Stretching coordinates

When looking at the snapshots in Figure A3.13.6 we see that the position of maximal probability oscillates back and forth along the stretching coordinate between the walls at = -20 and +25 pm, with an approximate period of 12 fs, which corresponds to the classical oscillation period r = 1 / v of a pendulum with... 

Fig. 1. Schematic one-dimensional cross section through the Gibbs free energy surface G(R) of a spin-state transition system along the totally symmetric stretching coordinate. The situation for three characteristic temperatures is shown (B = barrier height, ZPE = zero-point energy, 28 = asymmetry parameter, J = electronic coupling parameter, AG° = Gh — GJ...

Distances in are different from those in the usual space S. A weighted space can be represented graphically by means of stretched coordinate axes [2]. The latter result when the basis vectors of the space are scaled by means of the corresponding quantities in Vw, where the vector w contains the main diagonal elements of W. Figure 32.3 shows that a circle is deformed into an ellipse if one passes from usual coordinate axes in the usual metric I to stretched coordinate axes in the weighted metric W. In this example, the horizontal axis in 5, is stretched by a factor. l-6 = 1.26 and the vertical axis is shrunk by a factor Vo.4 = 0.63. 

In Section 29.3 it has been shown that a matrix generates two dual spaces a row-space S" in which the p columns of the matrix are represented as a pattern P , and a column-space S in which the n rows are represented as a pattern P". Separate weighted metrics for row-space and column-space can be defined by the corresponding metric matrices and W. This results into the complementary weighted spaces and S, each of which can be represented by stretched coordinate axes using the stretching factors in -J v and, where the vectors w and Wp contain the main diagonal elements of W and W. ... 

Figure 4.67 depicts the potential-energy curve for reaction (4.102) along an adiabatic reaction coordinate (R = /Oimc) obtained by stepping along the H—CH3 stretching coordinate with full optimization of geometries at each step. As shown in Fig. 4.67, the reaction exhibits a substantial barrier ( 20.5 kcal mol-1) and overall exothermicity. 

What does this time dependent friction look like To answer this, we describe the MD calculated results [5c] for (t) for the reactive asymmetric stretch coordinate for the Cl" + CH3C1 SN2 system in H20 solvent, and its associated Fourier spectrum. The latter is particularly illuminating, since it displays peaks clearly identifiable from the spectrum of the same pure H20 liquid. Thus contributions from the H20 bends and... 

A consequence of the symmetry of the molecule is that states must transform according to representations of the appropriate symmetry group. In terms of coordinates, this implies that one must form internal symmetry coordinates. These are linear combinations of the internal coordinates. For example, denoting in Fig. 6.1 by sx, s2, s3,, v4, j5, s6 the stretching coordinates of the six C-H bonds, the internal symmetry coordinates are linear combinations... 

For these vibrations, the quantization scheme of Section 4.2 can be carried over without any modification (Iachello and Oss, 1991a). The potentials in each stretching coordinate 5 are in an anharmonic force field approximation represented by Morse potentials. The boson operators (Ot,xt) correspond to the quantization of anharmonic Morse oscillators, with classical Hamiltonian... 

Figure 6.11 Structures of 1,3-didehydrobenzene (m-benzyne) from experiment and RHF calculations. Because of its tendency to overemphasize bonding interactions, RHF optimization results in a bicyclic structure. While the RHF error in bond length is very large, it should be noted that the bond-stretching coordinate is known to be very flat (for very detailed analyses on the sensitivity of this system to different theoretical levels, see Kraka et al. 2001 and Winkler and Sander 2001)...

Figure 1. Intramolecular vibrational density redistribution IVR of Na3

Several works have assigned M—O stretching frequencies with these ligands,117119 typically in the range 300-450 cm-1, but to what extent these are pure stretching coordinates has not been established. 

The origin of the tunneling efficiency can be traced to a specific feature of the PES, specifically, the presence of an extremely narrow barrier between the initial state (physisorbed N2 molecule) and the final state (chemisorbed N atoms). The model PES proposed by Haase et al. [1989] is characterized by two coordinates an N-N stretching coordinate (r) directed along the surface, and the distance (z) from the surface to the N2 center of mass. In the initial state, r0 = 1.10A, and z0 = 2.48A. The corresponding vibrational frequencies are 2180 and 355 cm-1. At the transition state r = 1.14 A, and z = 2.25 A (Figure 9.5). Comparing... 

Figure 12-1. Classification of O-H - O hydrogen bonding situations of different strengths and schematic representation of corresponding potential energy curves along the O-H stretching coordinate.

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