JOURNAL OF CHEMICAL PHYSICS
VOLUME 111, NUMBER 24
22 DECEMBER 1999
An ab initio molecular dynamics study of the S N 2 reaction ClⴚⴙCH3Br˜CH3ClⴙBrⴚ Simone Raugei, Gianni Cardini,a) and Vincenzo Schettino Laboratorio di Spettroscopia Molecolare, Dipartimento di Chimica, Universita` di Firenze, Via G. Capponi 9, 50121 Firenze, Italy and European Laboratory for Nonlinear Spectroscopy (LENS), Largo E. Fermi 2, 50125 Florence, Italy
共Received 30 June 1999; accepted 28 September 1999兲 An ab initio molecular dynamics study of the S N 2 reaction Cl⫺⫹CH3Br→CH3Cl⫹Br⫺ has been performed at the Becke, Lee, Yang, and Parr 共BLYP兲 level of theory by the blue-moon method. The potential energy and the free energy profile along the reaction coordinate have been determined and compared with the available experimental and calculated data. An analysis of the structural parameters along the reaction pathway is presented. Results of impact studies are also reported. It is shown that, depending on impact velocity, recrossing of the barrier can occur. Strong polarization effects are reported. © 1999 American Institute of Physics. 关S0021-9606共99兲50948-3兴
X⫺⫹RY→XR⫹Y⫺,
I. INTRODUCTION
Computer simulations of chemical reactions have been shown to be an invaluable tool for a full understanding of the chemical kinetics at a microscopic level.1–9 In many cases computer simulations offer a tool to obtain microscopic information from experimental kinetic data. In addition they can give a detailed picture of the energy flow among the degrees of freedom of reactants, intermediate species, and products. A description of chemical reactions at an atomic level, on the other hand, allows the calculation of the thermodynamic properties of the reaction in a controlled system, i.e., also in a situation that is not easily achieved experimentally. This information is essential to test the validity of the various theoretical models which are based on statistical assumptions and are normally used to compute reaction rates. Among the variety of molecular dynamics 共MD兲 simulation techniques the ab initio Car–Parrinello approach10 has progressed to a point that can now be applied to the study of gas phase chemical reactions. The advantage of the method lies essentially in the high quality of the interatomic potential at all phase space points. This is a substantial difference from traditional MD applications to chemical reactions based on semiempirical potential energy surfaces fitted to reproduce ab initio results at few selected phase space points. Since the method allows an exact consideration of the anharmonic effects at finite temperatures, a detailed study of the energy redistribution as a function of time among the degrees of freedom and of the polarization effects along the reaction pathway on the ground state potential energy surface are feasible. Choosing the reaction coordinate appropriately, the method allows the calculation of the free energy change as well. The ab initio MD has been used in only few cases for chemical reactions.1–9 The purpose of the present paper is to apply the method to the S N 2 nucleophilic substitution a兲
Electronic mail:
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0021-9606/99/111(24)/10887/8/$15.00
共1兲
representing a class of chemical reactions that has been most extensively studied, both experimentally and theoretically.11,12 These reactions can either occur directly or proceed through an intermediate that has a detectable lifetime. reactions have been studied Gas phase S N 2 experimentally11,12 by ab initio methods,11–16 by trajectory simulations,17–22 and quantum dynamical method.23 All these studies suggest significant deviations from the assumptions of statistical theories.24 In particular, in the present article an ab initio molecular dynamics10 study of the S N 2 reaction, Cl⫺⫹CH3Br→CH3Cl⫹Br⫺,
共2兲
