Local mode approximation

The spectrum of fundamental vibrations of acetylene, in the local-mode approximation (5.16) is shown in Figure 5.4. For acetylene, C2H2, bonds 1 and 2 are identical and thus one must have Nl = N2, A = A2. 

The general conclusions probably are independent of the details. The two higher-frequency peaks, derived from (o and can be reasonably well described in a local mode approximation, but the lowest mode, derived from (Oy, is so mixed with the acoustical modes that it should be treated together with the acoustical modes in a more complete calculation. 

The model of the anharmonic oscillator or local mode approximation more closely follows the actual condition for molecular absorption than that of the harmonic oscillator. Figure 1.15 illustrates the differences between the ideal harmonic oscillator case that has been discussed in detail for this chapter vs. the anharmonic oscillator model (better representing the actual condition of molecules). Unlike the ideal model illustrated by the harmonic oscillator expression, the anharmonic oscillator involves considerations such that, when two atoms are in close proximity (minimum distance), they repel each other when two atoms are separated by too large a distance, the bond breaks. The anharmonic oscillator potential energy curve is most useful to predict behavior of real molecules. 

Consider the step-profile tapers of Fig. 28-1, whose ends have fixed core radii Po and p but otherwise are of arbitrary, slowly decreasing radius. We are primarily interested in radiation loss when a local mode approximates the fundamental mode on the taper, but the analysis is easily modified to describe losses from higher-order local modes. This also provides criteria for the accuracy of the local-mode description. 

Starting from the normal mode approximation, one can introduce anharmonicity in different ways. Anharmonic perturbation theory [206] and local mode models [204] may be useful in some cases, where anharmonic effects are small or mostly diagonal. Vibrational self-consistent-field and configuration-interaction treatments [207, 208] can also be powerful and offer a hierarchy of approximation levels. Even more rigorous multidimensional treatments include variational calculations [209], diffusion quantum Monte Carlo, and time-dependent Hartree approaches [210]. 

Herein we present calculations [6] for liquid H20 that are similar in spirit but different in detail from those of Buch [71, 110] and Torii [97]. The MD simulations are of the SPC/E model [135]. Local-mode anharmonic frequencies are generated from our most recent map developed for the H0D/D20 system [98], as are our transition dipoles. The relatively small intramolecular coupling fluctuates with molecular environment, and is determined by a separate map parameterized from ab initio calculations on clusters. The form of the intermolecular couplings is transition dipole, which is tested and parameterized from additional ab initio calculations. The effects of motional narrowing are taken into account approximately with the TAA [99]. 

Specifically, the various papers working within both the adiabatic and the Condon approximations, and using the (frequent) assumption of harmonic vibrations, can still differ in how many and what type (optical, acoustic, or local) modes they consider and in how they approximate the four separate integrals on the right-hand side of Eq. (40). And the choice of modes applies to both the ground and the excited states (so does the choice of electronic wave functions, but this choice is implicit in the evaluation of the electronic integrals.) It is this choice regarding the two states that was emphasized in connection with Fig. 15 (Section 10b). It can be seen that even within the stated approximations (adiabatic, Condon, harmonic) there is an appreciable number of permutations and combinations. 

As an example, we consider the multiphonon relaxation of a local mode caused by an anharmonic interaction with a narrow phonon band. We suppose that the mode is localized on an atom and take into account two diagonal elements of the Green function which stand for the contribution of two nearest atoms of the lattice to the interaction the non-diagonal elements are usually much smaller [16] and approximate the density of states of the phonon band by the parabolic distribution... 

Fig. 10.5. Measured rotational state distributions of OH following the dissociation of the three lowest bending states of H2O (open circles). In addition to the bending quanta H20(X) also contains 4 respectively 3 quanta of OH stretching excitation. The local mode nomenclature nm k) is explained in Section 13.2. The total angular momentum is zero in all cases. The filled circles represent the harmonic oscillator approximation defined in the text. Reproduced from Schinke, Vander Wal, Scott, and Crim (1991).

H2O in its electronic ground state is best described by a local mode expansion (Child and Halonen 1984 Child 1985 Halonen 1989). For the purpose of this chapter it suffices to consider a simple two-dimensional model in which the bending angle is frozen at its equilibrium value 104° and the oxygen atom is assumed to be infinitely heavy. For an exact three-dimensional treatment see Bacic, Watt, and Light (1988), for example. The approximate two-dimensional Hamiltonian reads... 

The early work of Miyazawa [109] described the normal modes of vibration for a polypeptide backbone in terms of the normal modes of 77-methyl acetamide (NMA). This established the basis for understanding these complex spectra in terms of normal coordinate analysis (NCA) f 7/0]. A detailed review of the development of this methodology is given by Krimm [7/7]. The foundation for the use of NCA resides in the useful approximation that the atomic displacements in many of the vibrational modes of a large molecule are concentrated in the motions of atoms in small chemical groups, and that these localized modes are transferrable to other molecules. This concept of transferability is the basic principle for the use of spectroscopic techniques for studying problems associated with peptide structure [777],... 

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