Molecular dynamics simulation of liquid N2O4r2NO2

JOURNAL OF CHEMICAL PHYSICS

VOLUME 120, NUMBER 2

8 JANUARY 2004

Molecular dynamics simulation of liquid N2 O4 r2NO2 by orientation-sensitive pairwise potential. III. Reaction dynamics Toshiko Kato¯a) Seibo Jogakuin Jr. College, Fukakusa, Fushimi-ku, Kyoto, 612-0878, Japan

共Received 17 July 2003; accepted 7 October 2003兲 The dissociation and association dynamics of N2 O4 2NO2 in liquid state are studied by classical molecular dynamics simulations of reactive liquid NO2 . An OSPP⫹LJ potential between NO2 molecules, which is a sum of an orientation-sensitive pairwise potential 共OSPP兲 between N–N atoms proposed in Paper I 关J. Chem. Phys. 115, 10852 共2001兲兴 and Lennard-Jones potentials between N–O and O–O atoms, has been used in the simulation. The reaction dynamics is studied as a function of well depth D e and anisotropy factors of the OSPP potential: A ␪ (0⭐A ␪ ⭐1) for the rocking angle and A ␶ (0⭐A ␶ ⭐0.5) for the torsional angle of relative NO2 – NO2 orientation. The lifetime ␶ D of initially prepared NO2 dimers is found to increase as D e increases, A ␪ increases, and A ␶ decreases. Dissociation and association dynamics are studied in detail around the extreme limit of pure NO2 -dimer liquid: D e ⫽0.12⫻10⫺18 J, A ␪ ⫽0.5, and A ␶ ⫽0.1, which has been found to reproduce both the observed liquid phase equilibrium properties and Raman band shapes of the dissociation mode very well. The dissociation dynamics from microscopic reaction trajectories is compared with the potential of the mean force 共PMF兲 as a function of the N–N distance R. The PMF of reactive liquid NO2 shows a transition state barrier at R⫽2.3– 2.5 Å, and NO2 -trimer structure is found to be formed at the barrier. Two types of dissociation of the NO2 dimer—the dissociation by collisional activation of the reactive mode to cross the dissociation limit and the NO2 -mediated dissociation via bond transfer—are studied. The latter needs less free energy and is found to be much more probable. The dissociation trajectories and PMF in reactive liquid NO2 are compared with those of a reactive NO2 pair in inert solvent N2 O4 . © 2004 American Institute of Physics. 关DOI: 10.1063/1.1630291兴

I. INTRODUCTION

k TST⫽

The present series of papers aims to understand chemical equilibrium and reaction dynamics of N2 O4 2NO2 in the liquid state by considering all degrees of freedom of the liquid. Although we are in a stage to probe a chemical reaction in real time by femtosecond spectroscopy,1 the role of solvent in solution reactions is not thoroughly understood. Following the success in formulating the intermolecular potential between NO2 molecules and reproducing the observed equilibrium constant in Paper I,2 dissociation and association dynamics will be studied in this paper. Although many of the chemical reactions occur in solution, much less is known about the microscopic reaction dynamics in solution than ones in the gas phase. In contrast to the situation in the gas phase, where reactants follow their route to products in isolation, the solvent molecules in solution continuously perturb the reactants in their course.3,4 It is known that the rates of reaction in the gas and liquid phases can differ by many orders of magnitude, and the solvent molecules can play an important role in determining the basic reaction mechanisms. On the assumption that reactive trajectories cross the activation barrier once and only once on their way to the product, transition-state theory 共TST兲 gives the rate of classical barrier crossing as3,5

冊

共1兲

where ⌬G ‡ is the activation free energy. In the liquid phase, recrossing of the activation barrier is induced by solvent dynamics, and the rate constant is reduced below the TST value by a factor of the transmission coefficient ␬ (⭐1), 3,5 k⫽ ␬ k TST.

共2兲

Solvent can influence a chemical reaction in two ways:3– 6 the first way is via a modification of the activation free energy ⌬G ‡ and the second via the transmission coefficient ␬. Hynes reviewed radical recombination, atom transfer, isomerization, and charge transfer reactions in solution from experiments and theories.3 Molecular dynamics 共MD兲 simulations of high barrier reactions with weak reactant– solvent interactions showed that there are almost no recrossings and TST gives the correct reaction rate.3,7,8 For reactions including charges and electric dipoles, however, polar solvents often exert a dramatic influence on the reactions in solutions.6 The nucleophilic substitution SN 2 reaction Cl⫺ ⫹CH3 Cl→ClCH3 ⫹Cl⫺ is known to be experimentally sensitive to solvent polarity, and the rate constant in aqueous solution was up to 20 orders of magnitude smaller than in the gas phase.9 The solvent effect was interpreted in terms of the preferential stabilization of the reactants relative to the charge dispersed transition state by polar solvents.10 The role of polar solvent dynamics in affecting the transmission coef-