will be reported. This reaction is known to be characterized by two minima, attributed to dipole-charge interaction complexes, separated by a free energy barrier.11,12 Classical molecular dynamics simulations20 of the reaction 共2兲 have been performed and statistical rate theory calculations25 using a semiempirical potential, refined on the basis of ab initio calculations performed on a few points of the reaction coordinate, have allowed an evaluation of the anharmonicity effects within the assumed form of the potential, and of the energy flow between the degrees of freedom. Nuclear quantum effects have been considered in a recent wave packet simulation on a model system23 using a semiempirical potential surface derived from the full dimensional potential PES1 共Br兲26 adopted in previous calculations. Viggiano et al.24 and Graul and Bowers27 reported on experimental evidence for nonstatistical behavior. Several trajectory-sampling calculations with empirical26 and semiempirical20 potential energy surfaces have shown that this could arise from weak coupling between the intermolecular modes of the ion–dipole association complex and the intramolecular modes of the methyl bromide. The energy profile for the reaction has been computed by ab initio methods13,15 and the thermal effects have been derived by statistical theories both in the harmonic approxi-
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mation and including some terms of the anharmonicity through a perturbative approach. The results have shown the central barrier height to depend strongly on the adopted level of theory. From the experimental point of view, despite the considerable amount of work available, several quantitative details of the potential energy surface still remain unclear since there is only a limited amount of direct experimental data available for the barrier, and the values reported depend strongly on the statistical theory applied to interpret the experiments.11 Ab initio calculations20 indicate that the Cl⫺共CH3Br兲 complex corresponds to an energy minimum characterized by a dissociation energy of 44.89 kJ/mol. The Cl⫺共CH3Br兲 and (CH3Cl兲Br⫺ species are separated by a potential energy barrier. There is no consensus on the height of the barrier. While former experimental work 共see, for example, Li et al.兲28 led to a derived barrier height of the order of 40 kJ/mol, recently Seeley et al.29 obtained from their experimental data and using Rice–Ramsperger–Kassel– Marcus 共RRKM兲 theory a value of 18.8 kJ/mol. The issue could not be resolved by ab initio calculations. In fact while MP2/6-31⫹G(d) and G2共⫹兲 calculations30,31 gave a value of the barrier in the range of 40 kJ/mol later density functional theory calculations using the B3-LYP exchangecorrelation functional15 gave a value of the barrier of 20 kJ/mol. It can be noted that MP2 and density functional theory 共DFT兲 calculations disagree on the central barrier but are in good agreement as far as the enthalpies of formation of the ion–molecule complexes and the overall enthalpy change of the reaction are concerned. In the present article, we report on the free energy profile calculation for reaction 共2兲 using the blue moon ensemble32,33 and on some results on free trajectory calculations that allow a description of the structural changes as a function of the reaction coordinates and a description of the energy flow among the degrees of freedom during the reaction. Using the blue moon ensemble we assume for the reaction the validity of one of the most used statistical theories of chemical reactions, i.e., the transition-state theory 共TST兲. Therefore it should be possible to compare, at least qualitatively, the results of calculations with both the experimental thermal and free energy. It has been found that the molecular dipole changes strongly with the reaction coordinate and this points to the importance of the polarization forces also in the neighborhood of the so-called dipole-charge complexes. II. THE SIMULATIONS
Two different kinds of ab initio molecular dynamics simulations have been performed, the first devoted to the calculation of the free energy profile of the reaction selecting the Y¯C distance as reaction coordinate and the second devoted to impact studies. In all cases the simulations were carried out in a cubic box of 15 Å side, and the usual technique to consider the system isolated34,35 has been employed. Troullier–Martins pseudopotentials36 and BLYP exchange and correlation functionals37,38 have been adopted. The electronic wave-functions were expanded in a plane waves basis set with an energy cutoff of 50 Ry. With this choice the CH3Cl and CH3Br geometry converged within about 0.5%. The deuterium isotope has been chosen for hydrogen atoms.
Raugei, Cardini, and Schettino
FIG. 1. Evolution of the structural parameters for the CH3Br⫹Cl⫺ reaction. Dots: average values of the parameters; dashed lines: mean-square deviation from the average; the solid line is only a guide for the eye. 共a兲 C–Br distance; 共b兲 cosine of the Br–C–H angle. 共c兲 C–H bond length 共d兲 temperature.
