A Model Multidimensional Analytic Potential Energy Function for the

J. Phys. Chem. 1994,98, 1608-1619

1608

A Model Multidimensional Analytic Potential Energy Function for the C1Br- Reaction

+ CH3Br

-

ClCH3

+

Haobin Wang, Ling Zhu, and William L. Hase’ Department of Chemistry, Wayne State University, Detroit, Michigan 48202 Received: July 23, 1993’

Hartree-Fock calculations, utilizing the SV4PP split valence contracted Gaussian basis set for C1 and Br and the 6-31G* basis set for CH3, are used to determine potential energy surface properties for the C1- CH3Br ClCH3 B r sN2 nucleophilic substitution reaction. This ab initio information is used in concert with experimental data to develop a multidimensional analytic potential energy function for the C1- CH3Br reaction system, by modifying the analytic potential energy function developed previously for C1,- CH3Clb CIaCH3 clb-. Each atom and degree of freedom is treated explicitly in the analytic function. Transition-state theory calculations are performed with the derived analytic potential to determine rate constants for C1- CH3Br Cl--.CH3Br association, and for dissociation of the Cl--CH3Br prereaction complex to reactants C1- CH,Br, and isomerization of this complex to the postreaction complex ClCH3.-Br. R R K M calculations are also performed to determine rate constants for dissociation and isomerization of the complexes versus energy and angular momentum. The analytic function developed here for the C1- CH3Br ClCH3 B r system has been incorporated into the general chemical dynamics computer program VENUS.

-

+

+

+

+

+

-

+

Introduction Gas-phase bimolecular nucleophilic substitution (SN2) has been the subject of numerous experimentall-z5 and theoretical studies.ZM3 Valuable physical insight can be obtained by studying the simplest s N 2 reaction X-

+ CH3Y

-

XCH,

+ Y-

(1) Both experimental and theoretical studies show that this reaction is characterized by pre- and postreaction complexes X-.-CH,Y and XCH3.-Y- and a central barrier [X.-CH3.-Y]-. If it is assumed the complexes are long-lived and their internal energy is distributed statistically, Rice-Ramsperger-Kassel-Marcus (RRKM) theory44 can be used to calculate rate constants for the complexes to cross the central barrier and to form reactants and product^.^-^^ Several studies have indicated that RRKM theory may be an incomplete model for analyzing some s N 2 reactions. For the series of s N 2 reactions X- + CH3F XCH3 F- (X = H, F, OH) studied by quantum-dynamical and reaction-path Hamiltonian calculations, Basilevsky and R y a b o ~ ~ ~found 8 ~ 3 evidence that the reactions are direct without trapping in the prereaction complex and that the low reaction efficiency for s N 2 reactions with a central barrier lower than the reactant potential is due to a high probability of reflecting incoming trajectories back to reactants. For highly exothermic s N 2 reactions with a large exothermicity and a very low central barrier, an RRKM analysis predicts that forming the prereaction complex should be rate controlling,since the prereaction complex’s RRKM rateconstant for central barrier crossing is much larger than that for dissociating to reactants. However, Su et aLZ3in experimental studies of F- with CH3C1, CH3Br, and CH31 and Depuy et al.24 in experimental studies of HO-, CH30-, and F- with CH3Cl found that this model overestimates the reaction rate for these highly exothermic reactions, for which the central barrier should have no (or negligible) influence. These experimental results are consistent with the above theoretical predictions of Basilevsky and Ryaboy for similar s N 2 rea~tions.423~3 A significant number of deviations from the RRKM model for s N 2 reactions have been observed in trajectory simulations of the

-

@

-

+

+

Abstract published in Advance ACS Abstracts, January 15, 1994.

0022-365419412098-1608$04.50/0

+

-

+

+

C1,CH3Clb reactive sy~tem.3~-’~ The unimolecular rate constants for Cl,-.-CH&!lb complexes formed by C1,- + CH3Clb association do not agree with RRKM theory and are consistent with a model in which energy is only redistributed between the complex’s three intermolecular modes and not between the nine CH3Cl intramolecularmode~.3~.3* Exciting the C-Cl stretch mode of the CH3Cl reactant opens up a direct substitution mechan i ~ m 3 ~akin J ~ to the effect of vibrational excitation on a A + BC AB C displacement reaction with a late potential energy barrier.4s Trajectories initialized at the central barrier remain trapped in the vicinity of the barrier with frequent barrier re crossing^,^^ a result reflective of the bottleneck for energy transfer between the Cla-.-CH3Clb complex’s intermolecular and intramolecular modes.38 Recently, the non-RRKM dynamics for the C1,- + CH3Clb system has been analyzed by studying the dynamical stereochemistry and energy-transfer pathways for C1,--CH3Clb complex formation and direct substitution.@ In recent work,20.21J5,4~8 a number of experimental studies focused on the S Nreaction ~

-

+

C1-

+ CH,Br

-

ClCH,

+ Br-

(2)

Non-RRKM effects were observed in these studies. Viggiano et al.47found that the rate constant for this reaction is insensitive to the temperature of CH3Br. In contrast, an RRKM analysis, as has been performed for the ClCHzCN CI- C1- ClCH2CN would suggest a temperature dependence. Graul and Bowers48 formed the prereaction complex Cl--CH3Br with sufficient internal excitation to monitor its unimolecular decay into products. They observed a product translational energy considerably less than the prediction of statistical phase space theory, which, by energy conservation, indicates extensive internal excitation of the CH3Cl product. Drawing from the trajectory results for the C1,CH3Clb s y ~ t e mand ~ ~the~ principle ~ ~ of microscopic reversibility, they suggested that the C-Cl stretch and/or CH3 umbrella vibrational modes may be preferentially excited. Of particular assistance in analyzing the C1- + CH3Br reaction dynamics may be measurements of the rate constant’s pressure dependence,20J since they may reveal the lifetime of the Cl-...CH3Br complex, which can be compared with that of RRKM theory. The rate constant decreases with increase in pressure,20.21 and transition-state theory can be tested at high

+

+

0 1994 American Chemical Society

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+

Analytic Potential Energy Function

The Journal of Physical Chemistry, Vol. 98, No. 6, 1994 1609

pressure where there is a Boltzmann distribution of energy states at the [Cl--CH3-.Br]- central barrier.50 To interpret the above experimental results, a theoretical/ computational study would be very beneficial. In this paper we report three components of such a study. First, ab initio calculations are reported for the C1- + CH3Br ClCH3 + B r reaction. The results of these calculations are then combined with experimental data to derive an analytic potential energy function for the C1- + CH3Br ClCH3 B r reaction by modifying the Cia- + CH3Clb Cl,CH3 clb- analytical potential.36 This analytic potential is constructed from a bondenergy-bond-order (BEBO) and switching function formalism, which has yielded meaningful potentials for a number of chemical reaction^.^'-^^ In the final part of the paper, the analytic potential energy function developed here is used to calculate RRKM and transition-state theory rate constants for the C1- CH3Br ClCH3 B r reactive system. Detailed trajectory calculations utilizing this analytic potential will be reported later.57

-

--

+ +

+

+

-

TABLE 1: Ab Initio Energies and Structures for Statio~ry Points. property exptb this workC ref 58d ref 59e Reactant, CH3Br 1.934' 1.944 (1.939) 2.026 1.95, 1.97 RcBr RH-C