a兲

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冉

k BT ⌬G ‡ exp ⫺ , h RT

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Toshiko Kato¯

J. Chem. Phys., Vol. 120, No. 2, 8 January 2004

ficient has also been studied.11 The reaction in the transition barrier neighborhood was characterized by nonadiabatic solvation rather than the adiabatic equilibrium solvation assumed in TST.8 The SN 1 ionization reaction is another important reaction in which the solvent plays an essential role.12,13 The mechanism of solvent-induced electronic curve crossing was studied by a MD simulation of t-BuCl→t-Bu⫹ ⫹Cl⫺ in water, where t-Bu group was treated as a united atom.13 Significant stabilization by water was found to lower the energy of the ionic state and to allow dissociation into the ionic products t-Bu⫹ ⫹Cl⫺ . We have thus seen that solvents often play important roles in solution reactions with strong reactant– solvent interactions. However, detailed MD studies of condensed phase reaction dynamics have been confined to rather simple reactants 共less than several atoms兲 in simple solvents 共less than four interaction sites with the use of united atom assumption兲. The present series of papers aim to understand thermal dissociation–association equilibrium and dynamics of N2 O4 2NO2 in which the dissociated molecules have more than two internal degrees of freedom. The gas and liquid N2 O4 are in thermal equilibrium of N2 O4 2NO2 at room temperature. This reaction is one of the frequently studied examples of chemical equilibrium in the gas and liquid phases in classic textbooks.5,14 The degree of dissociation in the pure liquid phase was observed by absorption and electron spin resonance 共ESR兲 spectroscopy to be in the extreme limit of pure N2 O4 , 0.018% at the freezing point 共262.0 K兲 and 0.12% at the boiling point 共294.3 K兲,15–17 and pure liquid may be considered as a very dilute solution of NO2 in organic solvent N2 O4 . The degrees of dissociation in various solvents15,18,19 are similar to ones in pure liquid N2 O4 , and they are very much less than ones in the gas phase, 8.6% and 15.9% at the freezing point and boiling point, respectively.15,20 Hence N2 O4 2NO2 is a very interesting reaction to see how chemical equilibrium and dynamics change from the gas phase to the liquid phase. The kinetics of gas phase N2 O4 /NO2 system has been the subject of numerous investigations. The observed rate of dissociation of N2 O4 in the gas phase by shock waves21 and by ultrasonic absorption measurement22 showed that the dissociation proceeds according to Lindemann unimolecular mechanism.23 Temperature jump measurements of the highand low-pressure limiting rate constants for dissociation and association reactions were found to agree with those calculated with a statistical reaction rate theory.24 Ab initio calculations of N2 O4 have been performed at high levels of electron correlation to account for its equilibrium geometry, virational frequencies, thermodynamic properties, and reaction rates.25 Thermal dissociation and association properties of N2 O4 /NO2 have also been studied by molecular dynamics simulations.26,27 The transition state was defined as a point of no return in phase space, and the effect of internal energy redistribution was formulated. The historiography of the experimental and theoretical studies of the very rapid gas phase reaction N2 O4 2NO2 has been given by Bauer.28 Recently, the femtosecond ground-state dynamics of gas phase N2 O4 and NO2 have been studied by nonresonant femtosecond time-resolved four-wave mixing.29 Initial rotational dephas-

ing and vibrational dynamics of the NN dissociation mode were observed for N2 O4 . Contrary to the numerous studies of gas phase reaction kinetics, the reaction rates of N2 O4 2NO2 in liquid phase have not been measured, and only a few reports on the dissociation and association mechanisms have been published.26,30 In the present series of papers, the reactive liquid N2 O4 is modeled as liquid NO2 which interacts with an orientation-sensitive pairwise potential, and all degrees of freedom on the ground electronic state potential energy surface are included. All NO2 molecules in the liquid are reactants as well as products, and they also act as a solvent for an NO2 pair of interest. MD simulations of liquid phase reactions of such a great size have never been carried out so far. The dissociation dynanics will be studied in two ways: 共i兲 from atomic trajectories of reactive liquid NO2 , and 共ii兲 calculation of the potential of mean force as a function of the reaction coordinate, the N–N distance R of a reactant NO2 pair. In Paper I, the equilibrium properties of N2 O4 2NO2 in a liquid state have been studied by classical molecular dynamics simulations of liquid NO2 . 2 An ab initio MO calculation has been carried out to elucidate the NO2 – NO2 potential, and an orientation-sensitive pairwise potential 共OSPP兲 which can reproduce highly anisotropic character of covalent bond between N–N has been formulated. The OSPP potential between two atoms j and k belonging to different molecules is represented by V 共 r jk 兲 ⫽D e exp关 ⫺2a 共 r jk ⫺r eq jk 兲兴 ⫺2 f ORD e ⫻exp关 ⫺a 共 r jk ⫺r eq jk 兲兴 ,

共3兲

where the orientation factor f OR is a function of angular variables describing the relative orientation of the two molecules, and r eq jk is the equilibrium distance between j and k of the association product. By introducing an orientation factor in the attractive term of the Morse potential, the potential covers from the fully attractive Morse potential for f OR⫽1 to the completely repulsive potential for f OR⫽0. The orientation factor is formulated as a product of three subfactors: the first two factors f ␪ ( ␪ ) are a single-variable function of the rocking angle ␪ j or ␪ k between the NN bond and ONO direction. These are parametrized by rocking anisotropy A ␪ (0⭐A ␪ ⭐1) and the limiting rocking angle for suitable bonding, ␪ lim⫽60°. Another factor f ␶ ( ␾ i jk ’s, ␶ i jkl ’s), which takes into account the torsional angle ␶ i jkl of the two NO2 about the NN bond, is parametrized by torsional anisotropy A ␶ (0⭐A ␶ ⭐1). The reactive liquid N2 O4 is modeled as liquid NO2 which interacts with the OSPP⫹LJ potential: the OSPP potential between N–N atoms and Lennard-Jones 共LJ兲 potentials between N–O and O–O atoms. In this formulation, Morse⫹LJ in our preceding papers26,27,30 corresponds to OSPP⫹LJ for zero anisotropy A ␪ ⫽A ␶ ⫽0. Equilibrium properties of the OSPP⫹LJ liquid NO2 are very sensitive to the well depth D e and anisotropy factors of OSPP.2 The population of more than NO2 dimer 共3-mer, 4-mer, 5-mer, . . . 兲 is large when anisotropy factors of the N–N bond are small for D e ⭓0.0945⫻10⫺18 J. On the contrary, the equilibrium NO2 /NO2 -dimer liquid is formed; that


Reaction dynamics of N2 O4 2NO2


is, most NO2 form a monomer or dimer at high anisotropy factors A ␪ ⫹A ␶ ⭓0.4– 0.5. In equilibrium NO2 /NO2 -dimer liquid, the concentration of monomer NO2 increases as A ␶ increases. Real liquid N2 O4 was found to be a very dilute solution of dissociated NO2 in solvent N2 O4 . 15–17 The equilibrium constant for dissociation reaction has been derived by computing the potential of mean force as a function of the N–N distance R. The OSPP⫹LJ potential for D e ⫽0.12 ⫻10⫺18 J, A ␪ ⫽0.5, and A ␶ ⫽0.1 was found to reproduce the observed liquid phase equilibrium constant at 273 K very well. We express this potential by P0 . In order to understand the character of NN bond dissociation, the Raman spectrum of the NN stretching mode31 was analyzed in Paper II.32 The vibrational correlation function and Raman band shape were calculated from a MD simulation of OSPP⫹LJ liquid NO2 . The potential P0 was found to reproduce both the observed liquid phase equilibrium properties and Raman band shape of the dissociation mode very well. The observed broad and asymmetric Raman band shape of the NN dissociation mode was ascribed to the broad and asymmetric distribution of the dissociation mode energy E T . Highly excited molecular pairs, which give rise to the low-energy wing of the band, are the most probable candidates for dissociation. In the present paper, the reaction dynamics in OSPP ⫹LJ liquid NO2 will be studied in detail as a function of the well depth D e and anisotropy factors of the OSPP potential. The lifetime of initially prepared NO2 dimers will be compared with the TST rate constant derived from the potential of mean force 共PMF兲 as a function of the N–N distance R. Two types of dissociation of the NO2 dimer—D, the dissociation by collisional activation of the reactive mode to cross the dissociation limit, and T, the NO2 -mediated dissociation via bond transfer—are studied. The microscopic reaction trajectories in reactive liquid NO2 will be compared with ones of an activated reactant NO2 pair in the gas phase and in inert solvent N2 O4 on the same potential energy surface. These comparisons are useful to understand the similarity and difference between the reaction dynamics in solution and in pure liquid N2 O4 . II. MOLECULAR DYNAMICS SIMULATION

