Selection rules 50 algebra

The essential features of these equations can be represented in terms of gr h-theoretical/topological selection rules for peri-cydic reactions (see Table 3.2). By using these rules the results are equivalent to that of the well-known Dewar-Zimmermann and Woodward-Hoffmann rules (11,42,45). Note however that their derivation does not require any physical concept, such as orbital considerations or the geometrical structure of the molecular system, but is based only on a logical analysis of the reaction structure and the algebraic formalism of the (single-parameter) X-model. 

Not only do the experimental vibrational predissociation lifetimes require interpretation, so do the increasingly sophisticated theoretical calculations whose results often fall out of a web of coupled differential equations or the convoluted algebra of quantum mechanics. In order to offer a qualitative overview of dynamical processes in van der Waals molecules, we shall introduce a selection rule which can provide insight into possible relaxation channels of vibrationally excited molecules. This selection rule concerns the change in a quantum number, Anj., which is to remain small for efficient vibrational predissociation processes. It bears a close analogy to the selection rules of optical spectroscopy which require small changes in quantum numbers Au, AJ, AS, etc. for efficient transitions between molecular states. Let us review the origin of the vibrational predissociation selection rule which has been developed in more detail elsewhere. ... 

To confound in 4 blocks of 4, we need to select 3 terms to correspond to the 3 degrees of freedom that we confound with block differences. It is found that while we can select any two arbitrarily, the third is then uniquely determined. The third is determined by the rule that we multiply the first two together by the ordinary laws of algebra, but wherever a squared term appears we place it equal to unity. Thus if the two selected are PQ and RS, then the third is PQRS if the two selected are PQR and QRS, then the third is PS. 

So far we have stuck to least squares as the rule for selecting our fitted coefficients, even though we were aware of this rule s deficiencies. Most of the questions addressed so far (What series is physically meaningful What series is implied by the fitting process When to stop ) are all general and could as well be discussed in a least squares context as not indeed, the algebra is simplest there. It is time now to consider anecdotes about what other rules can do for us. 

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