Classical Studies of Reaction Mechanisms

Contents

1. Introduction
2. Quasiclassical Trajectories
3. The Hydrogen Exchange Reaction
   • The H₃ Potential Energy Surface
   • The Conical Intersection
   • The Direct Recoil Mechanism
4. New Studies into Reaction Mechanisms
   • New, Unexpected Mechanisms
   • Time-delayed Forward Scattering
   • Inelastic Scattering
   • The Geometric Phase Effect
5. References

1. Introduction

We use quasi-classical trajectory (QCT) calculations to model the dynamics of bimolecular reactions involving three atoms: A + BC → AB + C. Our aim is to understand how the individual atoms move during a collision and which common pathways (mechanisms) lead to reaction.

The current system under investigation is the hydrogen exchange reaction in its H + D₂ → HD + D isotopic form. This reaction was thought to be well understood, however, our recent detailed calculations have revealed new and unexpected reaction mechanisms.

2. Quasiclassical Trajectories

In quasi-classical trajectory calculations, the atoms are treated like classical particles, e.g. billiard or golf balls. The motion of the atoms is governed by the potential energy surface (PES) which describes the forces between the atoms and which can be thought of as the mounds and troughs of a golf course. However, the potential energy surface changes with the positions of the atoms (a really crazy golf course). In a way, the classical trajectories are like playing billiard with atoms on a pool table with ever changing mountains and valleys.

At the start of each trajectory, the positions and speeds of the atoms are set-up according to a designated quantum state of the BC molecule, i.e. with the specific vibrational and rotational motions. Other initial conditions are randomly sampled over the full range of orientations, distances, etc. The trajectory is then run without any quantum restrictions, i.e. like billiard balls in the mountain range of the PES, until the particles separate into a product molecule and atom. The positions and speeds of the atoms is then analysed to assign the quasi-quantum state of the product molecule and the direction in which the products fly apart (the scattering angle).

Using QCT we can 'look' inside the reaction to determine mechanisms, view the transition state behaviour (where chemical bonds are broken and formed), examine the possible presence of resonances, etc.

3. The Hydrogen Exchange Reaction

The hydrogen exchange reaction, H + H₂ → H₂ + H, has been the prototype for bimolecular reactions since 1929. It is the simplest chemical reaction, with only three electrons and three protons present in the system, and as such it has been studied intensively for the past 75 years. More powerful computers and more ingenius computational methods have made accurate quantum mechanical calculations on this system possible which can be compared with very detailed data from recent, highly sophisticated experiments. The very good agreement between these theoretical and experimental studies lead to the conclusion that this prototype chemical reaction is solved and well understood theoretically.

The quantum nature of the hydrogen atoms involved in this reaction requires a quantum mechanical treatment of the dynamics (i.e., the motion of the atoms) of this reaction. However, quasi-classical trajectory calculations have shown that the essential dynamics of this reaction can treated using classical (Newtonian) mechanics. The main advantage of classical mechanics is that the path (i.e. the trajectory) of the atoms can be traced and analysed during the reaction which is not possible by quantum methods.

The H₃ Potential Energy Surface

Both quantum mechanical and QCT calculations require the potential energy surface (PES) for the H₃ system. The PES does not dependent on the masses of the atoms, thus, the H₃ surface can be used for any isotopic variant of this reaction. As three nuclei (ABC) are involved, the PES depends on three coordinates. Thus, one coordinate needs to be fixed in order to plot the PES as a function of the remaining two.

A common way is to plot the potential in molecular coordinates, i.e., as a function of the two shortest distances (R_AB and R_BC) at a fixed A-B-C bending angle (α). This projection highlights the passage of the reactants over the transition state to products. Figure 1 shows the H₃ PES for the linear A-B-C geometry (bending angle α = 180°). In this geometry, atom A can approach BC at the bottom of the potential on the so called minimum energy path (MEP).