Details of the simulation of reactive liquid NO2 have been given in Paper I, and a brief outline is given here. MD simulations are carried out under (N,V,T) conditions in two kinds of liquids, reactive liquid NO2 which interacts with OSPP⫹LJ potential 共simulation A兲 and a reactant NO2 pair in inert solvent N2 O4 共simulation B兲. Here 125 N2 O4 molecules, 250 NO2 molecules in simulation A and 2NO2 ⫹124N2 O4 molecules in B, are contained in a periodic cube of length 23.40 Å, which corresponds to the density 1.491 g cm⫺3 of liquid N2 O4 at 273 K. The time step (⌬t) was set to 1/2048 ps. All atoms interact through intramolecular and intermolecular potentials V intra and V inter , and the force Fi on the ith atom is given by Fi ⫽⫺ⵜi V intra ⫺ⵜi V inter . The intramolecular potential of NO2 molecules and solvent N2 O4 molecules are assumed to be quadratic as to 3 and 12 vibrational degrees of freedom of each species.

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TABLE I. Atom–atom potential parameters of Lennard-Jones type for the NO2 – NO2 , NO2 – N2 O4 , and N2 O4 – N2 O4 intermolecular interactions used in the MD simulations. Interaction

NO2 – NO2 N–O O–O NO2 – N2 O4 and N2 O4 – N2 O4 N–N N–O O–O

Parameters A (10⫺18 J Å 6 )

B (10⫺18 J Å 12)

0.5978 1.4177

115.14 646.3

3.0509 3.0857 1.4177

3655.5 1722.2 646.3

Force constants of NO2 and N2 O4 in the internal coordinate representation are given in Table I of Ref. 26. The anisotropy factors of the OSPP potential between N–N atoms have been changed in the range 0⭐A ␪ ⭐0.5 and 0⭐A ␶ ⭐0.4 for D e ⫽0.0945– 0.154⫻10⫺18 J, and LJ potential parameters between N–O and O–O atoms are given in Table I. The simulation A for each OSPP⫹LJ potential is initiated from an equilibrium configuration and momenta of pure NO2 -dimer liquid of D e ⫽0.120⫻10⫺18 J, A ␪ ⫽0.5, and A ␶ ⫽0.1 at 273 K 共IN2 in Paper I兲. In simulation B of a reactant NO2 pair in inert solvent N2 O4 , the OSPP⫹LJ potential is used between the reactant NO2 pair, and LJ potentials of parameters given in Table I are used for the NO2 – N2 O4 and N2 O4 – N2 O4 interactions. Because no reaction occurs in the limited time of equilibrium MD simulations, the reactant pair must be excited above the dissociation limit by an initial condition. An NO2 pair in inert solvent N2 O4 does not have any free energy barrier as will be shown in Sec. III D. A pair may be excited to any N–N distance, R: from an inner turning point to infinite separation. It is important that E T —the sum of potential and kinetic energy of intermolecular motion—be positive: i.e., V OSPP⫹LJ(R)⫹E kin ⭓0, where V OSPP⫹LJ(R) is the OSPP ⫹LJ potential at R. Here the system was excited initially to an inner turning point as follows. During the molecular dynamics simulation of the equilibrium NO2 pair in solvent N2 O4 , a time was chosen when the oscillating N–N distance of the NO2 pair reached its minimum distance R s and excited the pair by compressing two NO2 fragments in the direction of N–N until the N–N distance became a selected value R ex , keeping the center of N–N unchanged. The reactant pair is given a potential energy V OSPP⫹LJ(R ex )⫺V OSPP⫹LJ(R s ) by the selected initial condition. In order to compare the reaction dynamics of an excited pair in inert solvent N2 O4 with the ones in the gas phase, molecular dynamics simulations of an isolated NO2 pair have been carried out from the same initial condition. From each trajectory of a pair of activated NO2 molecules, we stored R G , the distance between the centers of masses of two NO2 molecules: E V , the vibrational energy of NO2 molecules: E R , the sum of the rotational kinetic energies of two NO2 : E T , the sum of the kinetic and potential energies of intermolecular motion: and E⫽E V ⫹E R ⫹E T , total energy of two NO2 in the center-of-mass scheme as a function of time.


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FIG. 1. An example of time evolution of total energy E, the energy partitioning among V, R, and T modes, and intermolecular separation R G between centers of masses of an activated reactant NO2 pair in the gas phase 共dashed lines兲 and in inert solvent N2 O4 共solid lines兲 for an OSPP potential of D e ⫽0.0945⫻10⫺18 J, A ␪ ⫽0.5, and A ␶ ⫽0.1.

III. RESULTS AND DISCUSSION A. Dissociation in inert solvent N2 O4

The reaction dynamics of a reactant NO2 pair on the OSPP⫹LJ potential energy surface in inert solvent N2 O4 are studied as in our previous simulations for the Morse⫹LJ potential.26,30 By comparing the reaction mechanism of a reactant NO2 pair in inert solvent N2 O4 and the one in reactive liquid NO2 , the role of reactive solvent in equilibrium liquid N2 O4 2NO2 will be clarified. Figure 1 shows an example of time evolution of total energy of reactant, energy partitioning among V, R, and T modes and intermolecular separation R G between centers of masses of NO2 molecules of an activated reactant pair on the OSPP⫹LJ potential in the gas phase 共dashed lines兲 and in inert solvent N2 O4 共solid lines兲 for ⫾500 fs. The total energy E of the reactant is conserved in the gas phase, while it is not conserved because of the reactant–solvent interaction in the solvent. The dynamics in both phases are essentially the same for about ⫾30 fs, but afterwards the behavior becomes different. In the gas phase the potential energy given by the initial condition changes into the kinetic energy of separation as the pair rolls down the repulsive part of the OSPP⫹LJ potential. The pair leads to dissociation in many cases, keeping the reactive mode energy E T positive constant. In inert solvent N2 O4 , the activated molecules lose their kinetic energy through collisions with solvent molecules 共total energy E of the reactant transfers to solvent兲, E T becomes negative, and the reactant mol-

