Vibrational modes normal coordinates

Once the Hamiltonian has been transformed into normal form, the quantization of the nonreactive DOFs is straightforward. While complications can occur, the present example is free of the worst of these. The vibrational modes normal to the reactive coordinate are not in resonance. Consequently, the quantization is straightforward and accomplished by quantizing the classical action variables... 

Here the sum runs over all 3N-6 normal modes of vibration with normal coordinates Q. Furthermore it is assumed that only one state b is energetically close to state f, so the perturbation expansion can be restricted to a single term. 

The Hamiltonian will now be summarized for one overtone excited CH oscillator interacting with the N — 1 ring modes. Normal coordinates for the ring Q2,. . . , Q.v and for the overtone excited oscillator Q, are defined by uncoupling the overtone excited oscillator from the ring. The final form for the Hamiltonian then contains terms (both potential and kinetic) which couple the CH stretch mode to the ring modes. The derivation of the vibrational Hamiltonian was presented in Section II.C of Benzene I (103), and we will only summarize the final result here. The vibrational Hamiltonian may be partitioned into the terms... 

In the normal vibration whose normal coordinate is < , J5, JB21 B3, gives the ratio of the displacements. From the previous calculation, it is obvious that B, B21 3i = 1 -0 -1. Similarly, 1 ,2 22 32= 1 - -2Vm,/ 1 gives the ratio of the displacements in the normal vibration whose normal coordinate is < 2 Thus the mode of a normal vibration can be drawn if the normal coordinate is translated into a set of rectangular coordinates, as is shown above. 

The number of degrees of freedom equals the number of normal modes of vibration. The normal modes, also called fundamental modes, are a set of harmonic motions, each independent of the others and each having a distinct frequency. It is possible for two or more of the frequencies to be identical, and the corresponding modes are said to be degenerate. However, the total number of modes in the individual degenerate states are counted separately and still total 3A — 6 for nonlinear and 3 A — 5 for linear molecules. A set of coordinates can be defined, each of which gives the displacement in one of the normal modes of vibration. The normal coordinates can be expressed as combinations of the x-, y-, and z-coordinates of the individual nuclei. 

Polyatomic molecules vibrate in a very complicated way, but, expressed in temis of their normal coordinates, atoms or groups of atoms vibrate sinusoidally in phase, with the same frequency. Each mode of motion functions as an independent hamionic oscillator and, provided certain selection rules are satisfied, contributes a band to the vibrational spectr um. There will be at least as many bands as there are degrees of freedom, but the frequencies of the normal coordinates will dominate the vibrational spectrum for simple molecules. An example is water, which has a pair of infrared absorption maxima centered at about 3780 cm and a single peak at about 1580 cm (nist webbook). 

Normal modes of vibration, with their corresponding normal coordinates, are very satisfactory in describing the low-lying vibrational levels, usually those with u = 1 or 2, which can be investigated by traditional infrared absorption or Raman spectroscopy. For certain types of vibration, particularly stretching vibrations involving more than one symmetrically equivalent terminal atom, this description becomes less satisfactory as v increases. 

This is clearly a matrix eigenvalue problem the eigenvalues determine tJie vibrational frequencies and the eigenvectors are the normal modes of vibration. Typical output is shown in Figure 14.10, with the mass-weighted normal coordinates expressed as Unear combinations of mass-weighted Cartesian displacements making up the bottom six Unes. 

In general, the first excited state (i.e. the final state for a fundamental transition) is described by a wavefunction pt which has the same symmetry as the normal coordinate (Appendix). The normal coordinate is a mathematical description of the normal mode of vibration. 

Each normal mode of vibration can be described by a normal coordinate Qi which is a linear combination of nuclear displacement coordinates of the molecule. For the symmetric stretching vibration vi of C02, the normal coordinate is of the form... 

Summarizing, in the crystal there are 36 Raman active internal modes (symmetry species Ug, hig, 2g> and 26 infrared active internal modes (biw b2w hsu) as well as 12 Raman active and 7 infrared active external vibrations (librations and translations). Vibrations of the type are inactive because there appears no dipole moment along the normal coordinates in these vibrations of the crystal. 

The degree of vibrational excitation in a newly formed bond (or vibrational mode) of the products may also increase with increasing difference in bond length (or normal coordinate displacement) between the transition state and the separated products. For example, in the photodissociation of vinyl chloride [9] (reaction 7), the H—Cl bond length at the transition state for four-center elimination is 1.80 A, whereas in the three-center elimination, it is 1.40 A. A Franck-Condon projection of these bond lengths onto that of an HCl molecule at equilibrium (1.275 A) will result in greater product vibrational excitation from the four-center transition state pathway, and provides a metric to distinguish between the two pathways. 

At the other extreme is the associatively (a) activated associative (A) mechanism, in which the rate-determining step for substitution by 1/ proceeds through a reactive intermediate of increased coordination number, [M(H20) L](m x,+, which has normal vibrational modes and survives several molecular collisions before losing H20 to form [M(H20) 1L](m t,+, as shown in Eq. (8). Equation (9) indicates the linear variation with excess [I/-] anticipated for obs, which is similar in form to that of Eq. (5) when if0[I/ ] 1 and kohs + k. ... 

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