Dirac-Slater calculation

The Pauli operator of equations 2 to 5 has serious stability problems so that it should not, at least in principle, be used beyond first order perturbation theory (20). These problems are circumvented in the QR approach where the frozen core approximation (21) is used to exclude the highly relativistic core electrons from the variational treatment in molecular calculations. Thus, the core electronic density along with the respective potential are extracted from fully relativistic atomic Dirac-Slater calculations, and the core orbitals are kept frozen in subsequent molecular calculations. 

Oda, Y. Funasaka, H. Nakamura, Y. Adachi, H. Discreter variational Dirac-slater calculation of Urangl (VI) nitrate complexes, J. Alloys Comp. 255 (1997) 24-30. 

DS relativistic Dirac-Slater calculation MP2 second order Mpller-Plesset perturbation theory... 

The transition metal atom has a possibility to possess a magnetic moment in metaUic material, then an investigation of the spin polarization of the cluster from a microscopic point of view is very important in understanding the magnetism of the metallic materials. We try to explain the spin polarization and the magnetic interactions of the cluster in terms of the molecular orbital. For the heavy element in the periodic table whose atomic number is beyond 50, it is mentioned that the relativistic effects become very important even in the valence electronic state. We perform the relativistic DV-Dirac-Slater calculation in addition to the nonrelativistic DV-Xa calculation for the small clusters of the 3d, 4d and 5d transition elements to clarify the importance of the relativistic effects on the valence state especially for the 5d elements. 

The electronic structure of the alkoxide complexes Cp3U(OR) and Cp3Th(OR) has been investigated by He(i) and He(n) UV photoelectron spectroscopy combined with SGF Xa-DVM calculations. Full relativistic Dirac-Slater calculations were also carried out for the thorium complexes.67 Comparative relativistic effective core potential ab initio calculations have been reported for both Th(iv) and U(iv) Cp3AnL (L = Me, BH4) complexes.68... 

Self-consistent Dirac-Slater calculations of molecules and embedded clusters have been recently reviewed by Ellis and Goodman. Relativistic band structure calculations have also been carried out. Dirac scattered-wave calculations have been carried out on a number of inorganic complexes such as W(CO)fi and WjQg - The electronic structure and geometries of X2H2 (X = O,S, Sc and Te) have also been investigated recently. ... 

Fig. 7.29. Fourth-order crystal field parameter times the lattice constant, a, raised to the fifth power for a number of lanthanide monopnictides. The solid line is the point charge model prediction with Z = -1.2 and (r" ) equal to that obtained from a Dirac-Slater calculation (after Birgeneau et al., 1973).

SCF Dirac-Slater Calculations of the Translawrendum Elements. /. Chem. 

D. Sundhohn. Two-Dimensional, FuUy Numerical Solution of Molecular Dirac Equations. Dirac-Slater Calculations on liH, Ii2/ BH and CH+. Chem. Phys. Lett., 149(3) (1987) 251-256. 

In actual applications of the method, Snijders and Baerends used a frozen core and incorporated core relativistic effects by using atomic all-electron relativistic orbitals. This is because the Pauli Hamiltonian is not bounded from below. The orthogonality requirement against the core prevents the orbitals from collapsing. With this approach they were able to reproduce quite accurately the valence shell orbital energies from fully relativistic all-electron Dirac-Slater calculations for atoms. 

Waber, J.T., Cromer, D.T., Liberman, D. SCE Dirac-Slater calculations of the trans-lawrencium elements. J. Chem. Phys. 51, 664—668 (1969)... 

B. Ericke and G. Soff, Dirac-Slater calculations for the elements Z=100, fermium, to Z=173, At. Data Nucl. Data Tables 19, 83 (1977). 

Lo et a/,102 have calculated spin-orbit coupling constants for first- and second-row atoms and for the first transition series, results agreeing with the work of Blume and Watson. Karayanis103 has extended the calculation to triply ionized rare earths. However, with very heavy atoms relativistic effects on the part of the wavefunction near the nucleus become severe, leading to a breakdown of the conditions under which simple perturbation theory ought to be applied. Lewis and co-workers104 have used relativistic self-consistent Dirac-Slater and Dirac-Fock wavefunctions to evaluate spin orbit coupling... 

On the other hand, a large number of relativistic Slater calculations [23] (Dirac-Fock-Slater DFS), in which the RKS-equations are used with the nonrelativistic x-only LDA, can be found in the literature (see e.g. [8,9]). However, no attempt is made to review this extensive body of literature here. 

