Restricted Hartree-Fock wavefunctions

Neutral, positive, and negative atomic systems from H to Xe, and the Be iso-electronic series from B+ to Cr+20 gjnj series ions from Ne" " to Ne" " are investigated. A few first-row monohydrides from LiH to HE are examined as an example of the application to molecules. The restricted Hartree-Fock wavefunctions of Koga et al. [42] have been used for atomic systems. For molecules, calculations are performed at the B3PW91/6-31 l+-i-G(3d, 2p) level. 

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. 

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 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. 

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. 

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 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. 

The formal justification of this expansion form is analogous to that given by Hurley, Lennard-Jones and Pople in the development of a correlated-pair extension of the closed-shell restricted Hartree-Fock (RHF) wavefunction. 

When the Schrodinger equation is solved in the Hartree-Fock— Roothaan procedure, the coefficients c,> are obtained and the wavefunction is at hand.2 Unfortunately, all the chemical information is contained in this wave-function, and it is expressed as a (very) long list of coefficients. As an example, a restricted Hartree-Fock calculation of benzene using the 6-31G basis set will have 102 atomic orbitals and 21 doubly occupied MOs for a total of 2142 coefficients. For the chemist, the interesting and pertinent data are entangled in a series of numbers, and the question becomes how to extract the chemical concepts from these numbers. 

Eqs. (l)-(3), (13), and (19) define the spin-free CGWB-AIMP relativistic Hamiltonian of a molecule. It can be utilised in any standard wavefunction based or Density Functional Theory based method of nonrelativistic Quantum Chemistry. It would work with all-electron basis sets, but it is expected to be used with valence-only basis sets, which are the last ingredient of practical CGWB-AIMP calculations. The valence basis sets are obtained in atomic CGWB-AIMP calculations, via variational principle, by minimisation of the total valence energy, usually in open-shell restricted Hartree-Fock calculations. In this way, optimisation of valence basis sets is the same problem as optimisation of all-electron basis sets, it faces the same difficulties and all the experience already gathered in the latter is applicable to the former. 

Calculations such as the one discussed above do not involve imposed symmetry restrictions on the reference wavefunction. Hence this approach is referred to as the unrestricted Hartree-Fock (UHF) method. When symmetry restrictions are imposed upon the reference wavefunction the resulting calculation is denoted a restricted Hartree-Fock (RHF) calculation. When the simplest RHF type calculation is carried out for a closed-shell reference state (i.e., one having doubly occupied orbitals), the nondehned part of the Fock potential (the occupied-occupied) and (empty-empty) part is often chosen to have the same form as the (occupied-empty) part defined from the BT. We then would obtain for the entire Fock potential... 

Theoretical Results. A survey of ab initio SCF-MO (restricted Hartree-Fock) and configuration interaction (Cl) calculations for NF in the three lowest states X 2", a A, and b 2 is presented in Table 15, p. 273. It contains the total molecular energies Ej and the corresponding (calculated or experimental) bond lengths r, and, for sake of completeness, a listing of other molecular data obtained from the respective wavefunctions. 

Sometimes the term restricted Hartree-Fock (RHF) is used to emphasize that the wavefunction is restricted to be a single determinantal function for a configuration wherein electrons of a spin occupy the same space orbitals as do the electrons of P spin. When this restriction is relaxed, and different orbitals are allowed for electrons with different spins, we have an unrestricted Hartree-Fock (UHF) calculation. This refinement is most likely to be important when the numbers of a- and -spin electrons differ. We encountered this concept in Section 8-13, where we noted that the unpaired electron in a radical causes spin polarization of other electrons, possibly leading to negative spin density. 

The mathematical idea of Cl is quite obvious. Recall that we restricted our SCF wavefunction to be a single determinant for a closed-shell system. To go beyond the optimum (restricted Hartree-Fock) level, then, we allow the wavefunction to be a linear... 

Fig. 1 Diagonal elements of the density response function for water in the molecular plane. Coordinates are expressed in atomic units. The wavefunction is restricted Hartree-Fock with the 6-31G basis set. The figure shows raw values of the response function. These must be multiplied by 1.0012 x 10 (twice the inverse square of the volume element) to get the values in atomic units...

---