Anharmonicity, spectroscopic perturbation theory

Purely quantum studies of the fully coupled anharmonic (and sometimes nonrigid) rovibrational state densities have also been obtained with a variety of methods. The simplest to implement are spectroscopic perturbation theory based studies [121, 122, 124]. Related semiclassical perturbation treatments have been described by Miller and coworkers [172-174]. Vibrational self-consistent field (SCF) plus configuration interaction (Cl) calculations [175, 176] provide another useful alternative, for which interesting illustrative results have been presented by Christoffel and Bowman for the H + CO2 reaction [123] and by Isaacson for the H2 + OH reaction [121]. The MULTIMODE code provides a general procedure for implementing such SCF-CI calculations [177]. Numerous studies of the state densities for triatomic molecules have also been presented. 

In practice, the result of the perturbation treatment may be expressed as a series of formulae for the spectroscopic constants, i.e. the coefficients in the transformed or effective hamiltonian, in terms of the parameters appearing in the original hamiltonian, i.e. the wavenumbers tor, the anharmonic force constants , the moments of inertia Ia, their derivatives eft , and the zeta constants These formulae are analogous to equations (23)—(27) for a diatomic molecule. They are too numerous and too complicated to quote all of them here, but the various spectroscopic constants are listed in Table 3, with their approximate relative orders of magnitude, an indication of which parameters occur in the formula for each spectroscopic constant, and a reference to an appropriate source for the perturbation theory formula for that constant. 

Another manifestation of vibrational anharmonicity occurs in Fermi resonance [8]. When two vibrational states of the same overall symmetry are accidentally degenerate, they can become strongly mixed by the anharmonic coupling terms between them. Their energies may be repelled considerably (in the language of degenerate perturbation theory), and the intensities of the spectroscopic transitions to these levels may be redistributed by the mixing. 

The basic theory of vibration-rotation of polyatomic molecules has been worked out for many years, including the form of the vibration-rotation Hamiltonian, the quantum mechanical solution of certain anharmonic oscillators," and the relationship, in terms of perturbation theory,between experimental spectroscopic data and higher than quadratic terms in the potential function. Still, prediction of harmonic... 

Since in computations of electronic structure theory derivatives of the total energy of molecular systems with respect to geometrical coordinates are best obtained in Cartesian coordinates, transformation of these derivatives to coordinate systems of more spectroscopic use, e.g., internal or normal coordinates, needs to be discussed. Furthermore, it is noted that, due to the lack of analytic higher-derivative methods at correlated levels of computational quantum chemistry, in practice higher-order force constants are usually determined first in a convenient set of internal coordinates. Then, in order to employ varia-tional or perturbational approaches utilizing anharmonic force fields they may n6ed to be expressed in normal coordinates, never known a priori to the calculation. It is thus clear that these usually nonlinear and somewhat complicated transformation equations occupy a central role in anharmonic force field studies. 

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