Essentials of Computational Chemistry

10.3 ENSEMBLE PROPERTIES AND BASIC STATISTICAL MECHANICS 357
One can generalize the concept of ZPVE to non-stationary points on the PES (although
some convention must be adopted for dealing with the non-zero first derivatives of the energy
at such points). The new surface that is generated by summing the Born–Oppenheimer
surface with this generalized ZPVE is called the zero-point-including energy surface. This
surface thus includes the quantum mechanical character of the nuclear motion at 0 K, and can
be very useful in reaction dynamics simulations, as described in more detail in Chapter 14.
A key feature of the ZPVE is that it is isotope dependent, since the vibrational frequencies
themselves are isotope dependent (see Eq. (9.48) and recall that the reduced mass µ for any
mode is a function of the atomic masses for the nuclei involved in the motion). Thus, if one
is considering a large ensemble of molecules, it must be kept in mind that the computed
ZPVE refers to an ensemble of isotopically pure molecules, not to an ensemble composed
from isotopes at natural abundance. Most electronic structure programs default to using
the atomic isotopes of highest natural abundance, and permit use of other isotopes in some
keyword-driven way. Older versions of some semiempirical programs originally used atomic
masses derived from natural-abundance averaging over isotopes, but this is a fundamentally
flawed approach, since the influence of the atomic masses on the frequencies is not linear.
Typically, in part because many elements have a single dominant isotope, one does not
need to worry about the rather small inaccuracies introduced by assuming isotopically pure
samples (Jensen 2003). In those rare instances where true natural abundance results are
desired, it is a straightforward if somewhat tedious task to construct multiple ensembles
differing in isotopic composition and weight them appropriately in an overall mixture.
10.3 Ensemble Properties and Basic Statistical Mechanics
Statistical mechanics is, obviously, a course unto itself in the standard chemistry/physics
curriculum, and no attempt will be made here to introduce concepts in a formal and rigorous
fashion. Instead, some prior exposure to the field is assumed, or at least to its thermodynamical
consequences, and the fundamental equations describing the relationships between key
thermodynamic variables are presented without derivation. From a computational-chemistry
standpoint, many simplifying assumptions make most of the details fairly easy to follow, so
readers who have had minimal experience in this area should not be adversely affected.
In order to deal with collections of molecules in statistical mechanics, one typically requires
that certain macroscopic conditions be held constant by external influence. The enumeration
of these conditions defines an ‘ensemble’. We will confine ourselves in this chapter to the
so-called ‘canonical ensemble’, where the constants are the total number of particles N
(molecules, and, for our purposes, identical molecules), the volume V , and the temperature
T . This ensemble is also sometimes referred to as the (N, V , T ) ensemble.
Just as there is a fundamental function that characterizes the microscopic system in
quantum mechanics, i.e., the wave function, so too in statistical mechanics there is a fundamental
function having equivalent status, and this is called the partition function. For the
canonical ensemble, it is written as
Q(N,V,T)=
e −Ei(N,V )/kBT
(10.2)
i