The equations of motion have been integrated with a time step of 0.09 fs 共5 a.u.兲. In all the simulations, the system has been thermalized at 300 K by uniform scaling of the velocities. The phase space trajectories have been accumulated during simulations, ranging from 0.4 to 3.8 ps, in which each degree of freedom of the system was coupled to a Nose´ – Hoover chain of thermostats.39,40 The reference point for the reactants has been obtained adding the energy of CH3X and Y⫺ computed separately in the same conditions. All the simulations have been performed using the QMDCP code.41 III. RESULTS AND DISCUSSION A. The free energy calculation
From the MD simulation in the blue moon ensemble in addition to the free energy, the structural properties and the potential energy of reactants, intermediates, and products are available. The simulation was started either from Cl⫺⫹CH3Br or from Br⫺⫹CH3Cl and carried through the transition state. It is worth noting that in the present study the Cl–C or Br–C distance has been chosen as reaction coordinate, while generally the potential and free energy are reported as a function of a r Cl–Br⫽r C–Cl⫺r C–Br coordinate. The results of the two semireactions considered in the present article can be easily combined for comparison with literature results. Since, as discussed in the following, the bond lengths show large fluctuations the average values of the distances have been used for the definition of r Cl–Br . In Figs. 1 and 2 some structural parameters are shown as a function of the reaction coordinate for the two semireactions, respectively. It can first be noted that, despite the use of a massive thermostat in the simulation, the fluctuation of the temperature is large with a mean-square deviation of 100 K due to the small number of atoms that compose the system. It can also be seen from the figures that the C–H bond length attains a minimum for values of the C–Cl or C–Br distance of about 2.5 Å. This is useful to characterize the transition state that corresponds to an almost planar configuration of the CH3 group and an increased order of the C–H
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FIG. 2. Evolution of the structural parameters for the CH3Cl⫹Br⫺ reaction. Dots: average values of the parameters; dashed lines: mean-square deviation from the average; the solid line is only a guide for the eye. 共a兲 C–Cl distance; 共b兲 cosine of the Cl–C–H angle. 共c兲 C–H bond length 共d兲 temperature.
bond. The minimum C–H bond length is found to be 1.080 Å for a C–Cl separation of 2.595 Å or to be 1.079 Å for a C–Br separation of 2.381 Å. It has already been reported that at the transition state the C–H bond length is the shortest. The attainment of a planar structure for the CH3 group in the transition states and the Walden inversion are clearly monitored by the cosine of the Br–C–H or of the Cl–C–H bond angle in the direct and inverse reaction. The structural evolution along the reaction pathway is easily followed from the behavior of the C–Br or C–Cl bond lengths. When the Br⫺ ion is at large distance from the molecule the C–Cl bond length is already larger 共1.9897 Å兲 than the calculated free molecule values 共1.9742 Å兲. This could be due either to thermal effects or to an already active ion dipole interaction. From Figs. 1 and 2 it can be seen that the carbon–halogen distance does not change appreciably until the reaction coordinate approaches the transition state value. At this point the bond length increases abruptly 共e.g., in the direct reaction a decrease of 0.2 Å in the Cl–C approach produces an increase of 0.5 Å for the C–Br bond length兲. This abrupt increase of the bond length is accompanied by a growth of the meansquare deviation 共see Figs. 1 and 2兲, due to an increased amplitude of the carbon–halogen stretching mode. An abrupt change in the vicinity of the transition state is also observed for the cosine of the H–C–Cl and H–C–Br angle. In conclusion it is found that the transition state Cl¯CH3¯Br has a C 3 v symmetry with a difference of 0.26 Å between the C–Cl and C–Br bond lengths. The calculated potential energy profile along the reaction coordinate is shown in Fig. 3共b兲 at 300 K and Fig. 3共c兲 at 0 K. The Cl⫺共CH3Br兲 ion–dipole association complex occurs at higher energy than the (CH3Cl兲Br⫺ complex in agreement with experiments and previous reports and is characterized by a dissociation energy of about 45 kJ/mol at 0 K. This is in good agreement with the experimental values of 42 ⫾4 kJ/mol 共Ref. 42兲 and with the value of 42.3 kJ/mol proposed by Seeley et al.29 taking the zero point energies into account. The barrier of the transition state relative to the
The S N 2 reaction Cl⫺⫹CH3Br→CH3Cl⫹Br⫺
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FIG. 3. Free energy with respect to the transition state 共a兲, run average potential energy at 300 K 共b兲, and minimum potential energy 共c兲 along the reaction path with respect to the reactants. Circles: Cl⫺ attack; triangles: Br⫺ attack. The minimum potential energy up to the ion–dipole complexes was computed by impacts at almost zero velocity and the maximum determined by constrained energy minimization 共the dashed line is only a guide for the eye兲. The potential energies at 0 K and the free energies at 300 K for the stationary points calculated at BLYP/6-311G共d兲 level are reported with filled and empty squares, respectively.