Toshiko Kato¯

FIG. 2. An example of time evolution of total energy E, the energy partitioning among V, R, and T modes, and intermolecular separation R G between centers of masses of an activated reactant NO2 pair in the gas phase for an OSPP potential of D e ⫽0.0945⫻10⫺18 J, A ␪ ⫽0.5, and A ␶ ⫽0.1. A pair on the OSPP⫹LJ potential energy surface in the gas phase is found to be bound when E T ⬍0 and dissociates when E T ⬎0.

ecules remain trapped in a local cage in many cases 共the cage effect兲. Dissociation is less probable in solvent than in the gas phase because of the cage effect. An example of the gas phase reaction dynamics 共Fig. 2兲 shows the following: A reactant pair associates at an inner turning point 共ITP兲 A of intermolecular motion by transferring the reactive mode energy E T to vibrational mode, energy redistribution among T-R-V modes continues during the pair is bound, and the pair dissociates when V→T and R→T energy transfer changes the sign of E T from negative to positive at D: then, the molecules separate by transferring energy from intermolecular orbital motion to translation. The transition state in no-barrier reactions in the gas phase is thus defined by a phase space surface at E T ⫽0 or, more precisely, at E T ⫽V eff(Rl), where V eff(Rl) is a slightly positive orbitalangular-momentum-dependent barrier height.27 The reactions in inert solvent N2 O4 are influenced by strong reactant–solvent interactions. Time-dependent energy partitioning among modes and intermolecular separation R G in inert solvent N2 O4 are shown in Fig. 3. A pair excited at t⫽0 separates first but collides and associates at A1 by transferring energy from the T mode to solvent S 共total energy E of the reactant decreases兲. Energy redistribution continues in the bound state: then, E T is excited by V→T energy transfer at an ITP, D1 , and E T becomes positive by S→T energy transfer at D2 . The pair separates but cannot diffuse far apart and associates again at A2 by cooperative T→R, T→V, and T→S energy transfer. The pair is bound for about 3 ps until it starts to dissociate at D3 by V→T and S→T energy trans-



FIG. 3. An example of time evolution of total energy E, the energy partitioning among V, R, and T modes, and intermolecular separation R G between centers of masses of an activated reactant NO2 pair in inert solvent N2 O4 for an OSPP potential of D e ⫽0.0945⫻10⫺18 J, A ␪ ⫽0.5, and A ␶ ⫽0.2. A pair on the OSPP⫹LJ potential energy surface in inert solvent N2 O4 is found to be bound when E T ⬍0 and dissociates when E T ⬎0.

fer. Then the fragments diffuse in solvent N2 O4 and separate far apart. From molecular dynamics simulations of various reactant NO2 pairs on the OSPP⫹LJ potential energy surface, it is found that a pair in inert solvent N2 O4 is bound when E T ⬍0 and dissociates when E T ⬎0. The TS in inert solvent is hence defined by the phase space surface of E T ⫽0 as in the gas phase. Dissociation of an activated reactant pair is found to occur when the TS surface of E T ⫽0 is crossed from negative to positive by energy transfer from the reactant’s V and R modes to the reactive T mode in the gas phase and from the solvent mode as well in inert solvent. B. Rate of dissociation in reactive liquid NO2

In the present model of reactive liquid NO2 , the dissociation and association reactions occur without any excitation. MD simulations of reactive liquid NO2 were found to give equilibrium NO2 /NO2 -dimer liquid at A ␪ ⫹A ␶ ⭓0.4– 0.5. 2 From the trajectory of the MD initiated from the pure NO2 -dimer state IN2, the surviving number of NO2 dimers, N, was obtained as a function of the simulation time. We defined an NO2 pair as surviving if the pair keeps the N–N distance within 2.2 Å with the same partner and as dissociated if the pair is separated. The N-t plots for equilibrium NO2 /NO2 -dimer liquids of D e ⫽0.12⫻10⫺18 J and A ␪ ⫽0.5 at 273 and 373 K are shown in Fig. 4. It sometimes shows a small jump from 125 to a lower number N 0 at t


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FIG. 4. The number N of surviving NO2 dimers in a simulation initiated from pure NO2 -dimer state IN2 is plotted as a function of simulation time for an OSPP potentials of D e ⫽0.12⫻10⫺18 J, A ␪ ⫽0.5, and A ␶ ⫽0.1– 0.4 at 273 K 共a兲 and 373 K 共b兲. Dots show the simulation data and a solid line is the best fit of the data to single-exponential decay with ␶ D , which is indicated in the figure.

⫽0, which corresponds to the abrupt change of the OSPP potential parameters, but afterwards it decays monotonically. If the dissociation of NO2 dimer is represented by a singleexponential decay, we can derive the lifetime ␶ D of initially prepared NO2 dimers by fitting the simulation data to N ⫽N 0 exp(⫺t/␶D). The solid line shows our best fit of the data to single-exponential decay. The dissociation rate is found to increase as the temperature increases. The lifetimes ␶ D in equilibrium NO2 /NO2 -dimer liquids at 273 K are listed in Table II, and ␶ D for D e ⫽0.120⫻10⫺18 J is plotted in Fig. 5 as a function of anisotropy factors. A circle in the figure denotes ␶ D for the P0 potential, which gives the best fit to the observed equilibrium constant. The lifetime ␶ D of initially prepared NO2 dimers depends crucially on the anisotropy factors of the OSPP, and it changes in the range from 5 ps to over 1000 ns for A ␪ ⫽0.3– 0.5 and A ␶ ⫽0 – 0.4 at D e ⫽0.120⫻10⫺18 J. The lifetime ␶ D increases and the NO2 dimer stabilizes as the well depth D e increases, A ␪ increases, and A ␶ decreases. The liquid becomes almost static for very stable NO2 -dimer liquid at high A ␪ and low A ␶ . As a result, the shift to the equilibrium state is observed to be very slow (⭓10 ns), and ␶ D contains a considerable statistical error as shown by the error bars in Fig. 5. C. Microscopic reaction mechanisms in reactive liquid NO2

It has been shown that the dissociation of a reactant NO2 pair in the gas phase and in inert solvent N2 O4 must be activated artificially above the dissociation limit to dissociate. In reactive liquid NO2 , however, spontaneous dissocia-


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Toshiko Kato¯

J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 TABLE II. Dependence of lifetime ␶ D 共in units of ps兲 of initially prepared NO2 dimers on OSPP potential parameters in equilibrium NO2 /NO2 -dimer liquid at 273 K. A ␶ ⫽0

