Two-electron contributions

The self-consistent treatment of the SO interaction is an important aspect of the relativistic simulation of atomic and molecular systems. It is known that the electron-electron contributions are mq ortant for a quantitative description of these relativistic effects [19,80-86]. The SO terms derived from the electron-electron interaction can be even more important than the corresponding SR terms [87], e.g. when spectroscopic properties are of interest. SR corrections have larger effects on other properties, e.g. binding energies. 

Although the DKS Hamiltonian is linear in Vgff = F c + Vh, the second-order DK Hamiltonian 9- (1 )- Hence, during the SCF iterations, 

A proper relativistic description of the Hartree contribution has to be based on the true four-component density of the electrons. The two tasks, construction 

(22) represents the DKnuc strategy where the nonrelativistic form of the Hartree potential is added a-posteriori to the DK Hamiltonian with the relativistic approximation restricted to the nuclear potential. In that case, not only the Hartree potential remains in its nonrelativistic form, but also the electron density in its two-component (SchrOdinger) form. This is the DK approach used most often in self-consistent two-component relativistic DF models [16,18,19]. As next approximation, one can apply the fpFW transformation U= Uq to the Hartree potential in analogy to Eq. (18), to yield the approxi- 

In the LCGTO-FF strategy [20,32] where electron density and Hartree potential are expanded with the help of an auxiliary basis set, Eq. (29), one may account for this picture change [19] in an efficient way. 

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If one has no other information available, one can carry out one iteration of the SCF process in which the two-electron contributions to F y are ignored (i.e., take F >v = < Xp I h I Xv >) and use the resultant solutions to Ev F y CVjj = Ev S >v Cv j as initial guesses for the CV J. Using only the one-electron part of the Hamiltonian to determine initial values for the LCAO-MO coefficients may seem like a rather severe step it is, and the resultant CVJ values are usually far from the converged values which the SCF process eventually produces. However, the initial CV J obtained in this manner have proper symmetries and nodal patterns because the one-electron part of the Hamiltonian has the same symmetry as the full Hamiltonian. 

The Breit-Pauli Hamiltonian with an external field contains all standard one- and two-electron contributions as well as the magnetic interaction of the electrons and their interactions with an external electromagnetic field. We may group the various contributions in the Breit-Pauli Hamiltonian according to one-and two-electron terms,... 

When we explicitly consider a- and / -spins in Eq. (99) and reorder the resulting terms according to one and two-electron contributions, we obtain double sums over the number of a- and the / -electrons, AT and N13, as well as over the mixed terms where one sum runs over № and the other over N13,... 

The two-electron contribution is derived in the same way from the second term in equation (A. 18). Consider an arbitrary pair of electrons, i and j, and two permutations, the do-nothing permutation (p — 0) and the specific permutation, Pp, which interchanges just the /th andy th electron. For the second permutation, p is odd and (—1) will yield a minus sign. Thus,... 

All other permutations give identically zero terms due to the orthogonality of the orbitals. The total two-electron contribution to the electronic energy is... 

From Eq. 5.100 the two-electron contribution to each Fock matrix element is... 

The nuclear recoil correction is the sum of the one-electron and two-electron contributions. The one-electron contribution is the sum of the expressions (15) for the a and b states. The two-electron contributions are defined by the diagrams shown in Figs. 4-6. A simple calculation of these diagrams yields... 

In lithiumlike ions, in addition to the one-electron contributions, we must evaluate the two-electron contributions. In the case of one electron over the (Is)2 shell the total two-electron contribution to the zeroth order in 1/Z is given by the expression... 

The two-electron contribution to Eq. (245) may be combined with the first term in Eq. (244) before transforming the effective densities to the contravariant AO representation. In this way transformations of differentiated AO integrals are completely avoided (Rice and Amos, 1985). In addition, the second term in Eq. (244) contains contributions from differentiated overlap matrices -(S(1), iF<0)) (0), which are easily calculated in the AO basis using the techniques described in Appendix E. The last contribution to Eq. (244) is easily calculated having transformed the Fock matrix [Pg.223]

Computation of the spin-orbit contribution to the electronic g-tensor shift can in principle be carried out using linear density functional response theory, however, one needs to introduce an efficient approximation of the two-electron spin-orbit operator, which formally can not be described in density functional theory. One way to solve this problem is to introduce the atomic mean-field (AMEI) approximation of the spin-orbit operator, which is well known for its accurate description of the spin-orbit interaction in molecules containing heavy atoms. Another two-electron operator appears in the first order diamagnetic two-electron contribution to the g-tensor shift, but in most molecules the contribution of this operator is negligible and can be safely omitted from actual calculations. These approximations have effectively resolved the DET dilemma of dealing with two-electron operators and have so allowed to take a practical approach to evaluate electronic g-tensors in DET. Conventionally, DET calculations of this kind are based on the unrestricted... 

For the 2pi/2-2s transition in lithium-like high-Z ions, the gauge-invariant sets of the screened vacuum-polarization corrections and of the screened-self-energy have been evaluated recently by Artemyev etal. (1999) and Yerokhin etal. (1999), respectively. Finally, the complete two-electron contributions have been presented recently, even beyond the Breit level (Yerokhin et al. 2000, 2001). 

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