Hartree-Fock scheme, general

The idea of constructing a good wave function of a many-particle system by means of an exact treatment of the two-particle correlation is also underlying the methods recently developed by Brueck-ner and his collaborators for studying nuclei and free-electron systems. The effective two-particle reaction operator and the self-consistency conditions introduced in this connection may be considered as generalizations of the Hartree-Fock scheme. 

A common feature of the Hartree-Fock scheme and the two generalizations discussed in Section III.F is that all physical results depend only on the two space density matrices p+ and p, which implies that the physical and mathematical simplicity of the model is essentially preserved. The differences lie in the treatment of the total spin in the conventional scheme, the basic determinant is a pure spin function as a consequence of condition 11.61, in the unrestricted scheme, the same determinant is a rather undetermined mixture of different spin states, and, in the extended scheme, one considers only the component of the determinant which has the pure spin desired. 

In the bibliography, we have tried to concentrate the interest on contributions going beyond the Hartree-Fock approximation, and papers on the self-consistent field method itself have therefore not been included, unless they have also been of value from a more general point of view. However, in our treatment of the correlation effects, the Hartree-Fock scheme represents the natural basic level for study of the further improvements, and it is therefore valuable to make references to this approximation easily available. For atoms, there has been an excellent survey given by Hartree, and, for solid-state, we would like to refer to some recent reviews. For molecules, there does not seem to exist something similar so, in a special list, we have tried to report at least the most important papers on molecular applications of the Hartree-Fock scheme, t... 

Calculations based on the continuum dielectric model have been performed by the hydrated electron in the limit of zero cavity size (19). The general treatment is based on a variational calculation using hydrogenic type wave functions for the ground and the first excited states. This treatment is based on a Hartree Fock scheme, where the Coulomb and exchange interaction of the excess electron with the medium are replaced by the polarization energy of a continuous dielectric. The results obtained are summarized in Table V. The fair agreement obtained with... 

In this paper, the general theory developed in Part I is applied to the Hartree-Fock Scheme for a transformed many-electron Hamiltonian. It is shown that, if the transformation is a product of one-electron transformations, then the Fock-Dirac operator as well as the effective Hamiltonian undergo similarity transformations of the one-electron type. The special properties of the Hartree-Fock scheme for a real self-adjoint Hamiltonian based on the bi-variational principle are discussed in greater detail. 

The general stability problem for a pair of adjoint many-particle operators T and T has been discussed in a previous paper2, which will be referred to as reference B. The Hartree-Fock scheme for a pair of such operators has also been discussed, and this paper will be referred to as reference C. Some of the most important results in the references A, B, and C will be briefly reviewed here to make the presentation more self-contained. 

In Sec. 3.3, we have emphasized that the complex symmetric Hartree-Fock scheme usually does not reduce to the conventional one for the special case when q = 1. More generally, in the complex Hartree-Fock scheme, the results obtained from a starting value q i do not necessarily reduce to the results obtained from another starting value q2 - he. each starting value leads to a specific result. 

We note that the GHF-scheme is identical with the Hartree-Fock scheme originally derived by Fock and Slater, since in their derivations there were no restrictions imposed on the nature of the one-electron functions. This means that the GHF-scheme is still based on the equations (1.3)-(1.6). It should be observed that the functions yk.t-( ) Vk-(r) in general are of complex nature. 

It is somewhat surprising that methods based on the Hartree-Fock scheme such as GIAO or IGLO work so well, in view of the rather approximate character of Hartree-Fock. The reason is probably that the perturbating operator in a magnetic field is a one-electron operator, such that a one-electron approximation should not perform too poorly. The inclusion of electron correlation, however, is necessary in two situations (a) in special cases, even in order to get acceptable first order results (b) generally, if one strives at very high accuracy. 

Abstract This contribution reviews a selection of findings on atomic density functions and discusses ways for reading chemical information from them. First an expression for the density function for atoms in the multi-configuration Hartree-Fock scheme is established. The spherical harmonic content of the density function and ways to restore the spherical symmetry in a general open-shell case are treated. The evaluation of the density function is illustrated in a few examples. In the second part of the paper, atomic density functions are analyzed using quantum similarity measures. The comparison of atomic density functions is shown to be useful to obtain physical and chemical information. Finally, concepts from information theory are introduced and adopted for the comparison of density functions. In particular, based on the Kullback-Leibler form, a functional is constructed that reveals the periodicity in Mendeleev s table. Finally a quantum similarity measure is constructed, based on the integrand of the Kullback-Leibler expression and the periodicity is regained in a different way. 

Early determinations of RSE values employed unrestricted Hartree-Fock (UHF) theory in combination with 3-21G [9] or 4-31G [10] basis sets to evaluate the RSE according to Eq. 1. The appropriate consideration of correlation effects, the avoidance of spin contamination, and the treatment of thermochemical corrections have in detail been studied in the following, in particular by Bauschlicher [11], Coote [12-14], Morokuma [15-18], and Radom [19-25]. Highly accurate RSE and BDE results can be obtained with high level compound methods such as the G2 [26-30] and G3 [31-34] schemes (and variants thereof [11,15-18]), as well as extrapolation methods such as the CBS schemes [35,36], Wl, or W2 [37-39]. Generally, the accurate... 

Tupitsyn, I. I. and Mosyagin, N. S. grecp/hfj 1995. Program for atomic finite-difference two-component Hartree-Fock calculations with the generalized RECP in the //-coupling scheme. 

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