number free and consider operators that fix only the parity, but in applications to electronic structure theory one deals with fixed particle number and one may restrict A to have a definite action on the particle number N, so that A+A is particle conserving. There are then two cases for the one-body operator A consideration of A = with undetermined coefficients gives rise to the... 

In principle, we already have in our disposal the SRPA formalism for description of the collective motion in space of collective variables. Indeed, Eqs. (11), (12), (18), and (19) deliver one-body operators and strength matrices we need for the separable expansion of the two-body interaction. The number K of the collective variables qk(t) and pk(t) and separable terms depends on how precisely we want to describe the collecive motion (see discussion in Section 4). For K = 1, SRPA converges to the sum rule approach with a one collective mode [6]. For K > 1, we have a system of K coupled oscillators and SRPA is reduced to the local RPA [6,24] suitable for a rough description of several modes and or main gross-structure efects. However, SRPA is still not ready to describe the Landau fragmentation. For this aim, we should consider the detailed Iph space. This will be done in the next subsection. 

For a complete one-electron basis, the canonical commutators become proportional to the number operators. For finite basis sets, the canonical commutator becomes a general one-body operator. 

The a definition of Harris [134] represents a special case of (6.8) because it arises from a van Vleck formalism. Similar considerations apply to that of Westhaus et al. [103] since, as mentioned in Section V.C, they use mappings which are not canonical, but which can be written as in Eq. (5.17). Westhaus et al. introduce approximate mapping operators to obtain the one- and two-body parts of their a when A is a one-body operator. Finally, Johnson and Barranger [121,122] suggest an a definition of the form A for any one of the norm-preserving mappings possible from their formalism (see Section V.C). 

For the MP MBPT we choose for the unperturbed Hamiltonian Ho the following diagonal one-body operator... 

Let us group the operators with a pair of uncontrar ted operators to define a new one-body operator / ... 

The one-body operators appearing in H - can be reduced to simple radial integrals by inserting the definition (142) in the relevant formulae. Thus, from (100) we find that the elements of the Gram matrices reduce to radial integrals of the form ... 

As we have seen above, the well-known Wigner-Eckart theorem represents a powerful tool for the evaluation of MEs. Thus, the MEs of any U(2 ) tensor that may be decomposed into the irreducible tensors of U( ) and SU(2) can be expressed as a product of three factors (1) the RME that depends on the relevant tensors and irreps of U(2n), U( ), and SU(2), (2) the U(n) C-G coefficient, and (3) the SU(2) C-G coefficient. In a multiplicity-free case for U(n) irreps, the U(n) C-G coefficients can be further factorized into simple products of isoscalar factors, yielding the ME segmentation formalism for spin-dependent operators. We shall see that this is exactly the case for one-body operators (36). 

Finally, let us emphasize that in contrast to a spin-independent case, the U(n) irreps in the U(2n) D U(n) SU(2) bases are generally changed or shifted by U(2 ) operators. Thus, one-body spin-dependent operators can change the total spin [or U(n) irrep label] by A5 = 0, 1 (or Ah = 0, 2) and two-body ones by A5 = 0, 1, 2 (or Ah = 0, 2, 4) and thus take us out of the U(n) framework. Nonetheless, as we have indicated above, the U( )-adapted creation (C ) and annihilation (C) type operators— that represent very useful tensors serving as fundamental building blocks for various U(n) tensors—are also useful in the spin-dependent U(2n) case. Indeed, since xj (X, ) are vector (contragredient vector) operators when acting on the irrep modules of U(n), their MEs in the U(n) basis are clearly related to those of (Cf) operators. In view of this fact, the MEs of one-body operators must be related to those of cfCj. The latter were carefully examined in [36] and briefly reviewed above. 

Table 1 Coefficients required for the evaluation of MEs of spin-dependent, one-body operators, Eqs. (61), (62), and (65), for three possible shifts, as functions of the SU(2) labels...

The segmentation of MEs of spin-dependent one-body operators is very similar to that for spin-independent two-body operators. This is a very useful fact in view of an actual implementation of this formalism, since it enables an evaluation of MEs of spin-dependent one-body operators by exploiting presently available UGA or GUGA codes. We now compare both formalisms in greater detail. 

We have shown that one-electron spin-dependent terms, Eq. (36), in the electronic Hamiltonian, Eq. (33), may be efficiently handled in much the same way as the standard spin-independent two-electron (i.e., Coulomb) terms. Indeed, as clearly implied by Eqs. (70), (73), and (76), the MEs of spin-dependent one-body operators in the U(2 ) D U(n)(8> SU(2) basis may be evaluated as MEs of spin-independent two-body operators in a standard U( -1-1) electronic G-T basis (see also [37, 39]). Since the MEs of generator products within the spin-independent UGA approach are well known, the above presented development should facilitate the implementation of the spin-dependent UGA formalism. This opens a possible avenue enabling us to handle spin-dependent MBPT terms via a simple modification of the existing UGA and GUGA codes, similarly as done by Yabushita et al. [37]. We emphasize that all the required segment values may be found in [36], and a few additional ones that are specific to a spin-dependent case are given in Tables 1 and 2. 

Then a one-body operator may change 0 into cpj = a aicp,. A second-order correction may be viewed as... 

The are one-body operators, i.e. they act only on partons of type j and the partons scatter under the action of in a point-like fashion. 

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