Perturbation theory polyatomic molecules

Treating the full internal nuclear-motion dynamics of a polyatomic molecule is complicated. It is conventional to examine the rotational movement of a hypothetical "rigid" molecule as well as the vibrational motion of a non-rotating molecule, and to then treat the rotation-vibration couplings using perturbation theory. 

Btiilding on atomic studies using even-tempered basis sets, universal basis sets and systematic sequences of even-tempered basis sets, recent work has shown that molecular basis sets can be systematically developed until the error associated with basis set truncation is less that some required tolerance. The approach has been applied first to diatomic molecules within the Hartree-Fock formalism[12] [13] [14] [15] [16] [17] where finite difference[18] [19] [20] [21] and finite element[22] [23] [24] [25] calculations provide benchmarks against which the results of finite basis set studies can be measured and then to polyatomic molecules and in calculations which take account of electron correlation effects by means of second order perturbation theory. The basis sets employed in these calculations are even-tempered and distributed, that is they contain functions centred not only on the atomic nuclei but also on the midpoints of the line segments between these nuclei and at other points. Functions centred on the bond centres were found to be very effective in approaching the Hartree-Fock limit but somewhat less effective in recovering correlation effects. 

For polyatomic molecules with 3 N-6 internal degrees of freedom it has been generally agreed (JJ, J7)thatit would be possible to satisfy two conditions of the type established above and therefore that potential energy surfaces could cross even if they had the same symmetry. For example, a linear triatomic has two internal degrees of freedom and hence surfaces of the same symmetry could be expected to cross at a point. This supposition was called into question byNaqvi (38) who based his conclusion on an extension of the diatomic proof (which was based upon perturbation theory) given by himself and Byers-Brown (34). However, proof... 

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. 

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) ... 

In this chapter we review some of the developments that have been made over the past fifteen years with regard to the calculation of vibrational contributions to linear and nonlinear (NLO) optical properties. Despite a number of advances it is important to recognize that more are needed since there is still no fully satisfactory general treatment for either resonant or non-resonant NLO processes in polyatomic molecules. Two major intertwining approaches to practical computations that include electrical and mechanical anharmonicity have emerged. The older approach is from the viewpoint of ordinary sum-over-states perturbation theory and it is presented in Section 1. The other approach, discussed in Section 2, is from what may be called the nuclear relaxation/curvature point of view. Even though there is 101... 

The second common type of operationally defined structure is the so-called substitution or rt structure.10 The structural parameter is said to be an rs parameter whenever it has been obtained from Cartesian coordinates calculated from changes in moments of inertia that occur on isotopic substitution at the atoms involved by using Kraitchman s equations.9 In contrast to r0 structures, rs structures are very nearly isotopically consistent. Nonetheless, isotope effects can cause difficulties as discussed by Schwendeman. Watson12 has recently shown that to first-order in perturbation theory a moment of inertia calculated entirely from substitution coordinates is approximately the average of the effective and equilibrium moments of inertia. However, this relation does not extend to the structural parameters themselves, except for a diatomic molecule or a very few special cases of polyatomics. In fact, one drawback of rs structures is their lack of a well-defined relation to other types of structural parameters in spite of the well-defined way in which they are determined. It is occasionally stated in the literature that r, parameters approximate re parameters, but this cannot be true in general. For example, for a linear molecule Watson12 has shown that to first order ... 

Canonical Van Vleck Perturbation Theory and Its Application to Studies of Highly Vibrationally Excited States of Polyatomic Molecules... 

At higher levels of excitation anharmonicity has to be included to obtain accurate energy levels. Perturbation theory has been used to derive the following expression, often called a Dunham expansion (Hirst, 1985), for polyatomic anharmonic vibrational energy levels, which is similar to the Morse energy level expression Eq. (2.59), for a diatomic molecule ... 

The model described here is based on a series of approximations, but it has nonetheless been found to be quantitatively successful in fitting the observed vibrational levels of a considerable number of small molecules [6,7,9,15,16,17,18]. The most obvious approximations concern the use of perturbation theory and the implicit use of harmonic basis functions in deriving the vibrational hamiltonian matrices, and the neglect of matrix elements coupling different values of the total vibrational excitation quantum number V. However when the same approximations are applied to a diatomic Morse oscillator they yield the exact solution, as observed by Mills and Roblette [7], and some similar cancellation of approximations holds for the polyatomic examples. Another more serious problem concerns the mixing with other vibrations this has been discussed here only in connection with Fermi resonance, but its importance in other ways is only just being investigated. 

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