Energy potential, surfaces

Potential Energy Surfaces Barriers, Minima, and Funnels 

Most potential energy surfaces are extremely complex. Fiber and Karplus analyzed a 300 psec molecular dynamics trajectory of the protein myoglobin. They estimate that 2000 thermally accessible minima exist near the native protein structure. The total number of conformations is even larger. Dill derived a formula to calculate the upper bound of thermally accessible conformations in a protein. Using this formula, a protein of 150 residues (the approx- 

Karplus, M. Multiple conformational states of proteins a molecular dynamics analysis of myoglobin. Science 235 318-321, 1987. 

Theory for the folding and stability of globular proteins. Biochemistry 24 1501-1509, 1985. 

United atom force fields (see United versus All Atom Force Fields on page 28) are sometimes used for biomolecules to decrease the number of nonbonded interactions and the computation time. Another reason for using a simplified potential is to reduce the dimensionality of the potential energy surface. This, in turn, allows for more samples of the surface. 

Example Crippen and Snow reported their success in developing a simplified potential for protein folding. In their model, single points represent amino acids. For the avian pancreatic polypeptide, the native structure is not at a potential minimum. However, a global search found that the most stable potential minimum had only a 1.8 Angstrom root-mean-square deviation from the native structure. 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrddinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrddinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

People are interested in potential energy surfaces for a variety of reasons. They might want to 

If we were to calculate the potential energy V of the diatomic molecule AB as a function of the distance tab between the centers of the atoms, the result would be a curve having a shape like that seen in Fig. 5-1. This is a bond dissociation curve, the path from the minimum (the equilibrium intemuclear distance in the diatomic molecule) to increasing values of tab describing the dissociation of the molecule. It is conventional to take as the zero of energy the infinitely separated species. 

Most chemical reactions are more complicated than this one, and the system potential energy is a function of more than one variable. Consider this reaction, which is a generalized group-transfer reaction  

In the simplest case the AB and BC bond axes will remain colinear throughout the reaction then the potential energy can be expressed as a function of the bond 

The potential energy surface consists of two valleys separated by a col or saddle. The reacting system will tend to follow a path of minimum potential energy in its progress from the initial state of reactants (A + BC) to the final state of products (AB -F C). This path is indicated by the dashed line from reactants to products in Fig. 5-2. This path is called the reaction coordinate, and a plot of potential energy as a function of the reaction coordinate is called a reaction coordinate diagram. 

Before discussing the kinds of kinetic information provided by potential energy surfaces we will briefly consider methods for calculating these surfaces, without going into detail, for theoretical calculations are outside the scope of this treatment. Detailed procedures are given by Eyring et ah There are three approaches to the problem. The most basic one is purely theoretical, in the sense that it uses only fundamental physical quantities, such as electronic charge. The next level is the semiempirical approach, which introduces experimental data into the calculations in a limited way. The third approach, the empirical one, makes extensive use of experimental results. 

With this new approach we have been able to obtain full-dimensional potentials using high quality ah initio calculations of electronic energies and dipole moments. Some of these results for H5OJ will be presented below. 

The calculation of the PE surface is basically quantum mechanical. Accurate surfaces are used to show how the topography of the surface affects the reaction unit as it changes configuration across the surface. Predictions can be made, and these can be tested by molecular beams, spectroscopic techniques and chemiluminescence. 

In order to generate an electronic potential energy surface, Eq. (3.1) must be solved at a set of fixed values of the nuclear coordinates R, as indicated by in the electronic wave function. 

let us consider what is meant by internuclear coordinates and, in particular, how many of these coordinates are needed in order to specify the electronic energy. We consider a collection of N atomic nuclei, which in this context are considered as point particles. In the following, we will for convenience refer to any collection of nuclei and electrons as a molecule . The atomic nuclei and the electrons may form one or more stable molecules but this is of no relevance to the following argument. The internuclear coordinates are defined as coordinates that are invariant to overall translation and rotation. These coordinates can, for example, be chosen as internuclear distances and bond angles. 

