this we are assuming that if there were no interactions at all between A and B at the
geometry of the AB species, then the AB energy would be that of isolated A plus
that of isolated B. The problem is that when we do a calculation on the AB species
(say the dimer HOH...OH 2 ), in this “supermolecule” the basis functions (“atomic
orbitals”) of B are available to A so A in AB has a bigger basis set than does isolated
A; likewise B has a bigger basis than isolated B. When in AB each of the two
components can borrow basis functions from the other. The error arises from
“imposing” B’s basis set upon A and vice versa, hence the name basis set superpo-
sition error. Because of BSSE the separated species A and B are not being fairly
compared with AB, and we should use for the energies of separated A and of B
lower values than we get in the absence of the borrowed functions available in the
weak complex. Accounting for BSSE will thus give a smaller energy drop on AB
formation. The value for the hydrogen bond energy (or van der Waals’ energy, or
dipole–dipole attraction energy, or whatever weak interaction is being studied) will
be less than if BSSE were ignored.
There are two ways to deal with BSSE. One is to say, as we implied above, that
we should really compare the energy of AB with that of A with the extra basis
functions provided by B, plus the energy of B with the extra basis functions
provided by A. This method of correcting the energies of A and B with extra
functions is called thecounterpoise method[ 105 ], presumably because it balances
(counterpoises) functions in A and B against functions in AB. In the counterpoise
method the calculations on the components A and B of AB are done withghost
orbitals, which are basis functions (“atomic orbitals”) not accompanied by atoms
(spirits without bodies, one might say): one specifies for A, at the positions that
wouldbe occupied by the various atoms of B in AB, atoms of zero atomic number
bearing the same basis functions as the real atoms of B. This way there is no effect
of atomic nuclei or extra electrons on A, just the availability of B’s basis functions.
Likewise one uses ghost orbitals of A on B. A detailed description of the use of
ghost orbitals (in Gaussian 82, but still instructive) has been given by Clark [ 105 a].
The counterpoise correction is rarely applied to anything other thanweakly-bound
dimers, like hydrogen-bonded and van der Waals species: strangely, the correction
worsenscalculated atomization energies (e.g. covalent AB!AþB). and it is has
been said to be not uniquely defined for species of more than two components
[ 105 b]; however, see calculations on a ternary complex, ethene–water–ethene
[ 106 ]. A review of criticisms and a defence of the counterpoise method is given
in [ 105 e].
The second way to handle BSSE is to swamp it with basis functions. If each
fragment A and B is endowed with a really big basis set, then extra functions from
the other fragment won’t alter the energy much – the energy will already be near the
asymptotic limit. So if one simply carries out a calculation on A, B and AB with a
sufficiently big basis, the straightforward procedure of subtracting the energy of AB
from that of AþB should give a stabilization energy essentially free of BSSE.
Nevertheless, the counterpoise method is the standard way of overcoming BSSE.
The best experimental estimate of the binding enthalpy of the water dimer was
said to be#13.4 kJ mol#^1 (#3.2)0.5 kcal mol#^1 )[ 104 c]; this is the enthalpy, at
5.4 Post-Hartree–Fock Calculations: Electron Correlation 279