Classical perturbation

Consider a quantum center (i.e., a molecule or a subpart of a molecule) embedded in a classical molecular environment. Defining with rn the nuclear coordinates of the quantum center and with x the coordinates of the atoms providing the (classical) perturbing field we can expand [26] the perturbed Hamiltonian matrix H of the quantum center on the Born-Oppenheimer surface as... 

Zeleznik [54] has derived rotational collision numbers for pure polar gases from a classical perturbation theory in two dimensions, in which the polar... 

Why does more than og The classical perturbation theoretic measure of interaction ... 

The classical perturbation theory, which was not discussed here, has made tremendous practical progresses, allowing one to treat intricate Hamiltonians. The presence of complicated kinetic energies or resonances is no longer a hindrance for perturbation theory. 

Under the condition when one can use FO perturbation approach in quantum and quasiclassical calculations, the classical mean square eneigy transfer can also be calculated in the FO classical perturbation theory. This again opens a possibility for comparison of FOD and FOA (now classical) approximations. Fig.4 shows the... 

A similar approach has been pursued by Daudey, Claverie and Malrieu, who used the same orthogonalization procedure for the orbitals but determined the interaction energy by means of classical perturbation theory. 

Having in mind the dramatic effects the establishment of an H-bond has on the I s band-shape, we may anticipate that this anharmonic coupling is not small. It means that it cannot be handled by classical perturbation techniques. It may, however, be taken into account in the frame of the adiabatic separation (6) of rapid and slow motions. This adiabatic separation is already used to separate the motions of the electrons in the molecular complex from the vibrations of the atoms and is then called Bom-Oppenheimer separation. In this approximation applied to the separation of from the intermonomer modes, the rapid vibration I s, which is ruled by H(q,Q ) of eq. (5.2) and displays characteristic wavenumbers around... 

Classic perturbation theory considerations [55] predicts that changes in the effective refractive index of a guided mode A eff are related to the changes in the... 

The theory of vibration-rotation interactions has been developed over the last 50 years by many prominent researchers. It has been presented in many texts on the subject e.g. [10], among them a rather complete summary by Aliev and Watson [11]. It is based on classical perturbation theory in the form of a sequence of contact transformations. The results relevant to the rotational constants are summarized here. The effective rotational constant about the P axis in the vibrational state characterized by the vibrational quantum numbers v=(vi. .. vjt...) with degeneracies d. .. <4...), is given by [12]... 

Abstract These lectures are devoted to the main results of classical perturbation theory. We start by recalling the methods of Hamiltonian dynamics, the problem of small divisors, the series of Lindstedt and the method of normal form. Then we discuss the theorem of Kolmogorov with an application to the Sun-Jupiter-Saturn problem in Celestial Mechanics. Finally we discuss the problem of long-time stability, by discussing the concept of exponential stability as introduced by Moser and Littlewood and fully exploited by Nekhoroshev. The phenomenon of superexponential stability is also recalled. 

The problem of small divisors is related to another well known problem that shows up in classical perturbation theory, namely the problem of secular terms. Let us illustrate the problem with a very simple example. We consider the Duffing s equation... 

Giorgilli, A. (1995a). Quantitative methods in classical perturbation theory. In Roy, A. E. and Steves, B. A., editors. Proceedings of the Nato ASI school From Newton to chaos modern techniques for understanding and coping with chaos in N-body dynamical systems . Plenum Press, New York. 

Giorgilli, A. (1995b). Methods of complex analysis in classical perturbation theory. In Benest, D. and Froeschle, C., editors. Chaos and diffusion in Hamiltonian systems. Editions Frontieres. 

Giorgilli, A. and Locatelli, U. (1997a). Kolmogorov theorem and classical perturbation theory. ZAMP, 48 220-261. 

Hamilton s equations of motion Eq. (4)] then provide expressions for dhjdt that are nonzero due to the coupling y2. In the event that the coupling is small, one may approximate the solution for the time dependence of the angles as that of the time dependence in the absence of the perturbation. This approach, a classical perturbation theory, gives the following result for the time dependence of 7,(0 ... 

So far, we have discussed the extension of classical perturbation theory to the collision probability model and the generalization of perturbation theory to arbitrary characteristics in steady-state but subcritical systems having a source. Even this is not the most general characterization we might require of a reactor. We are, in addition, concerned to compute arbitrary ratios in a critical system. These might (typically) be the breeding ratio, although several others are of interest. 

Microscopic theory of lattice dynamics studies the response of electron-ion systems to displacements of nuclei the "direct" method generally means an approach in which the undistorted crystal and the crystal with displacements are treated from the very beginning as two distinct and unrelated systems. The "direct" treatment of phonons is an alternative to the classical perturbation approach, in which phonons are viewed as a small perturbation of the ground state to be treated by linear response theory. This method, based on the inverse dielectric matrix, is explained in this Volume in the lecture-notes by J. T. Devreese, R. Resta and A. Baldereschl. 

The random response of slightly nonlinear vibrating systems can also be obtained by applying the classical perturbation method (Crandall 1963). The method is based on the assumption that the nonlinearity is small enough to allow the solution of the stochastic differential equation of motion to be expressed as a power series. If the following SDOF system is considered... 

G. C. Schatz and T. Mulloney, Classical perturbation theory of good action-angle variables. Applications to semiclassical eigenvalues and to collisional energy transfer in polyatomic molecules, J. Phys. Chem. 83 989 (1979). 

To determine the H2O semiclassical vibrational eigenstates using the vibrational part of the molecular Hamiltonian, a second order classical perturbation theory method was used. This relates the vibrational energy to the three good constants of the motion Jjj2> the vibrational good actions. These actions... 

W. R. Gentry, Ion-dipole scattering in classical perturbation theory, J. Chem. Phys. 60 2547 (1974). 

The determination of the good actions describing vibration-rotation motion requires the solution of the molecular Hamilton-Jacobi equation, which is a nonlinear partial differential equation in 3Na"5 variables (including rotation), where is the number of atoms. Even for = 3 (a triatomic molecule) an exact solution to this equation is extremely complex computationally, and it is not practical for collisional applications. Several approximations can be used to simplify this treatment, however, including (i) the separation of vibration from rotation (valid in the limit of an adequate vibration-rotation time scale separation), and (ii) the use of classical perturbation theory (in 2nd and 3rd order) to solve the three-dimensional vibrational Hamilton-Jacobi equation which remains after the separation of rotation. Details of both the separation procedures and the perturbation-theory solution are discussed elsewhere. For the present application, the validity of the first... 

---