Distortion terms

Resorting to reasoning of Kochendorfer ([112] and [115], p. 463) an approximation for the breadth of the lattice distortion term due to small amounts of strain broadening... 

In a fundamental paper [265] Ruland develops an advanced method for the analysis of scattering patterns showing moderate anisotropy. The deduction is based on a 3D model and the concept of highly oriented lattices. The addition of distortion terms makes sure that the theory is applicable to distorted structures and their scattering. 

The relative magnitude of both second-order terms determines whether the molecule is stable with respect to distortion along the vibrational coordinate Qt. If the relaxation term is of larger magnitude than the distortion term then the geometry is unstable and the potential energy surface has negative curvature for mode Qi. 

The q = 1 components of the first term on the right-hand side of this equation represents a correction to the so-called rotational distortion term, whose matrix elements were given in equations (8.363) and (8.364). In the light of this term we should replace —2B by (—2B + y) in the off-diagonal elements (8.364), add nothing to the energy of the 2n3/2 component, and subtract y from the energy of the 2ni/2 component. 

An obvious improvement to the theory, which should be most significant for the J = 13/2 levels, would be the inclusion of centrifugal distortion terms. There is, however, another interaction which might be significant, namely, the nuclear spin-rotation term. This is represented by an additional term in the effective Hamiltonian,... 

Further improvement in the agreement between experiment and theory, particularly for the higher J levels, would be obtained by the inclusion of centrifugal distortion terms. We will, however, defer discussion of these terms until we come to describe the results for the OH radical, where centrifugal distortion is much more important. Our reduction in the number of parameters required for a satisfactory quantitative interpretation of the radiofrequency electric resonance spectrum arises because of a number... 

Note that a centrifugal distortion term has been included the resolution and accuracy of pure microwave spectroscopy warrants this inclusion, even for the lowest rotational levels. 

The effective Hamiltonian used by Wayne and Radford was essentially that described earlier for 3S SO, with the addition of a centrifugal distortion term ... 

Using the basis representation rj, A J, A) we see that the first, second and fourth terms in (10.99) are diagonal, whereas the third and fifth terms have matrix elements off-diagonal in A, and therefore mix other electronic states with the 1 Ag state. If we include the first rotational distortion term it is easy to show that, from (10.99), the rotational energies are given by... 

The four terms in X7, however, require closer attention. The second term, which in molecule-fixed cartesian coordinates may be expressed as L2X + L2y, affects all levels equally and is therefore usually omitted. The third term, which does not involve the orbital angular momentum, is known as the rotational distortion term. Its matrix elements are readily obtained ... 

The constant term B .Ia(.Ia + 1) is absorbed into the term value Tv( 2) and is not considered to be part of the rotational Hamiltonian. There are matrix elements off-diagonal in v but these are small and are taken into account as centrifugal distortion terms. The result of (10.145) is that the first-order rotational energies of the two electronic states (for Ja = 3 /2) are given by... 

The centrifugal distortion terms in (11.44) were incorporated rather simply by making the replacements... 

The centrifugal distortion term involving yd is remarkably large, so much so that at N = 60, the order of the spin components actually changes sign. This is probably a unique observation. 

When the layers of a lamellar block copolymer are distorted, the free energy density is augmented by a distortional term that can, like the smectic-A phase, be described as the sum of layer compression/dilation and layer-bending energies ... 

The latter will also introduce pseudocentrifugal distortion terms. 

In Eq- (4.32) we shall provisionally neglect the centrifugal distortion terms. 

The rotational spectra of the four CWA simulants are shown in Figure 2. The upper two panels show the rotational spectmm of pinacolone (PC) and pinacolyl alcohol (PCA), respectively and are precursors to the nerve agent soman. The rotational constants, distortion terms, and dipole moment components (determined by ab initio calculations ) have been reported elsewhere. The rotational constants and projection of the permanent dipole moment onto the inertial axes are reported in Table 1. 

Another complication arises from the fact that the rotational constants are usually obtained as effective fitting parameters of a reduced rotational Hamiltonian [14]. Not only do the numerical values of the rotational constants depend on the exact form of the reduced Hamiltonian, they also contain small contributions from quartic and higher order centrifugal distortion terms. Watson [14] has proposed to always determine the so-called determinable combinations of these constants. The values of these combinations are independent of the form of the reduction, although they still contain small contributions from the distortion terms. Up to the quartic centrifugal distortion terms, the determinable combinations of the rotational constants are... 

Figure 5. Potential-distortion model solid line, attractive potential only dashed lines, with distortion term. Shaded area gives the spreading pressure.

Table I Includes the results of ellipsometric adsorption studies of water on a similar (and similarly prepared) PE to that of Ref. 14. (18) In addition, the adsorption was determined for a surface partially oxidized by treatment with an acid dlchromate solution. As might be expected, the effect of the introduction of polar sites reduces 6, Increases and Increases both Cq nd B. The relaxation distance for Is Increased, while that for the distortion term Is decreased. Systematic studies of this type should lead to a much better understanding of the nature of adsorbed films near F°.

Although centrifugal distortion is not a perturbation effect, a derivation of the form of the centrifugal distortion terms in Heff provides an excellent illustration of the Van Vleck transformation. If the vibrational eigenfunctions of the nonrotating molecular potential, V(R) rather than [V(R) + J(J + 1)H2/2/j,R2, are chosen as the vibrational basis set, then the rotational constant becomes an operator,... 

It is evident from Eq. (4.2.22) that the correct form of the centrifugal distortion terms is not given by the intuitive prescription replace Bv everywhere it appears in Eq. (4.2.20) by BV—DVJ(J+1). Alternative methods for evaluating the Eq. (4.2.21) perturbation summation have been proposed by Tellinghuisen (1973) and Kirschner and Watson (1973). 

Fig. 7 Schematic illustration of the origin of catalysis by the enrjrme ODCase. Arrows indicate the free energy decomposition of Eqs. 2 and 3 that separate the binding free energy into a protein distortion term and an intrinsic substrate-protein interaction component. The figure shows that the protein has a smaller distortion energy than that in the reactant state, so the change in protein conformational energy provides the predominant contribution to the lowering of the activation free energy of the catalyzed reaction...

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