Subject: Re: CCL:BSSE (Quantum ab initio)

From: Joerg Hill <[email-masked]>
Date: Thu, 16 Mar 1995 11:08:45 -0800 (PST)

Peter Winn asks:
> What is the current consensus on the affect of
> basis set superposition errors on ab initio quantum 
> calculations???? In particular I would be interested
> in:-
> 
> (i) The affect it has on interatomic distances when
> optimising a group of molecules.
> 
> (ii) The overall expected error in interaction 
> energies one might expect from a group of optimised
> molecules.
> 
> (iii) Identifying you have a BSSE
> 
> (iv) Correcting for BSSE.
> 
> (v) A reference covering the above problems in
a consistent manner.
> 
> I have found many references that deal with BSSE
> but nothing that really gives an overview of the
> whole problem.

The BSSE is always present if you are doing calculations on intermolecular
interactions. In this case you will do a calculation on your molecules alone
(monomers) and then you will put these molecules together (complex, supermolecule
approach) and do another calculation. From the results of all these calculations
you will try to figure out what happens when the molecules interact. Unfortunately,
by putting all molecules together, you add additional basis functions to your
monomers (the basis functions of the other monomers), so that your basis set
becomes more flexible (and the energy therefore lower) compared to your monomer
calculations. Or, to put it another way, the electrons in your monomer are
described by basis functions of the other monomers. Therefore, your BSSE
depends on the quality (size) of your basis set. If you use a larger basis
set, your electrons will be described more complete in the monomer basis, thus
reducing the BSSE.
Since the BSSE affects the energy it will affect all properties calculated:
interaction energies, the geometry changes going from monomers to the complex
etc. The standard method of correcting for the BSSE is the counterpoise
correction method of Boys and Bernardi (S.F. Boys, F. Bernardi; Mol. Phys. 19,
553 (1970)). This method requires additional calculations to be performed for
the monomers: You take the basis functions as they are in the complex, but
remove all the atoms which do not belong to your monomer (creating so-called
ghost functions). Then you do a calculation for your monomer with this larger
basis set. Your BSSE for the interaction energy is then:

              BSSE = (E(a)-E(a*)) + (E(b)-E(b*))                        (1)

where E(a) and E(b) are the energies of the monomers without ghost functions
and E(a*) and E(b*) are the energies of the monomers with ghost functions.
(Note: This procedure in ambigious if you have more than two molecules;
L. Turi, J. J. Dannenberg; J. Phys. Chem. 97, 2488 (1993).) The number you
get from Eq. (1) tells you something about the quality of your basis set.
It should be significantly smaller than the effect you are looking at (can
be hard to achieve). If you are using a correlated method you will see
a much larger BSSE with the same basis set compared to SCF. This has to
do with having additional virtual orbitals in the complex (S. Saebø, W. Tong,
P. Pulay; J. Chem. Phys., 98, 2170 (1993)).
As mentioned above, the BSSE also affects the geometry. I haven't seen any
paper which has done a BSSE correction for geometries. In principle you
have to do the same correction at each step of your geometry optimization
calculating the gradient from the corrected energy.
The counterpoise correction also does not recover all of the BSSE, since
it does not allow the ghost functions to change their positions in space.

There are a lot of references for the BSSE (from a previous discussion
of this topic on the CCL, you may look in the CCL archives for more):
- "Gaussian Basis sets for Molecular Calculations", S. Huzinaga, Elsevier
  (1991)
- "Ab Initio Calculations", P. Carsky and M. Urban, Springer- Verlag,
  Lecture Notes in Chemistry, 1980.
- "Intermolecular Complexes", P. Hobza and R. Zahradnik, Academia, Prague
  (1988)

Joerg-R. Hill

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