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Bimolecular Recombination Reactions: K‑Adiabatic and K‑Active Forms of RRKM Theory, Nonstatistical Aspects, Low-Pressure Rates, and Time-Dependent Survival Probabilities with Application to Ozone. 2 Nima Ghaderi and R. A. Marcus* Noyes Laboratory of Chemical Physics, California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125, United States S Supporting Information *

ABSTRACT: We consider for bimolecular recombination reactions the K-adiabatic versus the K-active forms of RRKM theory, where K is the component of the total angular momentum along the axis of least moment of inertia of the recombination product. When that product is approximately a prolate symmetric top, with two moments of inertia of the product substantially larger than the third, K becomes a dynamically slowly varying quantity and the K-adiabatic form of RRKM theory is the appropriate version to use. Using classical trajectory results for the rate constant for ozone formation in the low-pressure region as an example, excellent agreement for the recombination rate constant krec with the K-adiabatic RRKM theory is observed. Use of a two transition state (inner, outer TS) formalism also obviates any need for assessing recrossings in the exit channel. In contrast, the K-active form of RRKM theory for this system disagrees with the trajectory results by a factor of about 2.5. In this study we also consider the distribution of the (E, J) resolved time-dependent survival probabilities P(E, J, t) of the intermediate O3* formed from O + O2. It is calculated using classical trajectories. The initial conditions for classical trajectories were selected using action-angle variables and a total J representation for (E, J) resolved systems, as described in Part I.1 The difference between K-active and K-adiabatic treatments is reflected also in a difference of the K-active RRKM survival probability P(E, J, t) from its trajectory-based value and from its often non-single-exponential decay. It is shown analytically that krec (K-active) ≥ krec (K-adiabatic), independent of the details of the TS (e.g., variational or fixed RRKM theory, 1-TS or 2-TS). Nonstatistical effects for O3* formation include a small initial recrossing of the transition state, a slow (several picoseconds) equipartitioning of energy among the two O−O bonds of the newly formed O3*, and a small nondissociation (a quasi-periodicity) of some trajectories originating in O3* (∼10%) and so, by microscopic reversibility, are not accessible from O + O2. An apparently new feature of the present results is the comparison of classical trajectories with K-adiabatic and K-active theories for rate constants of bimolecular recombinations. The quantum mechanical counterpart of classical K-adiabatic RRKM theory is also given, and its comparison with the experimental krec for O3 is given elsewhere.

1. INTRODUCTION Formation and dissociation of vibrationally hot molecules has been the subject of numerous studies of bimolecular recombination and unimolecular dissociation chemical reaction rates.2−10 Of particular interest has been the validity and the limitations of the RRKM theory of these reactions,11−15 a theory that usually assumes that all vibrational−rotational quantum states or, classically, all regions of the classical phase space of the molecule of a given total energy E and total angular momentum J are equally accessible to the long-lived vibrationally hot species.16,17 In the case of the triatomic molecule ozone there has been considerable interest in the “mass-independent” oxygen isotope fractionation (MIF) effect observed in formation of stratospheric18,19 and laboratory ozone.20−28 In the present case, classical trajectories can reveal whether or not there is a statistical redistribution in the vibrational−rotational classical © XXXX American Chemical Society

phase space of the ozone molecule formed from collision of an O and an O2. While we believe this MIF to be a quantum mechanical symmetry-based nonstatistical effect,29−35 a prerequisite for it, in one view, is a nonstatistical classical behavior. The present calculations provide information on the latter. In previous studies the recombination rate constant krec for forming O3 and its isotopomers has been studied using classical trajectories,36−40 including some comparison with experiment.41 Hitherto, the explicit time-dependent survival probability P(E, J, t) of the intermediate and its possible nonstatistical behavior have not been investigated in the direct trajectorybased determination of krec. Sometimes, the zero-point energies have been quasi-classically considered in the past by Received: July 8, 2014 Revised: September 12, 2014

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dx.doi.org/10.1021/jp506788z | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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exponential decay at different total energies E and total angular momenta J. Their net effect on the recombination rate constant is investigated using the weak collision deactivation approximation at low pressure at various temperatures. The integrand in krec provides information on the important E’s and J’s, e.g., for use in quantum mechanical computations. Nonstatistical effects due to recrossings of the transition state46−48 are included. The present focus is on the bimolecular reaction leading to a collision-stabilized product. The reverse reaction has its own subtleties and apart from a few comparisons will be treated in a separate article. The bimolecular trajectory-based krec was determined using the explicit P(E, J, t) of the intermediate, constructed from the trajectories. Several computational strategies (Supporting Information) were explored for determination of krec.49,50 The paper is organized as follows. The theoretical treatment1 for P(E, J, t), W(E, J), and the low-pressure krec given in Part I1 is summarized in section 2 for classical trajectories and in section 3 for the K-active RRKM theory for krec. In section 4, the K-adiabatic form of the theory is given together with its quantum form and an analytical comparison of the K-active and K-adiabatic expressions for krec as well as for the time-integrated survival probability. Results are given in section 5, illustrating different nonstatistical aspects, and explored for P(E, J, t) and W(E, J) as a function of E and J and for krec. A contour plot for the (E, J) contributions to krec is given there to highlight the (E, J) regions contributing significantly to krec. Trajectory-based results are compared with the K-adiabatic and K-active forms of RRKM theory in section 5. It is seen there that a result, one that we now understand dynamically because of the “slowness” of K, is that the K-adiabatic version is close to the trajectory results. The K-active result is about a factor of 2.5 larger than it. Key results are discussed in section 6. Concluding remarks are given in section 7.

incorporating them into the potential energy function as in a study of krec under a strong collision approximation.40 Apparently this may be the first time that K-adiabatic RRKM theory has been tested by comparison with classical trajectory results for 3-D reactions. An early test of a variational RRKM theory for unimolecular dissociations using classical trajectories was made by Bunker and Pattengill,2 who minimized the density of states of the TS rather than the number of TS states to determine the position of the TS. They treated reactions in a plane for which, therefore, |K| = J. Expressions for various alternative combinations in RRKM theory, including K-active and K-adiabatic, have been delineated by Hase42 and also explored elsewhere.17 An aim in the present paper is to make a choice among these possibilities based upon dynamical arguments about the behavior of a slowly varying (diffusing) dynamical variable, here K. K is the component of the total angular momentum along the body-fixed axis with the least moment of inertia. In fact, we do not require K to be a constant in this formulation (it is not43−45) but only slowly varying during the lifetime of the O3*. In this paper we term this form of RRKM theory K-adiabatic. For comparison, we also compare the results for the bimolecular rate constant obtained from another form of RRKM theory, K-active, in which K, like the molecular vibrations, is treated as a rapidly fluctuating variable. In the present study, the K-adiabatic formulation is generalized further by noting that in a bimolecular recombination at low pressures there exists a canonical ensemble equilibrium between the reactants, here O + O2, and the accessible states of the energetic molecule O3*. The same remarks, and its consequences, apply to the recombination A + B → AB, where A and B are atomic or molecular species and AB is a symmetric or near symmetric prolate top with two moments of inertia much larger than the third. As in discussions of equilibrated systems in general, it is unnecessary to focus on many detailed kinetic features. In such a case, one focuses on the equilibrium and, in the present case, on the value of K in the transition state region. It is included later by a step function Θ(E, J, K), that is, in this system, the trajectories can be classified not only by their value of E and J but also, in addition, by their value of K in the exiting TS region. It is anticipated that for a subset of the (E, J) systems the survival probability P(E, J, K, t), where K denotes the value of K in the exiting TS region, will be a single exponential, exp[−k(E, J, K, t)], with a rate constant k(E, J, K). A rate constant k(E, J, K), equal to N* (E, J, K)/hρ(E, J, K), can then be defined at any pressure, not necessarily the low-pressure region. This approach generalizes K-adiabatic RRKM theory so that it no longer means that K is constant throughout the lifetime of the molecule, but rather ρ(E, J, K) is interpreted statistically as the K contribution to ρ(E, J). The classical trajectory results in the present paper are compared with those from the classical version of K-active and K-adiabatic RRKM theory.16,17 The K-active treatment is considered with a reactive flux traversing 1 transition state (TS) and the K-adiabatic with a reactive flux traversing both a 1-TS and a 2-TS. In the present study, the nonstatistical contributions to krec, when O3 is formed from the collision of O and O2, are studied by looking for deviations from statistical theory, both for the reaction flux W(E, J) leading to formation of the O3* complexes and for the time-dependent survival probability P(E, J, t), such as deviation from a single-

2. TRAJECTORY-BASED BIMOLECULAR RECOMBINATION RATE CONSTANTS KREC: INDIVIDUAL COMPONENTS 2.1. Classical Trajectories for krec. For setting up a classical trajectory study for a bimolecular recombination rate constant in terms of the time-dependent survival probability, the low-pressure third-order rate constant for an atom−diatom system is given by eq 1 for the case of a weak deactivating collision (eq 1 of Part I1) 1 k rec = ∫ ... ∫ P(E , Γ10, t )Z(E′, E)Θ(E , Γ10)e−E / kT h dE dE′dΓ10 dt Q elecQ transQ rot − vib (1) where P(E, Γ10, t) is the survival probability for the intermediate complex at time t, the step function Θ(E, Γ10) is 1 or 0 according to if an intermediate is or is not formed from the trajectory of the colliding partners, Qelec, Qtrans, and Qrot−vib are the electronic, translational (per unit concentration) in the center of mass system of coordinates, and rotational− vibrational partition functions of the colliding partners,30,51 and dΓ10 = dl dml dj dmj dn dwl dwml dwj dwmj dwn/h5. One can reduce the number of variables and their sampling limits using the symmetry of the collision. The limits for the multiple integrals are (−l, l) for ml, (−j, j) for mj, (0,1) for w’s, (0, ∞) for E, l, j, n, t, and (−D,0) for E′ for a successful third-body collision. Here, D is the dissociation energy of the triatomic B



