Introducing Semi-Empirical Approximations

The second level of approximation in molecular orbital computations regards the various ways the Fock matrix elements of Eq. (4.281) are considered, namely the approximations of the integrals (4.282) and of the effective one-electron Hamiltonian matrix elements 

By neglecting the differential overlap (NDO) through the mono-atomic orbitalic constraint  

By neglecting the diatomic differential overlap (NDDO) of the bi-atomic orbitals  

For both groups of approximations specific methods are outlined below. 

The basic NDO approximation was developed by People and is known as the Complete Neglect of Differential Overlap CNDO semi-empirical 

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An approximate treatment for taking into account a—it interaction has been developed in the case of long polyenes 126>, and non-empirical calculations have been carried out for the various transitions of formaldehyde along the same lines as for ethylene 127,128,129)n) jn view of the intricacies of theoretical considerations concerning excited states, it is rather fortunate that calculations limited to the 71-electron systems can be forced to agree with experiment by introducing semi-empirical corrections on well-chosen matrix elements. 

To make matters worse, the use of a uniform gas model for electron density does not enable one to carry out good calculations. Instead a density gradient must be introduced into the uniform electron gas distribution. The way in which this has been implemented has typically been in a semi-empirical manner by working backwards from the known results on a particular atom, usually the helium atom (Gill, 1998). It has thus been possible to obtain an approximate set of functions which often serve to give successful approximations in other atoms and molecules. As far as I know, there is no known way of yet calculating, in an ab initio manner, the required density gradient which must be introduced into the calculations. 

The applicability of the Born-Oppenheimer approximation for complex molecular systems is basic to all classical simulation methods. It enables the formulation of an effective potential field for nuclei on the basis of the SchrdJdinger equation. In practice this is not simple, since the number of electrons is usually large and the extent of configuration space is too vast to allow accurate initio determination of the effective fields. One has to resort to simplifications and semi-empirical or empirical adjustments of potential fields, thus introducing interdependence of parameters that tend to obscure the pure significance of each term. This applies in... 

The term "semi-empirical" has been reserved commonly for electronic-based calculations which also starts with the Schrodinger equation.9-31 Due to the mathematical complexity, which involve the calculation of many integrals, certain families of integrals have been eliminated or approximated. Unlike ab initio methods, the semi-empirical approach adds terms and parameters to fit experimental data (e.g., heats of formation). The level of approximations define the different semi-empirical methods. The original semi-empirical methods can be traced back to the CNDO,12 13 NDDO, and INDO.15 The success of the MINDO,16 MINDO/3,17-21 and MNDO22-27 level of theory ultimately led to the development of AMI28 and a reparameterized variant known as PM3.29 30 In 1993, Dewar et al. introduced SAMI.31 Semi-empirical calculations have provided a wealth of information for practical applications. 

The Hartree-Fock approximation also provided the basis for what are now commonly referred to as semi-empirical models. These introduce additional approximations as well as empirical parameters to greatly simplify the calculations, with minimal adverse effect on the results. While this goal has yet to be fully realized, several useful schemes have resulted, including the popular AMI and PM3 models. Semi-empirical models have proven to be successful for the calculation of equilibrium geometries, including the geometries of transition-metal compounds. They are, however, not satisfactory for thermochemical calculations or for conformational assignments. Discussion is provided in Section n. 

As presented, the Roothaan SCF process is carried out in a fully ab initio manner in that all one- and two-electron integrals are computed in terms of the specified basis set no experimental data or other input is employed. As described in Appendix F, it is possible to introduce approximations to the coulomb and exchange integrals entering into the Fock matrix elements that permit many of the requisite F, v elements to be evaluated in terms of experimental data or in terms of a small set of fundamental orbital-level coulomb interaction integrals that can be computed in an ab initio manner. This approach forms the basis of so-called semi-empirical methods. Appendix F provides the reader with a brief introduction to such approaches to the electronic structure problem and deals in some detail with the well known Htickel and CNDO- level approximations. 

Before any computational study on molecular properties can be carried out, a molecular model needs to be established. It can be based on an appropriate crystal structure or derived using any other technique that can produce a valid model for a given compound, whether or not it has been prepared. Molecular mechanics is one such technique and, primarily for reasons of computational simplicity and efficiency, it is the most widely used. Quantum mechanical modeling of metal complexes with ab-initio or semi-empirical methods often remains prohibitive because these methods are so computationally intensive. The approximations that are introduced in order to reduce central processing unit (CPU) time and allow quantum mechanical calculations to be used routinely are often severe and such calculations are then less reliable. 

The FO approximation was introduced in 1952 when all calculations were done using the Hiickel method. In these conditions, the n frontier orbitals were also the chemically reactive MOs. Semi-empirical and ah initio calculations, which are frequently used now, take into account both o and n orbitals. With these methods, the orbitals to be considered in FO theory are not necessarily the formal frontier orbitals, i.e. the highest occupied and lowest unoccupied MOs Consider, for example, the reduction of acetone by LiAlH4. The chemically important MO is of course the Jt co of the ate complex, which is in fact the LUMO + 1. The formal LUMO is the empty s orbital of the lithium cation. 

In semi-empirical methods, the computational complexity of the ab-initio methods is reduced by making approximations in the computational procedures or by introducing constants derived from experimental data (such as ionization potentials). These methods are more generally applied to the calculation of relative conformational energies rather than to the calculation of quantitative molecular geometries. 

Cost of accuracy. At present with the appearance of the ZDO approximation that considerably simplifies calculations, theoreticians often prefer a semi-empirical approach in which most of the troublesome integrals are either neglected completely or expressed via the parameters found experimentally (orbital ionization potentials, electron affinity etc.). In this case the successful calibration of the empirical parameters can offset the loss in accuracy caused by various simplifications introduced into the Roothaan calculation method. 

Semi-empirical LCAO calculations for all azoles, introducing cr-electrons, indicate that charges are weak except those on NH nitrogen atoms, and that the cr-dipolar moments are close to those of lone pairs. It is therefore inappropriate to take cr-polarity into account in the approximations used in 7r-calculations for these heterocycles. A number of other quantum mechanical calculations have been applied to reactions of imidazoles (80AHC 27)241), while the nucleophilic substitution reactions at C-2 of benzimidazoles, and diazo coupling at C-2 of uncondensed imidazoles have been discussed from theoretical points of view. 

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