Perturbed stationary state method

As was already noted in [9], the primary effect of the YM field is to induce transitions (Cm —> Q) between the nuclear states (and, perhaps, to cause finite lifetimes). As already remarked, it is not easy to calculate the probabilities of transitions due to the derivative coupling between the zero-order nuclear states (if for no other reason, then because these are not all mutually orthogonal). Efforts made in this direction are successful only under special circumstances, for example, the perturbed stationary state method [64,65] for slow atomic collisions. This difficulty is avoided when one follows Yang and Mills to derive a mediating tensorial force that provide an alternative form of the interaction between the zero-order states and, also, if one introduces the ADT matrix to eliminate the derivative couplings. 

The perturbed stationary state method leads to the same conclusion. Here, the wave function of the colliding system (non-stationary state) is expressed in terms of the stationary state electronic wave functions Xx( ) of the Aj molecule (X denotes Eigen-value). Again, only two states, x (-R) and X (-S) with energy E R) and E R), respectively, are important (two-state approximation), and... 

The method of perturbed stationary state (PSS) was applied to the heavy particle collision of symmetrical resonance by Buckingham and Dalgarno, Matsuzawa and Nakamura, and Kolker and Michels. Although the essential idea of our method is the same as in these papers, except for the inclusion of four levels, we include a brief description of the PSS method for later discussion. 

Equation (190) is comparable to Eq. (178). The wave functions, however, are now stipulated by Eqs. (187) and (188), and the interaction potential is supplanted by terms from the small relative kinetic energy. One may treat Eq. (190) as we have Eq. (178), ignoring most of the nondiagonal elements in the series. It is important to understand, nevertheless, that even if such a procedure is adopted, the perturbed stationary-state wave functions still implicitly include some contribution from nondiagonal terms in the sense of Eq. (178). Thus, the method here is of greater accuracy than those based on Eq. (178), but note that Eq. (187) must be solved to apply this method. 

Green, T. A., Shipsey, E. J. and Browne, J. C. (1982) modified method of perturbed stationary states. IV. Electron-capture cross sections for the reaction... 

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. 

To apply described above non-perturbative method to the non-radiative transitions caused by the quadratic non-diagonal vibronic interaction qfli, one needs to find the phonon correlation functions Du(t, r) in the initial (non-stationary) state 11,0) for l > 0. To this end, we use the equation of motion xu + (ojxu + Veuqt j = 0 il, l > 0 / V l1) the integral form of it for t > 0 reads... 

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... 

The computationally viable description of electron correlation for stationary state molecular systems has been the subject of considerable research in the past two decades. A recent review1 gives a historical perspective on the developments in the field of quantum chemistry. The predominant methods for the description of electron correlation have been configuration interactions (Cl) and perturbation theory (PT) more recently, the variant of Cl involving reoptimization of the molecular orbitals [i.e., multiconfiguration self-consistent field (MCSCF)] has received much attention.1 As is reasonable to expect, neither Cl nor PT is wholly satisfactory a possible alternative is the use of cluster operators, in the electron excitations, to describe the correlation.2-3... 

There have been developed two essentially different wave-mechanical perturbation theories. The first of these, due to Schrodinger, provides an approximate method of calculating energy values and wave functions for the stationary states of a system under the influence of a constant (time-independent) perturbation. We have discussed this theory in Chapter VI. The second perturbation theory, which we shall-treat in the following paragraphs, deals with the time behavior of a system under the influence of a perturbation it permits us to discuss such questions as the probability of transition of the system from one unperturbed stationary state to another as the result of the perturbation. (In Section 40 we shall apply the theory to the problem of the emission and absorption of radiation.) The theory was developed by Dirac.1 It is often called the theory of the variation of constants the reason for this name will be evident from the following discussion. 

The linear variation method and the perturbation method for stationary states belong to the most important approximate methods of quantum mechanics. The latter exists in several variants. 

