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Dynamical interpretation of the low efficiency of gas-phase nucleophilic substitution reactions (SN2)
CHEMICAL PHYSICS LETTERS
Volume 129, number 1
OF THE LOW EFFICIENCY SUBSTITUTION REACTIONS
DYNAMICAL INTERPRETATION OF GAS-PHASE NUCLEOPHILIC M.V. BASILEVSKY
15 August 1986
(S,2)
and V.M. RYABOY
Karpov Institute of Physical Chemistry
107120, ul. Obukha, 10, Moscow B-120, USSR
Received 19 May 1986
Quantum dynamical calculations of gas-phase SN~ reactions X- + CH3F -+ CH3X + F- (X = H, F, OH) are carried out within the collinear collision approximation. For potential energy surfaces with an intrinsic energy barrier lower than the reactant energy level, the low reaction efficiency is probably conditioned by almost total reflection in the direct process due to the large reaction path curvature before the intrinsic barrier.
1. Introduction As can be seen from experiment [l-3], nucleophilic substitution reactions (SN2),
gas-phase
X- + CH3Y + CH,X t Y-,
(1)
may proceed with quite different efficiency. Reaction efficiency is defined as the ratiof= k(Z”)/k,, where k(T) is the measured thermal rate constant and kc is the ion-dipole molecule capture rate constant (calculated in the averaged dipole approximation [4]). The energy profile of reaction (1) shows [2,3] a double-well curve with the top of the central barrier either above (E. > 0) or below (&, < 0) the reactant energy level. On the basis of this idea, Brauman et al. [2,3 ] proposed a model, which starts from the assumption that the first reaction step represents the formation of a prereaction complex (I) which then decays both in the product and reactant directions. If the decay rate of the postreaction complex (II) in the reactant direction may be neglected, then the rate of exothermal reaction (1) becomes
W=kJ,
f= vi.1 +(k~/~2)1=ww,.
(2)
The constants k_, and k2 represent decay rates of complex I in the reactant and product directions, respectively. It is expected that their ratio, controlling the reaction efficiency, may be calculated from statistical unimolecular decay theory [S] . The low efflcien-
cy of several activationless reactions was attributed to the state density difference between “tight” and “loose” transition states corresponding to k, and k_, . The central barrier height represents the parameter of this model. Its variation enables one to fit the calculated and experimental efficiency data. In the present paper we investigate the role of dynamical factors in the kinetics of the gas phase reaction (1). The resonant character of this process, typical for its dynamics, is, generally, in accordance with the model of Brauman and co-workers [2,3]. However the direct processes are by no means negligible. It is likely that just these processes are responsible for the low efficiency when E, < 0. Moreover, the non-statistical deay of the intermediate complex may be another characteristic property of the reaction under consideration. If so, the ratio k-,/k,, as controlled by dynamical factors, cannot be consistently treated in statistical calculations.
2. The results of dynamical investigation Potential energy surfaces (PESs) of reaction (1) were constructed on the basis of ab initio SCF 4-31G calculations at five characteristic points: the minima I and II, the saddle point III and two points located in the reactant and product asymptotic regions (see ref. [6] for details). Vibrational analysis showed that 71
Volume 129, number 1
CHEMICALPHYSICS LETTERS
for the most part, the form and frequency values for the normal modes changed only slightly along the reaction coordinate. Consequently, the corresponding degrees of freedom may be treated adiabatically. In particular, this was assumed for the inversion of the methyl group configuration. By this means, only a pair of vibrations of the three-centre system X-CH3 -Y, with the CH, fragment treated as a structureless particle, need be considered as dynamical degrees of freedom. Such a separation allows us to perform dynamical calculations in a collinear collision approximation. These were incorporated in the subsequent calculation of the thermal rate constants k(T) by the method of Bowman et al. [7]. The natural reaction coordinate (NRC) Hamiltonian used in the dynamical study is defined by three functions of the translational coordinate s measured along the reaction path (RP): g(s) (RF’ curvature), J’(s) (PES profile along the RP) and w(s) (transversal vibration frequency) [8 1. These functions were obtained by the special interpolation procedure [6] from quantum chemical computations at the five characteristic PES points. As is known for SN2 reactions, the 4-3 1G approximation yields reasonable values for geometric parameters and vibrational frequencies, but errors in energies may exceed 10 kcal/mole. That is why ln the present work the energetic characteristics of the V(s) profiles were considered as adjustable parameters and their effect on reaction dynamics investigated. In doing so, both the results of more sophisticated quantum chemical calculations and experimental data on reaction enthalpies were taken into account. The functions g(s) and w(s) were left invariable in our dynamical treatment. Later we consider in detail the calculated results for the reaction H-+CH3F+CH4+F-.
