Essentials of Computational Chemistry

9.1 PROPERTIES RELATED TO CHARGE DISTRIBUTION 327
is, one can compute the energy in the absence of the perturbation, then modify the Hamiltonian
to include the perturbation (e.g., introduce an electric-field term), then compute the
property as
∂
〈|H|〉−〈|H
〈|H|〉 = lim
∂X X→0
(0) |〉
X
(9.34)
where H is again the complete Hamiltonian and H (0) is the perturbation-free Hamiltonian.
This procedure is called the ‘finite-field’ approach. In practice, one must take some care to
ensure that computed values are numerically converged (a balance must be struck between
using a small enough value of the perturbation that the limit holds but a large enough value
that the numerator does not suffer from numerical noise).
Note that Eq. (9.34) can be generalized for higher derivatives, but numerical stability now
becomes harder to achieve. Moreover, the procedure can be rather tedious, since in practice
one must carry out a separate computation for each component associated with properties that
are typically tensors. It is computationally much more convenient when analytic expressions
can be found that permit direct calculation of these higher-order derivatives in a fashion that
generalizes the procedure by which Eq. (9.33) is derived (not shown here).
As for the utility of different levels of theory for computing the polarizability and hyperpolarizability,
the lack of high-quality gas-phase experimental data available for all but the
smallest of molecules makes comparison between theory and experiment rather limited. As
a rough rule of thumb, ab initio HF theory seems to do better for these properties than
for dipole moments – at least there does not appear to be any particular systematic error.
Semiempirical levels of theory are less reliable. DFT and correlated levels of MO theory do
well, but it is not obvious for the latter that the improvement over HF necessarily justifies
the cost, at least for routine purposes.
9.1.6 ESR Hyperfine Coupling Constants
When a molecule carries a net electronic spin, that spin interacts with the (non-zero) spins of
the individual nuclei. The energy difference between the two possibilities of the electronic and
nuclear spins being either aligned or opposed in the z direction can be measured by electron
spin resonance (ESR) spectroscopy and defines the isotropic hyperfine splitting (h.f.s.) or
hyperfine coupling constant. If we were to pursue computation of this quantity using the
approach outlined in the last section, we would modify the Hamiltonian to introduce a spin
magnetic dipole at a particular nuclear position. The integral that results when Eq. (9.33) is
used to evaluate the necessary perturbation is known as a Fermi contact integral. Isotropic
h.f.s. values are determined as
aX = (4π/3)〈Sz〉 −1 ggXββXρ(X) (9.35)
where 〈Sz〉 is the expectation value of the operator Sz (1/2 for a doublet, 1 for a triplet,
etc.), g is the electronic g factor (typically taken to be 2.0, the approximate value for a free
electron), β is the Bohr magneton, gX and βX are the corresponding values for nucleus X,