- Email: [email protected]
Diatomic dissociation on aJ-averaged potential
Volume 128, number 1
CHEMlCAL PHYSICS LETTERS
11 July 1986
DIATOMIC DISSOCIATION ON A J-AVERAGED POTENTIAL Mark G. SCEATS Departmeni of Physic& Chemistry, Wniversity af Sydney, Sydney, NS W 2006, Australra Received 7 March 1986; in final form 21 May 1986
Illatomic dissociation rates are evaluated in the weak-cotlision low-pressure &nit from a ~~nd~rn~~~~~~~ impulsive stochastic model on the J-averaged effective potential Y(R) - 2&T h@?/T(T), and compared with the threedimensionat resuitf of Borkovec and Berne. The agreement within 10% over a wide range of Marse parameters and temperatures indicates that the contraction to one dimension is not only instructive, but also accurate.
The sitar advantage of stochastic ~~a~~~~~s of diatomic recombination and dissociation is that the one approach is applicable over the entire range of both gas densities [l-3] - from low pressures in which three-body collisions dominate to compressed hquids in which transport phenomena dominate. We have shown [4] that the coupled motion of a pair of particles in the Langevin picture can be reduced to the Brownian motion of a single particle on a onedimensional effective potential P(R) = Y(R) - 2kT In@&) , (1) where R, is t&e position of the tr~s~tion state of p(R), and this approach has been applied to analysis of experimental data for recombination and dissociation of iodine [ 1,23 in which the friction 0 describing the coupling of the atoms to the bath gas is reduced from the diffusion coefficient D, of the separated atoms by the Einstein re~a~o~~ip /3= kT~~~~~, where MB is the mass of the atom B. The agreement with experiment was very good, but the accuracy of the many aspects of the model is difficult to assess because of the uncertainty in the model parameters. The aspects of the approach which need to be independently considered are: (i) The validity of elimination of the orientational degrees of freedom [43 to reduce the problem to consideration of ordy the radial coordinate in (1). The important role of coupling between vibrational and rotational degrees of freedom has been considered by others C-7], and it is by no means clear that the use of me the effective potential of eq. (1) will incorporate these effects. 0 009-26 14/86/$03.50 0 Elsevier Science Publishers B.V. (Norm-Donald Physics ~b~~g Division)
(ii) The use of an untested model of ~yrnptot~~ curve hopping [8] or ad hoc combinations of efectronic degeneracy factors to account for the multiple potential surfaces (16 in iodine) presents the most formidable problem in judging the accuracy of the approach by comparison with experiments on halogens. (iii) The use of a frequency-~dependent friction implies that the timescales of collisions are shorter than the characteristic vibrational periods within kT of dissociation. This is not true for heavy bath gases or at high temperatures. In subsequent work we have developed a model of frequen~ydepend~~t collisional friction [9] and compared the results with the molecular dynamics simulations of Nordhohn et al. [lo] for highly vibrationally excited Br, in Ar, and have found good agreement for V-T energy transfer within 5 kT of dissociation [I 1]_ Nevertheless the cornerstone of our approach, namely the use of the effective potential of(f), eannot be tested with molecular dynamics simulations (which eliminate problems of (ii) by considering only one potential) because these simulations are not stochastic and an additional model of the friction is required to relate a stochastic model to the MD results, The only reasonable way to test the applicability of (1) is to compare the results from (1) with those from a full stochastic treatment in three dimensions recently obtained by Borkovec and Berne [12] for a Morse potential in which the dissociation rate is evaluated as tit average over a d~t~bution of~dependent rates related to dissociation over the centrifugal barrier of the rotational potentials
F&(R < J) = V(R) + J2/2/.tR2 .
(2)
For the Morse potential V(R) = De { exp [-2b(R
- Re)] - 2 exp [-b(R - R&l }
(3) it is convenient to put the dissociation in the form k =Z(T)P(2De/W
exp(-D,lkT)
(4)
,
where the factor Z(T) accounts for all effects associated with the three-dimensional aspects of the problem. The one-dimensional result of Kramers [ 13] is obtained if Z = 1. The numerical results of Borkovec and Berne could be fitted to the form Z(T) =: A lDe/kTl”-l
,
(9
where A and OLdepend only on the Morse parameter bR,.
