Polyatomic vibrational coordinates

The potential energy curve in Figure 6.4 is a two-dimensional plot, one dimension for the potential energy V and a second for the vibrational coordinate r. For a polyatomic molecule, with 3N — 6 (non-linear) or 3iV — 5 (linear) normal vibrations, it requires a [(3N — 6) - - 1]-or [(3A 5) -F 1]-dimensional surface to illustrate the variation of V with all the normal coordinates. Such a surface is known as a hypersurface and clearly cannot be illustrated in diagrammatic form. What we can do is take a section of the surface in two dimensions, corresponding to V and each of the normal coordinates in turn, thereby producing a potential energy curve for each normal coordinate. 

The harmonic oscillator energies and wavefunctions comprise the simplest reasonable model for vibrational motion. Vibrations of a polyatomic molecule are often characterized in terms of individual bond-stretching and angle-bending motions each of which is, in turn, approximated harmonically. This results in a total vibrational wavefunction that is written as a product of functions one for each of the vibrational coordinates. 

The first part of this chapter (Section II) deals with the SCF method for calculations of polyatomic vibrational energy levels. The aspect on which we focus is that the SCF approximation may strongly depend on the choice of the modes to which the SCF (generalized) separation is applied. Recent calculations have shown that a physically motivated choice of coordinates can... 

SCF is by no means the only framework in which good coordinates for coupled polyatomic vibrations can be defined, or pursued. Stefanski and Taylor31 stressed the importance of coordinate system choice for simple inter-... 

In general, theoretical studies of triatomic and tetra-atomic molecules employ analytical PESs carefully fitted to large grids of ab initio data points, and curvilinear vibrational coordinates, to take into account large-amplitude motions. On the other hand, larger polyatomic molecules are investigated with simple polynomial PES, whose parameters are obtained from ah initio data, and with normal coordinates, possibly considering only the active ones. Finite basis representations (FBR),... 

In this expression Q = 0 at the equilibrium configuration of the free polyatom. The first term gives the van der Waals potential at the free molecule equilibrium positions of the polyatom nuclei. This term cannot couple different polyatom vibrational states, however, so it does not contribute directly to vibrational predissociation. The second and third terms are written as derivatives of the van der Waals potential with respect to displacements in the polyatom coordinates, but they can also be viewed as changes in the intramolecular potential in the presence of the adatom. We can make the physical meaning of Eq. (II.6) clearer by rewriting it as... 

In a polyatomic molecule with N nuclei, 3N independent coordinates are required to specify all of the nuclear positions in space. We have already seen in the preceding chapter that rotations of nonlinear polyatomics about their center of mass may be described in terms of the three Euler angles < >, 0, and x- Three additional coordinates are required to describe spatial translation of a molecule s center of mass. Hence, there will be3N — 6 independent vibrational coordinates in a nonlinear polyatomic molecule. In a linear polyatomic molecule, the orientation may be given in terms of two independent angles and (f). Linear polyatomics therefore exhibit 3N — 5 rather than 3N — 6 independent vibrational coordinates. 

Our development of the normal mode description of polyatomic vibrations in Sections 6.1-6.4 rested on the assumption that the potential energy function (6.4) is harmonic in the nuclear coordinates. As in diatomics, this assumption... 

Thus far we mainly used a two-body point of view. From now on the discussion will emphasize the polyatomic nature of the dynamics of chemical reactions. This is the same transition that is made in books on spectroscopy. These go from boimd AB to boimd ABC, while we go from unbound AB to unbound ABC. There is more than one vibrational coordinate in ABC. Which one is to be imboimd Well, this is very much part of the discussion of Chapter 5. Nor is it only the stretch vibrations that are of interest. The bending vibration of ABC is the carrier of the steric preference during the collision... 

The selection rule for vibronic states is then straightforward. It is obtained by exactly the same procedure as described above for the electronic selection rules. In particular, the lowest vibrational level of the ground electronic state of most stable polyatomic molecules will be totally synnnetric. Transitions originating in that vibronic level must go to an excited state vibronic level whose synnnetry is the same as one of the coordinates, v, y, or z. 

In diatomic VER, the frequency Q is often much greater than so VER requires a high-order multiphonon process (see example C3.5.6.1). Because polyatomic molecules have several vibrations ranging from higher to lower frequencies, only lower-order phonon processes are ordinarily needed [34]- The usual practice is to expand the interaction Hamiltonian > in equation (03.5.2) in powers of nonnal coordinates [34, 631,... 

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). 

It is now fundamental to define the normal coordinates of this vihrational system - that is to say, the nuclear displacements in a polyatomic molecule. Again in the limit of small amplitudes of vibration, the normal coordinates in the form of the vector Q, are related to the internal coordinates by a linear transformation, viz. 

For polyatomic molecules, the stretching force constant for a particular bond cannot in general be obtained in an unambiguous manner because any given vibrational mode generally involves movements of more than two of the atoms, which prevent the expression of the observed frequency in terms of the force constant for just one bond. The vibrational modes of a polyatomic molecule can be analyzed by a method known a normal coordinate analysis to... 

So if the bond strength increases or reduced mass decreases, the value of vibrational frequency increases. Polyatomic molecules may exhibit more than one fundamental vibrational absorption bands. The number of these fundamental bands, is related to the degree of freedom in a molecule and the number of degrees of freedom is equal to the number of coordinates necessary to locate all atoms of a molecules in space. 

The formulation of the preceding section is very general. We are interested, however, in rotations and vibrations of polyatomic molecules. We therefore discuss now specific applications of the algebraic method beginning with the simple case of one-dimensional coupled oscillators, presented in Section 3.3 in the Schrodinger picture. In the algebraic theory, as mentioned, one associates to each coordinate, x, and related momentum, px = — iti d/dx, an algebra. For... 

---