Perturbation theories Terms Links

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

It is important to note that the Hamiltonian (2.120) contains the terms which produce both the adiabatic and non-adiabatic effects. In chapter 7 we shall show how the total Hamiltonian can be reduced to an effective Hamiltonian which operates only in the rotational subspace of a single vibronic state, the non-adiabatic effects being treated by perturbation theory and incorporated into the molecular parameters which define the effective Hamiltonian. Almost for the first time in this book, this introduces an extremely important concept and tool, outlined in chapter 1, the effective Hamiltonian. Observed spectra are analysed in terms of an appropriate effective Hamiltonian, and this process leads to the determination of the values of what are best called molecular parameters . An alternative terminology of molecular constants , often used, seems less appropriate. The quantitative interpretation of the molecular parameters is the link between experiment and electronic structure. 

We may recall that the desirability of ensuring size-extensivity for a closed-shell state was one of the principal motivations behind the formulation of the MBPT for the closed-shells. The linked cluster theorem of Bruckner/25/, Goldstone/26/ and Hubbard/27/, proving that each term in the perturbation series for energy can be represented by a linked (connected) diagram directly reflects the size-extensivity of the theory. Hubbard/27/ and Coester/30/ even pointed out immediately after the inception of MBPT/25,26/, that the size-extensivity is intimately related to a cluster expansion structure of the associated wave-operator that is not just confined only to perturbative theory. The corresponding non-perturbative scheme for the closed-shells was first described by Coester and Kummel/30,31/ in nuclear physics and this was transcribed to quantum chemistry... 

Coupled electron pair and cluster expansions. - The linked diagram theorem of many-body perturbation theory and the connected cluster structure of the exact wave function was first established by Hubbard211 in 1958 and exploited in the context of the nuclear correlation problem by Coester212 and by Coester and Kummel.213 Cizek214-216 described the first systematic application to molecular systems and Paldus et al.217 described the first ab initio application. The analysis of the coupled cluster equations in terms of the many-body perturbation theory for closed-shell molecular systems is well understood and has been described by a number of authors.9-11,67,69,218-221 In 1992, Paldus221 summarized the situtation for open-shell systems one must nonetheless admit... 

One important property of Eq. (23), which manifests itself very easily in the diagrammatic evaluation of the perturbation expansions, is the linked-diagram theorem (LOT). According to this theorem, only a selected class of terms survives in the perturbation expansion of the wave operator, while all unlinked diagrams cancel in each order of perturbation theory in addition, this theorem ensures the size consistency of MBPT and coupled-cluster theory. Based on LDT, Eq. (23) simplify to... 

We know that the Rayleigh-Schrodinger perturbation theory series leads directly to the many-body perturbation theory by employing the linked diagram theorem. This theory uses factors of the form Eq—Ek) as denominators. Furthermore, this theory is fully extensive it scales linearly with electron number. The second term... 

T/ie Goldstone expansion is rederived by elementary time-independent methods, starting from Brillouin-Wigner (BW) perturbation theory. Interaction energy terms AE are expanded out of the BW energy denominators, and the series is then rearranged to obtained the linked-cluster result. ... 

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