Contact transformations

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 anharmonicities of the potential contribute by the terms involving the constants x, g, y,. .. as well as the energy shifts AEx = 0(h2),. .. and the frequency shifts Aw, = 0(h2),. These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semi-classical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of h. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22], The energy shifts AEx,... 

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. 

Contact Transformation for the Effective Hamiltonian.—The vibration-rotation hamiltonian of a polyatomic molecule, expressed in terms of normal co-ordinates, has been discussed in particular by Wilson, Decius, and Cross,24 and by Watson.27- 28 It is given by the following expression for a non-linearf polyatomic molecule, to be compared with equation (17) for a diatomic molecule ... 

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 vibration-rotation theory, the /., / and contributions to the contact-transformed Hamiltonian are commonly evaluated directly from the relationships (7.59), (7.63), (7.65) and (7.66). This is because the particularly simple commutation relationships which exist between the normal coordinate operator Q, its conjugate... 

In conclusion, we note that thus far we have derived matrix elements of the transformed Hamiltonian Xfor a given block in the complete matrix labelled by a particular value of rj rather than an effective Hamiltonian operating only within the subspace of the state rj. It is an easy matter to cast our results in the form of an effective Hamiltonian for any particular case since the matrix elements involved in either the commutator bracket formulation (contact transformation) or the explicit matrix element formulation (Van Vleck transformation) can always be factorised into a product of a matrix element of operators involved in X associated with the quantum number rj and a matrix element of operators that act only within the subspace levels of a given rj state, associated with the quantum number i. This follows because the basis set can be factorised as in equation (7.47). The matrix element involving the rj quantum number can then either be evaluated or included as a parameter to be determined experimentally, while the... 

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. 

Brown, Colbourn, Watson and Wayne [7] have shown that this type of indeterminacy occurs for any 2,s 1 A state which conforms to Hund s case (a) coupling. This result can be established most easily by applying a contact transformation to the effective Hamiltonian. Let us divide 3Q,V into a principal part 3C and a remainder 3( ... 

Since P, differs from p,- only by an infinitesimal and the final derivative in eqn (8.62) already involves e, Pj in the derivative of G has been replaced by pj to retain only first-order terms. It is usual practice to refer to G as well as F as the generator, and eqns (8.62) define infinitesimal canonical or contact transformations with the generator G. 

Thus, in analogy with the quantum result for the change induced in an observable by an infinitesimal unitary transformation (eqn (8.36)), the change in a property A caused by an infinitesimal contact transformation is given by the Poisson bracket of A with the generator G. The Poisson bracket of a constant of the motion with H vanishes, as does its corresponding commutator in quantum mechanics. Thus, the equation... 

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. 

To extract from the effective Floquet Hamiltonian (185) an effective dressed Hamiltonian independent of 0, we can apply a contact transformation consisting... 

---