John E. Hulse, Ronald A. Jackson, and James S. Wright1
Carleton University Ottawa, Ontario, Canada
Energy Surfaces, Trajectories, and the Reaction Coordinate
Although the concepts of energy profile and reaction coordinate are widely used in chemistry teaching, their usage tends to remain schematic rather than precise. The familiar diagram showing an energy profile as a function of reaction coordinate often inspires the more perceptive student to ask: "Exactly what is the meaning of 'reaction coordinate?' Where does the energy profile come from? How can something as complex as a chemical reaction be reduced to such a simple description?" To give satisfactory answers to these fundamental questions requires a knowledge of the nature of energy surfaces and trajectories on these surfaces. By using computer illustrations of a rather new type we hope to make the correct answers more easily visualized. For a very clear and detailed treatment of the subject, the reader is referred to the recent book by Laidler, "Theories of Chemical Reaction Rates" (I), and to an old classic, "The Theory of Rate Processes," by Glasstone, Laidler, and Eyring (2). Also, some recent articles (3) have dealt with various aspects of the subject.
Figure 1. A crude piaster m o d e of the LEP surface.
Conventional Representationof Energy Surfaces
The relationship between chemical reactions and energy surfaces was developed by Eyring and Polanyi (4) in the early 1930's. They studied the exchange reaction H + Hz Hz + H, where in the simplest case a linear system was assumed, with an H atom approaching an Hz molecule. Since the system is linear the potential energy can be written in terms of the two internal coordinates shown below
-
where rAe = distance of nucleus A from nucleus B, and rBc = distance of nucleus B from nucleus C. T o obtain the potential energy, Eyring and Polanyi used a modification of the equation derived by London (5) for the Ha system. This function, since referred to as the LEP function (London-Eyring-Polanyi) is given by
where A is a Coulombic integral based on an isolated Hz molecule having distance RBC;B is a Coulombic integral based on an isolated Hz molecule having distance ~ A C= rAB r ~ and ~ 0;is an exchange integral for an Hz molecule a t distance rBc, and so on (2). It was assumed that the Coulomhic integral contrihuted 14% to the total energy and the exchange integral contributed 86% so that A = 0.14 E H ~a, = 0.86 E H ~for , an Hz molecule with distance r,,, and similarly for the other t e r r n ~ A . ~ Morse function for the energy of an Hz molecule completes the formula
+
EH?= De(l -
e-!,t,-r,.,
Y2
- D*
where D, = dissociation energy of Hz (kcal/mole), P = 1.945 A-1 r = H-H distance, A, and re = equilibrium value of Hz distance, 0.74 A. Since for any value of the two coordinates rABand rBc there is a corresponding energy of the Ha system, a three-dimensional representation of the energy is needed. Eyring and coworkers used two 78
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methods to represent the energy function-contour diagrams and plaster models. Figure 1 shows an attempt by one of our students at constructing a model of the LEP surface, and since referred to as the "plaster disaster.'' It shows both the virtues and defects of the modeling method. The viewer is able to get a rather good idea of the general features of the energy surface, and can see the two energy valleys (corresponding to HA + HB-Hc, and HA-HB + Hc) and the transition state region joining them (corresponding to HA---HE---Hc). Also visible are the contours of potential energy indicated along the walls, and the plateau corresponding to three dissociated atoms HA + HB + Hc. On the other hand, the difficulty of construction is obvious and for that reason the use of solid models has not been widespread. Instead, contour diagrams, showing the height of the surface in terms of equal-elevation contours, are commonly used. A contour diagram of the LEP surface showing the same features as the plaster model is given in Figure 2, with contour values of the potential energy in kcall mole. Such contour diagrams are easily generated fbr a function of two variables, using widely distributed contour plotting subroutines (available a t most computer centers). Although the contour diagram shows the same information as the plaster model, there is a significant loss in visual understanding of the nature of the energy surface. Clearly a three-dimensional representation of the energy surface which can be computer generated would combine the advantages of the easily obtained contour plot plus the visual appeal of the plaster model.
To whom correspondence should be addressed. The HZ energy function generated by application of these farmulas is simple to use but inexact, and is referred to as a "semiempirical potential energy function."