(la)
The functions V(s), g(s) and o(s) specifying the PES for this reaction are shown in fig. 1. Two peaks of g(s) occur on the slopes of the intrinsic barrier; the peak in region I being much larger than the other peak. In essence, the intrinsic barrier of reaction (la) in the product valley is a “late barrier” using Polanyi’s terms 193. This PES structure governs the dynamics of reaction (1 a). The method of quantum dynamical 1D calcula. tions of the transmission P&, and reflection flm,, 72
15 August 1986
3‘-1 2
1
(
-5/.
-11
-1:
u5
-2a
\
Fig. 1. Characteristics of the NRC Hamiltonian functions V(s), w(s) and g(s) (in dimensionless energy &CF and length lo = (A/2J&lCF)“2 units).
probabilities has been reported previously [lo]. The dynamical reaction efficiency may be characterized by the total reflection probability J$ for the process proceeding from the ground reactant state. The translational reactant energy was varied between the thresholds of the ,ground (m = 0) and fust excited (m = 1) states. Because of this $ = P&. The energy dependence of this value for reaction (la) is shown in fig. 2. Curve 1 (fig. 2a) corresponds to PES 1 (see table 1) with the 4.31G energy parameters unchanged. At a negative barrier height ofEo = -12.3 kcal/ mole, six vibrational channels are open at s = 0. Despite such an excess of translational energy, there is an appreciable reflection probability in the threshold region: $ = 0.15, due to excitation of vibrational states n > 7 which are closed in this region. The population of these states is a consequence of the strong interaction between translational and vibrational motions. More specifically, the maximum RP
CHEMICAL PHYSICS LETTERS
Volume 129, number 1
15 August 1986
PR
III
T
ofi-.
04..
02.i
IC ,
: 0.6
:
: 08
:
1: ill
:
:
:
I \J,
4.2 E/kw,,
curvature is located on the potential well slope where the translational energy is large, and this promotes considerable vibrational population inversion [ 111. As the central barrier height increases by the value of the transversal vibrational quantum ho(O) (PES 2), one more channel (n = 6) is closed resulting in an increase of the @ value. The function &!?) (fig. 2a) is more complicated than that on PES 1: it contains three resonance peaks due to the excitation of quasibound
Fig. 2. Dependence of the reflection probability Pp on the total reactant energy: (a) PESs 1 and 2; (b) PESs 3 (dashed line) and 4; (c) PESs 5 (dashed line) and 6.
