Relativistic Dirac-Hartree-Fock method

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. 

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. 

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) ... 

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. 

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. 

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 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. 

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. ... 

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

DHF (Dirac -Hartree-Fock) relativistic ah initio method DHF (derivative Hartree-Fock) a means for calculating nonlinear optical properties... 

Table 6 Matrix Dirac-Hartree-Fock (Edhf) and Hartree-Fock (Ehf) energies calculated using BERTHA. The Gaussian exponential parameters are those of the non-relativistic sets derived by van Duijenveldt and tabulated in Poirier et al [36]. Thejirst-order molecular Breit energy, Eb, v as calculated using methods described in [12] relativistic corrections to Ehf collected in the column labelled E energies are in atomic units.

The relativistic or non-relativistic random-phase approximation (RRPA or RPA)t is a generalized self-consistent field procedure which may be derived making the Dirac/Hartree-Fock equations time-dependent. Therefore, the approach is often called time-dependent Dirac/Hartree-Fock. The name random phase comes from the original application of this method to very large systems where it was argued that terms due to interactions between many alternative pairs of excited particles, so-called two-particle-two-hole interactions ((2p-2h) see below) tend to... 

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-... 

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. 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

The incorporation of electron correlation effects in a relativistic framework is considered. Three post Hartree-Fock methods are outlined after an introduction that defines the second quantized Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. Aspects that are considered are the approximations possible within the 4-component framework and the relation of these to other relativistic methods. The possibility of employing Kramers restricted algorithms in the Configuration Interaction and the Coupled Cluster methods are discussed to provide a link to non-relativistic methods and implementations thereof. It is shown how molecular symmetry can be used to make computations more efficient. 

An important problem in the application of QED methods to many-electron atoms is the choice of the zero-order approximation (actually the choice of the basis set of the one-electron relativistic wave functions in Eq(86). One natural choice is the approximation of noninteracting electrons when the potential V in Eq(2) is the Coulomb potential of the nucleus (28). This approximation is convenient for highly charged, few-electron ions. For a many-electron neutral atom a better choice is the Dirac-Hartree-Fock (DHF) approximation. 

Dirac-Hartree-Fock and Dirac-Kohn-Sham methods By an application of an independent-particle approximation with the DC or DCB Hamiltonian, the similar derivation of the non-relativistic Hartree-Fock (HF) method and Kohn-Sham (KS) DFT yields the four-component Dirac-Hartree-Fock (DHF) and Dirac-Kohn-Sham (DKS) methods with large- and small-component spinors. 

DFT-Based Pseudopotentials. - The model potentials and shape-consistent pseudopotentials as introduced in the previous two sections can be characterized by a Hartree-Fock/Dirac-Hartree-Fock modelling of core-valence interactions and relativistic effects. Now, Hartree-Fock has never been popular in solid-state theory - the method of choice always was density-functional theory (DFT). With the advent of gradient-corrected exchange-correlation functionals, DFT has found a wide application also in molecular physics and quantum chemistry. The question seems natural, therefore Why not base pseudopotentials on DFT rather than HF theory ... 

While the relativistically parameterised extended Hiickel approach to the calculation of molecular parity violating effects has the merit of simplicity, it suffers in particular from the non-self-consistent character of the extended Hiickel method. This problem is avoided in the four-component Dirac Hartree-Fock approaches to the computation of parity violating potentials in chiral molecules introduced by Quiney, Skaane and Grant [155] as well as Laerdahl and Schwerdtfeger [156]. These will be described in the following section. 

In Table 3.2 we compare results of various methods for Au2. Four-component Dirac-Hartree-Fock (DHF) [133] as well as nonrelativistic DFT-LDA [14] results remind one that correlation as well as relativistic effects have to be taken into account to achieve a reasonable description of a molecule containing heavy elements. DHF overestimates the Au2 bond length compared to experiment by 10 pm and underestimates the vibrational frequency by 30 cm (Table 3.2) the calculated binding energy is less than half of the experimental value, 223 kJ/mol [134]. Noiu elativistic DF VWN calculations overestimate the bond... 

---