Determination of the normal modes

Normal mode coordinates are linear combinations of the atomic displacements (x, yt z,, which are the components of a set of vectors Q in a 3/V-dimcnsional vector space called 

The MR of R, Tdisp(R), is a 3Nx3Nmatrix which consists of N 3x3 blocks labeled Tlm which are non-zero only when R transforms atom l into atom m, and then they are identical with the MR for an orthonormal basis ei e2 e3 in 3-D space. Since a 3x3 matrix Tlm occurs on the diagonal of rdisp(i ) only when / / /, it is a straightforward matter to 

are a set of normal coordinates, which are the components of Q(T7) referred to the new basis e(T7) in which T denotes one of the IRs and 7 denotes the component of the IR T when it has a dimension greater than unity. The particle masses do not appear in T and because they have been absorbed into the Qk by the definition of the normal coordinates. A displacement vector Q is therefore 

Here qy = M fxy, Mt is the mass of atom i, and xy is the /th component of the displacement of atom i. The procedure must be repeated for each of the IRs (labeled here by Tj N(T /) is a normalization factor. The projection needs to be carried out for a maximum of three times for each IR, but in practice this is often performed only once, if we are able to write down by inspection the other components Q(Xl) of degenerate representations. It is, in fact, common practice, instead of using eq. (5), to find the transformed basis 

Example 9.4-1 Determine the normal coordinates for the even parity modes of the ML6 molecule or complex ion with Oh symmetry. 

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The determination of the normal modes and their frequencies, however, depends upon solving the secular equation, a 3N X3N determinant. This rapidly becomes nontrivial as N increases. Methods do exist which somewhat simplify the computational problem. Thus, if the molecule has symmetry, the 3Ar X 3N determinant can be resolved into sub-determinants of lower order, each of which involves only normal frequencies of a given symmetry class. These determinants are of course easier to solve. (We will return shortly to the subject of symmetry considerations since they not only aid in the solution of the secular equation, but they permit the determination — without any other information about the molecule — of many characteristics of the normal modes, such as their number, activity in the infrared and Raman spectra, possibilities of interaction, and so on.) In addition, special techniques have been developed for facilitating the setting up and solving of the secular equation [Wilson, Decius, and Cross (245)]. Even these, however, become prohibitive for the large N encountered in complex molecules such as high polymers. 

A critical pre-requisite to using Raman and resonance Raman spectroscopy to examine the excited-state structural dynamics of nucleic acids and their components, is the determination of the normal modes of vibration for the molecule of interest. The most definitive method for determining the normal modes is exhaustive isotopic substitution, subsequent measurement of the IR and Raman spectra, and computational analysis with the FG method of Wilson, Decius, and Cross [77], Such an analysis is rarely performed presently because of the improvements in accuracy of ab initio and semi-empirical calculations. Ab initio computations have been applied to most of the nucleobases, which will be described in more detail below, resulting in relatively consistent descriptions of the normal modes for the nucleobases. 

A procedure similar to that outlined in the elementary theory of flexion allows the determination of the normal modes. However, this method is not only tedious but also has the inconvenience that some terms in the secular equation depend explicitly on the material properties, that is, on the modulus. Instead of developing a solution of Eq. (17.132) in the classical way, it is more convenient to establish a method based on comparison of the apparent and real viscoelastic moduli (11,12). The basic idea is to compare Eq. (17.132) with the Laplace transform of Eq. (17.85), which is... 

Detailed analyses of the vibrational spectra of raacromolecules, however, have provided a deeper understanding of structure and interactions in these systems (Krimm, 1960). An important advance in this direction for proteins came with the determination of the normal modes of vibration of the peptide group in A -methylacetamide (Miyazawa et al., 1958), and the characterization of several specific amide vibrations in polypeptide systems (Miyazawa, 1962, 1967). Extensive use has been made of spectra-structure correlations based on some of these amide modes, including attempts to determine secondary structure composition in proteins (see, for example, Pezolet et al., 1976 Lippert et al., 1976 Williams and Dunker, 1981 Williams, 1983). 

In the amplitude method, the motion of the ions over many oscillation periods (typically 10 periods) is integrated such that it is the modulation frequency-dependent amplitude that is observed. The most precise determinations of the normal mode frequencies are found through fitting the relevant expression for z (co) (that is Equation 10.14 for identical ions and Equation 10.16 for non-identical ions) to the measured amplitudes of the laser-cooled ion as a function of the drive frequency, as shown in... 

The determination of the normal modes of a membrane can be applied to membranes contained by any boundary. It can be extended to problems such as that of a kettledrum , in which air is trapped in the drum by a soap film membrane. The motion of the air in the drum must also be taken into account in this problem. 

Evidence exists that some of the softest normal modes can be associated with experimentally determined functional motions, and most studies apply normal mode analysis to this purpose. Owing to the veracity of the concept of the normal mode important subspace, normal mode analysis can be used in structural refinement methods to gain dynamic information that is beyond the capability of conventional refinement techniques. 

