Potential energy surfaces theory

B. H. Lengsfield and D. R. Yarkony, Nonadiabatic Interactions Between Potential Energy Surfaces Theory and Applications, in State-Selected and State to State Ion-Molecule Reaction Dynamics Part 2 Theory, M. Baer and C.-Y. Ng, eds., John Wiley Sons, Inc., New York, 1992, Vol, 82, pp. 1-71. 

OM Becker, M Karplus. The topology of multidimensional potential energy surfaces Theory and application to peptide stiaicture and kinetics. I Chem Phys 106 1495-1517, 1997. 

Lengsfield BH, Yarkony DR (1992) Nonadiabatic interactions between potential energy surfaces theory and applications. In Baer M, Ng CY (eds) State-selected and state-to-state ion-molecule reaction dynamics part 2 theory, Vol. 82 of Advances in Chemical Physics, John Wiley and Sons, New York, p 1-71. 

The result is that, to a very good approxunation, as treated elsewhere in this Encyclopedia, the nuclei move in a mechanical potential created by the much more rapid motion of the electrons. The electron cloud itself is described by the quantum mechanical theory of electronic structure. Since the electronic and nuclear motion are approximately separable, the electron cloud can be described mathematically by the quantum mechanical theory of electronic structure, in a framework where the nuclei are fixed. The resulting Bom-Oppenlieimer potential energy surface (PES) created by the electrons is the mechanical potential in which the nuclei move. Wlien we speak of the internal motion of molecules, we therefore mean essentially the motion of the nuclei, which contain most of the mass, on the molecular potential energy surface, with the electron cloud rapidly adjusting to the relatively slow nuclear motion. 

The theory coimecting transport coefficients with the intemiolecular potential is much more complicated for polyatomic molecules because the internal states of the molecules must be accounted for. Both quantum mechanical and semi-classical theories have been developed. McCourt and his coworkers [113. 114] have brought these theories to computational fruition and transport properties now constitute a valuable test of proposed potential energy surfaces that... 

Jeziorski B, Moszynski R and Szalewicz K 1994 Perturbation theory approach to intermolecular potential energy surfaces of van der Waals complexes Chem. Rev. 94 1887... 

The above discussion represents a necessarily brief simnnary of the aspects of chemical reaction dynamics. The theoretical focus of tliis field is concerned with the development of accurate potential energy surfaces and the calculation of scattering dynamics on these surfaces. Experimentally, much effort has been devoted to developing complementary asymptotic techniques for product characterization and frequency- and time-resolved teclmiques to study transition-state spectroscopy and dynamics. It is instructive to see what can be accomplished with all of these capabilities. Of all the benclunark reactions mentioned in section A3.7.2. the reaction F + H2 —> HE + H represents the best example of how theory and experiment can converge to yield a fairly complete picture of the dynamics of a chemical reaction. Thus, the remainder of this chapter focuses on this reaction as a case study in reaction dynamics. 

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. 

Schlegel H B 1995 Geometry optimization on potential energy surfaces Modern Electronic Structure Theory vo 2, ed D R Yarkony (Singapore World Scientific) pp 459-500... 

Hammes-Schiffer S and Tully J C 1995 Nonadiabatic transition state theory and multiple potential energy surfaces molecular dynamics of infrequent events J. Chem. Phys. 103 8528... 

E. Kracka, T. H. Dunning, Jr., Advances in Molecular Electronic Structure Theory Calculation and Characterization of Molecular Potential Energy Surfaces T. H. Dunning, Jr. Ed., 129, JAI, Greenwich (1990). 

Variational transition state theory (VTST) is formulated around a variational theorem, which allows the optimization of a hypersurface (points on the potential energy surface) that is the elfective point of no return for reactions. This hypersurface is not necessarily through the saddle point. Assuming that molecules react without a reverse reaction once they have passed this surface... 

Rather than using transition state theory or trajectory calculations, it is possible to use a statistical description of reactions to compute the rate constant. There are a number of techniques that can be considered variants of the statistical adiabatic channel model (SACM). This is, in essence, the examination of many possible reaction paths, none of which would necessarily be seen in a trajectory calculation. By examining paths that are easier to determine than the trajectory path and giving them statistical weights, the whole potential energy surface is accounted for and the rate constant can be computed. 

The low-temperature chemistry evolved from the macroscopic description of a variety of chemical conversions in the condensed phase to microscopic models, merging with the general trend of present-day rate theory to include quantum effects and to work out a consistent quantal description of chemical reactions. Even though for unbound reactant and product states, i.e., for a gas-phase situation, the use of scattering theory allows one to introduce a formally exact concept of the rate constant as expressed via the flux-flux or related correlation functions, the applicability of this formulation to bound potential energy surfaces still remains an open question. 

The semiempirical methods combine experimental data with theory as a way to circumvent the calculational difficulties of pure theory. The first of these methods leads to what are called London-Eyring-Polanyi (LEP) potential energy surfaces. Consider the triatomic ABC system. For any pair of atoms the energy as a function of intermolecular distance r is represented by the Morse equation, Eq. (5-16),... 

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