states in the adiabatic potentials for the n = 6,7 channels. Calculations with the 4-31G basis overestimate the enthalpy of reaction (1 a) by 10.5 kcal/mole [ 121. In a more advanced study [ 131 the energy level of the central barrier differs from the reactant level by only 2-3 kcal/mole. When constructing PESs 3-6 (table 1) the energy profile was corrected according to these data. PES 3 lowers the reactant energy level by 10.5 73
CHEMICAL PHYSICS LETTERS
Volume 129, number 1
Table 1 Energy parameters of the PES profile V(s) along the reaction path (la), used in dynamical calculations @al/mole) PES
El
&II
&I
AE
1 2 3 4 5 6
-19.7 -19.7 -9.2 -7.8 -7.8 -19.6
-12.3 -10.2 -1.8 -0.5 2.0 -1.9
-6.6 -6.6 -6.6 -6.6 -6.6 -6.6
-66.8 -66.8 -56.3 -56.3 -56.3 -56.3
kcal/mole. Consequently, the height of the central barrier increases and the well corresponding to the prereaction complex becomes shallower. The first factor decreases the translational energy in the maximum curvature region. With unchanged reaction-path curvature, this results in decreasing the degree of vibrational excitation. However, due to the second factor, channels with n > 2 turn out to be closed in the barrier region. Altogether the lower degree of vibrational excitation is sufficient to maintain approximately the same reflection probability as that found on the PESs 1 and 2. The resonance peaks in the PRQ dependence corresponding to PES 3 (fig. 2b) indicate the excitation of quasibound states in n = 2-4 channels. PES 4 differs from PES 3 by further levelling the reactant and the central barrier energies. In this case even the channel corresponding to n = 1 is closed in the barrier region, raising the reflection probability up to zo.5. Finally there is an activation barrier (E. > 0) on PES 5. Making allowance for the energy difference of zero-point vibrations, its height is 1.5 kcal/mole (fig. 2~). The direct process corresponds to total reflection in the underbarrier energy region, but the quasibound states in n = l-3 channels generate four transmission peaks. Above the energy barrier (arrow in fig. 2c), an interference of direct and resonance processes leads to a complicated energy dependence PRO, with an appreciable average reflection probability. PES 6 coincides with PES 3 except for the energy level of the prereaction complex. It was taken from the 4-31G calculations (cf. with PESs 1 and 2). Due to the increased depth of the potential well and relatively small value of fro, the vibrational excitation of the reactive system results in practically total reflection in the direct process (the reflection probability at the reaction threshold is 0.999) (fig. 2~). The reactive colli74
15 August 1986
sions on PES 6 may proceed solely by the resonance mechanism. We were unable to obtain considerable reflection probabilities generated through dynamical factors in the other S,2 reactions studied (X = F and X = OH). For them the curvature of the RP is not sufficiently large. So, for X = F two curvature peaks appear symmetrically on both slopes of the barrier, each being 4.5 times lower as the first curvature peak of reaction (la). For X = OH the value of the curvature peak located in the prebarrier region is also small. A low efficiency of these reactions may be achieved only by the trivial incorporation of a barrier with E. > 0. The reaction X = OH has a large RP curvature in the region of the postreaction complex II. This results in considerable excitation of product vibrational states. The details of these calculations will be published separately [ 141.
