Torsion angles, internal degrees freedom

Spanned by tbc atoms 4, 2, and 1, and 2, 1, and 3 (tlic ry-planc), Except of the first three atoms, each atom is described by a set of three internal coordinates a distance from a previously defined atom, the bond angle formed by the atom with two previous atoms, and the torsion angle of the atom with three previous atoms. A total of 3/V - 6 internal coordinates, where N is the number of atoms in the molecule, is required to represent a chemical structure properly in 3D space. The number (,3N - 6) of internal coordinates also corresponds to the number of degrees of freedom of the molecule. 

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. 

Here x,x denote two configurations of the system (specified, for instance, by the set of coordinates of all atoms r or the position of one chain end for all chains and all bond lengths, bond angles, and torsion angles rf, If, 0a, where a = 1,... M runs over all chains and the indices i,j, k run over all internal degrees of freedom of one chain). The transition rates W(x —> x ) are chosen to fulfill the detailed balance condition... 

One example of non-IRC trajectory was reported for the photoisomerization of cA-stilbene.36,37 In this study trajectory calculations were started at stilbene in its first excited state. The initial stilbene structure was obtained at CIS/6-31G, and 2744 argon atoms were used as a model solvent with periodic boundary conditions. In order to save computational time, finite element interpolation method was used, in which all degrees of freedom were frozen except the central ethylenic torsional angle and the two adjacent phenyl torsional angles. The solvent was equilibrated around a fully rigid m-stilbene for 20 ps, and initial configurations were taken every 1 ps intervals from subsequent equilibration. The results of 800 trajectories revealed that, because of the excessive internal potential energy, the trajectories did not cross the barrier at the saddle point. Thus, the prerequisites for common concepts of reaction dynamics such TST or RRKM theory were not satisfied. 

The most commonly used stochastic methods are the torsional Monte Carlo method11101 and the cartesian stochastic (or random kick) method11111. The two methods differ in the coordinate system in which they operate. The torsional Monte Carlo method uses internal coordinates, while the random kick method uses cartesian coordinates. The advantage of using internal coordinates is that the molecular degrees of freedom are reduced. The reason for choosing torsional angles as the vari-... 

Let us think again about the dehydrobutane molecule and its internal degrees of freedom. Evidently, there are three bond lengths that can stretch, two bond angles that can bend, and one torsion angle which that rotate. Thus, there are six internal degrees of freedom in the molecule, and, hence, we will need to solve six simultaneous... 

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