1.082 107.7 0

OHX-B~

energy

1.077 (1.087) 107.8 (107.8) 0

1.189 109.6 0

1.08, 1.08 107.7, 107.2

0

Complex, C1-- CH3Br RCIM Rc-B~ RH-C OH-C-Br

energy

-10f lg

3.221 1.991 1.071 106.8 -10.74

3.078 2.073 1.186 1 1 1.3 -8.82

Central Barrier Rci-c RC-Br RH-C

2.469 2.458 1.062 92.2 -2.91

OH-CBr

energy

2.410 2.418 1.178 93.2 -0.78

Complex, ClCH3.s . B r

Ab Initio Calculation

Structures and Energies. Ab initio calculations of reaction 2 have been previously r e p ~ r t e d . Hirao ~ ~ , ~and ~ Kebarles8performed Hartree-Fock calculations with a MINI basis set developed by Huzinaga et a1.60to study the minimum energy path for reaction 2. However, though the relative energies they calculated for stationary points on the potential energy surface agree fairly well with experiment, the geometries they found were quite poor. To fit an analytic potential energy function, one needs both good energies and geometries. To achieve this, we had to select a larger basis set for our calculations. However, in choosing the basis set we had to consider both the accuracy required and the expense of the calculations since numerous potential energy points are needed to fit an analytic function. After taking both of these features into account and comparing the results of different basis sets, we decided to use a split valence contracted Gaussian basis, i.e., the SV4PP basis set developed by Andzelm et a1.61for C1 and Br and the 6-31G* basis set for CH3. These two basis sets have similar sizes and are, thus, considered compatible for the calculation reported here. The calculations were performed with the Gaussian 92 series of programs.62 Geometries and relative energies calculated in this work for stationary points on the C1- CH3Br ClCH3 B r potential energy surface are listed in Table 1, where they are compared with the results of other ab initio calculation^,^^^^^ experimental relative energies for the complexesg and p r o d ~ ~ t s , and ~~,~ experimental geometries for CH3Cl and CH3Br.65,66The ab initio geometries reported here for CH3Cl and CH3Br are in good agreement with experiment and the ab initio calculations of Vetter and Z U l i ~ k e .The ~ ~ calculated relative energy for the Cl--CH3Br complex is in good agreement with experiment,I3 while the calculated energy for the ClCH3-Br complex and the calculated heat of reaction are lower than the experimental values.13,46,63,64 Transition-state theory and RRKM theory have been used to model the rate constant for reaction 2 and deduce a relative energy for the central barrier -1-2 kcal/mol below the reactant energy,i3,58,67which is similar to the ab initio value found here. However, the appropriateness of using the RRKM and transitionstate theories to model the reaction energies may be inappropriate, since, as discussed in the Introduction, a number of nonstatistical attributes have been observed for the reaction. Two higher level calculations were performed to test the sensitivity of the ab initio results to the treatment of electron correlation. The same basis set, as described above, was used for these calculations. The relative energy between products and reactants was calculated a t the MP2 level with CH3Cl and CH3Br geometries also optimized a t this level of theory (the resulting geometries are given in Table 1). The MP2 heat of reaction is -1 1.34 kcal/mol, in agreement with the HF result. Energies of

+

-

+

RCI-C &-Br

RH-C 0H-C-Br

energy

-168

Rcix RH-C

1.776h 1.085 71.4 -6,'-8,'-9k

1.825 3.517 1.073 72.2 -21.21

1.922 3.365 1.189 68.9 -19.50

Product, CH3C1 8H-C-Br

1.789 (1.793) 1.076 (1.088) 71.9 (71.9) -12.63 (-11.34)

1.815 1.190 70.5 -12.10

1.787, 1.79 1.079, 1.08 71.5,71.8

energy Energies are in kilocalories per mole, bond lengthsin angstroms,and angles in degrees. *The experimental relative energies are for 0 K. Hartree-Fock calculations with a SV4PP basis set for C1 and Br, and a 6-31G* basis set for CH3. Geometries and energies for the MP2 level calculationswith the same basis sets are given in parentheses. The listed relative energies are electronicenergy differenceswithout including zeropoint energies. With zero-point energy included, the relative energy differencesfor the prereaction complex, the central barrier, the postreaction complex, and products are -10.32, -3.18, -20.06, and -11.85, respectively. Zero-point energy is calculated from the HF harmonic vibrationalfrequencieswithout scaling. Hartree-Fock calculationswith a MINI basis set. The listed relative energies are electronic energy differenceswithout including zero-pointenergies. The first value is the result of an all electron (AE) Hartree-Fockcalculation, the second value is the result of a pseudopotential (PP) model for the nonvalenceelectrons. /The CH3Br experimental geometry is taken from ref 66. 8 Reference 13. TheCH3CIexperimentalgeometryistakenfromref65. References 46 and 63. References 13 and 64. The reported C-Cl and C-Br bond ~energies * ~ ~ and , ~ C1 ~ and Br electron affinities64give a 0 K heat of reaction of -9 kcal/mol. J

the stationary points were also calculated a t the MP4SDTQ level using the HF-optimized geometries. The resulting relative electronic energy differences, without including zero-point energy, are-12.34,-6.01,-21.37, and-1 1.34 kcal/molfor theprereaction complex, central barrier, postreaction complex, and products, respectively. Comparison with the H F energies (footnote c of Table 1) shows that, except for the central barrier energy, the HF and MP4SDTQ energies are similar. With such a high level of theory the basis set and treatment of electron correlation may not be balanced for all regions of the potential energy surface, and to determine an accurate central barrier energy with the MP4SDTQ level of theory, it may be necessary to add diffuse functions. Harmonic Vibrational Frequencies. Harmonic vibrational frequencies for stationary points on the potential energy surface were calculated a t the HF/SV4PP/6-31G* level of theory described above. The resulting frequencies for CH3Br and CH3C1 are listed in Table 2, and those for the complexes and central barrier are listed in Table 3. The frequencies for CH3Cl are similar to those calculated previously at the HF/6-31G* (Le.,

1610 The Journal of Physical Chemistry, Vol. 98, No. 6, 1994

TABLE 2: CH3Br and CH3Cl Harmonic Frequencies. mode

ab initiob

AI, C-Br str E, CH3 rock AI, CHI deform E, CH3 deform AI, C-H str E, C-H str

642 1066 1484 1620 3278 3392

AI, CI-C str E, CH3 rock AI, CHI deform E, CH3 deform AI, C-H str E, C-H str

774 1130 1528 1627 3268 3374

exptc

PESl(Br)d

PES2(Br)d

620 1065 1497 1457 3048 3183

617 927 1374 1442 3047 3182

739 1108 1550 1460 3050 3181

735 968 1423 1440 3050 3181

CH3Br 617 974 1333 1472 3082 3184

CH3Cl 740 1038 1383 1482 3074 3166

a Frequency units are reciprocal centimeters. Frequencies calculated in this work at the HF/SV4PP/6-3 1G* level of theory. See ref 68.d For PESl (Br) the H-C-CI and H-C-Br bending force constants are the ab initiovalues, while these a b initio force constants are scaled for PES2(Br) to obtain better agreement with experiment (see text).

E, CI- bend AI, CI-C str AI, C-Br str E, CH3 rock AI, CH3 deform E, CH3 deform AI, C-H str E, C-H str E, CI-C-Br bend AI, CI-C-Br str E, CH3 rock A2, out-of-plane bend E, CH3 deform AI, C-H str E, C-H str reaction coordinate

E, B r bend AI, C-Br str AI, CI-C str E, CH3 rock AI, CH3 deform E, CH3 deform AI, C-H str E, C-H str

a b initiob

scaleC PES1(Br)d -PES2(Br)d

Complex, CI-. CH3Br 71 64 94 90 518 498 1032 943 1420 1276 1595 1450 3334 3134 3471 3258 Central Barrier 183 165 172 164 974 895 1203 1089 1549 1411 3418 3215 3630 3406 380i Complex, CICH3. .Br65 58 70 67 674 644 1099 1010 1483 1342 1610 1467 3314 3117 3438 3226

experimental and ab initio frequencies for CH3Br and CH3Cl. Scale factors for CH3Br were used to scale the Cl--CH3Br complex and central barrier ab initio frequencies, and scale factors for CH3CI were used to scale the ClCH3m-Br ab initio frequencies. A scale factor for a particular mode of CH3Br or CH3Cl was used to scale the ab initio frequency of the complexesor central barrier for the same type of mode. The central barrier ab initio frequencies were scaled by CH3Br scale factors, since the central barrier structure resembles the reactants more than the products. However, this is a minor issue, since the CH3Br and CH3C1 scale factors are nearly identical. The scaled ab initio frequencies for the complexes and central barrier are expected to be more accurate than the unscaled frequencies.68 Reaction Path. To derive an analytic potential energy function for reaction 2 requires information regarding the reaction path in addition to the information given above for the stationary points. Reaction-path calculations were performed at the HF/SV4PP/ 6-3 l G * level of theory using Gaussian 92.62 The path of steepest descent in mass-weighted Cartesian coordinates was followed in the calculations, which is a standard option in Gaussian. These calculations were initiated at large C1- + CH3Br and ClCH3 B r separations to study the reaction path between the reactants and the Cl---CH3Br complex and the reaction path between the products and the CICH3-.Brcomplex. Thereaction path between the central barrier and the complexes was determined by initializing calculations at the central barrier with either positive or negative reaction coordinate motion. Internal coordinates and relative electronic energies along the reaction path are listed in Table 4. The reaction path potential is plotted in Figure 1 as a function of gb which is the difference in the C-Br and C-Cl bond lengths, i.e., g b = rC-Br - rcxI.

