Transformation normal<->local mode

Levine, R. D., and Kinsey, J. L. (1986), Anharmonic Local-Mode-Normal-Mode Transformations An Algebraic Treatment, J. Phys. Chem. 90, 3653. 

At this point it is useful to define creation (at) and annihilation (a) operators, which are analogous to angular momentum raising and lowering operators. These at, a operators profoundly simplify the algebra needed to set up the polyad Heff matrices, to apply some of the dynamics diagnostics discussed in Sections 9.1.4 and 9.1.7, and to transform between basis sets (e.g., between normal and local modes). They also provide a link between the quantum mechanical Heff model, which is expressed in terms of at, a operators and adjustable molecular constants (evaluated by least squares fits of spectra), and a reduced-dimension classical mechanical HeS model. 

Transformation Between Local and Normal Mode Limits... 

Transformation between 4-Parameter Forms of the Normal and Local Mode Basis Sets... 

The local mode basis functions and HfJoCAL may be transformed to the normal mode basis set by exploiting the properties of the creation and annihilation... 

The transformation from the normal mode usua)jv to the local mode vrvl)l basis set is accomplished by replacing the normal mode at, a operators by the corresponding local mode operators... 

It is straightforward to transform the normal mode basis states, i>s, vQ)n into local mode basis states, vr,vr)l (or trice versa). For example, the 0,4)at basis state, which corresponds to vs = 0, va = 4, is expressed in terms of local mode basis states,... 

Transformation between 6-Parameter Forms of the Normal Mode and Local Mode Heff... 

In Section 9.4.12.4 the simplest possible local mode HlqCAL, expressed in terms of four independently adjustable parameters (the Morse De and a parameters, and two 1 1 kinetic and potential energy coupling parameters, Grr and km,), is transformed to the simplest possible normal mode H )oRMAL, which is also expressed in terms of four independent parameters. However, the interrelationships between parameters, based on the 1 1 coupled local Morse oscillator model, result in only 3 independent fit parameters. This paradox is resolved when one realizes that the 4 parameter local-Morse model generates the Darling-Dennison 2 2 coupling term in the normal mode model. However, the full effects of this (A ssaa/16hc)[(at + as)2(a+ + aa)2] coupling term are not taken into account in the local mode model. 

The transformation between the standard spectroscopic normal mode model, expressed in terms of 6 independently adjustable fit parameters, to an equivalent 6 parameter local mode model, is considered here. The standard H oRMal model is... 

Since this transformation to normal coordinates is invertible, one can readily determine the functional dependencies of the terms in Eq. (1) using either the normal or internal coordinates. Interestingly, in our study of vibrational states of the well-known local mode molecule H20 and its deuterated analogs we found only minor differences between the results of CVPT in the internal and normal mode representations (46). The normal mode calculations, however, required significantly less computer time to run, since many terms in the Hamiltonian are constrained to zero by symmetry. For this reason we chose to use the normal mode coordinates for all subsequent studies. 

A normal mode is composed of the movement of many or even all atoms of a molecule, which is difficult to visualize. Because of this chemists try to simplify the description of a normal mode by focusing on the motions of just few atoms that seem to dominate the normal mode. This requires an appropriate measure that determines which atomic motion is dominant. Attempts in this direction have been made and it is common practice now to associate certain normal modes of a molecule with chemically interesting fragment modes even though this simplification is usually not justified. Hence the basic problem of vibrational spectroscopy is the transformation of the delocalized normal modes, which are difficult to visualize, to chemically more appealing localized modes that can be associated with particular fragments of a molecule. 

An analogy to molecular orbital (MO) theory may help to clarify further what is needed. Chemists prefer to discuss chemical problems in terms of localized MOs rather than in terms of (canonical) delocalized MOs resulting from Hartree-Fock (HF) based quantum chemical calculations. The localized MOs are obtained from the delocalized ones by a transformation ("localization"), which in most cases yields MOs directly related to the bonds of a molecule. The same should be true with regard to localized modes associated with a particular internal coordinate q. The question is only How can we transform from delocalized normal modes to localized internal modes To answer this question we will first summarize the basic theory of vibrational spectroscopy. 

This simple result, compared with the transformation (3.35), justifies the introduction of symmetric and antisymmetric combinations of local modes with respect to the aforementioned reflection operation. It is, of course, possible to write, in place of Eq. (3.26) or (3.28), the energy spectrum of the normal limit expressed as a function of the quantum numbers (for identical oscillators, N. = N2 = N) ... 

By considering two interacting normal vibrations Qj and Qj) to be related to two local modes qi, q2) (the normal modes of a smaller part of the system) by a proper unitary transformation [147], it follows that... 

The localized mode procedure [14] consists in performing a unitary transformation on a subset of k normal modes (gsub) belonging to a specific band in order to obtain localized modes. 

The diagonal elements of this matrix, / , are referred to as the intensities of the Zth localized modes, while the off-diagonal values, Ii , are the intensity coupling terms. The intensity coupling matrix I can be used to understand the shapes of the band (i.e., how the intensity of a band is spread over all the normal modes, see also below). Indeed, the intensity associated with each vibrational normal mode (Ip) of one band is related to the intensity coupling matrix I and to the unitary transformation U ... 

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

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