Spin-orbit operators relativistic effective core potential

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... 

Fig. 6. Average relativistic effective core potential and relativistic effective core potential energy curves for two states of Bi2. HF, Hartree-Fock GVB(pp), eight-configuration perfect-pairing generalized valence bond FVCI, full-valence Cl based on the GVB(pp) wave functions FV7R, full-valence Cl plus single and double promotions to virtual MOs relative to seven-dominant configurations. (The FVCI and FV7R calculations include the REP-based spin-orbit operator.)...

Abstract An implementation of a massively parallel spin-orbit configuration interaction (PSOCI) method is described. This is an extension of a conventional Cl method that explicitly includes one-electron spin-orbit operators and certain scalar relativistic effects extracted from relativistic effective core potentials. The performance of the PSOCI code is analyzed on several large-scale computing platforms. 

Ermler WC, TUson JL (2012) GeneraUy contracted valence-core/ valence basis sets for use with relativistic effective core potentials and spin-orbit coupUng operators. Comp Theor Chem 1002 24-30. doi 10.1016/j.comptc.2012.09.020... 

Hurley, M. M., Pacios, L. E, Christiansen, P. A., Ross, R. B., Ermler, W. C. (1986). Ab initio relativistic effective core potentials with spin-orbit operators. II. K through Kr. The Journal of Chemical Physics, 84, 6840-6853. 

The RECP method is an extension of the nomelativis-tic effective core potential approach [30], which has been reviewed by Krauss and Stevens [31]. RECPs are obtained by several algorithms, particularly from wave functions from relativistic atomic calculations [32-34] and from fitting the energy results from all-electron atomic calculations [20]. Corresponding spin-orbit operators are obtained as part of the same process [22, 33] or by a separate process [35]. 

There is no need to explicitly include terms for direct relativistic effects, such as the dependence of mass on velocity, which are important only in the core region, in the valence-electron Hamiltonian. These terms are included as a consequence of using the Diiac-Fock wave functions. Thus, the Hamiltonian for the valence electrons is composed of the nonrelativistic Hamiltoman for the valence electrons plus the RECPs, which include the effects of the core electrons as well as the relativistic effects on the valence electrons in the core region [37]. The RECPs thns represent, for the valence electrons, the dynamical effects of relativity from the core region, the repulsion of the core electrons, the spin-orbit interaction with the nucleus, the spin-orbit interaction with the core electrons, and an approximation to the spin-orbit interaction between the valence electrons [38], which has usually been found to be quite stnaU, especially for heavier element systems [39-41]. The REP operators can be written as a summation of spin-independent potential and the spin-orbit operator, as written below, and the readers are referred to reference [39] for details. 

With the core potentials and spin-orbit operators given in the forms of Eqs. (2.1) and (2.2), existing programs for nonrelativistic calculations can be adapted to include relativistic effects. In the literature, the coefficients of the spin-orbit potentials are not always defined in the same manner. In our implementation of the spin-orbit interaction in NWChem, the spin-orbit potential is defined as the A in (2.3) scaled by... 

L, F. Pacios and P. A. Christiansen, /. Chem. Phys., 82, 2664 (1985). Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. I. Li Through Ar. See also, P. A. Christiansen, Y, S. Lee, and K. S. Pitzer, /. Chem. Phys. 71, 4445 (1979). Improved Ab Initio Effective Core Potentials for Molecular Calculations. In this latter paper are found equations pertinent to inverting the Fock equations, i.e., solving for the effective potentials from the Fock equations of the atoms. 

In recent years, there has been an increasing interest in the inclusion of relativistic effects for molecules containing heavy atoms. One of the most practical yet reliable methods is to use relativistically derived effective core potentials. Major relativistic effects such as the Darwin and mass-velocity effects are easily taken into account in the form of a spin-free (SF) one-electron operator. The spin-orbit (SO) interaction is in general too strong to be considered as a small perturbation, and therefore should be treated explicitly as a part of the total Hamiltonian. 

Because u is constructed from a potential fit to the Dirac fovu-component wave functions, it contains all relativisitic effects, except for spin-orbit coupling. The spin-orbit coupling operator for the effective core potential is then given as the difference between the relativistic effective potential, and the averaged relativistic effective potential,... 

To formalize these concepts, we first develop the frozen-core approximation, and then examine pseudoorbitals or pseudospinors and pseudopotentials. From this foundation we develop the theory for effective core potentials and ab initio model potentials. The fundamental theory is the same whether the Hamiltonian is relativistic or nonrela-tivistic the form of the one- and two-electron operators is not critical. Likewise with the orbitals the development here will be given in terms of spinors, which may be either nonrelativistic spin-orbitals or relativistic 2-spinors derived from a Foldy-Wouthuysen transformation. We will use the terminology pseudospinors because we wish to be as general as possible, but wherever this term is used, the term pseudoorbitals can be substituted. As in previous chapters, the indices p, q, r, and s will be used for any spinor, t, u, v, and w will be used for valence spinors, and a, b, c, and d will be used for virtual spinors. For core spinors we will use k, I, m, and n, and reserve i and j for electron indices and other summation indices. 

The method works as follows. The mass velocity, Darwin and spin-orbit coupling operators are applied as a perturbation on the non-relativistic molecular wave-functions. The redistribution of charge is then used to compute revised Coulomb and exchange potentials. The corrections to the non-relativistic potentials are then included as part of the relativistic perturbation. This correction is split into a core correction, and a valence electron correction. The former is taken from atomic calculations, and a frozen core approximation is applied, while the latter is determined self-consistently. In this way the valence electrons are subject to the direct influence of the relativistic Hamiltonian and the indirect effects arising from the potential correction terms, which of course mainly arise from the core contraction. 

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