Hartree-Fock method relativistic

Computed using Hartree-Fock methods and Including an approximate relativistic correction (24). 

As was the case with lanthanide crystal spectra (25), we found that a systematic analysis could be developed by examining differences, AP, between experimentally-established actinide parameter values and those computed using Hartree-Fock methods with the inclusion of relativistic corrections (24), as illustrated in Table IV for An3+. Crystal-field effects were approximated based on selected published results. By forming tabulations similar to Table IV for 2+, 4+, 5+ and 6+ spectra, to the extent that any experimental data were available to test the predictions, we found that the AP-values for Pu3+ provided a good starting point for approximating the structure of plutonium spectra in other valence states. However,... 

In most atomic programs (5) is actually solved self-consistently either in a local potential or by the relativistic Hartree-Fock method. There is, however, an important time-saving device that is often used in energy band calculations for actinides where the same radial Eq. (5) must be solved If (5.a) is substituted into (5.b) a single second order differential equation for the major component is obtained... 

In this respect, the single-configurational Hartree-Fock method looks more promising and universal when combined with accounting for the relativistic effects in the framework of the Breit operator and for correlation effects by the superposition-of-configurations or by some other method (e.g. by solving the multi-configurational Hartree-Fock-Jucys equations (29.8), (29.9)). 

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) ... 

It was pointed out in chapter 1 that there exist alternative mean-field theories to the Hartree-Fock method. In particular, one of these, the <7-Hartree method, is a fully relativistic theory which determines the optimum mean field in such a way as to make the Lagrangian of quantum field theory stationary. This is a fundamental choice, but turns out [230] to be satisfied by a whole family of SCF potentials of the general form... 

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. 

The main relativistic effects come not from the terms arising from 1/r, being made Lorenz invariant, but from the one-electron terms (H and H4) in Eq. (2) of Reference 98 (see also Reference 96). The increase in the mass of an electron at higher velocities causes, as in mesonic atoms, (i) tighter binding to nuclei and ( ) contraction of the orbits. These effects that come from the one-electron terms are included in the relativistic Hartree-Fock method. > ... 

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. 

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. 

Another approach is to do a nonrelativistic calculation, using, for example, the Hartree-Fock method, and then use perturbation theory to correct for relativistic effects. For perturbation-theory formulations of relativistic Hartree-Fock calculations and relativistic KS DFT calculations, see W. Kutzelnigg, E. Ottschofski, and R. Franke, J. Chem. Phys., 102,1740 (1995) and C. van Wiillen, J. Chem. Phys., 103,3589 (1995) 105,5485 (1996). 

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... 

Calculations were performed with the ab initio Hartree-Fock method (4) and in some cases with methods incorporating electron correlation such as MP2 (5) or BLYP (6), Geometry optimizations were initally done with relativistic effective core potential polarized valence double-zeta bases (7) and then refined with standard 6-3IG bases (8). NMR shieldings were evaluated using the GIAO-SCF method (9) and 6-3IG bases. We used the quantum chemical software GAMESS (10) and GAUSSIAN (11). 

An interesting approach to the quantum mechanical description of many-electron systems such as atoms, molecules, and solids is based on the idea that it should be possible to find a quantum theory that refers solely to observable quantities. Instead of relying on a wave function, such a theory should be based on the electron density. In this section, we introduce the basic concepts of this density functional theory (DFT) from fundamental relativistic principles. The equations that need to be solved within DFT are similar in structure to the SCF one-electron equations. For this reason, the focus here is on selected conceptual issues of relativistic DFT. From a practical and algorithmic point of view, most contemporary DFT variants can be considered as an improved model compared to the Hartree-Fock method, which is the reason why this section is very brief on solution and implementation aspects for the underlying one-electron equations. For elaborate accounts on nonrelativistic DFT that also address the many formal difficulties arising in the context of DFT, we therefore refer the reader to excellent monographs devoted to the subject [383-385]. 

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

J. Karwowski, M. Szulkin. Relativistic calculations on the alkali atoms by a modified Hartree-Fock method. /. Phys. B, 14 (1981) 1915-1927. 

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