A ␶ ⫽0.1

A ␶ ⫽0.2

A ␶ ⫽0.3

A ␶ ⫽0.4

⬃106 ⭓106

⬃104 ⬃105

33 460 3000

10 86 200

5 14 17

⬃106 ⭓106

⬃106 ⬃106

1300 ⬃105

250 2000

36 130

⫺18

D e ⫽0.120⫻10 J A ␪ ⫽0.3 A ␪ ⫽0.4 A ␪ ⫽0.5 D e ⫽0.154⫻10⫺18 J A ␪ ⫽0.4 A ␪ ⫽0.5

tion occurs and the rate depends on the OSPP potential parameters. In this section the dissociation and association dynamics will be studied around the potential P0 which reproduces the observed equilibrium constant. No dissociation was observed during 1 ns of the MD simulation for P0 at 273 K initiated from IN2 共pure NO2 -dimer liquid兲. Hence the temperature was raised, and reactions were observed to occur at higher temperatures 298 –373 K. Figure 6 shows the N-t plot at each ps during 1024 ps of a MD for P0 at 298 K initiated from IN2, and Fig. 7 shows the expanded plot at 300–360 ps. Snapshots of the dissociation and association reactions are shown in Fig. 8. A pure NO2 -dimer state is maintained for 0–309 ps 共shown by a horizontal line兲 until an NO2 dimer 1-2 starts to dissociate at 310 ps. This dissociation is shown by a downward arrow and is named D1 . The snapshots are schematically shown in Fig. 8共a兲. The bond is found to break by changing mutual orientation as well as mutual distance, and the pair separates diffusively in NO2 -dimer liquid. A separated NO2 molecule 2 approaches to an NO2 dimer 3-4 关Fig. 8共b兲兴, and three molecules form a trimer at around 333 ps, which may correspond to the transition state of the NO2 ⫹NO2 dimer NO2 dimer ⫹NO2 reaction in reactive liquid NO2 : then, bond 3-4 is broken and bond 2-3 is formed by changing the relative orientation as well as mutual distance, and finally dimer 2-3 and monomer 4 separate. From the figure, we find that the dissociation–association reactions can occur in reactive liquid NO2 as a bond transfer from a dimer to a monomer, and this reaction is named T1 . It should be noted that the reactant

FIG. 5. Lifetime ␶ D of initially prepared NO2 dimers in equilibrium NO2 /NO2 -dimer liquid at 273 K is plotted as a function of anisotropy factors. D e ⫽0.120⫻10⫺18 J is assumed. An open circle shows ␶ D for the potential P0 which gives the best fit to the observed equilibrium constant.

experiences the transition state 共loose NO2 -trimer state兲 for a long time 共332–335 ps兲 in NO2 -mediated bond transfer reaction T1 , and recrossing of the activation barrier by solvent dynamics is often expected. Similar dissociation–association processes via bond transfers T2 , T3 , T4 , T5 , and T6 are observed at 344, 363, 376, 509, and 542 ps, respectively, and they are schematically shown by bonds and bond transfers 共arrows兲 among 16 NO2 molecules in Fig. 8共d兲. Solid lines show the original bonds at t⫽0, and dashed lines show new bonds formed by bond transfers and association reactions. A second dissociation D2 of another NO2 dimer 13-14 at 584 ps is followed by two association reactions A1 共588 ps兲 and A2 共597 ps兲 to form NO2 dimers 14-15 and 12-13, respectively. Snapshots of A1 are shown in Fig. 8共c兲. At this stage the liquid recovers pure NO2 -dimer structure as shown in Fig. 8共d兲. The structure is kept for 287 ps until the third dissociation D3 occurs at 884 ps. NO2 -mediated bond transfers T7 and T8 follow D3 . The overall lifetime ␶ D for P0 at 0–1024 ps is derived to be 11 ns, and ␶ D ⫽5.5 ns for intervals when the same liquid has two dissociated NO2 molecules in 124 NO2 dimers 共Fig. 6兲. This shows that the existence of monomer NO2 speeds up dissociation reactions via bond transfer. From the trajectory analysis of reactive liquid NO2 for potential P0 at 298 K and the one at higher temperatures 323–373 K, we find three types of reactions typically shown in Fig. 8: D, the dissociation by collisional activation of the reactive mode to cross the dissociation limit 关Fig. 8共a兲兴: T,

FIG. 6. The N-t plot during 1024 ps of a MD for P0 (D e ⫽0.12⫻10⫺18 J, A ␪ ⫽0.5, and A ␶ ⫽0.1) at 298 K initiated from pure NO2 -dimer state IN2. Dots show the simulation data at each ps and solid lines are the best fits of the data to exponential decay with ␶ D , which are indicated in the figure. Open squares show the points of reactions Di , Ti , and Ai . See text.




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FIG. 7. The expanded N-t plot of Fig. 6 for D1 , T1 , and T2 reactions at 300–360 ps. Dots below the main stairs of reactions show the recrossing events when the TS of dissociation is defined by the N–N distance R ⫽2.2 Å.

the NO2 -mediated dissociation–association via bond transfer from an NO2 dimer to a monomer NO2 关Fig. 8共b兲兴: and A, the association reaction by encounter of NO2 monomers 关Fig. 8共c兲兴. Dissociation by collisional activation seldom happens, while dissociation via bond transfer occurs much more often for P0 at 298 K. This is also true for P0 at 273 K because no reaction occurs in the MD initiated from pure NO2 -dimer state IN2, while T-type dissociation occurs often in MD simulations including dissociated NO2 species. It should be noted that the reactant stays at the TS neighborhood for a long time (⭓1 ps), and recrossing of the TS barrier occurs rather often. Dots below the main stairs of reactions in Figs. 6 and 7 show the recrossing events when the TS of dissociation is defined by the N–N distance R ⫽2.2 Å. A reactant pair passed the TS, but recrossed the TS to reform more stable chemical species on the main stairs of reactions. The rate will be reduced considerably from the TST value by solvent dynamics: i.e., the transmission coefficient ␬ may be low. At this stage, it is highly desirable to study experimentally the microdynamics of dissociation–association reactions in pure liquid N2 O4 . Above threshold dissociation can be controlled by a series of IR femtosecound–picosecond laser pulses.33 Femtosecond–picosecond real time probes of transition-state dynamics may reveal the existence of D and T dissociation types, and answer the question which is the major process in real liquid N2 O4 . This is, in other words, whether solvent N2 O4 molecules work just as a solvent for a reactant NO2 pair or work as reactants for the dissociation– association reactions. D. Potential of mean force and dissociation–association mechanisms

