Spin-orbit mean-field operator

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). 

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... 

The problem with this approximation is that we have only neglected the commutator for one electron coordinate. The commutator for the other electron coordinate is embedded in the mean-field equation, and it still gives rise to a spin-free and a spin-orbit operator. Neglecting for the moment the normalization terms, if we use the partitioning in (18.5) and substitute yCoui = for V, we get... 

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. 

Malkina OL et al (20(X)) Density functional calculations of electronic g-tensors using spin-orbit pseudopotentials and mean-field all-electron spin-orbit operators. J Am Chem Soc 122 9206-9218... 

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 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... 

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. 

As in general all the y-coefficients do not vanish one has to assume a more general reference state than the single determinant SCF state. This is the rather well-known problem of finding the consistent reference state for the Random Phase Approximation (RPA). It also means that the field operator basis can be enlarged and can for instance include the iV-electron occupation number operators (in this discussion, electron field operators and their adjoints are used referring to a basis of spin orbitals that are the natural spin orbitals of the reference state, as will be discussed below, i.e., the spin orbitals that diagonalize the one-matrix)... 

The basis for all wave function based ab initio methods is the Hartree-Fock (HF) approach. [11, 12] It makes use of a single-determinant ansatz constructed from one-electron spin orbitals. These orbitals describe the motion of each electron within the field of the nuclei and the mean field of the remaining n-1 electrons. The mean field is not known a priori, but depends on the orbitals which are determined self-consistently from the eigenvalue problem of the Fock operator. 

The occupied spin orbitals included in the operators J and K have to be the solutions of Equation 2.30. An iterative method is used, where the successive solutions of Eqnation 2.30 define J and K. This procedure is repeated until the energy eigenvalues are within a stipnlated convergence limit. This solution is called the self-consistent field (SCE) solution. Physically, SCF means that screening and penetration effects are taken into account in the best possible way within the one-electron approximation. 

An alternative approach is based on the application of these spin-orbit terms in a perturbation theory (PT) or configuration interaction (Cl) step following a scalar mean-field calculation. For such an application we will need to evaluate matrix elements over wave functions, expressed as linear combinations of Slater determinants. To do this, we apply the usual Slater-Condon rules. We start by looking at the one-electron term, where we may examine the elements of the operator... 

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