Unrestricted Hartree-Fock method dissociation

Corrections for Improper HF Asymptotic Behaviour.—There are two techniques which may be used to obtain results at what is essentially the Hartree-Fock limit over the complete range of some dissociative co-ordinate in those cases where the single determinants] approximation goes to the incorrect asymptotic limit. These techniques are (i) to describe the system in terms of a linear combination of some minimal number of determinantal wavefunctions (as opposed to just one) 137 and (ii) to employ a single determinantal wavefunction to describe the system but to allow different spatial orbitals for electrons of different spins - the so-called unrestricted Hartree-Fock method. Both methods have been used to determine the potential surfaces for the reaction of an oxygen atom in its 3P and 1Z> states with a hydrogen molecule,138 and we illustrate them through a discussion of this work. 

It is generally found that if one increases the flexibility of a single-determinantal wavefunction by allowing each space orbital to assume an independent form (rather than insisting on double occupation by an and a (1 electron for those orbitals which would otherwise be so occupied as dictated by the electronic configuration) that the asymptotic difficulties of the wavefunction are removed. Thus, the unrestricted Hartree-Fock method usually predicts the correct dissociation products of a molecular system.140 The symmetrical (C2 ,) insertion of Of3P) into Ha yields the 33i state of the HaO system. The electronic configuration of this state expressed in terms of the unrestricted set of orbitals is... 

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. 

There are a number of slightly more approximate methods for determining the electron affinity (EA) based on the restricted Hartree-Fock (RHF) and spin-unrestricted Hartree-Fock (UHF) methods. For closed shell anions, molecules which dissociate to... 

The Coulson-Fischer wave function for H2 can be considered as the start of the Unrestricted Hartree-Fock (UHF) approach in quantum chemistry, which is the most general single determinant method. We shall not proceed further along this line, but instead ask ourselves if there is a way to correct the simation such that we obtain a wave function that dissociates correctly while preserving the spin and space symmetry of the wave function. The CF wave function gives acmally a hint. What happens if we simply skip the trouble-some triplet term in Eq. (22). This gives rise to a wave function that is a linear combination of two closed shell determinants ... 

The Restricted Hatiree-Fock (RHF) method used to describe the dissociation of the chemical bond gives simply tragic results (cf. Chapter 8, p. 437), qualitatively wrong on the other hand, the Unrestricted Hartree-Fock (UHF) method gives a qualitatively correct description. 

Numerous steps have been undertaken in order to overcome these shortcommings of the standard CCSD method. The simplest way to achieve a proper dissociation limit (size-consistency) is to employ the unrestricted Hartree-Fock (UHF) reference. This often works rather well, except that UHF solution(s) exist(s) only in a limited range of internuclear separations and, at the onset of the RHF triplet instability the computed energies display a nonanalytic behavior. Of course, in more general situations, the UHF solution may dissociate to a wrong limit [cf., e.g. Refs. 4J0)]. not to mention the multiplicity and often haphazard behavior of various broken-spin-symmetry solutions, spin contamination, etc 4), Thus, this approach is usually reserved for computation of dissociation energies rather than for the generation of accurate PESs. 

The second approach to treating nondynamical correlation has an air of the ostrich about it ignore the spin symmetry of the wave function and use unrestricted Haxtree-Fock (UHF) theory as the single configuration description [7]. Since the UHF wave function comprises one spin-orbital for each electron, a molecular UHF wave function should dissociate to atomic UHF wave functions, for example. This is certainly not the case for spin-restricted Hartree-Fock (RHF) molecules and atoms in general. And there is an attractive simplicity about UHF — no active orbitals to identify, and so forth. However, where nondynamical correlation would be important in an RHF-based treatment, the UHF method will suffer from severe spin-contamination, while where nondynamical correlation is not important the RHF solution may be lower in energy than any broken-symmetry UHF solution, so potential curves and surfaces may have steps or kinks where the spin symmetry is broken in the UHF treatment. 

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