The potential of mean force will be useful to understand the dissociation and association mechanisms in reactive liquid NO2 . The PMF that represents the relative free energy of the system as a function of the N–N distance R of a reactant NO2 pair is given by G 共 R 兲 ⫽⫺kT ln g 共 R 兲 ,

共4兲

where g(R) represents the probability density for finding the system with each value of R. 10 Importance sampling has

FIG. 8. Snapshots of dissociation and association reactions in the MD for 1 ns for OSPP⫹LJ potential P0 at 298 K. 共a兲 Dissociation by collisional activation D1 : an NO2 dimer 1-2 dissociates orientationally as well as translationally, and the pair separates diffusively in NO2 -dimer liquid. A solid line shows an NN bond and a dashed line shows a long NN bond being formed or broken. 共b兲 Dissociation via bond transfer T1 : an NO2 molecule 2 approaches NO2 dimer 3-4, the three molecules form a 3-mer, which corresponds to the transition state of the NO2 ⫹NO2 dimer⫽NO2 dimer ⫹NO2 reaction: then, bond 3-4 is broken and bond 2-3 is formed by changing relative orientation as well as mutual distance. 共c兲 Association reaction A1 by encounter of NO2 monomers: monomers 14 and 15 approach and associate to form an NO2 dimer 14-15. 共d兲 Bonds and bond transfers 共arrows兲 among 16 NO2 molecules during 600 ps. Prompted by dissociation D1 of NO2 dimer 1-2 at ⬃310 ps, dissociation reactions via bond transfers T1 , T2 , T3 , T4 , T5 , and T6 follow during 330–550 ps. Second dissociation D2 of a dimer 13-14 is followed by two association reactions A1 and A2 to form NO2 dimers 14-15 and 12-13, respectively. Solid lines show the original bonds at t⫽0, and dashed lines show new bonds formed by bond transfers and association reactions. See text.

been used to cover an adequate range of R. An arbitrary NO2 pair was chosen as a reactant in the initial condition IN2, and a series of simulations is run with biasing functions u(R) that center the sampling in different overlapping regions of R 共windows兲.2 A distribution function was obtained by averaging over configurations during 64 ps after equilibration for 16 ps. The true distribution function g(R) in each range was recovered by removing the effects of biasing functions. The resultant potentials of mean force from different windows are then spliced together to yield the overall continuous free energy surface G(R) which satisfies G(⬁)⫽0. Calculated PMF are shown in Fig. 9 for D e ⫽0.120 ⫻10⫺18 J, A ␪ ⫽0.5, and A ␶ ⫽0.1– 0.4. In order to see the


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Toshiko Kato¯


FIG. 9. Potential of mean force for an NO2 pair in reactive liquid NO2 for D e ⫽0.120⫻10⫺18 J, A ␪ ⫽0.5, and A ␶ ⫽0.1– 0.4 at 273 K. The PMF for an NO2 pair in inert solvent N2 O4 is shown for comparison. The PMF of the OSPP⫹LJ liquid NO2 shows a minimum at R min⫽1.8– 2.0 Å and a barrier at R max⫽2.3– 2.5 Å, while the PMF in inert solvent N2 O4 does not have any barrier and has a deeper minimum.

effect of solvent NO2 dimer being able to dissociate, the PMF of an NO2 pair in inert solvent N2 O4 was also calculated and is shown for comparison. The PMF of the OSPP ⫹LJ liquid NO2 shows a minimum at R⫽1.8– 2.0 Å and a barrier at R⫽2.3– 2.5 Å, while PMF in inert solvent N2 O4 does not show any barrier and has a deeper minimum. The barrier in PMF observed in reactive liquid NO2 can be identified as the transition state of the dissociation–association reactions. Table III lists typical features of the calculated PMF: R min , R max , G(R min), G(R max), the positions and free energies of the minimum and maximum PMF, respectively, and ⌬G ‡ ⫽G(R max)⫺G(Rmin), the activation free energy of dissociation. The rates of dissociation k TST calculated from TST ⫽(k TST) ⫺1 , and lifetime ␶ D deTST 关Eq. 共1兲兴, lifetime ␶ D rived from trajectory analysis are also listed. Lifetimes ␶ D TST are considerably longer than ␶ D for A ␶ ⫽0.2– 0.4, and this may be because of the recrossing of the activation barrier by solvent dynamics. The PMF in reactive liquid NO2 is found

to be very sensitive to torsional anisotropy; the well becomes shallower and the barrier becomes lower as A ␶ increases. This shows that an NO2 dimer can easily pass over the barrier to dissociation with thermal energy at high A ␶ , leading to NO2 dominant liquid. The well of PMF in solvent N2 O4 also shallows as A ␶ increases. A reactant pair interacting with potential P0 in solvent N2 O4 needs an activation free energy (⌬G ‡ ) as much as 41.8 kJ/mol, and no dissociation occurs in equilibrium MD TST (␶D ⫽2⫻104 ns). Dissociation in OSPP⫹LJ liquid NO2 for P0 can hardly occur with thermal energy because of the TST ⫽490 ns). Equilibrium and smaller ⌬G ‡ ⫽33.7 kJ/mol ( ␶ D rate properties of NO2 -dimer dissociation in OSPP⫹LJ liquid NO2 and in inert solvent N2 O4 around potential P0 at 273 K are summarized in Table IV. The activation free energy cannot be compared with experimental data because no dissociation rate is observed in liquid phase. Instead the free energy of dissociation ⌬G * ⫽⫺G(R min) can be compared with ⌬G * derived from experimental equilibrium constant 共Table IV兲. It was found that the OSPP⫹LJ liquid NO2 at P0 reproduces the experimental equilibrium constant at 273 K very well,2 but calculated ⌬G * in inert solvent N2 O4 is too high. Observed ⌬G * in pure liquid N2 O4 is similar to ones in nonpolar solvents CS2 , CCl4 , and C6 H12 . Observed ⌬G * in aromatic solvent C6 H6 and ones in dipolar aprotic solvents CH3 CN and sulpholane are higher. The model of N2 O4 dissociation in inert solvent N2 O4 cannot explain ⌬G * in various organic solvents. The solvent effects on dissociation reactions can only be explained by considering the change of electron cloud of reactants—i.e., the change of intermolecular potential in various solvents. In order to see the effect of solvent molecules on the PMF, the average structure around a reactant NO2 pair at R ⯝2.4 Å 共TS in reactive liquid NO2 ) has been calculated by constrained dynamics in both solvents. The time-averaged distribution of solvent’s N atoms around a reactant NO2 pair (R i ⭐5 Å) is shown in Fig. 10, where R i is the distance between center C of the reactant N–N atoms and N atom of

TABLE III. Typical features of the calculated PMF of a reactant NO2 pair in reactive liquid NO2 and in inert solvent N2 O4 at 273 K. OSPP⫹LJ potential parameters between NO2 molecules are D e ⫽0.120,0.154 ⫻10⫺18 J, A ␪ ⫽0.5, and A ␶ ⫽0 – 0.4. Lifetime ␶ D from trajectory analysis is listed for comparison.

