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 )
However, we still need to integrate this over all values of ps that correspond to enough energy ps /2u to access the TSs energy, which we denote E*. Moreover, we have to account for the fact that it may be that not all trajectories with kinetic energy equal to e or greater pass on to form product molecules, some trajectories may pass through the Ts but later recross the TS and return to produce reactants. Moreover, it may be that some trajectories with kinetic energy along the reaction coordinate less than e* can react by tunneling through the barrier The way we account for the facts that a reactive trajectory must have at least E* in energy along s and that not all trajectories with this energy will react is to integrate over only values of ps greater than (2ue")and to include in the integral a so-called transmission coefficient k that specifies the fraction of trajectories crossing the ts that eventually proceed onward to products. Putting all of these pieces together, we carry out the integration over ps just described to obtain SS(/h)K exp(-P32/2ukT(p/u8 ) ds dp where the momentum is integrated from ps =(2uE*)to oo and the s-coordinate is integrated only over the small region 8s. If the transmission coefficient is factored out of the integral(treating it as a multiplicative factor), the integral over ps can be done and yields the following (kT/h)exp(E*/kT) 6
6 However, we still need to integrate this over all values of ps that correspond to enough energy ps 2 /2m to access the TS’s energy, which we denote E*. Moreover, we have to account for the fact that it may be that not all trajectories with kinetic energy equal to E* or greater pass on to form product molecules; some trajectories may pass through the TS but later recross the TS and return to produce reactants. Moreover, it may be that some trajectories with kinetic energy along the reaction coordinate less than E* can react by tunneling through the barrier. The way we account for the facts that a reactive trajectory must have at least E* in energy along s and that not all trajectories with this energy will react is to integrate over only values of ps greater than (2mE*)1/2 and to include in the integral a so-called transmission coefficient k that specifies the fraction of trajectories crossing the TS that eventually proceed onward to products. Putting all of these pieces together, we carry out the integration over ps just described to obtain: ò ò (1/h) k exp(-ps 2 /2mkT) (ps /mds ) ds dps where the momentum is integrated from ps = (2mE*)1/2 to ¥ and the s-coordinate is integrated only over the small region ds. If the transmission coefficient is factored out of the integral (treating it as a multiplicative factor), the integral over ps can be done and yields the following: k (kT/h) exp(-E*/kT)
The exponential energy dependence is usually then combined with the partition function of the TS species that reflect this speciesother 3N-7 vibrational coordinates and momenta and the reaction rate is then expressed as Rate=K(kT/h)AB*=K(kT/h)(gaB//(qA/V(qB/AJb] This implies that the rate coefficient krate for this bimolecular reaction is given in terms of molecular partition functions by krate K kT/h(qaB V)/(n/v(qB/v)i which is the fundamental result of TsT. Once again we notice that ratios of partition functions per unit volume can be used to express ratios of species concentrations(in number of molecules per unit volume), just as appeared in earlier expressions equilibrium constants as in Chapter 7 The above rate expression undergoes only minor modifications when unimolecular reactions are considered. For example, in the hypothetical reaction A>B via the tS(A"), one obtains krat K kT/hI(qA V(qav)) where again qa is a partition function of A* with one missing vibrational component
7 The exponential energy dependence is usually then combined with the partition function of the TS species that reflect this species’ other 3N-7 vibrational coordinates and momenta and the reaction rate is then expressed as Rate = k (kT/h) [AB*] = k (kT/h) (qAB* /V)/{(qA/V)( qB /V)} [A] [B]. This implies that the rate coefficient krate for this bimolecular reaction is given in terms of molecular partition functions by: krate = k kT/h (qAB*/V)/{(qA/V)(qB /V)} which is the fundamental result of TST. Once again we notice that ratios of partition functions per unit volume can be used to express ratios of species concentrations (in number of molecules per unit volume), just as appeared in earlier expressions for equilibrium constants as in Chapter 7. The above rate expression undergoes only minor modifications when unimolecular reactions are considered. For example, in the hypothetical reaction A ® B via the TS (A*), one obtains krate = k kT/h {(qA*/V)/(qA/V)}, where again qA* is a partition function of A* with one missing vibrational component