3. Discussion Our results suggest that the dynamical excitation of high vibrational states in the direct scattering process should be considered as the most probable reason for the low efficiency of gas phase S,2 reactions in the absence of an activation barrier. This effect is of considerable importance when the maximum RP curvature region is located on the potential well slope before the intrinsic barrier where the translational energy of the reactive system is large. In this respect the calculations on PES 6 are most convincing: they show practically total reflection in the direct process. According to our calculations of the 3D rate constants k(T) of reaction (la), the reaction efficiency defined by formula (2) takes values between 1 and 10T3 when varying the energy profile as mentioned above. When E. < 0 the low efficiency is always a dynamical effect. The rate constant for PES 6 k(T) = 2.5 X lo-l1 cm3 s-l molecule-’ agrees with the experimental value of 1.5 X lo-l1 cm3 s-l molecule-1 113. The reaction probability on PES 6 is completely associated with resonance processes. In this case the reaction kinetics may be described by a formula of the type (2). However, its interpretation may be unusual for two reasons. Firstly, the constants k, and k_, involve both resonant and direct processes. In the calculation of kc [4,15 ] both are taken into account. However, a reflection constant k_, conditioned com-
CHEMICAL PHYSICS LETTERS
Volume 129, number 1
Table 2 Energy Eh = EX - ird2 and partial width of the resonances with respect to decay in the reactant (7-r) and product (72) direction (in hWCF units). The value 72 represents a sum over all accessible product states. To convert partial widths to s-l units they must be multiplied by the frequency value WCF (in s-l units) x 1
2 3 4 5 6 7 8 9 10
Eh = EA - irk/2
7-l
72
0.6097 - i 0.00987
0.00013 0.00034 0.00194 0.00066 0.00030 0.00206 0.00179 0.00300 0.00221 0.00190
0.02379 0.00970 0.00120 0.02022 0.04710 0.00595 0.01007 0.01270 0.00387 0.00383
0.6364 0.7035 0.8511 0.9412 1.0276 1.2129 1.3327 1.4312 1.4844
-
i 0.00440 i 0.00166 i 0.00789 i 0.01733 i 0.00388 i 0.00491 i 0.00647 i 0.00288 i 0.00255
pletely by the direct process, cannot be treated as a result of the intermediate complex decay. A consistent kinetic treatment of such a situation will be reported elsewhere [ 161. Secondly, the decay rate constants of complex I in the product direction (contributing mainly to k2) and in the reactant direction (leaving k-1 unaltered) do not represent statistical values. The results of dynamical calculations of these constants are shown in table 2. Here partial decay rate constants are identified with partial 1D widths calculated by the direct search method for the S-matrix poles [ 171. Their values are a few orders of magnitude higher than those which one would expect from statistical theory. The ratio of the 1D constants does not satisfy the statistical law either. The question arises, how reliable are 1D calculations of partial decay rates in multidimensional chemical systems? The answer depends essentially on two characteristic time intervals. One of them is the lifetime TV of the resonance 1D states which in accordance with the data of table 2 equals 71 = 10-11-10-12
s.
(3)
The second time, ro, characterizes the role of energy redistribution between the two subsystems: a pair of 1D degrees of freedom treated explicitly in a dynamical calculation and the remaining degrees of freedom considered as a thermal bath. If the inequality 71170 Q 1
(4)
holds, then the 1D resonances decay before their ener-
15 August 1986
gy dissipates into a bath. In this case the decay kinetics may be adequately characterized by the time r1 and the estimations of table 2 are reasonable. Conversely, if inequality (4) is inverted, the energy exchange prevails. In this case the decay process proceeds statistically. The time r. is determined by the matrix element of the interaction between 1D degrees of freedom and the bath. Our quantum chemical calculations show that in the systems under consideration their 1D degrees of freedom differ insignificantly from the normal modes of the total system. It may be accounted for by, first, high symmetry and, second, considerable difference between the strengths of equatorial (C-H) and axial (C-X and C-Y) bonds, both in the intermediate complex I and in the transition state II. The time estimate for the energy change between normal modes commons. Comparison of this ly used are r. = 10-lo-lO-‘l value with rl (3) supports the assumption that the non-statistical decay process may actually occur, at least in some S,2 reactions.
