Perturbation theory, basic structure

Let us first inquire whether basic criteria for the validity of low-order perturbation theory are actually satisfied in the present case. As described in Section 1.4, the perturbative starting point is an idealized natural Lewis-structure wavefunction (t//,l )) of doubly occupied NBOs. The accuracy of this Lewis-type starting point may be assessed in terms of the percentage accuracy of the variational energy (E) or density (p(l ). as shown for each molecule in Table 3.20. 

Computer simulations, Monte Carlo or molecular dynamics, in fact appear to be the actual most effective way of introducing statistical averages (if one decides not to pass to continuous distributions), in spite of their computational cost. Some concepts, such as the quasi-structure model introduced by Yomosa (1978), have not evolved into algorithms of practical use. The numerous versions of methods based on virial expansion, on integral equation description of correlation functions, on the application of perturbation theory to simple reference systems (the basic aspects of these... 

Today we know that the HF method gives a very precise description of the electronic structure for most closed-shell molecules in their ground electronic state. The molecular structure and physical properties can be computed with only small errors. The electron density is well described. The HF wave function is also used as a reference in treatments of electron correlation, such as perturbation theory (MP2), configuration interaction (Cl), coupled-cluster (CC) theory, etc. Many semi-empirical procedures, such as CNDO, INDO, the Pariser-Parr-Pople method for rr-eleetron systems, ete. are based on the HF method. Density functional theory (DFT) can be considered as HF theory that includes a semiempirical estimate of the correlation error. The HF theory is the basie building block in modern quantum chemistry, and the basic entity in HF theory is the moleeular orbital. 

Many-body perturbation theory is based on the premise that the Hamiltonian from HF theory provides the basic foundation for the solution of the electronic structure and that configurational interactions can be treated as small perturbations to the Hamiltonian. The Hamiltonian is, therefore, written as the sum of the reference (HF) Hamiltonian (HO) and a small perturbation H ... 

There are two basic differences in [his] approach which permit all orders to be treated at once. First, the starting point is the Brillouin-Wigner BW) perturbation theory, whose formal structure is much simpler than that of the RS expansion. Secondly, we use a factorization theorem , which expresses the required energy-denominator identities in a simple and general form. ... 

Abstract The most promising approach for the calculation of polymer phase equilibria today is the use of equations of state that are based on perturbation theories. These theories consider an appropriate reference system to describe the repulsive interactions of the molecules, whereas van der Waals attractions or the formation of hydrogen bonds are considered as perturbations of that reference system. Moreover, the chain-like structure of polymer molecules is explicitly taken into account. This work presents the basic ideas of these kinds of models. It will be shown that they (in particular SAFT and PC-SAFT) are able to describe and even to predict the phase behavior of polymer systems as functions of pressure, temperature, polymer concentration, polymer molecular weight, and polydispersity as well as - in case of copolymers - copolymer composition. 

Now that we have some better understanding of where the H atom orbitals come from, the next topic should be the electronic structure of molecules and ways to treat problems for which we are unable to solve the Schrodinger equation exactly. Recall the difficulty of solving the Schrodinger equation for just one electron in the H atom. Then perhaps you may faint when you consider the notion of how one might treat the electronic structure of benzene with 12 atoms and 42 electrons Well, there is no known exact solution for even the He atom with only two electrons so do not faint but continue to wonder about how we are going to treat the multielectron case for molecules. There are two main methods the variation method and perturbation theory. In this chapter, we will emphasize the variation method, which is the most powerful mathematical approach, and give a few key examples. However, we will first mention the basic approach of perturbation theory but without much elaboration since it is the weaker of the two methods. 

In this chapter, we will do basically two things—examine in some detaii the form of the molecular orbitals for an AH2 molecule and secondly find out howto predict the form and energetic consequences of moiecular orbitals when a molecule is distorted via geometric perturbation theory. Furthermore, we see the physical underpinnings of why some molecules distort away from the most symmetrical structures and a technique of how to predict the sense of distortion. 

Schrodinger equation and the solutions supported by an independent electron model, scale linearly with electron number. Approximate treatments of electron correlation effects may not necessarily scale linearly with electron number. Approximations which display such a linear scaling are termed many-body methods. In the previous section, we described the many-body perturbation theory which is the basic ingredient of all many-body theories. It provided a fundamental tool for both the synthesis and the analysis of many-body methods. In this section, we consider some many-body theories of atomic and molecular structure and some theories which contain unphysical terms which scale nonlinearly with electron number. 

---