Second-order perturbations simplification

These simplifications underlie the usefulness of second-order perturbative analysis of NBO donor-acceptor interactions (Section 3.2.3) in which only the diagonal Lewis-type elements of are needed to form the unperturbed natural... 

In intermolecular perturbation theory one of the major problems concerns electron exchange between molecules. In the method described here exchange is limited to single electrons. This simplification is definitely a good approximation at large intermolecular distances. The energy of interaction between the molecules, AE (R), is obtained as a sum of first order, second order, and higher order contributions ... 

One may note that quasienergy derivatives dt listed above are symmetric with respect to the capital indices Ai... A . It must be emphasized that the indices must be thought of as a double index specifying a specific perturbation strength a, as well as a frequency cOa, and, as discussed previously, they must be permuted at the same time in order to retain symmetry. Further simplifications of the expressions for molecular properties are possible using the response equations, e.g. (48), (49) and (50), but generally at the expense of index symmetry and numerical precision [28,9]. As an example, using the response equation the expression for second-order molecular properties (52) can be rewritten as... 

In the Zimm theory, the flow perturbations and the co-operative hydrodynamic interactions between segments are treated using the Oseen tensor, pre-averaged for simplification. Pyun and Fixman (PF) avoided this approximation by a perturbation solution of the Kirkwood diffusion equation up to second order. One of the consequences was that [equation (3)] was re-evaluated (see Table 1). 

Using the perturbation theory proposed by Larsen et al. Karakatsani and Economou have extended the PC-SAFT equation to account for dipole-dipole, dipole-quadrupole, quadrupole-quadrupole and dipole-induced dipole interactions. The exact second- and third-order perturbation terms in the work of Larsen are however rather complex and so a simplified version of the two terms was also proposed based on the work of Nezbeda and co-workers. While simplifying the equation, in an effort to generate a more usable, engineering-type, approach, the simplification of the dipolar term in the model introduces an additional pure-component model parameter. We refer to these as the PC-SAFT-KE and truncated PC-SAFT-KE (denoted PC-PSAFT and tPC-PSAFT by the authors respectively) approaches. In both KE approaches, the multipoles are assumed to be uniformly distributed over all segments in the molecule. The truncated and full PC-SAFT-KE approaches have been applied to study a wide range of polar fluids and their mixtures, " with the truncated equation found to be as accurate as the full PC-SAFT-KE model, and the inclusion of polar interactions found to improve the theoretical predictions in most cases. ... 

First principles methods need less parametrisation, since the hamiltonian is built up from more fundamental principles, although simplifications and even some level of parametrisation remains necessary for practical use. For instance, in the case of DFT, that point is particularly evident in the construction of the exchange-correlation energy functional, which is not known a priori for an arbitrary molecular system. ° In the case of Hartree-Fock, the correlation effects are not considered and post-Hartree Fock adjustments are necessary to include such effects, for instance in the form of perturbative expansion terms of the type derived in the Moller-Plesset perturbation theory of second order (MP2). ... 

When one is far from resonance, then a further simplification is possible. Kleinman74 showed [15] that energy is simply exchanged among the fields E Ej, and Ek along the three axes i, j, and k, so that the suffixes can be interchanged freely this means that the three fields act independently and can be applied in arbitrary order (at resonance, one field will distort the electronic cloud in such a way that a second field will act on a severely perturbed electron configuration). Thus, when this is not a problem (i.e., when one is far from resonance) for SHG the 27 (or 18) coefficients reduce to 10 ... 

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