Fourth-order, generally perturbation theory

In this section, we shall use the degenerate perturbation theory approach to derive the form of the effective Hamiltonian for a diatomic molecule in a given electronic state. Exactly the same result can be obtained by use of the Van Vleck or contact transformations [12, 13]. The general expression for the operator up to fourth order in perturbation theory is given in equation (7.43). Fourth order can be considered as the practical limit to this type of approach. Indeed, even its implementation is very laborious and has only been used to investigate the form of certain special terms in the effective Hamiltonian. We shall consider some of these terms later in this chapter. For the moment we confine our attention to first- and second-order effects only. 

The operator H-mt in (15.4) contains, in general, terms of third and of fourth order with respect to the operators Ps and Pj. Terms of third order with respect to the operators Ps and Pj always lead to a weak exciton-exciton interaction. Since in such crystals the exciton bandwidth is much smaller than the energy required for the exciton formation, the third-order terms, which do not preserve the number of excitons, contribute to the exciton-exciton interation energy only in even orders of perturbation theory. 

G. Fitzgerald, R. J. Harrison, and R. J. Bartlett,/. Chem. Phys., 85, 5143 (1986). Analytic Energy Gradients for General Coupled-Cluster Methods and Fourth-Order Many-Body Perturbation Theory. 

All calculations in Ref. [22] were performed utilizing the Gaussian-98 code [30]. The potential energy scan was performed by means of the Mqller-Plesset perturbation theory up to the fourth order (MP4) in the frozen core approximation. The electronic density distribution was studied within the population analysis scheme based on the natural bond orbitals [31,32], A population analysis was performed for the SCF density and MP4(SDQ) generalized density determined applying the Z-vector concept [33]. 

McRae87 88 has derived an expression for the solvent-induced frequency shift, from the second order perturbation theory, taking into account all the types of interactions suggested by Bayliss and McRae86. On the basis of a simple electrostatic model, the frequency shift, Av, is related to the refractive index and the static dielectric constant of the solvent by an equation consisting of four terms. The first term in the equation represents the contribution from dispersive interactions which give rise to a general red shift the second term represents the contribution from the solute dipole-induced solvent dipole interactions the third term accounts for the solute dipole-solvent dipole interactions and the fourth term represents the contribution from the... 

Although small systematic changes in basis sets can frequently lead to erratic alterations of the quantitative nature of the H-bonding phenomenon, subtraction of the superposition error by the counterpoise procedure leads to much more uniform behavior. This argument applies to the correlated as well as SCF level of treatment. Perturbation theory of the Moller-Plesset type furnishes an efficient and accurate means to account for electron correlation. The literature indicates that MP2 calculations are quite reliable for H-bonds, due in large measure to the opposite effects generally observed for the third and fourth-order terms. 

Van der Waals attraction is due to a photon exchange between electrons located at different particles 1 and 2, i.e. the lowest order contribution to the attraction arises from the graph shown in Fig. 31. In order to obtain the energy of attraction, we have to apply at least fourth order perturbation theory. Since in Sections 3.3 and 7.1 we have seen how to generalize the results of finite perturbation theory by complex integration techniques, we shall he able to extend the present findings in a similar manner. 

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