Figure 3. 3-D plot of the LEP function, showing a view up one valley.
Figure 2. Contour diagram for the LEP potential energy function. Contours in kcai/rnole.
Three-Dimensional Computer Plots
To accomplish this, we have used computer plots which show a threedimensional perspective view o f ~ t h eenergy surface. The LEP surface in 3-D is shown in Figures 3 and 4. The program used allows the viewer to orient the surface so as to view along any direction desired, e.g., up one of the valleys, looking down from the plateau ( H + H + H region), and so on. The viewing distance of an "eye" can he changed so as to alter the perspective effect. Finally, the maximum and minimum values of TAB and rBc are entered; this allows one to "zoom in" on regions of interest such as the transition state. As is evident in the illustration, the 3-D plot has the appearance of a net draped over the surface. Values of the function are normally calculatFigure 4. 3-0 plat of the LEP function, viewed over the H i H + H plaed a t each point of intersection of the net. Contour lines teau. superimposed on the net allow one to relate position on the 3-D surface to the ordinary contour plot (Fig. 2). Figure 3 shows a view up one valley of the LEP surface. Minimum Reaction Path-the path requiring the least energy to The upper valley represents the reactant valley (HA + traverse from reactants to products, shown as a dashed line HB-Hc) and the valley projecting toward the viewer is the (i.e., the path of steepest descent fmm the saddle point ( 1 0 ) ) . product valley (HA-Hs Hc). The contours shown corThe concepts of reaction coordinate and energy profile are respond to -100, -96, and -90 kcal/mole. The famous more difficult to define. They depend on the particular "well a t the top of the pass," a slight energy basin which conditions of the encounter being studied, and will he deoccurs for the H3 molecule in the LEP approximation, is ferred for later discussion. clearly visible a t -96 kcal/mole. The Morse curve form is T o further illustrate the technique, the more accurate apparent in the foreground, where rgc = 3.0 A, and r A ~ quantum mechanical Ha energy function of Shavitt, Stevaries between 0.5 and 3.0 A. This region of the surface vens, Minn, and Karplus (7) will he used. In their treatthus approximates an H atom (Hc, which is rather far ment, approximate solution of the Schrodinger wave from HA-Hs) and an Hz molecule whose distance can equation for three interacting H atoms (again, linear) a t vary. various internuclear distances led to a table of energy Figure 4 is a view of the same surface, but rotated by values which was closely fitted by an analytical expression 45" to show the equivalence of reactant and product valof the form leys. Some conventional terms which may he defined with respect to this surface are
+
Transition State-the configuration at the col or saddle point separating reactants and products (there are two such saddle points here). Intermediate-a shallow energy well between transition states (H. H . HI. Barrier Height-the height of the potential energy barrier at the transition state (see reference (6) for a discussion of the close relationship between barrier height and activation energy).
The constants published far these expressions in reference (7) contained an error which was reported later. For the correction see Shavitt, I., J . Chem. Phvs., 49, 4048 (1968). For the most accurate potential surface yet published for Ha, see reference (10). Volume 57, Number 2, February 1974
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Figure 6.3-D view of the (truncated) SSMK surface
In these expressions the Cr, and Ci are constants, TAB and variable^.^ The analytical function for H3 SO oh: tained was found to he satisfactory for distances ~ A B . rBc > 1.0 a.u. (0.529 A). Figure 5 shows a contour plot of the SSMK function, with the contours labelled in kcal/mole. The "well a t the top of the pass," an artifact of the LEP function is absent and the function is very smooth. The minimum reaction path (MRP) is shown by ;dashed line. Firmre 6 shows a 3-D view of this surface, with some additio>al features. The surface has been truncated at a chosen value of the energy (upper grid) and lines have been added to simulate a cube. This helps the viewer to see the valley as a trough rather than a hill; otherwise optical illusions are unavoidable.4 Contours are shown a t -101, -96, and -92 kcal/mole. To further relate to the contour plot (Fig. 5) the MRP is shown hy a dashed line on the surface.
