Contact transformation perturbation

The onset of sudden variations in vibrational fine structure is one of the most sensitive indicators of a change in resonance structure. The magnitudes of fine-structure parameters are determined by second-order perturbation theory (a Van Vleck or contact transformation) [17]. The energy denominators in these second-order sums over states are approximately independent of vib as long as the <01 <02 - 3/v-6 resonance structure is conserved. 

The vibration-rotation hamiltonian of a polyatomic molecule is more complicated than that of a diatomic molecule, both because of the increased number of co-ordinates, and because of the presence of Coriolis terms which are absent from the diatomic hamiltonian. These differences lead to many more terms in the formulae for a and x values obtained from the contact transformation, and they also lead to various kinds of vibrational and rotational resonance situations in which two or more vibrational levels are separated by so small an energy that interaction terms in the hamiltonian between these levels cannot easily be handled by perturbation theory. It is then necessary to obtain an effective hamiltonian over these two or more vibrational levels, and to use special techniques to relate the coefficients in this hamiltonian to the observed spectrum. 

The perturbation calculation may also be described as a contact transformation. The original hamiltonian is transformed to a new effective hamiltonian which has the same eigenvalues but different eigenfunctions, to some carefully chosen order of magnitude. This contact transformation of the vibration-rotation hamiltonian was originally studied by Nielsen and co-workers. >33... 

In this section, we shall use the degenerate perturbation theory approach to derive the form of the effective Hamiltonian for a diatomic molecule in a given electronic state. Exactly the same result can be obtained by use of the Van Vleck or contact transformations [12, 13]. The general expression for the operator up to fourth order in perturbation theory is given in equation (7.43). Fourth order can be considered as the practical limit to this type of approach. Indeed, even its implementation is very laborious and has only been used to investigate the form of certain special terms in the effective Hamiltonian. We shall consider some of these terms later in this chapter. For the moment we confine our attention to first- and second-order effects only. 

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. 

The use of van Vleck s contact transformation method for the study of time-dependent interactions in solid-state NMR by Floquet theory has been proposed. Floquet theory has been used for studying the spin dynamics of MAS NMR experiments. The contact transformation method is an operator method in time-independent perturbation theory and has been used to obtain effective Hamiltonians in molecular spectroscopy. This has been combined with Floquet theory to study the dynamics of a dipolar coupled spin (I = 1/2) system. 

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]... 

An effective Hamiltonian is profoundly different from an exact Hamiltonian. This is a reason for imperfect communication between experimentalists and ab initio theorists. The two communities use the same symbols and language to refer to often quite different molecular properties. The main difference between effective and exact Hamiltonians is that the molecule gives experimentalists an empirical basis set that has been prediagonalized implicitly to account for the infinite number of remote perturbers . This is the Van Vleck or contact transformation, but it is performed by the molecule, not by a graduate student. The basis set is truncated and the dynamics occurs in a reduced-dimension state space. 

In this section we describe the required contact transformations. The section is structured in the following manner. It begins with a description of the perturbative transformations as they apply to the (J = 0) vibrational Hamiltonian. Select results are given for C02 and H2CO. The section concludes with a brief overview of how these ideas are extended to include rotations. Here results will be presented for H20, H2CO, and S02. More detailed accounts of these studies can be found elsewhere (43,44,47,49). 

In the field of rotational spectroscopy, the relevant terms in these perturbative treatments are those that multiply J, as these are the effective rotational constants which contain contributions beyond the rigid-rotator harmonic oscillator approximation. To first order there are no corrections to the simple rigid-rotor rotational constant (equilibrium structure). In second order, there are three contributions that expressed by means of the usual contact transformation method give the second-order result [31]... 

A perturbation theory treatment known as the method of contact transformations [162] provides a convenient approach in defining the anharmonic potential energy and transitional dipole moment [158-163]. Both mechanical and electrical anhaimonicities influence the intensities of overtone and combination bands. 

The method of contact transformations simplifies the perturbation theory treatment. The perturbation Hamiltonian can be expressed as... 

Two successive contact transformations remove from the expression of the Hamiltonian the first and second degree terms. Thus, the wave functions of die zero order term which, in our case, are the standard linear harmonic oscillator wave functions, are also eigenfunctions of the Hamiltonian H up to the second order. If a standard perturbation dieory is applied, there will be an extensive number of off-diagonal matrix elements of the first-order perturbation Hamiltonian appearing in the expressions for any molecular quantity estimated from second order matrix elements. By the contact transformations the matrix elements will be diagonal through second order which greatly simplifies the calculations. If the linear harmonic oscillator wave function is denoted by ( n i, die matrix element < n I H" I m) may be expressed as [Eq. (6.10)]... 

In the context of the Wigner-Seitz theory, in 1937 Brillouin [7] gave a formal analysis of the atomic energy variations under boundary deformations using contact coordinate transformations that transform the boundary modifications into Hamiltonian transformations for a fixed region and generate the associated commutation relations. This allows application of the usual form of perturbation theory for the problem. 

A convenient feature of the isomerization between the monolayer and stacked structures is the fact that the frequency shifts predicted for the fundamental V3 band of SFg are distinctly different for the two forms. In particular, the magnitude of this (negative) frequency shift is approximately proportional to the number of perturber atoms in direct contact with the SFg, so the magnitude of the shift becomes distinctly smaller when a cluster isomerizes from a monolayer to a stacked form. This provides both a simple property for monitoring the state of the system during a simulation, and a possible means of observing such transformations experimentally. 

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