Rotation, modes

For bimolecular reactions (i.e. where the reactant is two separate molecules) and contribute a constant —4 RT. The translational and rotational enttopy changes are substantially negative, —30 to —50 e.u., due to the fact that there are six translational and six rotational modes in the reactants but only three of each at the TS. The six remaining degrees of freedom are transformed into the reaction coordinate and five new vibrations at the TS. These additional vibrations usually make a few kcal/mol... 

By extending this to include vectors for all three translation and rotational modes, a projection matrix is obtained. 

Another way of removing the six translational and rotational degrees of freedom is to use a set of internal coordinates. For a simple acyclic system these may be chosen as 3N — I distances, 3N — 2 angles and 3N -3 torsional angles, as illustrated in the construction of Z-matrices in Appendix E. In internal coordinates the six translational and rotational modes are automatically removed (since only 3N — 6 coordinates are defined), and the NR step can be formed straightforwardly. For cyclic systems a choice of 3A — 6 internal variables which span the whole optimization space may be somewhat more problematic, especially if symmetry is present. 

FIGURE 6.17 The translational and rotational modes of atoms and molecules and the corresponding average energies of each mode at a temperature T. (a) An atom or molecule can undergo translational motion in three dimensions, (b) A linear molecule can also rotate about two axes perpendicular to the line of atoms, (c) A nonlinear molecule can rotate about three perpendicular axes. 

A linear molecule, such as any diatomic molecule, carbon dioxide, and ethyne (acetylene, HC=CH), can rotate about two axes perpendicular to the line of atoms, and so it has two rotational modes of motion. Its average rotational energy is therefore 2 X jkT = kT, and the contribution to the molar internal energy is NA times this value ... 

A nonlinear molecule, such as water, methane, or benzene, can rotate about any of three perpendicular axes, and so it has three rotational modes of motion. The average rotational energy of such a molecule is therefore 3 X jkT = ]kT. The contribution of rotation to the molar internal energy of a gas of nonlinear molecules is therefore... 

If the molecule adsorbs via a physisorbed precursor state in which it is free to move across the surface, while rotating and vibrating (with possibly modified frequencies and rotational modes), we obtain ... 

In general a nonlinear molecule with N atoms has three translational, three rotational, and 3N-6 vibrational degrees of freedom in the gas phase, which reduce to three frustrated vibrational modes, three frustrated rotational modes, and 3N-6 vibrational modes, minus the mode which is the reaction coordinate. For a linear molecule with N atoms there are three translational, two rotational, and 3N-5 vibrational degrees of freedom in the gas phase, and three frustrated vibrational modes, two frustrated rotational modes, and 3N-5 vibrational modes, minus the reaction coordinate, on the surface. Thus, the transition state for direct adsorption of a CO molecule consists of two frustrated translational modes, two frustrated rotational modes, and one vibrational mode. In this case the third frustrated translational mode vanishes since it is the reaction coordinate. More complex molecules may also have internal rotational levels, which further complicate the picture. It is beyond the scope of this book to treat such systems. 

List the nvunber of expected vibrational and rotational modes for the following gaseous molecules ... 

In second order perturbat i+PJf h e a r y with the perturbing Hamiltonian H = e E r cos u>t, and both the fundamental and created combined frequencies below electronic resonances but well above vibrational and rotational modes, can be expressed as... 

If A and B are atoms, the two rotational modes in the transition state add 70 J mol-1 K-1 to the entropy of the transition state. The total AS0 is therefore approximately -40 J mol-1 K-1, a value in agreement with the typical value given in Table 6.1. For each of the two rotational modes, the moment of inertia cited in Table 6.2 is / = l dAB the value above is calculated using dAB = 3 X10 10 m. 

Show that, for the bimolecular reaction A + B - P, where A and B are hard spheres, kTsr is given by the same result as jfcSCT, equation 6.4-17. A and B contain no internal modes, and the transition state is the configuration in which A and B are touching (at distance dAR between centers). The partition functions for the reactants contain only translational modes (one factor in Qr for each reactant), while the transition state has one translation mode and two rotational modes. The moment of inertia (/ in Table 6.2) of the transition state (the two spheres touching) is where p, is reduced mass (equation 6.4-6). 

The bosons 4 induce (spurious) rotational modes. Also... 

A. Kasprzak and G. Weber, Fluorescence depolarization and rotational modes of tyrosine in bovine pancreatic trypsin inhibitor, Biochemistry 21, 5924-5927 (1982). 

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