Spin-orbit mean-field approximation

The spin-orbit mean field (SOMF) operator (56-58) is used to approximate the Breit—Pauli two-electron SOC operator as an effective one-electron operator. Using second-order perturbation theory (59), one can end up with the working equations ... 

The one-center approximation allows for an extremely rapid evaluation of spin-orbit mean-field integrals if the atomic symmetry is fully exploited.64 Even more efficiency may be gained, if also the spin-independent core-valence interactions are replaced by atom-centered effective core potentials (ECPs). In this case, the inner shells do not even emerge in the molecular orbital optimization step, and the size of the atomic orbital basis set can be kept small. A prerequisite for the use of the all-electron atomic mean-field Hamiltonian in ECP calculations is to find a prescription for setting up a correspondence between the valence orbitals of the all-electron and ECP treatments.65-67... 

In order to test the validity of the inherent approximations in the spin-orbit mean-field and the DFT/MRCI approaches, electronic spectra and transition rates for spin-allowed as well as spin-forbidden radiative processes were determined for two thioketones, namely dithiosuccinimide and pyranthione (Tatchen 1999 Tatchen et al. 2001). In either case absorption and emission spectra as well as depletion rates for the first... 

The Breit-Pauli SOC Hamiltonian contains a one-electron and two-electron parts. The one-electron part describes an interaction of an electron spin with a potential produced by nuclei. The two-electron part has the SSO contribution and the SOO contribution. The SSO contribution describes an interaction of an electron spin with an orbital momentum of the same electron. The SOO contribution describes an interaction of an electron spin with the orbital momentum of other electrons. However, due to a complicated two-electron part, the evaluation of the Breit-Pauli SOC operator takes considerable time. A mean field approximation was suggested by Hess et al. [102] This approximation allows converting the complicated two-electron Breit-Pauli Hamiltonian to an effective one-electron spin-orbit mean-field form... 

To improve upon die mean-field picture of electronic structure, one must move beyond the singleconfiguration approximation. It is essential to do so to achieve higher accuracy, but it is also important to do so to achieve a conceptually correct view of the chemical electronic structure. Although the picture of configurations in which A electrons occupy A spin orbitals may be familiar and usefiil for systematizing the electronic states of atoms and molecules, these constructs are approximations to the true states of the system. They were introduced when the mean-field approximation was made, and neither orbitals nor configurations can be claimed to describe the proper eigenstates T, . It is thus inconsistent to insist that the carbon atom... 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

The Hartree Fock determinant describes a situation where the electrons move independently of one another and where the probability of finding one electron at some point in space is independent of the positions of the other electrons. To introduce correlation among the electrons, we must allow the electrons to interact among one another beyond the mean field approximation. In the orbital picture, such interactions manifest themselves through virtual excitations from one set of orbitals to another. The most important class of interactions are the pairwise interactions of two electrons, resulting in the simultaneous excitations of two electrons from one pair of spin orbitals to another pair (consistent with the Pauli principle that no more than two electrons may occupy the same spatial orbital). Such virtual excitations are called double excitations. With each possible double excitation in the molecule, we associate a unique amplitude, which represents the probability of this virtual excitation happening. The final, correlated wave function is obtained by allowing all such virtual excitations to happen, in all possible combinations. 

The mean-field approximation requires the proper occupation numbers of the valence orbitals (w = 2 for core and 111 = 0 for virtual molecular orbitals). These should be taken from a nonrelativistic ab initio calculation. Nevertheless, all reasonable choices proved to be suitable for practical purposes because they cause only negligible fluctuations in the calculated energy of spin-orbit coupling. 

As a result of the mean-field approximation, those pairs of Slater determinants that differ by more than one spin-orbital no longer contribute. This approximation is based on an idea similar to the conversion of the ordinary full electronic Hamiltonian into the one-electron Hartree-Fock operator and can be interpreted as describing electronic motion in an averaged field of the other electrons. 

It is better to go one step further and also neglect the multicenter one-electron integrals. The resulting atomic mean-field approximation seems to be good even for molecules composed of light atoms, because it appears that the one- and two-electron multicenter contributions to spin-orbit coupling par-tisdly compensate in a systematic manner. This approxima-... 

Vahtras et al. used the CASSCF(6,6)/DZP and atomic mean-field approximations to calculate the zero-field splitting parameters of benzene, including both spin-spin coupling to the first order and spin-orbit coupling to the second order of... 

Christiansen et al. s tested the coupled-cluster response theory on spin-orbit coupling constants of substituted silylenes (HSiX, X = F, Cl, Br) in the atomic mean-field approximation. Comparison with the full Breit-Pauli showed that the approximation is quite accurate. The calcu-... 

Theory can now provide much valuable guidance and interpretive assistance to the mechanistic photochemist, and the evaluation of spin-orbit coupling matrix elements has become relatively routine. For the fairly large molecules of common interest, the level of calculation cannot be very high. In molecides composed of light atoms, the use of effective charges is, however, probably best avoided, and a case is pointed out in which its results are incorrect. It seems that the mean-field approximation is a superior way to simplify the computational effort. The use of at least a double zeta basis set with a method of wave function computation that includes electron correlation, such as CASSCF, appears to be imperative even for calculations that are meant to provide only semiquantitative results. The once-prevalent degenerate perturbation theory is now obsolete for quantitative work but will presumably remain in use for qualitative interpretations. 

Mean-Field Approximations for Spin-Orbit Interaction... 

The spin-orbit operators for the model potential and pseudopotential approximations are one-electron operators. These operators include the effect of the two-electron spin-orbit interaction used in the mean-field approximation to derive the model potential or pseudopotential. Molecular calculations with these potentials therefore include, at least at the atomic level, the two-electron spin-orbit terms. This is just the kind of approximation we are looking for. 

Here, the mean field potential includes the phenomenological isoscalar part Uq x) along with the isovector U (x) and the Coulomb Uc(x) parts calculated consistently in the Hartree approximation Uo(r) and Uso(x) = Uso r)a l are the central and spin-orbit parts of the isoscalar mean field, respectively, and, SPot(r) is the potential part of the symmetry energy. 

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