Quantum mechanics methods approximation

The preferable theoretical tools for the description of dynamical processes in systems of a few atoms are certainly quantum mechanical calculations. There is a large arsenal of powerful, well established methods for quantum mechanical computations of processes such as photoexcitation, photodissociation, inelastic scattering and reactive collisions for systems having, in the present state-of-the-art, up to three or four atoms, typically. " Both time-dependent and time-independent numerically exact algorithms are available for many of the processes, so in cases where potential surfaces of good accuracy are available, excellent quantitative agreement with experiment is generally obtained. In addition to the full quantum-mechanical methods, sophisticated semiclassical approximations have been developed that for many cases are essentially of near-quantitative accuracy and certainly at a level sufficient for the interpretation of most experiments.These methods also are com-... 

The importance of FMO theory hes in the fact that good results may be obtained even if the frontier molecular orbitals are calculated by rather simple, approximate quantum mechanical methods such as perturbation theory. Even simple additivity schemes have been developed for estimating the energies and the orbital coefficients of frontier molecular orbitals [6]. 

Both molecular and quantum mechanics methods rely on the Born-Oppenheimer approximation. In quantum mechanics, the Schrodinger equation (1) gives the wave functions and energies of a molecule. 

The quantum mechanics methods in HyperChem differ in how they approximate the Schrodinger equation and how they compute potential energy. The ab initio method expands molecular orbitals into a linear combination of atomic orbitals (LCAO) and does not introduce any further approximation. 

The semi-empirical methods of HyperChem are quantum mechanical methods that can describe the breaking and formation of chemical bonds, as well as provide information about the distribution of electrons in the system. HyperChem s molecular mechanics techniques, on the other hand, do not explicitly treat the electrons, but instead describe the energetics only as interactions among the nuclei. Since these approximations result in substantial computational savings, the molecular mechanics methods can be applied to much larger systems than the quantum mechanical methods. There are many molecular properties, however, which are not accurately described by these methods. For instance, molecular bonds are neither formed nor broken during HyperChem s molecular mechanics computations the set of fixed bonds is provided as input to the computation. 

For small molecules, the accuracy of solutions to the Schrodinger equation competes with the accuracy of experimental results. However, these accurate ab initio calculations require enormous computation and are only suitable for the molecular systems with small or medium size. Ab initio calculations for very large molecules are beyond the realm of current computers, so HyperChem also supports semi-empirical quantum mechanics methods. Semi-empirical approximate solutions are appropriate and allow extensive chemical exploration. The inaccuracy of the approximations made in semi-empirical methods is offset to a degree by recourse to experimental data in defining the parameters of the method. Indeed, semi-empirical methods can sometimes be more accurate than some poorer ab initio methods, which require much longer computation times. 

Pauncz, R., Acta Phys. Hung 4, 237, Investigation of a new quantum-mechanical method of approximation."... 

This paper is dedicated to Gaston Berthier, from whom I have learned a lot. Although Berthier s publications have mostly dealt with applications of quantum mechanical methods to chemical problems, he never liked black boxes or unjustified approximations even if they appeared to work. The question why the quantum chemical machinery does so well although it often lies on rather weak grounds has concerned him very much. I am therefore convinced that he will appreciate this excursion to applied mathematics. 

Semi-empirical quantum-mechanical methods combine fundamental theoretical treatments of electronic behavior with parameters obtained from experiment to obtain approximate wavefunctions for molecules composed of hundreds of atoms20-22. Originally developed in response to the need to evaluate the electronic properties of organic molecules, especially those possessing unusual structures and/or chemical reactivity in organic chemistry,... 

MOPAC is a general-purpose semiempirical molecular orbital program for the study of chemical structures and reactions. It is available in desktop PC running Windows, Macintosh OS, and Unix-based workstation versions. It uses semiempirical quantum mechanical methods that are based on Hartree-Fock (HF) theory with some parameterized functions and empirically determined parameters replacing some sections of the complete HF treatment. The approximations in... 

L-A. With polarizable charges obtained by an approximate quantum mechanical method including electron correlation or by a Class IV charge model. 

