Hamiltonian vibration-rotational

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... 

Symmetry tools are used to eombine these M objeets into M new objeets eaeh of whieh belongs to a speeifie symmetry of the point group. Beeause the hamiltonian (eleetronie in the m.o. ease and vibration/rotation in the latter ease) eommutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "bloek diagonal". That is, objeets of different symmetry will not interaet only interaetions among those of the same symmetry need be eonsidered. 

Solutions to a Schrodinger equation for this last Hamiltonian (7) describe the vibrational, rotational, and translational states of a molecular system. This release of HyperChem does not specifically explore solutions to the nuclear Schrodinger equation, although future releases may. Instead, as is often the case, a classical approximation is made replacing the Hamiltonian by the classical energy ... 

This Hamiltonian is used in the Schrodinger equation for nuclear motion, describing the vibrational, rotational, and translational states of the nuclei. Solving the nuclear Schrodinger equation (at least approximately) is necessary for predicting the vibrational spectra of molecules. 

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. 

There exists no significant comprehensive fit of spectral data of H2 with which we might here make comparison. Our discussion above demonstrates that, as for GaH above, application of an algorithm based on Dunham s algebraic approach to analysis of vibration-rotational spectral data of H2, especially through implementation of hypervirial perturbation theory [30,72] that allows the term for the vibrational g factor in the hamiltonian in formula 29 to be treated directly in that form, proves extremely powerful to derive values of fitting parameters that not only have intrinsic value in reproducing experimental data of wave numbers of transitions but also relate to other theoretical and experimental quantities. 

That effective hamiltonian according to formula 29, with neglect of W"(R), appears to be the most comprehensive and practical currently available for spectral reduction when one seeks to take into account all three principal extramechanical terms, namely radial functions for rotational and vibrational g factors and adiabatic corrections. The form of this effective hamiltonian differs slightly from that used by van Vleck [9], who failed to recognise a connection between the electronic contribution to the rotational g factor and rotational nonadiabatic terms [150,56]. There exists nevertheless a clear evolution from the advance in van Vleck s [9] elaboration of Dunham s [5] innovative derivation of vibration-rotational energies into the present effective hamiltonian in formula 29 through the work of Herman [60,66]. The notation g for two radial functions pertaining to extra-mechanical effects in formula 29 alludes to that connection between... 

The vibration-rotation interaction term makes the Hamiltonian for nuclear motion of a polyatomic molecule difficult to deal with. Frequently, this term is small compared to the other terms. We shall make the initial approximation of omitting Tvib rot. The rotational kinetic energy TTOt involves the moments of inertia of the molecule, which in turn depend on the instantaneous nuclear configuration. However, the vibrational motions are much faster than the rotational motions, so that we can make the approximation of calculating the moments of inertia averaged over the vibrational motions. 

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... 

Here, Q is the projector on the bound subspace and P projects onto the open, continuum channels. The intramolecular coupling is written as V+ U so that, as before, U is any additional coupling brough about by external perturbations. The equation H = Hq + V+U, where Ho is the zero-order Hamiltonian of the Rydberg electron and so includes only the central part of the potential due to the core plus the motion (vibration, rotation) of the core, uncoupled to the electron. The perturbations V + U can act within the bound subspace, as the operator Q(V+l/)Q is not necessarily diagonal and is the cause of any intramolecular dynamics even in the absence of coupling to the continuum. The intramolecular terms can also couple the bound and dissociative states. 

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