Chapter 8. Chemical Dynamics Chemical dynamics is a field in which scientists study the rates and mechanisms of chemical reactions. It also involves the study of how energy is transferred among molecules as they undergo collisions in gas-phase or condensed-phase environments Therefore, the experimental and theoretical tools used to probe chemical dynamics must be capable of monitoring the chemical identity and energy content (i.e, electronic, vibrational, and rotational state populations) of the reacting species. Moreover, because the rates of chemical reactions and energy transfer are of utmost importance, these tools must be capable of doing so on time scales over which these processes, which are often very fast, take place. Let us begin by examining many of the most commonly employed theoretical models for simulating and understanding the processes of chemical dynamics L. Theoretical Tools for Studying Chemical Change and dynamics A. Transition State Theor The most successful and widely employed theoretical approach for studying reaction rates involving species that are undergoing reaction at or near thermal-equilibrium conditions is the transition state theory (Tst)of Eyring. This would not be a good way to model, for example, photochemical reactions in which the reactants do not reach thermal
1 Chapter 8. Chemical Dynamics Chemical dynamics is a field in which scientists study the rates and mechanisms of chemical reactions. It also involves the study of how energy is transferred among molecules as they undergo collisions in gas-phase or condensed-phase environments. Therefore, the experimental and theoretical tools used to probe chemical dynamics must be capable of monitoring the chemical identity and energy content (i.e., electronic, vibrational, and rotational state populations) of the reacting species. Moreover, because the rates of chemical reactions and energy transfer are of utmost importance, these tools must be capable of doing so on time scales over which these processes, which are often very fast, take place. Let us begin by examining many of the most commonly employed theoretical models for simulating and understanding the processes of chemical dynamics. I. Theoretical Tools for Studying Chemical Change and Dynamics A. Transition State Theory The most successful and widely employed theoretical approach for studying reaction rates involving species that are undergoing reaction at or near thermal-equilibrium conditions is the transition state theory (TST) of Eyring. This would not be a good way to model, for example, photochemical reactions in which the reactants do not reach thermal
equilibrium before undergoing significant reaction progress. However, for most thermal reactions, it is remarkably successful In this theory, one views the reactants as undergoing collisions that act to keep all of their degrees of freedom(translational, rotational, vibrational, electronic)in thermal equilibrium. Among the collection of such reactant molecules, at any instant of time some will have enough internal energy to access a transition state (ts)on the born Oppenheimer ground state potential energy surface. Within tsT, the rate of progress from reactants to products is then expressed in terms of the concentration of species that exist near the Ts multiplied by the rate at which these species move through the ts region of the energy surface The concentration of species at the Ts is, in turn, written in terms of the equilibrium constant expression of statistical mechanics discussed in Chapter 7. For example, for a bimolecular reaction A+B>C passing through a ts denoted ab one writes the concentration(in molecules per unit volume)of AB species in terms of the concentrations of A and of b and the respective partition functions as AB=(qAB/V(V(BVAjB
2 equilibrium before undergoing significant reaction progress. However, for most thermal reactions, it is remarkably successful. In this theory, one views the reactants as undergoing collisions that act to keep all of their degrees of freedom (translational, rotational, vibrational, electronic) in thermal equilibrium. Among the collection of such reactant molecules, at any instant of time, some will have enough internal energy to access a transition state (TS) on the BornOppenheimer ground state potential energy surface. Within TST, the rate of progress from reactants to products is then expressed in terms of the concentration of species that exist near the TS multiplied by the rate at which these species move through the TS region of the energy surface. The concentration of species at the TS is, in turn, written in terms of the equilibrium constant expression of statistical mechanics discussed in Chapter 7. For example, for a bimolecular reaction A + B ® C passing through a TS denoted AB*, one writes the concentration (in molecules per unit volume) of AB* species in terms of the concentrations of A and of B and the respective partition functions as [AB*] = (qAB*/V)/{(qA/V)( qB /V)} [A] [B]
There is, however, one aspect of the partition function of the Ts species that is specific to this theory. The qAb. contains all of the usual translational, rotational, vibrational, and electronic partition functions that one would write down, as we did in Chapter 7, for a conventional Ab molecule except for one modification. It does not contain a (exp(-hv, /2kT)(1-exp(-hv/kT)), vibrational contribution for motion along the one internal coordinate corresponding to the reaction path Second Order Saddle Point Transition structure a Transition Structure B Minimurn for Product A for Product B 05 Second Order o 0.5 Saddle paint Valley. Ridge for Reactant Infection Point Figure 8. 1 Typical Potential Energy Surface in Two Dimensions Showing Local Minima, Transition States and Paths Connecting them In the vicinity of the Ts, the reaction path can be identified as that direction along which the pes has negative curvature; along all other directions, the energy surface is positively curved. For example, in Fig 8. 1, a reaction path begins at Transition Structure B and is directed"downhill". More specifically, if one knows the gradients((aE/aSk))and
3 There is, however, one aspect of the partition function of the TS species that is specific to this theory. The qAB* contains all of the usual translational, rotational, vibrational, and electronic partition functions that one would write down, as we did in Chapter 7, for a conventional AB molecule except for one modification. It does not contain a {exp(-hnj /2kT)/(1- exp(-hnj /kT))} vibrational contribution for motion along the one internal coordinate corresponding to the reaction path. Figure 8.1 Typical Potential Energy Surface in Two Dimensions Showing Local Minima, Transition States and Paths Connecting Them. In the vicinity of the TS, the reaction path can be identified as that direction along which the PES has negative curvature; along all other directions, the energy surface is positively curved. For example, in Fig. 8.1, a reaction path begins at Transition Structure B and is directed "downhill". More specifically, if one knows the gradients {(¶E/¶sk) }and
