Essentials of Computational Chemistry

190 6 AB INITIO HARTREE–FOCK MO THEORY
MOs can be different, and this permits spin polarization. Equations (6.8) and (6.9) define
unrestricted Hartree-Fock (UHF) theory.
While UHF wave functions have the desirable feature of including spin polarization, they
are not, in general, eigenfunctions of S 2 . By allowing the spatial parts of the different spin
orbitals to differ, the final UHF wave function incorporates some degree of ‘contamination’
from higher spin states – specifically, states whose high-spin components would derive from
flipping the spin of one or more electrons. Thus, doublets are contaminated by quartets,
sextets, octets, etc., while triplets are contaminated by pentets, heptets, nonets, etc. The
degree of spin contamination can be assessed by inspection of 〈S 2 〉, which should be 0.0
for a singlet, 0.75 for a doublet, 2.00 for a triplet, 3.75 for a quartet, etc. Values that vary
from these proper eigenvalues by more than 5 percent or so should inspire great caution
in working with the wave function, since other expectation values will also be skewed by
differences between the property for the desired state and those for the contaminating states
(see Section 9.1.4 and Appendix C for details on the calculation of 〈S 2 〉).
Various techniques have been developed to reduce or eliminate the contribution of contaminating
states to the UHF wave function or expectation values derived from it. Some of these
are described in Appendix C, which contains a more detailed description of spin algebra in
general. In general, however, it should be noted that none of these approaches are convenient
for geometry optimization, which makes characterization of an open-shell PES quite difficult
when spin contamination effects are large. Thus, open-shell systems nearly always require
more care than closed-shell singlets, because both the ROHF and the UHF formalisms are
subject to intrinsically unphysical behavior. Depending on the nature of the system and the
properties being calculated, such behavior may or may not be manifest.
Finally, note that some open-shell systems cannot be described by a single determinant.
The classical example is an open-shell singlet, i.e., a system having electrons of α and β spin
in different spatial orbitals a and b. The wave function for such a system that is properly
antisymmetric and preserves the indistinguishability of particles is
1 1 = [a(1)b(2) + a(2)b(1)][α(1)β(2) − α(2)β(1)] (6.10)
2
which cannot be expressed as a single determinant. Because RHF and UHF are defined to use
single-determinantal wave functions, they are formally unable to address this wave function
(cf. Appendix C). In its most general form, ROHF is defined for multideterminantal systems,
but the more typical approach is to use multiconfiguration self-consistent field theory, as
described in Section 7.2.
6.3.4 Efficiency of Implementation and Use
We have emphasized up to this point the formal N 4 scaling of HF theory. However, in
practice, the situation is never so severe, and indeed linear scaling HF implementations have
begun to appear. Of course, one should remember that scaling behavior is different from
speed. Thus, for a system of a given size, a HF calculation using algorithms that scale
linearly may take significantly longer than conventional algorithms – it is simply true that at