reactants is about ⫺20 kJ/mol at 300 K and about ⫺30 kJ/ mol at 0 K. The present result is in perfect agreement with the thermal energy at 298 K proposed recently by Seeley et al.29 and with the enthalpy change calculated by Glukhovtsev et al.15 using density functional theory with the B3-LYP exchange–correlation functional. The agreement is not so good if the present result is compared with other calculated or experimental evaluations of the barrier height. In conclusion, it appears that it is presently difficult to decide on the best choice of a theoretical method to calculate the energy barrier for the reaction of interest in this work. It also appears that comparison with the experimental data is not a great help since, as discussed in the introduction, the actual value of the barrier is a matter of controversy. In any case, the experimental value of the barrier is indirect since it is obtained using statistical models that are not necessary valid. It is interesting to note that the thermal effect on the energy barrier between 0 and 300 K obtained in the present work is 10 kJ/mol 共i.e., ⫺30 kJ/mol at 0 K and ⫺20 kJ/mol at 300 K兲, while Glukhovtsev et al.15 estimated a difference of only 1.6 kJ/mol between 0 and 298 K. This remarkable difference can be explained as follows. Glukhovtsev et al.’s15 estimation of the thermal effect was based on a statistical mechanics approach using the equilibrium properties 共geometry and vibrational frequencies兲 at 0 K. In the present molecular dynamics simulation the potential energy surface along the reaction coordinates is explored more thoroughly since all the internal degrees of freedom are retained and, therefore, a complete structural rearrangement is monitored as a function of both the reaction coordinates and the temperature. It is worth noting, however, that due to large fluctuations of the temperature noted above, the thermal effect could be overestimated in the present simulation. The larger thermal effect obtained in the present work could, at least in part, be also due to the fact that the BLYP functional
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produces lower vibrational frequencies for the transition state. As already discussed in the introduction much larger values of the barrier have previously been estimated from experiments or calculated at MP2 level by Hu et al.13 and at the G2共⫹兲 level by Curtiss.30,31 So the controversy remains as to the actual height of the energy barrier for the transition state for this reaction. In general, the dependence of evaluated potential energy barriers for chemical reactions on the level of theory adopted has already been stressed. For the (CH3Cl兲Br⫺ complex a dissociation energy of 35 kJ/mol has been obtained in the present work at 0 K in good agreement again with the density functional calculation by Glukhovtsev et al.15 The energy change for the overall reactions is calculated to be ⫺18 kJ/mol, to be compared with DFT value of ⫺35 kJ/mol reported by Glukhovtsev et al.15 A possible reason for this large difference can be due to convergence problems. In the present work a 50 Ry cutoff for the plane waves expansion of the wave-functions has been used in the simulation and found appropriate for the convergence of the structural parameters of the isolated molecules. However, due to the selected pseudopotential, the cutoff could be inadequate for the Br⫺ ion. A larger cutoff was beyond our computational facilities. The free energy change has been computed separately for the Cl⫺⫹CH3Br→CH3Cl⫹Br⫺ and Br⫺⫹CH3Cl →CH3Br⫹Cl⫺ reactions imposing during the simulation a constraint on the X–C distance where X is Cl or Br, respectively. The holonomic constraint ( 兵 RX(t)⫺RC(t) 其 )⫽ 0 is included in the Car–Parrinello Lagrangian L⫽L CP⫹ 兩 共 兵 RX⫺RC其 兲 ⫺ 0 兩 , where is the Lagrange multiplier associated to the chosen constraint and corresponds to the so-called mean force. Exploring the constrained phase space allows to compute the mean constrained force and from this the change in free energy can be obtained according to the blue moon method32 as ⌬G⫽
冕
B
A
d 0具 共 0 兲典 ,
where A and B are the initial and final values of the reaction coordinate 0 . The constrained initial distance has been chosen for both series of simulations to be 5.41 Å. In Fig. 4 the calculated mean force and the change in free energy for the direct and inverse reactions as a function of the reaction coordinate are reported. The transition state value has been chosen as the reference point for the free energy. The overall free energy profile of the reaction is shown in Fig. 3共a兲. The obtained profile is in generally good qualitative agreement with experimental and theoretical reports as far as the presence of two asymmetric minima and of a barrier are concerned. The free energy difference between the two ion–dipole complexes is appreciably smaller than the potential energy difference at the same temperature. The calculated free energy barrier relative to the Cl⫺共CH3Br兲 complex is 15 kJ/mol. An estimate of the free energy barrier, based on a high pressure mass spectrometric analysis, has been reported by Li et al.28 Their value 共55 kJ/mol兲 is much
Raugei, Cardini, and Schettino
FIG. 4. Energy 共top panel兲 and average mean force of the constraint 共lower panel兲 for the Y⫺ attack as a function of the C–Y distance. The error bars indicate the standard deviation from the average. Left side: CH3Cl⫹Br⫺. Right side: CH3Br⫹Cl⫺.
