Subject: BSSE

From: Istvan Mayer <[email-masked]>
Date: Fri, 17 Mar 1995 13:47:49 +0100 (MET)

Let me make some comments on the BSSE problem, first about the
Boys-Bernardi scheme, then about an alternative (and perhaps
more accurate) method, called CHA.
 
1. J-R. Hill correctly summarizes the origin of BSSE and most
aspects of the Boys-Bernardi counterpoise correction. It seems
however worth to quote the working fomula of this method
as it is usually applied:
 
E(int.,corrected) = E(AB) - E(A*) - E(B*)
 
where E(AB) is the actual supermolecule energy, E(A*) and
E(B*) are the monomer energies calculated by using the whole
supermolecule basis. This way of calculation mislead a number
of people who have assumed that "this method is exact because
all the quantities are calculated in the same basis set". This
is, however, not the case; it is an (ingenious and often
accurate, but no ways exact ot mathematically rigorous) A
POSTERIORI correction scheme. Its real meaning is just in the
correction formula quoted by Hill, that is
 
E(int.,corrected) = E(int.,uncorr.) + BSSE(correction)
 
where E(int, uncorr.) is the interaction energy which one
obtains from a "straightforward" calculation:
 
E(int.,uncorr.)=E(AB)-E(A)-E(B)
 
E(A), E(B) being the free monomer energies calculated in their
own basis, and
 
BSSE(correction) = (E(A) - E(A*)) + (E(B) - E(B*))
 
(For a related discussion see e.g., P. Valiron, A. Vibok and
I. Mayer, J. Comp. Chem. 14, 401, 1993; and also: I. Mayer and
L. Turi, J. Mol.Struct.(THEOCHEM) 227, 43, 1991.)
 
It is important to call the attention that the Boys-Bernardi's
Counterpoise method becomes ambiguous not only if there are
more than two molecules but also if the geometry optimization
is considered. If the change of the internal monomer
geometries ("monomer relaxation") is also admitted, then the
only way to avoid the use of mathematically ill-defined
quantities is to calculate monomer relaxation in pure monomer
basis sets [I.Mayer and P.R.Surjan, Chem. Phys. Lett. 191, 497
(1992)]. Looking from other side, this means that one has to
use the correction formula also quoted in Hill's mail for the
RELAXED monomer geometries, which means that each point on the
energy hypersurface can be obtained by using as much as 7
different energy values.
 
In fact, the uncorrected interaction energy is
 
E(int.,uncorr)=E(AB)-E(A0)-E(B0)
 
where E(AB) is the supermolecule energy with the actual intra-
and intermolecular geometrical parameters, E(A0) and E(B0) are
the energies of the FREE monomers in their own optimized
geometries, respectively. Then the BSSE correction to this is,
similarly to that quoted above (and in Hill's mail)
 
BSSE(correction) = ( E(Ar)- E(Ar*)) + (E(Br) - E(Br*))
 
where "r' indicates the actual relaxed geometry and "*" the
use of the whole supermolecule basis in the monomer
calculations. Then the resulting corrected interaction energy
is
 
E(int.,corrected) = E(int.,uncorr) + BSSE(correction)
 
The geometry optimization would require simultaneous
calculation of 5 (!) gradient values (the energies E(A0)
and E(B0) do not depend on the actual coordinates in the
supermolecule).
 
The same result quoted above can be presented as the actual
BSSE-corrected interaction energy of the relaxed monomers:
 
E(int, BSSE-corr.,neglecting relax.) = E(AB) - E(Ar*) - E(Br*)
 
which is then corrected also for the monomer relaxation calculated
in the monomer basis sets:
 
E(int., corrected) = E(int, BSSE-corr., neglecting relax)
                   + E(relax, A) + E(relax(B)
 
where
 
E(relax, A) = E(Ar)-E(A0)  and  E(relax, B) = E(Br)-E(B0) .

2. In the last years we have started to develop an A PRIORI
scheme of excluding BSSE, in which the Hamiltonian (or
Fockian) has been subjected to special analysis, and the terms
responsible for BSSE are identified and then omitted from the
actual calculations. This is called "Chemical Hamiltonian
Approach" (CHA):
 
- Its development is essentially completed at the SCF level
(two slightly different variants, see e.g. I. Mayer and
P.R.Surjan Int. J. Quantum Chem. 36, 225 (1989); I. Mayer and
A. Vibok, Int. J. Quantum Chem. 40, 139 (1991) and 43, 801
(1992) and - especially - the paper Valiron et al. quoted
above; for the case of more than two molecules a preprint of a
fresh manuscript is available).
 
- It has been adopted to the LCAO-DFT theory (M. Kieninger, S.
Suhai and I. Mayer, Chem. Phys. Lett. 230, 485 (1994);
 
- Its application to correlated wave functions is underway;
for pivot calculations at the full CI level see I. Mayer et
al. Chem. Phys. Lett. 224, 166 (1994)
 
For CHA geometry optimization does not represent any
CONCEPTUAL difficulty: only one gradient is needeeed. (Its
calculation is, however, is not so trivial, as the "chemical
Hamiltonian" in not Hermitean - BSSE does not represent a
physical observable, a Hermitean operator could be assigned to
which.)
 
One can observe by comparing CHA and Boys Bernardi results,
that they converge to each other with the increasing basis
sets much faster than BSSE diapperas from the uncorrected
interaction energy, so these a priori and a posteriori methods
corroborate other to a great extent.

I would call the attention to the recent special issue of
Chem. Rev., devoted to the intermolecular interactions.
 
Finally it should be noted that BSSE may occur within a single
molecule as well - but this is a different story (cf. I.M.
IJQC 23, 341, 1983)
 
With best regards to all CCL,
 
Istvan Mayer
 
   Central Research Institute for Chemistry of the
          Hungarian Academy of Sciences
E-mail:
         [email-masked] (or [email-masked])
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