Hartree-Fock method restricted

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). 

So far, we have considered only the restricted Hartree-Fock method. For open shell systems, an unrestricted method, capable of treating unpaired electrons, is needed. For this case, the alpha and beta electrons are in different orbitals, resulting in two sets of molecular orbital expansion coefficients ... 

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... 

Whereas there are only two different bond lengths in Ceo, short between atoms 1 and 2 and long between atoms 2 and 3, there are seven different bond lengths in C. The Crobond lengths have been calculated here and previously [12] by the restricted Hartree-Fock method using an STO-3G basis set and are discussed in some detail... 

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... 

Restricted Hartree-Fock method, 34 136 Retinal pigment models, 32 451-452 Reverse spillover, 34 2, 4, 11, 16, 31, 44, 46,... 

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. 

All the calculations have been made with the Gaussian 94 [5] or the Gaussian 98 [6] program. Molecular structures for all molecules studied here have been calculated by geometry optimization within the procedures of the restricted Hartree-Fock method (HF) but good agreement with experimental structures is obtained only accidentally with HF when using small basis sets, so that Post-SCF... 

Another example is provided by the EA of the state of the Na atom to generate the Na anion. Because the state is open shell, one would have to employ the unrestricted Hartree-Fock method to evaluate its orbitals and orbital energies to use in an EOM or GF EA calculation. However, one could, alternatively, compute the EA of Na by evaluating the IP of Na . The advantage would be that the Na is closed shell, so one could employ restricted Hartree-Fock methods to compute the requisite orbitals and orbital energies. 

For a homonuclear diatomic molecule within the Restricted Hartree-Fock method one finds to be equal to Eq.(2.51). Equation (2.310) shows that... 

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 (RHI, 408 109,499 retarded potential,... 

Kobayashi, Sasagane and Yamaguchi" have developed the theory of the time-dependent spin-restricted Hartree-Fock method for application to open shell systems (TDROHF). The expression for the cubic hyperpolarizability is obtained from the quasi-energy derivative (QED) method. The theory is applied to the investigation of the frequency-dependent y susceptibility of the Li, Na, K and N atoms. 

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... 

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