In this paper, we review progress in the experimental detection and theoretical modeling of the normal modes of vibration of carbon nanotubes. Insofar as the theoretical calculations are concerned, a carbon nanotube is assumed to be an infinitely long cylinder with a mono-layer of hexagonally ordered carbon atoms in the tube wall. A carbon nanotube is, therefore, a one-dimensional system in which the cyclic boundary condition around the tube wall, as well as the periodic structure along the tube axis, determine the degeneracies and symmetry classes of the one-dimensional vibrational branches [1-3] and the electronic energy bands[4-12]. 

One way to do so is to look at the normal mode corresponding to the imaginary frequency and determine whether the displacements that compose it tend to lead in the directions of the structures that you think the transition structure connects. The symmetry of the normal mode is also relevant in some cases (see the following example). Animating the vibrations with a chemical visualization package is often very useful. Another, more accurate way to determine what reactants and products the transition structure coimects is to perform an IRC calculation to follow the reaction path and thereby determine the reactants and products explicity this technique is discussed in Chapter 8. 

This condition on the so-called secular determinant is the basis of the vibrational problem. The roots of Eq. (59), X, are the eigenvalues of the matrix product GF, while the columns of L, the eigenvectors, determine the forms of the normal modes of vibration. These relatively abstract relations become more evident with the consideration of an example. 

Here the a, and b, are a collection of constants that are uniquely determined by the initial positions and velocities of the atoms. This means that the normal modes are not just special solutions to the equations of motion instead, they offer a useful way to characterize the motion of the atoms for all possible initial conditions. If a complete list of the normal modes is available, it can be viewed as a complete description of the vibrations of the atoms being considered. 

Problem 7-13. Determine the symmetry species of the normal modes of vibration of the cyclopropilium cation, CsH. This molecule is planar and has Dsh symmetry. 

Problem 7-14. Determine the symmetry species of the normal modes of cyclopropane. 

There are two totally symmetric ( ,) normal modes and one b2 normal mode. (The convention is to use lowercase letters for the symmetry species of the normal modes.) The symmetry species of the normal modes have been found without solving the vibrational secular equation. Moreover, since there is only one b2 normal mode, the form of this vibration must be determined from symmetry considerations together with the requirement that the vibration have no translational or rotational energy associated with it. Thus (Fig 6.1), any bent XYX molecule has a b2 normal mode with the X atoms vibrating along the X—Y bonds and the Y atom vibrating in the plane of the molecule and perpendicular to the symmetry axis. On the other hand, there are two ax symmetry coordinates and the two ax normal vibrations are linear combinations of the ax symmetry coordinates, where the coefficients are dependent on the nuclear masses and the force constants. Thus the angles between the displacement vectors of the X atoms and the X—Y bonds for the ax modes of a bent XYX molecule vary from molecule to molecule. 

Each such vibration (6.32) is called a normal mode of vibration. For each normal mode, the vibrational amplitude Aim of each atomic coordinate is constant, but the amplitudes for different coordinates are, in general, different. The nature of the normal modes depends on the molecular geometry, the nuclear masses, and the values of the force constants ujk. The eigenvalues m of U determine the vibrational frequencies the eigenvectors of U determine the relative amplitudes of the vibrations of the q, s in each normal mode, since Ajm / A im = Ijm/L- For H2° here are 9-6-3 normal modes, and the solution of (6.17) and (6.18) yields the vibrational modes shown in Fig. 6.1. For some molecules, two or more normal modes have the same vibrational frequency (corresponding to two or more equal roots of the secular equation) such modes are called degenerate. For example, a linear triatomic molecule has four normal modes, two of which have the same frequency. See Fig. 6.2. The general classical-mechanical solution (6.30) is an arbitrary superposition of the normal modes. 

The two characteristic features of normal modes of vibration that have been stated and discussed above lead directly to a simple and straightforward method of determining how many of the normal modes of vibration of any molecule will belong to each of the irreducible representations of the point group of the molecule. This information may be obtained entirely from knowledge of the molecular symmetry and does not require any knowledge, or by itself provide any information, concerning the frequencies or detailed forms of the normal modes. 

Figure 10.3 The set of 3n = 12 Cartesian displacement vectors used in determining the reducible representation spanning the irreducible representation of the normal modes of COj".

It has been shown142 that, for long-chain polymers in an ordered conformation, the calculation of the normal modes is reduced by symmetry arguments to the determination of the vibrations of the repeat unit. The vibrations in a chemical unit are related to those in adjacent units by the secular equation through a phase angle 6, so that the form of the secular equation used in the previous calculations29-30 was... 

When XeF4 was first prepared it was thought to be highly symmetrical, but it was not known whether it was a tetrahedral or a square-planar molecule. The infra-red absorption spectrum of XeF4 consists of three fundamental bands and the vibrational Raman spectrum also has three bands. Determine the symmetry of the normal modes of a... 

When a sound source is turned on in an enclosure, it excites one or more of the normal modes of the room. When the source is turned off, the modes continue to resonate their stored energy, each decaying at a separate rate determined by the mode s damping constant, which depends on the absorption of the room. This is entirely analogous to an electrical circuit containing many parallel resonances [Beranek, 1986], Each mode has a resonance curve associated with it, whose inxquality factor (Q) depends on the damping constant. 

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