Second-order theory

The coupling of warping and torsion is manifested in the equilibrium equations, as they do not appear separately  

The above procedure also provides corresponding sets of equations for the electric loads of every group of electrically paralleled laminae  

First-order theories are bound to the equilibrium of the undeformed system and therefore are basically suitable for small deformations. Second-order theories consider the equilibrium of the slightly deformed system and are necessary to investigate tensioned flexible structures as well as buckling phenomena. Since the behavior in this context is frequently dominated by the normal force, it is commonly not accounted for the other initial internal loads, while 

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Several structural theories of piezoelectricity [72M01, 72M02, 72A05, 74H03] have been proposed but apparently none have been found entirely satisfactory, and nonlinear piezoelectricity is not explicitly treated. With such limited second-order theories, physical interpretations of higher-order piezoelectric constants are speculative, but such speculations may help to place some constraints on an acceptable piezoelectric theory. 

Second-Order Integral Equations for Associating Fluids As mentioned above in Sec. II A, the second-order theory consists of simultaneous evaluation of the one-particle (density profile) and two-particle distribution functions. Consequently, the theory yields a much more detailed description of the interfacial phenomena. In the case of confined simple fluids, the PY2 and HNC2 approaches are able to describe surface phase transitions, such as wetting and layering transitions, in particular see, e.g.. Ref. 84. 

The main conclusion which can be drawn from the results presented above is that dimerization of particles in a Lennard-Jones fluid leads to a stronger depletion of the proflles close to the wall, compared to a nonassociating fluid. On the basis of the calculations performed so far, it is difficult to conclude whether the second-order theory provides a correct description of the drying transition. An unequivocal solution of this problem would require massive calculations, including computer simulations. Also, it would be necessary to obtain an accurate equation of state for the bulk fluid. These problems are the subject of our studies at present. 

Ehrhardt, H., Knoth, G., Schlemmer, P. and Jung, K. (1985). Absolute H(e, 2e)p cross section measurements comparison with first and second order theory. Phys. Lett. A 110 92-94. 

Positive pair-binding energies, and hence the effective attractive interaction between electrons, arc necessarily a core polarization effect. If all of the 60 valence electrons were treated as an inert Fermi sea, the net interaction between two electrons added to a given molecule would necessarily be repulsive there is no bound-state solution to the Cooper problem with purely repttlsive interactions. It is the dynamic interactions with the valence electrons that are crucial in producing overscreening of the purely bare repulsive interaction. Although the first-order term does not involve any virtual excitations, the second-order theory includes important core-polarization effects. 

Taylor series at second order is a questionable approximation. This approximation is justified if the third- and higher-order direct correlation functions of the liquid phase are negligibly small, but this does not appear to be the case. In particular, Curtin [130] and Cerjan et al. [132] have studied the effect of including third-order terms in the perturbation series, and have found that the agreement with computer simulations is significantly worse than for the second-order theory. This clearly shows that third-order (and perhaps higher-order) terms are important, and that the convergence properties of the perturbation series are poor. 

While the first two terms contribute to first-order perturbation theory, the last three terms contribute to second-order theory and are exactly the same as already known from the second-order g-tensor calculations. 

In Chapter 7 we discussed the basics of the theory concerned with the influence of diffusion on gas-liquid reactions via the Hatta theory for flrst-order irreversible reactions, the case for rapid second-order reactions, and the generalization of the second-order theory by Van Krevelen and Hofitjzer. Those results were presented in terms of classical two-film theory, employing an enhancement factor to account for reaction effects on diffusion via a simple multiple of the mass-transfer coefficient in the absence of reaction. By and large this approach will be continued here however, alternative and more descriptive mass transfer theories such as the penetration model of Higbie and the surface-renewal theory of Danckwerts merit some attention as was done in Chapter 7. 

Electron correlation effects are known to be impx>rtant in systems with weak interactions. Studies of van der Waals interetctions have established the importance of using methods which scale linearly with the number of electrons[28] [29]. Of these methods, low-order many-body perturbation theory, in particular, second order theory, oflfers computational tractability combined with the ability to recover a significant firaction of the electron correlation energy. In the present work, second order many-body perturbation theory is used to account for correlation effects. Low order many-body perturbation theory has been used in accurate studies of intermolecular hydrogen bonding (see, for example, the work of Xantheas and Dunning[30]). 

Even in second-order theory some predictions may be reliable if only qualitative results suffice. For instance, if only the principal peaks in a photoelectron spectrum are needed, such as in the study of some simple donor-acceptor complexes, as borane (BH3) with donors such as H2O and CO. 

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