Semiempirical Methods

Semiempirical Methods are simplified versions of Hartree-Fock theory using empirical (= derived from experimental data) corrections in order to improve performance. These methods are usually referred to through acronyms encoding some of the underlying theoretical assumptions. The most frequently used methods (MNDO, AM1, PM3) are all based on the Neglect of Differential Diatomic Overlap (NDDO) integral approximation, while older methods use simpler integral schemes such as CNDO and INDO. All three approaches belong to the class of Zero Differential Overlap (ZDO) methods, in which all two-electron integrals involving two-center charge distributions are neglected. A number of additional approximations are made to speed up calculations (see below) and a number of parameterized corrections are made in order to correct for the approximate quantum mechanical model. How the parameterization is performed characterizes the particular semiempirical method. For MNDO, AM1, and PM3 the parameterization is performed such that the calculated energies are expressed as heats of formations instead of total energies (see earlier remarks on this difference).

New versions of the NDDO methods have recently been developed that include d-Orbitals for second-row and higher elements (MNDO/d and PM3(tm)). Also, a slightly extended and reparameterized version of PM3 termed PM5 has recently been made available in the program package MOPAC 2000. In the following we will concentrate on the three methods available in Gaussian (MNDO, AM1, PM3).

The first strategy used to reduce computational effort is to consider only valence electrons in the quantum mechanical treatment. This is immediately obvious when comparing Hartree-Fock with, for example, AM1 energy calculations on a molecule such as methanol:
 

#P AM1 scf=tight AM1 energy of CH3OH (Cs) 0 1 C1 H2 1 r2 O3 1 r3 2 a3 H4 3 r4 1 a4 2 180.0 H5 1 r5 2 a5 3 d5 H6 1 r5 2 a5 3 -d5 r2=1.11900473 r3=1.41043172 r4=0.9641002 r5=1.11868093 a3=105.12806298 a4=107.16494018 a5=110.03331541 d5=119.51560095

Even though methanol CH₃OH is composed of overall 6 nuclei and 18 electrons, only 14 of the electrons are treated explicitly. The electrons located in the 1s orbitals of oxygen and carbon are not considered explicitly, but combine with the nuclei to provide an effective core potential for the valence electrons:
 

Standard basis: VSTO-3G (5D, 7F)
.
.
 12 basis functions       36 primitive gaussians
  7 alpha electrons        7 beta electrons   
 nuclear repulsion energy        25.1935849033 Hartrees.

Standard basis: STO-3G (5D, 7F)
.
.
 14 basis functions       42 primitive gaussians
  9 alpha electrons        9 beta electrons
 nuclear repulsion energy        40.2200678489 Hartrees.

The basis sets used in semiempirical calculations are specially optimized minimal basis sets composed of Slater-type orbitals. As only valence electrons are considered and the core electrons are treated together with the nuclei as one effective core potential, it is clear that the simple point charge model used in Hartree-Fock theory to calculate the nuclear repulsion energies is inappropriate for semiempirical calculations. In order to compensate for some of the approximations made in calculating the attractive core-electron energies, the core-core potential used in NDDO methods goes beyond the use of a simple point charge model with reduced nuclear charges such as:

E_AB = Z'_AZ'_Be²/R_AB

with R_AB being the internuclear separation and Z'_A being the effective core charge including the nuclear charge and all core electrons. A general expression for calculation of the core-core repulsion energies between nuclei A and B at distance R_AB in NDDO methods is:

E_AB = Z'_AZ'_B<s_As_A,s_Bs_B>[1+ F(A) + F(B)]

The core repulsion energy is here a function of both the electron-electron repulsion integral <s_As_A,s_Bs_B> as well as atom-type dependent functions F(A) and F(B) which in turn depend on the internuclear separation R_AB. Functions F(A) and F(B) have a relatively simple form in MNDO:



including only one additional parameter. In order to improve some of the deficiencies of MNDO (especially those concerning hydrogen bonding), a slightly more complex function was chosen for AM1:



The sum over additional exponentials inlcudes either three or four terms and introduces three new parameters K_Ai, L_Ai, and M_Ai for each element constituting the main difference (aside from the actual fitting procedure) between AM1 and MNDO. Which set of parameters is used in a given semiempirical calculation is specified in the Gaussian output file as:

 References:
  H: (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985)
  C: (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985)
  O: (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985)

For some of the elements, the core repulsion functions F(A) for a given element depend on the interaction partner B. Taking the element boron as an example, there are four different sets of parameters describing the core of boron implying that F_boron is different in E_BB, E_BH, E_BC, and E_BX!! Please note that this functionality is not implemented in all programs, Gaussian being one of the problematic cases. It is therefore not possible to perform AM1 calculations on boron-containing compounds in Gaussian. Any attempt to do so anyway will lead to a calculation with mixed AM1 and MNDO parameters identified in the output file as follows:
 

 References:
  H: (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985)
  B: (MNDO):  M.J.S. DEWAR, M.L. MCKEE, J. AM. CHEM. SOC., 99, 5231, (1977)
 
 Warning.  AM1 has been requested, along with some elements
 for which only MNDO parameters are available.  Such mixtures
 of methods are very risky and have not been fully tested.

Indeed, these types of calculations are extremely unreliable and not worth doing under any circumstance.

The performance of NDDO methods for a large number of molecular systems has been studied repeatedly and the mean signed and unsigned errors for the heat of formation (kJ/mol) in a set of 194 typical organic systems containing the elements C, H, N, and O has been collected in the following table:
 

The performance is much worse for all three methods in cases involving second-row elements such as S or P, the description of hypervalent compounds being particularly problematic.

Another critical area of application concerns calculations of nitrogen-containing compounds. While the inversion barriers for trivalent nitrogen are usually too low with AM1, they are predicted to be too high with PM3. As a consequence, some non-planar nitrogen containing compounds are predicted to be flat by AM1, while some flat compounds are predicted to be pyramidal by PM3. This is particularly deplorable for peptide structures as a realistic description of conformational space is impossible without a proper description of the amide bond. The performance of the NDDO models might be appreciated better using the dipeptide system used before in Hartree-Fock geometry optmization. The planarity of the central amide bond can be characterized through the C(O)-N-H-C dihedral angle, being 180^o for a fully planar conformation:

Pyramidalization of the amide nitrogen is indeed largest with PM3 and smallest with AM1. In order to rescue the situation the PM3 method is also available in a PM3MM version, adding a simple molecular mechanics correction term to the amide linkage. It can be seen quite nicely in the above example, that the amide bond becomes more planar upon inclusion of this correction term. Whether or not this correction term is in use can be seen in the output file as:
 

 NNHCO=    2.
 References:
  H: (PM3): J. J. P. STEWART, J. COMP. CHEM.     10, 209 (1989).
  C: (PM3): J. J. P. STEWART, J. COMP. CHEM.     10, 209 (1989).
  N: (PM3): J. J. P. STEWART, J. COMP. CHEM.     10, 209 (1989).
  O: (PM3): J. J. P. STEWART, J. COMP. CHEM.     10, 209 (1989).

A reliable prediction of peptide conformations is, however, not possible even with these corrections.

One important point to consider when comparing the performance of semiempirical calculations with other theoretical methods such as Hartree-Fock or DFT is that the former have been parameterized to include all thermochemical corrections to yield heat of formations at 300K, while the latter have not. Also, through parameterization with reference to experimental data, semiempirical methods *might* be expected to recover some part of electron correlation effects. This is, of course, only true for the ground state systems included in the parameterization procedure and not necessarily true for transition states or electronically excited states.