For the purpose of this discussion, only the simplest of interacting species
will be considered, i.e., precisely two species, **A** and **
B**. These can be monatomic, e.g. Hg, monatomic ions, e.g. [Cl]^{-},
polyatomic molecules, e.g., aniline, C_{6}H_{5}NH_{2},
or polyatomic ions, e.g. hydrogen sulfate, [HSO_{4}]^{-}. There
are several types of intermolecular interactions involving **A**
and **B**:

- "Covalent" - arising from quantum chemical contributions. Since
intermolecular interactions imply the absence of covalent interactions (the
distance between the interacting species is too large, by definition) there
are no covalent bonds,
*per se*, but there are very small energy contributions resulting from the overlap of atomic orbitals. These contributions are small relative to the other intermolecular interactions, and indeed become vanishingly small outside about two times the covalent distance, and can thus also be ignored in this discussion. - Electrostatic - arising from the partial charge on each atom,
**a**, in**A**interacting with the partial charge on each atom,**b**, in**B**. Electrostatic interactions decrease only slowly with distance, the energy falling off as the reciprocal of the interatomic distance, i.e., as 1/(R**ab**). Although they can be large, electrostatic terms are simple to calculate and will not be considered further. - Dispersion, otherwise known as van der Waals or VDW interactions, or
London forces - arising from the instantaneous correlation of electrons.
Like most quantum chemical theoretical methods, NDDO-type semiempirical
methods use molecular orbitals to solve the Hartree-Fock equations.
This is an approximation to the correct wavefunction, a good approximation,
but not perfect. The use of M.O.s implies that the motion of the
electrons is not correlated. Correlation of electrons is a
time-dependent phenomenon, and results when one electron interacts with
another electron. At each instant of time, the two electrons will
endeavor to avoid each other, because of electrostatic repulsion.
Dispersion energies can be quite large for nearby atoms, but fall of as the
sixth power of the distance, i.e., as 1/(R
**ab**)^{6}.

- Hydrogen bonds - a unique type of interaction that typically involves
three atoms, with the middle atom being hydrogen. The commonest
hydrogen bonds are of the type O-H - O and O-H - N. Hydrogen bonds could be
regarded as a special case of electrostatics, dispersion and covalent
interactions, but because of their great importance, and because they are
not, in practice, well reproduced by these other terms, they are now added
in
*post-hoc*to the energy terms.

Dispersion terms are particularly hard to calculate *de novo*.
The strategy used in MOPAC is to use published benchmark calculations as a
source of specific intermolecular interaction energies. In PM7, these are
then used in parameterizing a simple function in the parameter optimization
procedure used in developing a new method. In PM6-DH2 and PM6-DH+ the method
described in "A Transferable H-bonding Correction For Semiempirical
Quantum-Chemical Methods" Martin Korth, Michal Pitonak, Jan Rezac and Pavel
Hobza, J Chem Theory and Computation 6:344-352 (2010) is used. In PM6-D3,
the method used is described in "A consistent and accurate ab initio
parametrization of density functional dispersion correction (DFT-D) for the 94
elements H-Pu" S. Grimme, J. Antony, S. Ehrlich, H. Krieg, THE JOURNAL OF
CHEMICAL PHYSICS 132, 154104 .2010. Published reports indicate that the
most accurate semiempirical method for predicting intermolecular interaction
energies is PM6-D3H4. This method has not been included in MOPAC, because
by the time all the other methods had been included, enthusiasm for yet another
method had vanished see also below.

The energy correction due to a hydrogen bond is quite large, in the order of several kcal/mol. Despite it being so important, the functional forms of the hydrogen bond have been developed using purely pragmatic reasoning rather than by deductive science. Thus it is known experimentally that the most important hydrogen bonds involve oxygen, then nitrogen, then other atoms and ions such as sulfur and chloride; that the bonds are more-or-less linear; and that they depend on the local net charge in the region of the bond. Thus a hydrogen bond between two acetic acid molecules would be weaker than between an acetic acid molecule and an acetate ion. The extra energy of a charged hydrogen bond is reflected in a reduced distance between the two non-hydrogen atoms in a hydrogen bond.

Several approaches have been proposed for modeling the hydrogen bond, with the best being the -DH2, -NH+, and -D3H4, with -D3H4 being the most accurate. The approach used in PM7 can best be described as a mixture of -DH2 and -DH+ with a special correction for charged hydrogen bonds. The resulting approximation is by no means optimal, but does capture most of the phenomena associated with hydrogen bonds. Because it is not optimal, it should not be regarded as definitive, in other words, when a better approximation is developed, the current approximation in PM7 should be replaced. Why was -D3H4 not used? Simply put, after implementing the earlier approximations, I ran out of enthusiasm to implement the final, best, approximation.

Approximations for modeling both dispersion and hydrogen bonding terms have been evolving rapidly. At the present time, 2013, a clear description of this topic appears to be appearing: Terms of the type developed by Grimme, i.e., the D3 terms, represent the dispersion effect with very good accuracy, and should be used, without modification, in the development of all future methods. This had several advantages, among these are: (A) Because the dispersion effect is completely separate from all other effects, it can be treated separately. (B) By not using any parameters for the dispersion effect, the number of parameters in a new method is reduced - always a desirable objective. (C) By using a "black box" to represent dispersion effects, the potential of introducing distance-dependent artifacts into a new method is reduced. Unfortunately, at the time PM7 was completed, this insight had not been developed.

The status of hydrogen bonding approximations is less clear. When PM6-DH2 was developed, a small theoretical error was introduced: the hydrogen bond energy was made a function of the partial charge on the hydrogen atom. This weakened the electronic variational principle, because now the energy minimum depended on the post-SCF hydrogen bond partial charge. A consequence of this violation was that the energy gradient norm no longer went to zero at the energy minimum, and as a result geometry optimization became ill-defined. The error was small, and for most systems could be neglected. The most important practical effect was that geometry optimization became less efficient and as a result took more cycles, i.e., more CPU time.

The error in PM6-DH2 was corrected in PM6-DH+, by making the hydrogen bond energy independent of partial charge. But now a new error was introduced in PM6-DH+ where for some highly specific systems there was a discontinuity in the second derivative of the energy with respect to geometry.

In PM7, both of the errors in PM6-DH2 and PM6-DH+ were removed, and the behavior of the hydrogen bond improved. In recent months, various faults in the shape of the hydrogen bond have been described.

These problems with the hydrogen bond illustrate the fact that approximations for the hydrogen bond are still evolving. The problem to be addressed is clear, it is only the approximation that describes the hydrogen bond that is not clear. of organic compounds.