For less symmetric surfaces the reaction coordinate can be expressed as a linear combination of normal coordinates of the saddle uoint confirmration ( I ) . However when the reaction coordinate is iefined relative to the triatomic com~lex. . . it clearlv has limitations if one wants to exuress the progress of an entire reaction, from reactants to pmducts. A type of coordinate frequently used in molecular dynamics is simply time. The encounter begins when reactants approach within a certain distance, called the reaction "shell," and ends when products (or reactants) leave the reaction shell.5 An energy profile/reaction coordinate diagram could then consist of a plot of the potential energy of the system (i.e., the height of a point on the potential surface) as a function of time. In fact, this type of plot is not commonly seen in chemical dynamics studies. Instead, one sees trajectory plots, showing the path followed on a contour diagram, or distance-time studies, showing variation of internuclear distances as a function of time (8,9).Neither approach is suitable for our purposes. A generally useful definition of reaction coordinate would seem to require a single-valued measure of progress along the true reaction path, which can be related to the distance traveled in the TAB,~ B Ccoordinate system. The simplest such definition is the following: "The reaction coordinate is the distance traveled in the rra. ..-, rar - - .lane along the true reaction path, relative to an arbitrary startine- uoint . which defines the reaction shell." To clarifv the meaning of true reaction path, what we require is a "motion picture" of relative positions of the atoms in the triatomic system.
Reaction Coordinate and Energy Profile
Trajectories on Energy Surfaces
In the literature of chemical kinetics, the meaning of reaction coordinate has depended on the theoretical framework being used. In activated complex theory, the reaction coordinate is defined in the region of the saddle point as the path of steepest descent from the saddle point toward products and reactants. For symmetric surfaces like Ha this corresponds to an antisymmetric stretching mode of the Ha complex
T o create such a motion nicture or traiectorv. .. the basic assumption is introduced &at the collision process can he described by classical mechanics. Then given initial posi-
Figure 5 . Contour diagram for the SSMK function, showing the minimum reaction path (MRP). for rAB< ~ B C . The energy function for Hz, fitted to the results of their calculation on the isolated molecule, has the form
~ B Care
where
rAB' = A-B 80
distance at the saddlepoint
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'Optical illusions result from the fact that the 3-D program does not delete hidden lines on the surface, i.e., those parts of the surface which would be invisible to the viewer if the net were solid. Programs are available which can delete hidden lines in a 3-D perspective drawing, but they require considerably mare computer time in operation. However, judicious choice of viewing angle and distance minimizes the loss in clarity due to overlapping lines. For a surface which contained more complex features than two valleys and a saddle point, this would be a serious problem. $For instance, one value for the reaction shell which has been used (11) for Hs is rac = 5.0 A. Unless reactants can approach this closely there is no possibility of reaction.
Figure 7. Trajectory on SSMK surface, assuming a relative translational energy of 18 kcai/mole for reactants (the barrier height is 11 kcai/ mole).
tions and momenta of the particles involved, and the form of the potential surface, solution of the classical equations of motion leads to the relative positions of the particles as a function of time. Such an approach has been thoroughly discussed (1, 2). For dynamical calculations for the HJ 'system, the r ~ and e rBc axes are skewed to 60°, to give a faithful representation of the motion of a mass point sliding on the surface, where the mass point is an appropriate reduced mass for the system. A typical trajectory on the SSMK surface is shown in the conventional way in Figure 7, where a trace on the plot indicates the actual path followed. In this trajectory atom A possesses 18 kcal/mole of translational energy relative to molecule B-C, assumed to he not vibrating. The trajectory thus proceeds smoothly up the reactant valley as A approaches B-C (and distance TAB decreases) until the saddle point A . . . B . . . C is reached, after which atom C retreats and A-B is left vibrating (increasing rec distance, fluctuating rAe distance). A better understanding of the trajectory is given by inspection of Figure 8, which is a 3-D plot of the SSMK surface with a sphere placed on the surface a t intervals of 10-l5 s to indicate the progress of the reaction. Motion of the "mass point" can now be seen to he straight up the reactant valley toward the saddle point, and banking off the walls of the product valley. Reaction is seen to have occurred since the trajectory passes from reactant valley to product valley. The net effect has been the convenion of pure translational energy of reactants into a mixture of translational and vibrational energy of products. The time behavior of the system can be inferred from the relative spacing of the mass points (spheres). One would expect the "marble" to slow down as it approaches the saddle point, and in fact, this can he seen by the crowding of spheres in that region. A diagram of this sort thus conveys a considerable amount of useful information about the progress of a reaction. Finally, then, what is the precise meaning of reaction coordinate and energy profile for this particular reaction, characterized by its own set of initial conditions? The distance along the actual path traced out on the surface (Fig. 7) is the reaction coordinate. This is clearly different from the distance along the minimum reaction path. A coordi-
Figure 8. Motion of a mass point an the SSMK surface, using a time interval of s.
