A Short Overview of Modeling Methods

The drawback of the classical mechanics approach is that it gives no information on the electronic structure of the system, and thus no information on chemical reactions. To simulate reactions, we must turn to quantum mechanics. Using QM we can obtain a full picture of the electronic and nuclear system, by solving the deceptively simple-looking Schrodinger equation (Eq. (6.2)), proposed by the Austrian physicist and 1933 Nobel laureate Erwin Schrodinger in 1926. 

In this equation, H is the Hamiltonian operator, P is the wavefunction that represents the state of the electrons and nuclei, and E is the system s energy. Although in theory it can be solved for any system, exact solutions are impracticable for anything more complicated than a hydrogen atom. Therefore we must find approximate 

Two commonly used approximations are the Hartree-Fock approach and density-functional theory (DFT). The Hartree-Fock approach approximates the exact solution of the Schrodinger equation using a series of equations that describe the wavefunc-tions of each individual electron. If these equations are solved explicitly during the calculation, the method is known as ab initio Hartree-Fock. The less expensive (i.e., less time-consuming) semi-empirical methods use preselected parameters for some of the integrals. DFT, on the other hand, uses the electronic density as the basic quantity, instead of a many-body electronic wavefunction. The advantage of this is that the density is a function of only three variables (instead of 3N variables), and is simpler to deal with both in concept and in practice. 

An interesting alternative that combines the advantages of both classical and quantum mechanics is to use hybrid QM/MM models, first introduced by Arieh Warshel for modeling enzymatic reactions [7]. Here, the chemical species at the active site are treated using high-level (and therefore expensive) QM models, which are coupled to a force field that describes the reaction environment. Hybrid models can thus take into account solvent effects in homogeneous catalysis, support structure and interface effects in heterogeneous catalysis, and enzyme structure effects in biocatalysis. 

For example, Samuel French and co-workers used a combined QM/MM approach for modeling the catalyst/substrate interactions in the methanol synthesis process [8]. The annual worldwide production of methanol exceeds 32 M tons, most of which is 

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