We discussed some aspects of the responses of chemical systems, linear or nonlinear, to perturbations on several earlier occasions. The first was the responses of the chemical species in a reaction mechanism (a network) in a nonequilibrium stable stationary state to a pulse in concentration of one species. We referred to this approach as the pulse method (see chapter 5 for theory and chapter 6 for experiments). Second, we studied the time series of the responses of concentrations to repeated random perturbations, the formulation of correlation functions from such measurements, and the construction of the correlation metric (see chapter 7 for theory and chapter 8 for experiments). Third, in the investigation of oscillatory chemical reactions we showed that the responses of a chemical system in a stable stationary state close to a Hopf bifurcation are related to the category of the oscillatory reaction and to the role of the essential species in the system (see chapter 11 for theory and experiments). In each of these cases the responses yield important information about the reaction pathway and the reaction mechanism. 

Orlov Rozonoer (1984a, b) present a general phenomenological approach to the macroscopic description of the dynamics of open systems. They prove theorems on existence, uniqueness and stability of the stationary slates. Singularly perturbed equations are also considered. The variational principle for the studied equations is formulated as well. As an application of the general results, nonisothermal kinetic differential equations are considered, detailed balanced and balanced systems are studied and the method of quasistationary concentrations is discussed. For open isothermal systems a theorem on the existence of a positive stationary state is proved providing a solution to an old basic problem. 

The response functions theory the PCM method [1] is an extension of the response theory for molecules in the gas phase [2, 3], This latter is based on a variational-perturbation approach for the description of the variations of the electronic wave function and of the changes of the observables properties at the various orders of perturbation with respect to the perturbing fields, and no restrictions are posed on the nature of the observables and on the nature of the perturbing fields, and the theory gives also access to a direct determination of the transition properties (i.e. transition energies and transition probabilities) associated with transitions between the stationary states of the molecular systems. The PCM response theory adds to this framework several new elements. 

For larger systems such as those typically fotmd in complex chemical problems, non-dimensionalisation may be impractical, and hence, numerical perturbation methods are generally used to investigate system dynamics and to explore timescale separation. By studying the evolution of a small disturbance or perturbation to the nonlinear system, it is possible to reduce the problem to a locally linear one. The resulting set of linear equations is easier to solve, and information can be obtained about the local timescales and stability of the nonlinear system. Several books on mathematics and physics (see e.g. Pontryagin 1962) discuss the linear stability analysis of the stationary states of a dynamical system. In this case, the dynamical system, described by an ODE, is in stationary state, i.e. the values of its variables are constant in time. If the stationary concentrations are perturbed, one of the possible results is that the stationary state is asymptotically stable, which means that the perturbed system always returns to the stationary state. Another possible outcome is that the stationary point is unstable. In this case, it is possible that the system returns to the stationary state after perturbation towards some special directions but may permanently deviate after a perturbation to other directions. A full discussion of stationary state analysis in chemical systems is given in Scott (1990). 

Equation (1-1-1) provides a basis for the discussion of the electronic structure and properties of molecules in their stationary states. In the next few chapters we shall deal almost exclusively with stationary states but in later chapters we develop theoretical procedures for calculating the effects of time-dependent perturbations, and we then have to start from (1.1.12)- This latter development will lead to the study of propagator and equation-of-motion methods, which now play an important role in molecular quantum mechanics. 

So far, attention has been concentrated on the calculation of (bound) stationary states and static electronic properties. In principle, a knowledge of the stationary-state wavefunctions provides a basis for the discussion of both static and dynamic properties but in Chapter 11 some of the difficulties of the usual approach, in which the wavefimction is expanded in terms of the eigenfunctions of some suitable unperturbed Hamiltonian, were pointed out. In particlar, exact unperturbed functions are never available, and even good approximate wavefunctions are usually available only for a few of the lowest states, certainly not in a sufficient number to provide a reliable expansion of a perturbed wavefunction. Such difficulties, which were avoided in Chapter 11 by using Unite-basis variational methods, prove to be even more serious in discussing dynamic properties and related quantities such as absorption frequencies and transition probabilities. 

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