Restricted HF

The most general version of Hartree-Fock (HF) theory, in which each electron is permitted to have its own spin and spatial wave function, is called unrestricted HF (UHF). Remarkably, when a UHF calculation is performed on most molecules which have an equal number of alpha and beta electrons, the spatial parts of the alpha and beta electrons are identical in pairs. Thus the picture that two electrons occupy the same MO with opposite spins comes naturally from this theory. A significant simplification in the solution of the Fock equations ensues if one imposes this natural outcome as a restriction. The form of HF theory where electrons are forced to occupied MOs in pairs is called restricted HF (RHF), and the resulting wave function is of the RHF type. A cal-... 

If we carry out a restricted HF calculation, one or other of the degenerate frontier pair will be chosen to be occupied, the calculation will optimize the shapes of all of the occupied orbitals, and we will end up with a best possible single-Slater-determinantal wave function formed from those MOs. But it should be fairly obvious that an equally good wave function... 

One way to deal with unpaired electrons is to use the unrestricted HF (UHF). Whereas regular ab initio calculations restrict the one-electron spatial orbitals to be identical for a- and (3-spin electrons (so-called restricted HF, RHF), in UHF the orbitals are allowed to be different in the SCF processes. Usually, the difference in the spatial orbitals for a and (3 electrons is only slight. Unfortunately, when applied to a radical, UHF stumbles in a pitfall (97). It is called spin contamination. Unrestricted HF wave functions cannot be trusted to correspond to pure spin states such as a doublet for radicals or a singlet or triplet for diradicals. Theoretically speaking, the UHF wave function may not be an eigenfunction of the spin operators. 

Figure 1 Single-particle spectra for a ring with four sites with U = 10.0. The solid line is the spectrum calculated from the restricted HF while the dashed line is the spectrum obtained with the 2HF, and the dotted line is the exact spectrum.

Hartree-Fock (theory) (FIF) (restricted HF, RFIF for closed shells unrestricted HF, UHF for open shells)... 

More recently, Marcellus et al. (47) have questioned the usefulness of dividing spin density distributions into direct spin delocalization and spin polarization mechanisms. Both are non-observables and not uniquely defined in most calculations. The authors show that, if both unrestricted and restricted HF INDO calculations are performed, the spin densities due to spin delocalization Pain) and spin polarization Pp(n) may be precisely defined respectively as ... 

Formally, Koopmans theorem applies to eigenfunctions of the Fock operator of a closed shell restricted HF wavefunction or an open shell unrestricted HF wavefunction... 

Let us say a little more about these methods. The HF methods are divided into spin-restricted HF (RHF) and spin-unrestricted HF (UHF) methods. Closed-shell systems are almost always calculated using RHF. In this procedure, one set of molecular orbitals is calculated and pairs of electrons are entered to the lowest-energy orbitals. If the molecule has an odd number of electrons, one orbital will be singly occupied and the species is a radical (spin = 0.5, expectation value of the spin-squared operator =0.75). Inmost cases, however, radicals are calculated using the UHF formalism. UHF calculations determine two sets of molecular orbitals, one for each type of spin named alpha and beta. These MO sets are similar but not identical. A radical, for instance, has one more a than P electron. The UHF procedure is more flexible than RHF because the paired a and P orbitals, which correspond to doubly occupied MOs in the RHF formalism, need not be identical. So UHF allows for spin polarization but, on the other hand, spin-contamination occurs (i.e., states of higher spin are mixed into the wave function). 

The method of calculating wavefimctions 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 HF method outlined here is based on closed-shell Slater determinants and is called the restricted HF 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 spin inspection of Eq. (5.12) shows that we do not have a set of a spatial orbitals and a set of spatial orbitals. If unqualified, a HF (i.e. an SCF) calculation means an RHF calculation. 

Poppinger et al. 32 explored the singlet potential-energy surface of the CHNO system with the aid of the restricted HF method and with STO-3G, 4-3IG, and 6-3IG basis sets. These authors found the structures of 40 and 44 to be minima on the singlet potential-energy surface by using an STO-3G basis... 

The effects of the use of the restricted HF model are associated with the idea of frozen cores and will become clear shortly. 

For the sake of simplicity, we assume in this chapter that our molecule of interest has an even number of electrons and is adequately represented, to a first approximation, by a closed-shell restricted HF determinant, j Fo)- Suppose we have solved Roothaan s equations in a finite basis set and obtained a set of 2K spin orbitals The determinant formed from the N lowest energy spin orbitals is I Fq)- As we have seen in Chapter 2, we can form, in addition to Po> large number of other N-electron determinants from the 2K spin orbitals. It is convenient to describe these other determinants by stating how they differ from I Fq) Thus the set of possible determinants include Fo> the singly excited determinants (which differ from I Fq) in having the spin orbital Xa replaced by xX the doubly excited determinants etc., up to and including N-tuply excited determinants. We can use these many-electron wave functions as a basis in which to expand the exact many-electron wave function Oo>. If I Fq) is a reasonable approximation to Oo>, then we know from the variation principle that a better approximation (which becomes exact as the basis becomes complete) is... 

We have seen that Cl rectifies this situation unfortunately, perturbation theory using restricted HF orbitals makes things even worse. In fact, the second-order energy in the limit 00 diverges. Recall that for minimal basis set H2 the second-order energy (see Eq. (6.77)) is... 

In MO theory, the network of chemical bonds is determined by the occupied MO in the system ground state. For reasons of simplicity, assume the closed-shell (cs) configuration of N = 2n electrons in the standard spin-restricted HF (RHF) description, which involves the n lowest (doubly occupied, orthonormal) MO. In the familiar LCAO MO approach, they are expanded as linear combinations (LC) of the (Lbwdin) orthogonalized AO x = iXoXi - -XJ = iXi) contributed by the system constituent atoms (x x)= 8, si, = ( 2,... ) = J = xC, where the rectangular matrix C = = (x ) groups the relevant expansion coefficients of MO (i.e., LC... 

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