Matrix elements Breit operator

Table 18.1. Matrix elements of the Breit-Bethe spin-orbit and spin-spin operators, Eqs. (18.13) and (18.14).

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

The creation/annihilation operators aj /a, denote the one-particle operators which diagonalize the Hamiltonian Hen. The summation indices i, j, k, l denote the usual set of one-electron quantum numbers and run over positive-energy states only. The quantities Vjju are two-electron Coulomb matrix elements and the quantities biju denote two-electron Breit matrix elements, respectively. We specify their static limit (neglecting any frequency dependence) ... 

Since the operators f and f2 occur only at the level of the calculation of the spatial spin-orbit integrals over atomic orbitals, Breit-Pauli spin-orbit coupling operators and DKH spin-orbit coupling operators can be discussed on the same footing as far as their matrix elements between multi-electron wave functions are concerned. These terms constitute, by definition, the spin-orbit interaction part of the operator H+ (Hess etal. 1995). The spin-independent terms characteristic of relativistic kinematics define the scalar relativistic part of the operator, and terms with more than one cr matrix (not considered here) contribute to spin-spin coupling phenomena. 

Judd, Crosswhite, and Crosswhite (10) added relativistic effects to the scheme by considering the Breit operator and thereby produced effective spin-spin and spin-other-orbit interaction Hamiltonians. The reduced matrix elements may be expressed as a linear combination of the Marvin integrals,... 

The decomposition mechanisms were studied by using B3LYP/6-31 l+G(d,p)//B3LYP/6-31+G(d).3 The SOC matrix elements between singlet and triplet states were estimated by using CASCI wave functions based on Boys localized orbitals with respect to the full Pauli-Breit SOC operator see ref 1 for details. 

Having defined our starting point, the second quantized no-pair Hamiltonian, we may now take a closer look at the relations between the matrix elements. For future convenience we will also change the notation of these matrix elements slightly. Due to hermiticity of the Dirac Hamiltonian and the Coulomb-Breit operator we have... 

It is understood that the matrix elements of this operator should be evaluated with the relativistic 4-component wave function. The approximation (163) is called also the low-frequency approximation, since it arises when energy differences tend to zero AE — 0. An expression for the Breit operator suitable for the evaluations with the two-component (nonrelativistic) wave functions follows when we expand also the relativistic wave functions using Eqs(24)-(26) ... 

The presence of two-electron operators in the Breit-Pauli and similar expressions for makes their use computationally quite demanding, because such operators have nonvanishing matrix elements even between Slater determinants that differ in two spin-orbital occupancies and because there are many two-electron integrals. This is especially true in studies of photochemical reaction paths, where information about spin-orbit coupling is needed at many geometries. Several simplifications have been quite popular. 

The (So H Tj matrix element is small, less than 0.08 cm using the full Breit—Pauli ifsoss qj. q q5 i using the one-electron operator with effective nuclear charge. ... 

Eq. (113). For frequency-dependent Breit interaction, these Breit matrix elements arc modified according to the recipe shown in Section 3.4. Furthermore, off-diagonal matrix elements are calculated with the frequency-symmetrized Breit operator shown in Eq. (87). 

The matrix elements of the spin-orbit coupling operator have been included in these works using empirically obtained or computed spin-orbit coupling constants for an effective one electron operator. The Breit-Pauli spin-orbit coupling operator (115) with all multi-center terms was employed for the first time by Kiyonaga, Morihashi and Kikuchi [125]. 

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