Spin-restricted Hartree-Fock RHF Method

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. 

There is a possibility for more than one solution of the Hartree-Fock equations if different electronic states come close on a potential energy surface. Within the spin-restricted Hartree-Fock (RHF) method, singlet and triplet instabilities are distinguished, The former involves the existence of another solution with lower energy and an electron distribution of lower symmetry, normally indicating that the initially assumed geometry is incorrect. Triplet instability involves rejection of the condition of double occupancy of molecular orbitals and a spin-unrestricted Hartree-Fock (UHF) method treatment is mandatory. The triplet instability is a necessary, but insufficient, condition for a biradical character of a ground state. 

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. 

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... 

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. 

Some caution should be exercised in the application of the size consistency concept to open-shell fragments, however. As Taylor has pointed out, a given method may be size consistent for some systems but not for others. For example, the spin-restricted Hartree-Fock (RHF) approach is size consistent for the dissociation of the hydrogen fluoride in its n excited state into atoms. 

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. 

The Roothaan-Hall equations are not applicable to open-shell systems, which contain one or more unpaired electrons. Radicals are, by definition, open-shell systems as are some ground-state molecules such as NO and 02. Two approaches have been devised to treat open-shell systems. The first of these is spin-restricted Hartree-Fock (RHF) theory, which uses combinations of singly and doubly occupied molecular orbitals. The closed-shell approach that we have developed thus far is a special case of RHF theory. The doubly occupied orbitals use the same spatial functions for electrons of both a and spin. The orbital expansion Equation (2.144) is employed together with the variational method to derive the optimal values of the coefficients. The alternative approach is the spin-unrestricted Hartree-Fock (UHF) theory of Pople and Nesbet [Pople and Nesbet 1954], which uses two distinct sets of molecular orbitals one for electrons of a spin and the other for electrons of / spin. Two Fock matrices are involved, one for each type of spin, with elements as follows ... 

Hartree-Fock theory employs a single Slater determinant. In the restricted Hartree-Fock (RHF) method, one spatial function 4>i is multiplied by an a (representing spin up, spin quantum number ms = +j) or P (representing spin down, nis = — ) spin function with the properties... 

It is convenient to have the Fockian in terms of spatial orbitals. The situation is simple in the closed-shell case where each orbital is required to be either doubly occupied or empty according to the restricted Hartree-Fock (RHF) method. Than, the spatial density matrix is given by Eq. (9.8), and the transcription of Eq. (10.55) can be performed in the usual manner. We find that the spatial Fock matrix is independent of spin ... 

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. 

The term Restricted Hartree-Fock (RHF) is applied to those cases in which all the possible spin pairing in a system is allowed for by having electrons of both and p spin occupy the same space orbital. If this restriction is relaxed in writing out the determinantal wavefunction, the method of calculation is referred to as the Unrestricted Hartree-Fock (UHF) method. Unless Otherwise stipulated, the calculations referred to in this chapter are of the RHF variety. 

Each spin orbital is a product of a space function fa and a spin function a. or ft. In the closed-shell case the space function or molecular orbitals each appear twice, combined first with the a. spin function and then with the y spin function. For open-shell cases two approaches are possible. In the restricted Hartree-Fock (RHF) approach, as many electrons as possible are placed in molecular orbitals in the same fashion as in the closed-shell case and the remainder are associated with different molecular orbitals. We thus have both doubly occupied and singly occupied orbitals. The alternative approach, the unrestricted Hartree-Fock (UHF) method, uses different sets of molecular orbitals to combine with a and ft spin functions. The UHF function gives a better description of the wavefunction but is not an eigenfunction of the spin operator S.2 The three cases are illustrated by the examples below. 

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... 

There is, however, a method which has not been fully exploited yet, in particular for extended systems, namely the single-determinant method without any restrictions on the spin orbitals. Thanks to very important work by Fukutome and collaborators we now have a clear picture of the various possible forms of such spin orbitals, which includes, in addition to the well-known doubly filled restricted Hartree-Fock (RHF) orbitals, alternant molecular orbitals and other forms of different orbitals for different spins, Overhauser s spin density waves, as well as others. It has been known for a long time that a sufficiently general single determinant can... 

Unfortunately, differences in the spatial orbitals for different spins may be impractical if Cl calculations are performed afterward to improve the Hartree-Fock calculation. Therefore, the variation is restricted, to keep the same spatial function for spin up and spin down. This method is called restricted Hartree-Fock (RHF). The original method with unlimited variation of the orbital functions is called unrestricted Hartree-Fock (UHF). Only the latter satisfies the BriUouin theorem. 

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). 

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