+

TABLE 3: Harmonic Vibrational Frequencies for the ComDlexes and Central Barrier. mode

Wang et al.

72 91 500 1089 1467 1452 3047 3293

72 91 499 956 1344 1434 3047 3293

169 161 1155 1185 1340 3215 3415 400i

146 161 995 1023 1339 32!5 3415 400i

64 68 646 1123 1548 1462 3101 3238

64 68 644 987 1418 1439 3101 3238

a Frequency units arereciprocal centimeters. The abinitiocalculations performed hereat theHF/SV4PP/6-31G* leveloftheory. Scale factors determined from the CH3Br and CH3Cl a b initio and experimental frequencies were used to scale ab initio frequencies for the complexes and central barrier; see text. For PESl(Br) the H-C-CI and H-C-Br bending force constants are the a b initio values, while these ab initio force constants are scaled for PES2(Br) to obtain better agreement with experiment (see text).

783, 1138, 1538, 1629, 3268, and 3372 and MP2/631G** (Le., 788,1082,1464,1545,3171,and 3289cm-l)32levels of theory. The normal situation is for ab initio calculations to overestimate harmonic frequencies,68 and comparing the experimentaF9 and ab initio frequencies in Table 2 shows that this is the case for both CH3Br and CH3Cl. Ratios between the ab initioand experimental frequencies can be used to derive factors for scaling ab initio frequencies at regions of the potential energy surface where experimental frequencies are unknown. The HF/SV4PP/6-3 lG* harmonic vibrational frequencies for the complexes and central barrier are listed in Table 3. These frequencies were scaled by factors derived from ratios of the

Analytic Potential Energy Function The analytic potential function for reaction 2 was constructed by extending the previous global function for the C1; + CH$&, Cl,CH3 + c l b - reaction.36 Most of the terms in this new potential are the same as those for the c1,- + CH3Clb potential, with only the parameters for the terms changed. The same procedure was used here for fitting this potential as that for the c1,- + CH3Clb system.36 Here differences between the two potentials are presented. To construct the complete potential for reaction 2, the reader is expected to carefully read ref 36 and then amend the potential given for the C1,- + CH3Clb system using the information provided below. Following the previous ~ o r k , an 3 ~ analytic potential energy function for reaction 2 can be written as

-.

I1 - sLR(ga)l + vBr(rb&) - sLR(&)l v@(ra3ga)I1 - sLR(ga)l + v+(rb&) i1 - sLR(gb)l + vC',,Br[l -sLIi(ga)l[l -sLRkb)l + v&ga) + v&&) +

Vtotal

= vCl(ra,ga)

+

vH,(g) + cRsLR(ga) + Vb,RsLRkb) + constant (3) where r, = reel, r b = rC-Br, g, = ra - rb, and g b = r b - r,. The latter coordinates, g, and gb, measure the extent of reaction. The potential is divided into terms for short-range interactions, C1+ CH3Br long-range interactions c R , and ClCH3 + B r longrange interactions CR.Two different analytic potentials PESl (Br) and PES2(Br) are developed here by v-rying the scaling of the H-C-Cl and H-C-Br bending force canstants. For PES1(Br) "rawn HF/SV4PP/6-31G* values are used for these force constants, while these ab initio values are scaled to obtain better agreement with experiment for PES2(Br). This is the only difference between PESl(Br) and PES2(Br). CH3Cl and CH3Br Potentials. At very large B r and C1separations the analytic potential energy function becomes that for CH3Cl and CH3Br, respectively. The potentials used here for these two molecules have the same analytic form as that used

Analytic Potential Energy Function

The Journal of Physical Chemistry, Vol. 98, No. 6, 1994 1611

-

TABLE 4: AB Initio Reaction Path Data for CI- + CHjBr CICHt + Br a m

7.000 4.000 3.500 3.447 3.394 3.342 3.289 3.237 3.216 3.186 3.133 3.081 3.028 2.977 2.926 2.878 2.833 2.792 2.753 2.716 2.680 2.645 2.610 2.576 2.541 2.506 2.469 2.433 2.396 2.358 2.320 2.281 2.241 2.201 2.161 2.121 2.080 2.040 2.001 1.964 1.930 1.900 1.877 1.861 1.851 1.845 1.840 1.837 1.833 1.831 1.828 1.826 1.823 1.822 1.820 1.819 1.818 1.817 1.816 1.815 1.814 1.804 1.796 1.789

1.944 1.952 1.969 1.982 1.984 1.986 1.989 1.992 1.995 1.997 2.000 2.004 2.010 2.018 2.028 2.043 2.064 2.091 2.123 2.157 2.194 2.231 2.269 2.307 2.346 2.384 2.421 2.458 2.495 2.530 2.566 2.600 2.634 2.667 2.700 2.733 2.765 2.797 $831 2.865 2.901 2.940 2.983 3.030 3.080 3.132 3.184 3.237 3.290 3.342 3.395 3.448 3.517 3.577 3.630 3.683 3.736 3.788 3.841 3.894 3.947 4.000 5.000 7.000 m

1.076 1.076 1.074 1.073 1.073 1.072 1.072 1.072 1.072 1.071 1.071 1.07 1 1.070 1.070 1.069 1.069 1.068 1.067 1.066 1.066 1.065 1.064 1.064 1.063 1.063 1.062 1.062 1.062 1.062 1.062 1.062 1.062 1.062 1.063 1.063 1.063 1.064 1.065 1.065 1.066 1.067 1.068 1.069 1.070 1.070 1.071 1.071 1.072 1.072 1.073 1.073 1.073 1.073 1.074 1.074 1.074 1.074 1.074 1.074 1.075 1.075 1.075 1.076 1.077 1.077

72.2 72.4 72.4 72.6 72.7 72.8 72.8 73.0 73.1 73.1 73.2 73.4 73.6 73.9 74.2 74.7 75.4 76.2 77.2 78.2 79.3 80.4 81.6 82.8 84.0 85.3 86.5 87.8 89.0 90.3 91.5 92.8 94.0 95.3 96.5 97.7 98.9 100.1 101.2 102.3 103.4 104.4 105.3 106.0 106.5 106.8 107.1 107.3 107.4 107.5 107.6 107.7 107.8 107.9 107.9 108.0 108.0 108.0 108.0 108.1 108.1 108.1 108.1 108.1 108.2

111.1 111.3 111.3 111.5 111.6 111.6 111.7 111.8 111.9 111.9 112.0 112.2 112.4 112.6 112.9 113.3 113.9 114.5 115.2 115.9 116.6 117.3 117.9 118.5 118.9 119.3 119.6 119.9 120.0 120.0 119.9 119.8 119.5 119.2 118.7 118.2 117.6 117.0 116.3 115.5 114.8 114.0 113.3 112.7 112.3 112.0 111.8 111.6 111.4 111.3 111.2 111.2 111.1 111.0 111.0 110.9 110.9 110.9 110.9 110.9 110.8 110.8 110.8 110.8 110.7

0.000 -2.766 -8.367 -10.219 -10.361 -10.492 -10.599 -10.680 -1 0.7 18 -10.736 -10.712 -10.655 -1 1.536 -10.342 -10.066 -9.696 -9.220 -8.63 1 -7.942 -7.177 -6.381 -5.592 -4.852 -4.194 -3,649 -3.242 -2.991 -2.903 -2.997 -3.267 -3.718 -4.357 -5.166 -6.143 -7.265 -8.518 -9.872 -1 1.295 -12.736 -14.152 -15.481 -16.672 -18.824 -1 8.520 -19.197 -19.743 -20.175 -20.513 -20.770 -20.9 58 -21.084 -21.153 -21.214 -21.165 -21.11 5 -2 1.046 -20.952 -20.846 -20.726 -20.601 -20.463 -20.325 -17.803 -15.310 -12.627

Internal coordinates are in angstroms and degrees, and the relative electronicenergy in kilocalories per mole. The reaction path is the path of steepest descent in mass-weighted Cartesian coordinates.

