Special Computational Techniques

Smart algorithms and efficient approximations that do not sacrifice accuracy are hey building blocks of tractable computational schemes. Although this has already been an issue in the previous chapters, a couple of important methods have not yet been discussed. It should be obvious that the options to combine the ingredients developed in chapters 8-13 lead to numerous methods. However, we shall now present the most important techniques not mentioned so far. Moreover, as effects of relativistic quantum chemistry are most prominent in the region close to the nucleus of a heavy atom, efficient approximations can be introduced for actual numerical calculations, which exploit the local nature of relativistic effects. 

In this chapter, we present further computationally efficient methods for relativistic calculations of the electronic structures of molecules and molecular aggregates. While the theory has been developed in detail in the preceding chapters, we now ask the question of how it can be transformed into computationally most feasible methods. 

Relativistic Quantum Chemistry. Markus Reiher and Alexander Wolf 

in this chapter, we proceed further on our way from the fundamental theory to different representations of first-quantized relativistic quantum chemistry — now guided mostly by questions of algorithmic technique and feasibility. For the sake of compactness, the focus in this chapter must be on techniques that are specific to the relativistic realm. In nonrelativistic theory numerous approaches have been devised to reduce the computational effort of quantum chemical calculations. Apart from the just mentioned density-fitting approach, specific linear-scaling techniques have been devised [715-717] that ensure a linear increase in the computational effort with system size (measured by the atom or electron number or directly by the number of basis functions). These employ, for example, localized orbitals or sparse-matrix operations. All these techniques apply directly to the relativistic variants. 

The preceding three chapters have already introduced Hamiltonians of reduced dimension. Particularly successful in variational calculations are the DKH and ZORA approaches. In their scalar-relativistic variant, they can easily be implemented in a computer program for nonrelativistic quantum chemistry so that spin remains a good quantum number leading to great computational advantages (if this approximation is justifiable). Already for these methods we have seen that numerous approximations can be made in order to increase their computational efficiency with little or even no loss of accuracy compared with a four-component reference calculation with the same type of total wave function. 

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These involve a reversible reaction followed by one or more other reactions. Analysis shows that if the equilibrium is not established very rapidly then the kinetics are complex and are not of simple order. Special computer techniques are required to analyse the results. 

Mathematical and Computational Implementation. Solution of the complex systems of partial differential equations governing both the evolution of pollutant concentrations and meteorological variables, eg, winds, requires specialized mathematical techniques. Comparing the two sets of equations governing pollutant dynamics (eq. 5) and meteorology (eqs. 12—14) shows that in both cases they can be put in the form ... 

In most cases the only appropriate approach to model multi-phase flows in micro reactors is to compute explicitly the time evolution of the gas/liquid or liquid/ liquid interface. For the motion of, e.g., a gas bubble in a surrounding liquid, this means that the position of the interface has to be determined as a function of time, including such effects as oscillations of the bubble. The corresponding transport phenomena are known as free surface flow and various numerical techniques for the computation of such flows have been developed in the past decades. Free surface flow simulations are computationally challenging and require special solution techniques which go beyond the standard CFD approaches discussed in Section 2.3. For this reason, the most common of these techniques will be briefly introduced in... 

The various methods used in quantum chemistry make it possible to compute equilibrium intermolecular distances, to describe intermolecular forces and chemical reactions too. The usual way to calculate these properties is based on the independent particle model this is the Hartree-Fock method. The expansion of one-electron wave-functions (molecular orbitals) in practice requires technical work on computers. It was believed for years and years that ab initio computations will become a routine task even for large molecules. In spite of the enormous increase and development in computer technique, however, this expectation has not been fulfilled. The treatment of large, extended molecular systems still needs special theoretical background. In other words, some approximations should be used in the methods which describe the properties of molecules of large size and/or interacting systems. The further approximations are to be chosen carefully this caution is especially important when going beyond the HF level. The inclusion of the electron correlation in the calculations in a convenient way is still one of the most significant tasks of quantum chemistry. 

Application of computaional methods to the enantioselective dihydroxylations of alkenes by osmium complexes have been reviewed with a special focus on methods used to study the origin of high enantioselectivity. The use of a vast number of computational techniques such as QM, MM, Q2MM, QM/MM, molecular dynamics, and genetic algorithms has been enumerated.98... 

In our short survey of the computational techniques available for investigating TM compounds we first mention molecular mechanics (Chapter 3). It may seem humble by the standards of the quantum mechanical ab initio, semiempirical and DFT methods (Chapters 5, 6 and 8, respectively) but MM is useful for obtaining input structures for submission to one of those calculations, may even provide in itself useful information, and it is, of course, extremely fast. Indeed, a recent book on the modelling of inorganic compounds, mainly TM species, is devoted very largely to molecular mechanics and a program specially parameterized for TM compounds, Momec3 [105],... 

In the volumes to come, special attention will be devoted to the following subjects the quantum theory of closed states, particularly the electronic structure of atoms, molecules, and crystals the quantum theory of scattering states, dealing also with the theory of chemical reactions the quantum theory of time-dependent phenomena, including the problem of electron transfer and radiation theory molecular dynamics statistical mechanics and general quantum statistics condensed matter theory in general quantum biochemistry and quantum pharmacology the theory of numerical analysis and computational techniques. 

To explore the relevance of symmetry allowed conformations to chemically real structures it is necessary to locate steric-energy minima in a potential energy surface that spans all possible conformations. A variety of computational techniques [210] are available, commonly combined with experimental results retrieved from structural databases [211]. Such procedures have revealed the occurrence of countless different rotamers and conformers, arising from pseudorotation and conformational inversion under special environmental conditions. In addition, situations of disorder in the crystalline state are symptomatic in many cases, of the stabilization of variable intermediate forms. 

Up to now our attention was mainly devoted to calculations of the energy and other quantities referring to free, isolated molecules. The computational techniques and their applications were demonstrated to be profitable in the exploration of physico-chemical properties of free molecules and their reactivity in the gas phase (thermodynamic functions, equilibrium and rate constants). However, the gas-phase processes represent only a special minor part of chemistry. Not only processes in biological systems, but also processes in laboratory conditions proceed typically in the liquid phase - or expressed more specifically - in the solution. It is therefore not surprising that the effort for applications of ab initio calculations is also still increasing in this very important field . ... 

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