Torsion coordinate

Figure 3. Schematic illustration of the exact potential, the harmonic aporoximation V, and quasiharmonic approximations at lOOK, and 300K, for the torsional coordinate Q of trans butane. The anharmonicity of the exact potential results in the decreasing curvature of the quasiharmonic potentials with increasing temperature. (Reproduced from Ref. 30. Copyright 1984 American Chemical Society.)...

We are now in a position to discuss the reaction profile outlined in Figure 9.17 in the full space of coordinates corresponding to the branching space Xj X2 of a conical intersection and the torsional coordinate X3. This discussion will be focused on four related concepts ... 

Here cp represents the torsional coordinate, I denotes the moment of inertia, and Vf is the first coefficient of the Fourier series expansion of the isomerization potential of periodicity nn. 

As mentioned above, the = 1 components of the T-state of ethylene (Figure 2) are single-determinantal at the ground state equilibrium geometry, and remain single-determinantal at all values of the twisting angle. Therefore, they can be accurately described by SR methods at all the torsional coordinates (22). 

Figure 3. "Molecule" W-X-Y-Z subject to internal rotation along the torsional coordinate 9. The vector r connects atoms W and Z. The mean vector terminates at z for rotation along 9 subject to any hindrance potential symmetric in 9.

In this paper, the effect of the pseudopotential term, arising from the quantum mechanical correction to classical mechanism (V ), on the torsional levels of hydrogen peroxide and deuterium peroxide is evaluated. The V operator, depends on the first and second derivatives with respect to the torsional coordinate of the determinant of the g inertia matrix and on the first derivatives of the B kinetic energy parameter of the vihrational Hamiltonian. V has heen determined for each nuclear conformation from the optimized coordinates obtained using MP2/AUG-cc-pVTZ ah initio calculations. 

Moreover, as discussed in more detail later, force field parameters are not statistically orthogonal, so optimized values can be skewed by coupling with other parameters. With all of these caveats in mind, however, there are still instances where valuable physical insights derive from a term-by-term analysis of the torsional coordinate. 

Fig. 19. Potential energy curves and energy relationships in rhodopsin. Curve I Excited state of rhodopsin and bathorhodopsin. Curve II Ground state of rhodopsin and bathorhodopsin. Curve 111 Ground stale of isolated chromophore. Symbols , and 2 are quantum yields for reaching the single potential minimum along the 11,12 torsional coordinate. From Rosenfeld et al. [201].

Since so (trans) < 0.005], it is possible to rule out a photosensitized isomerization mechanism involving the establishment of thermal equilibrium between 11-cis and all-trans forms during the lifetime of the triplet state. Also excluded is the quantitative population of a common minimum along the C q-C 2 torsional coordinate. This conclusion is consistent with recent studies (169,175) which, in variance with the previous investigation (171), have reported different T-T spectra upon excitation of all-trans and 11-cis retinal. 

Although a barrierless C -C 2 torsional coordinate is predicted for the -Ag state of PRSB, this does not seem to be the case for the aldehyde and nonprotonated Schiff base derivatives (93,330). 

The example above reflects a fundamental problem for force field models that make use of internal coordinates to describe the PES of a system. The problem arises because some of the internal coordinates (mainly the anglebending and torsional coordinates) can be represented as linear combinations of others, and thus the complete set of internal coordinates is redundant. One cannot say a priori which of the coordinates are more important for the proper description of the PES, and one is unable to separate the contributions of the corresponding terms to the steric energy in a simple way. 

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