Spin-orbit mean-field

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... 

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

Phys. Lett., 286, 267 (1998). On the Combination of ECP-Based Cl Calculations with All-Electron Spin-Orbit Mean-Field Integrals. 

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... 

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

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... 

The one-electron additivity of the mean-field Hamiltonian gives rise to the concept of spin orbitals for any additive bi fact, there is no single mean-field potential different scientists have put forth different suggestions for over the years. Each gives rise to spin orbitals and configurations that are specific to the particular However, if the difference between any particular mean-field model and the fiill electronic... 

By expressing the mean-field interaction of an electron at r with the N- 1 other electrons in temis of a probability density pyy r ) that is independent of the fact that another electron resides at r, the mean-field models ignore spatial correlations among the electrons. In reality, as shown in figure B3.T5 the conditional probability density for finding one ofA - 1 electrons at r, given that one electron is at r depends on r. The absence of a spatial correlation is a direct consequence of the spin-orbital product nature of the mean-field wavefiinctions... 

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... 

In most of the connnonly used ab initio quantum chemical methods [26], one fonns a set of configurations by placing N electrons into spin orbitals in a maimer that produces the spatial, spin and angular momentum syimnetry of the electronic state of interest. The correct wavefimction T is then written as a linear combination of tire mean-field configuration fimctions qj = example, to describe the... 

The single Slater determinant wavefunction (properly spin and symmetry adapted) is the starting point of the most common mean field potential. It is also the origin of the molecular orbital concept. 

The first term is characterized by a scalar, 7, and it is the dominant term. Be aware of a convention disagreement in the definition of this term instead of -27, some authors write -7, or 7, or 27, and a mistake in sign definition will turn the whole scheme of spin levels upside down (see below). The second and third term are induced by anisotropic spin-orbit coupling, and their weight is predicted to be of order Ag/ge and (Ag/ge)2, respectively (Moriya 1960), when Ag is the (anisotropic) deviation from the free electron -value. The D in the second term has nothing to do with the familiar axial zero-field splitting parameter D, but it is a vector parameter, and the x means take the cross product (or vector product) an alternative way of writing is the determinant form... 

Mean-Field Spin-Orbit Method Applicable to Correlated Wavefunctions. 

Relativistic mean field (RMF) models have been applied successfully to describe properties of rinite nuclei. In general ground state energies, spin-orbit splittings, etc. can be described well in terms of a few parameters ref. [18]. Recently it has lead to the suggestion that the bulk SE is strongly correlated with the neutron skin [19, 20] (see below). In essence the method is based upon the use of energy-density functional (EDF) theory. 

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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