Potential Energy Surface (PES) and Chemical Reactions

A chemical reaction invariably implies changes in the relative nuclear co-ordinates of the reactant-set supermolecule system. As an example, the reaction H₂ + I₂ = 2HI involves large increase in the H-H and I-I internuclear distances, and significant decrease in the H-I distances; here the reactant-set supermolecule is the H₂-I₂ molecular combination. As nuclear motions are governed by the nuclear Schrodinger equation as per the Born-Oppenheimer Approximation theory, so the nuclear potential energy function U (q_a) in this equation dictates nuclear motions. Thus we see that the course of a chemical reaction is governed by the nuclear P.E. function U (q_a) for the reaction's supermolecule.

This function U (q_a) depends only on certain relative nuclear co-ordinates: for example in a diatomic supermolecule it depends only on the single internuclear distance R, while in a non-linear polyatomic supermolecule it depends on the (3N – 6) relative nuclear coordinates (that excludes the 3 translational and 3 rotational coordinates out of the total 3N coordinates in the N-atomic supermolecule). So a plot of U versus these relative nuclear coordinates takes the form of a curve for diatomic reaction systems, while for cases of N > 2 it takes the form of a hypersurface in a (3N – 6 + 1 = 3N – 5) dimensional hyperspace. Such an imaginary plot of the function U versus its nuclear coordinate variables is known as a potential energy surface (PES), and it is the shape of this PES that governs the course of the chemical reaction.

The value of U for a fixed nuclear framework is obtained by solving the electronic Schrodinger equation including internuclear repulsion, in which U is the eigenvalue. To get an idea of the shape of PES, this Schrodinger equation must be solved for many, say 10 or more, different values of each of the (3N - 6) relative nuclear coordinates to get the eigenvalues U, and then the plot of U versus its coordinate variables attempted. Such plotting is generally done mathematically, by trying to fit analytical functions of U vs. its variables to the (10^3N-6 or more) points obtained in the above procedure.

From this plot of U (to be called the PES), the local minima are found out. Such a minimum may correspond to the reactant-set, the product-set or a reaction intermediate (set). Then the Minimum Energy Path (MEP) joining the reactants (reactant-set) and products (product-set) through a minimum barrier of energy is searched for: generally the reaction, approximately speaking, progresses through the MEP, as it either goes along the MEP, or along paths slightly different from the MEP. (If there are another path with similar barrier, the reaction to some extent may progress along that path also.) The point of highest U on this minimum energy path (which is a saddle-point) is the transition-state for the reaction. From the shape of the MEP and also of its close nearby region in the PES (along which the reaction may progress), it is theoretically possible to calculate the reaction rate constant as a function of temperature; such a function obviously offers a theoretical energy of activation, E_a. The experimental and the theoretical E_a are close to the barrier height along the MEP, the barrier height being the difference in U values of the transition-state and of the reactant-set. Thus from the PES, a lot of information about the course of a given reaction may be obtained.

It is found that the nature of the MEP is dependent on the choice of nuclear coordinates (though the location of the minima aren't). It is known that the most proper MEP is obtained by using the intrinsic reaction coordinate (IRC), which indicates nuclear Cartesian coordinates multiplied by the square root of nuclear masses (i.e., m_a^0.5x_a, m_b^0.5z_b etc.): this MEP is identical with the path of a classical imaginary particle freely sliding downhill along the PES, starting from the transition-state. The MEP obtained using the IRC is called the reaction path, which is used in familiar chemistry discussions of energy vs. reaction path, used to explain activation energy and catalytic action.

Discussions of more familiar reactions involving larger number of nuclei will be much more complicated, and so a discussion of the reaction H₂ + H = H + H₂ (i.e., H_a-H_b + H_c = H_a + H_b-H_c) is attempted. This is a triatomic reaction-system, with U depending on (3 x 3 – 6 = 3) relative nuclear coordinates, the PES being a hypersurface in (3 + 1 = 4) dimensional hyperspace. To have a 3-D PES, a simplification is made by considering only the linear reactive collisions with all the three nuclei remaining collinear. This reduces the number of independent effective variables to two, and the two internuclear distances H_a–H_b & H_b–H_c may be taken as those variables. Denoted as R_ab & R_bc, these distance variables may be drawn along the x & y perpendicular axes on paper, with U to be imagined as plotted along z-axis, perpendicular to the plane of paper. The (x, y) points having the same value of U may be joined on the paper to get iso-potential curves, as if on a geographical altitude-indicating map. 

From the iso-potential curves thus drawn, the existence of two 'low-lying valleys' is obvious; they are with nearly fixed, small value of R_ab or R_bc with large value of the other distance variable: these two valleys correspond to the reactants and the products. In between them, with intermediately high and equal values of R_ab & R_bc lies the point indicating the transition-state, with higher values of U along its two sides, and lower values along the other two sides. Joining the reactants and the products via the transition state, the minimum energy path is indicated by the dotted line.