Effective potentials, relativistic

Many of the effective potentials (relativistic or non-relativistic) are generated using the Phillips-Kleinman transformation. In this method, the explicit core-valence orthogonality constraints are replaced by a modified valence Hamiltonian. If one replaces the potential generated by core electrons by a potential Fj, then one can write the one-electron valence wave equation as... 

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. 

The heavier elements are affected by relativistic effects. This is most often accounted for by using relativistic core potentials. Relativistic effects are discussed in more detail in Chapters 10 and 33. 

An important advantage of ECP basis sets is their ability to incorporate approximately the physical effects of relativistic core contraction and associated changes in screening on valence orbitals, by suitable adjustments of the radius of the effective core potential. Thus, the ECP valence atomic orbitals can approximately mimic those of a fully relativistic (spinor) atomic calculation, rather than the non-relativistic all-electron orbitals they are nominally serving to replace. The partial inclusion of relativistic effects is an important physical correction for heavier atoms, particularly of the second transition series and beyond. Thus, an ECP-like treatment of heavy atoms is necessary in the non-relativistic framework of standard electronic-structure packages, even if the reduction in number of... 

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. 

Model potential methods and their utilization in atomic structure calculations are reviewed in [139], main attention being paid to analytic effective model potentials in the Coulomb and non-Coulomb approximations, to effective model potentials based on the Thomas-Fermi statistical model of the atom, as well as employing a self-consistent field core potential. Relativistic effects in model potential calculations are discussed there, too. Paper [140] has examples of numerous model potential calculations of various atomic spectroscopic properties. 

The main advantage of the effective potential method consists in the relative simplicity of the calculations, conditioned by the comparatively small number of semi-empirical parameters, as well as the analytical form of the potential and wave functions such methods usually ensure fairly high accuracy of the calculated values of the energy levels and oscillator strengths. However, these methods, as a rule, can be successfully applied only for one- and two-valent atoms and ions. Therefore, the semi-empirical approach of least squares fitting is much more universal and powerful than model potential methods it combines naturally and easily the accounting for relativistic and correlation effects. 

Stevens W. J. Krauss M. Basch H. Jasien P. G. Relativistic compact effective potentials and efficient, shared-exponent basis sets for the third-, fourth-, and fifth-row atoms. Can. J. Chem. 1992, 70, 612-630. 

M.M. Hurley, P.A. Christiansen, Relativistic effective potentials in quantum Monte Carlo calculations. J. Chem. Phys. 86, 1069-1070 (1987)... 

Dolg, M. Relativistic Effective Core Potentials, Relativistic Electronic Structure Theory - Part 1. Fundamentals , Ed. Schwerdtfeger, P. Elsevier Amsterdam, 2002, pp. 793-862. 

Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. II. K through Kr. 

J. Chem. Phys., 93, 6654 (1990). Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. IV. Cs through Rn. 

Relativistic Effective Potentials with Spin—Orbit Operators. VII. Am through Element 118. 

Accurate Relativistic Effective Potentials for the Sixth-Row Main Group Elements. 

At MP2/CEP (RCEP). Compact effective potentials (CEPs) were used for C, Si and Cl, and their relativistic analogs (RCEP) were used for Ge, Sn and Pb. The contribution of basis set superposition errors (BSSE) is included from Reference 471. 

In order to carry out a priori theoretical calculations of the potential energy for systems with a large number of electrons, the semitheoretical methods use effective potentials which simulate the core electrons.19 48 Note here that the inclusion of relativistic effects may be important in the description of the effective potentials in heavy atoms.49... 

Xis the complete non-relativistic Hamiltonian given in (6.130). The Hermitian matrix in (6.141) is diagonalised to obtain the coefficients in the wave hmctions (6.140) and effective potential curves for the coupled states using these wave functions as a function... 

IV. Spin-Orbit Coupling and Relativistic Effective Potentials—Applications... 

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.

Relativistic Effective Potentials in Quantum Monte Carlo Studies... 

Although all-electron heavy element QMC studies are at the present time out of the question/ we have recently shown that by replacing the core electrons (and the corresponding fraction of the nuclear charge) with an appropriate relativistic effective potential (REP) the QMC domain can be quite readily exctended to the lower portion of the periodic table (60-62). To our knowledge/ reference (60) is the first QMC study involving an element from below the first... 

Perhaps equally inportant, relativistic effective potentials allow one to introduce relativity in a particularly convenient form. 

Atomic Studies In Table I electron affinities for Li, Na and K computed using Equation 3 with either relativistic (60) or nonrelativistic ( ) effective potentials are compared with the respective experimental values (64-66). Only in the relativistic Li calculation do we see a significant discrepancy, and even then the error is well below 0.1 eV. In all of these calculations single determinant trial wavefunctions were employed. While this is no approxdmation for the one-electron neutral atoms we might see minor problems for the anions, and Li could be a case in point. 

Table I. Alkali Electron Affinities (in eV) obtained from Relativistic and Nonrelativistic Effective Potential QMC Simulations...

---