References [l] K. Tanaka, G.I. Mackay, J.D. Payzant and D.K. Bohme, Can. J. Chem. 54 (1976) 1643. 12] W.N.Olmstead and J.I. Brauman, J. Am. Chem. Sot. 99 (1977) 4219. [3] M.J. Pellerite and J.I. Brauman, J. Am. Chem. Sot. 102 (1980) 5993. [4] T. Su and M.T. Bowers, J. Chem. Phys. 58 (1973) 3027. [5] P.J. Robinson and K.A. Holbrook, Unimolecular reactions (Wiley, New York, 1972). [6] V.M. Ryaboy, Theor. Eksp. Khim. (1986), to be published. [7] J.M. Bowman, G.-Zh. Ju and K.T. Lee, J. Phys. Chem. 86 (1982) 2232. [8] R.D. Marcus, J. Chem. Phys. 45 (1966) 4493. [9] J.C. Polanyi, Accounts Chem. Res. 5 (1972) 161. [lo] M.V. Basilevsky and V.M. Ryaboy, Chem. Phys. 41 (1979) 461. [ 11) M.V. Basilevsky and V.M. Ryaboy, Chem. Phys. 41 (1979) 477. [12] D.J. Mitchell, A. Thesis, Queen’s University, Kingston, Canada (19811. (131 M. Urban, I. CernuGk and V. Kello, Chem. Phys. Letters 105 (1984) 625. [14] V.M. Ryaboy, Theor. Eksp. Khim. (1986), to be publish[15] ?Troe, Chem. Phys. Letters 122 (1985) 425. [16] M.V. Basilevsky and V.M. Ryaboy, Khim. Fiz., to be published. [17] M.V. Basilevsky and V.M. Ryaboy, Intern. J. Quantum Chem. 19 (1981) 611.
75
Volume 129, number 1
OF THE LOW EFFICIENCY SUBSTITUTION REACTIONS
DYNAMICAL INTERPRETATION OF GAS-PHASE NUCLEOPHILIC M.V. BASILEVSKY
15 August 1986
(S,2)
and V.M. RYABOY
Karpov Institute of Physical Chemistry
107120, ul. Obukha, 10, Moscow B-120, USSR
Received 19 May 1986
Quantum dynamical calculations of gas-phase SN~ reactions X- + CH3F -+ CH3X + F- (X = H, F, OH) are carried out within the collinear collision approximation. For potential energy surfaces with an intrinsic energy barrier lower than the reactant energy level, the low reaction efficiency is probably conditioned by almost total reflection in the direct process due to the large reaction path curvature before the intrinsic barrier.
1. Introduction As can be seen from experiment [l-3], nucleophilic substitution reactions (SN2),
gas-phase
X- + CH3Y + CH,X t Y-,
(1)
may proceed with quite different efficiency. Reaction efficiency is defined as the ratiof= k(Z”)/k,, where k(T) is the measured thermal rate constant and kc is the ion-dipole molecule capture rate constant (calculated in the averaged dipole approximation [4]). The energy profile of reaction (1) shows [2,3] a double-well curve with the top of the central barrier either above (E. > 0) or below (&, < 0) the reactant energy level. On the basis of this idea, Brauman et al. [2,3 ] proposed a model, which starts from the assumption that the first reaction step represents the formation of a prereaction complex (I) which then decays both in the product and reactant directions. If the decay rate of the postreaction complex (II) in the reactant direction may be neglected, then the rate of exothermal reaction (1) becomes
W=kJ,
f= vi.1 +(k~/~2)1=ww,.
(2)
The constants k_, and k2 represent decay rates of complex I in the reactant and product directions, respectively. It is expected that their ratio, controlling the reaction efficiency, may be calculated from statistical unimolecular decay theory [S] . The low efflcien-
cy of several activationless reactions was attributed to the state density difference between “tight” and “loose” transition states corresponding to k, and k_, . The central barrier height represents the parameter of this model. Its variation enables one to fit the calculated and experimental efficiency data. In the present paper we investigate the role of dynamical factors in the kinetics of the gas phase reaction (1). The resonant character of this process, typical for its dynamics, is, generally, in accordance with the model of Brauman and co-workers [2,3]. However the direct processes are by no means negligible. It is likely that just these processes are responsible for the low efficiency when E, < 0. Moreover, the non-statistical deay of the intermediate complex may be another characteristic property of the reaction under consideration. If so, the ratio k-,/k,, as controlled by dynamical factors, cannot be consistently treated in statistical calculations.