higher than the result of the present article. This discrepancy parallels that found in the potential energy barrier as discussed above. To analyze possible polarization effects in the ion– molecule interaction a set of doubly occupied localized orbitals has been obtained using the maximally localized Wannier functions technique.43,44 In this way the electronic density was partitioned between the ion and the molecule. Subsequently the dipole moments of the molecule along the reaction path were computed. The results are shown in Fig. 5 and are summarized in Table I. As it can be seen from Table I the calculated values of the isolated molecules dipole moment are in good agreement with experiments. It is observed that the polarization effects change the molecular dipole moment appreciably with an increase by a factor 2 on approaching the transition state. This is evident from Fig. 5, even though at shorter distances an effect of overlap of the electron densities of the two species makes the partitioning questionable. Polarization effects are also pronounced on the halide ion. While at large separation the centers of the charge
FIG. 5. CH3Cl 共triangles兲 and CH3Br 共circle兲 dipole moment as a function of the reaction coordinate. The arrows indicate the position of the charge– dipole complexes. The dashed lines are a guide for the eye.
The S N 2 reaction Cl⫺⫹CH3Br→CH3Cl⫹Br⫺
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TABLE I. Energies and structural parameters for stationary points. Lengths in Å, angles in degrees, dipole moment in Debye, potential energy 共with respect to the reactants兲 and free energy 共with respect to the transition state兲 in kJ/mol. Property r C–Br r C–H H–C–Br Dipole Energy r C–Br r C–Cl r C–H H–C–Br Dipole Energy Free energy r C–Br r C–Cl r C–H H–C–Br Dipole Energy Free energy r C–Br r C–Cl r C–H H–C–Br Dipole Energy Free energy r C–Cl r C–H H–C–Br Dipole Energy
Exp. 1.934c 1.082 107.7 1.81d 0
⫺42e ⫺51f
⫺18.8g 0
⫺67g
1.776h 1.085 108.6 1.87d ⫺25i, ⫺33e
PES1 共Br兲a
MP2/PTZ⫹b
CH3Br Reactant 1.944 1.938 1.077 1.083 107.6 108.1 1.87 0 共CH3Br兲Cl⫺ Dipole complex 1.991 3.221 1.071 107.1 ⫺45 Cl¯CH3¯Br⫺ Transition State 2.462 2.392 2.470 2.322 1.062 1.069 92.6 91.3 ⫺12
⫺4.2
共CH3Cl兲Br⫺ Dipole complex 3.527 1.819 1.074 71.9 ⫺89
BLYP/6-311G(d)
BLYP/PW
1.991 1.0924 107.64 1.9996 0
1.974 1.090 107.3 1.81 0
2.109 3.018 1.086 105.7
2.058 3.113 1.089 106.0 4.23 ⫺44 ⫺15
⫺43.2 ⫺12.85 2.547 2.426 1.078 90.2 ⫺33.13 0 3.239 1.925 1.088 73.2 ⫺51.31 ⫺17.85
CH3Cl Product 1.789 1.781 1.077 1.083 108.1 108.5 ⫺53
1.8292 1.0932 108.08 2.1741
2.533 2.416 1.075 90.1 ⫺32 0 3.294 1.921 1.086 74.5 3.86 ⫺53 ⫺20 1.851 1.090 107.4 2.10 ⫺18
a
Analytic potential energy surface from Ref. 26. Reference 13. c The experimental geometry for CH3Br is taken from Ref. 49. d The CH3Br and CH3Cl dipole moments are taken from Ref. 50. e Reference 42. f Reference 28. g Reference 29. h Reference 51. i References 52 and 27. b
of the localized wave-functions are uncorrelated with the position of the molecule, at shorter separation the center of charge of one of the four halide ion lone pairs is clearly displaced from the halogen nucleus toward the approaching molecule. B. Impact studies
In order to obtain information on the energy flow among the degrees of freedom of the system, ab initio impact studies were carried out with two different initial velocities of the chloride ion, corresponding to a translational temperature of about 4000 and 1000 K. The simulation was started with the system at rest and the velocity of the chloride ion lying on the C 3 axis of CH3Br in the direction of the C atom. An
