Semiempirical accuracy

A highly readable account of early efforts to apply the independent-particle approximation to problems of organic chemistry. Although more accurate computational methods have since been developed for treating all of the problems discussed in the text, its discussion of approximate Hartree-Fock (semiempirical) methods and their accuracy is still useful. Moreover, the view supplied about what was understood and what was not understood in physical organic chemistry three decades ago is... 

Direct dynamics attempts to break this bottleneck in the study of MD, retaining the accuracy of the full electronic PES without the need for an analytic fit of data. The first studies in this field used semiclassical methods with semiempirical [66,67] or simple Hartree-Fock [68] wave functions to heat the electrons. These first studies used what is called BO dynamics, evaluating the PES at each step from the elech onic wave function obtained by solution of the electronic structure problem. An alternative, the Ehrenfest dynamics method, is to propagate the electronic wave function at the same time as the nuclei. Although early direct dynamics studies using this method [69-71] restricted themselves to adiabatic problems, the method can incorporate non-adiabatic effects directly in the electionic wave function. 

Jones et al. [144,214] used direct dynamics with semiempirical electronic wave functions to study electron transfer in cyclic polyene radical cations. Semiempirical methods have the advantage that they are cheap, and so a number of trajectories can be run for up to 50 atoms. Accuracy is of course sacrificed in comparison to CASSCF techniques, but for many organic molecules semiempirical methods are known to perform adequately. 

Having the Slater atomic orbitals, the linear combination approximation to molecular orbitals, and the SCF method as applied to the Fock matrix, we are in a position to calculate properties of atoms and molecules ab initio, at the Hartree-Fock level of accuracy. Before doing that, however, we shall continue in the spirit of semiempirical calculations by postponing the ab initio method to Chapter 10 and invoking a rather sophisticated set of approximations and empirical substitutions... 

The PRDDO (partial retention of diatomic differential overlap) method is an attempt to get the optimal ratio of accuracy to CPU time. It has been parameterized for the periodic elements through Br, including the 3rd row transition metals. It was parameterized to reproduce ah initio results. PRDDO has been used primarily for inorganic compounds, organometallics, solid-state calculations, and polymer modeling. This method has seen less use than other methods of similar accuracy mostly due to the fact that it has not been incorporated into the most widely used semiempirical software. 

In order to obtain the best accuracy results as quickly as possible, it is often advantageous to do two geometry optimizations. The first geometry optimization should be done with a faster level of theory, such as molecular mechanics or a semiempirical method. Once a geometry close to the correct geometry has been obtained with this lower level of theory, it is used as the starting geometry for a second optimization at the final, more accurate level of theory. 

A basis set is a set of functions used to describe the shape of the orbitals in an atom. Molecular orbitals and entire wave functions are created by taking linear combinations of basis functions and angular functions. Most semiempirical methods use a predehned basis set. When ah initio or density functional theory calculations are done, a basis set must be specihed. Although it is possible to create a basis set from scratch, most calculations are done using existing basis sets. The type of calculation performed and basis set chosen are the two biggest factors in determining the accuracy of results. This chapter discusses these standard basis sets and how to choose an appropriate one. 

Another related issue is the computation of the intensities of the peaks in the spectrum. Peak intensities depend on the probability that a particular wavelength photon will be absorbed or Raman-scattered. These probabilities can be computed from the wave function by computing the transition dipole moments. This gives relative peak intensities since the calculation does not include the density of the substance. Some types of transitions turn out to have a zero probability due to the molecules symmetry or the spin of the electrons. This is where spectroscopic selection rules come from. Ah initio methods are the preferred way of computing intensities. Although intensities can be computed using semiempirical methods, they tend to give rather poor accuracy results for many chemical systems. 

It is possible to use computational techniques to gain insight into the vibrational motion of molecules. There are a number of computational methods available that have varying degrees of accuracy. These methods can be powerful tools if the user is aware of their strengths and weaknesses. The user is advised to use ah initio or DFT calculations with an appropriate scale factor if at all possible. Anharmonic corrections should be considered only if very-high-accuracy results are necessary. Semiempirical and molecular mechanics methods should be tried cautiously when the molecular system prevents using the other methods mentioned. 

The conductor-like screening model (COSMO) is a continuum method designed to be fast and robust. This method uses a simpler, more approximate equation for the electrostatic interaction between the solvent and solute. Line the SMx methods, it is based on a solvent accessible surface. Because of this, COSMO calculations require less CPU time than PCM calculations and are less likely to fail to converge. COSMO can be used with a variety of semiempirical, ah initio, and DFT methods. There is also some loss of accuracy as a result of this approximation. 