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species measured from the bottom of the O3 potential energy52 well, and 0 is the energy of the potential energy asymptote at large separation distances of the two reactants. The Z(E, E′)’s,53,54 the number of collisions per unit concentration of third bodies, per unit time and per unit E′ that lead to an energy transfer for the vibrationally excited intermediate from E to (E′, E′ + dE′) are discussed in Part I,1 and the parameters used for Z(E, E′) for ozone are given in the Supporting Information. For simplicity, we use here the single-exponential model for Z(E, E′).53,54 The unit of krec is cm6 molecule−2 s−1. The quantum version for krec in eq 1 was given in eq 8 of Part I.1 For the quantum version of the trajectory-based K-adiabatic krec, mj is merely replaced by K there and the other quantities, classical action variables, are replaced by discrete variables, quantum numbers. We introduced in Part I1 the total J representation which permitted a reduction in sampling variables and led from eq 1 to an expression (eq 5 of Part I1) k rec =

1 h

for a given O3 alphabetical isomer, say ABC, or by other equivalent (wmj, wml) pairs. One only selects one such pair, since one can show that there is a 1:1 correspondence in the Γ6 space between any two sets of such pairs.57 Arbitrarily we select (wmj = 0, wml = 1/2). If one of the above selections correspond to (ABC) in an A + BC collision then certain other pairs, such as (wmj = 0, wml = 0) or (wmj = 1/2, wml = 1/2), each correspond to the other isomer (ACB). For the other alphabetical isomer ACB we select (wmj = 0, wml = 0). The stability of O3* after a collision depends not only on E′ but also on J′ after the collision. As noted in Part I,1 in obtaining eq 2 we integrated over all J’s yielding unity, assuming a thermal population of the latter. Our focus is on possible deviations of O3* from statistical behavior. For implementing eq 2 we suppose that Nr reactive trajectories that formed the O3* molecule are selected out of the total N attempts for a given channel; then a formal expression for the P(E, J, t) in eq 2 is

∫ ... ∫ P(E , J , t )W (E , J)Z(E′, E)e−E/kT

dJ dE dE ′ dt Q elecQ transQ rot − vib

P(E , J , t ) = (2)

∫ ... ∫ P(E , J , Γ6, t )Θ(E , J , Γ6)dΓ6 ∫ ... ∫ Θ(E , J , Γ6)dΓ6

(3)

where Γ6 denotes the set of generalized initial rotational− orbital−vibrational action-angle variables (j, l, n, wj, wl, wn) at any given total J and E for the atom−diatom system.1,55,56 The planes of motion describing J in eq 2, its z component M, wM, and wJ are depicted in Figure 2 of Part I,1 and dΓ6 denotes dj dl dn dwj dwl dwn, where the action variables are expressed in units of h. The Γ6 variables are depicted in Figure 1 of Part I;1 when the vector corresponding to J is placed along the z axis, wj and wl canonically conjugate to j and l and are measured from the line of nodes. J denotes a quantum number but is also the total angular momentum in units of ℏ. The survival probability P(E, J, t) obtained from classical trajectories is given by P(E, J, t) = N(t)/Ntotal, where N(t) is the total number of complexes that have survived at time t and Ntotal is the total number of complexes formed at t = 0. In eq 2 we also have W(E, J), a dimensionless quantity related to the incident flux, the trajectory equivalent of N* (E, J), the number of states of the transition state in RRKM theory. It is given by eq 7 of Part I1 W (E , J ) = 2J

∫ ... ∫ Θ(E , J , Γ6)dΓ6

Nr

∑ [1 − S(t − ti)] i=1

(5)

where S(τ) is the unit step function, i.e., which equals 0 for τ < 0 and 1 for τ ≥ 0, and ti is the time of dissociation of the ith trajectory. 2.2. Expression for P(E, J, t) in the Presence of Collisions. The survival probability of O3* depends not only on its dissociation properties discussed in the previous section but also on collisions, particularly at longer trajectory times, since then the probability of O3* having undergone a collision is high. We denote the probability of survival of O3* in the presence of collisions by Pc(E, J, t). It is equal to the product of two independent probabilities for the low-pressure region (and so multiple collisions not included): the survival probability in the absence of collisions, P(E, J, t) multiplied by the probability Pnon(E, J, t) that the molecule has not undergone a collision. Pnon(E, J, t) satisfies the equation

where E and J each lie in the interval (0, ∞). P(E, J, t) is the time-dependent survival probability for systems at a given E and J, namely, the fraction of the intermediate complexes formed at t = 0 from a collision, here from O + O2, that survives at time t. In the total J representation it is given by eq 6 of Part I1 P(E , J , t ) =

1 N

−dPnon(E , J , t ) = Pnon(E , J , t )Z dt

(6)

where Z is the frequency of collisions with third bodies that depletes the energy of O3* to below the potential energy asymptote and is given later in this section. The solution of eq 6 is Pc(E , J , t ) = P(E , J , t )Pnon(E , J , t ) = P(E , J , t )e−Zt

(7)

We next introduce P(E′, E), the probability density (per unit E′) that a collision causes E to change to E′. The normalized P(E′, E) is, for a single-exponential transition probability,53,54 written as exp[−(E − E′)/γ]/(γ + γ′) for E′ ≤ E and as exp[−(E′ − E)/γ′]/(γ + γ′) for E′ ≥ E. The activation and deactivation constants γ′ and γ are related to each other by detailed balance, 1/γ′ = 1/γ + 1/kT.53,54 To incorporate this particular effect of collisions in a distribution function Pc(E, J, t), the P(E, J, t) in eq 2 (and in eq 5 of Part I1) is replaced by Pc(E, J, t) where

(4)

where W is defined separately for each incident channel (e.g., A + BC → ABC or ACB). The step function Θ(E, J, Γ6) is unity if an intermediate is formed from that trajectory of colliding partners and zero otherwise. The sampling and treatment of W(E, J) are discussed in the Supporting Information. Any pair of wmj and wml values may be selected for a given isomer, e.g., the pair (wmj = 0, wml = 1/2) or equivalent (wmj = 1/2, wml = 0)

Pc(E , J , t ) = P(E , J , t )e−Zt

(8)

We now have C


The Journal of Physical Chemistry A k rec =

1 h

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classical action variable in units of ℏ and so is dimensionless. The classical eq 12 is seen to be independent of h, as it should be. The limits of integration in eq 12 are, as in Part I,1 E′ = −D to 0, E = 0 to ∞, J = 0 to ∞. In the present calculation we treat all atoms as distinguishable so as to distinguish more easily between various effects. Any comparison with the experimental value of krec would incorporate symmetry effects and indistinguishability as well as introduce some quantum corrections, but the present focus is instead in part a comparison and study of two forms of RRKM theory and nonstatistical effects. Rapid algorithms have been introduced6−10 for use with RRKM theory-type expressions, based on an analytic integration over the linear and angular momenta. They can be adapted to the present equations. In the present article the theoretical focus is mainly on derivations. Similar to the treatment of eq 10, using the same expression for Z(E′, E) we again integrate over E′ and obtain

∫ ... ∫ P(E , J , t )W (E , J)Z(E′, E)e−Zt e−E/kT

dJ dE dE ′ dt Q elecQ transQ rot − vib

(9)

When the typical lifetime of P(E, J, t) is short in comparison with 1/Z, eq 9 reduces to eq 2. For Z(E′, E), we use Z0P(E′, E), i.e., Z0 exp[−(E′, E)/γ]/(γ + γ′), where Z0 is the collision number (units of cm3 molecule−1 s−1). If we integrate over E′ from 0 to −D and replace the D by −∞, a very minor approximation, we have k rec =

Zoγ ... P(E , J , t )W (E , J )e−Zt e−E / γ ′ h(γ + γ ′) dJ dE dt Q elecQ transQ rot − vib (10)

∫ ∫

where Z0 is set to be the Lennard−Jones collision number, ZLJ (Supporting Information). The W∫ P dt in eq 10 corresponds to the ρ in the RRKM eq 12 and so is only a slowly varying function of E. Thus, one sees that E’s within an energy γ′ of stabilizing the excited O3* are important contributors to the integral. When γ = 85 cm−141 and at 298 K, γ′ obtained53,54 from 1/γ′ = 1/γ + 1/kT equals 60 cm−1, i.e., 0.29 kT and so not far from the peak at E ≈ 0.25 kT in the contour plot given later in Figure 6. We note that the units of krec in eq 10 are those of a three-body rate constant, particularly appropriate for the lowpressure region, cm6 molecule −2 s−1. The Z in the exponent in eqs 9 and 10 equals Z0M, where M is the concentration of third bodies, multiplied by the fraction of collisions that form a nondissociating O3*. Hence 0