Hessian matrix elements Hik=8E/as, Os of the energy surface at the Ts, one can express the variation of the potential energy along the 3N Cartesian coordinates s, of the molecule as follows: E(SK=E(0)+2x(aE/OSk)Sk +1/2 Ei kS;Hi kSk where E(O) is the energy at the Ts, and the is,i denote displacements away from the ts geometry. Of course, at the ts, the gradients all vanish because this geometry corresponds to a stationary point. As we discussed in the Background Material, the Hessian matrix Hk has 6 zero eigenvalues whose eigenvectors correspond to overall translation and rotation of the molecule. This matrix has 3N-7 positive eigenvalues whose eigenvectors correspond to the vibrations of the TS species, as well as one negative eigenvalue. The latter has an eigenvector whose components(s, along the 3N Cartesian coordinates describe the direction of the reaction path as it begins its journey from the ts backward to reactants(when followed in one direction) and onward to products( when followed in the opposite direction). Once one moves a small amount along the direction of negative curvature, the reaction path is subsequently followed by taking infinitesimal steps "downhill along the gradient vector g whose 3n components are(aE/as ) Note that once one has moved downhill away from the ts by taking the initial step along the negatively curved direction, the gradient no longer vanishes because one is no longer le stationary point Returning to the tsT rate calculation, one therefore is able to express the concentration [AB"of species at the ts in terms of the reactant concentrations and a
4 Hessian matrix elements { Hj,k = ¶ 2E/¶sj¶sk}of the energy surface at the TS, one can express the variation of the potential energy along the 3N Cartesian coordinates {sk} of the molecule as follows: E (sk) = E(0) + Sk (¶E/¶sk) sk + 1/2 Sj,k sj Hj,k sk + … where E(0) is the energy at the TS, and the {sk} denote displacements away from the TS geometry. Of course, at the TS, the gradients all vanish because this geometry corresponds to a stationary point. As we discussed in the Background Material, the Hessian matrix Hj,k has 6 zero eigenvalues whose eigenvectors correspond to overall translation and rotation of the molecule. This matrix has 3N-7 positive eigenvalues whose eigenvectors correspond to the vibrations of the TS species, as well as one negative eigenvalue. The latter has an eigenvector whose components {sk} along the 3N Cartesian coordinates describe the direction of the reaction path as it begins its journey from the TS backward to reactants (when followed in one direction) and onward to products (when followed in the opposite direction). Once one moves a small amount along the direction of negative curvature, the reaction path is subsequently followed by taking infinitesimal “steps” downhill along the gradient vector g whose 3N components are (¶E/¶sk ). Note that once one has moved downhill away from the TS by taking the initial step along the negatively curved direction, the gradient no longer vanishes because one is no longer at the stationary point. Returning to the TST rate calculation, one therefore is able to express the concentration [AB*] of species at the TS in terms of the reactant concentrations and a
ratio of partition functions. The denominator of this ratio contains the conventional partition functions of the reactant molecules and can be evaluated as discussed in Chapter 7. However, the numerator contains the partition function of the TS species but with one vibrational component missing (i.e, vib=Ik-13N-7exp(-hv /2KT)(1-exp(-hv /kT))) Other than the one missing vib, the Ts's partition function is also evaluated as in Chapter 7. The motion along the reaction path coordinate contributes to the rate expression in terms of the frequency (i.e, how often) with which reacting flux crosses the ts region given that the system is in near-thermal equilibrium at temperature t. pute the frequency with which traje ross the Ts and proceed onward to form products, one imagines the TS as consisting of a narrow region along the reaction coordinate S; the width of this region we denote ds. We next ask what the classical weighting factor is for a collision to have momentum ps along the reaction coordinate. Remembering our discussion of such matters in Chapter 7, we know that the momentum factor entering into the classical partition function for translation along the reaction coordinate is(1/h)exp(- P /2ukT)dps. Here, u is the mass factor associated with the reaction coordinate s. We can express the rate or frequency at which such trajectories pass through the narrow region of width Ss as(p/uds), with p /u being the speed of passage(cm s" )and 1/8s being the inverse of the distance that defines the ts region. So (p /uos has units of s". In summary, we expect the rate of trajectories moving through the Ts region to be (/h)exp(-ps 12ukt)dps(p/uos)
5 ratio of partition functions. The denominator of this ratio contains the conventional partition functions of the reactant molecules and can be evaluated as discussed in Chapter 7. However, the numerator contains the partition function of the TS species but with one vibrational component missing (i.e., qvib = Pk=1,3N-7 {exp(-hnj /2kT)/(1- exp(-hnj /kT))}). Other than the one missing qvib, the TS's partition function is also evaluated as in Chapter 7. The motion along the reaction path coordinate contributes to the rate expression in terms of the frequency (i.e., how often) with which reacting flux crosses the TS region given that the system is in near-thermal equilibrium at temperature T. To compute the frequency with which trajectories cross the TS and proceed onward to form products, one imagines the TS as consisting of a narrow region along the reaction coordinate s; the width of this region we denote ds. We next ask what the classical weighting factor is for a collision to have momentum ps along the reaction coordinate. Remembering our discussion of such matters in Chapter 7, we know that the momentum factor entering into the classical partition function for translation along the reaction coordinate is (1/h) exp(-ps 2 /2mkT) dps . Here, m is the mass factor associated with the reaction coordinate s. We can express the rate or frequency at which such trajectories pass through the narrow region of width ds as (ps /mds), with ps /m being the speed of passage (cm s-1) and 1/ds being the inverse of the distance that defines the TS region. So, (ps /mds) has units of s-1. In summary, we expect the rate of trajectories moving through the TS region to be (1/h) exp(-ps 2 /2mkT) dps (ps /mds )