Restricted open-shell Hartree-Fock approximation

All the calculations of F2 are carried out with a simple basis set of double-zeta polarization type, the standard 6-31G(d) basis set, and are performed at a fixed interatomic distance of 1.44 A, which is approximately the optimized distance for a full Cl calculation in this basis set. Only the corresponding orbitals are referred to as the active orbitals , while the orbitals representing the lone pairs, so-called spectator orbitals , remain doubly occupied in all calculations. A common point to the various VB methods we use, except the VBCI method, is that at the dissociation limit, the methods converge to two F fragments at the restricted-open-shell Hartree Fock (ROHF) level. 

Typical structures are specified in Table 1 which uses the labelling of carbon atoms in Cjo defined in Fig. 1. The restricted open-shell Hartree-Fock (ROHF) method was used in all geometry optimizations using a minimal basis set of orbitals (STO-3G) [13]. These calculations are therefore exploratory in nature. Here we have chosen to use the standard ab initio ROHF method since it is well-known that the UHF method (as used in the PRDDO approximation [9]) does not give wave functions which are eigenstates of the total spin operator S. The effect of spin contamination on molecular properties is uncertain, particularly if the contamination is high (the... 

The restricted open-shell Hartree-Fock (ROHF) and the unrestricted Hartree Fock Method ( UHF) approximations permit, however, open-shell systems to be described, while maintaining the simplicity of the single-determinant approximation. This is made at the stage of self-consistent electronic-structure calculations. Afterwards, the obtained spin-orbitals can be used to get the correct total spin many-determinant wavefunction and to calculate the corresponding electron energy. 

The electronic structure is being described by a special case of the singledeterminant model the spin-eigenfunction-restricted, spherical-approximation open-shell Hartree-Fock model. 

The poor behavior of restricted closed-shell Hartree-Fock calculations upon dissociation to open-shell products does not detract from their utility in the region of equilibrium. The calculated equilibrium geometry is that at which tot is a minimum with respect to the coordinates of the nuclei. Table 3.2 shows the value of this energy for intemuclear distances in the vicinity of the experimental bond length of 1.4 a.u. The calculated minimum energy occurs at 1.346 a.u. This is in error by 4% and errors of similar magnitude can be expected for equilibrium geometries of other molecules at this level of approximation. 

Closed-shell (diamagnetic) systems can be investigated using a restricted Hartree-Fock (RHF) calculation, while unrestricted Hartree-Fock (UHF) calculations are able to accommodate open-shell (paramagnetic) systems as well. The Hartree-Fock approximation is also important in serving as a foundation for a variety of more accurate quantum chemical calculations that account for electron correlation. 

In this subsection, we describe restricted closed-shell calculations on the ground state of H2. As we will see, there is a very basic deficiency in such calculations at long bond lengths. Later in this chapter, when we describe unrestricted open-shell calculations, we will return to minimal basis H2 and partially correct this deficiency. Some of the results obtained here will also be used in later chapters when we use the minimal basis H2 model to illustrate procedures that go beyond the Hartree-Fock approximation. 

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

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. 

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. 

G bases, except for the anions for which the MP2 energies were calculated in the former basis set only. Excitations from the core electrons were not included in the MP2 treatment. The restricted Hartree-Fock (RHF) method was used for the closed-shell molecules (parents and anions) and the unrestricted Hartree-Fock (UHF) method was applied to the spin doublet open-shell species (radicals and cations). Both methods are variants of the SCF approximation. A fuller description and explanation of the basis set and methods has been given previously8. 

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