Dirac-Hartree-Fock-Slater method

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]. 

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). 

Methods DHF, Dirac-Hartreer-Fock DHFS, Dirac-Hartree-Fock-Slater HF, Hartree-Fock OCE, one-center expansion MS, multiple-scattering DV, discrete-variational QR, quasirelativistic INDO, intermediate neglect of differential overlap WHT, Wolfsberg-Helmholz QR-EHT, quasirelativistic two-component extended Httckel EHT, extended Hllckel. 

Numerical discretization methods pose an interesting consequence for fully numerical Dirac-Hartree-Fock calculations. These grid-based methods are designed to directly calculate only those radial functions on a given set of mesh points that occupy the Slater determinant. It is, however, not possible to directly obtain any excess radial functions that are needed to generate new CSFs as excitations from the Dirac-Hartree-Fock Slater determinant. Hence, one cannot directly start to improve the Dirac-Hartree-Fock results by methods which capture electron correlation effects based on excitations that start from a single Slater determinant as reference function. This is very different from basis-set expansion techniques to be discussed for molecules in the next chapter. The introduction of a one-particle basis set provides so-called virtual spinors automatically in a Dirac-Hartree-Fock-Roothaan calculation, which are not produced by the direct and fully numerical grid-based approaches. 

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...

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. 

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). 

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. 

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. 

The matrix form of the atomic Dirac-Hartree-Fock (DHF) equations was presented by Kim [37,95], who used a basis set of modified radial Slater-type functions, without the benefit of a balancing presciption for the small component set. A further presentation of the atomic equations was made by Kagawa [96], who generalized Kim s work to open shells and discussed matrix element evaluation. An extension to include the low-ffequency form of the Breit interaction self-consistently in an S-spinor basis was presented by Quiney [97], who demonstrated that this did not produce variational collapse. Our presentation of the DHFB method is based on [97-99]. 

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. 

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. 

In approximate Cl methods, the set of many-particle basis functions is restricted and not infinitely large, i.e., it is not complete. Then, the many-particle basis is usually constructed systematically from a given reference basis function. (such as the Slater determinant, which is constructed to approximate the ground state of a many-electron system in (Dirac-)Hartree-Fock theory). 

So far, we have only discussed the four-component basis-set approach in connection with the simplest ab initio wave-function model, namely for a single Slater determinant provided by Dirac-Hartree-Fock theory. We know, however, from chapter 8 how to improve on this model and shall now discuss some papers with a specific focus on correlated four-component basis-set methods. 

Approximation to Dirac-Hartree-Fock method, using Slater exchange to model the exchange term. 

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. 

About the same time, Douglas Hartree, along with other members of the informal club for theoretical physics at Cambridge University called the Del-Squared Club, began studying approximate methods to describe many-electron atoms. Hartree developed the method of the self-consistent field, which was improved by Vladimir Fock and Slater in early 1930, so as to incorporate the Pauli principle ab initio.37 Dirac, another Del-Squared member, published a paper in 1929 which focused on the exchange interaction of identical particles. This work became part of what soon became called the Heisenberg-Dirac approach.38... 

The momentum wave functions in various atomic models are calculated for arbitrary atomic orbitals. The nonrelativistic hydrogenic, the Hartree-Fock, the relativistic hydrogenic, and the Dirac-Fock models are considered. The momentum wave functions are obtained as a Fourier transform of the wave function in the position space. The Hartree-Fock and the Dirac-Fock wave functions in atoms are given in terms of Slater-type orbitals (STO s), i.e. the Hartree-Fock-Roothaan (HFR) method and the relativistic HFR (RHFR) method. All the wave functions in the momentum space can be expressed analytically in terms of hypergeometric functions. 

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. 

For molecules, Hartree-Fock approximation is the central starting point for most ab initio quantum chemistry methods. It was then shown by Fock that a Slater determinant, a determinant of one-particle orbitals first used by Heisenberg and Dirac in 1926, has the same antisymmetric property as the exact solution and hence is a suitable ansatz for applying the variational principle. 

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