Theory of normal modes

Suppose we have at our disposal an analytical expression for V(R) (e.g., the force field), where R denotes the vector of the Cartesian coordinates of the N atoms of the system (it has 3N components). Let us assume (Fig. 7.7) that the function V(R) has been minimized in the configurational space, starting from an initial position Ri and going downhill until a minimum position Rq has been reached, the Rq corresponding to one of many minima the V function may possess (we will call the minimum the closest to the Ri point in the configurational space). All the points Ri of the configurational space that lead to Rq represent the basin of the attractor Ro. 

From this time on, all other basins of the function V(R) have disappeared from the theory - only motion in the neighbourhood of Rq is to be considered. If someone is aiming to apply harmonic approximation and to consider small displacements from Rq (as we do), then it is a good idea to write down the Taylor expansion of V about Rq [hereafter instead of the symbols X, Y, Z, X2, Y2, Z2. for the atomic Cartesian coordinates we will use a slightly more uniform notation  

In Rq all the first derivatives vanish. According to the harmonic approximation, the higher order terms denoted as H-- are neglected. In effect we have 

The Newton equations of motion for all the atoms of the system can be written in matrix form as (Jf means the second derivative with respect to time t) 

When we insert the proposed solution (7.8) in (7.7), we immediatefy obtain, that CO and L have to satisfy the following equation 

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For certain quantitative descriptions the introduction of massweighting of the RP may be an urgent problem (the mass-weighting, being formally analogous to that in the Wilson theory of normal modes at minima of a PES (Ref.23), admits kinetic effects through a back-... 

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

Marcus attempted to calculate the minimum energy reaction coordinate or reaction trajectory needed for electron transfer to occur. The reaction coordinate includes contributions from all of the trapping vibrations of the system including the solvent and is not simply the normal coordinate illustrated in Figure 1. In general, the reaction coordinate is a complex function of the coordinates of the series of normal modes that are involved in electron trapping. In this approach to the theory of electron transfer the rate constant for outer-sphere electron transfer is given by equation (18). 

The C18 phases have emerged as the overwhelming first choice when developing new methods because, as noted in the first section of this chapter, developing a reverse-phase separation is straightforward. As a result, many separation successes are easily attained on a reverse-phase system. Unfortunately, this has led many to forget about using the normal-phase mode. In fact, because the theory of normal phase on silica gel predates the introduc-... 

The simplest way to combine electronic stnicture calculations with nuclear dynamics is to use harmonic analysis to estimate both vibrational averaging effects on physico-chemical observables and reaction rates in terms of conventional transition state theory, possibly extended to incorporate tunneling corrections. This requires, at least, the knowledge of the structures, energetics, and harmonic force fields of the relevant stationary points (i.e. energy minima and first order saddle points connecting pairs of minima). Small anq)litude vibrations around stationary points are expressed in terms of normal modes Q, which are linearly related to cartesian coordinates x... 

As early as 1939 Slater proposed an alternative approach that related the rate of a unimolecular reaction to the vibrations of the reacting molecule [55]. This theory was developed over the succeeding years, and is explained in Slater s famous 1959 book [56]. The theory is based on the familiar description of molecular vibrations in terms of normal modes [57]. If the vibrations of a molecule are assumed to be harmonic, they can be reduced to a set of independent harmonic... 

During the past few years the various theories of unimolecular reactions have been applied to a considerable number of reactions, and the main conclusions may be summarized as follows. The rrk theories are the easiest to apply, since little detailed information is required about the structure and vibrations of the reactant molecules. The variation of the first-order rate coefficient k with pressure can always be interpreted reasonably satisfactorily by the rrk theories, except that the value of s that must be taken to get satisfactory agreement is usually only about half of the number of normal modes of vibration in the molecule. This may be in part due to the fact that there is free flow of energy only between a limited number... 

Diamond, R. On the use of normal modes in thermal parameter refinement theory and application to the bovine pancreatic trypsin inhibitor. Ada Cryst. A46, 425-435 (1990). 

