H–H bond energy of 435 kJ mol#^1 ) above the accepted exact energy of#1.17439 h
(Fig.5.18). Variational behavior is helpful because it serves as a guide to the quality
of our wavefunction – the lower the energy the better the function.
If we can’t have both, it is more important for a method to be size-consistent than
variational. Of the methods we have seen in this book:
Hartree–Fock is size-consistent and variational.
MP (MP2, MP3, MP4, etc.) is size-consistent but not variational.
Full CI, including its full MCSCF and MRCI variants, are size-consistent and
variational.
Straightforward truncated CI (CIS, CISD, etc.) is not size-consistent but is
variational.
CASSCF, a kind of truncated CI, can be size-consistent: if the active space is
chosen properly so that the MOs correspond throughout the process being exam-
ined. CASSCF is not variational.
CC and its QCI variants (QCISD, QCISD(T), QCISDT, etc.) are size-consistent
but not variational.
We could use one of the size-consistent methods to compare the energies of, say,
water and the water dimer, but only with HF or some version of CI can we besure
that the calculated energy is an upper bound to the exact energy, i.e. that the exact
energy is really lower than the calculated (only a very high correlation level and
basis set are likely to give essentially theexactenergy; seeSection 5.5.2). There is
however another thing to consider in connection with the energy of water compared
to its dimer, and similar problems: basis set superposition error, below.
5.4.3.3 Basis Set Superposition Error
This is not associated with a particular method, like HF or CI, but rather is a basis
set problem. Consider what happens when we compare the energy of the hydrogen-
bonded water dimer with that of two noninteracting water molecules. Here is the
result of an MP2(fc)/6–31G* calculation; both structures were geometry-optimized,
and the energies are corrected for ZPE:
Energy of H 2 O¼#76.27547 h
2 'Energy of H 2 O¼#152.55094 h
Energy of H 2 O dimer¼#152.55658 h
(2'Energy of H 2 O)#(Energy of H 2 O dimer)
¼#152.55094#(#152.55658) h¼0.00564 h¼14.8 kJ mol#^1
The straightforward conclusion is that at the MP2(fc)6–31G* level the dimer is
stabler than two noninteracting water molecules by 14.8 kJ mol#^1. If there are no
other significant intermolecular forces, then we might say the H-bond energy in the
water dimer [ 104 ] is 14.8 kJ mol#^1 (that it takes this energy to break the bond – to
separate the dimer into noninteracting water molecules). Unfortunately there is a
problem with using this simple subtraction approach to compare the energy of a
weak molecular association AB with the energy of A plus the energy of B. If we do
278 5 Ab initio Calculations