2. The results of dynamical investigation Potential energy surfaces (PESs) of reaction (1) were constructed on the basis of ab initio SCF 4-31G calculations at five characteristic points: the minima I and II, the saddle point III and two points located in the reactant and product asymptotic regions (see ref. [6] for details). Vibrational analysis showed that 71
Volume 129, number 1
CHEMICALPHYSICS LETTERS
for the most part, the form and frequency values for the normal modes changed only slightly along the reaction coordinate. Consequently, the corresponding degrees of freedom may be treated adiabatically. In particular, this was assumed for the inversion of the methyl group configuration. By this means, only a pair of vibrations of the three-centre system X-CH3 -Y, with the CH, fragment treated as a structureless particle, need be considered as dynamical degrees of freedom. Such a separation allows us to perform dynamical calculations in a collinear collision approximation. These were incorporated in the subsequent calculation of the thermal rate constants k(T) by the method of Bowman et al. [7]. The natural reaction coordinate (NRC) Hamiltonian used in the dynamical study is defined by three functions of the translational coordinate s measured along the reaction path (RP): g(s) (RF’ curvature), J’(s) (PES profile along the RP) and w(s) (transversal vibration frequency) [8 1. These functions were obtained by the special interpolation procedure [6] from quantum chemical computations at the five characteristic PES points. As is known for SN2 reactions, the 4-3 1G approximation yields reasonable values for geometric parameters and vibrational frequencies, but errors in energies may exceed 10 kcal/mole. That is why ln the present work the energetic characteristics of the V(s) profiles were considered as adjustable parameters and their effect on reaction dynamics investigated. In doing so, both the results of more sophisticated quantum chemical calculations and experimental data on reaction enthalpies were taken into account. The functions g(s) and w(s) were left invariable in our dynamical treatment. Later we consider in detail the calculated results for the reaction H-+CH3F+CH4+F-.
(la)
The functions V(s), g(s) and o(s) specifying the PES for this reaction are shown in fig. 1. Two peaks of g(s) occur on the slopes of the intrinsic barrier; the peak in region I being much larger than the other peak. In essence, the intrinsic barrier of reaction (la) in the product valley is a “late barrier” using Polanyi’s terms 193. This PES structure governs the dynamics of reaction (1 a). The method of quantum dynamical 1D calcula. tions of the transmission P&, and reflection flm,, 72
15 August 1986
3‘-1 2
1
(
-5/.
-11
-1:
u5
-2a
\
Fig. 1. Characteristics of the NRC Hamiltonian functions V(s), w(s) and g(s) (in dimensionless energy &CF and length lo = (A/2J&lCF)“2 units).
probabilities has been reported previously [lo]. The dynamical reaction efficiency may be characterized by the total reflection probability J$ for the process proceeding from the ground reactant state. The translational reactant energy was varied between the thresholds of the ,ground (m = 0) and fust excited (m = 1) states. Because of this $ = P&. The energy dependence of this value for reaction (la) is shown in fig. 2. Curve 1 (fig. 2a) corresponds to PES 1 (see table 1) with the 4.31G energy parameters unchanged. At a negative barrier height ofEo = -12.3 kcal/ mole, six vibrational channels are open at s = 0. Despite such an excess of translational energy, there is an appreciable reflection probability in the threshold region: $ = 0.15, due to excitation of vibrational states n > 7 which are closed in this region. The population of these states is a consequence of the strong interaction between translational and vibrational motions. More specifically, the maximum RP
CHEMICAL PHYSICS LETTERS
Volume 129, number 1
15 August 1986
PR
III
T
ofi-.
04..
