Dirac-Hartree-Slater

The analysis can also be carried out using the standard computer code. However, the procedure for analysis is the same as described earlier. PIXEF (for PIXE-fit) the Livermore PIXE spectrum analysis package has been developed by Antolak and Bench (1994). This software initially computes an approximation to the background continuum, subtracts from the raw spectral data and the resulting X-ray peaks are then fitted to either Gaussian or Hy > ermet distributions. The energy dependent ionization cross-sections for each element s K-shell or L-subshell are procured from the analytical functional fit, while the subshell and total photoelectric cross-sections are determined directly from the Dirac-Hartree-Slater calculations of Scofield. Schematic of a typical PIXE spectrum is as shown in Fig. 1.16. 

Desclaux [11] performed true relativistic Dirac-Fock calculations with exchange to obtain orbital binding energies for every atom. Relativistic Hartree-Fock-Slater calculations were made by Huang et al. [12] and later improved by ab initio Dirac-Hartree-Slater wave functions for elements with Z = 70 to 106 in [13]. 

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...

As an approach analogous of nonrelativistic Hartree-Fock theory, the four-component Dirac-Hartree-Fock wave function is described with a Slater determinant of one-electron molecular functions ( aX l= U Nelec, ... 

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). 

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). 

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. 

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).]... 

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. 

The proportionality constant k is usually assigned a value of approximately 1.75. The overlap matrix elements Sy are calculated with respect to a set of two component basis functions with lsjm) quantization. The radial parts were chosen to be one or two Slater functions yielding (r ) (k=-l,0,1,2) expectation values as close as possible (Lohr and Jia 1986) to the Dirac-Hartree-Fock or Hartree-Fock results tabulated by Desclaux (1973) for the relativistic and nonrelativistic case, respectively. The diagonal Hamiltonian matrix elements Hu were set equal to the corresponding orbital energies from Desclaux s tables. Due to the use of a two-component lsjm) basis set the matrices H and S are generally complex and of dimension 2nx2n, when is the number of spatial orbitals. 

The Slater determinant represents the simplest approximation to an electronic ground state, and its orbitals may be optimized to approximate the ground state as closely as possible within the independent particle model. We shall see later in this chapter how the optimization of the orbitals can be done within (Dirac-)Hartree-Fock theory by relying on the variational principle. 

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). 

In accordance with the equivalence restriction a shell is classified by one pair of quantum numbers n and k. A shell comprises 2 k atomic spinors and is therefore 2 x -fold degenerate. A closed shell is characterized by the fact that all of these degenerate spinors enter the single Slater determinant in Dirac-Hartree-Fock theory. 

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. 

In short Cl expansions, one may set up explicitly those CSFs which allow us to assign a correlating radial function to a given radial function of the Dirac-Hartree-Fock Slater determinant. This correlating function has got some well-defined properties for instance, the virtual radial function and the one to be correlated should live in the same spatial region. However, this can create additional technical difficulties for the guess of those radial functions which have been introduced to account for the correlation of a particular shell in the Cl expansions with more than one CSF. The above-mentioned model potentials are not the best choice, and additional adjustments to them are necessary so that it can be certain that the shells to be correlated live in the same radial space. 

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. 

An efficient approach to improve on the Hartree-Fock Slater determinant is to employ Moller-Plesset perturbation theory, which works satisfactorily well for all molecules in which the Dirac-Hartree-Fock model provides a good approximation (i.e., in typical closed-shell single-determinantal cases). The four-component Moller-Plesset perturbation theory has been implemented by various groups [519,584,595]. A major bottleneck for these calculations is the fact that the molecular spinor optimization in the SCF procedure is carried out in the atomic-orbital basis set, while the perturbation expressions are given in terms of molecular spinors. Hence, all two-electron integrals required for the second-order Moller-Plesset energy expression must be calculated from the integrals over atomic-orbital basis functions like... 

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. 

The exchange part, ex, which represents the exchange energy of an electron in a uniform electron gas of a particular density is, apart from the pre-factor, equal to the form found by Slater in his approximation of the Hartree-Fock exchange (Section 3.3) and was originally derived by Bloch and Dirac in the late 1920 s ... 

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... 

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