To elucidate the nature of chemical bonding in metal carbides with the NaCl structure, the valence electronic states for TiC and UC have been calculated using the discrete-variational (DV) Xa method. Since relativistic effects on chemical bonding of compounds containing uranium atom become significant, the relativistic Hamiltonian, i.e., the DV-Dirac-Slater method, was used for UC. The results... 

The relativistic DV-Xa calculations are based on the one-electron Hamiltonian for the Dirac-Slater MO method which is given as... 

Relativistic molecular orbital calculations have been performed for the study of the atomic-number dependence of the relativistic effects on chemical bonding by examining the hexafluorides XFg (X=S, Se, Mo, Ru, Rh, Te, W, Re, Os, hr, Pt, U, Np, Pu) and diatomic molecules (CuH, AgH, AuH), using the discrete-variational Dirac-Slater and Hartree-Fock-Slater methods. The conclusions obtained in the present work are sununarized. 

For the same reasons as in the nonrelativistic case the availability of a numerical solver of the DHF equations for molecules would be very much desired. One possible way to proceed would be to deal with the DHF method cast in the form of the second-order equations instead of the system of first-order coupled equations and try to solve them by means of techniques used in the FD HF approach. The FD scheme was used by Laaksonen and Grant (50) and Sundholm (51) to solve the Dirac equation. Sundholm used the similar approach to perform Dirac-Hartree-Fock-Slater calculations for LiH, Li2, BH and CH+ systems (52,53). 

The most successful truly ab initio calculation is the Dirac-Slater Discrete Variational Method of Walch and Ellis [67]. This handles the relativistic part of the Hamiltonian more rigorously than other approaches, and illustrates the importance of the equatorial ligands in determining the energy of the first optical transitions. Furthermore, the use of an optical transition state calculation makes... 

The 1998 Nobel Prize for Chemistry, awarded to a physicist for inventing modem Density Functional Theory (DFT), signaled widespread recognition of DFT as the pre-eminent many-electron theory for predictive, materials-specific (chemically specific) calculation of extended and molecular systems. The original papers of modem DFT are those of Hohenberg and Kohn [1] and Kohn and Sham [2] (preceded by seminal work of Thomas, Fermi, Dirac, Slater, Caspar, Gombas, and others not of direct relevance). General references include [3-16]. 

Table 3 presents relativistic effects on several properties calculated as the difference (A) obtained in calculations which included the quasirelativistic correction, and corresponding calculations that excluded the correction, and used Hartree-Fock-Slater core orbitals rather than Dirac-Slater. The method finds significant relativistic Pt-C bond shortening, and little effect on the CO bond. The effect on adsorption energy is dramatic. Eads increases by about 50% when relativity is included. There is also an increase in the Pt-C force constant and frequency. The shortened Pt-C bond results in an increase in CO frequency through a wall effect, a Pauli repulsion effect. Ref. 34 ascribed the anomalously small shift in CO frequency from gas phase to adsorbed on Pt to the relativistic effect. 

Atomic calculations. Most atomic calculations for the heaviest elements were performed by using Dirac-Fock (DF) and Dirac-Slater (DS) methods [20-24,58] and later by using multiconfiguration Dirac-Fock (MCDF) [64-72] and Dirac-Coulomb-Breit Coupled Cluster Single Double excitations (DCB CCSD) [73-85] methods, with the latter being presently the most accmate one. 

The relativistic form of the one-electron Schrodinger equation is the Dirac equation. One can do relativistic Hartree-Fock calculations using the Dirac equation to modify the Fock operator, giving a type of calculation called Dirac-Fock (or Dirac-Hartree-Fock). Likewise, one can use a relativistic form of the Kohn-Sham equations (15.123) to do relativistic density-functional calculations. (Relativistic Xa calculations are called Dirac-Slater or Dirac-Xa calculations.) Because of the complicated structure of the relativistic KS equations, relatively few all-electron fully relativistic KS molecular calculations that go beyond the Dirac-Slater approach have been done. [For relativistic DFT, see E. Engel and R. M. Dreizler, Topics in Current Chemistry, 181,1 (1996).]... 

In many quantum-mechanical calculations, use is made of the wave functions obtained by the Dirac—Slater and the Hartree—Fock methods for the approximate solution of the Schrddinger equation for free atoms. It woiild be very interesting to determine whether these functions could be refined specifically for crystals and whether the problem could be solved using relatively simple analytic approximations to the calculated functions. In particular, the approximation by Gaussian functions demands attention. 

---