Dirac-Fock-Slater 2503 method

The experimental XANES spectrum has been reported by Li et al. [13] as shown in Fig 7, where the horizontal scale was calibrated by the data obtained by O Brien et al. [28]. As in the case of a-A Oa, the splitting of peak A is attributable to the spin orbit splitting. The calculated splitting of 2p orbital for a free Si atom using the Dirac-Fock-Slater method is 0.66 eV, which agrees well with the experimental splitting (0.6 eV). 

From a very general point of view every ion-atom collision system has to be treated as a correlated many-body time-dependent quantum system. To solve this from an ab initio point of view is still impossible. So, one has to rely on various approximations. Nowadays the best method which can be applied to realistic collision systems (which we discuss here) is on the level of the non-selfconsistent time-dependent Hartree-Fock-Slater or, in the relativistic case, the Dirac-Fock-Slater method. Up-to-now no correlation beyond this approximation can be taken into account in the case of 3 or more electrons. (This is in accordance with the definition of correlation given by Lowdin [1] in 1956) In addition no QED contributions, i.e. no correction to the 1/r Coulomb interaction between the electrons, ever have been taken into account, although in very heavy collision systems this effect may become important. This will be discussed in section 5. A short survey of the theory used is followed by our results on impact parameter dependent electron transfer and excitation calculations of ion-atom and ion-solid collisions as well as first results of an ab initio calculation of MO X-rays in such complicated many particle scattering systems. 

Relativistic wave equation for a particle with spin 1/2. Dirac-Fock-Slater method... 

Figure4.7 Relativistic bond contractions A re for Au2 calculated in the years from 1989 to 2001 using different quantum chemical methods. Electron correlation effects Acte = te(corn) — /"e(HF) at the relativistic level are shown on the right hand side of each bar if available. From the left to the right in chronological order Hartree-Fock-Slater results from Ziegler et al. [147] AIMP coupled pair functional results from Stbmberg and Wahlgren [148] EC-ARPP results from Schwerdtfeger [5] EDA results from Haberlen and Rdsch [149] Dirac-Fock-Slater...

The so-called Hartree-Fock-Slater method is much more widely utilized, and is a hybrid of the Hartree and Thomas-Fermi-Dirac methods. In this method the direct part of the potential is calculated using the Hartree-Fock approach, whereas the exchange part is approximated by some statistical expression of the model of free electrons. The Slater potential is given by ... 

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]. 

These very complicated inhomogeneous coupled differential equations can again be simplified by using Slater s approximation. This method is therefore called the relativistic Hartree-Fock-Slater or Dirac-Fock-Slater (DFS) 52—53) calculations, and they have also been done by several authors for the superheavy elements 54-56). 

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. 

In all the calculations for the electronic and geometric structures of the system, the density functional method (6-9) was used. The total energy, E, in the Dirac-Fock-Slater approximation is expressed as a functional of charge density... 

The aim of the present work is to perform a detailed theoretical study of the electronic structures of actinyl nitrates. Relativistic effects are remarkable in the electronic structure and chemical bonding of heavy atoms such as actinide elements[6j. In our previous study, we applied the relativistic discrete variational Dirac-Fock-Slater(DV-DFS) method to study of the electronic structure of uranyl nitrate dihydrate[7]. The accuracy of the DV-DFS method was demonstrate by its ability to reproduce the uranyl nitrate dihydrate experimental X-ray photoelectron spectrum. 

The DV-DFS molecular orbital(MO) method is based on the Dirac-Fock-Slater approximation. This method provides a powerful tool for the study of the electronic structures of molecules containing heavy elements such as uranium[7,8,9,10]. The one-electron molecular Hamiltonian in the Dirac-Fock-Slater MO method is written as... 

The best-known and widely-quoted tabulation of atomic Dirac-Hartree-Fock energies was published by Desclaux [11], covered elements in the range Z=1 to Z=120 using finite difference methods. A number of computer packages are available to perform MCDHF calculations [19]. Published DHF and Dirac-Fock-Slater (DFS) calculations for atoms are now too numerous to construct a comprehensive catalogue. It is, however, possible to sort the purposes for which these calculations have been performed into general classes. 

The number of reported molecular DHF calculations is sufficiently small that a fairly complete account is possible. The cases which have been studied in the DHF model all involve small molecules, or molecules which exhibit high spatial symmetry. Larger molecules have been studied using more approximate schemes, ranging from semi-empirical and pseudopotential methods to Dirac-Fock-Slater and density functional methods. These are discussed elsewhere in this book. 

Finite difference methods Benchmark calculations have been performed for a number of diatomic species containing heavy elements, for one-electron systems [195-197], and in the Dirac-Fock-Slater approximation [198,199]. 

In Fig. 7 the purely repulsive Moliere potential is compared with the (partly) attractive Morse potential. It is to be noted from this comparison that the Morse (like the Lennard-Jones) potential, while providing a realistic description of the attractive part of the interaction, becomes insufficient at low intemuclear separations where the purely repulsive potentials are more adequate. While more sophisticated potentials calculated, for example, by employing the Dirac-Fock-Slater (DFS) method (Eckstein etai, 1992), have recently become available, such potentials are generally more complicated and are available only in numerical form, and are thus not very handy for the calculations of interest here. For Si-Si collisions, the interaction potential, being repulsive for small and attractive for... 

C. Diisterhoft, D. Heinemaim, D. Kolb. Dirac-Fock-Slater calculations for diatomic molecules with a finite element defect correction method (FEM-DKM). Chem. Phys. Lett., 296 (1998) 77-83. 

DF=Dirac-Fock DFS = Dirac-Fock-Slater DS = Dirac-Slater DVM = discrete variational method EC = electron capture MCDF = multi-configuration DF NR = nonrelativis-... 

More approximate four-component schemes of solution of the relativistic electronic structure problem have been used to obtain insight in chemical properties connected with relativistic effects. This comprises semiempirical methods such as the Relativistic Extended Hiickel (REX) method as well as the Dirac-Fock-Slater (DFS) method, the relativistic analogue of the Hartree-Fock-Slater (HFS) approach. 

Historically16 it is worthy of note that if one resorts in equation (51) to the TF approximation (18) for tr, then the Euler equation of the Thomas-Fermi-Dirac method results. We shall not go into the solutions of the Thomas-Fermi-Dirac equation in this review, though there has been recent interest in this area. Suffice it to say that in the full form of the Euler equation (51), we are working at the customary Hartree-Fock-Slater level. However, we shall content ourselves, until we come to Section 17 below, with understanding in a more intuitive, but inevitably less detailed, way how the corrections to the TF energy in equation (48) arise. 

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. 

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). 

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. 

---