Semiempirical, DFT, and ah initio methods also work well. Correlation effects are sometimes included for the sake of increased accuracy, but are not always necessary. One particular case for which correlation is often necessary is fluorine compounds. 

This study is particularly noteworthy in the evolution of QM-MM studies of enzyme reactions in that a number of technical features have enhanced the accuracy of the technique. First, the authors explicitly optimized the semiempirical parameters for this specific reaction based on extensive studies of model reactions. This approach had also been used with considerable success in QM-MM simultation of the proton transfer between methanol and imidazole in solution. 

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. 

In most cases the chemist only needs differences of values and/or relative estimations in comparison with a standard. The inaccuracies introduced by semiempirical methods with its relatively drastic approximations can be limited by applying the differences causing the calculated values to possess suitable accuracy. 

The usefulness of quantum-chemical methods varies considerably depending on what sort of force field parameter is to be calculated (for a detailed discussion, see [46]). There are relatively few molecular properties which quantum chemistry can provide in such a way that they can be used directly and profitably in the construction of a force field. Quantum chemistry does very well for molecular bond lengths and bond angles. Even semiempirical methods can do a good job for standard organic molecules. However, in many cases, these are known with sufficient accuracy a C-C single bond is 1.53 A except under exotic circumstances. Similarly, vibrational force constants can often be transferred from similar molecules and need not be recalculated. 

Equation (4-5) can be directly utilized in statistical mechanical Monte Carlo and molecular dynamics simulations by choosing an appropriate QM model, balancing computational efficiency and accuracy, and MM force fields for biomacromolecules and the solvent water. Our group has extensively explored various QM/MM methods using different quantum models, ranging from semiempirical methods to ab initio molecular orbital and valence bond theories to density functional theory, applied to a wide range of applications in chemistry and biology. Some of these studies have been discussed before and they are not emphasized in this article. We focus on developments that have not been often discussed. 

Section 2 mainly focuses on the current efforts to improve the accuracy of quantum calculations using simplified empirical model forms. McNamara and Hillier, in Chapter 5, summary their work on improving the description of the interactions in biological systems via their optimized semiempirical molecular models. Piquemal and co-workers present recent advances in the classical molecular methods, aiming at better reproduction of high-level quantum descriptions of the electtostatic interactions in Chapter 6. In Chatper 7, Cui and Elstner describe a different semiempir-... 

A useful compromise between speed and accuracy is provided by semiempirical methods. In this context, semiempirical methods can be used as fitting tools rather than predictive methods [76], Optimization of the semiempirical parameters to reproduce experimental or high-level ah initio results, for a specific reaction, can yield an accurate potential for the problem of interest at relatively low cost [77]. This approach has been shown to be successful in a QM/MM framework also [78] and is a powerful tool in the accurate study of enzyme reactions [57, 79] (see also Section 6.5.2). 

One of the main aims of such computations is the prediction and rationalization of the optoelectronic spectra in various steric and electronic environments by either semiempirical or ab initio methods or a combination of these, considering equilibrium structures, rotation barriers, vibrational frequencies, and polarizabilities. The accuracy of the results from these calculations can be evaluated by comparison of the predicted ionization potentials (which are related to the orbital energies by Koopman s theorem) with experimental values. 

The level of accuracy that can be achieved by these different methods may be viewed as somewhat remarkable, given the approximations that are involved. For relatively small organic molecules, for instance, the calculated AGsoivation is now usually within less than 1 kcal/mole of the experimental value, often considerably less. Appropriate parametrization is of key importance. Applications to biological systems pose greater problems, due to the size and complexity of the molecules,66 156 159 161 and require the use of semiempirical rather than ab initio quantum-mechanical methods. In terms of computational expense, continuum models have the advantage over discrete molecular ones, but the latter are better able to describe solvent structure and handle first-solvation-shell effects. 

If quantum theory is to be used as a chemical tool, on the same kind of basis as, say, n.m.r. or mass spectrometry, one must be able to carry out calculations of high accuracy for quite complex molecules without excessive cost in computation time. Until recently such a goal would have seemed quite unattainable and numerous calculations of dubious value have been published on the basis that nothing better was possible. Our work has shown that this view is too pessimistic semiempirical SCF MO treatments, if properly applied, can already give results of sufficient accuracy to be of chemical value and the possibilities of further improvement seem unlimited. There can therefore be little doubt that we are on the threshold of an era where quantum chemistry will serve as a standard tool in studying the reactions and other properties of molecules, thus bringing nearer the fruition of Dirac s classic statement, that with the development of quantum theory chemistry has become an exercise in applied mathematics. 

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