Atomic mean-field spin-orbit integrals

Just as for the Cowan-Griffin operator, the potential is the atomic SCF potential and so includes both one- and two-electron spin-orbit effects. In this respect the integrals over this potential resemble the atomic mean-field spin-orbit integrals of Hess et al. (1996). 

Since the mean-field spin-orbit integrals have been defined and implemented to molecular codes (Dalton, Molcas etc.), hundreds or more successful molecular and atomic applications have been performed with the accuracy comparable to the experimental results. 

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 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... 

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-... 

In the previous section we discussed how to calculate the spin-orbit splittings of atomic and molecular states. The effect of spin-orbit coupling was introduced via the restricted active space interaction (RASSI) method with the use of the atomic mean-field AMFI integrals. It appears however, that the discrepancies between the experimental and calculated values of energies can be still quite big. 

State interaction, RASSI [36], method combined with the atomic mean field integrals method, AMFI [37], has been used to take spin-orbit (SO) effects into account. One electron part of AMFI code has been used only. It gives approximate spin-dependent... 

B. Schimmelpfennig, Atomic Spin-Orbit Mean-Field Integral Program AMFI, developed at... 

Schimmelpfennig, B. (1996) Atomic spin-orbit Mean-Field Integral program AMFI. Stock-holms Universitet. 

In Eq. (1.13) the orbitals > are taken from spin-free atomic or molecular calculations. It was additionally assumed, in order to make the mean-field method efficient, that all two-electron integrals between basis functions centred on different centres could be neglected. 

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