Hartree-Fock method purpose

The purpose of the Hartree-Fock method is, of course, to find approximate eigenfunctions Ca and Da and eigenvalues I and... 

Some Applications. - Looking at the literature, one finds that, in atomic and molecular physics, comparatively little Interest has so far been devoted to the GHF-scheme, and the purpose of this paper is to try to focus the interests of some quantum chemists to this rather interesting problem of looking for the "absolute minimum" of the ori nal Hartree-Fock method. 

In this paper we have studied the original Hartree-Fock method - here referred as the GHF-scheme - for some simple atomic and molecular systems using general spin-orbitals (1.7) of a complex character, and we have found that there exist GHF-solutions of TSDW type which give lower energies than the RHF- and UHF-schemes, but also that quite a few SCF solutions of different type may exist simultaneously. The purpose of the... 

In general, the Hartree-Fock method indicates this SCF-based method. Despite the simplicity of the procedure, it soon became clear that solving this equation is not-trivial for usual molecular electronic systems. The Hartree-Fock equation essentially cannot be solved for molecules without computers. Actually, solving the Hartree-Fock equation for molecules had to await the appearance of general-purpose computers. 

Maroulis and Haskopoulos calculated the interaction electric dipole moment and polarizability for the C02-Rg systems, Rg = He, Ne, Ar, Kr and Xe. The potential minimum is very well defined for all these systems. In Fig. 19 is shown the potential energy surface for the C02-He interaction calculated at the MP2 level of theory. The most stable configuration corresponds to a T-shaped structure. The two local minima for the linear configuration of C02-He are also clearly visible. All interaction induced properties were extracted from finite-filed Moller-Plesset perturbation theory and coupled-cluster calculations with purpose-oriented basis sets. CCSD(T) values were calculated for the dipole moment pim of C02-He and C02-Ne the corresponding results are 0.0063 and 0.0107 eao, respectively. All post-Hartree-Fock methods yield stable values for this important property. For C02-He, = 0.0070 (SCF), 0.0063 (MP2), 0.0063 (MP4),... 

The aim of this chapter is to show, on the basis of several examples, how the location of the bond path may be useful to characterize, define and/or verify the specific, considered interaction. Mainly the QTAIM approach [4-7] is considered here however sometimes there are also references to other methods and concepts as for example the Namral Bond Orbitals (NBO) method [21, 22] or the o-hole concept [25-27]. This is worth to note that the results presented hereafter are mainly based on the MP2/aug-cc-pVTZ level of calculations those results are taken from earlier studies or the calculations were carried out especially for the purposes of this chapter. Consequently the QTAIM calculations were performed on the MP2/aug-cc-pVTZ wave functions. The binding energies (Ebin s) were calculated as differences between the energy of the complex and the sum of energies of monomers optimized separately and they were corrected for the basis set superposition error (BSSE) by the counterpoise method [28]. Since the NBO method is based on the Hartree-Fock method thus the corresponding NBO results, i.e. orbital-orbital interactions or atomic charges, if presented, are based on the HF/aug-cc-pVTZ//MP2/aug-cc-pVTZ level. Hence there is rather not indicated the level of calculations for the next systems discussed hereafter unless the results presented were obtained within other levels of calculations. 

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. 

While in principle all of the methods discussed here are Hartree-Fock, that name is commonly reserved for specific techniques that are based on quantum-chemical approaches and involve a finite cluster of atoms. Typically one uses a standard technique such as GAUSSIAN-82 (Binkley et al., 1982). In its simplest form GAUSSIAN-82 utilizes single Slater determinants. A basis set of LCAO-MOs is used, which for computational purposes is expanded in Gaussian orbitals about each atom. Exchange and Coulomb integrals are treated exactly. In practice the quality of the atomic basis sets may be varied, in some cases even including d-type orbitals. Core states are included explicitly in these calculations. 

MOPAC is a general-purpose semiempirical molecular orbital program for the study of chemical structures and reactions. It is available in desktop PC running Windows, Macintosh OS, and Unix-based workstation versions. It uses semiempirical quantum mechanical methods that are based on Hartree-Fock (HF) theory with some parameterized functions and empirically determined parameters replacing some sections of the complete HF treatment. The approximations in... 

For many ionization energies and electron affinities, diagonal selfenergy approximations are inappropriate. Methods with nondiagonal self-energies allow Dyson orbitals to be written as linear combinations of reference-state orbitals. In most of these approximations, combinations of canonical, Hartree-Fock orbitals are used for this purpose, i.e. 

With the development of powerful computers, these methods have been restricted in practice to the application in specific problems where no other calculations are available. For non-relativistic atomic systems, they have been replaced by quantum mechanical calculations like Monte Carlo or multiconfigurational Hartree-Fock ones. Nevertheless, Thomas-Fermi estimates can be easily evaluated by non-specialists in theoretical calculations and in some problems they provide a starting point for more sophisticated procedures. Moreover, they are interesting for theoretical purposes such as finding relationships among different average quantities [4]. 

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