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MOLECULAR ORBITAL THEORY CALCULATIONS 207

medium size of molecular model as a starting molecular system, and then per-
forms a series of calculations with several different levels of model chemistries.
After comparing the calculated results with experimental data, usually vibration
frequencies, and if they match fairly well, one can expect that the similar/other
molecular system (slightly enlarged) + model chemistry (a slightly higher level)
can be considered valid and well representative. Subsequently, higher-level cal-
culations may be performed to obtain more accurate results. Obviously, lowering
the calculation level and decreasing the molecular system size lead to invalid
results. Finally, there is a general notation for a given series of calculations
(Foresman and Frisch, 1996):

Energy Method/Energy Basis Set//Geometry Method/Geometry Basis Set

where the model to the left of the double slash is the one at which the energy
is computed, and the model to the right of the double slash is the one at which
the molecular geometry was first optimized. For example, RHF/6-31+G//RHF/6-
31G(d) denotes that the energy calculation was performed by using Hartree-Fock
theory and the 6-31+G basis set on a structure previously optimized with Hartree-
Fock theory and the 6-31G(d) basis set.
The saturation of boundaries for a model extracted from an infinite solid has
been a confusing issue. There is no standard rule or criterion for boundary satu-
ration. Point charge has been used to balance the boundary charges, but without
any geometric and chemical meanings for the boundaries. Hydrogen replacement
is widely used both to balance the charge and to terminate the boundaries, though
it can introduce somewhat incorrect geometric and chemical environment to the
boundaries. Regardless of the method that is used to saturate the boundaries, the
final goal is to achieve the calculation results closest to the experimental data,
usually the vibrational frequency data (Foresman and Frisch, 1996).

8.2.8. Natural Bond Orbital
The local atomic properties in a molecular system are often the most impor-
tant properties from a chemical point of view, although they are not quantum
mechanically observable. Usually there are several built-in schemes in a given
molecular orbital calculation package, including the Gaussian 98 package, to
partition the electron density among atoms in a molecular system and ultimately
obtain certain atomic properties. The most popular scheme is Mulliken popu-
lation analysis. Some other schemes, such as natural bond orbital (NBO) and
Merz-Kollman-Singh analysis have also been used.
Atomic charge, orbital energy, and population are important pieces of infor-
mation for determining electronic configuration, net charge association, and the
nature of the bond. Mulliken population analysis is a widely used method in
most ab initio molecular orbital calculations. However, reports about Mulliken
population analysis that fail to yield reliable characterization of molecular sys-
tems have appeared. A more accurate method for population analysis, NBO, was