University of Malta CH237 - Chemical Thermodynamics and Kinetics

Dr. Joseph N. Grima, Department of Chemistry
University of Malta, Msida, MSD 06, MALTA


A theoretical investigation of chemical kinetics - (v)

Potential energy surfaces

(1) Introduction
(2) Some results from experiments and calculations
       (2-1) The direction of attack and separation
       (2-2) Attractive and repulsive surfaces
       (2-3) Classical trajectories (i.e. trajectories obtained through equations of classical mechanics)

[ TOP ]
Potential energy surfaces 

(1) Introduction

The potential energy surface of a reaction may be defined as 'the potential energy as a function of the relative positions of all the atoms taking part in the reaction'

For example, a collision between a hydrogen atom (H) and a hydrogen molecule (H2), the potential energy surface is the plot of the potential energy for all relative locations of the three hydrogen nuclei. Detailed calculations show that the approach of an atom along the H-H axis requires less energy for reaction than any other approach, so initially we confine our attention to a collinear approach where a hydrogen atom HA approaches an HB-HC molecule in the direction of HB-HC, heading towards HB. Two parameters are required to define the nuclear separations: 

  • The HA-HB separation, RAB, (= infinite at the start of the of the encounter, equilibrium bond length at the end of a successful of the encounter)
  • The HB-HC separation, RBC, (= equilibrium bond length at the start of the of the encounter, infinite at the end of a successful encounter.)
as illustrated in Fig. 1 below.

Fig. 1: The definition of RAB and RBC.

The total energy of the three-atom system depends on their relative separations, and can be found through molecular modelling techniques (e.g. by doing a molecular orbital calculations). The plot of the total energy of the system against RAB and RBC is the potential energy surface of this collinear reaction, and is illustrated in Fig. 2a below. 

Note that:

  • This surface is normally depicted as a contour diagram, as illustrated in Fig. 2b. 
  • When RAB is very large, the variation in potential energy represented by the surface is that of an isolated H2 molecule as its bond length is altered, as illustrated in Fig. 2c. Similarly for when RBC is very large at the end of a successful encounter. 

Fig. 2 (a) The potential energy surface for the H + H2 -> H2 + H reaction when the atoms are constrained to be collinear.

Fig. 2(b) The contour diagram (with contours of equal potential energy) corresponding to the surface in Fig. 2(a). Re marks the equilibrium bond length of an H2 molecule (strictly, it relates to the arrangement when the third atom is at infinity).

Fig. 2(c) The molecular potential energy curve for the hydrogen molecule showing the variation of the energy of the molecule as the bond length is changed. Re marks the equilibrium bond length of an H2 molecule. 

The paths that are available to the atoms may be deduced from the shape of the potential energy surface, as these corresponds to paths of least potential energy. (The actual path of the atoms in the course of the encounter depends on their total energy, the sum of their kinetic and potential energies.)

Let us illustrate this for the simple case of a hydrogen atom HA is approaching a hydrogen molecule, HB-HC, to form HA-HB and HC. Referring to fig. 3a, let us investigate three way how this (at least theoretically) can be accomplished: 

(Path A): One approach could be to let the HB-HC bond length remain constant during the initial approach of HA. In this path, the potential energy rises to a high value as HA is pushed into the molecule, and then decreases sharply as HC breaks off and separates to a great distance. 

(PATH B): An alternative reaction path can be imagined in which the HB-HC bond length increases at an early stage of the approach of HA (i.e. while HA is still far away). This path takes the system through a region of a high potential energy in the course of the encounter (compare with path C).

(PATH C): This is the path of least potential energy, and corresponds to RBC lengthening as HA approaches and begins to form a bond with HB. The HB-HC bond relaxes at the demand of the incoming atom, and the potential energy climbs only as far as the saddle-shaped region of the surface, to the saddle point marked C. In this path of least potential energy, the atoms take a route up the floor of the valley, through the saddle point, and down the floor of the other valley as HC recedes and the new HA-HB bond achieves its equilibrium length. This path is the reaction coordinate we met in the earlier section discussing the activated complex. 

Thus, through Path C, we can now make contact with the activated complex theory of reaction rates. In terms of trajectories on potential surfaces, the transition state can be identified with a critical geometry such that every trajectory that goes through this geometry goes on to react, i.e. the saddle point C.

