Essentials of Computational Chemistry

9.4 NMR SPECTRAL PROPERTIES 345
The magnetic field is independent of the choice of the gauge origin. So too are the
computed magnetic properties if the wave function used is exact. Regrettably, we are not
often afforded the opportunity to work with exact wave functions. For HF wave functions,
one can also achieve independence of the gauge by using an infinite basis set, but that is
hardly a practical option either.
To reduce artifacts associated with the gauge origin, two different approaches have seen
extensive use in the literature. The older method employs gauge-including atomic orbitals
(GIAOs) as a basis set (London 1937). By a clever incorporation of the gauge origin into the
basis functions themselves, all matrix elements involving the basis functions can be arranged
to be independent of it. An alternative is the ‘individual gauge for localized orbitals’ (IGLO)
method, where different gauge origins are used for each localized MO in order to minimize
error introduced by having the gauge origin far from any particular MO (Schindler and
Kutzelnigg 1982). Of the two methods, modern implementations of GIAO are probably
somewhat more robust, but it is possible to obtain good results with either.
Much of the benchmark work in the area of NMR calculations has been carried out
with very large basis sets, and recommendations have tended to call for at least triple-ζ
quality with diffuse and polarization functions aplenty. Of course, such basis sets are simply
not practical for larger molecules, even when used solely in the context of a single-point
calculation following geometry optimization with some more economical basis (note that the
single-point calculation, being a second-derivative property, has timing requirements rather
similar to the more routinely carried out calculation of vibrational frequencies). Some early
work has begun to appear aimed at identifying scale factors, or linear regressions, that may
be applied to computational results from less well-converged calculations, this work being
very similar in spirit to the scaling of IR frequencies discussed in Section 9.3.2.2.
A separate basis set issue is associated with calculations for molecules including heavy
atoms. If the core electrons of the heavy atom are represented by an ECP, then it is not
in general possible to predict the chemical shift for that nucleus, since the remaining basis
functions will have incorrect behavior at the nuclear position (note that it is mostly the ‘tails’
of the valence orbitals at the nucleus that influence the chemical shift, not the core orbitals
themselves, since they are filled shells). However, ECPs may be an efficient choice if the
only chemical shifts of interest are computed for other nuclei.
A different issue associated with NMR chemical shifts for heavy atoms is the influence
of relativistic effects. In terms of computing absolute chemical shifts, relativistic effects
can be very large in heavy elements. For relative chemical shifts, since relativistic effects
are primarily associated with core orbitals, and core orbitals do not change much from one
chemical environment to the next, the effect is typically markedly reduced. Nevertheless,
accurate calculations involving atoms beyond the first row of transition metals are still a
particular challenge.
9.4.2 Chemical Shifts and Spin–spin Coupling Constants
Experimental chemical shifts are reported in parts per million (ppm) so as to make them independent
of the external magnetic field strength. Moreover, they are usually not reported as