analysis of the time evolution of the structural properties and of the energy flow for high and low Cl⫺ velocity is reported in Figs. 6 and 7, respectively. When the Cl⫺ ion reaches a distance of about 2.3 Å from the C atom a Walden inversion of the CH3 group occurs 关Figs. 6共b兲 and 7共b兲兴 and the C–Br distance increases rapidly with a breaking of the bond 关Figs. 6共a兲 and 7共a兲兴. The C atom starts to move in an opposite direction with respect to Br and a Cl–C bond can be formed. If the initial velocity of the chloride ion is not too high, the bond is formed, otherwise the Cl⫺ ion is bounced back. It is interesting to note that in both trajectories, a large part of the kinetic energies 关Figs. 6共a兲 and 7共d兲兴 is transferred from the chlorine to the bromine atom. The different behavior of the system as the impact velocity is changed is clearly monitored
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Raugei, Cardini, and Schettino
FIG. 8. Potential energy as a function of time for the CH3Br⫹Cl⫺ reaction with impact velocity corresponding to a relative translational temperature of 4000 共top panel兲 and 1000 K 共lower panel兲. FIG. 6. Results of trajectory study for the CH3Br⫹Cl⫺ reaction with impact velocity corresponding to a relative translational temperature of 4000 K. 共a兲 C–Br distance as a function of C–Cl distance; 共b兲 cosine of the Br–C–H angle as a function of the C–Cl distance; 共c兲 C–H bond length as a function of the C–Cl distance; 共d兲 kinetic energy of the atoms as a function of time.
samplings20 that have shown that when too much energy is transferred to these modes the reaction does not occur. IV. AB INITIO STATIC CALCULATIONS
by the cosine of the Br–C–H angle. At the higher impact velocity the Walden inversion is not stabilized and the CH3 umbrella, after the impact, is again toward the chlorine 关Fig. 6共b兲兴. For the lower velocity trajectory the umbrella is stabilized toward the bromine 关Fig. 7共b兲兴. The potential energy profile for the two trajectories is reported in Fig. 8. It is evident that in the hotter trajectory a much stronger deformation of the activated complex is produced. The much higher energy is essentially located in the Br–C stretching and in the umbrella motion of the CH3 group. This is in agreement with previous trajectory
FIG. 7. Results of trajectory study for the CH3Br⫹Cl⫺ reaction with impact velocity corresponding to a relative translational temperature of 1000 K. 共a兲 C–Br distance as a function of the C–Cl distance; 共b兲 cosine of the Br–C–H angle as a function of the C–Cl distance; 共c兲 C–H bond length as a function of the C–Cl distance; 共d兲 kinetic energy of the atoms as a function of time.
As has been discussed in a previous section, the results of the present work 共obtained using plane waves and the BLYP exchange–correlation functional兲 differ from previous ab initio calculations. In one case 关the dissociation energy of the (CH3Br兲Cl⫺ complex兴 they also differ from DFT results based on the B3-LYP functional in conjunction with a Gaussian basis set. In order to understand the source of these discrepancies a series of additional calculations were carried out using the BLYP functional and a Gaussian basis set. The equilibrium geometry and the vibrational frequencies were calculated for the free molecules, for the ion–molecule complexes, and for the transition state using a triple-zeta basis set 关 6-311G(d) 兴 . Calculations were carried out using the 45 GAUSSIAN98 program. The results are summarized in Table I. A comparison of the isolated molecule properties computed with plane waves or Gaussian basis set should give an idea on the importance of using the pseudoatom approach. From Table I it can be seen that carbon–halogen bond length is overestimated in the DFT calculation both relative to the experimental and to the MP2 calculation of Hu et al.13 This seems to be due not so much to the basis set choice but most likely to the exchange–correlation functional 共either BLYP or B3-LYP兲. The difference in the carbon–halogen bond lengths relative to the MP2 result is larger for the transition states 共about 0.1 Å兲. The effect of the basis set has been further checked performing a geometry optimization with the smaller 6-31G(d) basis set. An increase of carbon–halogen bond lengths smaller than 0.002 Å was observed. The DFT calculated vibrational frequencies are reported in Table II. The vibrational frequencies of the values for the transition state are compared with the MP2 results in in Table III. It can be seen that the DFT frequencies are consistently smaller, the ratio of the DFT to MP2 values increasing almost steadily with decreasing vibrational frequencies. This is not surprising by comparing with other DFT and ab initio