Z = ZoM

∫−D P(E′, E)dE′ =

ZoMγ −E / γ e γ + γ′

k rec =

(11)

(13)

∫ ... ∫ δ(E − E′)δ(J − J′)dΓ12 = 2πJ ∫ ... ∫ (1/PP 1 θ )dΓ9

ρ (E , J ) =

(14)

the second equality arising after δ(J − J′) and δ(E − E′) are each expressed in terms of δ(Pθ − Pθ′ ) and δ(P1 − P1′ ), and after integrating over ϕ and omitting primes in the symbols in the final expression. In eq 14 dΓ9 = dQ1 dQ2 dQ3 dθ dχ dP2 dP3 dPϕ dPχ/h6, Pθ = [J2 − (K2 + Pϕ2)csc2 θ + 2PϕK cot θ csc θ]1/2 and P1 defined by E = 1/2(P12 + P22 + P32) + Erot + V(Q1, Q2, Q3) with Erot = 1/2((J12/I1) + (J22/I2) + (J32/I3)). The relation between (J1, J2, J3) and the Euler momenta is given in ref 58. The limits of integration for the Q’s are selected such as to lie on the O3* side of the transition state. The normal coordinates for O3 were obtained using Wilson’s FG matrix method.58 When eq 14 is multiplied by ΔJ = 1, ρ(E, J) becomes the density of states per unit energy at the given J. The K-adiabatic case arises when the molecule has two moments of inertia substantially larger than the third (I1 and I2 ≫ I3). The number of states at the TS when R is the reaction coordinate is given by eq 19 of Part I1

3. KREC BASED ON K-ACTIVE RRKM THEORY We consider here the K-active form of RRKM theory for comparison with the dynamically based form of RRKM theory of the O + O2 recombination reaction. The RRKM theory counterpart of P(E, J, t) in eq 2 in Kactive RRKM theory is exp(−kRRKMt), where kRRKM = N* (EJ)/ hρ(EJ),2−10,16,17 N* being the number of states (related to the corresponding volume of phase space) of the transition state and ρ the density of states of the energetic intermediate. Upon integrating eq 2 over t we have, as in Part I1

∫ ∫ ∫ ρ(E , J)Z(E′, E)Θ(E , J)e−E/kT dE dE ′ dJ Q elecQ transQ rot − vib

∫ ∫

where Θ(E, J) = 1 when N* (E, J) > 0 and zero otherwise. Since the nonexponential factors in eq 13 vary slowly with E, one sees that E’s within an energy γ′ of stabilizing the excited O3* are important contributors to the integral. Phase space variables for a triatomic molecule are the Euler angular momenta (Pθ, Pϕ, Pχ), their conjugate58,59 coordinates (θ, ϕ, χ), the vibrational normal coordinates, (Q1, Q2, Q3), and their conjugate momenta (P1, P2, P3). The dimensionless phase space volume-element, dΓ12, is dQ1 dQ2 dQ3 dθ dϕ dχ dP1 dP2 dP3 dPθ dPϕ dPχ/h6. The (E, J) resolved ρ(E, J), a density per unit E and per unit J, can be written as

since −D is −∞ to a good approximation. The focus in the present paper is on the Z = 0 case. Later we plan to explore the use of eq 11. Equation 10 is applicable not only to atom− diatom collisions but also to collision of any species. The values of the functions W and P vary with the number of coordinates. Equation 10 is intended as a simple “bird’s eye view” of the pressure dependence of krec that in any accurate comparison with experiment would be replaced by a master equation, and similarly for an analogous equation for krec (adiabatic).

k rec =

Zoγ ... ρ(E , J )Θ(E , J )e−E / γ ′ (γ + γ ′) dE dJ Q elecQ transQ rot − vib

N *(E , J ) = min 2J

(12)

R

where Θ(E, J = 1) if N* (E, J) > 0 and 0 otherwise. A 1/h present in ρ(E, J) cancels the 1/h3 from Qrot−vib and the 1/h3 from Qtrans. In eq 12, J denotes the total angular momentum

∫ ... ∫ Θ(E − H′)dΓ4J dpr dr /h

(15)

where H′ is the Hamiltonian for the TS, apart from the PR /2 μ along the reaction coordinate, so E − H′ equals PR2/2 μ at the TS and Θ(E − H′) is a step function equal to unity for E ≥ H′

6

2

D



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K’s beyond ∼5. In eq 16 a canonical ensemble equilibrium is seen to be present in the form of the factor ρ(E, J, K) exp(−E/ kT)/QelecQtransQrot−vib. The factor Θ(E, J, K) serves as a constraint on the equilibrium, permitting the sorting of O3* molecules by the value they have for the slow variable K in the TS. For a K resolved ρ(E, J, K), occurring when the molecule is a prolate or near prolate symmetric top, we have, in the notation given earlier, I1 ≈ ≫ I3, and so K = J3 = Pχ. ρ(E, J, K) may be written as

at the TS and to zero for E < H′ there. dΓJ4 denotes dj dl dwj dwl, each action variable being in units of h, and pr in eq 15 is the momentum conjugate to r, the diatom internuclear distance. N*(E, J) is determined by the reduced number of variables appearing in eq 15. Implementation of the equations for the various coordinates used in Part I1 was made after assigning sampled values for mj, ml, wmj, and wml to give the specified J, as discussed there, and additionally setting wmj = 0 and wml = 1/2, similar to the treatment of W(E, J) there. A similar result is obtained for the other isomer, e.g., setting wmj = 0 and wml = 0. A numerical implementation of eq 15 is discussed in the Supporting Information. The transition state in variational RRKM theory16,17 occurs at the minimum of N* along the reaction coordinate. While R may not be the optimum choice for the reaction coordinate, it is simple and any shortcoming in its choice is corrected by counting the recrossings of the TS. N* is also used in eq 12 to determine whether the flux at the TS is zero or not via a unit step function Θ(E, J) there. The transition state equivalent of the trajectory-calculated W(E, J) is N*(E, J), the latter serving as its own Θ(E, J).

ρ(E , J , K ) = 2πJ

with dΓ8 = dQ1 dQ2 dQ3 dθ dχ dP2 dP3 dPϕ/h , where the angle ϕ has been integrated over to yield 2π since the integrand does not depend on it. P1Pθ in eq 18 arose from δ(E − E′) and δ(J − J′), as before in eq 14. We next consider N* (E, J, K). With the introduction of K as a state variable, providing a sharper distribution in the TS, it becomes increasingly important to introduce a two TS form of TS theory, which for RRKM theory is written as N*(E, J, K)eff, given by60−63 N *(E , J , K )eff =

N*EJK1

(16)

where Θ(E, J, K) = 1 if N*(E, J, K) > 0 and 0 otherwise. For simplicity, we again use the single exponential model for Z(E′, E),53,54 though more complex models54 can be used instead. We then have k rec =

Zoγ ... ρ(E , J , K )Θ(E , J , K )e−E / γ ′ (γ + γ ′) dE dJ dK Q elecQ transQ rot − vib

*1 N EJK *2 N EJK max *1 + N EJK *2 − N EJK *1 N EJK *2 /NEJK N EJK

(19)

N*EJK2

where and represent the number of states at the two minima in a plot of N(E, J, K) vs reaction coordinate R in our case and Nmax EJK represents the maximum in the number of states in between these two minima. In a treatment of the NEJK * ’s using variational RRKM theory in eq 19, we introduce for the TS, first for convenience, an approximation of using the line of centers of the atom and diatom as the body-fixed principal axis associated with the least moment of inertia. The action mj is then identified as K. To obtain an expression for N*(E, J, K), we again use R as the reaction coordinate, choosing PR so that the energy not contained in the other coordinate equals PR2/2 μabc at the given E. We use as variables the diatomic bond length and momentum, r, Pr, and the angular action-angle variables of the diatomic and of the orbital motion of the colliding pair, jlmjmlwjwlwmjwml, and note that R is fixed as the value that minimizes N* and PR is chosen as above, so neither is an integration variable. We then introduce a canonical transformation from the angular variables to a total J representation with variables that include J and its space-fixed z-component M, jlmjmlwjwlwmjwml → jlJMwjwlwJwM, with a Jacobian of the canonical transformation equal to unity. We now have, after integrating over M, wJ and wM to yield 2J, unity and unity, respectively, and approximating K by mj

∫ ∫ ∫ ρ(E , J , K )Z(E′, E)Θ(E , J , K )e−E/kT dE dE ′ dJ dK Q elecQ transQ rot − vib

(18) 6

4. KREC BASED ON K-ADIABATIC RRKM THEORY 4.1. K-Adiabatic RRKM krec. We consider the low-pressure recombination rate constant for K-adiabatic RRKM theory krec. As seen earlier, the theory in the present paper treats K as a special variable, either constant, like E and J, or slowly varying, so unlike all the other variables which are everywhere rapidly fluctuating. An equation for K-adiabatic RRKM theory can be derived in a way identical to the derivation of K-active RRKM theory in eq 14 in Part I1 by replacing Θ(E, J) there by Θ(E, J, K), replacing PPRKM(E, J, t) by PPRKM(E, J, K, t) = exp[−kPRKM(E, J, K)t], where kPRKM(E, J, K) = N*(E, J, K)/hρ(E, J, K) and replacing dt dE dJ by dt dE dJ dK. In this way, instead of the present eq 12 for K-active RRKM theory, krec for K-adiabatic RRKM theory is given by k rec =

