Essentials of Computational Chemistry

188 6 AB INITIO HARTREE–FOCK MO THEORY
is to be preserved. Typically the difference in energy between these two possible configurations
is so large that no reasonable means for guessing the initial wave function generates
the higher energy possibility. This is one of the advantages of closed-shell states compared
to open-shell ones. Certain other aspects of dealing with open-shell systems also merit
attention.
6.3.3 Open-shell Systems
The presentation of the HF equations in Chapter 4 assumed a closed-shell singlet for simplicity,
but what if there are one or more singly occupied orbitals? Let us proceed with an
example to guide the discussion, in this case, the methyl radical, which is planar in its
equilibrium structure (Figure 6.10). The most intuitive description of the wave function for
this system (ignoring symmetry for ease of discussion) would be
2
= C1s 2 σ 2 CHaσ 2 CHbσ 2 CHcC2p1
z
(6.6)
Thus, there is a doubly occupied carbon 1s core, three C–H bonding orbitals, and the
unpaired electron in a carbon 2p orbital. Given this configuration, it might seem natural to
envision an extension of HF theory where all of the orbitals continue to be evaluated using
essentially the restricted formalism (RHF) for closed-shell systems, but the density matrix
elements for the singly occupied orbital(s) are not multiplied by the factor of two appearing
in Eq. (4.57). In essence, this describes so-called restricted open-shell HF theory (ROHF).
In its completely general form, certain complications arise for systems whose descriptions
require more than a single determinant (i.e., unlike Eq. (6.6)), so we will not extend this
qualitative description of the nature of the theory to specific equations (such details are
available in Veillard (1975)). It suffices to note that most electronic structure packages offer
ROHF as an option for open-shell calculations.
Besides being intuitively satisfying, ROHF theory produces wave functions that are eigenfunctions
of the operator S 2 (just as the true wave function must be), having eigenvalues
S(S + 1) where S is the magnitude of the vector sum of the spin magnetic moments for all
of the unpaired electrons. However, ROHF theory fails to account for spin polarization in
H
H
C H
ROHF UHF
Figure 6.10 In the absence of spin polarization, which corresponds to the ROHF picture, there is zero
spin density in the plane containing the atoms of the methyl radical. Accounting for spin polarization,
which corresponds to the UHF picture, results in a build-up of negative spin density (represented as a
shaded region) in the same plane
H
H
H