3N coordinates are needed in order to completely specify the position of the nuclei. Three coordinates are needed in order to specify the position of the center of mass.1 Thus, 3N — 3 coordinates account for the internal degrees of freedom, that is, overall orientation and internuclear coordinates. The overall orientation can be specified by two coordinates for a linear molecule, say by the two polar angles (6, f ). For a nonlinear molecule three coordinates are needed in order to specify the orientation. These coordinates are often chosen as the so-called Euler angles. Thus, for a molecule with N atomic nuclei, 

Note that in a non-linear molecule, one of the vibrational modes of the linear molecule has been replaced by a rotational coordinate. As an illustration, let us consider two examples. For the stable linear triatomic molecule CO2, there are 3 x 3 — 5 = 4 internuclear coordinates, which corresponds to the vibrational degrees of freedom, namely the symmetric and antisymmetric stretch and two (degenerate) bending modes (see Appendix E). For the three atoms in the reaction D + H — H— D — H + H, there are 3 x 3 — 6 = 3 internuclear coordinates. These coordinates can, for example, be chosen as a D H distance, the H H distance, and the I) II H angle. 

The problem of the motion of the nuclei is next solved by letting the slow nuclear subsystem move in a potential field that is determined by the fast electronic subsystem the electronic wavefunction is considered to respond instantaneously to the changes in the nuclear coordinates. 

It is evident that the approximation is expected to work well in situations (such as that near to equilibrium distances in Fig. 1.2) in which the electronic wave-functions are slowly changing functions of the nuclear coordinates, but to be much less valid when (as in the region of the crossing dashed lines of Fig. 1.2) the electronic wavefunctions change abruptly with nuclear motion. In the latter case, there may be some dynamic tendency for the states to preserve their electronic identity instead of following the changes predicted by the adiabatic BO approximation. 

Computing Reaction Pathways on Molecular Potential Energy Surfaces 

Reviews in Computational Chemistry, Volume IV Kenny B. Lipkowitz and Donald B. Boyd, Editors VCH Publishers, Inc. New York, 1993 

Since stable isomers of a molecule are represented as local minima on the PES, it is reasonable to expect that properties of these species are determined by the shape of the PES in the vicinity of these minima. The local minima are fundamental points on the PES. They determine the molecular structure and the moments of inertia by which the rotational spectra can be estimated. The curvature of the many-dimensional PES about the local minima, the steepness of the walls, determines the vibrational properties of the molecule including the normal modes and harmonic vibrational frequencies. 

In Section 8.2 we found that the reaction cross section provides the link between the rate constant and collision dynamics. However, Q , i,J) is just a function of the collision energy and the internal states of the reactants. The outcome of a single collision also depends upon the impact parameter, as demonstrated for the simple hard-sphere model of reaction cross section discussed in Section 8.4. The probability of a specific result p(b, e, i, j) is dependent on all these variables. In Section 8.4 we showed that the relative importance of a collision with impact parameter between b and b + db % proportional to the annular area Inb db. The reaction cross section is found by multiplying the weighting factor for a particular b by the probability of a specific result and integrating over all values of the impact parameter 

The reaction probability measures the outcome of all possible collisions with specified initial values of b, e, i, and j. It is particularly simple for the hard-sphere model of Section 8.4, where 

Recent calculations on the H + Hg — H2 + H exchange reaction come close to carrying out this program see A. Kupperman and G. C. Schatz, /. Chem. Phys, 62, 2502 (1975) A. B. Elkowitz and R. E. Wyatt, J, Chem. Phys. 62, 2504 (1975). 

The calculation of a potential-energy surface is itself not easy. To describe the surface for the exchange reaction 

Many other ways of specifying the ABC triangle are possible but this is the most convenient choice. 

