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