-

previously for CH3Cl in developing the C1,- + CH3Clb C1,CH3 + clb- analytic potential. Though the CH3Cl potential developed here is similar to the previous one, there are some minor differences. First, the previous CH3Cl potential had a HF/6-3 lG* equilibrium geometry while the one developed here has a HF/SV4PP/6-31G* equilibrium geometry. These two

-

0.0

I

I

h

P

e $

-5.0 -10.0

U

W -15.0

w

3 er;

-20*o -25.0

-6.0

-4.0

-2.0

0.0

2.0

6.0

4.0

gb (A)

Figure 1. Plot of reaction path potential versus gb = rc-er - rc41: Solid points, HF/SV4PP/6-3 l G * calculation: solid line, the analytic function between the two complexes.

calculations give slightly different geometries, primarily for the H-C-H and H-C-Cl equilibrium valence angles (compare the geometries in Table I and Table VI11 of ref 36). Except for the quadratic H-C-Cl bending force constant and quadratic H-C stretch force constant, all the CH3Cl force constants for the potential developed here are the same as those chosen previously for the c1,- + CH3Clb ClaCH3 + clb- potential. The HF/ SV4PP/6-3 1G* calculation gives a H-C-Cl bending quadratic force c o n s t a n t t c = 0.8424 m d y d / r a d * that is 1.5% smaller than that from the previous HF/6-31G* calculation. To obtain better agreement with experiment, this force constant is scaled by F F = 0.7415 for the potential energy surface identified as = 1.OOO. PES2(Br). For PES 1(Br) is not scaled and For the c1,- CH3Clb ClaCH3 + Clb- potential, the same H-C “equilibrium” bond length and quadratic force constant were used for all regions of the potential energy surface. Values were chosen for these parameters intermediate between the values for CH3Cl and the central barrier. For the potential developed here, these two parameters are allowed to smoothly vary between stationary points. However, since the HF/SV4PP/6-31G* calculations give nearly the same H-C equilibrium bond length for the CH3C1and CH3Br asymptotic limits of the potential energy surface, the average of these two values rHC = 1.076 A was used for the equilibrium geometry of both of these molecules. Likewise, for both molecules the H-C bond strength was set to 110.00 kcal/mol, as before, to match experiment. To fit the experimental H-C stretching frequencies for CH3Cl and CH3Br, and Morse parameter f l is set ~ to ~ 1.876 A-l, a value 0.5% higher than that used for the cia-+ CH3Clb ClaCH3 + clb- potential. For PES2(Br), quadratic force constants were chosen for the non-HC stretching internal coordinates of CH3Br to match the experimental non-HC stretch vibrational frequencies. This involved scaling the HF/SV4PP/6-3 lG* H-C-Br quadratic bending force constant = 0.7822 mdyn.A/rad* by = 0.7415, which is the same as F F . For PESl(Br) is not = 1.000. CH3Cl and CH3Br have the same scaled and H-C-H and H-C-X (X = C1, Br) cubic and quartic bending force constants. These force constants are the same for PES1(Br) and PES2(Br). For the model analytic potential energy function developed here the classical exothermicity for reaction 2 is given by the difference in the C-C1 and C-Br classical bond energies. To fit the ab initio exothermicity without zero-point energies added, the classical C-Cl and C-Br bond energies are set to 84.976 and 72.349 kcal/mol, respectively. A complete list of potential parameters for CH$l and CH3Br is given in Table 5 . Long-Range Potentials. The long-range potentials are written as the sum of an ion-dipole attraction, a term describing increased anisotropy in the potential as the ion and the molecule approach, and a term to describe attractions between the anion X-and the

tc

-+

+

-

(F

-

tB

Fy

tB

Wang et al.

1612 The Journal of Physical Chemistry, Vol. 98, No. 6, 1994 TABLE 5: Nonswitcbing-Function Potential Energy Surface Parameters for PES2(Br). CH,Br, CH3CI Parametersb rMC 1.944,1.7890 A BMC 1.7217, 1.7368 A-I DMC 72.349,84.976 kcal/mol

x;2

110.8224, 11 1.2768' 107.5990, 108.0837' 0.5001,0.5001 mdyn.A/radz -0.1536, -0.1536 mdyneAlrad3

eMC

F:ay

*C

BHC

DHC

Complex Cl-* CH3Br Parameters

0.1655,0.1655 mdyn.A/rad4 0.7822,0.8424 mdyn.A/radZ -0.2305, -0.2305 mdyn-A/rad' 0.741 5 1.0770 A 1.8757 .&-I 110.00 kcal/mol

+

3.2157 A 1.2190 A 0.9530 A-I 10.7364 kcal/mol

rc gc BC

Dc

A

B c+

Saddle Point Parameters aMAX

120.0' 0.3549 mdyn.A/rad2

OMAX

flu

3.5166 A 1.691 1 A 1.0089 A-l 8.5874 kcal/mol

rc gc

Be Dc

-

bFAX

Complex ClCHy . B r Parameters A B c4

Long-RangeParameters for C1- + CH3Br D ba

0.017 077 34 mdyn.A/rad' 1.647 862 08 A-2 1.81 D

a€

b€ Pd

Long-RangeParameters for ClCH3 + B r

0.017 077 34 mdyn-Alrad3 1.647 862 08 A-2 1.87 D

at Pd

D

ba

4655.416 A6, kcal/mol 1403.114 A4.kcal/mol 0.631 382 88 rad-2

-0.0734 mdyn-br/rad3 0.0017 mdyn.A/rad4 4655.416 A6-kcal/mol 1403.114 A4.kcal/mol 0.631 382 88 rad-z

207.968 35 A'.kcal/mol 72.4010'

207.968 35 A4.kcal/mol 71.9163'

0 The only difference between PESl(Br) and PES2(Br) is that F+ = 1.000 for PESl(Br). The parameters for CHpCl, listed second, are identified by MC instead of MB, e.g., rMC. F+ and the HC Morse parameters mc, BHC, and &C are the same for CHpBr and CH3Cl.

CI-C and C-Br Stretching Terms. These two terms, which aredenoted by Vcl(r,,g,) and vBr(rbgb),have the Morse functional form:

H-atoms; i.e.

With the use of the expressions Here 5 is the Cl---C-Br (or Cl-C-Br) angle, the 4, are the three X--.C-H valence angles for the anion and the H-atoms, R is the distance between the anion and the molecule's (CH3Cl or CHpBr) center of mass, and rc is the X--C distance a t the complex. For X = c1, p d = 1.81 D,r = r,, and g = g,. For x = Br, p d = 1.87 D,r = rb, and g = gb. The parameters in fE(r) and the parameters in the switching function SLR, which is used to connect the long-range and short-range potentials, are kept the same as in ref 36. SLR is given by

eR

cR

s L , ( g ) = 1.0

- e*R&gJ2

g 2 gc

However, since the reaction is asymmetric, the critical value for g denoted by gc,which identifies the transition between the longrange and short-range potentials, is different for C1- + CHpBr, and ClCHp + B r . The term gc is g, = 1.2190 A and gcb = 1.691 1 A for and respectively. Short-Range Potential. The short-range potential energy terms are similar to those in ref 36. The major modification is that, since the potential is now asymmetric, parameters for terms which depend on r, and ga are different from the parameters for terms which depend on rb and gb. Also, some modifications have been made in the switching functions to make them more flexible.

eR, cR cR,

(9)

cR,

the Morse parameters are allowed to vary smoothly between their values for CH3C1given by DMC= 84.976 kcal/mol, BMC = 1.7368 A-l, rMC = 1.789 A, values for CH3Br given by DMB= 72.349 kcal/mol, = 1.7217 A-1, rMB = 1.944 A, values for the Cl--CH3Br complex given by Dm = 10.7364 kcal/mol, Pa = O.9530A-l,rm = 3.2157A,andvaluesfor theClCH3-Brcomplex given by D c b = 8.5874 kcal/mol, & = 1.0089 A-', and r, = 3.5165 A. Da and DCbare equal to the cluster ab initio binding energies, and Bcb were chosen so that the cluster Cl--.C and C - B r stretching frequencies for the analytic surface agree with the scaled ab initio values for the frequencies (Table 3). SD,Sfla, Sfi, S,,, and S r b are switching functions that are fit (see below)

