Computational Chemistry

mode [ 28 ] (all this is done by standard programs, which print out the ZPE after the
frequencies). Adding the HF electronic energy and the internuclear repulsion gives
what we mightEtotalHF, the total “frozen-nuclei” (no ZPE) energy:

EtotalHF ¼EHFþVNN¼

Xn

i¼ 1

eiþ

1

2

Xm

r¼ 1

Xm

s¼ 1

PrsHcorers þVNN ð 5 : 93 Þ

from Eqs.5.90and (5.92¼5.16).EtotalHF, the energy usually displayed at the end of a
Hartree–Fock calculation is, in ordinary parlance, “the Hartree–Fock energy”.
An aggregate of such energies, plotted against various geometries, represents
an HF Born–Oppenheimer PES (Section 2.3). The zero of energy for the
Schr€odinger equation for an atom or molecule is normally taken as the energy
of the electrons and nuclei at rest at infinite separation. The Hartree–Fock energy
(any ab initio energy, in fact) of a species in thus relative to the energy of
the electrons and nuclei at rest at infiniteseparation, i.e. it is the negative of the
minimum energy required to break up the molecule or atom and separate the
electrons and nuclei to infinity. We are normally interested inrelativeenergies,
differencesin absolute ab initio energies. Ab initio energies are discussed in
Section 5.5.2.
In a geometry optimization (Section 2.4) a series of single-point calculations
(calculations at a single point on the potential energy surface, i.e. at a single
geometry) is done, each of which requires the calculation ofEtotalHF, and the geometry
is changed systematically until a stationary point is reached (one where the poten-
tial energy surface is flat;ideally EtotalHF should fall monotonically in the case of
optimization to a minimum). The ZPE calculation, which is valid only for a
stationary point on the potential energy surface (Section 2.5; discussion in connec-
tion with Fig. 2.19), can be used to correctEtotalHF of the optimized structure for
vibrational energy; adding the ZPE gives the total internal energy of the molecule at
0 K, which we could callEtotal0K:

Etotal0K ¼EtotalHF þZPE $ð 5 : 94 Þ

The relative energies of isomers may be calculated by comparingEtotalHF, but for
accurate work the ZPE should be taken into account, even though the required
frequency calculations usually take significantly longer than the geometry optimi-
zation – seeSection 5.3.3, Table5.3). Fortunately, it is valid to correctEtotalHF with a
ZPE from a lower-level optimization-plus-frequency job (not a lower-level fre-
quency job on the higher-level geometry). Figure 2.19, Section 2.5 compares
energies for the species in the isomerization of HNC to HCN. The relative energies
with/without the ZPE correction for HCN, transition state, and HNC are 0/0, 202/
219, and 49.7/52.2 kJ mol#^1. The ZPEs of isomers tend to be roughly equal and
so to cancel when relative energies are calculated (less so where transition states
are involved), but, as implied above, in accurate work it is usual to compare the
ZPE-corrected energiesEtotal0K.

5.2 The Basic Principles of the ab initio Method 213