Essentials of Computational Chemistry

328 9 CHARGE DISTRIBUTION AND SPECTROSCOPIC PROPERTIES
and ρ(X) is the Fermi contact integral which, when the wave function can be expressed as
a Slater determinant, can be computed as
ρ(X) =
µν
P α−β
µν ϕµ(rX)ϕν(rX) (9.36)
where P α−β is the one-electron spin-density-difference matrix (computed as the difference
between the two separate density matrices for the α and β electrons), and evaluation of the
overlap between basis functions ϕµ and ϕν is only at the nuclear position, rX.
We have previously defined the one-electron spin-density matrix in the context of standard
HF methodology (Eq. (6.9)), which includes semiempirical methods and both the UHF
and ROHF implementations of Hartree–Fock for open-shell systems. In addition, it is well
defined at the MP2, CISD, and DFT levels of theory, which permits straightforward computation
of h.f.s. values at many levels of theory. Note that if the one-electron density matrix
is not readily calculable, the finite-field methodology outlined in the last section allows
evaluation of the Fermi contact integral by an appropriate perturbation of the quantum
mechanical Hamiltonian.
For Eq. (9.35) to be useful the density matrix employed must be accurate. In particular,
localization of excess spin must be well predicted. ROHF methods leave something to be
desired in this regard. Since all doubly occupied orbitals at the ROHF level are spatially
identical, they make no contribution to P α−β ; only singly occupied orbitals contribute. As
discussed in Section 6.3.3, this can lead to the incorrect prediction of a zero h.f.s. for all
atoms in the nodal plane(s) of the singly occupied orbital(s), since their interaction with the
unpaired spin(s) arises from spin polarization. In metal complexes as well, the importance
of spin polarization compared to the simple analysis of orbital amplitude for singly occupied
molecular orbitals (SOMOs) has been emphasized (Braden and Tyler 1998).
UHF, on the other hand, does optimize the α and β orbitals so that they need not be
spatially identical, and thus is able to account for both spin polarization and some small
amount of configurational mixing. As a result, however, UHF wave functions are generally
not eigenfunctions of the operator S 2 , but are contaminated by higher spin states.
The challenge with unrestricted methods is the simultaneous minimization of spin ‘contamination’
and accurate prediction of spin ‘polarization’. The projected UHF (PUHF, see
Appendix C) spin density matrix can be employed in Eq. (9.36), usually with somewhat
improved results.
A complicating factor is that each spin density matrix element is multiplied by the
corresponding basis function overlap at the nuclear positions. The orbitals having maximal
amplitude at the nuclear positions are the core s orbitals, which are usually described with
less flexibility than valence orbitals in typical electronic structure calculations. Moreover,
actual atomic s orbitals are characterized by a cusp at the nucleus, a feature accurately
modeled by STOs, but only approximated by the more commonly used GTOs. As a result,
there are basis sets in the literature that systematically improve the description of the core
orbitals in order to improve prediction of h.f.s., e.g. IGLO-III (Eriksson et al. 1994) and
EPR-III (Barone 1995).