Figure 9. Energy profile along the reaction coordinate.
nate of the type q = A ~ A B- A ~ B C is now seen to be unsuitable, since this coordinate need not be single valued for the real path. The energy profile along the true reaction path (TRP) will show the energy of the surface as a function of the distance travelled along the TRP. Since the motion shown in Figure 8 includes banking off the walls of the energy surface as well as progress up the valley, the true energy profile will have the general form of the profile along the MRP, but with superimposed oscillations representatiue of motion not directed along the MRP. The true energy harrier surmounted during the reaction will be greater than or equal to the energy a t the saddle point, since the reaction (crossing the saddle point area) does not necessarily occur at the lowest possible energy. These arguments are summarized in Figure 9, Volume51, Number 2,February 1974 / 81
which shows an energy profile for the MRP and TRP (from Fig. 8) on the same plot. The energy profile along the MRP is a smooth curve which just passes over the saddle point. This occurs a t 11.0 kcal/mole for the SSMK surface, relative to the energy of Hz + H. For the conditions chosen for the example of Figure 8, the excess energy of reactants carries them above the harrier and imparts vibrational energy to the product HZ molecule (periodic oscillations in the energy profile). It should he emphasized that the chosen trajectory represents only one of the infinite number of possible encounters (remember also that we have assumed a collinear H + Hz collision, but in fact collisions at any angle are possible). T o go from trajectories to rate constants ( I ) requires statistical treatment of the results of many trajectories on the surface. Summary
T o use the concepts of reaction coordinate and energy profile for the entire course of a chemical reaction, reference needs to be made to an actual trajectory on an energy surface, given a set of initial conditions which determine the encounter. The curves shown in the typical in-
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tnductory treatment of chemical kinetics are an idealimtion which neglerts the dynamics of a real reaction. The programs used to generate these results are compatible with CAL-COMP plotters, and may he obtained on request from the authors. The authors would like to thank the Carleton University Computer Center for computer time and assistance with the plotting routines, and John Sims for help with programming. Literature Cited (1) Laidler, K. J., "TheoneaoichrmicalReaction Rates," McGrav-Hill, N.Y., 1969. (2) Glaaatone, S., Laidler. K. J., and Eylng. H., "The Theory of Rate Processas? McGrav-Hill. N.Y.. 1941. (3) (a1 Bender. C. P., Peauson, P. K.. O'Neil, S. V., and Schaefer, H. F.. J. C h m . Phvs.. 56, 4626 119721. (bl Hemphill, G. L., and White, J. M.. J. CHEM. EDUC., 49.121 (1972).(c) Shahan, W . F., J. CHEM. EDUC.. 47,254(19701. I41 (a1 Eyling, H., and Polsnyi, M., Z Phva. C h m . 8, 12, 279 (19311. lbl See also Eying, H.. J. Amer. Chsm. Sor.. 53. 2531 (19311:54. 3191 (1932): Chsm R m , I" m m~ a m (5) London, F . , Z E l ~ k l m e h m .35,552(19291. , (6) Menzinger, M.. and Wolfgang, R.,Angew. C h z m 1°C. Ed.,8.438 (19691. , J. Chem. Phus.. 48, 6 Ill Shsvift. I., Stevens, R. M., Minn, F. L., and ~ a r p l u s M., (19681. (8) Wall. P.T., Hiller, L. A,. and Mazur, J , J Cham. Phys.. 2% 255 11958). (9) Karplus. M., Porter. R. N., and Shama. R. O . , J C h m . Phvs. 34,3259(19651 (10) Liu, B.. J Chsm Phva. 58.1925 (19731. (11) Wall. F.T.,Hiller.L.A..andMazur. J.. J C h e m . P h p 29.255 (19581.