NOTE: Paths A and B are theoretically feasible, provided, the molecules have sufficient initial kinetic energy to take the three atoms past the regions of high potential energy in the course of the encounter.

Fig. 3a: Various trajectories through the potential energy surface shown in Fig. 2. Path A corresponds to a route in which RBC is held constant as HA approaches; path B corresponds to a route in which RBC lengthens at an early stage during the approach of HA; path C is the route along the floor of the potential valley.

Fig. 3b: The transition state is a set of configurations (here, marked by the line across the saddle point) through which successful reactive trajectories must pass.

[ TOP ]
(2) Some results from experiments and calculations

To travel successfully from reactants to products the incoming molecules must possess enough kinetic energy to be able to climb to the saddle point of the potential surface. 

The shape of the surface can be explored experimentally by changing:

  1. The relative speed of approach (by selecting the beam velocity), and, 
  2. The degree of vibrational excitation 
and observing whether reaction occurs and whether the products emerge in a vibrationally excited state. 

Such work allows us to answer some important questions: For example, what is better, to smash the reactants together with a lot of translational kinetic energy or to ensure instead that they approach in highly excited vibrational states? (In other words, is trajectory Fig. 4c, where the HBHC molecule is initially vibrationally excited, more efficient at leading to reaction than the trajectory in Fig. 4a, in which the total energy is the same but has a high translational kinetic energy?)

Fig.. 4: An illustration of some successful (marked with an *) and unsuccessful encounters: (a) C1* corresponds to the path along the foot of the valley; (b) C2* corresponds to an approach of HA to a vibrating HBC molecule, and the formation of a vibrating HAB molecule as HC departs. (c) C3 corresponds to HA approaching a non-vibrating HBC molecule, but with insufficient translational kinetic energy; (d) C4 corresponds to HA approaching a vibrating HBC molecule, but still the energy, and the phase of the vibration, is insufficient for reaction.

[ TOP ]

(2-1) The direction of attack and separation

(i) Direction of attack: 

  • Fig. 5a shows the results of a calculation of the potential energy as an H atom approaches an H2 molecule from different angles, the H2 bond being allowed to relax to the optimum length in each case. The potential barrier is least for collinear attack, as we assumed earlier. (But we must be aware that other lines of attack are feasible and contribute to the overall rate.) 
  • In contrast, Fig. 5b shows the potential energy changes that occur as a Cl atom approaches an HI molecule. The lowest barrier occurs for approaches within a cone of half-angle 30o surrounding the H atom. The relevance of this result to the calculation of the steric factor of collision theory should be noted: not every collision is successful, because not every one lies within the reactive cone.
Fig. 5a: An indication of the anisotropy of the potential energy changes as H approaches H2 with different angles of attack. The collinear attack has the lowest potential barrier to reaction. The surface indicates the potential energy profile along the reaction coordinate for each configuration.

Fig. 5b: The potential energy barrier for the approach of Cl to HI. In this case, successful encounters occur only when Cl approaches within a cone surrounding the H atom.

(ii) Direction of separation: 

  • If the collision is sticky, so that when the reactants collide they orbit around each other the products can be expected to emerge in random directions because all memory of the approach direction has been lost. 
  • A rotation takes about 1 ps, so if the collision is over in less than that time, the complex will not have had time to rotate and the products will be thrown off in a specific direction. 
For example, in the collision of K and I2, most of the products are thrown off in the forward direction. This product distribution is consistent with the harpoon mechanism because the transition takes place at long range. In contrast, the collision of K with Cl2 leads to reaction only if the molecules approach each other very closely. In this mechanism, K effectively bumps into a brick wall, and the KI product bounces out in the backward direction. The detection of this anisotropy in the angular distribution of products gives an indication of the distance and orientation of approach needed for reaction, as well as showing that the event is complete in less than 1 ps.

[ TOP ]

(2-2) Attractive and repulsive surfaces

Some reactions are very sensitive to whether the energy has been predigested into a vibrational mode or left as the relative translational kinetic energy of the colliding molecules. For example: 

  • If two HI molecules are hurled together with more than twice the activation energy of the reaction, then no reaction occurs if all the energy is translational. 
  • For F + HC1 -> Cl + HF, the reaction is about five times as efficient when the HCl is in its first vibrational excited state than when, although HCl has the same total energy, it is in its vibrational ground state.
The origin of these requirements can be found by examining the potential energy surface: Figure 6a shows an attractive surface in which the saddle point occurs early in the reaction coordinate. Figure 6b shows a repulsive surface in which the saddle point occurs late. A surface that is attractive in one direction is repulsive in the reverse direction.