If it is assumed that (a) the force constants are the same in both ground and the excited states, (b) the potential surfaces are harmonic, (c) the transition dipole moment, n, is constant and (d) the normal coordinates are not mixed in the excited state, then the overlaps, << in absorption and ( < (t)> in Raman, have simple forms (Sections III.A and III.B). None of these assumptions are requirements of the time-dependent theory. Assumption (a) introduces at most an error of 10% if the distortions are very large [25]. When the vibrational frequencies in the excited state are not known, this assumption must be used but does not introduce serious error. The harmonic approximation is used because the number of parameters to be fitted is thereby reduced and because it allows the simple expression for the overlap to be used. Because there is no distinct evidence of normal mode mixing in the molecules studied in this chapter, all of the fits were done without including... 

Gilbert, F. Dziewonski, A. M. 1975. An application of normal mode theory to the retrieval of structural parameters and source mechanism from seismic spectra. Philosophical Transactions of the Royal Society of London, Series A, 278, 187-269. 

An isolated n-atom molecule has 3n degrees of freedom and in—6 vibration degrees of freedom. The collective motions of atoms, moving with the same frequency and which in phase with all other atoms, give rise to normal modes of vibration. In principle, the determination of the form of normal modes for any molecule requires the solution of equation of motion appropriate to the n-symmetry. Methods of group theory are important in deriving the symmetry properties of the normal modes. With the aid of the character tables for point groups and the symmetry properties of the normal modes, the selection rules for Raman and IR activity can be derived. For a molecule with a center of symmetry, e.g. AXe, octahedral molecule, a non-Raman active mode is also IR active, whereas for the BX4 tetrahedral molecule, some modes are simultaneously IR and Raman active. 

Group Theory and Normal Modes by A. Nussbaum, Am. J. Phys., 36,529 (1968) provides an interesting tutorial introduction to the use of symmetry to reduce the computational overhead associated with a given calculation of the normal modes. 

Chemistry students leam the theory and technique of normal mode analysis of molecules in introductory courses in quantum chemistry or spectroscopy. Perhaps the simplest way to describe this approach is to start with a Taylor series expansion of the potential about a stationary point, i.e. where the first derivatives vanish. In this case the lowest order... 

Experimental studies have had an enormous impact on the development of unimolecular rate theory. A set of classical thermal unimolecular dissociation reactions by Rabinovitch and co-workers [6-10], and chemical activation experiments by Rabinovitch and others [11-14], illustrated that the separability and symmetry of normal modes assumed by Slater theory is inconsistent with experiments. Eor many molecules and experimental conditions, RRKM theory is a substantially more accurate model for the unimolecular rate constant. Chemical activation experiments at high pressures [15,16] also provided information regarding the rate of vibrational energy flow within molecules. Experiments [17,18] for which molecules are vibrationally excited by overtone excitation of a local mode (e.g. C-H or O-H bond) gave results consistent with the chemical activation experiments and in overall good agreement with RRKM theory [19]. 

The discussion that follows is necessarily selective and is pitched at a simplistic level. Although in this section we derive the number of vibrational modes for some simple molecules, for more complicated species it is necessary to use character tables. The reading list at the end of the chapter gives sources of detailed discussions of the relationship between group theory and normal modes of vibration. 

There has been a great deal of work [62, 63] investigating how one can use perturbation theory to obtain an effective Hamiltonian like the spectroscopic Hamiltonian, starting from a given PES. It is found that one can readily obtain an effective Hamiltonian in terms of normal mode quantum numbers and coupling. 

In order to stndy the short time vibrational energy transfer behavior of a vibra-tionally excited system, we employ a non-Markovian time-dependent perturbation theory [83]. Onr approach builds on the successful application of Markovian time-dependent pertnrbation theory by Leitner and coworkers to explore heat flow in proteins and glasses, and Tokmakoff, Payer, and others, in modeling vibrational population relaxation of selected modes in larger molecules. In a separate chapter in this volnme, Leitner provides an overview of the development of normal mode-based methods, snch as the one employed here, for the study of energy flow in solids and larger molecnlar systems. 

But a new difficulty arose from the apparent insufficiency of the collisions to provide energy at the required absolute rate. The way out was provided by the now very natural idea that multiple internal degrees of freedom can be drawn upon to contribute to the activation process. The theory of reaction rates now becomes correlated with the study of normal modes of vibration of complex molecules. Fresh questions about the dependence of transformation probability on energy excess or energy distribution arise and the subject enters its specialized phase— where there are still some unsolved problems. 

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