energy of the electrons, whether calculated by the Hartree–Fock or by a correlation
method) plus the internuclear repulsion (cf. Eq.5.93):
Ab initio energy (Hartree–Fock)
EtotalHF ¼EHFþVNN $ð 5 : 176 ÞAb initio energy (correlated method)Etotalcorrel¼EcorrelþVNN $ð 5 : 177 ÞIf the ab initio energy has been corrected by adding the zero-point energy (cf.
Eq.5.94), giving the total internal energy at 0 K, this should be pointed out: ab initio
energy, corrected for ZPE:
Etotal0K ¼EtotalþZPE $ð 5 : 178 ÞAs has been pointed out, the ZPE-corrected ab initio energy is preferred over the
uncorrected for calculating relative energies. At the end of a calculationEtotal(HF
or correlated) is given; if we wish to include ZPE and getEtotal0K a frequency
calculation is necessary. The format in which these quantities appear at the end of
a calculation depends on the program.
What we actually want is rarely these “absolute” ab initio energies, because
chemistry really deals withrelativeenergies;allenergies are relative to something
of course, but in this context it is useful to restrict the term to the energy difference
between reactants and products or between reactants and transition states (the
energy difference between isomers is a special case reactants/products). We are
thus interested in the reaction energy (product energy minus reactant energy) and
what we might call the activation energy (transition state energy minus reactant
energy; note however – see below and Eq.5.175–that above 0 K the well-known
Arrhenius activation energy is not exactly simply the difference in calculated
energies of transition state and reactants).
Figure5.24shows what Coulson meant when he said that calculating the relative
stabilities of isomers by subtracting absolute energies is like finding the weight of the
captain by weighing the ship with and without him [ 137 ]. The absolute ab initio
energies of the two isomers shown are each about 407,000 kJ mol#^1 , and the difference
in their energies is only about 9 kJ mol#^1 , which is one part in 45,000, and these figures
are quite typical. If we conservatively assign a captain a weight of 100 kg, the analogy
corresponds to a smallship weighing 4,500,000 kgorabout 5,000 t. Yetthe astonishing
thing is that modern ab initio calculations can, as we shall see, accurately and reliably
predict relative energies. Comprehensive accounts of energy calculations by ab initio
and other methods are given by Irikura and Frurip [ 138 ] and by Cramer [ 139 ].
Reaction energies belong to the realm of thermodynamics, and activation ener-
gies to that of kinetics: the energy difference between the products and the reactants
5.5 Applications of the Ab initio Method 297