RHF method

The UIIF wnive fimction can also apply to singlet molecules. F sn-ally, the results are the same as for the faster RHF method. That is, electron s prefer to pair, with an alpha electron sh arin g a m olecu lar space orbital with a beta electron. L se the L lIF method for singlet states only to avoid potential energy discontinuities when a covalent bond Is broken and electron s can impair (see Bond Breaking on page 46). 

For systems with unpaired electrons, it is not possible to use the RHF method as is. Often, an unrestricted SCF calculation (UHF) is performed. In an unrestricted calculation, there are two complete sets of orbitals one for the alpha electrons and one for the beta electrons. These two sets of orbitals use the same set of basis functions but different molecular orbital coefficients. 

In the RHF method, the FSGO describing the electron pairs is doubly occupied and the wave-function has the form ... 

Table 1. shows the total energies obtained using the RHF method for 1. LCAO minimal basis set STO-IG for the sake of comparison with FSGO, 2. FSGO in its symmetric and broken symmetry solutions and, 3. LCAO minimal basis set STO-3G in order to allow a safer comparison with the quality of the subminimal basis used in the FSGO technique. The dissociation curves are given in Figure 1. 

The charge densities and bond orders of the hypervalent bond of these telluranes 159, 160, and 161 were calculated by the ab initio RHF method on the basis of 3-21G. The results are shown in Figure 8. Interestingly, these ab initio calculations reveal that the positive charges at the hyperavalent bond 159 and 160 are distributed... 

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,... 

A very important aspect of the results described above for De is that the error obtained at a certain level of approximation is systematic. This fact combined with the fact that the results improve as the method improves are aspects of ab initio methods which are at least as important as the final accuracy of the results. So far the only property discussed is De. It is clear that the most important chemical information, such as reaction pathways and thermochemistry, is obtained from relative energies, but the accuracy of other properties is also of interest. If we look at the equilibrium bond distance Re and the harmonic vibrational frequency we, these properties also display a systematic behaviour depending on the method chosen and this systematic behaviour is easy to understand. Since the RHF method dissociates incorrectly, the potential curves tend to rise too fast as the bond distance is increased. At the RHF level this leads to too short equilibrium bond distances and vibrational frequencies that are too high. When proper dissociation is included at the MCSCF level, the opposite trend appears. Since the dissociation energies are too small at this level the potential curves rise too slowly as the bond distance increases. This leads to too long bond distances and too low frequencies. These systematic trends are nicely illustrated by the results for three of the previously discussed diatomic molecules. For H2 the experimental value for Re is 1.40 ao and for uje it is 4400 cm-1. At the RHF level Re becomes too short, 1.39 ao, and we becomes too high, 4561 cm-1. At the two configuration MCSCF level Re becomes... 

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. 

The common way to treat free radicals is with the unrestricted Hartree-Fock method or UHF method. In this method, we employ separate spatial orbitals for the oc and the jl electrons, giving two sets of MO s, one for oc and one for fj electrons. Less commonly, free radicals are treated by the restricted open-shell Hartree-Fock or ROHF method, in which electrons occupy MO s in pairs as in the RHF method, except for the unpaired electron(s). The theoretical treatment of open-shell species is discussed in various places in references [1] and in [12]. 

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. 

In the restricted Hartree-Fock (RHF) method, two restrictions are placed on the molecular orbitals u< in equation (11). The first is chat each ui transform according to one of the irreducible representations of the point group of the molecule. The second restriction is that the space functions u come in identical pairs one with spin function a and the other with spin function /S. These are called, respectively, the symmetry and equivalence restrictions.190... 

Most calculations reported above have used various forms of the RHF method,... 

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. 

The many-electron wave function of a molecular system is taken as the antisymmetrized product of (pt, and for closed-shell systems it is convenient to represent it by a Slater determinant. Such an approach is known as the restricted Hartree-Fock (RHF) method and is the most widely used method in chemisorption calculations. Its principal drawback is the neglect of Coulomb electron correlation, which is of crucial importance for adequate treatment of chemical rearrangements with varying numbers of electron pairs. 

A somewhat modified MO LCAO scheme, without restriction on the identity of spin orbitals (p and

unrestricted Hartree-Fock (UHF) method and is usually used to treat open-shell systems (free radicals, triplet states, etc.). Electron correlation is partially taken into account in this method, and therfore it can be expected to be more efficient than the RHF method when applied to calculate potential energy surfaces of chemical rearrangements whose intermediate or final stages may involve the formation of free- or bi-radical structures. The potentialities of the UHF method are now under active study in organic reaction calculations. Also, it is successfully coming into use in chemisorption computations (6). 

It should be observed that, above R - 2.1 A, onfy the UHF/1 solution could be obtained. Similarly, below R = 1.82 A. the numerical effects cause a "Jump" from the UHF/1 branch to the UHF/2 one In the course of the SCF procedture, since UHF/2 has a lower energy at that distance. This means that there is an interval of a length of about 0.2 A. in which all the six different SCF solutions exist simultaneously. The situation is best characterized by Figs. 2a and 2b, in which the energy lowering with respect to the RHF method given by the different UHF and... 

The preceding development of the HF theory assumed a closed-shell wavefunction. The wavefunction for an individual electron describes its spatial extent along with its spin. The electron can be either spin up (a) or spin down (P). For the closed-shell wavefunction, each pair of electrons shares the same spatial orbital but each has a different spin—one is up and the other is down. This type of wavefunction is also called a (spin)-restricted wavefunction since the paired electrons are restricted to the same spatial orbital, leading to the restricted Hartree-Fock (RHF) method. 

In general, the relative spin state energies calculated for all the model heme complexes studied are consistent with and help explain their observed electromagnetic properties. Thus the INDO-RHF method used appears to be sensitive to the effect of the varying axial ligands and predicts the correct energy order of spin states produced by each of them. 

The CCSD results of Salek et al (2005) and the MCSCF results of Luo et al (1993) compute the frequency dependence as an intrinsic part of the correlated calculation. The work of Sim et alP (1993) and Reiss et al. (2005) takes the static value obtained at the MP2 level and scales it using the RPA method to get the frequency dependence. Luo et alP (1993) also report the result of an RHF/RPA calculation where the frequency dependence is the natural extension of the RHF method. The plotted points are at the four readily available laser frequencies that have been used in almost all experimental work. The most popular of these has been the YAG frequency corresponding to 1.17 eV or 1064 nm. At this frequency the spread of results ranges from about 1550 to 2600 au. If only the two fully frequency-dependent correlated calculations are considered the range is from about 1700 to 2600 au. Salek et al, using the CCSD method find that as the frequency is increased from zero to 1.17 eV, increases from 1736 to 2667 au and Luo et al, using MCSCF, from 1373 to 1898 au. 

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