Accuracy, Efficiency and Choice

In this chapter, we compare only those methods most likely to be used, out of all the methods shown in previous chapters, using abbreviations for them  

CN Crank-Nicolson without the implicit boundary algorithm  

These are considered the most accessible methods, although there are much more modern techniques that compare favourably (see Chapt. 5). 

We must define what is meant by accuracy and efficiency and we must decide what we want. A choice of technique cannot, of course, be made since this is a highly individual matter but we can provide here some basis for such a choice. 

Finally, for two-dimensional (UMDE) simulations, Gavaghan found [42] an for errors in concentration at edge points. 

Very briefly, this rather large subject in the general area of chemical kinetics [43-45] was carried into electrochemistry in the studies by Bieniasz et al. [46-48]. It asks the question, when fitting some parameter to a proposed mechanism by means of simulation using some simulation output (concentrations or current or some other result), how sensitive to the changes in the output is the value of the fitted parameter. This is expressed in the form of a sensitivity function s. If the simulation yields, for example, an array of concentrations c x,t,p), where x are positions in space, t the time (which may enter the problem) and the parameter(s) p, then the function is defined [46] as 5 = dc/dp, which is an expression of the sensitivity to changes in concentration. This can be useful in estimating the reliability of fitted parameters by a series of simulations. This subject will not be persued further here. 

We come now to the choice of method. There are no hard and fast rules here, the final choice depending to a large extent on personal preference and the inclination towards programming. Computers are now so fast that all but hard simulation problems such as CVs of, say, 2D problems or 3D problems execute in a very short time—usually just a few seconds. In such a case, the main bottleneck will 

The numerical solution produces concentration values, and one must therefore strive to obtain as accurate values for these as possible, so that currents calculated from them might also be accurate. Bieniasz now makes a practice of showing errors across the whole concentration profile, when reporting a new simulation method [53-55], or at least a few samples from the profile [56]. 

The two methods that stand out in terms of efficiency and convenience are BDF and BI with extrapolation. Both require minimal programming effort, and can be extended to higher-order spatial derivatives. However, in the case of BDF, a limit is encountered. For the most convenient start-up methods such as the simple or the rational start, the accuracy from BDF is limited to 0(8T ). This means for one thing that one need not go beyond three-point BDF (which is 0(8T ) in itself), but that no marked improvement can be gained from higher-order spatial derivative approximations, because there will then be a mismatch between the accuracy orders with respect to the time and spatial intervals. 

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The fundamental issue in implementing importance sampling in simulations is the proper choice of the biased distribution, or, equivalently, the weighting factor, q. A variety of ingenious techniques that lead to great improvement in the efficiency and accuracy of free energy calculations have been developed for this purpose. They will be mentioned frequently throughout this book. 

The choice between Equations 12-17 and 12-18 is strictly arbitrary either works with equal efficiency. And either equation requires a trial-and-error solution. Normally, successive trial values of either n g or ft L are selected until the summation equals 1.0. The value of h g or n L which causes the summation to equal 1.0 is the correct value of n g or n L. Then, the terms in the summation represent the composition of either the liquid or the gas, depending upon the equation used. The accuracy of the results depends strictly on the accuracy of the values of equilibrium ratios used. 

In recent years, higher orders of the DK transformation were formulated and explored in benchmark calculations on small molecules. Furthermore, it was shown that highly accurate transformed two-component Hamiltonians can be generated via the DK transformations of higher orders. These Hamiltonians converge quite well for the known elements of the periodic table limits of accuracy become noticeable only for elements with Z > 120. Higher orders of DK transformed Hamiltonians yield only small corrections for molecular observables thus, for most applications with normal demands of accuracy, DK2 is a reasonable, efficient, and well established choice. A valuable alternative is provided by the ZORA scheme, as comparison of available results shows. On the other hand, in the near future, accurate four-component approaches are expected to be essentially restricted to benchmark calculations due to their computational requirements. 

Because of its relative efficiency and accuracy, DFT remains popular for modeling systems up to a few hundred atoms (65). Correlated ab initio methods are also used and may offer higher accuracy than DFT methods (159). The choice of the QM method will of course also be influenced by the size of the QM system. As stated above, the QM subsystem should be selected with care, comprising all reactive groups. 

In the last decade methods based on CC theory have moved to the forefront of quantum chemistry. In particular when high accuracy is required, CC methods nowadays represent the preferred choice. Their application is facilitated by their general availibility in several popular program packages as well as established methods for efficient and routine determination of... 

The accuracy with which a system can measure lifetimes depends on a number of different factors including calibration of the instrument, the number of detected photons and also the efficiency of the analysis routines. In addition, sources of background and scattered light should be eliminated. Emission filters should be chosen with great care to make sure that no scattered laser light reaches the detector. Detection of scattered excitation light results in a spurious fast component in the decay and complicates the interpretation of the data. The choice of emission filters is much more critical in FLIM than in conventional fluorescence intensity imaging methods. 

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