Breit matrix elements

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) ... 

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 integral for the second coordinate set is of the same form. Finally, we obtain for the general Breit matrix element over one-electron spinors [201]... 

The matrices J, K and B are direct, exchange and Breit interaction matrices, of which only the first is block diagonal. Their matrix elements are linear combinations of interaction integrals over G-spinors. 

The corrections of order (Za) are just the first order matrix elements of the Breit interaction between the Coulomb-Schrodinger eigenfunctions of the Coulomb Hamiltonian Hq in (3.1). The mass dependence of the Breit interaction is known exactly, and the same is true for its matrix elements. These matrix elements and, hence, the exact mass dependence of the contributions to the energy levels of order (Za), beyond the reduced mass, were first obtained a long time ago [2]... 

In the Breit Hamiltonian in (3.2) we have omitted all terms which depend on spin variables of the heavy particle. As a result the corrections to the energy levels in (3.4) do not depend on the relative orientation of the spins of the heavy and light particles (in other words they do not describe hyperfine splitting). Moreover, almost all contributions in (3.4) are independent not only of the mutual orientation of spins of the heavy and light particles but also of the magnitude of the spin of the heavy particle. The only exception is the small contribution proportional to the term Sio, called the Darwin-Foldy contribution. This term arises in the matrix element of the Breit Hamiltonian only for the spin one-half nucleus and should be omitted for spinless or spin one nuclei. This contribution combines naturally with the nuclear size correction, and we postpone its discussion to Subsect. 6.1.2 dealing with the nuclear size contribution. 

As discovered by Breit [1] an exact calculation of this matrix element is really no more difficult than calculation of the leading binding correction of relative order After straightforward calculation one obtains a closed... 

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 evaluation of matrix elements of the Breit interaction requires the calculation of even more difficult singular integrals, and this remained an unsolved problem until the recent development of new algorithms [70,71]. With these results in hand, it is now possible to include all the relativistic and QED terms as in the helium case. The resulting theoretical ionization energy for the ground state of 0.19814209(2) a.u. is larger than the experimental value by... 

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. 

The matrix form of the atomic Dirac-Hartree-Fock (DHF) equations was presented by Kim [37,95], who used a basis set of modified radial Slater-type functions, without the benefit of a balancing presciption for the small component set. A further presentation of the atomic equations was made by Kagawa [96], who generalized Kim s work to open shells and discussed matrix element evaluation. An extension to include the low-ffequency form of the Breit interaction self-consistently in an S-spinor basis was presented by Quiney [97], who demonstrated that this did not produce variational collapse. Our presentation of the DHFB method is based on [97-99]. 

Similarly, multi-centre matrix elements of the Breit interaction (91) are given by... 

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