∫ ... ∫ (1/PP1 θ)dΓ8

∫ ∫

N *(E , J , K ) = min 2J R

∫ ... ∫ δ(K − mj)Θ(E − H′)dΓ6J (20)

(17)

where dΓJ6 = dwj dwl dj dl dr dpr/h and minus PR2/2 μ. We note that mj can be the variables in ΓJ6 by l2 = j2 + J2 − 2mjJ.

where Θ(E, J, K) = 1 when N*(E, J, K) > 0 and 0 otherwise. Since the nonexponential factors in eq 17 vary slowly with E, one sees that E’s within an energy γ′ of stabilizing the excited O3* are important contributors to the integral. Θ(E, J, K) in eqs 16 and 17 have been determined separately for both the 1-TS and the 2-TS case using variational RRKM theory to identify the flux as a function of (E, J, K) and then considered in eqs 16 and 17. Θ(E, J, K) provides a substantial reduction on the acceptable states for O3*, leading to a cutoff of

H′ is the total energy expressed1 in terms of A transformation from

l to mj has a Jacobian of J/l, yielding N *(E , J , K ) = min 2J R

∫ ... ∫ Θ(E − H′)(J /l)dΓ5J

(21)

dΓJ5

Here, the dimensionless quantity = dwj dwl dj dr dpr/h with each angular momentum action variable being in units of h and with the w’s lying in the interval (0,1). E



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To show that the K-active value of ∫ P dt is greater than the K-adiabatic value, we next compare these two results for ∫ P dt using Schwarz’s inequality.66 We introduce vectors A and B with components (PKkK)1/2 and (PK/kK)1/2, respectively. By Schwarz’s inequality, we have |A•B| ≤ |A∥B|. Since |A•B| = ∑KPK = 1, |A| = ∑K((PKkK)1/2)2 = ∑KPKkK, and |B| = ∑K((PK/ kK)1/2)2 = ∑KPK/kK, we have 1 ≤ ∑KPKkK∑KPK/kK and thus 1/∑KPKkK ≤ ∑KPK/kK. Thus, we have krec(adiabatic) ≤ krec (active). The above remarks apply, in general, to reactions A + B → AB, where A and B are not restricted to being atom− diatom and AB is a symmetric or near-symmetric prolate top with two moments of inertia much larger than the third. An inequality between the K-active and the K-adiabatic systems was discussed by Hase,42 who considered a TS whose position is fixed. The present results extend that finding, being valid independently of whether or not the position of the TS depends on K and/or J (variational form of RRKM theory) and whether a 1-TS or a 2-TS expression is used for krec (adiabatic). By considering the time-dependent behavior of the survival probabilities P(t)’s, this latter proof compliments the previous one in this section.

Since the actual triatomic potential is being used in N*(E, J, K), eqs 20 and 21, and for krec in eqs 16 and 17, γ, the angle bisecting r and R which appears as one of the Jacobi variables (r, R, γ) in the potential of the Hamiltonian,64 needs to be written in terms of the action-angle variables via eqs A1−A12 in Appendix A of Part I.1 In using these equations ml is obtained from ml = J − mj, since M = mj + ml, where mj = K, and after the integration over M, J was placed along the space-fixed z axis. Both N*EJK1 and N*EJK2 in eq 19 are calculated in this way. 4.2. Quantum Version of krec (K-Adiabatic RRKM). For the quantum version of K-adiabatic RRKM theory we can again use eqs 16 and 17, but now all quantities, ρ(E, J, K), the N* (E, J, K) that is used to determine Θ(E, J, K), and the partition functions Q, are quantum calculated. Either J, K, and vibrational states of O3* that contribute to eqs 16 and 17 can be written in a discretized form and the 3-fold sum evaluated (Supporting Information) or one or more can be treated as continuous variables. For example, if the vibrational states are counted and, because of their relatively high density, approximated by a continuous distribution, the density of vibrational states could be written as ρvib(E − EJK), where EJK is the energy of a JK rotational state. For any J and K there are 2J + 1 rotational states, so in eqs 16 and 17 we have ρ(E , J , K ) = 2Jρvib (E − EJK )

5. RESULTS FOR BIMOLECULAR TRAJECTORY CALCULATED P(E, J, T)’S, W(E, J)’S, KREC TO FORM O3, AND COMPARISON WITH K-ADIABATIC AND K-ACTIVE RRKM THEORY 5.1. Trajectory Results for P(E, J, t). The collision A + BC can form either ABC or ACB, while ABC can form the products A + BC and AB + C, and similarly for ACB which can form A + CB and AC + B. We distinguish these atoms, so as to study differences in dissociation via the entrance channel, denoted by en, and dissociation via the other channel, the exchange channel, ex. Thereby, we write Pen(E, J, t) and Pex(E, J, t). Examples of these P(E, J, t) results for typical E’s and J’s relevant for the low-pressure region are shown in Figures 1−3. They yield long-lived complexes with an average half-time of ∼50 ps for a thermal sampling. The lower energies of up to ∼1 kT contribute approximately 90% to krec at these pressures, as

(22)

where EJK for a near-symmetric top is given by EJK ≅

(J 2 − K 2 ) K2 + 2I 2I3

(23)

where I is the harmonic mean of I1 and I2, i.e., I−1 = (I1−1 + I2−1)/2.65 J and K are now angular momenta in units of ℏ or angular momentum action variables in units of h, but elsewhere in the article they are quantum numbers. 4.3. Analytic Comparison of K-Active and K-Adiabatic Forms of RRKM Theory. An inequality between two forms of RRKM theory, K-active and K-adiabatic, can be seen in the following relationship. The K-active krec contains the term ρ(E, J)Θ(E, J), while the K-adiabatic krec contains ∫ Kρ(E, J, K)Θ(E, J, K)dK. Since Θ(E, J, K)dK ≤ Θ(E, J), we have ∫ KΘ(E, J, K)ρ(E, J, K)dK ≤ Θ(E, J)∫ Kρ(E, J, K)dK = Θ(E, J)ρ(E, J) showing thereby that K-adiabatic krec ≤ K-active krec. There is another way of establishing the inequality and providing added insight into the time dependence of P(E, J, t) and P(E, J, K, t). With ρ(E, J) and ρ(E, J, K) written in terms of ∞ Θ(E, J)∫ ∞ 0 P(E, J, t)dt and Θ(E, J, K)∫ 0 P(E, J, K, t)dt and in particular, in their RRKM forms, N* (E, J)∫ ∞ 0 P(E, J, t)dt and N* (E, J, K)∫ ∞ 0 P(E, J, K, t)dt after introducing the RRKM expressions for the P’s. To this end, we classify for any given E and J the states of O3* for dissociation according to their value of K for the exit (dissociation) TS. We denote the probability of finding any K in O3* at the given E and J by PK and denote its (assumed) single-exponential decay rate constant by kK. In Kactive RRKM theory there is equilibration among the different K states in the TS region. Thus, the survival probability P(E, J, t) then equals a single-exponential exp(−∑KPKkKt). The integral over t, ∫ P(E, J, t)dt, then yields 1/∑KPKkK. Instead, when there is no equilibration between the K states in and near the TS region, the survival probability P(E, J, t) equals ∑KPK exp(−kKt). Thus, in this case the K-adiabatic RRKM theory is applicable, P versus t is a sum of exponentials, and so a ln P versus t plot is curved instead of being linear. Integration over t yields ∫ P(E, J, t)dt = ∑KPK/kK.

Figure 1. ln P(E, J, t) of O3* versus time (ps) for an excess total energy E = 0.2 kT and J = 15 using a bimolecular trajectory sampling. The entrance channel ln Pen(E, J, t) with its least-squares fit to a polynomial (red) and the exchange channel ln Pex(E, J, t) with its fit (black) are displayed. ln Pen(E, J, t) of the entrance channel is offset by +0.03 relative to ln Pex(E, J, t) on the y axis to accentuate the drop-off in early times in the entrance channel and the overlapping of the offset curves after the early drop-off of en. The time of the complete merging of both P(E, J, t)’s is seen to be 7 ps. F



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ps for the lower energies, and the curvature is typically enhanced as J increases for any given E. At higher energies, above those of interest in the low-pressure recombination at room temperature, this initial drop off for the entrance channel diminishes, as indicated by the merge time of the survival probability curves for each channel as reported in Table 1. Table 1. Vertical Offset of the ln P(E, J, t) for the en Channel Relative to the ex Channel for Dissociation of O3* from a Bimolecular Trajectorya E (kT)

J

offset

merge time (ps)

E (kT)

J

offset

merge time (ps)

0.1 0.1 0.2 0.2 0.2 0.5 0.5 0.5 1

5 10 5 10 15 5 10 20 5

0.02 0.02 0.04 0.05 0.03 0.14 0.11 0.05 0.10

2.0 1.0 3.0 1.4 7.0 5.0 2.0 2.0 5.0

1 1 2 2 2 3 3 6 6

10 20 5 10 20 5 20 5 20

0.05 0.09 0.05 0.03 0.03 0.25 0.05 0.04 0.10

1.0 2.0 0.6 0.5 0.5 2.5 0.2 0.2 0.3

Figure 2. ln P(E, J, t) of O3* versus time (ps) for an excess total energy E = 0.2 kT and J = 15 using a bimolecular trajectory sampling for times much longer than those in Figure 1. The entrance channel, ln Pen(E, J, t) with its least-squares fit to a polynomial (red), and exchange channel ln Pex(E, J, t) with its fit (black) are displayed.

a

The offset is the (+) vertical displacement of the ln P(E, J, t) curve of the entrance (en) channel relative to the exchange (ex) channel to make all but the drop-off region of the curves to overlap for clarity. The merge time identifies the time when the displaced curves begin to overlap.

