3 Electronic Structure
3.14 Limitations and Advantages of Semi-Empirical Methods
The neglect of all three- and four-centre two-electron integrals reduces the construc- tion of the Fock matrix from a formal order of M4basis to M2basis. However, the time required for diagonalization of the Fmatrix grows as the cube of the matrix size, thus semi-empirical methods formally scale as the cube of the number of basis functions in the limit of large molecules. Diagonalization of a matrix becomes significant when the size exceeds ~10 000 ×10 000. Several iterations are required for solving the SCF equa- tions, and usually the geometry is also optimized, requiring several calculations at dif- ferent geometries. This places the current limit of semi-empirical methods at around 1000 atoms. It should be noted that the conventional method of solving the HF equa- tions by diagonalizing the Fock matrix rapidly becomes the rate-limiting step in semi-
′ = + ( − ) aA aA w nA rA b
rA A
=MO
∑
n ci ii 2
a a b
b b
A C A CC
AB AB CC
= +
= h k F
F F
m m m m m m
a b
A A A B A B
A
AB A and B are neighbours A and B are not neighbours
=
= ( )
=0 ( )
empirical methods. Recent developments have therefore concentrated on formulating alternative methods for obtaining the SCF orbitals without the need for diagonaliza- tion.67Such methods display linear scaling with the number of atoms, allowing calcu- lations to be performed for systems containing several thousand atoms.
The parameterization of MNDO/AM1/PM3 is performed by adjusting the constants involved in the different methods such that the results of HF calculations fit experi- mental data as closely as possible. This is in a sense wrong. We know that the HF method cannot give the correct result, even in the limit of an infinite basis set and without approximations. The HF results lack electron correlation, as will be discussed in Chapter 4, but the experimental data of course include such effects. This may be viewed as an advantage, the electron correlation effects are implicitly taken into account in the parameterization, and we need not perform complicated calculations to improve deficiencies in the HF procedure. However, it becomes problematic when the HF wave function cannot describe the system even qualitatively correctly, as with for example biradicals and excited states. In such cases, additional flexibility can be intro- duced in the trial wave function by adding more Slater determinants, for example by means of a CI procedure (see Chapter 4 for details). But electron correlation is then taken into account twice, once in the parameterization at the HF level, and once explic- itly by the CI calculation.
Semi-empirical methods share the advantages and disadvantages of force field methods: they perform best for systems where much experimental information is already available but they are unable to predict totally unknown compound types. The dependence on experimental data is not as severe as for force field methods, owing to the more complex functional form of the model. The NDDO methods require only atomic parameters, not di-, tri- and tetra-atomic parameters as do force field methods.
Once a given atom has been parameterized, all possible compound types involving this element can be calculated. The smaller number of parameters and the more complex functional form has the disadvantage compared with force field methods that it is very difficult to “repair” a specific problem by re-parameterization. The lack of a reason- able rotational barrier in amides, for example, cannot be attributed to an “improper”
value for a single (or a few) parameter(s). Too low a rotational barrier in a force field model can easily be fixed by increasing the values of the corresponding torsional parameters. The clear advantage of semi-empirical methods over force field techniques is the ability to describe bond breaking and bond forming reactions.
Semi-empirical methods are zero-dimensional, just as force field methods are. There is no way of assessing the reliability of a given result within the method. This is due to the selection of a minimum basis set. The only way of judging results is by calibration, i.e. by comparing the accuracy of other calculations on similar systems with experi- mental data.
Semi-empirical models provide a method for calculating the electronic wave func- tion, which may be used for predicting a variety of properties. There is nothing to hinder the calculation of say the polarizability of a molecule (the second derivative of the energy with respect to an external electric field), although it is known from ab initio calculations that good results require a large polarized basis set including diffuse func- tions, and the inclusion of electron correlation. Semi-empirical methods such as AM1 or PM3 only have a minimum basis (lacking polarization and diffuse functions), elec- tron correlation is only included implicitly by the parameters, and no polarizability
data have been used for deriving the parameters. Whether such calculations can produce reasonable results, as compared with experimental data, is questionable, and careful calibration is certainly required. Again it should be emphasized:The ability to perform a calculation is no guarantee that the results can be trusted!
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