Averaged relativistic effective potentials

AQCC averaged quadratic coupled-cluster AREP averaged relativistic effective potential CAS complete active space... 

The relativistic effective potentials can be averaged with respect to spin. The averaged relativistic effective potentials can be written as... 

The spin-orbit operator derived this way can be ab initio if it is derived from relativistic ab initio potentials. It can be introduced in molecular calculations. Pacios and Christiansen have published Gaussian analytic fits of averaged relativistic effective potentials and spin-orbit operators for Li through Ar. The relativistic potentials of other elements are also being tabulated. ... 

The relativistic effective potential 17 is the j (total angular momentum) dependent core potential fit to the large component of the Dirac four-component wave functions. The average relativistic effective potential is the average relativisitic potential of the j states, given as... 

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,... 

Some scalar relativistic effects are included implicitly in calculations if pseudopotentials for heavy atoms are used to mimic the presence of core electrons there are several families of pseudopotentials available the effective core potentials (ECP) (Cundari and Stevens 1993 Hay and Wadt 1985 Kahn et al. 1976 Stevens et al. 1984), energy-adjusted pseudopotentials (Cao and Dolg 2006 Dolg 2000 Peterson 2003 Peterson et al. 2003), averaged relativistic effective potentials (AREP) (Hurley et al. 1986 Lajohn et al. 1987 Ross et al. 1990), model core potentials (MCP) (Klobukowski et al. 1999), and ab initio model potentials (AIMP) (Huzinaga et al. 1987). 

Fig. 4. All-electron, effective potential, and average relativistic effective core potential configuration-interaction potential-energy curves of Xe2 and Xe2+. Dashed curves are from allelectron calculations and AREP curves are less repulsive than EP.

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.)...

The RECPs are recast into the form of angular momentum-averaged relativistic effective core potentials (ARECPs) that may be used in standard non-relativistic electronic structure procedures based on atomic LS-coupling. 

Laskowski and Langhoff ° have carried out calculations on CrI using averaged relativistic efiective potentials. Similar calculations have been carried out on CsO as well as CsH . Krauss and Stevens carried out SCF calculations on UO, UH, UF and their ions. Krauss and Stevens S have investigated the electronic structure of FeO and RuO using relativistic effective potentials. Relativistic configuration-interaction calculations of low-lying states of BiF have been completed. 

Relativistic Effective Potentials (REPs) and Averaged REPs... 

The so-called Hartree-Fock (HF) limit is important both conceptually and quantitatively in the quantum mechanical theory of many-body interactions. It is based upon the approximation in which one considers each particle as moving in an effective potential obtained by averaging over the positions of all other particles. The best energy calculated from a wavefunction having this physical significance is called the Hartree-Fock energy and the difference between this and the exact solution of the non-relativistic wave equation is called the correlation energy. 

In order to improve the theoretical description of a many-body system one has to take into consideration the so-called correlation effects, i.e. to deal with the problem of accounting for the departures from the simple independent particle model, in which the electrons are assumed to move independently of each other in an average field due to the atomic nucleus and the other electrons. Making an additional assumption that this average potential is spherically symmetric we arrive at the central field concept (Hartree-Fock model), which forms the basis of the atomic shell structure and the chemical regularity of the elements. Of course, relativistic effects must also be accounted for as corrections, if they are small, or already at the very beginning starting with the relativistic Hamiltonian and relativistic wave functions. 

In Table 6.3, the values of De for RfCU are compared with those obtained within various approximations using relativistic effective core potentials (RECP) Kramers-restricted Hartree-Fock (KRHF) (Han et al 1999), averaged RECP including second-order M0ller-Plesset perturbation theory (AREP-MP2) for the correlation part (Han et al. 1999), RECP coupled-cluster single double (triple) [CCSD(T)] excitations (Han et al. 1999), and a Dirac-Fock-Breit (DFB) method (Malli and Styszynski 1998). The AREP-MP2 calculation of De gives 20.4 eV, while the RECP-CCSD(T) method with correlation leads to 18.8 eV. Our value of De of 19.5 eV is just between these calculated values. 

The total relativistic and QED energy shift for many-electron atoms consists of two parts. The first part contains the Bethe logarithm and the other is the average value of some effective potential. Throughout the exact nonrelativistic (Schrddinger) wave functions for the many-electron atom are used. The energy shift is [50] ... 

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