For the special case J = 0 for all relevant E’s, the curvature of ln P(E, J, t) is absent as a function of time and so largely is the offset, as seen in Figure 3. Some sample plots comparing ln P(E, J , t) versus time for the bimolecular trajectories and K-active RRKM theory are given in Figure 4. Typically, as J increases for a given energy E, the two plots, for the various E’s and J’s of interest, tend to converge. The ratio of the time-integrated P(E, J, t) from RRKM theory to the trajectory is found to be about 2.5, e.g., E = 0.2 kT and J = 15. 5.2. Results for Equilibration Time between O−O Bonds. The drop-off time data for P(E, J, t)’s for the en and ex channels in section 5.1 reflects some equilibration time between the two bonds. A different measure of a bond equilibration time is the time dependence of the difference in the mean length of the old and the newly formed O−O bond at early times for O3*. A histogram of Δr for O3* formed from O + O2 at E = 0.2 kT and J = 15 is plotted in Figure 5, where Δr = r1 − r2 is the length of the original O2 bond r1 minus the length of the newly formed O2 bond r2 as a function of time for early times. A bin size of 0.3 ao (ao is the Bohr radius) was selected for tallying these Δr’s, and then each tally was divided by the total tally from all bins to yield the ordinate of the plot in Figure 5. The time for approximate equilibration is seen to be ∼3 ps and thus a shorter than the time ∼5 ps for the drop off to dissociation in Figure 1 for the same E and J. 5.3. Trajectory Results for W(E, J). The trajectory results for W(E, J) for various (E, J)’s have been determined for an isomer of O3*, (ABC or ACB) and as in section 5.1 for the P(E, J, t) were partitioned according to the departure for each exit channel (en, ex), yielding Wen(E, J) and Wex(E, J). Wen(E, J) and Wex(E, J) are each the reaction flux times the fraction of complexes dissociated via either the entrance or the exchange channel, respectively, for a given isomer. Selected values are given in Table 2. Wex(E, J) is typically smaller than Wen(E, J),

Figure 3. Same as Figure 2 but for E = 0.2 kT, J = 0. The initial drop off for the entrance channel and the usual deviation from singleexponential decay for both channels are mostly absent.

seen in the contour plot given later in Figure 6. Selection of the microcanonical initial conditions and their sampling are given in the Supporting Information. Further details on the numerical method for propagation of trajectories, checking the conservation of E and J as a function of time for a given trajectory, the potential energy surface52 of O3, and the relatively few van der Waals complexes that occur are also described there. A plot of ln Pen(E, J, t) and ln Pex(E, J, t) versus time (ps) is given in Figure 1 for J = 15 and a total excess energy of E = 0.2kT above the asymptote. The number of trajectories used in obtaining the various figures, tables, and other data are given in the Supporting Information. In Figure 1 it is observed there that ln Pen(E, J, t) decreases more rapidly than ln Pex(E, J, t) during the first few picoseconds, causing a corresponding difference in ∫ ∞ 0 P(E, J, t)dt, and thereafter the instantaneous dissociation rate “constant” k(t) for en, the instantaneous slope ln Pen(E, J, t) versus t, becomes the same as that of ln Pex(E, J, t) versus t at each t. The initial “drop off” in P(E,J,t) versus t for the entrance channel is most pronounced at the lowest energies and typically decreases as E increases. For comparison, plots for some other E’s and J’s are given in the Supporting Information. Typically, at low excess energies (E = 0.2, 0.5, 1 kT) above the asymptote the plots of ln P(E, J, t) versus time, even after an initial drop off for “en”, are not precisely single-exponential decays. There is a curvature of the ln P(E, J, t) vs t plot that diminishes with time but persists to longer times such as 1000 G



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Figure 5. A typical histogram of Δr in Bohr radii for O3* formed from O + O2 using a classical trajectory at E = 0.2 kT and J = 15. Δr = r1 − r2, where r1 is the length of the existing bond, and r2 is that of the newly formed bond for O3*. Shown for the noted time intervals. The distribution is seen to become symmetric as time elapses.

5.4. Results on Comparison of the K-Active Quantity W(E, J)∫ ∞ 0 P(E, J, t)dt/h with ρ(E, J). It is seen from the comparison of eq 2 for krec based on trajectories with eq 12 for K-active RRKM that the trajectory-calculated W(E, J)∫ ∞ 0 P(E, J, t)dt/h has ρ(E, J) as its counterpart. In RRKM ρ(E, J), only those (E, J)’s are considered whose associated flux across the transition state is able to reach O3*’s density of states region. A comparison of selected ∫ W(E, J)P(E, J, t)dt/h and ρ(E, J), is given as a ratio in Table 3 for the en and ex channels. The deviations from unity in the last column reflect a small nonstatistical effect. A typical value for ρ(E, J), namely, at excess E = 0.2 kT and J = 15, is ∼20 quantum states per (cal mol−1). One can show analytically that although the numerator and the denominator in the third column of Table 3 each approach zero, the ratio approaches a finite value for a small J. J = 1 was arbitrarily selected, but any small J would suffice to reduce Coriolis effects on K. We also note that in the entry near the region of greatest interest (near J = 15, E = 0.2 kT) the deviation in the third column from unity is large, reflecting the deviation of the Kactive krec from its RRKM counterpart. 5.5. Results for Position of the TS. The minimum of N(E, J), N*(E, J), for O3* typically occurs near R = 5.2 ao. When the total excess energy increases from 1 to 6 kT, the R of the transition state decreases to 5.1 ao. At low energies, e.g., E = 0.1 and 0.2 kT, N(E, J) about the transition state changes more slowly as R changes when compared to the value when as the energy increases up to 6 kT; then N about the TS has a steeper minimum as a function of R. At a given energy, as J increases, typically, N changes more slowly about the TS. These results are given in the Supporting Information. At J’s near the maximum J, for any given energy, two transition states are observed at the inner R ≅ 5.2 ao and outer R ≅ 7.5 ao. In some cases, roaming trajectories extended beyond 7 ao. 5.6. Results for Contours of krec(E, J). The contributions to the recombination rate constant for a given isomer of O3, calculated from bimolecular classical trajectories, are shown as a contour plot in (E, J) space in Figure 6. The region centered around E = 0.25 kT, J = 15 is seen to be a principal contributor to krec. The maximum occurs near J ≈ [2 × 0.25(I1I2)1/2]1/2 = 11, which is close to the observed value of 15, with the greatest moment of inertia I1, where I1 ≅ I2. As noted earlier (comment

Figure 4. ln P(E, J, t) of O3* versus time (ps) based on K-active RRKM theory and bimolecular classical trajectory, exchange channel shown with its least-squares fit for (a) E = 0.1 kT, J = 5, (b) E = 0.2 kT, J = 15, (c) E = 0.5 kT, J = 15.

the ratio Wex(E, J)/Wen(E, J) at the lower energies being about 0.8−0.9 as seen in Table 2, and at energies of 1 kT and greater, a ratio greater than unity is also observed. The incoming flux toward the intermediate when passing the transition state, prior to any recrossings state, is defined as W(E, J)TS, was also determined and reported in Table 2. In this table N* should be compared with WTS. In principle, they should be equal when TS theory for incident flux is accurate. Similarly, Ncorr* should be compared with W. As an illustration of the recrossing results for the TS, when krec from the trajectory is calculated only based on the inner TS with the thermally averaged W’s in Table 2 krec drops by approximately about 20%, reflecting the difference due to the recrossing of the inner TS. H



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Table 2. Comparison of Ncorr* (E, J) with W(E, J) for O3* a E (kT)

J

N*

Ncorr*

WTS

W

Wex

Wen

Wex/Wen

0.1 0.2 0.2 0.2 0.5 0.5 0.5 1 1 1 2

10 5 10 20 5 10 20 10 15 20 10

0.22 0.65 0.86 0.40 2.54 4.15 4.01 14.7 16.5 15.5 51.9

0.17 0.56 0.70 0.32 2.03 3.65 3.28 11.8 12.7 11.6 45.1

0.20 0.64 0.83 0.38 2.56 3.88 3.79 14.7 16.1 15.0 52.0

0.16 0.57 0.67 0.32 2.09 3.54 3.42 12.0 12.4 11.5 45.6

0.08 0.26 0.31 0.16 0.98 1.65 1.59 5.75 5.91 5.38 21.8

0.08 0.31 0.36 0.16 1.11 1.89 1.83 6.25 6.47 6.11 23.8

1.00 0.85 0.85 1.00 0.89 0.87 0.85 0.92 0.89 0.89 0.95

N*(E, J) and W(E, J) for a given isomer of O3* were determined from eqs 15 and 4, respectively. N*(E, J)’s are corrected for the recrossings at the TS to yield N*(E, J)corr. Wex(E, J) and Wen(E, J) are defined in the text. A relative difference typically of about 3% is observed between the comparison of N*(E, J) with W(E, J)TS and also for Ncorr* with W(E, J). a