Before bringing this discussion of Tst to a close, I need to stress that this theory is not exact. It assumes that the reacting molecules are nearly in thermal equilibrium, so it is less likely to work for reactions in which the reactant species are prepared in highly non- equilibrium conditions. Moreover, it ignores tunneling by requiring all reactions to proceed through the ts geometry. For reactions in which a light atoms (i.e, an H or d atom)is transferred, tunneling can be significant, so this conventional form of TST can provide substantial errors in such cases. Nevertheless, TST remains the most widely used and successful theory of chemical reaction rates and can be extended to include tunneling and other corrections as we now illustrate B. Variational Transition State Theory Within the tST expression for the rate constant of a bi-molecular reaction, krate=K kT/h(qab/V(qv(aB for of a uni-molecular reaction, krate =K kT/h i(qav)/(qa vi, the height(E*)of the barrier on the potential energy surface ppears in the TS species'partition function gaB, or ga, respectively. In particular, the ts partition function contains a factor of the form exp(E*/kT)in which the Born- Oppenheimer electronic energy of the ts relative to that of the reactant species appears. This energy E* is the value of the potential energy e(S)at the Ts geometry, which we denote So It turns out that the conventional TS approximation to krate over-estimates reaction rates because it assumes all trajectories that cross the ts proceed onward to products unless the transmission coefficient is included to correct for this In the variational transition state theory (VTST), one does not evaluate the ratio of partition functions 8
8 Before bringing this discussion of TST to a close, I need to stress that this theory is not exact. It assumes that the reacting molecules are nearly in thermal equilibrium, so it is less likely to work for reactions in which the reactant species are prepared in highly nonequilibrium conditions. Moreover, it ignores tunneling by requiring all reactions to proceed through the TS geometry. For reactions in which a light atoms (i.e., an H or D atom) is transferred, tunneling can be significant, so this conventional form of TST can provide substantial errors in such cases. Nevertheless, TST remains the most widely used and successful theory of chemical reaction rates and can be extended to include tunneling and other corrections as we now illustrate. B. Variational Transition State Theory Within the TST expression for the rate constant of a bi-molecular reaction, krate = k kT/h (qAB*/V)/{(qA/V)(qB /V)}or of a uni-molecular reaction, krate = k kT/h {(qA*/V)/(qA/V)}, the height (E*) of the barrier on the potential energy surface appears in the TS species’ partition function qAB* or qA*, respectively. In particular, the TS partition function contains a factor of the form exp(-E*/kT) in which the BornOppenheimer electronic energy of the TS relative to that of the reactant species appears. This energy E* is the value of the potential energy E(S) at the TS geometry, which we denote S0 . It turns out that the conventional TS approximation to krate over-estimates reaction rates because it assumes all trajectories that cross the TS proceed onward to products unless the transmission coefficient is included to correct for this. In the variational transition state theory (VTST), one does not evaluate the ratio of partition functions
appearing in kate at So, but one first determines at what geometry(s")the ts partition function (i.e, AB, or qa)is smallest. Because this partition function is a product of (i)the exp(-E(S)kT) factor as well as(1i)3 translational, 3 rotational, and 3N-7 vibrational partition functions(which depend on S), the value of s for which this product is smallest need not be the conventional TS value So. What this means is that the location(s")along the reaction path at which the free-energy reaches a saddle point is not the same the location So where the born-Oppenheimer electronic energy E(S)has its saddle. This interpretation of how S* and so differ can be appreciated by recalling that partition functions are related to the helmholtz free energy a by q exp(-A/kT); so determining the value of s where g reaches a minimum is equivalent to finding that s where a is at a maximum So, in VTSt, one adjusts the "dividing surface"(through the location of the reaction coordinate S)to first find that value S" where krate has a minimum. One then evaluates both e(s")and the other components of the Ts species partition functions at this value S*. Finally, one then uses the kate expressions given above, but with S taken at s". This is how vast computes reaction rates in a somewhat different manner than does the conventional tst. as with tst. the vtst in the form outlined above, does not treat tunneling and the fact that not all trajectories crossing proceed to products. These corrections still must be incorporated as an"add-on to this theory (i.e., in the K factor) to achieve high accuracy for reactions involving light species(recall from the Background Material that tunneling probabilities depend exponentially on the mass of the tunneling particle)
