Effective one-electron spin-orbit Hamiltonians

Since spin-orbit coupling is very important in heavy element compounds and the structure of the full microscopic Hamiltonians is rather complicated, several attempts have been made to develop approximate one-electron spin-orbit Hamiltonians. The application of an (effective) one-electron spin-orbit Hamiltonian has several computational advantages in spin-orbit Cl or perturbation calculations (1) all integrals may be kept in central memory, (2) there is no need for a summation over common indices in singly excited Slater determinants, and (3) matrix elements coupling doubly excited configurations do not occur. In many approximate schemes, even the tedious four-index transformation of two-electron integrals ceases to apply. The central question that comes up in this context deals with the accuracy of such an approximation, of course. 

One of the simplest and least demanding approaches is to take the electron-electron interactions into account through screening of the nuclear potential. 

In this one-electron one-center spin-orbit operator, I denotes an atom and // an electron occupying an orbital located at center L Likewise, labels the angular momentum of electron ij with respect to the orbital origin at atom L The 

Blume and Watson33 showed in 1962 that only parts of the two-electron spin-orbit interaction may be expressed in the form [122]. According to these authors, the neglect of the remaining terms leads to a dependence of the effective nuclear charge from a particular state. For dn configurations, for example, they found that Zf can vary by as much as 20%.34 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from 

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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... 

A number of computational procedures have been proposed, which differ mainly in the relative order of including the perturbations. Nakatsuji et al. add the one-electron spin-orbit term (or an effective one-electron spin-orbit operator) to the unperturbed Hamiltonian and take the ground state of this new Hamiltonian as the corresponding ground state solution. Then,... 

Hamiltonian matrix including terms arising from an effective one-particle spin-orbit operator was set up and diagonalized in the basis of correlated many-electron states derived without spin-orbit interaction. Hay gives a detailed analysis of the electronic states with and without spin-orbit interaction up to term energies of 10 eV. 

The main effect of taking spin-orbit interaction into account will be an admixture of singlet character to triplet states and triplet character to singlet states. The spin-orbit coupling Hamiltonian can to a good approximation be described by an effective one-electron operator Hso ... 

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 effects of spin orbit coupling in the H2 case (b)3 n state have been discussed in detail by Fontana [39] and Chiu [40,41]. Chiu [41] starts by writing the Ml spin orbit Hamiltonian for the two-electron, two-nucleus system as a sum of one-electron terms,... 

Even though the spin orbitals obtained from (2.23) in general do not have the full symmetry of the Hamiltonian, they may have some symmetry properties. In order to study these Fukutome considered the transformation properties of solutions of (2.24) with respect to spin rotations and time reversal. Whatever spatial symmetry the system under consideration has, its Hamiltonian always commutes with these operators. As we will see, the effective one-electron Hamiltonian (2.25) in general only commutes with some of them, since it depends on these solutions themselves via the Fock-Dirac matrix. 

Electron-electron repulsion can have a profound effect on the electronic structure of a system. For a closed-shell system described by one Slater determinant, in which the up-spin and down-spin electrons of a given MO are restricted to have an identical spatial function, the effective one-electron Hamiltonian employed in Section 26.2 is given by the Fock operator [3]. When one Slater determinant is used to describe the electronic structure of an open-shell system, the up-spin and down-spin electrons are allowed to have different spatial functions. For a certain open-shell system (e.g. diradical), a proper description of its electronic structures even on a qualitative level requires the use of a configuration interaction (Cl) wave function [6], i.e. a linear combination of Slater determinants. In this section, we probe how electron-electron repulsion affects the concepts of orbital interaction, orbital mixing and orbital occupation by considering a dimer that is made up of two identical sites with one electron and one orbital per site (Fig. 26.3). 

It can be shown (Veseth, 1970) that all electron-nuclear distances, r ) can be referred to a common origin, and, neglecting only the contribution of spin-other-orbit interactions between unpaired electrons, the two-electron part of the spin-orbit Hamiltonian can be incorporated into the first one-electron part as a screening effect. The spin-orbit Hamiltonian of Eq. (3.4.2) can then be written as... 

Finally, Zaitsevskii et al [95] proposed a DGCF method differing mainly from the previous ones by the scalar effective Hamiltonian used and by the content of the spin-orbit interaction. In this work a dressed intermediate Hamiltonian is constructed using the spin-adapted many-body multipartitioning perturbation theory (MPPT) up to the second order. The MPPT theory is based on the simultaneous use of several quasi-one-electron zero-order Hamiltonians (see for... 

In most cases, however, the relativistic effects are rather weak and may be separated into spin-orbit coupling effects and scalar effects. The latter lead to compression and/or expansion of electron shells and can rather accurately be treated by modifying the one-electron part of the non-relativistic many-electron Hamiltonian. With this scalar-relativistic Hamiltonian the (modified) energies and wave functions are computed and subsequently an effective spin-orbit part is added to the Hamiltonian. The effects of the spin-orbit term on the energies and wave functions are commonly estimated using second-order perturbation theory. More information for the interested reader can be found in excellent textbooks on relativistic quantum chemistry [2, 3]. 

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