Dirac-Slater

Coulomb exchange effects are commonly introduced by means of the Dirac-Slater expression for the exchange energy of a electron gas ... 

The integral of the first term in square brackets gives the non-relativistic Dirac-Slater exchange energy, the second giving the relativistic correction ... 

While Dirac [3] chose to solve Eq. (4) as a quadratic equation for in terms of the Hartree potential yHC "), it was Slater in 1951 ([6] see also [4]) who chose an alternative, and more fruitful, route by regarding Eq. (4) as demonstrating that it could be viewed as a modified Hartree equation, with the Hartree potential Unfr) now supplemented by the exchange n -potential (the so-called Dirac-Slater (DS) exchange potential), to yield a total one-body potential energy... 

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... 

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... 

Cromer, D. T., J. T. Waber Scattering Factors Computed from Relativistic Dirac-Slater Wave Functions. Acta Cryst. 18, 104 (1965). 

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. 

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... 

We used the DV Hartree-Fock-Slater method for TiC, while for UC we used the DV Dirac-Slater (DV-DS) method taking fully relativistic effects into account. The basis functions used were ls-2p for C atom, ls-4p for Ti atom, and ls-7p for U atom. The bond nature of TiC and UC compounds were studied by Mulliken population analysis [6,7]. The details of the nonrelativistic and relativistic DV-Xa molecular orbital methods have been described elsewhere [7,8,9]. 

Fig. 11. Dirac-Slater (DFS) energy eigenvalues of the outer electrons for the 5d, d and Id elements. This shows the strong relativistic increase of the binding of the last s shell and the increase in the splitting of the d subshells (85)...

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

The potential surrouding each atom in a molecule is not the same as that for the free atom, because electron transfer occurs between atoms in the molecule. This means that atomic orbitals in the molecule are distinct from those in the free atom. Accordingly, it is necessary to use atomic orbitals optimized for each atomic potential in the molecule, as basis functions. In the present methods, the molecular wave functions were expressed as linear combinations of atomic orbitals obtained by numerically solving the Dirac-Slater or Hartree-Fock-Slater equations in the atomic-like potential derived from the spherical average of the molecular charge density around the nuclei [15]. Thus the atomic orbitals used as basis functions were automatically optimized for the molecule and thus the minimum size of the present basis set has enough flexibility to form accurate molecular orbitals. 

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. 

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... 

The fuUy relativistic Dirac-Slater atomic code, DIRAC, is one of the auxiliary programs distributed with the ADF code. 

A suitable computational approach for the investigation of electronic and geometric structures of transactinide compounds is the fully relativistic Dirac-Slater discrete-variational method (DS-DVM), in a modem version called the density functional theory (DFT) method, which was originally developed in the 1970s (Rosdn and Ellis 1975). It offers a good compromise between accuracy and computational effort. A detailed description can be found in Chapter 4 of this book. 

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... 

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