Table 3. W(E, J)P(E, J)/hρ(E, J) for the en and ex Channelsa

a

E (kT)

J

[(WP)en + (WP)ex]/2hρ

(WP)ex/(WP)en

0.2 0.1 0.5 0.2 0.1 0.5 0.2 0.1 0.5 0.2

1 1 5 5 5 10 10 10 15 15

1.02 0.78 0.92 0.80 0.59 0.72 0.59 0.30 0.54 0.33

0.94 0.90 0.81 0.82 0.86 0.85 0.91 0.94 0.87 0.85

The classical trajectory-based krec equals 4.8 × 10−34 cm6 molecule−2 s−1 at 298 K for O3, where the rate constant is for the classically distinguishable reaction A + BC → AB + C. Similar looking contours were also seen at lower and higher temperatures. Integration over (E, J) space for the given contour plot yields krec for each channel. The contribution to krec of O3 from van der Waals complexes was also calculated. It was extremely small, about 1.4 × 10−7 of the main contribution (Supporting Information). 5.7. Results for krec. K-active and K-adiabatic RRKM-based krec for a given isomer of O3 (e.g., ABC from the A + BC reaction) are compared for both channels and compared with the sum of the channel-resolved trajectory-based krec in Table 4. Two corrections are also given there to yield its krec(RRKM)corr: (1) “recrossings” refer to the recrossings of the transition state from reactant to product, including the roaming in the lifetime of complex. It is used to correct only the K-active krec(RRKM), where, in contrast, the K-adiabatic form used 2-TS and intrinsically accounted for any recrossings, (2) “quasi” is the average correction to the density of states for a given isomer which arises from the inaccessible region of phase space that is calculated from the plateaus of the P(E, J, t) of the quasiperiodic trajectories from the unimolecular sampling. It is seen in Table 4 to be only a ∼10% correction. The en and ex are the bimolecular trajectory-based krec when dissociation for a given isomer occurs through either the entrance channel or the exchange channel, respectively. Their sum ex + en is also given together with the ratio krec,corr(adiabatic)/krec(trajectories) in the last column. The energy-down collision energy transfer is γ, set to 85 cm−1 in the calculation of krec, which was obtained from experiment41 for N2 as a bath gas, making several approximations that remain to be tested. Should another value of γ apply as refinement of that work41 then an approximate integration of the equation for krec reveals that it is proportional to Γ, where Γ is the harmonic mean 1/Γ = 1/γ + 1/γ′. The bimolecular trajectory and K-adiabatic RRKM krec for O3 are 4.8 × 10−34 and 5.4 × 10−34 cm6 molecule−2 s−1, respectively at 298 K and in the low-pressure region, where the rate constant is for the classically distinguishable reaction A + BC → AB + C. Correcting the latter for the nonaccessible quasi-periodic trajectories (∼10%) reduces K-adiabatic 5.4 × 10−34 down to 4.8 × 10−34, which agrees with 4.82 × 10−34 cm6 molecule−2 s−1 from the trajectory-calculated krec. For a symmetric O + O2 reaction, its classical bimolecular trajectory krec would be twice this value, 9.6 × 10−34 cm6

WP denotes the integral of W(E, J)P(E, J, t) over t.

Figure 6. Contour plot of (E, J) resolved bimolecular trajectory-based krec, eq 10, for O3 formation per isomer is shown at 298 K, with a multiplication factor of ×10−35 cm6 molecule−2 s−1, with a maximum peak at 5.3. States shown are each weighted by the 2J factor.

after eq 10), the maximum in the contour plot occurs near E ≈ γ′ = 0.29 kT. I



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Table 4. K-Active, K-Adiabatic RRKM Theory and Trajectory-Based krec (×10−34) for O3a krec (RRKM)

krec (RRKM)corr

corrections (%)

trajectory krec

krec,corr/krec

T (K)

activeb

adiabaticc

recrossingsd

quasie

active

adiabatic

ex + en

exf

enf

adiabatic

130 200 298 373

269 53.3 11.2 4.50

132 25.4 5.40 2.12

23 21 20 21

11 11 11 11

184 37.5 7.97 3.16

118 22.6 4.80 1.89

113 22.4 4.82 1.83

54.0 10.7 2.26 0.86

59.2 11.7 2.56 0.97

1.04 1.01 1.00 1.03

a krec’s are reported for a reaction with distinguishable atoms, A + BC → AB + C, a single isomer of O3, ABC in the low-pressure region. The unit of krec is cm6 molecule−2 s−1. bUsed eq 13. cUsed eq 17. dCorrection from recrossings is performed only on the K-active krec(RRKM), the 2nd column. e Correction to the density of states from quasi-periodic trajectories. fUsed eq 10 with z = 0.

molecule−2 s−1. Although not strictly comparable, since the system is “quantum”, the experimental41 krec = 5.5 × 10−34 cm6 molecule−2 s−1. This correction for recrossings can be made in one of two ways, yielding similar results: One, the conventional way,46,47 is that using trajectories starting at the TS one counts the number of recrossings and so corrects for the “wasted” phase space of the TS. The second way to estimate the correction is to count the lifetime to begin when the coordinate R crosses the TS into the O3* region and ends when R f irst recrosses the inner TS. This result for krec is then compared with the value when recrossings are allowed for the same trajectory. This description in effect decreased trajectory-based krec, by decreasing ∫∞ 0 P(t)dt, by about 20% at 298 K. This value is consistent with the 20% recrossing correction in Table 4. The relatively high value of the transmission coefficient,46 and hence very few recrossings, also indicates that the simple selection of R as a reaction coordinate is satisfactory at room temperature for the present system. 5.8. Partial Compensation of Two Effects in Formation of O3*. In the O + O2 → O3* step, complexes associated with the entrance channel typically dissociate faster than those from the exchange channel; as seen in the initial drop off of the survival probability for the entrance channel, there is also a partial compensating effect in the form of the product W(E, J)∫ P(E, J, t)dt for the two channels: Wen(E, J) is typically greater than Wex(E, J) by about 15%. This result is seen in the ratio Wex(E, J)/Wen(E, J) in Table 2. In summary, the ratio W(E, J)exP(E, J)ex/W(E, J)enP(E, J)en, where P(E, J) denotes ∫ ∞ 0 P(E, J, t)dt, is typically ∼0.9 for the relevant (E, J)’s at 298 K (Supporting Information), so the net compensating effect result for the trajectory-based krec as seen in Table 4 is that the krec(ex) is about 10% less than krec(en) at room temperature and similarly for other temperatures in Table 4. That there is some cancellation is not surprising: On comparing eqs 3 and 4 one sees that in the product W(E, J)∫ P(E, J, t)dt there is a cancellation of W with the denominator of P, leaving 2J∫ ...∫ P(E, J, Γ6, t)Θ(E, J, Γ6)dΓ6 as the important quantity. We discuss in the next section a tentative explanation of the initial drop off of P(E, J, t) in the en channel, compared with the ex channel, seen in Table 1 and Figure 4, and absent in the rotationless case.

for ozone ζ is in this case ∼0.5 cm−1,67 and the splitting of two coupled vibrations at resonance is 2ζK. Coriolis coupling coefficients are available for higher vibrational states, e.g., ref 68. The fundamental ozone vibration stretching frequencies are around 1000 cm−1, each frequency being red shifted with increasing ozone energy, as seen, for example, in the power spectrum for each mode. The period of a local Coriolis-based oscillation is, from the above ζ, ∼1000 times longer than the vibrational periods, with K being a “slow coordinate” with a characteristic time of 1/2ζK for the resonant interchange, so with a characteristic time of 30 ps for this particular oscillation, K = 1. During the time between the collisions of O3*, about 100 ps at room temperature and low pressures (1 atm), one question is whether there would be time for K to explore the available phase space were it all accessible. Phase space exploration is quite different from localized energy oscillation that occurs in an isolated resonance, such as the one described above. Kdif f usion is related to the overlap of such resonances. Analysis for the diffusion is given in refs 44 and 45, and we plan to explore this topic in detail in a subsequent paper. Since K is a slow variable, shortly prior to the O3* entering the TS region K is approximately constant during many vibrational periods of the other variables. Thus, in that region contributions to krec come from species with different K and hence are controlled in passing through the TS by a restriction governed by Θ(E, J, K) in eq 17 rather than one governed by Θ(E, J) in eq 13. Thus, it is not perhaps surprising that Kadiabatic RRKM theory agrees well with the trajectory-based krec, in contrast with K-active RRKM. Typically, the smaller K’s contribute to the K-adiabatic RRKM krec because of Θ(E, J, K). For J ≈ 20, because of the energy locked up in the K motion, K2(1/2I1 − 1/2I2), Θ(E, J, K) is zero for K’s greater than ∼5 for the relevant E’s and J’s. The K dynamics may provide an explanation for the offset in the en and ex trajectories in Table 1 and Figure 1. Because of the K barrier, the entering trajectories are expected to be low in K, and so some can redissociate more quickly than in the ex channel. After some time, the merge time in Table 1 in this interpretation, the average K in the two channels, en and ex, becomes the same. We plan to test this interpretation, which also explains why the effect is absent when K is small (∼0 or 1 in the present study), in a subsequent article. 6.2. Differences between krec Based on Trajectories and K-Active RRKM Theory. We note from the entries in Table 3 that the difference between K-active ρ(E, J) and the trajectories W(E, J)∫ P(E, J, t)dt/h is larger near the peak of the contribution, where J is appreciable, and small when J is small (and hence when K is small). These findings are consistent with the corresponding agreement of k rec (trajectories) and