On the basis of the adiabatic approximation, one can calculate, in principle, the nuclear potential energy V(x) of any system of 5 iteracting atoms and molecules. This was demonstrated in the preceding section for the simplest chemical reaction H + H --- Hg  

It is clear that reaction (25a.I) does not require any activation energy, while reaction (25b.I) can occur only if the molecule AB is excited untill an energy level corresponding to its dissotiation energy is reached. 

V (r) on Fig 2 The two curves (x) and Y2M where x = r-j, denoted in Pig.3 by 1 and 2, cross at a point x x at which the elec-tronic energy of reactants equals that of products. They describe to a zero approximation the potential energy change of the whole system during reaction if the electronic coupling of both reactants and and products and is neglected. Moving from 

In the usual approximate treatment, we can describe the electronic state of the whole system A-B-C ) by the wave func- 

Using the variational method, the coefficients c and C2 can be determined based on the condition that the total electronic energy 

So far, we ve considered calculations which investigate a molecular system having a specified geometric structure. As we ve seen, structural changes within a molecule usually produce differences in its energy and other properties. 

There are three minima on this potential surface. A minimum is the bottom of a valley on the potential surface. From such a point, motion in any direction—a 

Exploring Chemistry with Electronic Structure Methods 

All of the above considerations can be illustrated in the course of a discussion of the potential energy surfaces of ground and excited state, and their interaction. 

A particularly clear example is that of the reactions of SO and NO with O3 studied by Thrush [45]. 

Both of these reactions are chemiluminescent and the potential energy surfaces which result in either ground state or excited state formation are shown. 

Bulky substituents can also stabilize RC=SiR relative to RR/C=Si and simultaneously can also stabilize the triply-bonded species toward dimerization to 1,3-disilacyclobutadiene, a process which is highly exothermic [e.g. by 91.3 kcalmol-1, at QCISD(T)/6-31G(d,p) for 70, M = Si, R = H], The effect of several substituents on the RC=SiR vs. RR C=Si energy difference and on the dimerization energy of RC=SiR to the corresponding 1,3-disilacyclobutadiene was recently studied by Kami and Apeloig 

FIGURE 22. Potential energy surfaces for the isomerization of HMCH to H2C=M (energies in kcalmoK1) (a) M = Si, calculated at QCISD(T)/6-31G(d,p)258 340 (b) M = Ge, calculated at QCISD/6-311G(d)356 

Miriam Kami, Yitzhak Apeloig, Jurgen Kapp and Paul von R. Schleyer 

FIGURE 24. Potential energy surfaces of M2H2, M = Si, Ge, Sn and Pb, calculated at MP2/6-311G(2d,p) energies are given in kcalmol-1. Adapted from Reference 343a 

The most satisfactory treatment of the reactions of interest in this chapter is in terms of classical trajectories on potential energy surfaces. They provide a detailed consideration of the reactive interaction (for which the kinematic models are limiting cases7), and provide ample scope for the theoretician to apply his intuition in explaining reactive molecular collisions. Reactions are naturally divided into those which take place on a single surface, usually leading to vibrational excitation, and those which involve two or more surfaces, often leading to electronic excitation. 

The reader should be warned that the calculation of a realistic surface is far from easy, since many crucial approximations have to be made. The interpretation of the results of the study (computed trajectories, efficiency of reaction, distribution of energy) requires a dedicated surface watcher. We shall emphasize qualitative conclusions, in the firm belief that these are the important end products. This use of surfaces is entirely analogous to the use of mechanisms in macroscopic kinetics a model is chosen, its results are determined and these are compared with those derived by the experimenter from his raw data. 

The concept of a potential energy curve or surface is reasonably useful when the coupling between electronic and vibrational motion is small. The potential energy function gives the electronic energy of the system for arbitrary fixed positions of the nuclei. 

All of the surfaces for reactions have more than three dimensions. For a tri-atomic system there are three independent coordinates (3N—6) and the potential energy function V(rlt r2, r3) is a surface in a four dimensional space. The potential function usually shown for a triatomic system ABC is a three dimensional projection of this four dimensional space, the ABC angle being held fixed. Motion restricted to such a projected surface allows no rotation of BC relative to A at large distances and no bending vibration of ABC at short distances. 

