Dirac-Fock approximation, quantum

The Pauli approximation may be used in conjunction with this method by neglecting the small component spinors Q) of the Dirac equation, leading to RECPs expressed in terms of two-component spinors. The use of a nonrelativistic kinetic energy operator for the valence region, and two-component spinors leads to Hartree-Fock-like expressions for the pseudoorbitals. Note that the V s (effective potentials) in this expression are not the same for pseudo-orbitals of different symmetry. Thus the RECPs are expressed as products of angular projectors and radial functions. In the Dirac-Fock approximation, the orbitals with different total j quantum numbers, but which have the same / values are not degenerate, and thus the potentials derived from the Dirac-Fock calculations would be y-dependent. Consequently, the RECPs can be expressed in terms of the /y-dependent radial potentials by equa-... 

The inclusion of relativistic effects is essential in quantum chemical studies of molecules containing heavy elements. A full relativistic calculation, i.e. based upon Quantum Electro Dynamics, is only feasible for the smallest systems. In the SCF approximation it involves the solution of the Dirac Fock equation. Due to the four component complex wave functions and the large number of basis functions needed to describe the small component Dirac spinors, these computations are much more demanding than the corresponding non-relativistic ones. This limits Dirac Fock calculations, which can be performed using e.g. the MOLFDIR package [1], to small molecular systems, UFe being a typical example, see e.g. [2]. 

Relativistic quantum chemistry is currently an active area of research (see, for example, the review volume edited by Wilson [102]), although most of the work is beyond the scope of this course. Much of the effort is based on Dirac s relativistic formulation of the Schrodinger equation this results in wave functions that have four components rather than the single component we conventionally think of. As a consequence the mathematical and computational complications are substantial. Nevertheless, it is very useful to have programs for Dirac-Fock (the relativistic analogue of Hartree-Fock) calculations available, as they can provide calibration comparisons for more approximate treatments. We have developed such a program and used it for this purpose [103]. 

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

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. 

Aspects of the relativistic theory of quantum electrodynamics are first reviewed in the context of the electronic structure theory of atoms and molecules. The finite basis set parametrization of this theory is then discussed, and the formulation of the Dirac-Hartree-Fock-Breit procedure presented with additional detail provided which is specific to the treatment of atoms or molecules. Issues concerned with the implementation of relativistic mean-field methods are outlined, including the computational strategies adopted in the BERTHA code. Extensions of the formalism are presented to include open-shell cases, and the accommodation of some electron correlation effects within the multi-configurational Dirac-Hartree-Fock approximation. We conclude with a survey of representative applications of the relativistic self-consistent field method to be found in the literature. 

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. 

In molecular quantum chemistry Gaussian-fxmction-based computations, effective core potentials were originally derived from a reference calculation of a single atom within the nonrelativistic Hartree-Fock or relativistic Dirac-Fock (see Sect. 8.3) approximations, or from some method including electron correlations (Cl, for instance). A review of these methods, as well as a general theory of ECPs is provided in [480,481]. 

Based on the Dirac-Coulomb-Breit operator, most known methods of quantum-chemical ab initio electronic structure determination have been implemented by now also for four-component spinors. This comprises time-honoured pioneering work on atoms in the Dirac-Hartree-Fock framework, using numerical techniques and basis set expansion techniques, " as well as work for molecules in Dirac-Har tree-Fock approximations with global basis sets " or finite elements and elaborate techniques to treat relativity and correlation on the same footing. " ... 

Aerts, P. J. C. (1986) Towards relativistic quantum chemistry—on the ab initio calculation of relativistic electron wave functions for molecules in the Hartree-Fock-Dirac approximation. PhD thesis, Rijksuniversiteit te Groningen, Netherlands. 

Eq. 20 can be solved iteratively and yields the same one-particle energies as the corresponding Dirac-equation. The radial functions P K r) correspond to the large components. In the many-electron case the correct nonlocal Hartree-Fock potential is used in Eq. 21, but a local approximation to it in Eqs. 22. Averaging over the relativistic quantum number k leads to a scalar-relativistic scheme. 

An overview of the salient features of the relativistic many-body perturbation theory is given here concentrating on those features which differ from the familiar non-relativistic formulation and to its relation with quantum electrodynamics. Three aspects of the relativistic many-body perturbation theory are considered in more detail below the representation of the Dirac spectrum in the algebraic approximation is discussed the non-additivity of relativistic and electron correlation effects is considered and the use of the Dirac-Hartree-Fock-Coulomb-Breit reference Hamiltonian demonstrated effects which go beyond the no virtual pair approximation and the contribution made by the negative energy states are discussed. 

I started this manuscript with a quotation from Professor Hartree s famous book on numerical integration. With the development of the methodology of the Hartree-Fock-Slater approximation on a spreadsheet, perhaps to finish, I can be allowed the temerity to quote a well-known comment by one of the giants of quantum mechanics. Professor P.A.M. Dirac (69),... 

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. 

CPD=Chang - Pelissier- Durand DCB = Dirac - Coulomb -Breit DHF = Dirac-Hartree-Fock DK = Douglas-Kroll FORA = first-order regular approximation MVD = mass-velocity-Darwin term QED = quantum electrodynamics ZORA = zero-order regular approximation. 

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