Vibrational energy polyatomic molecules

Vibrational states can be described in terms of the normal mode (NM) [50, 51] or the local mode (LM) [37, 52, 53] model. In the former, vibrations in polyatomic molecules are treated as infinitesimal displacements of the nuclei in a harmonic potential, a picture that naturally includes the coupling among the bonds in a molecule. The general formula for the energies of the vibrational levels in a polyatomic molecule is given by [54]... 

Doorway vibration decay is particularly interesting because it is the one situation where polyatomic molecule VER looks just like diatomic molecule VER. The doorway vibrations of polyatomic molecules decay by exactly the same multiphonon mechanism as the VER of a diatomic molecule. Diatomic molecules have been extensively studied (7). One prediction for diatomic molecules is an exponential energy-gap law (2). As the vibrational frequency is increased, with everything else held constant, the number of emitted phonons increases (the order of the multiphonon process increases) and the VER rate should decrease exponentially with increasing vibrational frequency. 

The importance of the Raman spectrum lies especially in the fact that it also occurs for homonuclear molecules, which, according to sections 22 and 23, have no rotation and vibration-rotation spectra. Hence, it may be used to supplement the evidence derived from electronic bands, regarding the energy of vibrational and rotational levels in the ground state, and for a confirmation of the values of and thus obtained. Researches of this sort have actually been carried out on HCl by Wood and on Hg, Ng, Og, CO by Rasetti (for literature see G) and (lO)) and, more recently, on CO by Amaldi(is). Really essential, however, is the Raman effect in analysing the possible vibrations of polyatomic molecules, as we shall see in the next chapter. For such molecules very rarely have sharply defined electronic bands, while the rotation and vibration-rotation data usually are insufficient to arrive at a unique description of the molecular behaviour. 

At low vibrational excitations polyatomic molecules exhibit a well-defined vibrational spectrum (particularly so if it is initially quite cold). We discuss in this section at what energy a large molecule changes its behavior to become de facto its own heat bath. This change involves a sudden increase in the density of vibrational states as the energy of the molecule approaches the dissociation threshold. Even... 

In the case of polyatomic molecules, one may consider separately the accommodation coefficients for translational and for vibrational energy. Values of the latter, civ, are discussed by Nilsson and Rabinovitch [7]. 

For a RRKM calculation without any approximations, the complete vibrational/rotational Flamiltonian for the imimolecular system is used to calculate the reactant density and transition state s sum of states. No approximations are made regarding the coupling between vibration and rotation. Flowever, for many molecules the exact nature of the coupling between vibration and rotation is uncertain, particularly at high energies, and a model in which rotation and vibration are assumed separable is widely used to calculate the quantum RRKM k(E,J) [4,16]. To illustrate this model, first consider a linear polyatomic molecule which decomposes via a linear transition state. The rotational energy for tire reactant is assumed to be that for a rigid rotor, i.e. 

Haarhoff P C 1963 The density of vibrational energy levels of polyatomic molecules Mol. Phys. 7 101-17... 

As in classical mechanics, the outcome of time-dependent quantum dynamics and, in particular, the occurrence of IVR in polyatomic molecules, depends both on the Flamiltonian and the initial conditions, i.e. the initial quantum mechanical state I /(tQ)). We focus here on the time-dependent aspects of IVR, and in this case such initial conditions always correspond to the preparation, at a time of superposition states of molecular (spectroscopic) eigenstates involving at least two distinct vibrational energy levels. Strictly, IVR occurs if these levels involve at least two distinct... 

Weitz E and Flynn G W 1981 Vibrational energy flow in the ground electronic states of polyatomic molecules Adv. Chem. Rhys. 47 185-235... 

Orr B J and Smith I W M 1987 Collision-induced vibrational energy transfer in small polyatomic molecules J. Rhys. Chem. 91 6106-19... 

Seilmeier A and Kaiser W 1988 Ultrashort intramolecular and intennolecular vibrational energy transfer of polyatomic molecules in liquids Ultrashort Laser Pulses and Applications (Topics in Applied Physics 60) ed W Kaiser (Berlin Springer) pp 279-315... 

Infrared Spectra for Molecules and Polyatomic Ions The energy of infrared radiation is sufficient to produce a change in the vibrational energy of a molecule or polyatomic ion (see Table 10.1). As shown in Figure 10.14, vibrational energy levels are quantized that is, a molecule may have only certain, discrete vibrational energies. The energy for allowed vibrational modes, Ey, is... 

As for diatomic molecules, there are stacks of rotational energy levels associated with all vibrational levels of a polyatomic molecule. The resulting term values S are given by the sum of the rotational and vibrational term values... 

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. 

Figure 7.46 States Sq, Si and Tj of a polyatomic molecule showing regions of low density of vibrational states and, at higher energy, pseudo-continua...

Another area of research ia laser photochemistry is the dissociation of molecular species by absorption of many photons (105). The dissociation energy of many molecules is around 4.8 x 10 J (3 eV). If one uses an iafrared laser with a photon energy around 1.6 x 10 ° J (0.1 eV), about 30 photons would have to be absorbed to produce dissociation (Eig. 17). The curve shows the molecular binding energy for a polyatomic molecule as a function of interatomic distance. The horizontal lines iadicate bound excited states of the molecule. These are the vibrational states of the molecule. Eor... 

Fig. 11. (a) Diagram of energy levels for a polyatomic molecule. Optical transition occurs from the ground state Ag to the excited electronic state Ai. Aj, are the vibrational sublevels of the optically forbidden electronic state A2. Arrows indicate vibrational relaxation (VR) in the states Ai and Aj, and radiationless transition (RLT). (b) Crossing of the terms Ai and Aj. Reorganization energy E, is indicated. 

For a polyatomic molecule the total vibrational energy may be written as a sum of energies for each vibration, and the partition function as a product of partition functions. 

For a polyatomic molecule, the complex vibrational motion of the atoms can be resolved into a set of fundamental vibrations. Each fundamental vibration, called a normal mode, describes how the atoms move relative to each other. Every normal mode has its own set of energy levels that can be represented by equation (10.11). A linear molecule has (hr) - 5) such fundamental vibrations, where r) is the number of atoms in the molecule. For a nonlinear molecule, the number of fundamental vibrations is (3-q — 6). 

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