9 appearing in krate at S0 , but one first determines at what geometry (S*) the TS partition function (i.e., qAB* or qA*) is smallest. Because this partition function is a product of (i) the exp(-E(S)/kT) factor as well as (ii) 3 translational, 3 rotational, and 3N-7 vibrational partition functions (which depend on S), the value of S for which this product is smallest need not be the conventional TS value S0 . What this means is that the location (S*) along the reaction path at which the free-energy reaches a saddle point is not the same the location S0 where the Born-Oppenheimer electronic energy E(S) has its saddle. This interpretation of how S* and S0 differ can be appreciated by recalling that partition functions are related to the Helmholtz free energy A by q = exp(-A/kT); so determining the value of S where q reaches a minimum is equivalent to finding that S where A is at a maximum. So, in VTST, one adjusts the “dividing surface” (through the location of the reaction coordinate S) to first find that value S* where krate has a minimum. One then evaluates both E(S*) and the other components of the TS species partition functions at this value S*. Finally, one then uses the krate expressions given above, but with S taken at S*. This is how VTST computes reaction rates in a somewhat different manner than does the conventional TST. As with TST, the VTST, in the form outlined above, does not treat tunneling and the fact that not all trajectories crossing S* proceed to products. These corrections still must be incorporated as an “add-on” to this theory (i.e., in the k factor) to achieve high accuracy for reactions involving light species (recall from the Background Material that tunneling probabilities depend exponentially on the mass of the tunneling particle)
C Reaction Path Hamiltonian Theory Let us review what the reaction path is as defined above. It is a path that 1. begins at a transition state (ts)and evolves along the direction of negative curvature on the potential energy surface(as found by identify ing the eigenvector of the Hessian matrix H k =aE/as, as, that belongs to the negative eigenvalue); i1. moves further downhill along the gradient vector g whose components are g =aE/as iii terminates at the geometry of either the reactants or products( depending on whether one began moving away from the ts forward or backward along the direction of negative curvature) The individual"steps" along the reaction coordinate can be labeled So, S, S,,... Spas they evolve from the ts to the products(labeled Sp)and SR,SR+I,. So as they evolve from reactants(S-g)to the Ts. If these steps are taken in very small (infinitesimal) lengths, they form a continuous path and a continuous coordinate that we label s At any point S along a reaction path, the Born-Oppenheimer potential energy surface E(S), its gradient components g (S)=(aE(S)as)and its Hessian components HK (S)=(OE(SyOS, Os ) can be evaluated in terms of derivatives of E with respect to the 3N Cartesian coordinates of the molecule. However, when one carries out reaction path dynamics, one uses a different set of coordinates for reasons that are similar to those that arise in the treatment of normal modes of vibration as given in the back ground material In particular, one introduces 3N mass-weighted coordinates x;=sj(mj)/2 that are related to the 3N Cartesian coordinates s, in the same way as we saw in the background material
10 C. Reaction Path Hamiltonian Theory Let us review what the reaction path is as defined above. It is a path that i. begins at a transition state (TS) and evolves along the direction of negative curvature on the potential energy surface (as found by identifying the eigenvector of the Hessian matrix Hj,k = ¶ 2E/¶sk¶sj that belongs to the negative eigenvalue); ii. moves further downhill along the gradient vector g whose components are gk = ¶E/¶sk ’ iii. terminates at the geometry of either the reactants or products (depending on whether one began moving away from the TS forward or backward along the direction of negative curvature). The individual “steps” along the reaction coordinate can be labeled S0 , S1 , S2 , … SP as they evolve from the TS to the products (labeled SP ) and S-R, S-R+1, …S0 as they evolve from reactants (S-R) to the TS. If these steps are taken in very small (infinitesimal) lengths, they form a continuous path and a continuous coordinate that we label S. At any point S along a reaction path, the Born-Oppenheimer potential energy surface E(S), its gradient components gk (S) = (¶E(S)/¶sk ) and its Hessian components Hk,j(S) = (¶ 2E(S)/¶sk¶sj ) can be evaluated in terms of derivatives of E with respect to the 3N Cartesian coordinates of the molecule. However, when one carries out reaction path dynamics, one uses a different set of coordinates for reasons that are similar to those that arise in the treatment of normal modes of vibration as given in the Background Material. In particular, one introduces 3N mass-weighted coordinates xj = sj (mj ) 1/2 that are related to the 3N Cartesian coordinates sj in the same way as we saw in the Background Material