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TABLE II. Vibrational frequencies 共 in cm⫺1兲 and infrared IR and Raman 共R兲 intensity 共in km/M and Å4/amu, respectively兲 computed at the BLYP/6-311G(d) level. 共CH3Br兲Cl⫺
CH3Br
IR
552.0 940.2 940.6 1307.0 1454.0 1455.0 3010.0 3112.0 3112.0
12 8 8 25 7 7 20 7 7
Cl¯CH3¯Br⫺
共CH3Cl兲Br⫺
CH3Cl
R
IR
R
IR
R
IR
R
IR
R
20 6 6 1 11 11 120 62 62
93.0 98.6 108.2 347.2 857.4 860.2 1173.8 1419.5 1420.4 3068.0 3201.0 3202.8
8 9 57 351 2 2 48 7 7 2 1 1
0 0 1 386 1 1 7 5 5 134 28 28
157.2 161.7 164.9 ⫺236.7 842.5 844.7 932.5 1376.0 1377.0 3120.0 3311.0 3312.0
8 5 8 748 0 0 96 5 5 0 0 0
0 30 0 0 2 2 0 4 4 87 19 19
81.5 85.7 87.0 433.6 918.6 923.8 1241.8 1426.9 1428.8 3058.0 3180.9 3181.8
8 17 9 429 3 3 17 6 6 6 3 3
0 0 0 842 1 1 5 6 6 142 27 26
664.4 1010.0 1011.0 1361.0 1459.0 1459.0 3003.0 3098.0 3099.0
29 7 7 18 6 6 29 11 11
14 6 6 3 13 13 122 63 63
calculations of vibrational frequencies.46–48 The difference in vibrational frequencies implies a different contribution of the vibrational partition function in the evaluation of the free energy. It is found that the free energy change between the transition-state and the molecule–ion complexes, as computed using the partition function, turns out to be not much different from the energy change. This is in agreement with the direct evaluation of the free energy as it is evident from Fig. 3. V. CONCLUSIONS
Using the Car–Parrinello method and blue moon ensemble the evaluation of the structural parameters, of the energy, and of the free energy along the reaction pathway has been calculated for the CH3Br⫹Cl⫺→CH3Cl⫹Br⫺ reaction. The energy barrier obtained in this work turns out to be appreciably lower than the MP2 estimate, even if allowance is made for the different computational approaches. A similar observation has been previously reported by Meijer and Sprik3 in their study of formaldehyde reduction in sulphuric acid solution. The results of the present work are in good agreement with the experimental data of Seeley et al.29 It has been found that the polarization effects are important for all TABLE III. Comparison between the transition-state vibrational frequencies 共in cm⫺1兲 obtained at BLYP/6-311G(d) level 共this work兲 and at MP2/PDZ⫹level 共Ref. 49兲. In the third column the ratio of the frequency obtained with the two methods is also shown.
BLYP
MP2
MP2 / BLYP
⫺236.7 157.2 161.7 164.9 842.5 844.7 932.5 1376.0 1377.0 3120.0 3311.0 3312.0
⫺490 195 199 199 955 955 1029 1426 1426 3225 3430 3430
2.07013 1.24046 1.23067 1.20679 1.13353 1.13058 1.10349 1.03634 1.03558 1.03365 1.03594 1.03563
significant values of the reaction coordinate and should be included in empirical modeling of the potential energy surface. Ab initio trajectory studies were carried out to obtain the energy redistribution among the various degrees of freedom. It has been found that for high impact velocities a recrossing of the barrier occurs. It is to be remarked that the accuracy that can be reached in the evaluation of the central barrier using the present approach is not sufficient to allow a significant evaluation the rate constants.2 This deficiency is shown by other ab initio methods as well. Therefore, no attempts were made to calculate the rate constant. ACKNOWLEDGMENTS
The authors would like to thank Professor M. Parrinello for making available the QMDCP program.41 This work was supported by the Italian Ministero dell’Universita´ e della Ricerca Scientifica e Tecnologica 共MURST兲, by the Consiglio Nazionale delle Ricerche 共CNR兲, and by the European Union 共Contract No. ERBFMGECT950017兲. 1
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