Reliability of Semiempirical Methods

The most reliable quantum mechanical method is no doubt ab initio. There is a systematic way of correcting for the deficiencies of any level applied by progressing to a higher level of theory, and if this systematic progression is not successful, the failure becomes of theoretical interest. With a large enough basis set and sufficient correlation, ab initio methods should approach the solution of the time-independent Schrodinger equation. 

In using any method often enough, one develops a strong feel for the method s strengths and weaknesses. One develops, perhaps, a model chemistry, that with experience can even be accurately corrected for not only shortcomings, but also failures. 

In this section we will examine some of the strengths of a few of the more popular methods we have described above. We have very little doubt that we will step on toes. The reason for this is that most methods, with enough experience, can yield satisfactory results. We, however, will look at those methods that yield satisfactory results without enough experience. 

In addition, we will concentrate on results from programs that are generally available and therefore have reproducible results. There does, however, exist a potential problem with the reproducibility of results quoted here. With the passage of time, new parameters do arise in semiempirical methods updates. These updates are not always documented, and modified versions of computer code do circulate containing parameters that have not been carefully examined. Sometimes these new parameters repre.sent improvements, and sometimes they do not. 

The accuracy of MINDO/3 in predicting geometry and heats of formation was far higher than that of contemporary semiempirical methods, and this together with its speed and general availability immediately made MINDO/3 popular. Soon, however, unacceptable limitations became apparent. 

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Reynolds CA, Ferenczy GG, Richards WG (1992) Methods for determining the reliability of semiempirical electrostatic potentials and potential derived charges. Theochem J Mol Struct 88,... 

A major weakness of semiempirical methods is that they must be assumed to be unreliable outside molecules of the kind used for their training set (the set of molecules used to parameterize them), until shown otherwise by comparison of their predictions with experiment or with high-level ab initio (or probably DFT) calculations. Although, as Dewar and Storch pointed out [125], the reliability of ab initio calculations, too, should be checked against experiment, the situation is somewhat different for these latter, at least at the higher levels studies of exotic species, in particular, are certainly more trustworthy when done ab initio than semiempirically. Semiempirical heats of formation are subject to errors of tens of kJ mol-1, and thus heats (enthalpies) of reaction and activation could be in error by scores of kJ mol-1. AMI and PM3 underestimate steric repulsions, overestimate basicity and underestimate nucleophilicity, and can give unreasonable charges and... 

At the same time, a conclusive and sufficiently reliable answer is frequently required. We may be interested in, for example, the question of the possibility of dissociative adsorption, or the problem of the existence of some chemisorption structures as in the discussion (see below) on the coordinative binding of water molecules by silicon atoms, etc. Ab initio calculations are required in these cases. They are needed as well to check some principal conclusions based on semiempirical schemes. Also, they are useful in providing the basis for proper choice and improving the parametrization of semiempirical methods. Therefore the nonempirical approach is finding ever-increasing application to the surface problems. 

Another limit of semiempirical methods is due to the fact that they do not provide a series of results of increasing accuracy, as ab initio methods do, from which the researcher often derives information about the reliability... 

A straight forward application of approximation IV to calculate W (r) maps is quite exacting, because the calculation of the potential contribution due to the couple distributions xt li 1S time consuming when directly performed on the Slater functions. This fact clashes with the basic philosophy of semiempirical methods, which is to sacrifice some reliability to speed up the calculations. It has been shown40) that expansion of each Slater-type orbital into three Gaussian functions (3G expansion41)) gives a substantial improvement of the computational times of W (r), without an appreciable reduction in the quality of the results. 

When selecting the strategy for a computational study, we advocate the use of accurate ab initio calculations whenever this is feasible, because reliable theoretical predictions can be obtained at high theoretical levels (ideally through a convergence toward the solution of the time-independent Schrodinger equation). We recommend the use of semiempirical methods in cases where a complete ab initio or DFT study is impractical. It is essential in such cases to establish the expected accuracy of the semiempirical results, either by comparisons with the available literature data or by separate calibrations. In this context it is usually helpful to perform some checks against ab initio or DFT calculations (if possible). The next section includes examples of such studies. 

