Polyatomic systems anharmonic

The effects of anharmonicity in polyatomic systems are similar to the diatomic case the zero-point level drops in energy, energy levels close up. 

Chaban, G. M., Jung, J. O., 8c Gerber, R. B. (1999). Ab initio calculation of anharmonic vibrational states of polyatomic systems Electronic structure combined with vibrational self-consistent field. The Journal of Chemical Physics, 111, 1823-1829. 

It has been shown that numerical precision of anharmonic force fields determined by methods of electronic structure theory does not deteriorate if the force field up to quartic terms is determined from well-selected, accurate energy points. Force constants of even higher order, on the other hand, should be determined from analytic derivative information. For larger polyatomic systems with a large number of anharmonic force constants either fully analytic determination of the force field... 

R. J. Harter, E. B. Alterman, and D. J. Wilson, Anharmonic effects in unimolecular rate theory. Vibrations and collisions of simple polyatomic systems, J. Chem. Phys. 40 2137 (1964). 

We elected to study coherent up-pumping dynamics in solution-phase metal-hexacarbonyl systems because of their strong vibrational infrared absorption cross sections, relatively simple ground-state spectra, and small (ca. 15 cm ) anharmonic overtone shifts. It was felt that these systems are ideal candidates to demonstrate that population control could be achieved for polyatomic species in solution because the excited state population... 

An alternative approach widely used in polyatomic molecule studies is based on the Golden Rule and a perturbative treatment of the anharmonic coupling (57,62). This approach is not much used for diatomic molecules. In the liquid O2 example cited above, the Hamiltonian must be expanded to 30th order or so to calculate the multiphonon emission rate. But for vibrations of polyatomic molecules, which can always find relatively low-order VER pathways for each VER step, perturbation theory is very useful. In the perturbation approach, the molecule s entire ladder of vibrational excitations is the system and the phonons are the bath. Only lower-order processes are ordinarily needed (57) because polyatomic molecules have many vibrations ranging from higher to lower frequencies and only a small number of phonons, usually one or two, are excited in each VER step. The usual practice is to expand the interaction Hamiltonian (qn, Q) in Equation (2) in powers of normal coordinates (57,62) ... 

The structure of this review is as follows. In Section 9.2, we briefly discuss methods for computing vibrational states of systems having several coupled vibrational degrees of freedom. This will also cover methods that were not yet adapted for direct use with ab initio potentials, since in our view, such extensions may be possible in the future, at least for some of the algorithms. The focus will be on methods that seem potentially applicable to large polyatomics, rather than those of great accuracy for small systems. Section 9.3 also deals with computational methods for anharmonic vibrational spectroscopy that are applicable to potential surfaces from electronic structure calculations. Our main focus will be on the Vibrational Self-Consistent Field (VSCF) approach in several variants and extensions. The performance of the available method in the present state of the art is discussed in Section 9.4. Future directions are outlined in Section 9.5. 

For polyatomic molecules, the problem is more complicated. For small-amplitude motion, certainly one can decompose the motion into independent harmonic (normal mode) motion and treat each of these as was done for a single oscillator. If the system has significant anharmonicity, then the good action-angle variables must first be found. Such techniques are available in the literature (8,21-25). 

The vibrations of a polyatomic molecule can be considered as a system of coupled anharmonic oscillators. If there are N atomic nuclei in the molecule, there will be a total of 3N degrees of freedom of motion for all the nuclear masses in the molecule. Subtracting the pure translations and rotations of the entire molecule leaves (3N-6) vibrational degrees of freedom for a non-linear molecule and (3N-5) vibrational degrees of freedom for a linear molecule. These internal degrees of freedom correspond to the number of independent normal modes of vibration. Note that in each normal mode of vibration all the atoms of the molecule vibrate with the same frequency and pass through their equilibrium positions simultaneously. 

As already discussed in Section 13.1, the multiphonon pathway for vibrational relaxation is a relatively slow relaxation process, and, particularly at low temperatures the system will use other relaxation routes where accessible. Polyatomic molecules take advantage of the existence of relatively small frequency differences, and relax by subsequent medium assisted vibrational energy transfer between molecular modes. Small molecules often find other pathways as demonstrated in Section 13.1 for the relaxation ofthe CN radical. When the concentration of impurity molecules is not too low, intermolecular eneigy transfer often competes successfully with local multiphonon relaxation. For example, when a population of CO molecules in low temperature rare gas matrices is excited to the v = 1 level, the system takes advantage of the molecular anharmonicity by undergoing an intermediate relaxation of the type... 

It is possible to obtain the results (5.13,14) by directly evaluating the thermodynamical averages of a and the force constant f (Problem 5.6.1). By starting with the equation of motion (5.4) and its solution (5.7), we have, in addition, gained some insight into the dynamics of the system. The dynamics of a polyatomic anharmonic molecule with several fundamental frequencies. .. (A). .. . is more complicated. Its vibrational motion will not only con-... 

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