6. DISCUSSION OF RESULTS We focus on several key results of the present paper. 6.1. Adiabatic K and Coriolis Coupling. The angular momentum component K in O3* varies because of Coriolis coupling in the vibrations, coupling, for example, the symmetric and antisymmetric stretching vibrations which are about 60 cm−1 apart at the lowest ozone energies. The coupling constant J



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incoming colliding pair, A + BC, initially has some helicity, mj, that corresponds in the O3* molecule to K, since the line of centers at large distances serves as a principal axis for the definition of mj. To form O3* in K-adiabatic theory, there is a lower barrier when K2/2μR2 is small in the TS, i.e., when K (and hence its counterpart mj) is small. Because of the slowness of K as a dynamical variable in O3*, dissociation to reform the reactants A + BC has a memory at short times that the initially small mj, small before any equilibration, leads to a faster than expected dissociation at short times because the K barrier is small. On the other hand, in the major energy redistribution needed for the system to dissociate via the other channel, ex, leading to AB + C, the relevant K has become equilibrated and so is not as low as it was in the initial channel, and so there is no immediate drop in the survival probability for the ex channel. In this way the offset in Figure 1 and Table 1 has a simple interpretation. A corollary is that when J = 0 and so K = 0 this difference between the en and the ex channels should disappear, and indeed, the plot in Figure 3 shows that the offset does disappear when J = 0. The present explanation for these various results seems reasonable and remains to be tested further by comparison with computations.

k rec (adiabatic) compared with the disagreement with krec(active). The largest contribution to the difference between the Kadiabatic and the K-active krec’s is the more restrictive nature of Θ(E, J, K) compared with Θ(E, J) for the flux passing through the TS, leading to a substantially smaller K. This difference is reflected in the difference between W(E, J)∫ P(E, J, t)dt/h and ρ(E, J) in Table 3. It also causes krec(RRKM) for K-adiabatic to be less than that of K-active by a factor of about 2.5. The fraction of trajectories from reactant to products that include recrossings, given in Table 2, typically reduces the Kactive RRKM krec rate by 20%, as noted earlier. In contrast, when both TS are considered for the case of K-adiabatic RRKM krec in Table 4 the 20% recrossing is naturally accounted. The correction of recrossings to K-active krec(RRKM) in Table 4 was made at the inner TS,with it being the predominant one. There are several additional aspects: (1) the curvature in some ln P(E, J, t) versus t plots (e.g., Figures 2 and 4) shows immediately that a key assumption of the K-active RRKM theory is not fulfilled, namely, the assumption that there is an equilibration among all states of O3*, apart from those of the two principal coordinates associated with J, (2) the K-active W(E, J)∫ P(E, J, t)dt/h is substantially smaller than ρ(E, J), and (3) when K = 0 the ln P(E, J, t) versus t plot is approximately linear rather than curved as in Figure 3. These observations point to the possible explanation based on one of the several versions of RRKM theory described in the original 1952 article,16 the one termed here as K-adiabatic. In particular, in that article,16 the various degrees of freedom of the dissociating or isomerizing molecule were classified as “active” or “adiabatic” according to if they were or were not equilibrated in the energysharing process with the other coordinates prior to dissociation or isomerization. The K coordinate was singled out as a candidate for possibly being adiabatic rather than active. K has since been identified as a slow coordinate.44,45 The fine agreement of K-adiabatic RRKM theory for krec with the trajectory results (the ratio of the two, seen in Table 4, is ∼1.02) confirms this supposition, and so each of the three deviations of the trajectory results noted above from the Kactive RRKM theory can be understood in terms of K-adiabatic RRKM theory. 6.3. Initial Equilibration Time between the Two O−O Bonds. An equilibration time between the two O−O bonds is seen in the Δr plot in Figure 5 and in the initial offset plot in Figure 1 and Table 1. The time scale for each is ∼3 and ∼5 ps, respectively, much longer than the vibrational periods of 48, 32, and 30 fs for the vibrational modes of O3*, the bend and antisymmetric and symmetric stretch modes, respectively. For a small J such as J = 0, and hence K = 0, the equilibration time, from the ln P(E, J, t) vs time plots, is substantially less than at higher J’s, as evidenced by the lack of a drop off and curvature such as that seen in Figure 1. For Δr at E = 0.2 kT, J = 0, the plot (Supporting Information) exhibits an attenuated symmetry in Δr during the first picosecond compared to the finite J case, and as E increases to 1 kT, J = 0 a near-symmetric profile is observed at the earliest 0−0.5 ps. The bond length equilibration during the onset of O3* formation, as informed by the Δr plots, is faster than the drop-off time and appears to have a nonCoriolis origin. 6.4. Offset at Short Times for the en versus ex Channels. The offset between the en and the ex channels, seen in Figure 1 and Table 1, can be understood in terms of Kadiabatic RRKM theory and dynamical features as follows: An

7. CONCLUDING REMARKS In the present investigation of K-adiabatic and K-active formulation of RRKM theory using classical trajectories the focus has been to evaluate the behavior in the low-pressure region (the region most sensitive to assumptions on intramolecular energy transfer) for the recombination rate constant for O3. The trajectory-based krec agreed well with K-adiabatic and differed from K-active by the latter being too large by a factor of about 2.5. The excellent agreement for K-adiabatic is now understood in terms of K being a slow variable. A difference between K-adiabatic and K-active RRKM values is the “passage function” Θ(E, J, K) compared with Θ(E, J). Elsewhere, we will present results comparing experiments and K-adiabatic theory for different temperatures and pressures. Different small nonstatistical effects were found and described, with a role played by two TS’s in the K-adiabatic case and a small inaccessibility in the O3* phase space. Although the focus in the present paper has been on the O + O2 recombination, similar remarks apply to more general systems where the recombination product is a symmetric or near-symmetric top with two moments of inertia substantially larger than the third.

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ASSOCIATED CONTENT

S Supporting Information *

(1) The method of bimolecular classical trajectory with actionangle variables; (2) the potential energy surface for O3; (3) identification of O3* and its lifetime for the 1-TS and 2-TS cases from the bimolecular classical trajectory; (4) ln P(E, J, t) of O3* versus time; (5) histograms of ΔR for O3* at J = 0; (6) the initial conditions for trajectory recrossings at the transition state for O3*, using action-angle variables for the canonical and microcanonical sampling at the TS; (7) the determination of the transmission coefficient κ for forming O3* and recrossings of the TS; (8) the determination of N(E, J), N*(E,J), W(E, J) and ρ(E,J) for O3*; (9) weak collision parameters for Z(E′,E); (10) bimolecular RRKM krec for O3; (11) bimolecular trajectory krec for O3; (12) bimolecular trajectory krec for O3 from vdW complex. This material is available free of charge via the Internet at http://pubs.acs.org. K



