Restricted Hartree-Fock method RHF

A variation on the HF procedure is the way that orbitals are constructed to reflect paired or unpaired electrons. If the molecule has a singlet spin, then the same orbital spatial function can be used for both the a and P spin electrons in each pair. This is called the restricted Hartree-Fock method (RHF). 

The Hartree-Fock description of the hydrogen molecule requires two spinorbitals, which are used to build the single-determinant two-electron wave function. In the Restricted Hartree-Fock method (RHF) these two spinorbitals are created from the same spatial... 

In the former, electrons are assigned to orbitals in pairs, the total spin is zero, so the multiplicity is 1. In this case, the restricted Hartree-Fock method (RHF) can be applied. For radicals with doublet or triplet states, the unrestricted Hartree-Fock (UHF) has to be applied. In this method, a and, 3 electrons (spin up and spin down) are assigned to different spatial orbitals, so there are two distinct sets I and FJf... 

In the Restricted Hartree-Fock method (RHF) for closed shell systems, we assume double occupancy of orbitals i.e., we form two spinorbitals out of each MO (by multiplying either by a or fi). 

Restricted Hartree-Fock method (RHF), 408-409, 499 retarded potential,... 

The wave function of Eq. (2.1) corresponds to the so-called restricted Hartree-Fock method (RHF) for the closed electron shells when 2n electrons... 

The great speed and known properties of RHF calculations are not sufficient justification for a limitation to RHF methods when they are inherently inappropriate. It is worth remarking that most potential-energy surfaces describing reactions, and many describing dissociations are inappropriate for RHF methods. Restricted Hartree-Fock methods are also of limited validity in many situations where two or more surfaces are at nearly the same energy. 

The method of calculating wavefunctions and energies that has been described in this chapter applies to closed-shell, ground-state molecules. The Slater determinant we started with (Eq. 5.12) applies to molecules in which the electrons are fed pairwise into the MO s, starting with the lowest-energy MO this is in contrast to free radicals, which have one or more unpaired electrons, or to electronically excited molecules, in which an electron has been promoted to a higher-level MO (e.g. Fig. 5.9, neutral triplet). The Hartree-Fock method outlined here is based on closed-shell Slater determinants and is called the restricted Hartree-Fock method or RHF method restricted means that the electrons of a spin are forced to occupy (restricted to) the same spatial orbitals as those of jl spin inspection of Eq. 5.12 shows that we do not have a set of a spatial orbitals and a set of [l spatial orbitals. If unqualified, a Hartree-Fock (i.e. an SCF) calculation means an RHF calculation. 

I. Flamant et al. successfully applied the FGSO basis set in Fourier Space Restricted Hartree Fock (FS-RHF) in a study of identification of conformational signatures in valence band of polyethylene. In 1998, they used a distributed basis set of s-type Gaussian function (DSGF) in FS-RHF. The method briefly is to use RHF-Bloch states (p (k,r), which are doubly occupied up to the Fermi energy Ep and orthonormalized. k and n the wave number and the band index, respectively. 

As several density functionals became available in the Gaussian program (19, 20), it became possible to study their effect on the computed value of electron affinity with systematically enlarged basis sets. The plan of the present work is as follows (a) initial calculations were done using the Hartree-Fock methods (RHF, the restricted Hartree-Fock ROHF, the restricted open-shell Hartree-Fock UHF, the unrestricted open-shell Hartree-Fock), and various non-local density functionals, defined in Table 2, with several standard basis sets (Section 2) (b) a large, saturated sp basis set was systematically enlarged by adding polarization functions with exponents optimized in molecular Hartree-Fock and density-functional cal-... 

Introductory descriptions of Hartree-Fock calculations [often using Rootaan s self-consistent field (SCF) method] focus on singlet systems for which all electron spins are paired. By assuming that the calculation is restricted to two electrons per occupied orbital, the computation can be done more efficiently. This is often referred to as a spin-restricted Hartree-Fock calculation or RHF. 

RHF (restricted Hartree-Fock) ah initio method for singlet systems ROHF (restricted open-shell Hartree-Fock) ah initio method for open-shell systems... 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

Such kind of calculations with a precise self-consistent account of crystal surrounding were performed by, at least, four scientific groups Baetzold [16], Das with coworkers [22,23], Winter etal. [28] and Ladik with coworkers [29]. Winter etal. [28] performed cluster calculation by restricted Hartree-Fock (RHF) method, so they did not take into account the electron correlation. The others groups used the umestricted Hartree-Fock (UHF) method for cluster calculations which allows to some extent the electron correlation. The strong covalent C-O bonding in planes and chains was revealed (in accordance with results obtained in Refs. [20,25,26]). For covalent systems,... 

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