Distortional perturbations

It is convenient to start the MO analysis from the model [(PH3)3Co(//-X)2-Co(PH3)3], Scheme la, which has a planar M2X2 skeleton. Many pieces of chemical information are obtainable by applying distortional perturbations to the basic MO picture of the latter. In partieular, it may be inferred how the formation of M-M and/or X-X tran.v-annular bonds in either model a or b in Scheme 1 depends on the electron count. Moreover, some hints can be gained about the potential reactivity of these species. 

Deep-level defects cannot be described by EMT or be viewed as simple perturbations to tlie perfect crystal. Instead, tlie full crystal-plus-defect problem must be solved and tlie geometries around tlie defect optimized to account for lattice relaxations and distortions. The study of deep levels is an area of active research. 

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p, being thus mostly pronounced in H electronic state. Renner developed the system of two coupled Schrbdinger equations and solved it for H states in the harmonic approximation by means of the perturbation theory. 

AH distortions of the nematic phase may be decomposed into three basic curvatures of the director, as depicted in Figure 6. Liquid crystals are unusual fluids in that such elastic curvatures may be sustained. Molecules of a tme Hquid would immediately reorient to flow out of an imposed mechanical shear. The force constants characterizing these distortions are very weak, making the material exceedingly sensitive and easy to perturb. 

Note that the structural form of each compound is implied by the presence (T-shaped structure) or absence (distorted structure) of the perturbed asymmetric stretching mode at about 550 cm . 

These concepts play an important role in the Hard and Soft Acid and Base (HSAB) principle, which states that hard acids prefer to react with hard bases, and vice versa. By means of Koopmann s theorem (Section 3.4) the hardness is related to the HOMO-LUMO energy difference, i.e. a small gap indicates a soft molecule. From second-order perturbation theory it also follows that a small gap between occupied and unoccupied orbitals will give a large contribution to the polarizability (Section 10.6), i.e. softness is a measure of how easily the electron density can be distorted by external fields, for example those generated by another molecule. In terms of the perturbation equation (15.1), a hard-hard interaction is primarily charge controlled, while a soft-soft interaction is orbital controlled. Both FMO and HSAB theories may be considered as being limiting cases of chemical reactivity described by the Fukui ftinction. 

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

For the present 12-6-4 potential we then have C 2 = 4Da12, C6 = —4Da, and C4 = — ae2/2. The WKB evaluation of the perturbation integral of the method of distorted waves results in the (temperature) average transition probability in the form (29) ... 

Hitherto it has been assumed that Tg corresponds to the classical equilibrium (or quantum-mechanical average) distance between the non-bonded atoms in the absence of interaction. It is inherent in the proper application of first-order perturbation theory that the perturbation is assumed to be small. In the case of the hindered biphenyls, however, it is known from the calculations cited in the introduction that the transition state is distorted to a considerable extent. The hydrogen atom does not occupy the same position relative to the bromine atom that it... 

It may be shown that when the polymer concentration is large, the perturbation tends to be less. In particular, in a bulk polymer containing no diluent a = l for the molecules of the polymer. Thus the distortion of the molecular configuration by intramolecular interactions is a problem which is of concern primarily in dilute solutions. In the treatment of rubber elasticity—predominantly a bulk polymer problem—given in the following chapter, therefore, the subscripts may be omitted without ambiguity. 

The concerns we have expressed are bound to get even more acute if the problem under study demands that we are able to adequately describe distortion effects induced in the electron distribution by external fields. The evaluation of linear (and, still more, non linear) response funetions [1] by perturbation theory then forces one to take care also of the nonoccupied portion of the complete orbital spectrum, which is entrusted with the role of representing the polarization caused by the external fields in the unperturbed electron distribution [4], ... 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

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