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(20) Heidenreich, J. E., III; Thiemens, M. H. A Non-MassDependent Isotope Effect in the Production of Ozone from Molecular Oxygen. J. Chem. Phys. 1983, 78, 892−895. (21) Thiemens, M. H.; Heidenreich, J. E., III The Mass-Independent Fractionation of Oxygen: A Novel Isotope Effect and its Possible Cosmochemical Implications. Science 1983, 219, 1073−1075. (22) Yang, J.; Epstein, S. The Effect of the Isotopic Composition of Oxygen on the Non-Mass-Dependent Isotopic Fractionation in the Formation of Ozone by Discharge of O2. Geochim. Cosmochim. Acta 1987, 51, 2011−2017. (23) Thiemens, M. H.; Jackson, T. Production of Isotopically Heavy Ozone by Ultraviolet Light Photolysis of O2. Geophys. Res. Lett. 1987, 14, 624−627. (24) Heidenreich, J. E., III; Thiemens, M. H. A Non-MassDependent Oxygen Isotope Effect in the Production of Ozone from Molecular Oxygen: The role of Molecular Symmetry in Isotope Chemistry. J. Chem. Phys. 1986, 84, 2129−2136. (25) Thiemens, M. H.; Jackson, T. New Experimental Evidence for the Mechanism for Production of Isotopically Heavy O3. Geophys. Res. Lett. 1988, 15, 639−642. (26) Bains-Sahota, S. K.; Thiemens, M. H. Mass-Independent Oxygen Isotopic Fractionation in a Microwave Plasma. J. Phys. Chem. 1987, 91, 4370−4374. (27) Thiemens, M. H.; Jackson, T. Pressure Dependency for Heavy Isotope Enhancement in Ozone Formation. Geophys. Res. Lett. 1990, 17, 717−719. (28) Morton, J.; Barnes, J.; Schueler, B.; Mauersberger, K. Laboratory Studies of Heavy Ozone. J. Geophys. Res. 1990, 95, 901−907. (29) Hathorn, B. C.; Marcus, R. A. An Intramolecular Theory of the Mass-Independent Isotope Effect. for Ozone.I. J. Chem. Phys. 1999, 111, 4087−4100. (30) Hathorn, B. C.; Marcus, R. A. An Intramolecular Theory of the Mass-Independent Isotope Effect for Ozone. II. Numerical Implementation at Low Pressures Using a Loose Transition State. J. Chem. Phys. 2000, 113, 9497−9509. (31) Gao, Y. Q.; Marcus, R. A. Strange and Unconventional Isotope Effects in Ozone Formation. Science 2001, 293, 259−263. (32) Gao, Y. Q.; Marcus, R. A. On the Theory of the Strange and Unconventional Isotopic Effects in Ozone Formation. J. Chem. Phys. 2002, 116, 137−154. (33) Gao, Y. Q.; Chen, W.-C.; Marcus, R. A. A Theoretical Study of Ozone Isotopic Effects Using a Modified Ab Initio Potential Energy Surface. J. Chem. Phys. 2002, 117, 1536−1543. (34) Gao, Y. Q.; Marcus, R. A. An Approximate Theory of the Ozone Isotopic Effects: Rate Constant Ratios and Pressure Dependence. J. Chem. Phys. 2007, 127, 244316/1−8. (35) Marcus, R. A. Theory of Mass-Independent Fractionation of Isotopes, Phase Space Accessibility, and a Role of Isotopic Symmetry. Proc. Natl. Acad. Sci. U.S.A. 2013, 110, 17703−17707. (36) Gross, A.; Billing, G. D. Rate Constants for Ozone Formation and for Isotopic Exchange Reactions. Chem. Phys. 1993, 173, 393−406. (37) Gross, A.; Billing, G. D. Rate Constants for Ozone Formation. Chem. Phys. 1994, 187, 329−335. (38) Gross, A.; Billing, G. D. Isotope Effects on the Rate Constants for the Processes O2 + O → O + O2 and O2 + O + Ar → O3 + Ar on a Modified Ground-State Potential Energy Surface for Ozone. Chem. Phys. 1997, 217, 1−18. (39) Baker, T. A.; Gellene, G. I. Classical and Quasi-Classical Trajectory Calculations of Isotope Exchange and Ozone Formation Proceeding Through O + O2 Collision Complexes. J. Chem. Phys. 2002, 117, 7603−7613. (40) Schinke, R.; Fleurat-Lessard, P. The Effect of Zero-Point Energy Differences on the Isotope Dependence of the Formation of Ozone: A Classical Trajectory Study. J. Chem. Phys. 2005, 122, 094317/1−9. (41) Hippler, H.; Rhan, R.; Troe, J. Temperature and Pressure Dependence of Ozone Formation Rates in the Range 1−1000 bar and 90−370 K. J. Chem. Phys. 1990, 93, 6560−6569 Additional references of experimental krec as a function of temperature with an N2 bath gas are given in there.

AUTHOR INFORMATION

Corresponding Author

*Phone: (626) 395-6566, E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS It is a pleasure to acknowledge the support of this research by the NSF.

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REFERENCES

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(42) See the following reference for a discussion of K-adiabatic and K-active RRKM theory: Zhu, L.; Chen, W.; Hase, W. L.; Kaiser, E. W. Comparison of Models for Treating Angular Momentum in RRKM Calculations with Vibrator Transition States: Pressure and Temperature Dependence of Chlorine Atom + Acetylene Association. J. Phys. Chem. 1993, 97, 311−322 and references cited therein. (43) Grebenshchikov, S. Y.; Schinke, R.; Fleurat-Lessard, P.; Joyeux, M. van der Waals States in Ozone and their Influence on the Threshold Spectrum of O3 (X1 A1). I. Bound States. J. Chem. Phys. 2003, 119, 6512−6523. (44) Kryvohuz, M.; Marcus, R. A. Coriolis Coupling as a Source of Non-RRKM Effects in Triatomic Near-Symmetric Top Molecules: Diffusive Intramolecular Energy Exchange Between Rotational and Vibrational Degrees of Freedom. J. Chem. Phys. 2010, 132, 224304/1− 14. (45) Kryvohuz, M.; Marcus, R. A. Coriolis Coupling as a Source of Non-RRKM Effects in Ozone Molecule: Lifetime Statistics of Vibrationally Excited Ozone Molecules. J. Chem. Phys. 2010, 132, 224305/1−10. (46) Wigner, E. Calculation of the Rate of Elementary Association Reactions. J. Chem. Phys. 1937, 5, 720−725. (47) Anderson, J. B. Statistical Theories of Chemical Reactions. Distributions in the Transition Region. J. Chem. Phys. 1973, 58, 4684− 4692. (48) The usual terminology, the transmission coefficient, has been used which is the same as the conversion coefficient defined in ref 47, which refers to those fraction of trajectories whose origin is strictly the reactant crossing the transition state onto the product side, out of the set of all possible trajectories crossing the TS into the product region, as illustrated in Figure 1 of ref 47. (49) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes: The Art of Scientific Computing; Cambridge University Press: New York, 2007. (50) Krommer, A. R.; Ueberhuber, C. W. Computational Integration; SIAM Publications: Philadelphia, PA, 1998. (51) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976 and references cited therein. (52) Varandas, A. J. C.; Murrell, J. N. Dynamics of the 18O + 16O2 (υ = 0) Exchange Reaction on a New Potential Energy Surface for Ground-State Ozone. Chem. Phys. Lett. 1982, 88, 1−6. (53) Troe, J. Theory of Thermal Unimolecular Reactions at Low Pressures. I. Solutions of the Master Equation. J. Chem. Phys. 1977, 66, 4745−4757. (54) Zhu, Z.; Marcus, R. A. On Collisional Energy Transfer in Recombination and Dissociation Reactions: A Wiener-Hopf Problem and the Effect of a Near Elastic Peak. J. Chem. Phys. 2008, 129, 214106/1−10. (55) Cohen, A. O.; Marcus, R. A. Classical Mechanics of RotationalTranslational and Other Energy Transfer. I. A Hamtilton-Jacobi(Action-Angle) Treatment. J. Chem. Phys. 1968, 49, 4509−4514. (56) Miller, W. H. Semiclassical Theory of Atom-Diatom Collisions: Path Integrals and the Classical S Matrix. J. Chem. Phys. 1970, 53, 1949−1959. (57) For an arbitrary selection of initial variables for canonical or microcanonical initial conditions and interchanging the fixed wmj and

(59) Klippenstein, S. J.; Marcus, R. A. Unimolecular Reaction Rate Theory for Highly Flexible Transition States: Use of Conventional Coordinates. J. Phys. Chem. 1988, 92, 3105−3109. (60) Miller, W. H. Unified Statistical Model for “Complex” and “Direct” Reaction Mechanisms. J. Chem. Phys. 1976, 65, 2216−2223. (61) Hirschfelder, J. O.; Wigner, E. Some Quantum-Mechanical Considerations in the Theory of Reactions Involving an Activation Energy. J. Chem. Phys. 1939, 7, 616−628. (62) Chesnavich, W. J.; Bass, L.; Su, T.; Bowers, M. T. Multiple Transition States in Unimolecular Reactions: A Transition State Switching Model. Application to the C4H8+. System. J. Chem. Phys. 1981, 74, 2228−2246. (63) Klippenstein, S. J.; Marcus, R. A. Application of Unimolecular Reaction Rate Theory for Highly Flexible Transition States to the Dissociation of CH2CO into CH2 and CO. J. Chem. Phys. 1989, 91, 2280−2292. (64) The Jacobi action-angle variable Hamiltonian is H = j2/2 μbcr2 + 2 l /2 μabcR2 + P2r /2 μbc + P2R/2 μabc + V(r,R,γ), where μbc and μabc are the reduced masses for the diatom and triatom, respectively. (65) Townes, C. H.; Schawlow, A. L. Microwave Spectroscopy; Dover Publications: New York, 1975, Eq 4-4. (66) For two vectors A and B with elements ai...an and bi...bn, respectively, it states that |A•B|2 ≤ (A•B)(B•B). In the present case ai = (ki/N)1/2 and bi = 1/(ki/N)1/2. (67) Clough, S. A.; Kneizys, F. X. Coriolis Interaction in the ν1 and ν3 Fundamentals of Ozone. J. Chem. Phys. 1966, 44, 1855−1861. (68) De Backer, M.-R.; Barbe, A.; Starikova, E.; Tyuterev, Vl. G.; Kassi, S.; Campargue, A. Detection and Analysis of Four New Bands in CRDS 16O3 Spectra Between 7300 and 7600 cm−1. J. Mol. Spectrosc. 2012, 272, 43−50; see Tables 4 and 5 and references cited therein.

wml whose difference in angle is either 0 or π, then rotating each by a π rotation, it is found by inspection of eqs A1−A12 from Part I of ref 1 that the coordinates and momenta undergo an inversion symmetry for the same isomer, preserving the apex angle and the bond lengths. This is also confirmed numerically in the presence of the actual potential for both N*(E,J) and ρ(E,J) by checking the position of the middle atom, bond lengths, and the apex angle or the angle γ between r and R for the integral region to exclusively correspond to either ABC or ACB. (58) Wilson Jr., E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra; Dover Publications: New York, 1955. M


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