Dirac Hartree-Fock method

The expression for the lowest order contribution to the parity violating potential within the Dirac Hartree-Fock framework is identical to that within the relativistically parameterised extended Hiickel approach in eq. (146). The difference is, however, that in DHF typically atomic basis sets with fixed radial functions are employed (see [161]) and that the molecular orbital coefficients are obtained in a self-consistent Dirac Hartree-Fock procedure. Computations of parity violating potentials along these lines have occasionally been called fully relativistic, although this term is rather unfortunate. In the four-component Dirac Hartree-Fock calculations by Quiney, Skaane and Grant [155] as well as in those by Schwerdtfeger, Laerdahl and coworkers [65,156,162,163] the Dirac-Coulomb operator has been employed, which is for systems with n electrons given by 

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These (see Chapter 2) may be obtained utilizing the relativistic analogue of the Hartree-Fock method, normally called the Dirac-Hartree-Fock method [176-178], The relevant equations may be found in an analogous manner to the non-relativistic case, therefore here we shall present only final results (in a.u. let us recall that X = nlj, X = nl j) ... 

An overview of the development of the finite difference Hartree-Fock method is presented. Some examples of it axe given construction of sequences of highly accurate basis sets, generation of exact solutions of diatomic states, Cl with numerical molecular orbitals, Dirac-Hartree-Fock method based on a second-order Dirac equation. 

Post Dirac-Hartree-Fock Methods - Electron Correlation... 

The Dirac-Hartree-Fock iterative process can be interpreted as a method of seeking cancellations of certain one- and two-body diagrams.33,124 The self-consistent field procedure can be regarded as a sequence of rotations of the trial orbital basis into the final Dirac-Hartree-Fock orbital set, each set in this sequence forming a basis for the Furry bound-state interaction picture of quantum electrodynamics. The self-consistent field potential involves contributions from the negative energy states of the unscreened spectrum so that the Dirac-Hartree-Fock method defines a stationary point in the space of possible configurations, rather that a variational minimum, as is the case in non-relativistic theory. 

In the following subsections I will outline the methodologies used for the calculation of P-odd effects in chiral molecules, namely the methodology of the pioneering four-component study by Barra, Robert and Wiesenfeld, the Dirac Hartree-Fock method and the four-component coupled cluster approach. 

Spin-dependent operators are required when we wish to account for relativistic effects in atoms and molecules [118, 119]. These effects can roughly be classified as strong and weak ones. The relativistic corrections are especially important in heavy atoms where they play a particularly significant role when describing the inner shells. In those cases, they have to be accounted for from the start, usually relying on Dirac-Hartree-Fock method. Fortunately, in most chemical phenomena, only valence electrons play a decisive role and are satisfactorily... 

T. Saue, H. J. A. Jensen. Quaternion symmetry in relativistic molecular calculations The Dirac-Hartree-Fock method. /. Chem. Phys., 111(14) (1999) 6211-6222. 

T. Yanai, T. Nakajima, Y. Ishikawa, K. Hi-rao. A new computational scheme for the Dirac-Hartree-Fock method employing an efficient integral algorithm. /. Chem. Phys., 114(15) (2001) 6526-6538. 

Relativistic calculations of NMR properties of RgH ion (where Rg = Ne, Ar, Kr, Xe), Pt shielding in platinum complexes, and Pb shielding in solid ionic lead(II) halides have been reported in this review period. For the Rg nucleus in the RgH ions, the following methods were used and results compared with each other non-relativistic uncorrelated method (HF), relativistic uncorrelated methods, four component Dirac Hartree-Fock method (DHF) and two-component zeroth order regular approach (ZORA), non-relativistic correlated methods using second order polarization propagator approach SOPPA(CCSD), SOPPA(MP2), respectively coupled cluster singles and doubles or second order Moller-Plesset, and... 

The non-relativistic Hartree-Fock and relativistic Dirac-Hartree-Fock methods have been applied by Cukras and Sadlej to calculate the NMR shielding constants and, for the first time, the spin-spin couplings in noble gas hydride cations RgH, where Rg = Ne, Ar, Kr, Xe. 

Computational Scheme for the Dirac-Hartree-Fock Method Employing an Efficient Integral Algorithm. 

An alternative to the operator approach is to start from the matrix equations (Filatov 2002). Then the elimination the small-component, the construction of the transformation and the transformed Fock matrix are all straightforward. There is no difficulty with interpretation because the inverse of a matrix is well defined. The matrix to be inverted is positive definite so it presents no numerical problems. The drawback of a matrix method is that the basis set for the small component must be used, at least to construct the potentials that appear in the inverse. In that case, the same number of integrals is required as in the full Dirac-Hartree-Fock method, and there is no reduction in the integral work or the construction of the Fock matrix. 

The above values are obtained using non-relativistic values calculated at the CCSD level of theory combined with relativistic corrections derived from the Dirac-Hartree-Fock method, plus dynamic and solvent effects estimated from cluster calculations involving water molecules that have been generated from Car-Parrinello molecular dynamics simulations. 

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

Dirac s one-particle equation was soon generalized to an equation for an electron in the self-consistent field of the other electrons in an atom - the relativistic analogue of the Hartree-Fock approximation for many-electron atoms. These equations - together with the proper bookkeeping to account for the notion of the Dirac sea - - are the basis of what is nowadays called Dirac-Hartree-Fock methods in relativistic electronic structure theory. ... 

The purpose of this contribution is to give an overview of the results which center around the atomic density function and the recovery of the periodicity. Since all the calculations are based on atomic density functions, it is appropriate to revisit the construction of these densities in some depth. First a workable definition of the density function is established in the framework of the multi-configuration Hartree-Fock method (MCHF) [2] and the spherical harmonic content of the density function is discussed. A spherical density function is established in a natural way, by using spherical tensor operators. The proposed expression can be evaluated for any multi-configuration state function corresponding to an atom in a particular well-defined state and a recently developed extension of the MCHF code [3] is used for that purpose. Three illustrative examples are given. In the next section relativistic density functions for the relativistic Dirac-Hartree-Fock method [4] are defined. The latter will be used for a thorough analysis of the influence of relativistic effects on electron density functions later on in this paper. 

Saue T. Post Dirac-Hartree-Fock Methods - Properties. In Schwerdtfeger P, editor. Relativistic Electronic Structure Theory. Part 1. Fundamentals. Amsterdam Elsevier 2002. p. 332. 

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