Perturbation theory criticism

The first section summarizes simply the essential features of the different types of molecular orbital calculations currently used to solve theoretical problems in organometallic chemistry. A critical comparison of these calculations is also given. The second section discusses the bonding in organometallic complexes and draws on recent computational results and develops the chemical and structural implications of bonding models based on perturbation theory arguments. 

In the construction of the RGf dimension d = 4 plays a special role as upper critical dimension of the thebry. This for instance shows up in the estimate of the nonuniversal corrections to the theorem of renormalizability, or in the feature that the nontrivial fixed point u merges with the Gaussian fixed point for d — 4. It naturally leads to the e-expansion. However, the RG mapping constructed in minimal subtraction only trivially depends on e. Also results of renormalized perturbation theory do not necessarily ask for further expansion in e. Equation (12.25) gives an example. We should thus consider the practical implications of the -expansion in some more detail. 

We should not close this section without touching upon some delicate technical point. A priori even for a discrete chain the critical chemical potential g s, defined as chemical potential per segment of an infinitely long chain, does not exist outside the -expansion. We already encountered this problem in Sect. 7.2, where we found that rC proportional to p, for d < 4 in naive perturbation theory suffers from infrared singularities. That problem has been considered in the field theoretic framework. The results, expressed in polymer language, show that the u-expansion is not invalidated, as long as we consider quantities which do not involve explicitly. Almost all the quantities of interest to us are of this type. Only in the equation of state relating gp(n) and cp(n) does an explicit contribution g n occur. But even... 

A non-perturbative theory of the multiphonon relaxation of a localized vibrational mode, caused by a high-order anharmonic interaction with the nearest atoms of the crystal lattice, is proposed. It relates the rate of the process to the time-dependent non-stationary displacement correlation function of atoms. A non-linear integral equation for this function is derived and solved numerically for 3- and 4-phonon processes. We have found that the rate exhibits a critical behavior it sharply increases near a specific (critical) value(s) of the interaction. 

In this paper, the multiphonon relaxation of a local vibrational mode and the non-radiative electronic transitions in molecular systems and in solids are considered using this non-perturbative theory. Results of model calculations are presented. According to the obtained results, the rate of these processes exhibits a critical behavior it sharply increases near specific (critical) value(s) of the interaction. Also the usual increase of the non-radiative transition rate with temperature is reversed at certain value of the non-diagonal interaction and temperature. For a weak interaction, the results coincide with those of the perturbation theory. 

A second type of eigenstates, illustrated in Fig. 3.10, is the band of states formed by the exciton-contaminated photon continuum. Far from the critical area cK co0, this band presents a lorentzian resonance (cf. Fig. 3.9), whose temporal instability (cf. Fig. 3.8) is described by an exponential decay. Thus, the exact solution leads back to that of second-order perturbation theory, obtained in Section III.A.2.b above. 

Micellar aggregates are considered in chapter 3 and a critical concentration is defined on the basis of a change in the shape of the size distribution of aggregates. This is followed by the examination, via a second order perturbation theory, of the phase behavior of a sterically stabilized non-aqueous colloidal dispersion containing free polymer molecules. This chapter is also concerned with the thermodynamic stability of microemulsions, which is treated via a new thermodynamic formalism. In addition, a molecular thermodynamics approach is suggested, which can predict the structural and compositional characteristics of microemulsions. Thermodynamic approaches similar to that used for microemulsions are applied to the phase transition in monolayers of insoluble surfactants and to lamellar liquid crystals. 

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