Perturbation theory, general methods

In general, the results of the perturbation theory based methods, SOPPA, SOPPA(CCSD), MPn and CCSD become smaller with increasing level of theory, whereas the results of the CASSCF/RASSCF calculations go through a maximum for the or °°°CAS wavefunctions depending on the molecule. 

The plan of this chapter is as follows. In the next section an overview of the history of the weak interactions in general, and atomic PNC in particular, is given. In section 3, Furry representation is introduced and applied to a calculation of a transition energy of a highly charged ion, Bi ". Section 4 describes the theory of cesium PNC, starting with low-order many-body perturbation theory (MBPT) methods, and then generalizing to all-orders methods based on coupled cluster theory. Section 5 closes the chapter with a brief description of the closely related field of atomic electric dipole moments. 

A number of types of calculations begin with a HF calculation and then correct for correlation. Some of these methods are Moller-Plesset perturbation theory (MPn, where n is the order of correction), the generalized valence bond (GVB) method, multi-conhgurational self-consistent held (MCSCF), conhgu-ration interaction (Cl), and coupled cluster theory (CC). As a group, these methods are referred to as correlated calculations. 

In developing perturbation theory it was assumed that the solutions to the unpermrbed problem formed a complete set. This is general means that there must be an infinite number of functions, which is impossible in actual calculations. The lowest energy solution to the unperturbed problem is the HF wave function, additional higher energy solutions are excited Slater determinants, analogously to the Cl method. When a finite basis set is employed it is only possible to generate a finite number of excited determinants. The expansion of the many-electron wave function is therefore truncated. 

This theory proves to be remarkably useful in rationalizing the whole set of general rules and mechanistic aspects described in the previous section as characteristic features of the Diels-Alder reaction. The application of perturbation molecular orbital theory as an approximate quantum mechanical method forms the theoretical basis of Fukui s FMO theory. Perturbation theory predicts a net stabilization for the intermolecular interaction between a diene and a dienophile as a consequence of the interaction of an occupied molecular orbital of one reaction partner with an unoccupied molecular orbital of the other reaction partner. 

Quantum mechanical calculations of intermolecular forces generally start from wave functions of the isolated particles. With regard to the actual treatment of the interaction, however, there is some competition between perturbation theory and MO methods. 

Some authors have described the time evolution of the system by more general methods than time-dependent perturbation theory. For example, War-shel and co-workers have attempted to calculate the evolution of the function /(r, Q, t) defined by Eq. (3) by a semi-classical method [44, 96] the probability for the system to occupy state v]/, is obtained by considering the fluctuations of the energy gap between and 11, which are induced by the trajectories of all the atoms of the system. These trajectories are generated through molecular dynamics models based on classical equations of motion. This method was in particular applied to simulate the kinetics of the primary electron transfer process in the bacterial reaction center [97]. Mikkelsen and Ratner have recently proposed a very different approach to the electron transfer problem, in which the time evolution of the system is described by a time-dependent statistical density operator [98, 99]. 

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