Essentials of Computational Chemistry

C.3 UHF Wave Functions
SPIN ALGEBRA 571
In evaluating Eq. (C.23), we invoked orthonormality between the spatial orbitals a and b,
each of which contains an electron of different spin. However, in a UHF wave function, the
α and β orbitals are not necessarily orthogonal to one another (only within each set, either
α or β, are all of the orbitals mutually orthogonal to one another). In that case, the second
and third terms on the r.h.s of Eq. (C.23) survive as −〈a|b〉 2 . In general, one can show that
for a UHF wave function where the number of α electrons is greater than or equal to the
number of β electrons, the expectation value of S 2 may be computed as
〈 UHF |S 2 | UHF 〉= nα − nβ
2
nα − nβ
2
occupied
+ 1 + nβ − 〈φi|φj〉
i∈α,j∈β
2
(C.29)
where nζ is the number of electrons of spin ζ and {φ} is the set of UHF molecular orbitals.
Consider the behavior of Eq. (C.29) in certain idealized limits. If there are no β electrons,
then Eq. (C.29) reduces to the correct eigenvalue for a system of all parallel spins (cf.
Eq. (C.21)). If there are β electrons, and for every occupied β MO there is a spatially
identical occupied α MO, then the sum on the r.h.s. of Eq. (C.29) is equal to nβ (there is
one overlap integral value of unity for each occupied β MO with its partner α MO, and
all other overlap integrals must be zero because since other α MOs must be orthogonal to
the partner α MO, so too they must be orthogonal to the spatially identical β MO), and
the expectation value is again the correct eigenvalue for a high-spin system with excess α
electrons. Note, however, that this expectation value can also be achieved to within arbitrary
accuracy without requiring every occupied β MO to have a spatially identical occupied α
MO: all that is required is that the sum of the squares of the overlap integrals approach its
limiting value, nβ.
Finally, consider the case where the overlap between the α and β orbitals is exactly zero
(which could happen, for instance, if all α MOs were on one atom and all β MOs on another
atom with the two atoms infinitely far apart). In that case, the expectation value will be larger
than the pure spin state where only the excess α electrons are unpaired, but smaller than
the value expected for the pure spin state where all electrons are unpaired (i.e., a low-spin
(n+1)-multiplet where n is the total number of electrons, for which the expectation value
would be computed using s = (nα + nβ)/2 instead of s = (nα − nβ)/2). Such a system is
said to be ‘spin-contaminated’ because it is a mixture of the lowest spin state and varying
contributions from states of higher spin multiplicity. Obviously, such wave functions are of
limited utility, since expectation values of other properties will also represent an admixture
of the properties of the different states.
C.4 Spin Projection/Annihilation
When a spin-contaminated wave function is obtained from a UHF calculation, the desired
spin state is inevitably the one of lower spin (otherwise one would have constructed the high-
Sz component of the higher spin state). The contaminated wave function can be improved