Chapter 5. Quantum chemistry in Molecular Modeling


5.7 Semi-empirical quantum chemistry

Ab initio quantum chemical methods are limited in their practical applicability because of their heavy demands of cpu-time and storage space on disk or in the computer memory.
At the Hartree-Fock level the problem is seen to be in the large number of two-electron integrals that need to be evaluated. Without special tricks this is proportional to the fourth power of the number of basis functions. In practice this can be reduced to something close to the third power for larger molecules, e.g. because use is made of the fact that integrals between orbitals centered on distant atoms need not be calculated because they will be zero anyway.

Still, the size of systems that can be treated is limited, and this holds much more strongly for correlated treatments.
MP2 for example formally scales with the fifth power of the number of basis functions. Therefore there is a place for more approximate methods that retain characteristics of the quantum-chemical approach, in particular the calculation of a wavefunction from which electronic properties can be derived. In this section we will present a brief overview of commonly used semi-empirical methods [6, 18].

The semi-empirical methods are based on the Hartree-Fock approach. A Fock-matrix is constructed and the Hartree-Fock equations are iteratively solved. The approximations are in the construction of the Fock matrix, in other words in the energy expressions. Recall how the Fock matrix elements are expressed as integrals over atomic basis functions :


in which P is the density matrix :


To simplify matters drastically, the Zero Differential Overlap (ZDO) approximation assumes :

which implies that

This can be justified when the atomic basis orbitals are orthogonalized (Löwdin orthogonalization).
As a result of the ZDO approximation many two-electron integrals vanish :

Another common feature of semi-empirical methods is that they only consider the valence electrons.
The core electrons are accounted for in a core-core repulsion function, together with the nuclear repulsion energy.
In the most popular semi-empirical methods used today (MNDO, AM1 and PM3) the ZDO approximation is only applied to basis functions on different atoms. This is called the NDDO approximation (Neglect of Diatomic Differential Overlap). The resulting Fock matrix elements are given in ref. 6 and discussed in detail in ref. 18.
The next step is to replace many of the remaining integrals by parameters, which can either have fixed values, or depend on the distance between the atoms on which the basis functions are located. At this stage empirical parameters can be introduced, which can be derived from measured properties of atoms or diatomic molcules. In the modern semi-empirical methods the parameters are however mostly devoid of this physical significance: they are just optimized to give the best fit of the computed molecular properties to experimental data. For more technical details see references 6 and 18.
Different semi-empirical methods differ in the details of the approximations (e.g. the core-core repulsion functions) and in particular in the values of the parameters. Note that in contrast to molecular mechanics, only parameters for single atoms and for atom pairs are needed. The number of published parameters increases steadily.
The semi-empirical methods can be optimized for different purposes. The MNDO, AM1 and PM3 methods were designed to reproduce heats of formation and structures of a large number of organic molecules. Other semi-empirical methods are specifically optimized for spectroscopy, e.g. INDO/S or CNDO/S, which involve CI calculations and are quite good at prediction of electronic transitions in the UV/VIS spectral region.

Some even more approximate methods are still quite useful. In the Hückel and Extended Hückel methods the whole sum over two-electron integrals is replaced by a single diatomic parameter (the resonance integral), so that no search for a self-consistent field is necessary (nor possible). These methods have proven extremely valuable in qualitative and semi-quantitative MO theories of pi-electron systems and of organometallic systems [3].
For pi-electron systems ZDO treatments have been developed that take only pi-centers (p-atomic orbitals) into account, but do perform the SCF calculation. An example is the Pariser-Parr-Pople method, which involves a CI calculation as well. This method is very successfully used to predict the optical absorption spectra of conjugated organic molecules [19].
In the MM2 and MM3 programs pi-electron calculations are used to adjust the force constants and equilibrium values of bond lengths to the prevailing bond order. The pi-bond order between two atoms is simply the sum over MOs of the product of the coefficients of the basis functions on the atoms in the MO, multiplied by the occupation number of the MO :

For a given geometry the pi-electron calculation is done, and the bond-orders computed. Then the force field is adjusted : the force constants for stretching and torsion are scaled and the equilibrium bond length for the bonds between the pi-centers are calculated.

When the geometry changes too much, the pi-electron treatment is repeated to adjust the force field to the new situation. For the pi-electron calculation the pi-system is treated as if it is planar. Otherwise the bond order for a twisted semi-single bond would become smaller as the bond is twisted more, and the "restoring force" towards planarity (conjugation) would vanish.

References:

[3] G. Nàray-Szabò, P.R. Surjàn and J.G. Angyàn
Applied Quantum Chemistry, Reidel, 1987
[6] Spartan User's Guide, version 3.0,
Wavefunction, Inc., 1993.
[9] Brouwer, A.M.; Bezemer, L.; Jacobs, H.J.C.,
Recl. Trav. Chim. Pays-Bas 1992, 111, 138-143
[18] Stewart, J.J.P.,
in Reviews in Computational Chemistry, K.B. Lipkowitz and D.B. Boyd, eds., VCH, Vol. 1, 1990, 45 - 82.

Next paragraph, 5.8 Quality of semi-empirical results
Previous paragraph 5.6 Quality of ab initio results
Chapter 5 MM Syllabus 1995 MODIFIED November 8, 1995
Fred Brouwer, Lab. of Organic Chemistry, University of Amsterdam.