Breit Interaction Matrix Element

Within the second quantization formahsm, Hmc is expressed as in Eq. (21), where F is the second-quantized form of Ad uik are the SCF potential matrix elements, and hyu are the Breit interaction matrix elements. 

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]... 

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 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... 

Then the 5-matrix element for the first-order Breit interaction can be written in the form ... 

While there have been a number of papers written on the basic structure problem since Ref. [40] appeared, [55], none of them go qualitatively further than the calculations of that work. Curiously, an experimental paper, [5], claims to have reduced the theoretical error through comparison with new measurements of transition matrix elements. It is the author s opinion, however, that this essentially semiempirical approach is dangerous, and prefers to leave the 1 percent error estimate unchanged at present. However, there are three places in which considerable activity has taken place that we address in turn, the vector polarizability / , the Breit interaction, and radiative corrections. 

Nakajima and Kato used the CASSCF(8,6)/DZP approximation to examine intersystem crossing from the state of glyoxal (Structure 3.4) induced by collisions with argon atoms. They evaluated the and interaction potentials and the spin-orbit coupling matrix elements between these two states at each geometry by using the full Breit—Pauli A semi-... 

In the direct matrix element babab, where fco = 0, the frequency-dependent Breit interaction reduces to its limiting static form ... 

As shown in detail in Ref. [31], the matrix elements of the first term in (86), the unretarded Breit interaction. 

Matrix elements of the frequency-dependent Breit interaction 612( 0) are somewhat more complicated they can be evaluated using the formulas given above with the following... 

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). 

Matrix elements of two-particle operators arise primarily either from the Coulomb or the Breit (or Gaunt) interaction. 

Since the difference e ween bx2(k) and the Breit interaction, hi2(0)> is of order a Z, which is a small number in many applications, it often suffices to replace bx2(h) by bx2(0) when computing exchange or off-diagonal.matrix elements. 

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