2.17 Combined Quantum/Classical (QM/MM) Methods
41
The first term corresponds to the bonded interactions, the second one to the van der
Waals interactions and the third one, to the electrostatic interactions between the two
subsystems Q and M. The treatment of the first two terms is easy because they can
be handled at the MM level. The case of the electrostatic coupling is more difficult
and several different levels of sophistication may be used. The simplest method is
called mechanical embedding. It treats the electrostatic interactions at the MM level.
It is simple but has severe shortcomings. At present, one of the best methods is the
electrostatic embedding: The MM charges are inserted in the QM Hamiltonian as oneelectron operators. The polarization of the Q system by the electrostatic interactions
with the M system is accounted for.
If the two subsystems are connected by a chemical bond, the bond is cut and
the dangling bond of the Q system must be capped by a link atom (for instance, H)
bound to Q. This link atom is only used for the QM calculation, the bond Q-M being
described at the MM level.
Even though QM/MM methods are often very efficient, they are still rather tricky
to handle. The Q region has to be carefully selected. Furthermore, the potential energy
surface generally has many local minima.
2.18 Quantum Theory of Atoms in Molecules (QTAIM
or AIM) (Bader 1990; Gillespie and Popelier 2001)
Ab initio methods can deliver accurate interatomic distances but do not give direct
information on the bonds and their properties. Nevertheless, this information can be
obtained by analyzing either the wavefunction or the electronic density, ρ. However,
it is much easier to use the electronic density because ρ is a function of only three
variables (x, y, and z) and has a direct experimental relevance (it can also be obtained
by X-ray diffraction).
The AIM method, developed by Richard F. W. Bader, uses the electronic density,
ρ, derived from the wavefunction as starting point and provides a simple quantum
definition for an atom in a molecule as well as for a bond. The AIM is also a method
for analyzing the electron density distribution that determines all the properties of a
molecule. The maximum of ρ corresponds to a nuclear position. Along the internuclear axis, ρ has a minimum value, and in a direction perpendicular to the internuclear
axis, the density at this critical point is a maximum, and it is a saddle point called
bond critical point. The magnitude of the electron density at this bond critical point
ρ b serves as a parameter that can be used in evaluating the corresponding bond
order (Cioslowski and Mixon 1991). It also gives the amount of electron density
shared between the two bonded atoms. It is roughly proportional to the bond length.
The strength of a bond increases and its length decreases as the bond critical point
density ρ b increases, and the increasing charges on the bonded atoms have the same
effect. There is still another factor affecting the bond lengths: They increase with
41
The first term corresponds to the bonded interactions, the second one to the van der
Waals interactions and the third one, to the electrostatic interactions between the two
subsystems Q and M. The treatment of the first two terms is easy because they can
be handled at the MM level. The case of the electrostatic coupling is more difficult
and several different levels of sophistication may be used. The simplest method is
called mechanical embedding. It treats the electrostatic interactions at the MM level.
It is simple but has severe shortcomings. At present, one of the best methods is the
electrostatic embedding: The MM charges are inserted in the QM Hamiltonian as oneelectron operators. The polarization of the Q system by the electrostatic interactions
with the M system is accounted for.
If the two subsystems are connected by a chemical bond, the bond is cut and
the dangling bond of the Q system must be capped by a link atom (for instance, H)
bound to Q. This link atom is only used for the QM calculation, the bond Q-M being
described at the MM level.
Even though QM/MM methods are often very efficient, they are still rather tricky
to handle. The Q region has to be carefully selected. Furthermore, the potential energy
surface generally has many local minima.
2.18 Quantum Theory of Atoms in Molecules (QTAIM
or AIM) (Bader 1990; Gillespie and Popelier 2001)
Ab initio methods can deliver accurate interatomic distances but do not give direct
information on the bonds and their properties. Nevertheless, this information can be
obtained by analyzing either the wavefunction or the electronic density, ρ. However,
it is much easier to use the electronic density because ρ is a function of only three
variables (x, y, and z) and has a direct experimental relevance (it can also be obtained
by X-ray diffraction).
The AIM method, developed by Richard F. W. Bader, uses the electronic density,
ρ, derived from the wavefunction as starting point and provides a simple quantum
definition for an atom in a molecule as well as for a bond. The AIM is also a method
for analyzing the electron density distribution that determines all the properties of a
molecule. The maximum of ρ corresponds to a nuclear position. Along the internuclear axis, ρ has a minimum value, and in a direction perpendicular to the internuclear
axis, the density at this critical point is a maximum, and it is a saddle point called
bond critical point. The magnitude of the electron density at this bond critical point
ρ b serves as a parameter that can be used in evaluating the corresponding bond
order (Cioslowski and Mixon 1991). It also gives the amount of electron density
shared between the two bonded atoms. It is roughly proportional to the bond length.
The strength of a bond increases and its length decreases as the bond critical point
density ρ b increases, and the increasing charges on the bonded atoms have the same
effect. There is still another factor affecting the bond lengths: They increase with