Semiempirical Theory - Introduction

There are six distinct methods available within MOPAC:  MNDO, AM1, PM3, PM6, RM1, and MNDO-d. All are semiempirical, and have roughly the same structure. A complete knowledge of these methods is not necessary in order to use MOPAC; however, a superficial understanding of these methods and their relationship to ab initio methods is important for using MOPAC and particularly for interpreting the results.

The six methods within MOPAC have many features in common. They are all self-consistent field (SCF) methods, they take into account electrostatic repulsion and exchange stabilization, and all calculated integrals are evaluated by approximate means. Further, they all use a restricted basis set of one s orbital and three p orbitals (px, py, and pz) and sometimes  five d orbitals per atom and ignore overlap integrals in the secular equation. Thus, instead of solving

|H-ES| = 0,

the expression

|H-E| = 0,

in which H is the secular determinant, S is the overlap matrix, and E is the set of eigenvalues, is solved. These approximations considerably simplify quantum mechanical calculations on systems of chemical interest. As a result, larger systems can be studied. Computational methods are only models, and there is no advantage in rigorously solving Schrödinger's equation for a large system if that system has had to be abbreviated in order to make the calculations tractable. Semiempirical methods are thus seen to be well balanced: they are accurate enough to have useful predictive powers, yet fast enough to allow large systems to be studied.

All the semiempirical methods contain sets of parameters. For PM6 atomic and diatomic parameters exist, while MNDO, AM1, PM3, RM1, and MNDO-d use only single-atom parameters. Not all parameters are optimized for all methods; for example, in  MNDO and AM1 the two electron one center integrals are normally taken from atomic spectra. In the list given in the Table, parameters optimized for a given method are indicated by '*'. A '+' indicates that the value of the parameter was obtained from experiment (not optimized). Where neither symbol is given, the associated parameter is not used in that method.


 

Table:

 Parameters used in Semiempirical Methods

Parameter

Description

MINDO/3

MNDO

AM1

PM3

RM1

PM6

Uss and Upp

s and p atomic orbital
one-electron one-center integrals

+

*

*

*

*

*

βs and βp

s and p atomic orbital one-electron
two-center resonance integral terms

 

*

*

*

*

*

Is

s atomic orbital ionization potential
for two center resonance integral term

+

 

 

 

 

 

Ip

p atomic orbital ionization potential
for two center resonance integral term

+

 

 

 

 

 

βAB

Diatomic two center one-electron
resonance integral multiplier

*

 

 

 

 

 

ξs

s-type Slater atomic orbital exponent

*

*

*

*

*

*

ξp

p-type Slater atomic orbital exponent

*

*

*

*

*

*

αA

Atom A core-core repulsion term

 

*

*

*

*

*

αAB

Atoms A and B core-core repulsion term

*

 

 

 

 

*

Gss

s-s atomic orbital one center
two electron repulsion integral

+

+

+

*

*

*

Gsp

s-p atomic orbital one center
two electron repulsion integral

+

+

+

*

*

*

Gpp

p-p atomic orbital one center
two electron repulsion integral

+

+

+

*

*

*

Gp2

p-p' atomic orbital one center
two electron repulsion integral

+

+

+

*

*

*

Hsp

s-p atomic orbital one-center
two-electron exchange integral

+

+

+

*

*

*

KnA or anA

A Gaussian multiplier for nth
Gaussian of atom A

 

 

*

*

*

*

LnA or bnA

A Gaussian exponent multiplier
Gaussian of atom A

 

 

*

*

*

*

MnA or cnA

A Radius of center of nth
Gaussian of atom A

 

 

*

*

*

*

All the semiempirical methods also use two experimentally determined constants per atom: the atomic weight and the heat of atomization.