In terms of spectroscopic observables, the potential energy function is that function V(r) which, when inserted into the quantum mechanical formulation of the vibration problem, gives the observed vibrational levels. In beam experiments, it is the potential which gives the observed scattering. In chemical excitation processes, it is the surface which predicts the observed total cross section and the observed distribution of products over internal energy states. Potential energy functions may be calculated from first principles14, or they may be constructed 

For polyatomic molecules the electronic potential is a function of more than one internuclear distance. It cannot be graphically represented in the plane of V(r) and r, but requires a space of higher dimensionality. For a molecule of N atoms, 3N Cartesian coordinates are required to specify the position of each atom. The electronic potential energy is independent of location in space and depends only on the relative positions of the atoms, which for any pair x and y is given by the vector Thus, only N— vector distances, or 3N—3 Cartesian coordinates, are required, the discarded three Cartesian coordinates being those that locate the (unnecessary) position of the molecule in space. 

Now consider a reaction between an atom and a diatomic molecule, A + BC — AB + C, where a stable triatomic molecule ABC does not exist. The PES can be described with the same coordinates used for the stable triatomic, but it will have a different shape because the minimum 

The first step to making the theory more closely mimic the experiment is to consider not just one structure for a given chemical formula, but all possible structures. That is, we fully characterize the potential energy surface (PES) for a given chemical formula (this requires invocation of the Born-Oppenheimer approximation, as discussed in more detail in Chapters 4 and 15). The PES is a hypersurface defined by the potential energy of a collection of atoms over all possible atomic arrangements the PES has 3N — 6 coordinate dimensions, where N is the number of atoms 3. This dimensionality derives from the three-dimensional nature of Cartesian space. Thus each structure, which is a point on the PES, can be defined by a vector X where 

Particularly interesting points on PESs include local minima, which correspond to optunal molecular structures, and saddle points (i.e., points characterized by having no slope in any direction, downward curvature for a single coordinate, and upward curvature for all of the other coordinates). Simple calculus dictates that saddle points are lowest energy barriers 

Finally, sometimes slices are chosen so that all structures in the slicing surface belong to a particular symmetry point group. The utility of synunetry will be illustrated in various situations throughout the text. 

With the complete PES in hand (or, more typically, with the region of the PES that would be expected to be chemically accessible under the conditions of the experimental system being modeled), one can take advantage of standard precepts of statistical mechanics (see Chapter 10) to estimate equilibrium populations for situations involving multiple stable molecular structures and compute ensemble averages for physical observables. 

Whereas the equilibrium structure itself is pretty much agreed on for the water dimer, there are a number of other geometries that one might suppose should be comparable in energy. For example, the notion of a cyclic structure has been advanced over the years as have various types of bifurcated geometries. Smith et al. recently completed the most comprehensive survey of the ab initio potential energy surface of the water dimer to date. They examined the question of a number of possible minima on the surface and potential transition states that connect them. 

We address as a preliminary matter the characterization of points on the potential energy surface. A stationary point is one for which the gradient is zero. That is, there are no forces acting on any of the nuclei. A minimum is one such point. A minimum is characterized not only by a zero gradient, but also in terms 

The title reaction is also of considerable interest to combustion [48] and laser chemistries [49]. Theoretically it is one of the best known complex-forming reactions from a fundamental point of view [50]. 

A general overview of the different potential energy surfaces relevant to the reactions involving an oxygen atom and the hydrogen molecule has been sketched in the work of Durand and Chapuisat [51]. 

The first excited state, A A , correlates with the ground state products OH (X /7) -I- H and must also be considered for energies higher than 8.4 kJ mol [22], or temperatures above 500 K. The second excited state, B A, adiabatically correlates with excited state products, OH (A X+) - - H. At collinear geometries these two surfaces correspond to a double degenerate 77 state. 

The other two upper PESs (B A and C A states) are repulsive and correspond to a double degenerate A state at collinear geometries and correlate with excited state products. Their contribution to the title reaction is negligible [22]. 

