Level renormalization

It is well known that in bulk crystals there are inversions of relative stability between the HCP and the FCC structure as a fxmction of the d band filling which follow from the equality of the first four moments (po - ps) of the total density of states in both structures. A similar behaviour is also expected in the present problem since the total densities of states of two adislands with the same shape and number of atoms, but adsorbed in different geometries, have again the same po, pi, P2/ P3 when the renormalization of atomic levels and the relaxation are neglected. This behaviour is still found when the latter effects are taken into account as shown in Fig. 5 where our results are summarized. 

We will describe a systematic approach to renormalize the intrachain interactions towards a coarser level for three different modifications of polycarbonates. The advantage of examinig three modifications of the same polymer gives a first hint of the sensitivity of the method. The three modifications of the polycarbonate are BPA-PC, BPZ-PC, and TMC-PC. The structures are given in Fig. 6.1. Although the backbone sequence is the same they have re-... 

The isomerization process interconverting the three isomers depicted in Figure 4, with and without ammonia ligands, has been studied recently,54 58 using the completely renormalized coupled cluster level of theory, including... 

To solve the full problem of finding an approximate ground state to Hamiltonian (13), one is faced to a self-consistent loop which can be proceeded in two steps. First one can get the occupations nia)o from a HWF, and a set of bare levels. Then one obtains a set of configuration parameters, the probabilities of double occupation, di by minimizing (18) with respect to these probabilities. Afterwards the on-site levels are renormalized according to (21) and the next loop starts again for the new effective Hamiltonian He// till convergence is achieved. 

Then e is associated with a level shift (or mass renormalization) of the unperturbed level e ° while e" gives us the decay time. 

Having at hand the techniques of perturbation thry and the renormalization group we are prepared to consider once more a question of fundamental importance can we justify our simple model Are we really allowed to ignore many-body interactions or other features of a microscopically realistic description of the polymer solution On a very superficial level we discussed this problem in Seet. 2.2. Let us critically reconsider that argument. 

This argument is nothing but the additive renormalization discussed in Sect 7.2 in the context of the continuous chain limit, now interpreted on the level of the interaction. From the field theoretic formulation we easily see that it holds to all orders. We thus may split the two-body interaction into a one-body part which effectively takes care of interactions within small fractions of the chains, and a remainder conforming to the estimate (10.1). 

On a deeper level we observe that the e-expansion does not properly respect the structure of the theory. As discussed before (see Sects. 8.3 or 10.2, in particular), we should first use the RG to map the system to an uncritical manifold, which for dilute systems is determined by Nr = 0(1), for instance. In a second step the scaling functions on the uncritical manifold can be calculated by renormalized perturbation theory. These two steps are well separated conceptually. The exponents, for instance, are properties of the RG, independent of specific scaling function. Strict e-expansion mixes these two steps since it simultaneously expands exponents and scaling functions. The expected scaling structure then has to be put in by hand at the end of the calculation. 

Thus the role of the high-frequency oscillators is to suppress the transverse field component (in other words, the transverse g-factor). If we are interested only in the contribution to the level spacing (the Lamb shift), one should consider only the longitudinal ( B) part of the renormalization, i.e. multiply the result by sin0, to obtain Eq. (13). 

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