Restricted Hartree-Fock approximation

To account for the magnetic hyperfine interaction in, e.g., the ground state of a lithium atom, it is necessary in the core model to go beyond the usual, restricted Hartree-Fock approximation, which assumes that the spatial parts of one-electron orbitals are independent of mB (the spin-... 

THE UNRESTRICTED VERSUS THE RESTRICTED HARTREE-FOCK APPROXIMATION. 

We will study the solution to Eq.(2.26lb) first in the restricted and then in the unrestricted Hartree-Fock approximation. According to the restricted Hartree-Fock approximation ... 

It must be remarked that in Blumberg s calculations, a restricted Hartree-Fock approximation has been used in computing the spin density at various nuclei. However, in the uiurestricted Hartree-Fock approximation where occupied orbitals with opposite spins do not have the same spatial wave function, there would be an additional contribution to the spin density at the nuclei and hence to the Knight shift. This additional contribution could also... 

Such a representation corresponds to the restricted Hartree-Fock approximation for open shells the functions of the tpi, orbitals with the a... 

In all cases where the question concerning the relative stabilities of equidistant versus bond alternating structures arises (polyyne [20,21, polyacetylene 22-27, polymethineimine 28,29 ) the latter are more stable within the framework of the restricted Hartree Fock approximation. For polyyne and polyacetylene this issue is in accord with the well known concept of a Peierls distortion jsoj. The occurence of Hartree Fock instabilities (see e.g. refs. 31,32 ) in the case of the equidistant, metallic structures of polyyne (cumulene) and all-trans polyacetylene points, however, to the need for improved methods going beyond the independent particle model. First efforts in this direction 27 show that at the level of second order Moller-Plesset perturbation theory the alternant configuration of polyacetylene is still preferred energetically although as expected the energy difference to the equidistant structures becomes smaller. 

Below is a brief review of the published calculations of yttrium ceramics based on the ECM approach. In studies by Goodman et al. [20] and Kaplan et al. [25,26], the embedded quantum clusters, representing the YBa2Cu307 x ceramics (with different x), were calculated by the discrete variation method in the local density approximation (EDA). Although in these studies many interesting results were obtained, it is necessary to keep in mind that the EDA approach has a restricted applicability to cuprate oxides, e.g. it does not describe correctly the magnetic properties [41] and gives an inadequate description of anisotropic effects [42,43]. Therefore, comparative ab initio calculations in the frame of the Hartree-Fock approximation are desirable. 

We review in this Section some recent work by Ludena, Lopez-Boada and Pino [113] on the construction of energy functionals that depend explicitly upon the one-particle density, but which are generated in the context of the local-scaling-transformation version of density functional theory. This work does not consider the general case involving exchange and correlation, but restricts itself to the exchange-only Hartree-Fock approximation. 

Pople refers to a specific set of approximations as defining a theoretical model. Hence the ab initio or Hartree-Fock models employ the Born-Oppenheimer, LCAO and SCF approximations. If the system under study is a closed-shell system (even number of electrons, singlet state), the constraint that each spatial orbital should contain two electrons, one with a and one with P spin, is normally made. Such wavefunctions are known as restricted Hartree-Fock (RHF). Open-shell systems are better described by unrestricted Hartree-Fock (UHF) wavefunctions, where a and P electrons occupy different spatial orbitals. We have seen that Hartree-Fock (HF) models give rather unreliable energies. 

In solving Eq. (2), an iterative process is used to adjust the until the best wavefunction is found [self-consistent field (SCF) theory]. For the open shell case where incompletely filled orbitals exist, spin-restricted Hartree-Fock (RHF) methods or unrestricted Hartree-Fock (UHF) methods may be used to calculate the energies.41 The extent of calculation, approximation, or neglect of the two-electron integral terms largely defines the computation method. 

When the second of the equivalence restrictions is removed, a single determinant wavefunction of lower energy is usually obtained. In fact, it is possible for a wave-function obtained in this way, a so-called unrestricted Hartree-Fock (UHF) wavefunction191 (perhaps more properly called a spin-unrestricted Hartree-Fock wavefunction) to go beyond the Hartree-Fock approximation and thus include some of the correlation energy. Lowdin192 describes this as a method for introducing a Coulomb hole to supplement the Fermi hole already accounted for in the RHF wavefunction. 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

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