When the effective potential v(R) is plotted, a transition state appears at R,, as shown in fig. 1 for the Mofse potential of izerest in this work. The resulting barrier height of V(R) can be significantly less than the dissociation energy. The effective potential gives the exact equilibrium distribution in the R co-
I
D
11 July 1986
CHEMICALPHYSICS LETTERS
Volume 128, number 1
b&=1
ordinate, namely R2 exp [-V(R)/kT], but like any effective potential, its use in dynamical applications is valid only in two limits depending on whether the relaxation of the property of interest in the retained coordinate (in this case the vibrational energy in R) is fast or slow compared to those of the coordinates which have been eliminated (rotational energy relaxation in 0, $). The adiabatic limit is obtained if rotational relaxation is fast relative to vibrational relaxation, such that the effective potential is a potential of mean force. The dynamics take place in this potential. If v(R) for R
= P[2QVW-‘l
exP NUW’l
,
(6)
giving ZAD =
[Q(TJ/oel w4 [De- QVII /kTl+
(7)
An exact result could be obtained by sol-$ng the energy diffusion equation for the potential Y(R) with an absorbing boundary at E = e(T). The diabatic limit is obtained if vibrational relaxation is fast relative to rotational relaxation. In this case the factor exp { [De - Q(T)] /kT} is a ratio of partition functions as in RRKM theory of unimolecular reactions for internal degrees of freedom, and the dynamics takes place on the bare potential V(R). The absorbing boundary is placed at E = De and the dissociation rate kD for kT Q De is given by (4) with ZD = expl[De - Q(T)]Ml .
(8)
The evaluation of ZAD and ZD requires only the trivial determination of the maximum and minimum values of v(R) to give Q(T). Thus we can compare our model with the numerical results of Borkovec and Berne [ 121 for a Morse potential. For kT < De the problem can be further simplified because the repulsive limb of the Morse potential does not contribute significantly to r(R) at R,, so that R, can be found by solution of bR, exp(-bRT) 2
4
3 DISTANCE
6
‘+,
Fig. 1. The Javeraged effective potential P(R) for a Morse oscillator is plotted for several values of DdkT and bRe in the vicinity of the transition state. 56
= (kT/De) exp(-bR,)
,
(9)
which explains why the numerical results depend only on bR, and kT/De. The results in terms ofR, are ZD = IR#,12 exp12bRTI and
(10)
Volume 128, number 1
CHEMICAL PHYSICS LETTERS
one-dimensional model are exhibited by the large values of Z, especially for sloppy potentials. The excellent agreement between the simple model and numerical results confirms that the use of the effective, or J-averaged, potential incorporates the primary contributions from the three-dimensional aspects of dissociation reactions, and by equilibrium arguments, recombination reactions within the framework of weak impulsive collisions at low densities. This confirms the basis of our previous work [ 1,2 ] and suggests that the next improvement of our model will be to remove the constraint of impulsive collisions by use of a frequency-dependent friction to account for the finite timescale for collisions. It remains to be seen wheter the simple expressions of (7) and (8) are applicable to the rotational contributions in polyatomic dissociation.
bR,=l
0
I
bR,=5
40
2o
6
20
I
40
11 July 1986
1 80
D/kT Fig. 2. The three-dimensional dissociation factor Z is plotted for several values of bRe against D,/kT. The numerical results of Borkovec and Berne (---), compared with the adiabatic (- - -) and the diabatic (...) approximations of this work.
The author wishes to thank Dr. M. Borkovec and Professor B.J. Berne for sending a preprint of their work [12], without which the comparisons with our model could not be made. This work was supported by the Australian Research Grants Scheme.
References ZA,
= ] 1 - (2kT/D) In@-@,)
X IRT/R,I~ exp12/bRTI .
