Function-Based Methods

To date the majority of QM-MM applications have employed density functional methods ab initio or semiempirical methods in the quantum region. The energy tenns evaluated in these methods are generally similar, but there are specific differences. The relevant equations for the density functional based methods are described first, and this is followed by a description of the specific differences associated with the other methods. 

It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

Before we start looking at possible approximations to Exc we need to address whether there will be some kind of guidance along the way. If we consider conventional, wave function based methods for solving the electronic Schrodinger equation, the quality of the... 

In regular wave function based methods J is determined through the four-center-two-... 

During the last years, more and more researchers have applied density functional theory to small transition-metal complexes and benchmarked the results against either high level wave function based methods or experimental data. A particular set of systems for which reasonably accurate benchmark data are available are the cationic M+-X complexes, where X is H, CH3 or CH2. Let us start our discussion with the cationic hydrides of the 3d transition-metals. 

A rigorous mathematical formalism of chemical bonding is possible only through the quantum mechanical treatment of molecules. However, obtaining analytical solutions for the Schrodinger wave equation is not possible even for the simplest systems with more than one electron and as a result attempts have been made to obtain approximate solutions a series of approximations have been introduced. As a first step, the Bom-Oppenheimer approximation has been invoked, which allows us to treat the electronic and nuclear motions separately. In solving the electronic part, mainly two formalisms, VB and molecular orbital (MO), have been in use and they are described below. Both are wave function-based methods. The wave function T is the fundamental descriptor in quantum mechanics but it is not physically measurable. The squared value of the wave function T 2dT represents probability of finding an electron in the volume element dr. 

A second fundamental classification of quantum chemistry calculations can be made according to the quantity that is being calculated. Our introduction to DFT in the previous sections has emphasized that in DFT the aim is to compute the electron density, not the electron wave function. There are many methods, however, where the object of the calculation is to compute the full electron wave function. These wave-function-based methods hold a crucial advantage over DFT calculations in that there is a well-defined hierarchy of methods that, given infinite computer time, can converge to the exact solution of the Schrodinger equation. We cannot do justice to the breadth of this field in just a few paragraphs, but several excellent introductory texts are available... 

Before giving a brief discussion of wave-function-based methods, we must first describe the common ways in which the wave function is described. We mentioned earlier that the wave function of an /V-particle system is an tV-dimension al function. But what, exactly, is a wave function Because we want our wave functions to provide a quantum mechanical description of a system of N electrons, these wave functions must satisfy several mathematical properties exhibited by real electrons. For example, the Pauli exclusion principle prohibits two electrons with the same spin from existing at the same physical location simultaneously. We would, of course, like these properties to also exist in any approximate form of the wave function that we construct. 

In order to use wave-function-based methods to converge to the true solution of the Schrodinger equation, it is necessary to simultaneously use a high level of theory and a large basis set. Unfortunately, this approach is only feasible for calculations involving relatively small numbers of atoms because the computational expense associated with these calculations increases rapidly with the level of theory and the number of basis functions. For a basis set with N functions, for example, the computational expense of a conventional HF calculation typically requires N4 operations, while a conventional coupled-cluster calculation requires N7 operations. Advances have been made that improve the scaling of both FIF and post-HF calculations. Even with these improvements, however you can appreciate the problem with... 

This situation changed drastically when it was discovered in the 1990s that density functional (DF) methods do a much better job of modeling force fields than (affordable) wave function based methods. Already within the local density approximation (LDA) of DF theory, vibrational frequencies were predicted with... 

In practice, extrapolations of p fR in water have usually used the older acidity function based method, for example, for trityl,61,62 benzhydryl,63 or cyclopropenyl (6) cations.66,67 These older data include studies of protonation of aromatic molecules, such as pKSi = —1.70 for the azulenium ion 3,59 and Kresge s extensive measurements of the protonation of hydroxy- and methoxy-substituted benzenes.68 Some of these data have been replotted as p fR or pKa against XQ with only minor changes in values.25,52 However, for more unstable carbocations such as 2,4,6-trimethylbenzyl, there is a long extrapolation from concentrated acid solutions to water and the discrepancy is greater use of an acidity function in this case gives pA 2° = —17.5,61 compared with —16.3 (and m = 1.8) based on X0. Indeed because of limitations to the acidity of concentrated solutions of perchloric or sulfuric acid pICs of more weakly nucleophilic carbocations are not accessible from equilibrium measurements in these media. 

In contrast to molecular mechanics force fields, modern semiempirical methods are classified as an SCF electron-structure theory (wave function-based) method [8]. Older (pre-HF)... 

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