Analytic Potential Energy Function

The Journal of Physical Chemistry, Vol. 98, No. 6, 1994 1613

to the reaction path data listed in Table 4. The value of g, varies from gaC = 1.2190 A at the cluster to for the product, whereas the value of gb varies from gb, = 1.6911 A a t the cluster to --OD for the reactant. For large negative values of g, and gb, respectively, Vcl and VB, reduce to their Morse forms for CH3Cl and CH3Br. Along the reaction path, the bond distances and bend deformation angles are equal to their “equilibrium” values given in Table 4. As a result, the V,, V C I BV,, ~ , and VHCterms in eq 3 are zero. Thus, in the short-range interaction region between the two complexes, the minimum energy path potential, which is equated to the reaction path potential in Table 4 for fitting, can be written as --o)

VMEP

= vCl(ra,ga) + vBr(rb,&)

+ DMB

(14)

Since the C1-C and C-Br bond lengths r, and rb are their equilibrium values ro,(g,) and rob(&) along the reaction path, Vcl(rayga) = -&(ga) and VBdrbgb) = -DBr(gb). Inserting eqs 8 and 11 into eq 14 gives VMEP

= DMB -

- Dm - [(DMC - Dm)sD(ga)

Values for rh(ga)and rob(&) (eqs 10 and 13) are fit by the switching functions

+ (DMB -

Equation 15 is then fit to the a b initio reaction path data in Table 4 with the following function and parameters for the& switching function

where

a, = 0.950 364 41 1 c, = 0.852 632 165 b, = 1.091 14826 LIZ

= -1.487 003 45

CZ

= 0.222 973 332 b, = 2.423 550 37

a3 = 1.536 639 039 c3 = 0.141 861 632 b, = 3.063 316 82

and sD(g)

= O.0

g

gc

(16)

f,= 2.410 027 98 k, = 0.555 458 486 h, = 1.299 499 27

aI = 0.429 582 451 c, = 0.025 809 482

d , = 5.463 687 42

f,= -2.720 496 18 k, = 0.297 049 016 h, = 1.856 279 61

a, = 0.618 888 557 c2 = 0.221 944 809 d, = 3.920 137 64 f3

= 1.310 468 20 k3 = 0.111 322 239 h, = 2.997 654 91

a, = -0.048 471 008 c3 = 3.270 175 930 d, = 2.307 193 04 The sum of the ai(s equals unity. In this fit the same SDfunction is used for both SI,(&)and SI,(gb). As shown in Figure 1, it gives an excellent fit to the a b initio reaction path between the two complexes. Different functions were tried for SI,(&)and SD(gb), but they did not give significantly better results than that obtained here with one function. Hence, to simplify the problem and retain accuracy, this single SI,function was used to fit VMEP. The Sp functions have a similar form as before and are given by

where cpa = 0.061 096 2883 A-3 and cab = 0.082 459 9639 A-3. Sp(g,) was fit to values of &(g,) for the Cl--.CH3Br complex, central barrier, and CH3C1, Le., 0.9530, 1.0330, and 1.7368 A-1, respectively. Sp(gb) was fit t o v a l u e s o f ~ B r (for ~ )t h e C I C H r B r complex,central barrier, and CHsBr, i.e., 1.0089,1.2471,1.7217 A-I, respectively.

The accuracy of these fits to the C-Cl and C-Br “equilibrium” bond lengths along the minumum energy path are of the same high quality as the previous fit to “equilibrium” C-Cl, bond length for the C1,- + CH3Clb Cl,CH3 + clb- reaction system (Figure 3 of ref 36).

-

Angular Deformations. As for the C1,- + CH3Clb system, there are both 4 and 0 valence angles. There are three ClCH 4, angles, three BrCH 4 b angles, and three H C H fl angles. The &, and fl along the reaction path “equilibrium” values for (Table 4) are given by

where dma= 72.4010’, 4 - b = 71.9163’, ~ M = c 108.0837’, 4 M B = = 107.5990’, eMAX = 120.0’, 6M.C = 111.2768’, and 110.8224’- The value g a = -0,1347 A is the position on the reaction path where eo has its max~mumvalue of 120’. The fitted switching functions are

1614

Wang et al.

The Journal of Physical Chemistry, Vol. 98, No. 6, 1994 and

3

(31) with c8 = 0.45 A-2

gi = 0.0112 A

-

The above are all the changes made to the Cia- + CH3Clb Cl,CH3 Clb-potentialof ref 36 todevelopa potential for reaction 2. A list of nonswitching function parameters for this latter potential is given in Table 5. Two potentials are developed here by using the "raw" HF/SV4PP/6-3 1G* force constants for the quadratic H-C-Cl and H-C-Br bending force constants to give PES1(Br) and scdling these two a 6 initio force constants to obtain better agreement with experimental frequencies to give PES2(Br). Geometries and energies are the same for PESl(Br) and PES2(Br).

+

with

a, = -0.269 056 737

b, = 0.274 189 264

a, = 0.404 086 620

b, = 0.295 055 717

a3 = 1.0 - a, - a2

Properties of the CI- + CH3Br Potential Function

b3 = 0.025 042 0179

go, = -2.186 c, = -0.240 745 306 c, = 1.953 097 46

c3 = 1.0 - b, - b,

d, = 0.101 824 671 d3 = 0.063 521 2585

go, = -2.031

-- e

~ & t 0.1347)'l

A

sO(gb)

--

- 0.1347)';

a, = 1.447 351 10

bo = 1.353 601 57

e, = 2.343 370 68

e, = 2.348 000 05

vHc(r,g) = DHc[ l_e-8HC(g)k-Ofc(g)lI 2 -DHc

(27) where r is the H-C bond length and the "equilibrium" H-C bond length rHC and Morse parameter BHC are

In these equations gcan be either gaor g b because of the following quadratic form of the switching functions DHC= 110.0 kcal/mol, r: = 1.061 67 A, r i = 1.077 A, S,' = 2.0033 A-1, and = 1.8757 A-I. The switching functions are given by

with c, = 0.55 k2 g: = -0.061 88 A

-

+

C-H Stretching. One improvement to the potential of ref 36 is the incorporation of changes to the parameters for the C-H stretching modes along the minimum energy path. The C-H "equilibrium" bond length and vibrational frequencies are now functions of the reaction coordinates Le., the C-H Morse function is written as

s,(g) = e4-g,'l*

+ +

+

and sha) with

ClCH3 + B r Analytic

The analytic potential energy function developed here for the C1- CH3Br ClCH3 B r reaction is a modification of the analytic potential energy function presented previously for the Cia- + CH&lb ClaCH3 clb- reaction.36 For the most part, it was only necessary to change parameters in switching functions to fit the a6 initio calculations for C1- + CH3Br ClCH3 B r ; however, in a few cases slight changes in the form of the switching functions were also required. In the previous work for the C1,CHjClb system,36various functional forms were tested for each switching function and the functional form adopted for a particular switching function was the one that gave the best fit to the a b initio energies. It was initially hoped that the same simple functional form could be used for each switching function. However, it was found that this was not possible if good fits were to be obtained to the ab initio energies. It is noteworthy that the functional form of some of the switching functions for the C1,+ CH3Clb system had to be altered in developing the analytic potential energy function for the C1- CH3Br system. This attests to the lack of generality and portability of various switching functions. The quality of the fits for the C1,- + CH3Clb system is shown in Figures 2-6 of ref 36 and, as illustrated in Figure 1, the same type of fits are obtained for the C1- + CH3Br system. The switching functions and potential terms in eqs 6-31 fit the a b initioenergies given here for the C1- CH3Br system to within 0.7% on average, with a largest error for any point of 1.7%. It is not known how accurate an analytic potential energy function is needed to represent the dynamics of S Nreactions. ~ It certainly would be of interest to see if trajectories performed on a rather simple analytic function with a few parameters can reproduce the interesting dynamics observed from experiment47948 and from trajectories performed on surfaces accurately fit to ab initio data.3ss3740 In another vein, it would be of interest to test semiempirical quantum chemistry theory70by comparing the latter trajectories with those of direct dynamics calculations utilizing semiempirical quantum chemistry method~.~63~1 The analytic potential energy function developed above for reaction 2 was incorporated into the general chemical dynamics computer program VENUS.72 This involved writing algorithms for the potential energy andderivativesof the potential with respect to Cartesian coordinates for each term in the analytic function. With use of the parameters derived in the previous section, VENUS was used to determined the following properties of reaction 2 furnished by the analytic function. Potential Energy ContourDiagram. A contour diagram of the potential energy function in terms of the C-Cl and C-Br distances, with the remaining coordinates set a t their optimized values at very point on the contour, is shown in Figure 2. This contour map shows that the flexibility of movement in the cluster regions

+

d , = 0.597 150 028

-

-

(30)

+

+

The Journal of Physical Chemistry, Vol. 98, No. 6, 1994 1615

Analytic Potential Energy Function

----- - - . --- _- _-_

5.0

-. ,. ... ... /

4.5

-8

2500

5

2000

.