Aguilar, M. A., Olivares del Valle, F. J. and Tomasi, J. Nonequilibrium solvation an ab initio quantum-mechanical method in the continuum cavity model approximation, J.Chem.Phys., 98 (1993), 7375-7384... 

Quantum mechanical methods can now be applied to systems with up to 1000 atoms 87 this capacity is not only from advances in computer technology but also from improvements in algorithms. Recent developments in reactive classical force fields promise to allow the study of significantly larger systems.88 Many approximations can also be made to yield a variety of methods, each of which can address a range of questions based on the inherent accuracy of the method chosen. We now discuss a range of quantum mechanical-based methods that one can use to answer specific questions regarding shock-induced detonation conditions. 

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. 

This theory proves to be remarkably useful in rationalizing the whole set of general rules and mechanistic aspects described in the previous section as characteristic features of the Diels-Alder reaction. The application of perturbation molecular orbital theory as an approximate quantum mechanical method forms the theoretical basis of Fukui s FMO theory. Perturbation theory predicts a net stabilization for the intermolecular interaction between a diene and a dienophile as a consequence of the interaction of an occupied molecular orbital of one reaction partner with an unoccupied molecular orbital of the other reaction partner. 

The various types of successful approaches can be classified into two groups empirical model calculations based on molecular force fields and quantum mechanical approximations. In the first class of methods experimental data are used to evaluate the parameters which appear in the model. The shape of the potential surfaces in turn is described by expressions which were found to be appropriate by semiclassicala> or quantum mechanical methods. Most calculations of this type are based upon the electrostatic model. Another more general approach, the "consistent force field method, was recently applied to the forces in hydrogen-bonded crystals 48> 49>. 

During the last decade MO-theory became by far the most well developed quantum mechanical method for numerical calculations on molecules. Small molecules, mainly diatomics, or highly symmetric structures were treated most accurately. Now applicability and limitations of the independent particle, or Hartree-Fock (H. F.), approximation in calculations of molecular properties are well understood. An impressive number of molecular calculations including electron correlation is available today. Around the equilibrium geometries of molecules, electron-pair theories were found to be the most economical for actual calculations of correlation effects ). Unfortunately, accurate calculations as mentioned above are beyond the present computational possibilities for larger molecular structures. Therefore approximations have to be introduced in the investigation of problems of chemical interest. Consequently the reliability of calculated results has to be checked carefully for every kind of application. Three types of approximations are of interest in connection with this article. 

Fig. 7.10. Approximate charge distribution in methyl acetate and S-methyl thioacetate, as calculated by the INDO quantum-mechanical method [144]...

To summarize, the results presented for five representative examples of nonadiabatic dynamics demonstrate the ability of the MFT method to account for a qualitative description of the dynamics in case of processes involving two electronic states. The origin of the problems to describe the correct long-time relaxation dynamics as well as multi-state processes will be discussed in more detail in Section VI. Despite these problems, it is surprising how this simplest MQC method can describe complex nonadiabatic dynamics. Other related approximate methods such as the quantum-mechanical TDSCF approximation have been found to completely fail to account for the long-time behavior of the electronic dynamics (see Fig. 10). This is because the standard Hartree ansatz in the TDSCF approach neglects all correlations between the dynamical DoF, whereas the ensemble average performed in the MFT treatment accounts for the static correlation of the problem. 

Quantum Mechanics. Methods based on approximate solution of the Schrodinger Equation. 

Semi-Empirical Models. Quantum Mechanics methods that seek approximate solutions to the many electron Schrodinger Equation, but which involve empirical parameters. 

A third matter to mention here is that the WKB approximation outlined above is limited in the realm in which it is valid. It is more applicable to protons than to electrons (Bockris and Sen, 1973). Other quantum mechanical methods of a quite different nature can be used13 (D. Miller, 1995) and have been applied to make numerical quantal calculations of the rate of redox reactions (Khan, Wright, and Bockris, 1977 Newton, 1986), but they depend on a knowledge of wave functions which, for electron levels in hydrated ions in solution, may still be too primitive for calculations of rate. 

---