Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.4 Two-body interaction energy 433
example given in the figure) only introduces quantitative modifications, sufficient however
to show by simple visual inspection if there is a hydrogen bond or not.
Analogous trends in the decomposition of ΔE are present in the interactions involving
charged-neutral species (also in the case of apolar molecules). A special case is given by the
interaction of a molecule with the bare proton: in this case there is no EX contribution.
If the partners have no permanent charge, or dipole, the interaction at large-medium
distances is dominated by DIS, and by EX at small values of R.
8.4.2 BASIS SET SUPERPOSITION ERROR AND COUNTERPOISE
CORRECTIONS
Calculations of the interaction energies are affected by a formal error that may have important
consequences on the final value of the energy and on its decomposition. We shall consider
here the case of variational calculations for dimers, but the basic considerations can be
extended to larger clusters and to other computational methods. The origin of the error is a
non-perfect balance in the quality of the calculation of dimer energy, EAB and of energies of
the two monomers, EA and EB. In fact, there are more computational degrees of freedom
available for the dimer than for each monomer separately. The number of degrees of freedom
corresponds to the number of basis functions available for the optimization of the electronic
structure of the molecule, and hence for the minimization of the energy. Let us
consider, to clarify the concept, the case of two water molecules giving origin to a dimer;
each water molecule has ten electrons, while the quality and number of expansion functions
is selected at the beginning of the calculation. This is called the expansion basis set (just basis
set, or BS, for brevity) and it will be indicated for the molecule A with {χA}. The number
of these basis functions is fixed, for example, 30 functions. The second molecule will be described
by a similar basis set {χB} containing in this example expansion functions of the
same quality and number as for molecule A (the two molecules are in this example of the
same chemical nature). The wave function of the dimer AB and its energy will be determined
in terms of the union of the two basis sets, namely {χAB}= {χA⊕χB}, composed of 60
functions. It is evident that it is easier to describe 20 electrons with 60 parameters than 10
electrons with 30 only. The conclusion is that the dimer is better described than the two
monomers, and so the dimer energy is relatively lower than the sum of the energies of the
two monomers.
This is called the basis set superposition (BSS) error. Why superposition error? When
the two components of the dimer are at large distance, {χA} and {χB} are well separated, i.e.,
they have small superposition (or overlap), and so the relative error we are considering is
modest, zero at infinity. When the two monomers are at shorter distances the superposition
of the two basis sets increases (the basis functions are always centered on the pertinent nuclei)
as well as the error.
There is a simple recipe to correct this error: it consists of performing all the necessary
calculations with the same basis set, the dimeric basis {χAB} which depends on the geometry
of the dimer. 9 This means that the energy of the monomers must be repeated for each position
in the {R 6 } configuration space (see Section 8.4 for its definition). We add a superscript
CP (counterpoise) to denote quantities modified in such a way and we also add the specification
of the basis set. We replace eq. [8.11] with the following one:
[ AB B AB ]
CP
CP
( χ ; ) ( χ ; ) ( χ ; ) ( χ ; )
CP
ΔE R = E R − E R + E R [8.23]
AB
AB AB AB A