Barriers potential energy surfaces

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

In the statistical description of ununolecular kinetics, known as Rice-Ramsperger-Kassel-Marcus (RRKM) theory [4,7,8], it is assumed that complete IVR occurs on a timescale much shorter than that for the unimolecular reaction [9]. Furdiemiore, to identify states of the system as those for the reactant, a dividing surface [10], called a transition state, is placed at the potential energy barrier region of the potential energy surface. The assumption implicit m RRKM theory is described in the next section. 

To calculate N (E-Eq), the non-torsional transitional modes have been treated as vibrations as well as rotations [26]. The fomier approach is invalid when the transitional mode s barrier for rotation is low, while the latter is inappropriate when the transitional mode is a vibration. Hamionic frequencies for the transitional modes may be obtained from a semi-empirical model [23] or by perfomiing an appropriate nomial mode analysis as a fiinction of the reaction path for the reaction s potential energy surface [26]. Semiclassical quantization may be used to detemiine anliamionic energy levels for die transitional modes [27]. 

The vibrationally excited states of H2-OH have enough energy to decay either to H2 and OH or to cross the barrier to reaction. Time-dependent experiments have been carried out to monitor the non-reactive decay (to H2 + OH), which occurs on a timescale of microseconds for H2-OH but nanoseconds for D2-OH [52, 58]. Analogous experiments have also been carried out for complexes in which the H2 vibration is excited [59]. The reactive decay products have not yet been detected, but it is probably only a matter of time. Even if it proves impossible for H2-OH, there are plenty of other pre-reactive complexes that can be produced. There is little doubt that the spectroscopy of such species will be a rich source of infonnation on reactive potential energy surfaces in the fairly near future. 

Similar to the case without consideration of the GP effect, the nuclear probability densities of Ai and A2 symmetries have threefold symmetry, while each component of E symmetry has twofold symmetry with respect to the line defined by (3 = 0. However, the nuclear probability density for the lowest E state has a higher symmetry, being cylindrical with an empty core. This is easyly understand since there is no potential barrier for pseudorotation in the upper sheet. Thus, the nuclear wave function can move freely all the way around the conical intersection. Note that the nuclear probability density vanishes at the conical intersection in the single-surface calculations as first noted by Mead [76] and generally proved by Varandas and Xu [77]. The nuclear probability density of the lowest state of Aj (A2) locates at regions where the lower sheet of the potential energy surface has A2 (Ai) symmetry in 5s. Note also that the Ai levels are raised up, and the A2 levels lowered down, while the order of the E levels has been altered by consideration of the GP effect. Such behavior is similar to that encountered for the trough states [11]. 

The origin of a torsional barrier can be studied best in simple cases like ethane. Here, rotation about the central carbon-carbon bond results in three staggered and three eclipsed stationary points on the potential energy surface, at least when symmetry considerations are not taken into account. Quantum mechanically, the barrier of rotation is explained by anti-bonding interactions between the hydrogens attached to different carbon atoms. These interactions are small when the conformation of ethane is staggered, and reach a maximum value when the molecule approaches an eclipsed geometry. 

Chapter 12, to symmetry-caused energy barriers on the H2CO ==> H2 + CO reaction potential energy surface. 

It should be stressed that although these symmetry considerations may allow one to anticipate barriers on reaction potential energy surfaces, they have nothing to do with the thermodynamic energy differences of such reactions. Symmetry says whether there will be symmetry-imposed barriers above and beyond any thermodynamic energy differences. The enthalpies of formation of reactants and products contain the information about the reaction s overall energy balance. 

In principle, energy landscapes are characterized by their local minima, which correspond to locally stable confonnations, and by the transition regions (barriers) that connect the minima. In small systems, which have only a few minima, it is possible to use a direct approach to identify all the local minima and thus to describe the entire potential energy surface. Such is the case for small reactive systems [9] and for the alanine dipeptide, which has only two significant degrees of freedom [50,51]. The direct approach becomes impractical, however, for larger systems with many degrees of freedom that are characterized by a multitude of local minima. 

The previous treatment relied on the assumption that the transition occurs on a single potential energy surface V(x) characterized by a barrier separating two wells. This potential is actually created from the terms of the initial and final electronic states. The separation of electron and nuclear coordinates in each of these states gives rise to the diabatic basis with nondiagonal Hamiltonian matrix... 

Fig. 13.11. A schematic drawing of the potential energy surfaces for the photochemical reactions of stilbene. Approximate branching ratios and quantum yields for the important processes are indicated. In this figure, the ground- and excited-state barrier heights are drawn to scale representing the best available values, as are the relative energies of the ground states of Z- and E -stilbene 4a,4b-dihydrophenanthrene (DHP). [Reproduced from R. J. Sension, S. T. Repinec, A. Z. Szarka, and R. M. Hochstrasser, J. Chem. Phys. 98 6291 (1993) by permission of the American Institute of Physics.]...

On the isopotential map three minima (III, IV, V) are separated by barriers. They can be reached by decreasing of the distance R between the educts (I) via an activated complex (II). A detailed discussion of this potential energy surface also under the influence of a solvent will be given in part 4.3.1. 

At R > 400 pm the orientation of the reactants looses its importance and the energy level of the educts is calculated (ethene + nonclassical ethyl cation). For smaller values of R and a the potential energy increases rapidly. At R = 278 pm and a = 68° one finds a saddle point of the potential energy surface lying on the central barrier, which can be connected with the activated complex of the reaction (21). This connection can be derived from a vibration analysis which has already been discussed in part 2.3.3. With the assistance of the above, the movement of atoms during so-called imaginary vibrations can be calculated. It has been attempted in Fig. 14 to clarify the movement of the atoms during this vibration (the size of the components of the movement vector... 

Figure 2.4. Reaction coordinate diagram for a simple chemical reaction. The reactant A is converted to product B. The R curve represents the potential energy surface of the reactant and the P curve the potential energy surface of the product. Thermal activation leads to an over-the-barrier process at transition state X. The vibrational states have been shown for the reactant A. As temperature increases, the higher energy vibrational states are occupied leading to increased penetration of the P curve below the classical transition state, and therefore increased tunnelling probability.

In qualitative terms, the reaction proceeds via an activated complex, the transition state, located at the top of the energy barrier between reactants and products. Reacting molecules are activated to the transition state by collisions with surrounding molecules. Crossing the barrier is only possible in the forward direction. The reaction event is described by a single parameter, called the reaction coordinate, which is usually a vibration. The reaction can thus be visualized as a journey over a potential energy surface (a mountain landscape) where the transition state lies at the saddle point (the col of a mountain pass). 

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