Multidimensional potential energy

Flammer B, Scheffler M, Jacobsen K W and Norskov J K 1994 Multidimensional potential energy surface for H2 dissociation over Cu(111) Phys.Rev. Lett. 73 1400... 

KD Ball, RS Beii y, RE Kunz, E-Y Li, A Proykova, DJ Wales. Erom topographies to dynamics of multidimensional potential energy surfaces of atomic clusters. Science 271 963-966, 1996. RS Berry, N Elmaci, JP Rose, B Vekhter. Linking topography of its potential surface with the dynamics of folding of a protein model. Proc Natl Acad Sci USA 94 9520-9524, 1997. Z Guo, D Thii-umalai. J Mol Biol 263 323-343, 1996. 

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. 

The semiempirical nature of the methods used to construct multidimensional potential energy surfaces makes the quantitative validity of the results questionable. Hence the present state of the theoretical calculation of activation energies is unsatisfactory. 

Before we do this, though, we point out that for a simple diatomic molecule, assuming ideal conditions, one can in principle calculate the rate of the uni-molecular process. This is so because the lower excited states of the ion are (relatively) few and well separated. If the potential curves are then given, the value of the rate can be provided. For a polyatomic molecule, two great complications immediately arise (1) the number of lower excited states increases tremendously and (2) multidimensional potential energy surfaces make trajectory calculations intractable. 

Figure 11 Simplified two-dimensional schematic of a multidimensional potential energy surface as a function of its configurational degrees of freedom. The landscape topology is specified by the density, whereas the system s elevation on the landscape is dictated by temperature. Reprinted with permission from Ref. 6.

At the end of this section, it is worthwhile to point out that resonances in quantum mechanics are intimately related to the existence of trapped classical trajectories. The smaller the classical forces between r and 7 on one hand and R on the other, the longer is the lifetime and vice versa. In this sense it might be helpful for understanding the complex quantum dynamics by imagining the trajectories of a classical billiard ball moving on multidimensional potential-energy surfaces (see, e.g., Chapter 5 of Ref. 4). 

By associating the reaction graph with a surface we obtain a minimal crosssection of the actual multidimensional potential energy hypersurface, which still has... 

A saddle point is a stationary point on the multidimensional potential energy surface. It is a stable point in all dimensions except one, where the second-order derivative of the potential is negative (see Appendix E). The classical energy threshold Eci or barrier height of the reaction corresponds to the electronic energy at the saddle point relative to the electronic energy of the reactants. 

In unimolecular reactions, initially, there is only a single stable molecule. This configuration corresponds to a minimum, that is, a well on a multidimensional potential energy surface see Figs 3.1.5 and 3.1.6. 

The multidimensional potential energy surface was written as the sum of a gas-phase (LEPS) energy surface incorporating the main features of the one-dimensional double-well potential in Example 10.1, solvent-solute interactions described by Lennard-Jones potentials with added (Coulomb) interactions corresponding to point charges, and solvent solvent interactions including intermolecular degrees of freedom. The solvent consisted of 64 water molecules. 

A large number of elementary molecular collision processes proceeding via (or in) excited electronic states are known at present. A prominent feature of all these is that as a rule they can not be interpreted (even at a very low kinetic energy of nuclei) in terms of the motion of a representative point over a multidimensional potential-energy surface. The breakdown of the Born-Oppenheimer approximation, which manifests itself in the so-called nonadiabatic coupling of electronic and nuclear motion, induces transitions between electronic states that remain still well defined at infinitely large intermolecular distances. 

T. HoUebeek, T.S. Ho, H. Rabitz, A fast algorithm for evaluating multidimensional potential energy surfaces, /. Chem. Phys. 106 (17) (1997) 7223-7227. 

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