A␶

R min 共Å兲

R max 共Å兲

G(R min) 共kJ/mol兲

NO2 pair in reactive liquid NO2 D e ⫽0.120⫻10⫺18 J A ␪ ⫽0.5 0.0 1.80 2.38 ⫺29.3 0.1 1.84 2.48 ⫺27.7 0.2 1.88 2.49 ⫺13.8 0.3 1.94 2.49 ⫺3.6 0.4 1.99 2.37 ⫺1.9 D e ⫽0.154⫻10⫺18 J A ␪ ⫽0.5 0.1 1.84 2.46 ⫺30.8 NO2 pair in inert solvent N2 O4 D e ⫽0.12⫻10⫺18 J A ␪ ⫽0.5 0.0 1.80 ⫺61.1 0.1 1.84 ⫺41.8 0.2 1.86 ⫺35.9 0.3 1.86 ⫺23.5 0.4 2.02 ⫺13.1

␶D 共ps兲

G(R max) 共kJ/mol兲

⌬G ‡ 共kJ/mol兲

7.0 6.0 5.1 4.6 2.8

36.3 33.7 18.9 8.2 4.7

6.6⫻105 2.1⫻106 1.4⫻109 1.6⫻1011 7.2⫻1011

1.5⫻106 4.9⫻105 740 6.3 1.4

⭓106 ⬃105 3000 200 17

7.2

38.0

3.0⫻105

3.4⫻106

⭓106

61.1 41.8 35.9 23.5 13.1

1.2⫻101 5.5⫻104 7.7⫻105 1.8⫻108 1.8⫻1010

9⫻1010 2⫻107 1.3⫻106 5.5⫻103 56

k TST (s⫺1 )

␶ DTST 共ps兲



J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 TABLE IV. Equilibrium and rate properties of OSPP⫹LJ liquid NO2 at 273 K around potential P0 : the free energy of dissociation ⌬G * , equilibrium constant K c , the activation free energy of dissociation ⌬G ‡ for D e ⫽0.120⫻10⫺18 J and A ␪ ⫽0.5. Experimental ⌬G * and K c are listed for comparison. A␶

T 共K兲

Reactive liquid NO2 0.0 273 0.1 273 0.2 273 NO2 pair in inert solvent N2 O4 0.1 273 Experimental System Pure N2 O4

N2 O4 /CS2 N2 O4 /C6 H6 N2 O4 /CCl4 N2 O4 /C6 H12 N2 O4 /CH3 CN N2 O4 /sulfolane

T 共K兲 273 293 298 293 293 298 298 298 298

⌬G * 共kJ/mol兲 K c 共mol/l兲 ⌬G ‡ 共kJ/mol兲 29.3 27.7 13.8

2.5⫻10⫺6 5.0⫻10⫺6 2.3⫻10⫺3

36.3 33.7 18.9

41.8

1.0⫻10⫺8

41.8

⌬G * 共kJ/mol兲 K c 共mol/l兲

Ref.

27.9 22.3 21.3 21.7 26.1 21.4 21.4 25.5 28.9

⫺6

4.60⫻10 1.05⫻10⫺4 1.8⫻10⫺4 1.78⫻10⫺4 2.23⫻10⫺5 1.78⫻10⫺4 1.77⫻10⫺4 3.80⫻10⫺5 8.6⫻10⫺6

16 15 17 15 15 19 19 19 19

the solvent, and ␪ i is the angle between the reactant’s N–N axis and the reactant–solvent C–N axis 关Fig. 10共a兲兴. The angular distribution of solvent’s N atoms P( ␪ i ) for R i ⭐5 Å shows a peak at ␪ i ⫽90° in both solvents 关Fig. 10共b兲兴, and this is presumably because of the exclusion effect of the reactant NO2 pair along the N–N axis. In reactive liquid NO2 , the radial distribution P(R i )/R 2i of solvent’s N atoms around the reactant’s center C shows a maximum at R i ⯝2.1 Å inside the shell of R i ⭓4.5 Å 关Fig. 10共c兲兴. The reactant solvent N–N distance at the TS barrier, R⫽2.4 Å, is around 2.4 Å, and this suggests that the NO2 -trimer structure is formed at the TS barrier. It can be concluded that

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NO2 -mediated dissociation T via bond transfer through the TS of the NO2 trimer takes less free energy and is much more probable than dissociation D by collisional activation. In inert solvent N2 O4 , however, solvent’s N atoms are not observed at R i ⭐4 Å 关Fig. 10共d兲兴. A shallower PMF with a barrier in reactive liquid NO2 is considered to come from the preferential attraction between reactant–solvent NO2 molecules even in very stable NO2 -dimer liquid. It should be noted here that the dissociation via bond transfer through the TS of the NO2 trimer is not the result of a specific potential between three NO2 molecules but the result of OSPP⫹LJ potential between two molecules, which reproduces the observed equilibrium properties and hence includes the effect of interaction between more than two molecules on average. IV. CONCLUDING REMARKS

Molecular dynamics simulations of reactive liquid NO2 have been carried out by using the OSPP⫹LJ potential between NO2 molecules. Dissociation and association dynamics are studied around the potential P0 : D e ⫽0.12 ⫻10⫺18 J, A ␪ ⫽0.5, and A ␶ ⫽0.1, which has been found to reproduce both the observed liquid phase equilibrium properties and Raman band shapes of the dissociation mode very well. A bound pair in reactive liquid NO2 is not stable and dissociates with a characteristic lifetime. The lifetime ␶ D increases and the NO2 dimer stabilizes as the well depth D e increases, A ␪ increases, and A ␶ decreases. The PMF of a reactant NO2 pair in reactive liquid NO2 has been calculated as a function of the N–N distance R. It shows a transition state of dissociation–association reactions at R⫽2.3– 2.5 Å, and NO2 -trimer structure is found to be formed at the barrier. Two types of dissociation of the NO2 dimer were observed: D, the dissociation by collisional activation of the reactive mode to cross the dissociation limit, and T, the NO2 -mediated dissociation via bond transfer from a dimer to a monomer through the TS of the NO2 trimer. It is concluded that NO2 -mediated dissociation via bond transfer takes less free energy and is much more probable. It is considered that a reactant pair can rarely but spontaneously pass over the dissociation barrier with thermal energy because the activation barrier of an NO2 pair is considerably lowered by the attractive potential from neighboring NO2 molecules. Dissociation and association mechanisms of types D, T, and A in reactive liquid NO2 will be studied from detailed analysis of the reaction trajectory and energy evolution in a forthcoming paper. 1