- (2kT/DbRT) I
WI M.G. &eats, J.M. Dawes and D.P. Millar, Chem. Phys. (11)
We note that in the limit of rigid potentials where bR, + 00, the solution of (9) gives Z, = Z,, = 1, which is the one-dimensional result. This was also the result obtained numerically in ref. [12]. In fig. 2 the estimates of Z, and ZA, from (9)-(11) are plotted against D/kT for representative values of bR, = 1, 5, 10 and compared with the numerical results of Borkovec and Berne using their values of A and (Yin (5). From fig. 2 we note that the diabatic approximation slightly overestimates 2, but only by at most 10% for the sloppy case of bR, = 1. On the other hand the adiabatic approximation underestimates Z significantly for high temperatures and sloppy potentials, but gives excellent agreement at low temperatures for all cases. Generally, the diabatic approximation works best at high temperatures and the adiabatic approximation at high temperatures, although the diabatic approximation is always good. The large deviations from a
Letters 114 (1985) 63; M.G. &eats, J.M. Dawes, P.M. Rodger and D.P. Millar, Ber. Bunsenges. Physik. Chem. 89 (1985) 233. PI J.M. Dawes and M.G. Sceats, Chem. Phys. 96 (1985) 315. [31M. Borkovec and B.J. Beme, J. Phys. Chem. 89 (1985) 3994. 141P.M. Rodger and M.G. Sceats, J. Chem. Phys. 83 (1985) 3358. 151J. Troe, J. Chem. Phys. 66 (1977) 4745. 161AP. Penner and W. Forst, Chem. Phys. 11 (1975) 243. [71H.O. Pritchard, in: “Reaction kinetics, Vol. 1, Chem. Sot. Specialist Periodical Report, Senior Reporter P.G. Ashmore (The Chemical Society, London, 1975) p. 243. PI M.G. Sceats, Chem. Phys. 96 (1985) 299. PI M.G. Sceats, G.E.C. Fell and D.P. Millar, to be published. WI S. Nordholm, D.L. Jolly and B.C. Freasier, Chem. Phys. 23 (1977) 135; b.L. Jolly, B.C. Freasier and S. Nordholm, Chem. Phys. 21 (1977) 211. VI M.G. Sceats and D.P. Millar, to be published. WI M. Borkovec and B.J. Beme J. Chem. Phys. 84 (1986) 4327. iI31H.A. Kramer% Physica 7 (1940) 284.
57
CHEMlCAL PHYSICS LETTERS
11 July 1986
DIATOMIC DISSOCIATION ON A J-AVERAGED POTENTIAL Mark G. SCEATS Departmeni of Physic& Chemistry, Wniversity af Sydney, Sydney, NS W 2006, Australra Received 7 March 1986; in final form 21 May 1986
Illatomic dissociation rates are evaluated in the weak-cotlision low-pressure &nit from a ~~nd~rn~~~~~~~ impulsive stochastic model on the J-averaged effective potential Y(R) - 2&T h@?/T(T), and compared with the threedimensionat resuitf of Borkovec and Berne. The agreement within 10% over a wide range of Marse parameters and temperatures indicates that the contraction to one dimension is not only instructive, but also accurate.
