Spin-restricted Hartree-Fock method

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. 

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). 

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. 

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... 

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. 

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. 

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. 

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. 

Larsson [95] who subsequently lifted all restrictions to the wave function ansatz when performing atomic calculations. As a first step he allowed the radial functions to be different for different mg values of the participating orbitals (the spin-polarized Hartree-Fock method SPHF) leading to a core orbital contribution to magnetic effects. Furthermore the radial parts of functions possessing different mi values are... 

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 ... 

Technically, it is possible to define methods in which other restrictions than those on spin are lifted, but the most common unrestricted method is the spin-unrestricted method, and we follow the common usage of UHF for the spin-unrestricted Hartree-Fock 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. 

Another way of constructing wave functions for open-shell molecules is the restricted open shell Hartree-Fock method (ROHF). In this method, the paired electrons share the same spatial orbital thus, there is no spin contamination. The ROHF technique is more difficult to implement than UHF and may require slightly more CPU time to execute. ROHF is primarily used for cases where spin contamination is large using UHF. 

M. Urban, P. Neogrady, and I. Hubac, Spin Adaptation in the Open-Shell Coupled-Cluster Theory with a Single Determinant Restricted Hartree-Fock Reference. In R. J. Bartlett (Ed.) Recent Advances in Coupled-Cluster Methods. Recent Advances in Computational Chemistry, Vol. 3. (World Scientific, Singapore, 1997), pp. 275-306. 

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. 

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... 

When the second of the equivalence restrictions is removed, a single determinant wavefunction of lower energy is usually obtained. In fact, it is possible for a wave-function obtained in this way, a so-called unrestricted Hartree-Fock (UHF) wavefunction191 (perhaps more properly called a spin-unrestricted Hartree-Fock wavefunction) to go beyond the Hartree-Fock approximation and thus include some of the correlation energy. Lowdin192 describes this as a method for introducing a Coulomb hole to supplement the Fermi hole already accounted for in the RHF wavefunction. 

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. 

PDDO PRDDO RHF SAMO SCF SOGI STO STO-nG UA UHF VB VIP Projectors of Diatomic Differential Overlap Partial Retention of Diatomic Differential Overlap Restricted Hartree-Fock Simulated ab initio Method Self Consistent Field Spin Optimized GVB method Slater Type Orbital Slater Type Orbital expanded in terms of nGTO United Atom Unrestricted Hartree-Fock Valence Bond Vertical Ionization Potential... 

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