Recursion formula, definition

A power series solution of equation (G.27) yields a recursion formula relating i+4, Ui+2, and ak, which is too complicated to be practical. Accordingly, we make the further definition... 

In this appendix we shall provide exphdt expressions for the recursion formulas of Eq. (3.13) for the case of a nitroxide spin probe and a copper complex in solution. To this end we need these preliminary definitions ... 

This recursion formula is identical to the definition of Fibonacci numbers ... 

The associated Legendre functions are most easily tabulated by the use of the recursion formula 19-2 and the definition 19-7, together with the value P< (z) = 1 as the starting point. A detailed discussion of the functions is given in Section 21. 

Thus Eq. (8.454) is used as a recursion formula to generate the W s of higher index m. Eventually an interface between two regions is reached, and it is necessary to invoke the conditions of continuity of flux and current in order to cross the boundary and proceed with the recursion formula (8.454) as applied to the second region. Let the interface be at radial position R = Ar then, using the definitions above, we have... 

DEFINITION The operation of primitive recursion defines from functions h and g the function f by the formula ... 

An advantage of the Combination Rules is, that one is not restricted to only one definition for the calculation of the hardness kernels. Other, more refined approximations can be used (e.g. the Mataga-Nishimoto [34] formula or the formulas by Pariser and Ohno) which may enhance the accuracy. This is not the case in EEM, at least when we want to retain its internal consistency. For larger systems, EEM has the advantage that it enables the direct calculation of global and local properties while the Combination Rules have to be applied recursively. Also EEM is directly applicable to the solid state. 

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