02.i
IC ,
: 0.6
:
: 08
:
1: ill
:
:
:
I \J,
4.2 E/kw,,
curvature is located on the potential well slope where the translational energy is large, and this promotes considerable vibrational population inversion [ 111. As the central barrier height increases by the value of the transversal vibrational quantum ho(O) (PES 2), one more channel (n = 6) is closed resulting in an increase of the @ value. The function &!?) (fig. 2a) is more complicated than that on PES 1: it contains three resonance peaks due to the excitation of quasibound
Fig. 2. Dependence of the reflection probability Pp on the total reactant energy: (a) PESs 1 and 2; (b) PESs 3 (dashed line) and 4; (c) PESs 5 (dashed line) and 6.
states in the adiabatic potentials for the n = 6,7 channels. Calculations with the 4-31G basis overestimate the enthalpy of reaction (1 a) by 10.5 kcal/mole [ 121. In a more advanced study [ 131 the energy level of the central barrier differs from the reactant level by only 2-3 kcal/mole. When constructing PESs 3-6 (table 1) the energy profile was corrected according to these data. PES 3 lowers the reactant energy level by 10.5 73
CHEMICAL PHYSICS LETTERS
Volume 129, number 1
Table 1 Energy parameters of the PES profile V(s) along the reaction path (la), used in dynamical calculations @al/mole) PES
El
&II
&I
AE
1 2 3 4 5 6
-19.7 -19.7 -9.2 -7.8 -7.8 -19.6
-12.3 -10.2 -1.8 -0.5 2.0 -1.9
-6.6 -6.6 -6.6 -6.6 -6.6 -6.6
-66.8 -66.8 -56.3 -56.3 -56.3 -56.3
kcal/mole. Consequently, the height of the central barrier increases and the well corresponding to the prereaction complex becomes shallower. The first factor decreases the translational energy in the maximum curvature region. With unchanged reaction-path curvature, this results in decreasing the degree of vibrational excitation. However, due to the second factor, channels with n > 2 turn out to be closed in the barrier region. Altogether the lower degree of vibrational excitation is sufficient to maintain approximately the same reflection probability as that found on the PESs 1 and 2. The resonance peaks in the PRQ dependence corresponding to PES 3 (fig. 2b) indicate the excitation of quasibound states in n = 2-4 channels. PES 4 differs from PES 3 by further levelling the reactant and the central barrier energies. In this case even the channel corresponding to n = 1 is closed in the barrier region, raising the reflection probability up to zo.5. Finally there is an activation barrier (E. > 0) on PES 5. Making allowance for the energy difference of zero-point vibrations, its height is 1.5 kcal/mole (fig. 2~). The direct process corresponds to total reflection in the underbarrier energy region, but the quasibound states in n = l-3 channels generate four transmission peaks. Above the energy barrier (arrow in fig. 2c), an interference of direct and resonance processes leads to a complicated energy dependence PRO, with an appreciable average reflection probability. PES 6 coincides with PES 3 except for the energy level of the prereaction complex. It was taken from the 4-31G calculations (cf. with PESs 1 and 2). Due to the increased depth of the potential well and relatively small value of fro, the vibrational excitation of the reactive system results in practically total reflection in the direct process (the reflection probability at the reaction threshold is 0.999) (fig. 2~). The reactive colli74
15 August 1986
sions on PES 6 may proceed solely by the resonance mechanism. We were unable to obtain considerable reflection probabilities generated through dynamical factors in the other S,2 reactions studied (X = F and X = OH). For them the curvature of the RP is not sufficiently large. So, for X = F two curvature peaks appear symmetrically on both slopes of the barrier, each being 4.5 times lower as the first curvature peak of reaction (la). For X = OH the value of the curvature peak located in the prebarrier region is also small. A low efficiency of these reactions may be achieved only by the trivial incorporation of a barrier with E. > 0. The reaction X = OH has a large RP curvature in the region of the postreaction complex II. This results in considerable excitation of product vibrational states. The details of these calculations will be published separately [ 141.
