Normal Mode of Molecular Vibrations

The normal mode of molecular vibrations ensures that a molecule vibrates at its equilibrium position without shifting its center of gravity. Each type of molecule has a defined number of vibration modes and each mode has its own frequency. We can examine the... 

The determination of the form and of the frequency of normal modes of molecular vibrations is beyond the scope of this present book. The reader interested in this topic is referred to the relevant books listed in the bibliography. 

Whole-chain motions, A much more elaborate model is required for any further interpretation of overall chain motions. Its derivation is well beyond the scope of this chapter, but may be followed in [11]. The authors of this review concentrate on the Doi-Edwards reptation model [12] for tangled polymer chains, either in relatively concentrated solutions or in mobile bulk states. Such a chain portion is represented in a simplified way in Figure 4.6. There are essentially four levels of angular motion available to the chain. The normal modes of molecular vibration constitute the most rapid motions, but their angular extent, a few degrees, is insufficient to contribute significantly and directly to the correlation function. 

The quantities Rx, Ry, and Rz approximate instantaneous rotational motions around the three orthogonal principal axes of the water molecule, and Qi, Q2, and Q3 approximately describe the three normal modes of molecular vibrations usually referred to as symmetric stretch, bend, and asymmetric stretch, respectively (schematically illustrated on the inserts in Fig. 19). 

The value of the frequency parameter controlling the temperature of nuclei should enable an efficient coupling of the thermostat to the molecular vibrations. For chain-thermostats the frequency for each item in the chain should be specified. In general, a larger number of thermostats in a chain makes it possible to couple to the normal modes of different vibrational frequencies. In the extreme case, each normal mode can be separately thermostated in the MD terminology such a procedure is called massive thermostating.33... 

Assuming that a reasonable force field is known, the solution of the above equations to obtain the vibrational frequencies of water is not difficult However, in more complicated molecules it becomes very rapidly a formidable one. If there are N atoms in the molecule, there are 3N total degrees of freedom and 3N-6 for the vibrational frequencies. The molecular symmetry can often aid in simplifying the calculations, although in large molecules there may be no true symmetry. In some cases the notion of local symmetry can be introduced to simplify the calculation of vibrational frequencies and the corresponding forms of the normal modes of vibration. 

For a given normal mode of frequency v we may write the polarizability as the sum of the polarizability in the equilibrium position Mq and the induced polarizability due to molecular vibrations. 

This model has the advantage that the atomic polar tensor elements can be determined at the equilibrium geometry from a single molecular orbital calculation. Coupled with a set of trajectories (3R /3G)o obtained from a normal coordinate analysis, the IR and VCD intensities of all the normal modes of a molecule can be obtained in one calculation. In contrast, the other MO models require a separate MO calculation for each normal mode, since the (3p,/3G)o contributions for each unit are determined by finite displacement of the molecule along each normal coordinate. Both the APT and FPC models are useful in readily assessing how changes in geometry or refinements in the vibrational force field affect the frequencies and intensities of all the vibrational modes of a molecule. 

Every example of a vibration we have introduced so far has dealt with a localized set of atoms, either as a gas-phase molecule or a molecule adsorbed on a surface. Hopefully, you have come to appreciate from the earlier chapters that one of the strengths of plane-wave DFT calculations is that they apply in a natural way to spatially extended materials such as bulk solids. The vibrational states that characterize bulk materials are called phonons. Like the normal modes of localized systems, phonons can be thought of as special solutions to the classical description of a vibrating set of atoms that can be used in linear combinations with other phonons to describe the vibrations resulting from any possible initial state of the atoms. Unlike normal modes in molecules, phonons are spatially delocalized and involve simultaneous vibrations in an infinite collection of atoms with well-defined spatial periodicity. While a molecule s normal modes are defined by a discrete set of vibrations, the phonons of a material are defined by a continuous spectrum of phonons with a continuous range of frequencies. A central quantity of interest when describing phonons is the number of phonons with a specified vibrational frequency, that is, the vibrational density of states. Just as molecular vibrations play a central role in describing molecular structure and properties, the phonon density of states is central to many physical properties of solids. This topic is covered in essentially all textbooks on solid-state physics—some of which are listed at the end of the chapter. 

In order to apply group-theoretical descriptions of symmetry, it is necessary to determine what restrictions the symmetry of an atom or molecule imposes on its physical properties. For example, how are the symmetries of normal modes of vibration of a molecule related to, and derivable from, the full molecular symmetry How are the shapes of electronic wave functions of atoms and molecules related to, and derivable from, the symmetry of the nuclear framework ... 

Projection operators are a technique for constructing linear combinations of basis functions that transform according to irreducible representations of a group. Projection operators can be used to form molecular orbitals from a basis set of atomic orbitals, or to form normal modes of vibration from a basis of displacement vectors. With projection operators we can revisit a number of topics considered previously but which can now be treated in a uniform way. 

Sidebar 10.3 outlines the useful analogy to normal-mode analysis of molecular vibrations, where the null modes correspond to overall translations or rotations of the coordinate system that lead to spurious alterations of coordinate values, but no real internal changes of interatomic distances. For this reason, the internal metric M( of (10.29) is the starting point for analyzing intrinsic state-related (as opposed to size-related) aspects of a given physical system of interest. 

In view of the Hessian character (10.20) of the thermodynamic metric matrix M(c+2), the eigenvalue problem for M(c+2) [(10.23)] can be usefully analogized with normal-mode analysis of molecular vibrations [E. B. Wilson, Jr, J. C. Decius, and P. C. Cross. Molecular Vibrations (McGraw-Hill, New York, 1955)]. The latter theory starts from a similar Hessian-type matrix, based on second derivatives of the mechanical potential energy Vpot (cf. Sidebar 2.8) rather than the thermodynamic internal energy U. 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

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. 

Up to this point the treatment of molecular vibrations has been purely classical. Quantum mechanics does not allow the specification of the exact path taken by a particle hence the picture of each nucleus executing the appropriate simple harmonic motion for a given normal mode should not be taken literally. On the other hand, nuclei are relatively heavy (compared to electrons), so that the classical picture of the motion is not totally lacking in validity. 

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. 

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