The sitar advantage of stochastic ~~a~~~~~s of diatomic recombination and dissociation is that the one approach is applicable over the entire range of both gas densities [l-3] - from low pressures in which three-body collisions dominate to compressed hquids in which transport phenomena dominate. We have shown [4] that the coupled motion of a pair of particles in the Langevin picture can be reduced to the Brownian motion of a single particle on a onedimensional effective potential P(R) = Y(R) - 2kT In@&) , (1) where R, is t&e position of the tr~s~tion state of p(R), and this approach has been applied to analysis of experimental data for recombination and dissociation of iodine [ 1,23 in which the friction 0 describing the coupling of the atoms to the bath gas is reduced from the diffusion coefficient D, of the separated atoms by the Einstein re~a~o~~ip /3= kT~~~~~, where MB is the mass of the atom B. The agreement with experiment was very good, but the accuracy of the many aspects of the model is difficult to assess because of the uncertainty in the model parameters. The aspects of the approach which need to be independently considered are: (i) The validity of elimination of the orientational degrees of freedom [43 to reduce the problem to consideration of ordy the radial coordinate in (1). The important role of coupling between vibrational and rotational degrees of freedom has been considered by others C-7], and it is by no means clear that the use of me the effective potential of eq. (1) will incorporate these effects. 0 009-26 14/86/$03.50 0 Elsevier Science Publishers B.V. (Norm-Donald Physics ~b~~g Division)
(ii) The use of an untested model of ~yrnptot~~ curve hopping [8] or ad hoc combinations of efectronic degeneracy factors to account for the multiple potential surfaces (16 in iodine) presents the most formidable problem in judging the accuracy of the approach by comparison with experiments on halogens. (iii) The use of a frequency-~dependent friction implies that the timescales of collisions are shorter than the characteristic vibrational periods within kT of dissociation. This is not true for heavy bath gases or at high temperatures. In subsequent work we have developed a model of frequen~ydepend~~t collisional friction [9] and compared the results with the molecular dynamics simulations of Nordhohn et al. [lo] for highly vibrationally excited Br, in Ar, and have found good agreement for V-T energy transfer within 5 kT of dissociation [I 1]_ Nevertheless the cornerstone of our approach, namely the use of the effective potential of(f), eannot be tested with molecular dynamics simulations (which eliminate problems of (ii) by considering only one potential) because these simulations are not stochastic and an additional model of the friction is required to relate a stochastic model to the MD results, The only reasonable way to test the applicability of (1) is to compare the results from (1) with those from a full stochastic treatment in three dimensions recently obtained by Borkovec and Berne [12] for a Morse potential in which the dissociation rate is evaluated as tit average over a d~t~bution of~dependent rates related to dissociation over the centrifugal barrier of the rotational potentials
F&(R < J) = V(R) + J2/2/.tR2 .
(2)
For the Morse potential V(R) = De { exp [-2b(R
- Re)] - 2 exp [-b(R - R&l }
(3) it is convenient to put the dissociation in the form k =Z(T)P(2De/W
exp(-D,lkT)
(4)
,
where the factor Z(T) accounts for all effects associated with the three-dimensional aspects of the problem. The one-dimensional result of Kramers [ 13] is obtained if Z = 1. The numerical results of Borkovec and Berne could be fitted to the form Z(T) =: A lDe/kTl”-l
,
(9
where A and OLdepend only on the Morse parameter bR,.
When the effective potential v(R) is plotted, a transition state appears at R,, as shown in fig. 1 for the Mofse potential of izerest in this work. The resulting barrier height of V(R) can be significantly less than the dissociation energy. The effective potential gives the exact equilibrium distribution in the R co-
I
D
11 July 1986
CHEMICALPHYSICS LETTERS
Volume 128, number 1
b&=1
ordinate, namely R2 exp [-V(R)/kT], but like any effective potential, its use in dynamical applications is valid only in two limits depending on whether the relaxation of the property of interest in the retained coordinate (in this case the vibrational energy in R) is fast or slow compared to those of the coordinates which have been eliminated (rotational energy relaxation in 0, $). The adiabatic limit is obtained if rotational relaxation is fast relative to vibrational relaxation, such that the effective potential is a potential of mean force. The dynamics take place in this potential. If v(R) for R
= P[2QVW-‘l
exP NUW’l
,
(6)
giving ZAD =
[Q(TJ/oel w4 [De- QVII /kTl+
(7)
An exact result could be obtained by sol-$ng the energy diffusion equation for the potential Y(R) with an absorbing boundary at E = e(T). The diabatic limit is obtained if vibrational relaxation is fast relative to rotational relaxation. In this case the factor exp { [De - Q(T)] /kT} is a ratio of partition functions as in RRKM theory of unimolecular reactions for internal degrees of freedom, and the dynamics takes place on the bare potential V(R). The absorbing boundary is placed at E = De and the dissociation rate kD for kT Q De is given by (4) with ZD = expl[De - Q(T)]Ml .