As a conclusion, in addition to the role of the lowest (X A ) PES the contribution of the two first excited states should be considered. 

The theory of reactive collisions is dealt with in several recent texts. In particular, the works of Levine [118] and Child [119], two volumes edited by Miller [142] and a very comprehensive volume edited by Bernstein [120]. 

The topographical features of a reaction potential energy surface, such as the magnitude of a well or barrier that separate reactants and 

A wide variety of different types of calculations are included in these 

Other potential surface forms have been used for certain specific reactions. The Rittner potential for alkali halides has formed the basis for several calculations of surfaces for reactions involving alkali halides such as [134] 

The Rittner method has also been used as the basis for pseudo-potential calculations for reactions such as 

Most computations begin with a search for chemically interesting structures. Typically, this means locating the structures of ground states and transition states and 

In general, interesting structures on the PES are critical points, points where the gradient of the energy vanishes. Critical points are characterized by the eigenvalues of the Hessian matrix evaluated at the point of interest. The matrix elements of the Hessian are defined as 

Let s consider what happens after the molecule passes over TSl. The molecule continues to follow the reaction path downhill until it reaches the valley ridge inflection (VRI) point. At this point, the gradient in one direction perpendicular to the reaction path becomes zero, and after this point, the PES actually falls downhill faster in the direction off of the reaction path than continuing on the reaction path to TS2. It is important to note that the VRI is not a critical point the gradients are not all zero at a VRI point. So at the VRI, the pathway diverges off of the ridge that leads to TS2, in one direction toward Product 1 and in a second direction toward Product 2. What is so unusual about this type of surface is that the reaction paths that lead from the reactant over TSl then proceed onward to two products (Product 1 and Product 2), without crossing any more transition states. In other words, this surface has a TS that leads to two products  

A contour diagram for unimolecular dissociation of the ethyl radical, C2H5 H 

A potential energy surface may also be represented by a perspective. Such a perspective is given in figure 3.4 for CH4 — CHj + H dissociation. The coordinates used in this perspective arc r the H—C bond length and the angle x which is a 

The coordinates of the CH3 group are varied to minimize the potential energy versus r, but not versus x- 

An important property of a potential energy surface is the reaction path s which connects reactants and products by a path which may pass across a saddle point on the surface. The reaction path for A + BC — AB + C is depicted by the dashed line in figure 3.2. In evaluating reaction paths one finds that they depend on the coordinates used in the analysis. An intrinsic (i.e., unique) reaction path may be found by starting 

Equation (3.1) comprises 3N first-order differential equations which are solved numerically. 

The usual application is to reactions for which the crossing occurs in the region where the reactants are interacting strongly, i.e., during an intimate collision in the vicinity of closest approach. In particular, this has been applied to that challenging reaction 0 (N2,N)N0 , which is truly baroque in its complexity, and, most recently, to the simplest chemical reaction 

Related to this type of approach, but conceptually distinct, is the use of the two-state model, familiar in its application to symmetric charge-transfer reactions. Implicit in the model is the idea that the transition between the initial reactant and final product states may occur at large reactant separations. Bohme have extended this model to consider 

The exothermic reaction 0 (N2,N)N0 has received the principal attention. The experimental data, which demand a unified description, may be summarized as follows (i) The rate constant decreases from 2x10 to 5 X 10 cm molecule sec as the temperature is raised from 80 to indicating the absence of an activation energy, (ii) At 

vibrational excitation of the N2 reactant increases the rate constant by orders of magnitude. (iii) Measurement of the excitation function reveals a cross section of at the lowest collision energy attainable 

O Malley has also tackled the problem of this reaction in terms of a crossing between surfaces, obtaining good quantitative agreement 

The process we have just gone through to obtain the irreducible representations for a number of properties of H2O was a little bit messy, and will quickly get more complicated for bigger molecules. A more rigorous approach will therefore be required. We can, however, finish our introductory discussions here, and return to these points in Section 8.5.3 (for molecular vibrations) and Section 9.3 (for orbitals). 

