Essentials of Computational Chemistry

6.3 KEY TECHNICAL AND PRACTICAL POINTS OF HARTREE–FOCK THEORY 189
the doubly occupied orbitals. To appreciate this point, let us return to the methyl radical.
Note that because the unpaired spin in this molecule is in the carbon 2pz orbital, the plane
containing the atoms, which is the nodal plane for the 2pz orbital, must have zero spin
density. This being the case, electron spin resonance experiments should detect zero hyperfine
coupling between the magnetic moments of the hydrogen atoms (or of a 13 C nucleus) and
the unpaired electron. However, even after correcting for the effects of molecular vibrations,
it is clear that there is a coupling between the two.
Spin density is found in the molecular plane because of spin polarization, which is an effect
arising from exchange correlation. The Fermi hole that surrounds the unpaired electron allows
other electrons of the same spin to localize above and below the molecular plane slightly
more than can electrons of opposite spin. Thus, if the unpaired electron is α, we would expect
there to be a slight excess of β density in the molecular plane; as a result, the 1 H hyperfine
splitting should be negative (see Section 9.1.3), and this is indeed the situation observed
experimentally. An ROHF wave function, because it requires the spatial distribution of both
spins in the doubly occupied orbitals to be identical, cannot represent this physically realistic
situation.
To permit the α and β spins to occupy different regions of space, it is necessary to treat
them individually in the construction of the molecular orbitals. Following this formalism,
we would rewrite our methyl radical wave function Eq. (6.6) as
2 =
C1s α C1s ′β α ′β
σ σ CHa CHaσ α ′β
σ CHb CHbσ α ′β
σ CHc CHcC2pα
z
(6.7)
where the prime notation on each β orbital emphasizes that while it may be spatially similar
to the analogous α orbital, it need not be identical. The individual orbitals are found by
carrying out separate HF calculations for each spin, with the spin-specific Fock operator
now defined as
F ξ µν =
µ
−1 2 ∇2
ν
+
λσ
µ
1
ν
nuclei
− Zk
rk
k
P α
β
λσ + Pλσ (µν|λσ ) − P ξ
λσ (µλ|νσ)
where ξ is either α or β, and the two spin-density matrices are defined as
P ξ
λσ =
ξ−occupied
i
a ξ
λi aξ
σi
(6.8)
(6.9)
where the coefficients a are the usual ones expressing the MOs in the AO basis, but there are
separate sets for the α and β orbitals. Notice, then, that in Eq. (6.8), the Coulomb repulsion
(the first set of integrals in the double sum) is calculated with both spins, but exchange (the
second set of integrals) is calculated only with identical spins. Because the SCF is being
carried out separately for each spin, the two density matrices can differ, which is to say the