,

...

...

E

8 4*0 5 3.5

En

t z

3.0

4 u 2.5 -5.0

2.0

-3.0

-1.0

1.0 g,

1.5 1.5

2.0

-

3.0

3.5

4.0

4.5

5.0

C-CIDISTANCE

TABLE 6 Comparison of ab Initio and Potential Energy Surface Stationary Points’ RCI-C Rc-B~ Rc-H ~ H - c - B ~ 0H-C-H energy Reactants, CI- + CH3Br ab initio PES

-

1.944 1.944

ab initio PES

3.216 3.221

1.997 1.991

ab initio PES

2.469 2.470

2.458 2.462

abinitio PES

1.825 1.819

3.517 3.527

0)

1.076 1.077

.

107.8 107.6

111.1 111.3

0.0 0.0

111.9 111.5

-10.736 -10.731

119.9 119.8

-2.908 -2.785

111.9 111.8

-21.214 -21.209

Complex, CI-. sCH3Br 1.071 1.071

106.8 107.1

Central Barrier [CI-.CH3.-Br]1.062 1.062

Complex, CICH,.

--

1.073 1.074

92.2 92.6

.Br *

72.2 71.9

Products, ClCH3 + B r 71.9 110.8 -12.627 1.077 ab initio 1.789 71.9 110.8 -12.647 1.789 1.077 PES 0 Distances are in angstroms, angles are in degrees, and energies are in kilocalories per mole. is in sharp contrast to restricted passage through the central barrier region. Also evident are the considerable extensions of the C-Cl and C-Br bonds at the central barrier. The structure of the central barrier is more like that of the reactants than the products. Stationary Points. Geometries and energies of the reactants, complexes, central barrier, and products determined by the ab initio HF/SV4PP/6-3 1G* calculation, and the analytic potential are compared in Table 6. The exact locations of the complexes on the potential energy surface were found by following the negative of the gradient vector.73-75 The coordinate driving method76 was used to find the structure of the central barrier on the analytic function. Parameters for the analytic function give theCH3Cland CH3Br geometries. AsseeninTable6, theanalytic potential energy surface gives a good representation of the ab initio stationary point geometries and energies. Normal mode calculations were performed at the stationary points on the analytic function to calculate harmonic vibrational frequencies, which are compared in Table 3 with the HF/SV4PP/ 6-3 1G*, scaled, and experimental harmonic vibrational frequencies. The frequencies calculated from PES2(Br) are in good agreement with the experimental and scaled frequencies. For CH3Cl and CH3Br, the average difference between the experimental and PES2(Br) harmonic frequencies is 2.4%. The average difference between the scaled ab initio and PES2(Br) frequencies for the Cl--CH3Br complex, central barrier, and the C l C H r - B r complex is 4.5%. The frequencies for three X--C stretch and

5.0

Figure3. Harmonicfrequenciesorthogonalto thereaction path for PESl(Br) versus a.The descriptionsof the vibrational modes in descending CH stretch (E), CH stretch (A), (- -) frequency are (--e-)

ClCH3 + B r contour diagram in terms of the C-CI and C-Br distances in angstroms. The remaining coordinates are set equal to their optimized values as a function of g, = rc-ci - rc-Br. The contour lines are at 5 kcal/mol intervals. Figure 2. CI-

+ CH3Br

2.5

3.0

(A)

CH3 deformation(A), (-) (E).

-

(--a-)

CH3 deformation(E), and (-) CH3 rocking

bending intermolecular modes of a complex are much smaller than the molecule’s (Le., either CH3Cl or CH3Br) intramolecular modes. This disparity is larger forthe C l C H y B r complex than the Cl-.-CH3Br complex. ReactionPath. The reaction path in mass-weighted coordinates and its propertie~~’-~9 for the C1- CH3Br ClCH3 B r PESl (Br) analytic function were determined in four steps. First, the path was followed from the reactants C1- + CH3Br to complex a, Cl---CH3Br. Second, the path was followed from the central barrier to complex a. Third, the path was followed from the central barrier to complex b, ClCH3-.Br. Last, the path was followed from the products ClCH3 B r to complex b. The differential equations for integrating the reaction path were numerically integrated with a fourth-order Runge-Kutta algorithm, with the stepsize 0.000 002 amul/*-A. This stepsize was 1/10 that for the previous,work for the C1,- + CH3Clb system because of the incorporation of changes in the C-H stretching mode along the reaction path. The 3 N - 7 harmonic frequencies orthogonal to the reaction path were determined by diagonalizing the projected Cartesian force constant m a t r i ~ . ~ ~ - ~ ~ Figure 3 shows the variation of the five highest frequency vibrational modesalong thereaction path for PESl (Br). [Except for the E,CH3 rock and A,CH3 deformation, the frequencies along the reaction path are very similar for PESl(Br) and PES2(Br); see Tables 2 and 3, where it is seen that the frequencies for the E,CH3 rock and A,CH3 deformation modes are 1.09-1.1 6 times larger for PESl(Br) than PES2(Br).] These frequencies are plotted versus the value of g b on the reaction path instead of the reaction coordinate. Since there are three degenerate modes here-CH3 deformation (E), CH3 rocking (E), and C-H stretching (E)-the five frequencies in Figure 3 comprise eight vibrational modes along the reaction path. Frequencies for two modes, which change their character dramatically along the reaction path, are plotted in Figure 4. One mode is a symmetric C-Br stretch for the reactants, a symmetric Cl-C-Br stretch for the central barrier, and a C-Cl stretch for the products. The other is a Cl--CH3Br transitional mode bendno in the reactant channel, a Cl-C-Br bend a t the central barrier, and a ClCH3-Br transitional mode bend in the product channel.

+

-

+

+

Rate Constant Calculations Using the PESl(Br) and PES2(Br) analytic potential energy functions developed in the previous section, transition-state theory rate constants versus temperature and Rice-Ramsperger-KasselMarcus (RRKM) rate constants versus energy and angular momentum were calculated for the C1- + CHlBr ClCH3 + B r reaction system. These calculations are presented here not because we are confident the RRKM and transition-state theories

-

Wang et al.