The fourth aspect of molecular structure, after connectivity, symmetry and geometry, is electron density. This is at the heart of the concept of chemical bonding, and is important in the interpretation of data from both experiments and quantum mechanical calculations. On the one hand, high-resolution X-ray diffraction of well diffracting crystals can provide us with three-dimensional electron density maps (Section 10.9), and on the other ab initio molecular orbital theory and density functional theory allow us to simulate them directly using first-principles calculations (Section 3.6). Either way, we get information with a real physical meaning. 

Morse potential and harmonic (parabolic) potential-energy surfaces for a diatomic molecule. The dissociation energy, Df, represents the energy required to sever the chemical bond. 

Also shown on this plot are the vibrational energy levels (labeled v = 0 — 6) which are discussed in Section 8.2. For now note the v = 0 level this is the vibrational ground-state energy level that all molecules occupy unless they have been vibrationally excited (normally to v = 1) by the absorption of infrared radiation. It is not possible for a molecule to be at the very bottom of the well, because here both the relative positions of the two atoms (separated by and their energy (zero) would be exactly defined. This would be in violation of the Uncertainty Principle, which states that there is a limit to the accuracy with which it is possible to determine simultaneously both the position and momentum (i.e. the energy) of a particle. Note that this applies to the experimental PES only within the mathematically-pure confines of a computational study we can compute both simultaneously to give the vibration-free OK equilibrium structure. 

When faced with a multiple conformer problem, Boltzmann statistics allow us to calculate the relative occupancies of two states, using equation 2.3. 

Yi — 0, pi oo. Therefore, the spin-orbit operators in the Douglas-Kroll Hamiltonian are stable in the variational calculations. 

New transformation formalisms for obtaining two component equations are currently being investigated and being applied to atomic systems. Such studies and others together with implementation of the Douglas-Kroll spin-orbit Hamiltonian, give special importance to this active field of relativistic quantum chemistry. 

---

The result is that, to a very good approxunation, as treated elsewhere in this Encyclopedia, the nuclei move in a mechanical potential created by the much more rapid motion of the electrons. The electron cloud itself is described by the quantum mechanical theory of electronic structure. Since the electronic and nuclear motion are approximately separable, the electron cloud can be described mathematically by the quantum mechanical theory of electronic structure, in a framework where the nuclei are fixed. The resulting Bom-Oppenlieimer potential energy surface (PES) created by the electrons is the mechanical potential in which the nuclei move. Wlien we speak of the internal motion of molecules, we therefore mean essentially the motion of the nuclei, which contain most of the mass, on the molecular potential energy surface, with the electron cloud rapidly adjusting to the relatively slow nuclear motion. 

Figure Al.4.6. A cross-section of the potential energy surface of PH. The coordinate p is defined in figure Al.4.5.

For a molecule that has no observable tiumelling between minima on the potential energy surface (i.e., for a... 

This section discusses how spectroscopy, molecular beam scattering, pressure virial coeflScients, measurements on transport phenomena and even condensed phase data can help detemiine a potential energy surface. 

The theory coimecting transport coefficients with the intemiolecular potential is much more complicated for polyatomic molecules because the internal states of the molecules must be accounted for. Both quantum mechanical and semi-classical theories have been developed. McCourt and his coworkers [113. 114] have brought these theories to computational fruition and transport properties now constitute a valuable test of proposed potential energy surfaces that... 

The complete intemiolecular potential energy surface depends upon the intemiolecular distance and up to five angles, as discussed in section Al.5,1.3. 

Jeziorski B, Moszynski R and Szalewicz K 1994 Perturbation theory approach to intermolecular potential energy surfaces of van der Waals complexes Chem. Rev. 94 1887... 

Hu C H and Thakkar A J 1996 Potential energy surface for interactions between N2 and He ab initio calculations, analytic fits, and second virial coefficients J. Chem. Phys. 104 2541... 