Fig.6a: An attractive potential energy surface. A successful encounter (C*) involves high translational kinetic energy and results in a vibrationally excited product.

Fig. 6b: A repulsive potential energy surface. A successful encounter (C*) involves initial vibrational excitation and the products have high translational kinetic energy. A reaction that is attractive in one direction is repulsive in the reverse direction.

(i) The attractive surface (Fig. 6a)

  • Observations:
    • If the original molecule is vibrationally excited, then a collision with an incoming molecule takes the system along C. This path is bottled up in the region of the reactants, and does not take the system to the saddle point. 
    • If, however, the same amount of energy is present solely as translational kinetic energy, then the system moves along C* and travels smoothly over the saddle point into products. 
  • Conclusions: 
    • Reactions with attractive potential energy surfaces proceed more efficiently the energy is in relative translational motion.
    • Moreover, the potential surface shows that once past the saddle point the trajectory runs up the steep wall of the product valley, a then rolls from side to side as it falls to the foot of the valley as the products separate. In other words, the products emerge in a vibrationally excited state.
(ii) The repulsive surface (Fig. 6b). 
  • Observations:
    • On trajectory C, the collisional energy is largely translational. As the reactants approach, the potential energy rises. Their path takes them up the opposing face of the valley, and they are reflected back into the reactant region. This path corresponds to an unsuccessful encounter, even though tl energy is sufficient for reaction. 
    • On C* some of the energy is in the vibration of the reactant molecule and the motion causes the trajectory to weave from side to side i the valley as it approaches the saddle point. This motion may be sufficient to tip the system round the corner to the saddle point and then on to products. In this case, the product molecule is expected to be in an unexcited vibrational state. 
  • Conclusions:
    • Reactions wit repulsive potential surfaces can therefore be expected to proceed more efficiently if the excess energy is present as vibrations. 

[ TOP ]

(2-3) Classical trajectories (i.e. trajectories obtained through equations of classical mechanics)

A clear picture of the reaction event can be obtained using classical mechanics to calculate the trajectories of the atoms taking place in a reaction. 

  • Fig. 7a shows the result of such a calculation of the positions of the three atoms in the reaction H + H2 -> H2 +H, the horizontal coordinate now being time and the vertical coordinate the separations. This illustration shows clearly the vibration of the original molecule and the approach of the attacking atom. The reaction itself, the switch of partners, takes place very rapidly and is an example of a direct-mode process. The newly formed molecule shakes, but quickly settles down to steady, harmonic vibration as the expelled atom departs. 
  • In contrast, Fig. 7b shows an example of a complex-mode process, in which the activated complex survives for an extended period. The reaction in the illustration is the exchange reaction KCl + NaBr -> KBr + NaCl. The tetraatomic activated complex survives for about 5ps, during which time the atoms make about 15 oscillations before dissociating into products.
Although this kind of calculation gives a good sense of what happens during a reaction, its limitations must be kept in mind. 
  1. In the first place, a real gas-phase reaction occurs with a wide variety of different speeds and angles of attack. 
  2. In the second place, the motion of the atoms, electrons, and nuclei is governed by quantum mechanics. The concept of trajectory then fades and is replaced by the unfolding of a wavefunction that represents initially the reactants and finally the products.
Nevertheless, recognition of these limitations should not be allowed to obscure the fact that recent advances in molecular reaction dynamics have given us a first glimpse of the processes going on at the core of reactions. Also, one is to note that such studies are now extermely feasable using very basic computer software and hardware (see CH407). 

Fig. 7a:  The calculated trajectories for a reactive encounter between HA and a vibrating HBHC molecule leading to the formation of a vibrating HAHB molecule. This direct-mode raection is between H and H2 (M. Karplus et al., J. Chem. Phys., 43 (1965) p.3258)

Fig. 7b: An example of the trajectories calculated for a complex-mode raection, KCl + NaBr -> KBr + NaCl in which the collision cluster has a long lifetime. (P. Brumer and M. Karplus, Faraday Disc. Chem. Soc., 55 (1973) p.80).



E-mail me at