Parameter dependence of renormalized perturbation theory

Here we analyze to which extent the choice of the parameters no,qp,go in Eq. (13.21) fixing the uncritical manifold 

1 Choice of The parameter determines the momentum region where the observables cross over from being governed by scales r Rg or r 1 to being dominated by r l/q Rg, R B. The radius of gyration Rg is defined by the small momentum behavior of the (normalized) density correlations of an isolated chain 

To determine qo we consider the large momentum behavior. In the excluded volume limit the scaling theory predicts (cf. Eq. (9.20)) 

This result is also bom out by the renormalized perturbation theory as will be shown in Chap. 15. Roo/2 is a universal ratio connecting the limits q — 0 or q —3 oo. This is clearly exhibited by using Eqs. (A 13.7), (A 13.6) to write 

By construction TZoo/2 is particularly sensitive to the choice of q . A first-order calculation, evaluated in three dimensions and using no = -53 yields 

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As perturbation theory on the basis of (222) yields the irreducible functions in terms of the bare parameters the expressions on the right-hand sides of (228), (229), (237) and (238) are obtained by use of (220) and (221) and thus depend themselves on the renormalization constants. The renormalization thus has to proceed in a recursive fashion, i.e. order by order in perturbation theory. 

At the ZORA level, the sum of the A, B and X operators vanishes, thus only the first term and half of the second and the third terms in equation (62) are nonzero, whereas at the lORA level also the two last and energy dependent renormalization terms must be considered. At the general quasi-relativistic level of theory all terms contribute to the magnetic properties. Expressions for second-order magnetic properties can be obtained by differentiating the energy expression (59) twice with respect to the perturbation-strength parameters of the vector potentials. 

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