and extended H€uckel calculations). We saw that all these energies wererelative to
something: that of a species on a potential energy surface (PES) can be taken as
being relative to the energy of the global minimum, the MM energy is relative to
that of some hypothetical unstrained isomer, and the energy of a molecular orbital
is, with qualifications, the energy of an electron in it compared to the energy of the
electron infinitely distant from the orbital, at rest. Before considering the ab initio
calculation of energy, it is worthwhile to look briefly further into the meaning of
“energy”, because this entity manifests itself in several ways and in favorable cases
all of them can be calculated by ab initio methods. We will take cognisance of seven
kinds of energy: potential, kinetic, internal, “heat energy” or enthalpy, Gibbs free
energy, Helmholtz free energy, and Arrhenius activation energy. The reader may
wonder why we need so many kinds of energy (we could add even more, like
electrical energy and nuclear energy). The answer is, partly because in different
situations energy appears in different guises, and partly because although some
kinds are really composites of others with thermodynamic concepts like tempera-
ture and entropy (thus the Gibbs free energy is enthalpy minus the product of
temperature and entropy), it is neater to have one word and symbol for the
composite. I present the seven kinds of energy in the approximate order in which
some build conceptually on others. Five are of considerable importance in chemis-
try: potential energy, internal energy, enthalpy, Gibbs free energy, and, in experi-
mental studies of reaction rates, Arrhenius activation energy. In this short
preliminary to the calculation of energies, we consider the subject from the view-
point of molecular chemistry, rather than that of classical thermodynamics, which,
albeit elegant, knows nothing of atoms and molecules. The connection between
the two stances is made in the subject of statistical mechanics. Besides the many
standard texts on these subjects, one may recommend Atkin’s graceful, compact,
and masterful book on the four laws of thermodynamics [ 128 ]. We can safely
ignore here relativity theory, which requires conservation of “mass–energy”.
1.Potential energyis the work obtainable from a body that “temporarily” resists a
restoring force, so that if the body is allowed to submit to the force it will do
work. We use here Newton’s concept of a force: something that acting on a body
produces an acceleration. An example is a stone at the edge of a cliff, temporar-
ily resisting the gravitational force; a kick submits it to gravity and it will gain
kinetic energy, which could be converted into useful work by a machine. In
chemistry the relevant potential energy is the energy of a molecule on a
Born–Oppenheimer surface (a potential energy surface,Chapter 2). In this,
granted, more abstract, situation, a molecule not at the global minimum resists
the electromagnetic force – chemistry’s only force – that will eventually
(delayed by kinetic barriers) pull it downhill to that minimum. In this process
energy is released as heat or light. On the usual Born–Oppenheimer surface,
which includes simple two-dimensional potential energy curves such as plots of
energy against torsional (dihedral) angle, as well as hypersurfaces, the energy at
various points can be taken as being relative to the global minimum. The units
of this energy could be from molecular mechanics or some kind of quantum
292 5 Ab initio Calculations