LeRoy R J and Carley J S 1980 Spectroscopy and potential energy surfaces of van der Waals molecules Adv. Chem. Phys. 42 353... 

LeRoy R J and Hutson J M 1987 Improved potential energy surfaces for the interaction of H2 with Ar, Kr and Xe J. Chem. Phys. 86 837... 

Figure Al.6.10. (a) Schematic representation of the three potential energy surfaces of ICN in the Zewail experiments, (b) Theoretical quantum mechanical simulations for the reaction ICN ICN [I--------------...

CN] —> I + CN. Wavepacket moves and spreads in time, with its centre evolving about 5 A in 200 fs. Wavepacket dynamics refers to motion on the intennediate potential energy surface B. Reprinted from Williams S O and lime D G 1988 J. Phys. Chem.. 92 6648. (c) Calculated FTS signal (total fluorescence from state C) as a fiinction of the time delay between the first excitation pulse (A B) and the second excitation pulse (B -> C). Reprinted from Williams S O and Imre D G, as above. 

Figure Al.6.20. (Left) Level scheme and nomenclature used in (a) single time-delay CARS, (b) Two-time delay CARS ((TD) CARS). The wavepacket is excited by cOp, then transferred back to the ground state by with Raman shift oij. Its evolution is then monitored by tOp (after [44])- (Right) Relevant potential energy surfaces for the iodine molecule. The creation of the wavepacket in the excited state is done by oip. The transfer to the final state is shown by the dashed arrows according to the state one wants to populate (after [44]).

Figure Al.6.26. Stereoscopic view of ground- and excited-state potential energy surfaces for a model collinear ABC system with the masses of HHD. The ground-state surface has a minimum, corresponding to the stable ABC molecule. This minimum is separated by saddle points from two distmct exit chaimels, one leading to AB + C the other to A + BC. The object is to use optical excitation and stimulated emission between the two surfaces to steer the wavepacket selectively out of one of the exit chaimels (reprinted from [54]).

Figure Al.6.27. Equipotential contour plots of (a) the excited- and (b), (c) ground-state potential energy surfaces. (Here a hamionic excited state is used because that is the way the first calculations were perfomied.) (a) The classical trajectory that originates from rest on the ground-state surface makes a vertical transition to the excited state, and subsequently undergoes Lissajous motion, which is shown superimposed, (b) Assuming a vertical transition down at time (position and momentum conserved) the trajectory continues to evolve on the ground-state surface and exits from chaimel 1. (c) If the transition down is at time 2 the classical trajectory exits from chaimel 2 (reprinted from [52]).

It is the relationship between the bound potential energy surface of an adsorbate and the vibrational states of the molecule that detemiine whether an adsorbate remains on the surface, or whether it desorbs after a period of time. The lifetime of the adsorbed state, r, depends on the size of the well relative to the vibrational energy inlierent in the system, and can be written as... 

Rosker M J, Rose T S and Zewail A 1988 Femtosecond real-time dynamics of photofragment-trapping resonances on dissociative potential-energy surfaces Ghem. Phys. Lett. 146 175-9... 

Because of the general difficulty encountered in generating reliable potentials energy surfaces and estimating reasonable friction kernels, it still remains an open question whether by analysis of experimental rate constants one can decide whether non-Markovian bath effects or other influences cause a particular solvent or pressure dependence of reaction rate coefficients in condensed phase. From that point of view, a purely... 

There are significant differences between tliese two types of reactions as far as how they are treated experimentally and theoretically. Photodissociation typically involves excitation to an excited electronic state, whereas bimolecular reactions often occur on the ground-state potential energy surface for a reaction. In addition, the initial conditions are very different. In bimolecular collisions one has no control over the reactant orbital angular momentum (impact parameter), whereas m photodissociation one can start with cold molecules with total angular momentum 0. Nonetheless, many theoretical constructs and experimental methods can be applied to both types of reactions, and from the point of view of this chapter their similarities are more important than their differences. 

---