3. Discussion Our results suggest that the dynamical excitation of high vibrational states in the direct scattering process should be considered as the most probable reason for the low efficiency of gas phase S,2 reactions in the absence of an activation barrier. This effect is of considerable importance when the maximum RP curvature region is located on the potential well slope before the intrinsic barrier where the translational energy of the reactive system is large. In this respect the calculations on PES 6 are most convincing: they show practically total reflection in the direct process. According to our calculations of the 3D rate constants k(T) of reaction (la), the reaction efficiency defined by formula (2) takes values between 1 and 10T3 when varying the energy profile as mentioned above. When E. < 0 the low efficiency is always a dynamical effect. The rate constant for PES 6 k(T) = 2.5 X lo-l1 cm3 s-l molecule-’ agrees with the experimental value of 1.5 X lo-l1 cm3 s-l molecule-1 113. The reaction probability on PES 6 is completely associated with resonance processes. In this case the reaction kinetics may be described by a formula of the type (2). However, its interpretation may be unusual for two reasons. Firstly, the constants k, and k_, involve both resonant and direct processes. In the calculation of kc [4,15 ] both are taken into account. However, a reflection constant k_, conditioned com-
CHEMICAL PHYSICS LETTERS
Volume 129, number 1
Table 2 Energy Eh = EX - ird2 and partial width of the resonances with respect to decay in the reactant (7-r) and product (72) direction (in hWCF units). The value 72 represents a sum over all accessible product states. To convert partial widths to s-l units they must be multiplied by the frequency value WCF (in s-l units) x 1
2 3 4 5 6 7 8 9 10
Eh = EA - irk/2
7-l
72
0.6097 - i 0.00987
0.00013 0.00034 0.00194 0.00066 0.00030 0.00206 0.00179 0.00300 0.00221 0.00190
0.02379 0.00970 0.00120 0.02022 0.04710 0.00595 0.01007 0.01270 0.00387 0.00383
0.6364 0.7035 0.8511 0.9412 1.0276 1.2129 1.3327 1.4312 1.4844
-
i 0.00440 i 0.00166 i 0.00789 i 0.01733 i 0.00388 i 0.00491 i 0.00647 i 0.00288 i 0.00255
pletely by the direct process, cannot be treated as a result of the intermediate complex decay. A consistent kinetic treatment of such a situation will be reported elsewhere [ 161. Secondly, the decay rate constants of complex I in the product direction (contributing mainly to k2) and in the reactant direction (leaving k-1 unaltered) do not represent statistical values. The results of dynamical calculations of these constants are shown in table 2. Here partial decay rate constants are identified with partial 1D widths calculated by the direct search method for the S-matrix poles [ 171. Their values are a few orders of magnitude higher than those which one would expect from statistical theory. The ratio of the 1D constants does not satisfy the statistical law either. The question arises, how reliable are 1D calculations of partial decay rates in multidimensional chemical systems? The answer depends essentially on two characteristic time intervals. One of them is the lifetime TV of the resonance 1D states which in accordance with the data of table 2 equals 71 = 10-11-10-12
s.
(3)
The second time, ro, characterizes the role of energy redistribution between the two subsystems: a pair of 1D degrees of freedom treated explicitly in a dynamical calculation and the remaining degrees of freedom considered as a thermal bath. If the inequality 71170 Q 1
(4)
holds, then the 1D resonances decay before their ener-
15 August 1986
gy dissipates into a bath. In this case the decay kinetics may be adequately characterized by the time r1 and the estimations of table 2 are reasonable. Conversely, if inequality (4) is inverted, the energy exchange prevails. In this case the decay process proceeds statistically. The time r. is determined by the matrix element of the interaction between 1D degrees of freedom and the bath. Our quantum chemical calculations show that in the systems under consideration their 1D degrees of freedom differ insignificantly from the normal modes of the total system. It may be accounted for by, first, high symmetry and, second, considerable difference between the strengths of equatorial (C-H) and axial (C-X and C-Y) bonds, both in the intermediate complex I and in the transition state II. The time estimate for the energy change between normal modes commons. Comparison of this ly used are r. = 10-lo-lO-‘l value with rl (3) supports the assumption that the non-statistical decay process may actually occur, at least in some S,2 reactions.
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