In Ref. [56] the NLO properties of Qo covalently bound to benzothiadiazole, triphenylamine-benzothiazole and carbazole derivatives of different electron-donating power were investigated (Pig. 5.3). Geometries were optimized at the semiempirical PM3 level. To access the reliability of this method, the geometry of molecule 1 was additionally optimized at the B3LYP76-31G level, and found to be in good agreement with the PM3 optimized structure. 

Of the semiempirical methods, only MINDO," MNDO, and AM-1 are claimed to reliably estimate energies, and the range of reliability is open to some discussion. With the ab initio method, the 4-3 IG and 6-3 IG basis sets achieve a level of accuracy that permits comparison of energy data. Users of MO data, however, must critically assess the reliability of the method being applied in the particular case under study. 

As mentioned in the introduction a number of QM/MM methods have been implemented using different QM and MM approximations and different QM/MM interaction schemes. In principle, there is no restriction on the types of potential that can be coupled. In practice, certain combinations have proved more popular than others, in large part due to the computational expense of using ab initio QM methods. Thus, it is true that while combined potentials with ab initio HF or DFT methods would be preferable due to their greater reliability than semiempirical methods, the semiempirical methods have proved more popular because they can be used to study relatively large systems, with up to 100 QM atoms, using molecular dynamics or Monte Carlo simulation techniques. 

Not so many works have made use of MO methods to study the CD-guest minimum energy conformations [26-40]. It is not the purpose of this work a full analysis of the results for the different complexes but to present the capabilities of the MO methods in this field and analyze the limitations and future trends. Whereas the first works were almost restricted to the use of semiempirical methods, recently the more reliable ab initio or DFT methods have begun to be applied to such large molecular systems. A recent comparative study has shown that the more popular semiempirical methods AMI and PM3 may give unphysical very short H - H distances between the host and guest molecules. 

The relative merits of various MO methods have been discussed in die literature. In general, the ab initio type of calculations will be more reliable, but the semiempirical calculations are faster in terms of computer time. The complexity of calculation also increases rapidly as the number of atoms in the molecule increases. The choice of a method is normally made on the basis of evidence that the method is adequate for the problem at hand and the availability of appropriate computer programs and equipment. Results should be subjected to critical evaluation by comparison widi experimental data or checked by representative calculations using higher-level mediods. Table 1.12 lists some reported deviations from experimental AHf for some small hydrocarbons. The extent of deviation gives an indication of the accuracy of the various types of MO calculations in this application. 

Molecular orbital calculations (ah initio or semiempirical methods) are also often used to provide a description of radical species and their reactions. High levels of theory are required to provide reliable data. However, rapid advances in computer power and computational methods are seeing these methods more widely used and with greater success (for leading references on the application of theory to describe radical addition reactions, see Section 1.2.7). 

Molecular orbital calculations, whether by ab initio or semiempirical methods, can be used to obtain structures (bond distances and angles), energies (such as heats of formation), dipole moments, ionization energies, and other properties of molecules, ions, and radicals—not only of stable ones, but also of those so unstable that these properties cannot be obtained from experimental measurements." Many of these calculations have been performed on transition states (p. 279) this is the only way to get this information, since transition states are not, in general, directly observable. Of course, it is not possible to check data obtained for unstable molecules and transition states against any experimental values, so that the reliability of the various MO methods for these cases is always a question. However, our confidence in them does increase when (1) different MO methods give similar results, and (2) a particular MO method works well for cases that can be checked against experimental methods. ... 

A critical comparison between experiment and theory is hindered by the range of experimental values reported in the literature for each molecule. This reflects the difficulty in the measurement of absolute ionization cross sections and justifies attempts to develop reliable semiempirical methods, such as the polarizability equation, for estimating the molecular ionization cross sections which have not been measured or for which only single values have been reported. The polarizability model predicts a linear relationship between the ionization cross section and the square root of the ratio of the volume polarizability to the ionization potential. Plots of this function against experimental values for ionization cross sections for atoms are shown in Figure 7 and for molecules in Figure 8. The equations determined... 

Unless there is some obvious flaw such as those mentioned above, semiempirical methods still give a more reliable answer than qualitative MO arguments which we are about to discuss, and it certainly makes sense to use them if a program and computer time are available, provided that limitations such as the above are recognized. However, what is really needed most is a cheap semiempirical method free of the above disadvantages. 

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