Potential energy surface

A potential energy surface (PES) is the energy of a system, especially a collection of atoms, in terms of certain parameters, normally the positions of the atoms. The surface might define the energy as a function of one or more coordinates; if there is only one coordinate, the surface is called a potential energy curve. An example is the Morse potential.

It is helpful to use the analogy of a landscape: for a system with two degrees of freedom (e.g. two bond lengths), the value of the energy (analogy: the height of the land) is a function of two bond lengths (analogy: the coordinates of the position on the ground).[1]

The PES concept finds application in fields such as chemistry and physics, especially in the theoretical sub-branches of these subjects. It can be used to theoretically explore properties of structures composed of atoms, for example, finding the minimum energy shape of some molecule and computing the rates of a chemical reaction.

Mathematical Definition and Computation

The geometry of a set of atoms can be described by a vector, r, where the elements in the vector come from the atoms' positions. Vector r could be the set of the Cartesian coordinates of the atoms and could also be a set of inter-atomic distances and angles.

Given r, one can introduce the concept of the energy as a function of the positions, E(r). It is the value of E(r) for all r of interest that, using the landscape analogy from the introduction, gives height of the "land" on the "energy landscape" and it is from here that the concept of a potential energy surface arises.

To study a chemical reaction using the PES of the constituent atoms, it must be possible to calculate the energy for every arrangement of the atoms that one is interested in. Methods of calculating the energy of a particular arrangement of atoms is well described in the Computational Chemistry article, and the emphasis here will be on forming approximations of E(r) for when fine-grained energy-position information is desired.

For very simple chemical systems or when significant assumptions are made about inter-atomic interactions, it is sometimes possible to use an analytically derived expression for the energy as a function of the atomic positions, for example the London-Eyring-Polanyi-Sato potential[2][3][4] for the system H + H2 as a function of the three H-H distances.

Often, for more complicated systems, calculation of the energy of a particular arrangement of atoms is too computationally expensive for large scale representations of the surface to be feasible. For these systems a possible approach is to calculate only a reduced set of points on the PES and then use a computationally cheaper interpolation method, for example Shepard interpolation, to fill in the gaps.[5]

Applications of a Potential Energy Surface

In one sense a PES can be thought of as a conceptual tool for aiding the analysis of molecular structure and chemical reaction dynamics, and such an analysis makes use of the concept of a PES in the following way. Once it is possible to evaluate the necessary points on a PES, the points can be further classified according to the first and second derivatives of the energy with respect to position, which respectively are the gradient and the curvature. Stationary points, that is, those points with a zero gradient, have some physical meaning: energy minima correspond to physically stable chemical species and saddle points correspond to transition states, the highest energy point on the reaction pathway, that is, the lowest energy pathway connecting a chemical reactant to a chemical product.

See also

References

  1. ↑ IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997)
  2. ↑ Sato, S. (1955). "A New Method of Drawing the Potential Energy Surface". Bulletin of the Chemical Society of Japan 28 (7): 450. doi:10.1246/bcsj.28.450."On a New Method of Drawing the Potential Energy Surface". The Journal of Chemical Physics 23 (3): 592. 1955. doi:10.1063/1.1742043.
  3. ↑ Keith J. Laidler, Chemical Kinetics (3rd ed., Harper & Row 1987) p.68-70 ISBN 0-06-043862-2
  4. ↑ Steinfeld J.I., Francisco J.S. and Hase W.L. Chemical Kinetics and Dynamics (2nd ed., Prentice-Hall 1998) p.201-2 ISBN 0-13-737123-3
  5. ↑ Moving least-squares enhanced Shepard interpolation for the fast marching and string methods, Burger SK1, Liu Y, Sarkar U, Ayers PW, J Chem Phys. 2009 130(2) 024103. doi: 10.1063/1.2996579.