Figure 1: Contour plot of the H3 potential energy surface in linear geometry (A-B-C bending angle α = 180°). The minimum energy path (MEP) from the reactant (RAB large) to the product valley (RBC large) is indicated. The diagonal line separates the reactant and product regions of the potential. Both lines cross at the transition state (saddle point on the potential).

Often, we find it more instructive to fix the internuclear distance of the reactant molecule (R_BC) and plot the potential in as a function of the position of the third atom (A) with respect to the BC centre-of-mass in Cartesian (x,y) coordinates. This way, the "shape" of the reactant BC molecule, as experienced by the incoming atom A, is more apparent: the atom is repelled by rising hills and attracted by lower lying wells on the potential. The H₃ PES is plotted in Cartesian coordinates in Fig. 2 below.

Figure 2: Contour plot of the H3 potential energy surface in Cartesian coordinates. The H-H distance of the reactant H2 molecule is fixed at RBC = 1.044 Å and the potential is plotted as a function of the (x,y) position of the third H atom. If the third H atom drops into the well at linear geometry it can form a new bond with the nearest atom of the former molecule while the bond to the remaining atom is broken. Note that the potential is cylindrically symmetric around the x axis.

The Conical Intersection

One striking feature of the H₃ PES is the conical intersection. At equilateral triangular geometry (D_3h symmetry) the energies of the ground and first excited electronic states of the H₃ system are equal, i.e., both potential surfaces touch. Fig. 3 below shows a 3-dimensional surface plot of both potentials. Around the intersection point, the two surfaces form a double cone, hence the name.

Figure 3: 3D surface plot of the ground and first excited state of the H3 potential energy surface. At equilateral triangular geometry (D3h symmetry), the ground state energy surface touches the surface of the first excited state and forms a conical intersection (cf. Fig. 1).

The Direct Recoil Mechanism

The shape of the PES dictates how the atoms will move during their reactive encounter.

4. New Studies into Reaction Mechanisms

New, Unexpected Mechanisms

[under construction]

Time-delayed Forward Scattering

[under construction]

Inelastic Scattering

In contrast to reactions, where the product molecule is different from the reactant molecule, e.g. H + D₂ → HD + D, inelastic collisions do not change the reactant species. However, the internal motions of the outgoing molecule may change. For H + D₂ collisions this means

H + D₂(v = 0, j = 0) &rarr H + D₂(v', j')

[under construction]

The Geometric Phase Effect

[under construction]

5. References

New, unexpected and dominant mechanisms in the hydrogen exchange reaction.
S. J. Greaves, D. Murdock, E. Wrede, and S. C. Althorpe.
J. Chem. Phys. 128 (2008), 164306.

A quasi-classical trajectory study of the time-delayed forward scattering in the hydrogen exchange reaction.
S. J. Greaves, D. Murdock, and E. Wrede.
J. Chem. Phys. 128 (2008), 164307.

Observation and interpretation of a time-delayed mechanism in the hydrogen-exchange reaction.
S. C. Althorpe, F. Fernández-Alonso, B. D. Bean, J. D. Ayers, A. E. Pomerantz, R. N. Zare, and E. Wrede.
Nature 416 (2002), 67.

Vibrational excitation through tug-of-war inelastic collisions.
S. J. Greaves, E. Wrede, N. T. Goldberg, J. Zhang, D. J. Miller, and R. N. Zare.
Nature 454 (2008), 88.

Theoretical Study of Geometric Phase Effects in the Hydrogen-Exchange Reaction.
J. C. Juanes-Marcos, S. C. Althorpe and E. Wrede.
Science 309 (2005), 1227.

Effect of the geometric phase on the dynamics of the hydrogen-exchange reaction.
J. C. Juanes-Marcos, S. C. Althorpe, and E. Wrede.
J. Chem. Phys. 126 (2007), 044317.

Eckart Wrede, 11 Apr 2008 [under construction]

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