1616 The Journal of Physical Chemistry, Vol. 98, No. 6, 1994

difference decreasing with increase in temperature. The same result was found for C1,- + CH3Clb association (Table VI of ref 38). Several models, based on the ion-dipole/ion-induced dipole potential V ( r , t ) = -aq2/2r4 - qpd cos([/?) (32) have been developed for calculating ion-molecule association rate 200 constants. In this workC1- CH3Br -Cl---CH3Br rateconstants ..... 300 were determined for this model potential using the pCVTST 100 ....... ...... .............. theory,85 the statistical adiabatic channel model (SACM),86and ................... 0 the trajectory capture model.85987These rate constants are listed in Table 7, where they are compared with the CVTST and -5.0 -3.0 -1.0 1.0 3.0 5.0 pCVTST rate constants calculated for the analytic potential g, (A) derived here. The relationship between the pCVTST, SACM, Figure4. Harmonicfrequenciesorthogonaltothereaction path for P S I and trajectory capture rate constants has been discussed previ(Br) versus gb: (-) stretch (A) and (- - -) bend (E). The modes are ously,88 and values similar to those found here are expected. The defined in the text. analytic potential derived here has the same long-range iondipole potential as does the model potential (eq 32). Thus, when represent the dynamics/kinetics of this system (see discussion in a particular kinetic model is applied to these potentials, they are the Introduction) but because they are useful for comparison expected to yield similar C1- CH3Br association rate constants. with experiment and dynamical calculation^.^^ In this work the pCVTST rate constants for these two potentials Transition-State Theory Rate Constants. Cl- CHjBr are found to be similar. Another approach for performing CVTST Cl--.CHjBr Association. Variational transition-state theory calculations for PESl(Br) and PES2(Br) is to use the flexible (VTST)8144wasused tocalculatetheC1-+ CH3Br+Cl--CH3Br transition-state mode1.89.90 For the C1,CHSClb analytic association rate constant versus temperature. The calculation potential this model gives nearly the same CVTST rate constant involves determining the association reaction and its as do the hindered rotor and harmonic oscillator models.g1 Thus, potential energy, and calculating as a function of the reaction for C1,- CH3Clb association the harmonic oscillator, hindered path the principal moments of inertia for the reactive system and rotor, and flexible CVTST rate constants are nearly identical. A frequencies for the 3N- 7 normal modes orthogonal to the reaction similar relationship is expected between the pCVTST rate path.77,78 Two different canonical variational transition-state constants for these models.80 The flexible model is also expected theory (CVTST) calculations were performed by treating the to give pCVTST and CVTST rate constants similar to those of two transitional bending modes, formed as C1- and CH3Br the harmonic oscillator and hindered rotor models when applied associate, as either harmonic oscillators (ho) or hindered rotors to C1- CH3Br association. This is because the long-range (hr). The transition state is located at the maximum in the free potential of PESl(Br) and PES2(Br) for C1- CH3Br is almost energy along the reaction path, and the rate constant is determined thesameas the long-range potential for theC1,-+ CH3Clb analytic from the free energy difference between the transition state and potential. reactants. Details of these two types of CVTST calculations are It should be emphasized that the trajectorycapturerateconstant described in the previous work for C1; + CH3Clb C1,CHs + in Table 7 assumes that association occurs when the ion molecule Clb- association38and other ion-dipole molecule a s s o ~ i a t i o n s . ~ ~ J ~ separation becomes less than a critical value.85J7 The actual Since PES1 (Br) and PES2(Br) have the same long-range potential energy-transfer processes which are required for association are (eq 4), these two potentials have the same CVTST rate constant not treated in this model. A trajectory study which includes the for C1- CH3Br C1--CH3Br association, which is listed in effect of this energy transfer is expected to give a smaller rate Table 7. constant.38.40*92 The CVTST rate constants for C1- CH3Br association are Cl--CH~Br Dissociation and Isomerization. VTST rate nearly the same as those determined previously for C1,- + CH3constants were calculated for decomposition of the Cl--CH3Br clb association (Table IX of ref 36). Similarly, the small complex. Dissociation of the complex to the reactants CIdifference between the hindered rotor and harmonic oscillator CH3Br proceeds through the same canonical and microcanonical CVTST rate constants for C1- CH3Br association is nearly variational transition state as does association of the reactants. identical to what was found for C1,- CH3Clb association. Not Thus, the CVTST/pCVTST rate constant ratio for complex given here are the Cl-...C distances, transitional mode bending dissociation to reactants is the same as this ratio for association frequencies, and classical potential energies for the [Cl--CH3Br] * of the reactants. The harmonic oscillator pCVTST rate constant canonical variational transition states. These properties are nearly for Cl-.-CH3Br C1- + CH3Br dissociation is listed in Table the same as those reported for C1,- CH3Clb association, i.e., 8. For the same reason as given above the CI- + CH3Br Table IX of ref 36. association, this dissociation rate constant is almost the same (within 0.5% difference) for PESl(Br) and PES2(Br). Microcanonical variational transition-state theory (pCVTST) rate constants81-84 were also calculated for C1CH3Br For Cl---CH3Br isomerization to CICH3-.Br, a VTST calassociation using the harmonicoscillator model for the transitional culation places the transition state at the central barrier. Thus, modes. In this theory the transition-state sum of states is conventional TST can be used to calculate the isomerization rate constant, i.e., with a fixed transition state the pCVTST and calculated versus total energy ( E ) and total angular momentum CVTST rate constants are the same and equal to that of (J),and these N*(E,J) are then Boltzmann weighted to obtain a thermal association rate constant. Since transition states are conventional TST. The isomerization rate constants for PES 1(Br) are listed in Table 8. The isomerization rate constants for found versus energy and angular momentum for a microcanonical PES2(Br) are 45-120% larger than those of PESl(Br), with the ensemble in pCVTST, the pCVTST rate constant is by definition82 difference decreasing with increase in temperature. The khm/ less than or equal to that of CVTST, since only one transition kdis ratio in Table 8 shows that isomerization is the preferred state is found versus temperature for CVTST. As shown in Table unimolecular pathway for the complex at low temperature, while 7, the pCVTST harmonic oscillator rate constants are smaller dissociation becomes more important at higher temperature. than the CVTST harmonic oscillator rate constants, with the

LI

i

"

"

"

"

"

"

+

"

"

"

+

-

+

+

+

+

+

-

+

+

-

+

+

+

+

-

+

+

The Journal of Physical Chemistry, Vol. 98, No. 6, 1994

Analytic Potential Energy Function

TABLE 7: C1- + CH3Br

-

-

CI-. vCH3Br Association Rate Constants. complete potential functionb CVTST,hrC CVTST,hod uCVTST,hod

T (K) 100 150 200 250 300 500 750 1000

1617

3.22 2.68 2.37 2.16 2.01 1.70 1.56 1.50

3.40 2.73 2.41 2.19 2.03 1.68 1.47 1.42

1.78 1.53 1.41 1.34 1.29 1.18 1.03 0.85

two-body potential (eq 32)

SACMC

uCVTSTf

trai c a v

2.03 1.78 1.63 1.53 1.46 1.29 1.18 1.11

2.13 1.80 1.61 1.49 1.40 1.20 1.08 1.02

1.89 1.62 1.46 1.35 1.27 1.08 0.97 0.91

a Rate constants are given in units of 10l2 Lmol-l s-l. The association rate constants for PESl(Br) and PES2(Br). The transitional modes are treated as classical hindered rotors. The transitional modes are treated as quantum harmonicoscillators. Calculated from eq 5 of ref 86a. fcalculated from eqs 3.19-3.21 of ref 85. f Calculated from eq 3 of ref 87.

-

TABLE 8: Transition-StateTheory Rate Constants for the C1- C H a r ClCH3 + B r System.

+

T(K) 100 150 200 250 300 500 750 1000

kasb 1.78 1.53 1.41 1.34 1.29 1.18 1.03 0.85

h a C

1.17 X 2.15 X 8.13 x 1.05 X 2.54 X 1.16 X 1.81 X 5.52 X

lW7 loo 103 lo6 lo7 1Olo

10" 10"

kis" 3.71 X lW5 1.09 X 10' 6.09 x 103 2.76 x 105 3.55 x 106 6.16 X lo8 8.46 X lo9 3.19 X 1Olo

ksN2' 1.77 1.28 0.60 0.28 0.16 0.060 0.046 0.046

The rate constants are calculated for PES1(Br). ksN2for PES2(Br) is 1.72 at 100 K and 0.065 at 1000 K. Bimolecular rate constants are in units of 1012 L-mol-1 s-1. Unimolecular rate constants are in units of reciprocal seconds. The pCVTST, ho rate constant from Table 7. The

.

