Watson renormalization

In order to answer this question, a significant source of statistical correlation arising from mutation paths that visit a particularly advantageous mutant more than once must be considered. In the perturbation theory these paths are represented by products of factors involving the mutant replication rates, and it is necessary to remove the strong correlation that arises between these factors where repeated indices are present in order to obtain a tractable statistical analysis of convergence. The Watson renormalization procedure [29], the application of which to the steady-state quasi-species is summarized in Appendix 7, accomplishes just this [30]. The cost is a consecutive modification of the denominator, which may however be simplified to good approximation, as in Eqn. (A7.5). 

An approach that is very closely related to the Atomic Sphere Approximation is the Renormalized Atom Theory, introduced first by Watson, Ehrenreich, and Hodges (1970) (sec also Watson and Ehrenreich, 1970, Hodges et al., 1972, and particularly Gelatt, Ehrenreich, and Watson, 1977). The name derives from the way the potential is constructed a charge density for each atom is constructed on the basis of atomic wave functions that are truncated at the Wigner-Seitz, or atomic, sphere. The charge density from each state is then scaled up (renormalized) to make up for that density beyond the sphere which has been dropped. 

The analytic properties of A (z) can be studied conveniently by using the renormalized perturbation series (RPS) expansion (Anderson (1958) Watson (1957)) for this quantity,... 

The calculation of 4f promotion energy requires the combination of an atomic calculation for the 4f shell and a band calculation for the 5d band. The two calculations are based on such drastically different approximations that the combination of the two under one algorithm is a seemingly impossible task. Herbst et al. (1972) overcame this difficulty by using the renormalized atom method first proposed for the d band metals by Watson et al. (1970) and reported in detail by Hodges et al. (1972). We will review here the philosophy of the method, with particular emphasis on the meaning of the various approximations. The computational details are found in the original article. The relativistic version of the calculation has been published recently by Herbst et al. (1976). 

---