Perturbed wave functions, definitions

Our definition of molecular properties in the previous section is restricted to static and periodic perturbations V t) =V t + T) where T is the period. The perturbed wave function then has the form [17]... 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

In order to leam more about the nature of the intermolecular forces we will start with partitioning of the total molecular energy, AE, into individual contri butions, which are as close as possible to those we defined in intermolecular perturbation theory. Attempts to split AE into suitable parts were undertaken independently by several groups 83-85>. The most detailed scheme of energy partitioning within the framework of MO theory was proposed by Morokuma 85> and his definitions are discussed here ). This analysis starts from antisymmetrized wave functions of the isolated molecules, a and 3, as well as from the complete Hamiltonian of the interacting complex AB. Four different approximative wave functions are used to describe the whole system ... 

The parameter X has been embedded in the definition of Hp. The wave function from perturbation theory [equation (A.109)] is not normalized and must be renormalized. The energy of a truncated perturbation expansion [equation (A.110)] is not variational, and it may be possible to calculate energies lower than experimental. ... 

Let us consider two stationary states n and m of an unperturbed system represented by the wave function V and such that Em > Let us assume that at / = 0, the system is in the state n. At this time, the system comes under the perturbing influence of the radiation of a range of frequencies in the neighbourhood of vm of a definite field strength E. 

The expression for J is derived via the general quantum mechanical definition (32), introducing the perturbation expansion for the current density and the a-state molecular wave-function (depending on n-electron space-spin coordinates ), yb—Zd e-... 

Hamiltonian proposed by Muller and Plesset gives rise to a very successful and efficient method to treat electron correlation effects in systems that can be described by a single reference wave function. However, for a multireference wave function the Moller-Plesset division can no longer be made and an alternative choice of B(0> is needed. One such scheme is the Complete Active Space See-ond-Order Perturbation Theory (CASPT2) developed by Anderson and Roos [3, 4], We will briefly resume the most important definitions of the theory one is referred to the original articles for a more extensive description of the method. The reference wave function is a CASSCF wave function that accounts for the largest part of the non-dynamical electron correlation. The zeroth-order Hamiltonian is defined as follows and reduces to the Moller-Plesset operator in the limit of zero active orbitals ... 

Turning on the perturbation A W produces the correction to the wave functions (eigenvectors) of the system described by eq. (1.62). Inserting it into the definition of the expectation value of A yields ... 

A spin-correct description of a wave function perturbed by a one-electron operator leads to a many-determinant function containing certain single excitations in addition to the RHF determinant. A definite reference state for the discussion of dynamic correlation effects would be one which includes all singly excited configurations coupled directly to the RHF function by a perturbing Hamiltonian, which may be the exchange operator of an unpaired electron as well as an external field or a nuclear moment. 

The most popular way of including dynamic correlation upon a CASSCF reference wave function [54-57] is the second-order perturbation theory (CASPT2) developed by Roos and coworkers [58]. However, in contrast to the single-configurational case, where the definition of the zeroth-order Hamiltonian is universal and taken as the sum of the one-electron Fock operators, the generalization of the zeroth-order Hamiltonian to the multiconfigurational case is not straightforward [59, 60]. A different, theoretically more justified approach is to... 

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