Calculation of potential energy surfaces (PESs)
1. The conventional supermolecule approach
Choose a size-extensive quantum chemistry model. Calculate separately the energy for the supersystem and the different subsystems comprizing
the supersystem. For the subsystem calculations use preferably the supersystem basis in order to avoid basis set superposition
errors. The intermolecular potential for a fixed configuration of the nuclei in the system, is
then the difference between the energy of the supersystem and the sum of the energies for the isolated
subsystems.
Advantage:
Standard quantum chemistry programs can be used.
Disadvantage:
(i) Several calculations/runs must be performed in order to obtain a point on the potential energy
surface.
(ii) Interpretation/analysis of the results are usually lacking.
2. Calculations based on the EXGEM model
EXGEM calculations are performed within the supermolecule framework.
Advantage:
(1) Localized orbitals.
When considering intermolecular interactions it is convenient to use a RHF (restricted Hartree-Fock ) wave function as root function in the EXGEM
model. Hence, doubly occupied orbitals are assumed for all subsystems. The RHF wave function is invariant with respect to any unitary transformation
of the occupied orbitals. Due to this invariance we can define localized orbitals which can be associated with the appropriate subsystems or fragments.
In the EXGEM program these localized orbitals are defined by a minimal distortion
procedure, i.e. the localized orbitals of the supersystem are determined such that the deviation from the corresponding orbitals of the
isolated subsystems, is minimal. The localized orbitals are essential in an interpretation of the
results.
(2) Elimination of BSSE ( basis set superposition error ).
Subsystem calculations are performed with the supersystem basis for the appropriate supersystem nuclear configuration ( zero charges on all atoms
not belonging to the subsystem in question ). This is the Boys-Bernardi
counter point procedure. Accordingly, there is no basis set superposition error in
the calculation.
(3)The EXGEM model is size-extensive.
A model is size-extensive if, for two non-interacting systems A and B, the energy satsifies the relation
 
where
is the energy when A and B are considered as a combinded system, and
and
are the energies of the isolated subsystems. The size-extensivity property is of paramount importance in order to obtain
reliable results.
(4) Interpretative features.
For interpretative purposes two slightly different energy partitioning schemes
are introduced in the EXGEM model. In the primary scheme the intermolecular potential is written as a sum of distortion energies for the subsystems and
the interaction between destorted subsystems:
In the preceeding equation
denotes the distortion energy, i.e. the change in energy of subsytem s due the presence of the other
subsystems,
, 
and
denote respectively the Coulombic, exchange and correlation interaction between the distorted subsystems s and t.
If
is the Coulombic interaction between the undistorted subsystems s and t, we define an induction energy as
A modified distortion energy is defined as
As a result we have the following modified partitioning scheme:
Another interpretative feature is related to the description of the changes in the one-electron density during complex
formation. Due to the adopted localization procedure, these changes can be monitored by the changes in charge centroids and charge ellipsoids
of the subsystem orbitals.
(5) Computational simplisity.
One point on the PES is obtained in one run of the EXGEM program. This is to be compared with (n+1) runs with a standard approach for
a supersystem comprising n subsystems.
Disadvantage:
(1) Unconventional basis set.
The integral program is based on familiy type GTF (Gaussian type functions). This choice was motivated by computational requirements in
the construction of the Cholesky decomposition of the two-electron matrix.
Hence, standard basis sets cannot be adopted. A family type basis is characterized by having the same exponent for all GTFs in the
family. An L-type family ( L - angular momentum quantum number ) includes GTFs
with angular momentum from the set { o, 1, ...., L }. For example, the f-type
family includes s-, p-, d- and f-fype sets of GTFs. The construction of family
type functions is discussed in the section The EXGEM
program.
(2) Not intirely reliable for very large distances between subsystems.
In the present implementation of the EXGEM
model, triple and higher order pair corrections are neglected. As a consquence there is a small numerical
instability in the calculation of the intra-subsystem correlation energy.
Hence, for large distances between subsystems, neglect the change in intra-subsystem
energies ( These terms can easily be picked out from the partitioning scheme).
This weakness will be remedied when the intra-subsystem energy is approximated by a coupled cluster
approach.
(3) Only ground state potentials can be calculated.
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