- -

pCVTST, ho rate constant for C1-. eCH3Br C1- + CH3Br. The ClCH3. * - B r . conventional TST rate constant for C1-. eCH3Br Calculated from eq 33 using the k,, kdi,. and ki,,,,,,rate constants in this table. High-pressure s ~ 2 Rate Constants. The sN2 rate constant for reaction 2 depends on pressure. At low pressures the energy of the Cl--.CH3Br complex is equal to the energy of the C1- + CH3Br reactants. However, at high pressures, where all complexes are collisionally stabilized before isomerizing or decomposing, it becomes more appropriate to assume a thermal energy distribution for the complex. This collisional stabilization will lower the average energy of the Cl--CH3Br complexes below that of the reactants, which should increase the kisom/kdis ratio, since isomerization has a lower unimolecular threshold than does dissociation. As a result, the sN2 rate constant is pressure dependent and increases with increase in pressure. This has been observed in experiments.21 With a thermal distribution of energies for the C1- CH3Br reactants, the Cl-.-CH3Br complex, and the [Cl.-CH3-.Br]central barrier, which is the case a t high pressure, the sN2 rate constant can be written as

+

ksNz

kaskisom = kdis+ kisom

(33)

Listed in Table 8 arevalues of kSNzcalculated from this equation using the k,,, kisom,and kdi, rate constants also listed in Table 8. This calculated ksN2rate constant decreases with increase in temperature. From experimental studies, Knighton and co-workers21have estimated the high-pressure S Nrate ~ constant for reaction 2 as 6.0 and 3.3 X 1010 Lsmol-1 at 300 and 500 K, respectively. In comparison, the high-pressure sN2 rate constants calculated here at these two temperatures (Table 8) are 2.1 X 10" and 8.0 X 1O1O Lemol-I s-1. If it is assumed that RRKM and transition-state theoriescomprise a proper model (eq 33) for analyzing the kinetics of reaction 2, the temperature-dependent high-pressure sN2 rate constant calculated here can be brought into agreement with the experimental estimate by raising the classical central barrier for PESl(Br) by 0.9 kcal/mol so that it lies 2.0 kcal/mol below the

-

+ +

classical energy of the C1- CH3Br reactants. In an analysis of C1- CH3Br ClCH3 B r rate constants at intermediate pressures, with a transition-state theory model which assumes that central barrier crossing is rate determining, Viggiano et al.46950 estimated the central barrier to lie 1.5 kcal/mol below the reactants, which is similar to our findings. RRKM Rate Constant Calculations. To analyze the lifetime of Cl-.-CH3Br and ClCH3-.Br complexes formed by C1- CH3Br association, quantum harmonic RRKM calculations were performed using the PES 1(Br) analytic potential derived above. As described above, for isomerization the transition state is at the central barrier, while the transition state is chosen variationally for complex dissociation. The transitional bending modes are treated as quantum harmonic oscillators in the variational RRKM (Le., P C V T S T ) ~calculation. ~ The K-dependent term of the rotational energy is treated as an active degree of freedom so that all values of K from -J to +J are a l l o ~ e d . ~For ~ ?this ~ ~model the RRKM rate constant is written as

+

+

where the summation in the numerator is the total sum of states for the transition state, the sum in the denominator is the total density of states for the complex, and Eo is the difference between the zero-point energy levels of the transition state and complex. The rotational energies for the complex and transition state, E,( J , K ) and E:(J,K), are calculated by treating them as symmetric tops. Values of k(E,J) were calculated by choosing the total energy E between 0 and 4.0 kcal/mol above the zero-point energy of the reactants. The total angular momentum quantum number Jwas equated to zero and to a value of J representative of Cl--CH3Br complexes formed by C1- + CH3Br collisions. This latter value, J = 300, was estimated from I,,,,, = J(J + l)l/Zh,where I,, was taken from a classical trajectory study of C1- + CH3Br Cl--CH3Br complex forming c0llisions.5~ It is approximately the largest value of orbital angular momentum which resulted in complex formation. The RRKM rate constants for Cl--CH3Br C1- + CH3Br decomposition and Cl--CH3Br ClCHy-Br isomerization are plotted in Figure 5 versus E(..), the total energy of the C1CH3Br reactants above their zero-point level. At J = 300, the isomerization rate constant is zero for E ( - ) less than 4.5 kcal/ mol. This is because the isomerization transition state has moments of inertia smaller than those of the Cl--CH3Br complex, and for large J , reaching the isomerization transition state with its large rotational energy is not energetically possible unless E ( - ) is of sufficient size. For J of 0 the decomposition and

-

-

-

+

1618 The Journal of Physical Chemistry, Vol. 98, No.6,1994

12.000

-

-e

11.500

-

s

11.000

1

_._.-.-*

_.-.--.

u

10.000

~

~

0.0

"

'

"

"

0.5

'

"

"

'

'

1.0

"

'

" " " " " " " "

"

1.5

2.0

2.5

3.0

3.5

1 4.0

E (-) Figure 5. CI-.-CH3Br unimolecular rate constants (s-I): (-) CI-.-CH3Br CI- CHaBr dissociation with J = 0; (-) same dissociation with J = 300; (- -) CI-.-CHpBr C I C H y B r isomerization with J = 0.

-

+

-

-

isomerization rate constants become equal a t E( a)less than 0.5 kcal/moLg6 At higher E ( - ) thedissociation rateconstant becomes significantly larger than that for isomerization. ClCH3-Br complexes formed by C1- + CH3Br association and then isomerization are highly energized and have a RRKM rate constant for dissociation to ClCH3 + B r which equals 1.6 X 1012 s-1 and is independent of energy for E ( - ) = 0-4 kcal/mol. The RRKM rate constant for isomerization of the C1CH3-.Br complex is orders of magnitude smaller than this dissociation rate. The dissociation rate constants given in Figure 5 can be used to estimate at what pressures collisional stabilization of the Cl--.CH3Br complex becomes important. For a methane bath gas,21 the collision frequency for the complex is estimated as w = 1.7 X lo7T o r r 1 s-l X P(Torr).g7 For C1- CHsBr collisions forming Cl--CH3Br at 300 K, representative values for E( m) and J are 1.8 kcal/mol and 300, respectively. From Figure 5 , the C1--CH3Br unimolecular rate constant for these conditions is 5 X 10" s-I. The collisional stabilization rate constant equals this unimolecular rate constant at 2.9 X lo4Torr. Comparisons of this type between RRKM theory and experiment will in future studies be a meaningful way to test the validity of RRKM theory for the C1- + CH3Br reaction system. However, for the comparisons to be unambiguous, anharmonicity, which is not treated here, will have to be included in the RRKM calculations. Also, more accurate treatment of vibrational/rotational coupling than that used here may be necessary.

+

Acknowledgment. This research was supported by theNationa1 Science Foundation. References and Notes (1) Ingold, C. K. Structure and Mechanism in Organic Chemistry, 2nd ed.; Cornell University Press: Ithaca, NY, 1969. (2) Bohme, D. K.; Young, J. Am. Chem. Soc. 1970, 92, 7354. (3) Olmstead, W. M.; Brauman, J. I. J . Am. Chem. SOC.1977,99,4219. (4) Bohme, D. K.; Raskit, A. B. J . Am. Chem. SOC.1984, 106, 3447. (5) Hierl, P. M.; Ahrens, A. F.;Henchman, M.; Viggiano,A. A.; Paulson, J. F. J. Am. Chem. Soc. 1986, 108, 3142. (6) Henchman, M.; Hierl, P. M.; Paulson, J. F. J . Am. Chem. SOC.1985, 107, 2812. (7) DePuy, C. H.; Della, E. W.; Filley, J.; Grabowski, J. J.; Bierbaum, V. M. J. Am. Chem. SOC.1983, 105, 5509. (8) Bierbaum, V. M.; Grabowski, J. J.; Depuy, C. H. J . Phys. Chem. 1984,88, 1389. (9) Pellerite, M. J.; Brauman, J. I. J . Am. Chem. SOC.1980,102, 5993. (10) Pellerite, M. J.; Brauman, J. I. J . Am. Chem. SOC.1983, 105, 2672. (11) Dodd, J. A.; Brauman, J. I. J . Am. Chem. SOC.1984, 106, 5356. Dodd, J. A.; Brauman, J. I. J . Phys. Chem. 1986, 90, 3559. (12) Shaik,S.S.;Schlegel,H. B.; Wolfe, S. TheoreticalAspectsofPhysical Organic Chemistry: The S NMechanism; ~ Wiley: New York, 1992. (13) Caldwell, G.; Magnera, T. F.; Kebarle, P. J. Am. Chem. SOC.1984, 106. 959. (14) Tanaka, K.; Mackay, G. I.; Payzant, J. D.; Bohme, D. K. Can. J. Chem. 1976, 54, 1643.

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+

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