Spin-Electronic Homogeneous Perturbations

In contrast, the spin-orbit operator only includes a summation JT Its involving the same electron for both 1+ and s-. The terms in HSE, l sj, are two-electron operators and can cause nonzero matrix elements between wave-functions that differ by two spin-orbitals. 

Moreover, the operator B(R) = H2/2(j,R2 acts on the vibrational part of the wavefunction, 

When the two potential curves are well-characterized, the Bvv matrix elements may be calculated numerically. For nearly identical potentials, the Bvv matrix elements behave much less like Svv functions than do (v v ) vibrational overlaps. 

As an example, consider the interaction between the 3IIi state (cnr4Tr configuration) and the 3 + state ( r27r37r configuration). For B(R)L S, as for the Hso operator, the interacting basis wavefunctions must have the same value of ft. For this example, the ft = 1 matrix elements of the L+S part of HSE will be evaluated. 

The 3IIi 3Ej matrix element of Hso is much easier to evaluate because Hso is a one-electron operator, 

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QMC methods were first applied to the case of the electron gas by Ceper-ley in the late 1970s,and the results have been widely used in density functional theory. Only recently have these early calculations been extended by others to provide greater detail. Pickett and Broughton carried out VQMC calculations for the spin-polarized gas. Ortiz and Ballone used both VQMC and fixed-node DQMC for the spin-polarized gas in the density range most important to density functional theory. Kenny et al. performed VQMC and DQMC calculations for the nonpolarized homogeneous electron gas, incorporating relativistic effects via first-order perturbation theory. 

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