Vibrational-rotational Hamiltonian

We find it convenient to reverse the historical ordering and to stait with (neatly) exact nonrelativistic vibration-rotation Hamiltonians for triatomic molecules. From the point of view of molecular spectroscopy, the optimal Hamiltonian is that which maximally decouples from each other vibrational and rotational motions (as well different vibrational modes from one another). It is obtained by employing a molecule-bound frame that takes over the rotations of the complete molecule as much as possible. Ideally, the only remaining motion observable in this system would be displacements of the nuclei with respect to one another, that is, molecular vibrations. It is well known, however, that such a program can be realized only approximately by introducing the Eckart conditions [38]. 

An alternative form of exact nonrelativistic vibration-rotation Hamiltonian for triatomic molecules (ABC) is that used by Handy, Carter (HC), and... 

The most consequent and the most straightforwaid realization of such a concept has been carried out by Handy, Carter, and Rosmus (HCR) and their coworkers. The final form of the vibration-rotation Hamiltonian and the handling of the corresponding Schrddinger equation in the absence of the vibronic... 

The complete vibration-rotation Hamiltonian for acetylene-like tetraatomic molecules has been derived by Handy et al. by hand [155] and using a computer algebra program [156]. (Note that in both of the mentioned papers there are some minor errors, see also [144,157,158]). Handy uses as bending coordinates... 

Jahn, H. A. (1938), A New Coriolis Perturbation in the Methane Spectrum. I. Vibrational-Rotational Hamiltonian and Wave Functions, Proc. Roy. Soc. A 168,469. 

Sadovskii, D. A., and Zhilinskii, B. I. (1988), Qualitative Analysis of Vibration-Rotation Hamiltonians for Spherical Top Molecules, Mol. Phys. 65, 109. 

What is a polyad A polyad is a subset of the zero-order states within a specifiable region of Evib (typically a few hundred reciprocal centimeters) that are strongly coupled by anharmonic resonances to each other and negligibly coupled to all other nearby zero-order states. If approximate constants of motion of the exact vibration-rotation Hamiltonian exist, then the exact H can be (approximately) block diagonalized. Each subblock of H corresponds to one polyad and is labeled by a set of polyad quantum numbers. For the C2H2S0 state, a procedure proposed by Kellman [9, 10] identifies the three polyad quantum numbers... 

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... 

Watson JKG (1968) Simplification of themolecular vibration-rotation Hamiltonian. Mol Phys 15 479-490... 

We shall see later how, with the appropriate expressions for the magnetic vector potentials, the effects of an external magnetic vector potentials, the effects of an external magnetic field can be introduced into the vibration rotation Hamiltonian. 

The vibration—rotation Hamiltonian is valid irrespective of the magnitude of the amplitudes of vibrations, Le. it could be used in principle to non-rigid molecules as well. There are, however, two main sources of difficulty which arise if we wish to apply to non-rigid molecules. 

RECENT ADVANCES IN THE THEORY OF VIBRATION-ROTATION HAMILTONIANS... 

We give a brief overlook to the derivation of the vibration-rotation Hamiltonian with the coordinate chain rules of differentiation (e.g., Refs. 22-28). For example, in Ref. 23, is written as... 

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