FIG. 10. The time-averaged distribution of N atoms of the solvent around a reactant NO2 pair at R⫽2.4 Å 共the TS in reactive liquid NO2 ): 共a兲 R i is the distance between center C of the reactant N–N atoms and N atom of the solvent, and ␪ i is the angle between the reactant’s N–N axis and the reactant–solvent C–N axis; 共b兲 angular distribution of solvent N atoms P( ␪ i ) for R i ⭐5 Å in reactive liquid NO2 ; 共c兲 radial distribution P(R i )/R 2i of solvent’s N atoms around the reactant’s center C in reactive liquid NO2 ; 共d兲 radial distribution in inert solvent N2 O4 . Maximum distributions are normalized to 1. A maximum in 共c兲 at R i ⯝2.1 Å in reactive liquid NO2 suggests that an NO2 -trimer structure is formed at the TS barrier.

A. H. Zewail, Femtochemistry, Ultrafast Dynamics of the Chemical Bonds 共World Scientific, Singapore, 1994兲. 2 ¯ , S. Hayashi, and K. Machida, J. Chem. Phys. 115, 10852 共2001兲. T. Kato 3 J. T. Hynes, Annu. Rev. Phys. Chem. 36, 573 共1985兲. 4 G. R. Fleming and P. G. Wolynes, Phys. Today 43共5兲, 36 共1990兲. 5 P. W. Atkins, Physical Chemistry 共Oxford University Press, New York, 1990兲. 6 M. Maroncelli, J. MacInnis, and G. R. Fleming, Science 243, 1674 共1989兲. 7 J. T. Bergsma, J. R. Reimers, K. R. Wilson, and J. T. Hynes, J. Chem. Phys. 85, 5625 共1986兲; I. Benjamin, B. J. Gertner, N. J. Tang, and K. R. Wilson, J. Am. Chem. Soc. 112, 524 共1990兲. 8 G. van der Zwan and J. T. Hynes, J. Chem. Phys. 76, 2993 共1982兲; 78, 4174 共1983兲. 9 A. J. Parker, Chem. Rev. 共Washington, D.C.兲 69, 1 共1969兲; W. M.


838


Olmstead and J. I. Brauman, J. Am. Chem. Soc. 99, 4219 共1977兲; D. K. Bohme and G. I. Mackay, ibid. 103, 978 共1981兲. 10 J. Chandrasekhar, S. F. Smith, and W. L. Jorgensen, J. Am. Chem. Soc. 106, 3049 共1984兲; 107, 2974 共1985兲; W. L. Jorgensen, Acc. Chem. Res. 22, 189 共1989兲. 11 J. P. Bergsma, B. J. Gertner, K. R. Wilson, and J. T. Hynes, J. Chem. Phys. 86, 1356 共1987兲. 12 R. A. Ogg and M. Polanyi, Trans. Faraday Soc. 31, 604 共1935兲; D. A. Zichi and J. T. Hynes, J. Chem. Phys. 88, 2513 共1988兲; S. Lee and J. T. Hynes, ibid. 88, 6853 共1988兲; 88, 6863 共1988兲. 13 W. P. Keirstead, K. R. Wilson, and J. T. Hynes, J. Chem. Phys. 95, 5256 共1991兲; 88, 6853 共1988兲; 88, 6863 共1988兲. 14 W. J. Moore, Physical Chemistry 共Prentice Hall, Englewood Cliffs, NJ, 1972兲; J. E. Murrell and E. A. Boucher, Properties of Liquids and Solutions 共Wiley, New York, 1982兲. 15 P. Gray and P. Rathbone, J. Chem. Soc. 1955, 3550. 16 C. M. Steese and A. G. Whittaker, J. Chem. Phys. 24, 776 共1956兲; C. M. Steese, ibid. 24, 780 共1956兲. 17 A. J. Vosper, J. Chem. Soc. A 1970, 2191. 18 T. F. Redmond and B. B. Wayland, J. Phys. Chem. 72, 1626 共1968兲. 19 A. Boughriet, M. Wartel, and J. C. Fischer, J. Electroanal. Chem. Interfacial Electrochem. 190, 103 共1985兲. 20 W. F. Giauque and J. D. Kemp, J. Chem. Phys. 6, 40 共1938兲; H. K. Roscoe

Toshiko Kato¯ and A. H. Hind, J. Atmos. Chem. 16, 257 共1993兲; A. J. Vosper, J. Chem. Soc. A 1970, 625. 21 T. Carrington and N. Davidson, J. Phys. Chem. 57, 418 共1953兲. 22 M. Cher, J. Chem. Phys. 37, 2564 共1962兲. 23 F. A. Lindemann, Trans. Faraday Soc. 17, 598 共1922兲. 24 B. Markwalder, P. Gozel, and H. van den Bergh, J. Chem. Phys. 97, 5472 共1992兲. 25 S. S. Wesolowski, J. T. Fermann, T. D. Crawford, and H. F. Schaefer, J. Chem. Phys. 106, 7178 共1997兲; F. R. Ornellas, S. M. Resende, F. B. C. Machado, and O. Roberto-Neto, ibid. 118, 4060 共2003兲. 26 ¯ , S. Hayashi, M. Oobatake, and K. Machida, J. Chem. Phys. 100, T. Kato 2777 共1994兲. 27 ¯ , J. Chem. Phys. 105, 4511 共1996兲; 105, 9502 共1996兲; 108, 6611 T. Kato 共1998兲. 28 S. H. Bauer, Chem. Rev. 共Washington, D.C.兲 102, 3893 共2002兲. 29 I. Pastirk, M. Comstock, and M. Dantus, Chem. Phys. Lett. 349, 71 共2001兲. 30 ¯ , S. Hayashi, and K. Machida, J. Mol. Liq. 65Õ66, 405 共1995兲. T. Kato 31 ¯ and H. Katoˆ, Mol. Phys. 66, 1183 共1989兲. T. Kato 32 ¯ , J. Chem. Phys. 117, 6629 共2002兲. T. Kato 33 M. V. Kololkov, J. Manz, and G. K. Paramonov, Chemical reactions and their control on the femtosecond time scale, Advances in Chemical Physics, Vol. 101 共Wiley, New York, 1997兲, p. 327.
