Potential energy surfaces

One of the most important concepts for discussing beam results and calculating the state-to-state collision cross-section is the potential energy surface of a reaction, the potential energy as a function of the relative positions of all the atoms taking part in the reaction. Potential energy surfaces may be constructed from experimental data, with the techniques described in Section 24.6, and from results of quantum chemical calculations (Section 11.7). The theoretical method requires the systematic calculation of the energies of the system in a large number of geometrical arrangements. Special computational techniques are used to take into account electron correlation, which arises from instantaneous interactions between electrons as they move closer to and farther from each other in molecule or molecular cluster. Techniques that incorporate electron correlation are very time-consuming and, consequently, only reactions between relatively small particles, such as the reactions H + H₂ → H₂ + H and H+H₂O→OH+H₂, are amenable to this type of theoretical treatment. An alternative is to use semi-empirical methods, in which results of calculations and experimental parameters are used to construct the potential energy surface.

To illustrate the features of a potential energy surface we consider the collision between an H atom and an H2molecule. 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. Two para meters are required to define the nuclear separations: one is the HA-HB separation RAB, and the other is the HB-HC separation RBC. At the start of the encounter RAB is infinite and RBC is the H₂ equilibrium bond length. At the end of a successful reactive encounter RAB is equal to the equilibrium bond length and RBC is infinite. The total energy of the three-atom system depends on their relative separations, and can be found by doing a molecular orbital calculation. The plot of the total energy of the system against RAB and RBC gives the potential energy surface of this collinear reaction (Fig. 24.14). This surface is normally depicted as a contour diagram (Fig. 24.15). When RAB is very large, the variations in potential energy represented by the sur face as RBC changes are those of an isolated H₂ molecule as its bond length is altered. A section through the surface at RAB =∞, for example, is the same as the H2 bonding potential energy curve drawn in Fig. 11.16. At the edge of the diagram where RBC is very large, a section through the surface is the molecular potential energy curve of an isolated H_AH_B molecule.

Fig. 24.14 The potential energy surface for the H + H₂ →H₂+H reaction when the atoms are constrained to be collinear.

Fig. 24.15 The contour diagram (with contours of equal potential energy) corresponding to the surface in Fig. 24.14. Re marks the equilibrium bond length of an H2 molecule (strictly, it relates to the arrangement when the third atom is at infinity).

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. However, we can obtain an initial idea of the paths available to the system for paths that correspond to least potential energy. For example, consider the changes in potential energy as HA approaches HBHC. If the HB-HC bond length is constant during the initial approach of HA, then the potential energy of the H₃ cluster rises along the path marked A in Fig. 24.16. We see that the potential energy reaches a high value as HA is pushed into the molecule and then decreases sharply as HC breaks off and separates to a great distance. An alternative reaction path can be imagined (B) in which the HB-HC bond length increases while HA is still far away. Both paths, although feasible if the molecules have sufficient initial kinetic energy, take the three atoms to regions of high potential energy in the course of the encounter.

Fig. 24.16 Various trajectories through the potential energy surface shown in Fig. 24.15. 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.

The path of least potential energy is the one marked C, corresponding 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‡. The encounter of least potential energy is one in which the atoms take route C 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 Section 24.4. We can now make contact with the transition state 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 (Fig. 24.17).

Fig. 24.17 The transition state is a set of configurations (here, marked by the line across the saddle point) through which successful reactive trajectories must pass.

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مواضيع ذات صلة

Quantum mechanical scattering theory

Classical trajectories

Attractive and repulsive surfaces

The direction of attack and separation

State-to-state dynamics

Experimental probes of reactive collisions

Photosensitization

Rate laws of complex photochemical reactions

The overall quantum yield of a photochemical reaction

Electron transfer reactions

Resonance energy transfer

The primary quantum yield