Essentials of Computational Chemistry

324 9 CHARGE DISTRIBUTION AND SPECTROSCOPIC PROPERTIES
to be one of a large number of possible types, and a fixed correction determined from
having fit these parameters on a 2700-molecule test set of HF/6-31G(d) ESP charges is
applied. While this protocol is robust for most molecules, it cannot be readily applied to
structures not characterized by standard bonding, like transition states or structures along a
reaction pathway.
9.1.4 Total Spin
Well-behaved wave functions are eigenfunctions of the total spin operator S 2 , having eigenvalues
of s(s + 1), where the quantum number s is 0 for a singlet, 1/2 for a doublet, 1 for
a triplet, etc. One sometimes sees it written that s is equal to the sum of the sz values for
all of the electrons, where sz is the expectation value of the corresponding operator Sz (spin
angular momentum along the z coordinate) and takes on values in a.u. of +1/2 for an α
electron and −1/2 foraβ electron. This is incorrect, however. In fact, s is equal to the
magnitude of the vector sum of the individual electronic angular momenta, and thus s can
take on values according to
s = |nα − nβ |
,
2
|nα − nβ |
+ 1,...,
2
nα + nβ 2
(9.29)
where n ξ is the number of unpaired electrons of spin ξ. Thus, for instance, a system having
an α and a β electron that are not paired with one another in the same MO can be either a
singlet or a triplet (the so-called Sz = 0 triplet), reflecting the ability of s to take on values
of either 0 or 1.
As described in more detail in Appendix C, the Sz = 0 triplet cannot be expressed as a
single determinant over spin orbitals, so it cannot be represented in HF or KS theory. Of
course, this is not usually a concern, since it is trivial to construct one of the other two
degenerate representations of the triplet state (having an excess of either two α or two β
electrons), and these can be approximated as single-determinantal wave functions, so we
work with them instead. The point, however, is that one cannot arbitrarily sum together
the sz eigenvalues for all the unpaired electrons in a single determinant and assign to that
determinant a unique spin state with s equal to the sum. Unless all of the unpaired electrons
have the same spin (in which case inspection of Eq. (9.29) indicates that s can only take on
one value), a single determinant is usually a mixture of states, and any properties determined
as expectation values over that determinant reflect this mixing.
In UHF theory, the expectation value of the total spin operator over the singledeterminantal
UHF wave function is computed as
〈S 2 〉=
α β α β |N − N | |N − N |
2
2
+ 1 + min{N α ,N β αocc. βocc.
}− 〈φ α i |φβ j 〉 (9.30)
i=1 j=1
where N ξ is the total number of electrons of spin ξ and the φ ξ are the UHF MOs for spin
ξ. Note that if all of the ‘doubly’ occupied orbitals are identical in shape for the α and