Examples of Calculations and Practical Issues

In the following sections some examples will be given of the calculation of the electromagnetic molecular properties introduced in Chapters 4 to 8 with some of the ab initio methods described in Chapters 10 to 12. The examples are neither meant to give an exhaustive overview of the performance of the different ab initio methods nor the molecular properties. But before doing so we have to discuss one important practical issue in all quantum chemical calculations, the one-electron basis set, and the more technical question of how the response functions or propagators are evaluated in actual calculations, i.e. the reduced linear equations algorithm. 

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The 1998 Nobel prize for chemistry was awarded to two scientists whose principal contribution was to devise methods that brought approximate quantum theory calculations for the medium-sized molecules within the realms of practicality. The suite of programs that the modern chemist has available for calculating molecular structures is extensive and sophisticated. But, in practice, a compromise always has to be made in terms of the computational effort versus the level of approximation, and some issues of approximation cannot be avoided within the framework of the suite. One such example is the assumption of stationary nuclei another is the problem of relativistic velocity effects, which become significant for the electrons of elements heavier than about iron (that is, the heavier two thirds of the elements). The time-independent Schrbdinger equation is based on Newtonian rather than relativistic mechanics. 

A final caveat that must be applied to phase diagrams determined using DFT calculations (or any other method) is that not all physically interesting phenomena occur at equilibrium. In situations where chemical reactions occur in an open system, as is the case in practical applications of catalysis, it is possible to have systems that are at steady state but are not at thermodynamic equilibrium. To perform any detailed analysis of this kind of situation, information must be collected on the rates of the microscopic processes that control the system. The Further Reading section gives a recent example of combining DFT calculations and kinetic Monte Carlo calculations to tackle this issue. 

In principle, the broad range of functional monomers eurrently available makes it possible to design an MIP specific for any type of stable ehemical compound. Currently the selection of the best monomers for polymer preparation is one of the most crucial issues in molecular imprinting. Thermodynamic calculations and combinatorial screening approaches olfer possible solutions, and have already been used successfully for predicting polymer properties and for the optimization of polymer compositions (see Ref. 9,57,58, and Chapter 8), however, in practical terms, application of these methods is not trivial. The problem lies in the technical difiiculty of performing detailed thermodynamic calculations on multicomponent systems and the amount of time and resources required for the combinatorial screening of polymers. To check a simple two-component combination of 100 monomers, for example, one has to synthesize and test more than 5000 polymers, a very difiicult task. This task will be further complicated by the possibility that these monomers could be used in monomer mixtures in dilferent ratios. 

Thermodynamic state models are normally written as explicit functions in either T,p,Ni or T,V,Ni, but in the lack of a universal phase description the model may require contributions from several (empirical) state functions (equations of state, activity models, Cp-functions, etc.). Given a combined model of this kind, the computation of thermodynamic phase and reaction equilibrium is, at least in principle, routine, but it remains a practical issue how to construct a consistent thermodynamic framework, and in particular so when dealing with general computer interfaces. One example is the calculation of H = U + pV. Everything works smoothly as long as the variables are taken from the same equation of state but not so if e.g. V is replaced by V taken from an external correlation. The user must then decide to use either // or t/ -1- E in the calculations. 

The interaction energies between molecules is a complex issue and is most frequently satisfied by approximations. For example, we have seen that (see section 6.6) the interaction between two molecules does not disturb their internal organizations and the mteractions forces only depend on the distance between the molecules (see section 6.10). Between two molecules, the corrective energetic term hnked to the interaction ab will appear, which is practically zero when both molecules A and B are far enough as in the example of the Letmard-Jones potential (see relations [6.100] or [6.103]). Calculating the partition function of a fluid, in particular the calculation of the second coefficient of the virial, is tricky many examples in the literature are approximate or full of poorly justified shortcuts. This is why we chose to... 

Finally, we note that Jkl is very sensitive to variations in the molecular geometry. It has been repeatedly observed that the best results are obtained when the same approximation (for instance DFT functional and basis set) is used for geometry optimization and for the following calculation of spin-spin coupling constants. We refer to the recent reviews discussing the calculation of spin-spin coupling constants (Aucar 2008 Helgaker et al. 2008 Krivdin and Contreras 2007) for a discussion of many practical issues and numerous examples of successful applications. 

Beyond its ability to account for what is known, the second important consideration in the selection of an appropriate molecular mechanics or quantum chemical model is its cost . It is really not possible to estimate precisely how much computer time a particular calculation will require, as many factors remain uncertain. In addition to the size of the system at hand and the choice of model (both of which can be precisely defined), there are issues the quality of the guess (which in turn relates to the experience of the user) and the inherent difBculty of the problem (some things are easier than others). It is possible, however, to provide representative examples to help distinguish applications which are practical from those which are clearly not. 

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