Stationary perturbation theory

Both the initial- and the final-state wavefunctions are stationary solutions of their respective Hamiltonians. A transition between these states must be effected by a perturbation, an interaction that is not accounted for in these Hamiltonians. In our case this is the electronic interaction between the reactant and the electrode. We assume that this interaction is so small that the transition probability can be calculated from first-order perturbation theory. This limits our treatment to nonadiabatic reactions, which is a severe restriction. At present there is no satisfactory, fully quantum-mechanical theory for adiabatic electrochemical electron-transfer reactions. 

The interest aroused by the field of radiationless transitions in recent years has been enormous, and several reviews have been published 72-74) Basically, the ideas of Robinson and Frosch 75) who used the concepts on non-stationary molecular states and time-dependent perturbation theory to calculate the rate of transitions between Born-Oppenheimer states, are still valid, although they have been extended and refined. The nuclear kinetic energy leads to an interaction between different Born-Oppenheimer states and the rate of radiationless transitions is given by... 

Importantly, the anti-Hermitian CSE may be evaluated through second order of a renormalized perturbation theory even when the cumulant 3-RDM is neglected in the reconstruction. The anti-Hermitian part of the CSE [27, 31, 63] is the stationary condition for two-body unitary transformations of the A-particle wave-function [31, 32], and hence the two-body unitary transformations may easily be evaluated with the anti-Hermitian CSE and RDM reconstruction without the many-electron Schrodinger equation. The contracted Schrodinger equation in conjunction with the concepts of reconstruction and purification provides a new, important approach to computing the 2-RDM directly without the many-electron wavefunction. 

Bardeen considers two separate subsystems first. The electronic states of the separated subsystems are obtained by solving the stationary Schrodinger equations. For many practical systems, those solutions are known. The rate of transferring an electron from one electrode to another is calculated using time-dependent perturbation theory. As a result, Bardeen showed that the amplitude of electron transfer, or the tunneling matrix element M, is determined by the overlap of the surface wavefunctions of the two subsystems at a separation surface (the choice of the separation surface does not affect the results appreciably). In other words, Bardeen showed that the tunneling matrix element M is determined by a surface integral on a separation surface between the two electrodes, z = zo. 

The polarization, or the van der Waals interaction, can be accounted for by a stationary-state perturbation theory, effectively and accurately. The exchange interaction or tunneling can be treated by time-dependent perturbation theory, following the method of Oppenheimer (1928) and Bardeen (1960). In this regime, the polarization interaction is still in effect. Therefore, to make an accurate description of the tunneling effect, both perturbations must be considered simultaneously. This is the essence of the MBA. 

There is another way of looking at this coupled ion system, namely, in terms of stationary states. From this point of view, one considers that the excitation belongs to both ions simultaneously. To determine the wave functions of the two-ion system, one resorts to degenerate perturbation theory. The coupling H can be shown to remove the degeneracy, and two new states that are mixtures of X20 and X11 are formed. For each the excita-... 

Unfortunately, the stationary Schrodinger equation (1.13) can be solved exactly only for a small number of quantum mechanical systems (hydrogen atom or hydrogen-like ions, etc.). For many-electron systems (which we shall be dealing with, as a rule, in this book) one has to utilize approximate methods, allowing one to find more or less accurate wave functions. Usually these methods are based on various versions of perturbation theory, which reduces the many-body problem to a single-particle one, in fact, to some effective one-electron atom. 

In the previous chapter we have briefly discussed the use of the stationary many-body perturbation theory of the effective Hamiltonian, sketched in Chapter 3, to account for correlation effects. Here we shall continue such studies for electronic transitions on the example of the oxygen isoelectronic sequence. Having in mind (29.32) and the approximation Q(1) iiji the matrix element of the 1-transition operator Oei between the initial state... 

A non-perturbative theory of the multiphonon relaxation of a localized vibrational mode, caused by a high-order anharmonic interaction with the nearest atoms of the crystal lattice, is proposed. It relates the rate of the process to the time-dependent non-stationary displacement correlation function of atoms. A non-linear integral equation for this function is derived and solved numerically for 3- and 4-phonon processes. We have found that the rate exhibits a critical behavior it sharply increases near a specific (critical) value(s) of the interaction. 

Using expansion (16.2) for the wavepacket in terms of the stationary wavefunctions we can derive a set of coupled equations for the expansion coefficients au(t) similar to (2.16). In the limit of first-order perturbation theory [see Equation (2.17)] the time dependence of each coefficient is then given by... 

A modified effective Hamiltonian Goep is defined by replacing vxc by a model local potential vxc(r). The energy functional is made stationary with respect to variations of occupied orbitals that are determined by modified OEL equations in which Q is replaced by Goep- 84>i is determined by variations 8vxc(r) in these modified OEL equations. To maintain orthonormality, <5, can be constrained to be orthogonal to all occupied orbitals of the OEP trial state , so that (r) = J]a(l — na)i). First-order perturbation theory for the OEP Euler-Lagrange equations implies that... 

Kutzelnigg, W., Stationary perturbation theory. Theor. Chim. Acta (1992) 83 263-312. 

Spectroscopy is concerned with the observation of transitions between stationary states of a system, with the accompanying absorption or emission of electromagnetic radiation. In this section we consider the theory of transition probabilities, using time-dependent perturbation theory, and the selection rules for transitions, particularly those relevant for rotational spectroscopy. 

A review of the semidassical method is given by Cottrell and McCoubrey [9] and by Rapp and Kassal [13]. In this method, the translational motion is treated classically, while the molecule BC is assumed to have quantized vibrational levels. By converting the force V (x) on the oscillator due to the incident atom to V (t) by utilization of the classical trajectory x(t), one may apply time dependent perturbation theory. The wave function for the perturbed system is written as a sum of the stationary-state wave functions Y (y)exp( —icojf), with coefficients ck given by... 

On the theoretical side the H20-He systems has a sufficiently small number of electrons to be tackled by the most sophisticated quantum-chemical techniques, and in the last two decades several calculations by various methods of electronic structure theory have been attempted [77-80]. More recently, new sophisticated calculations appeared in the literature they exploited combined symmetry - adapted perturbation theory SAPT and CCSD(T), purely ab initio SAPT [81,82], and valence bond methods [83]. A thorough comparison of the topology, the properties of the stationary points, and the anisotropy of potential energy surfaces obtained with coupled cluster, Moller-Plesset, and valence bond methods has been recently presented [83]. 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

Since in the Floquet representation the Hamiltonian K defined on the enlarged Hilbert space is time-independent, the analysis of the effect of perturbations (like, e.g., transition probabilities) can be done by stationary perturbation theory, instead of the usual time-dependent one. Here we will present a formulation of stationary perturbation theory based on the iteration of unitary transformations (called contact transformations or KAM transformations) constructed such that the form of the Hamiltonian gets simplified. It is referred to as the KAM technique. The results are not very different from the ones of Rayleigh-Schrodinger perturbation theory, but conceptually and in terms of speed of convergence they have some advantages. 

For very small field amplitudes, the multiphoton resonances can be treated by time-dependent perturbation theory combined with the rotating wave approximation (RWA) [10]. In a strong field, all types of resonances can be treated by the concept of the rotating wave transformation, combined with an additional stationary perturbation theory (such as the KAM techniques explained above). It will allow us to construct an effective Hamiltonian in a subspace spanned by the resonant dressed states, degenerate at zero field. 

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