PRISM algorithm

Recently, however, Johnson et al. [63] have found that, if one wishes to minimize the number of Mops (as opposed to Flops) in the transformation of (mOInO) to (ablcd), it is often preferable to dispense with the HRR entirely and, in lieu of it, employ RR s from a novel family which these authors term "nth-order transfer relations". We will say more about these later in the context of the PRISM algorithm. 

In addition to stimulating a number of variations on the HGP theme, the seminal paper by Head-Gordon and Pople [55] also served to catalyse the development of a completely different approach to the Contraction Problem called the PRISM algorithm. 

After developing the mathematical foundations of the PRISM algorithm, we will discuss its implementation within the Gaussian 92 computer program [84]. 

The MD-PRISM and HGP-PRISM algorithms are most easily understood when presented in terms of diagrams which resemble rectangular prisms - whence their names. To simplify the discussion, we will confine our attention to the front face of each prism. The generalizations to the complete prisms are neither difficult nor especially interesting they are outlined in [61c]. 

The front face of the MD PRISM is shown below. As the arrows indicate, the MD-PRISM algorithm consists of a set of highly interrelated pathways from shell-pair data to the desired brakets. 

The first step (To), the generation of [0](m) integrals from shell-pair data, is common to both the MD- and HGP-PRISM algorithms. We will discuss To in detail in Section 4.3. 

Thus, by using (59), (89), (92), (94) and (95) in judicious combinations, one can traverse any of the 10 paths in Figure 1 (and, indeed, with only trivial extensions, any of the 20 paths in Figure 3). By choosing always to go by the cheapest (in a Flops or Mops sense) path, one gains the full benefit of the PRISM algorithm. 

At first glance, there is not much more that can be said about this transformation. The RR (70) is extremely simple and is easy to use and it might appear that our analysis can probe no further. However, as Ryu, Lee and Lindh have shown [59b], if one wishes to apply (70) in a way that minimizes the number of Flops involved, a complicated tree-search problem must first be solved. These authors were unable to solve the general problem but gave heuristic solutions which clearly indicated that substantial savings were available. However, this is not the approach which is followed in the HGP-PRISM algorithm... 

Let us demonstrate work of the algorithm on a typical example, a prism of reaction that consists of two connected cycles (Figures 2 and 3). Such systems appear in many areas of biophysics and biochemistry (see, e.g. the paper of Kurzynski, 1998). 

During the 1960s further improvements made infrared spectroscopy a very useful tool used worldwide in the analytical routine laboratory as well as in many fields of science. Grating spectrometers replaced the prism instruments due to their larger optical conductance (which is explained in Sec. 3 of this book). The even larger optical conductance of interferometers could be employed after computers became available in the laboratory and algorithms which made Fourier transformation of interferograms into spectra a routine. The computers which became a necessary component of the spectrometers made new powerful methods of evaluation possible, such as spectral subtraction and library search. 

We also remark that the language needed to express quasi-inverses requires disjunction. As a result, PRISM uses an extension of the chase and backchase algorithm that is able to handle disjunctive dependencies this extension was developed as part of MARS [Deutsch and Tannen 2003], Finally, we note that we may not always succeed in finding equivalent reformulations, depending on the input query, the evolution mappings and also on the quasi-inverses that are chosen. Hence, PRISM must still rely on a human DBA to solve exceptions. 

The most recent integral algorithms evolved with, and were influenced by, the advent of supercomputer technologies - if a new method cannot be "vectorized" and/or "parallelized", it faces a cool reception these days - and, of these, the Obara-Saika-Schlegel (OS) [53, 54], Head-Gordon-Pople (HGP) [55] and PRISM [61] algorithms are the most significant. 

The last factor in (75), which scales it according to the angular momentum (a + b) of the shell-pair, is termed the principal scaling and is included only if the MD-PRISM is used. For reasons which will become clearer below, its presence reduces the Flop- and Mop-costs of the algorithm. 

As we indicated in Section 3.3, modem integral algorithms invariably employ recurrence relations to build complicated brakets from simple ones. Their use permits algorithms to deal (in principle) with brakets of arbitrarily high angular momentum and, additionally, to make good use of the intermediates that are shared by fraternal brakets. In the next four subsections, we will discuss the various recurrence relations which are used to move vertically on the prisms. 

Yang, Honig [204] PrISM Multiple structures are aligned, and the most appropriate template is used for each segment of the target to be built. Loops are built ah initio and side chains are built using the template or based on mainchain torsions and a neural network algorithm. 

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