(8)
The evaluation of ZAD and ZD requires only the trivial determination of the maximum and minimum values of v(R) to give Q(T). Thus we can compare our model with the numerical results of Borkovec and Berne [ 121 for a Morse potential. For kT < De the problem can be further simplified because the repulsive limb of the Morse potential does not contribute significantly to r(R) at R,, so that R, can be found by solution of bR, exp(-bRT) 2
4
3 DISTANCE
6
‘+,
Fig. 1. The Javeraged effective potential P(R) for a Morse oscillator is plotted for several values of DdkT and bRe in the vicinity of the transition state. 56
= (kT/De) exp(-bR,)
,
(9)
which explains why the numerical results depend only on bR, and kT/De. The results in terms ofR, are ZD = IR#,12 exp12bRTI and
(10)
Volume 128, number 1
CHEMICAL PHYSICS LETTERS
one-dimensional model are exhibited by the large values of Z, especially for sloppy potentials. The excellent agreement between the simple model and numerical results confirms that the use of the effective, or J-averaged, potential incorporates the primary contributions from the three-dimensional aspects of dissociation reactions, and by equilibrium arguments, recombination reactions within the framework of weak impulsive collisions at low densities. This confirms the basis of our previous work [ 1,2 ] and suggests that the next improvement of our model will be to remove the constraint of impulsive collisions by use of a frequency-dependent friction to account for the finite timescale for collisions. It remains to be seen wheter the simple expressions of (7) and (8) are applicable to the rotational contributions in polyatomic dissociation.
bR,=l
0
I
bR,=5
40
2o
6
20
I
40
11 July 1986
1 80
D/kT Fig. 2. The three-dimensional dissociation factor Z is plotted for several values of bRe against D,/kT. The numerical results of Borkovec and Berne (---), compared with the adiabatic (- - -) and the diabatic (...) approximations of this work.
The author wishes to thank Dr. M. Borkovec and Professor B.J. Berne for sending a preprint of their work [12], without which the comparisons with our model could not be made. This work was supported by the Australian Research Grants Scheme.
References ZA,
= ] 1 - (2kT/D) In@-@,)
X IRT/R,I~ exp12/bRTI .
- (2kT/DbRT) I
WI M.G. &eats, J.M. Dawes and D.P. Millar, Chem. Phys. (11)
We note that in the limit of rigid potentials where bR, + 00, the solution of (9) gives Z, = Z,, = 1, which is the one-dimensional result. This was also the result obtained numerically in ref. [12]. In fig. 2 the estimates of Z, and ZA, from (9)-(11) are plotted against D/kT for representative values of bR, = 1, 5, 10 and compared with the numerical results of Borkovec and Berne using their values of A and (Yin (5). From fig. 2 we note that the diabatic approximation slightly overestimates 2, but only by at most 10% for the sloppy case of bR, = 1. On the other hand the adiabatic approximation underestimates Z significantly for high temperatures and sloppy potentials, but gives excellent agreement at low temperatures for all cases. Generally, the diabatic approximation works best at high temperatures and the adiabatic approximation at high temperatures, although the diabatic approximation is always good. The large deviations from a
Letters 114 (1985) 63; M.G. &eats, J.M. Dawes, P.M. Rodger and D.P. Millar, Ber. Bunsenges. Physik. Chem. 89 (1985) 233. PI J.M. Dawes and M.G. Sceats, Chem. Phys. 96 (1985) 315. [31M. Borkovec and B.J. Beme, J. Phys. Chem. 89 (1985) 3994. 141P.M. Rodger and M.G. Sceats, J. Chem. Phys. 83 (1985) 3358. 151J. Troe, J. Chem. Phys. 66 (1977) 4745. 161AP. Penner and W. Forst, Chem. Phys. 11 (1975) 243. [71H.O. Pritchard, in: “Reaction kinetics, Vol. 1, Chem. Sot. Specialist Periodical Report, Senior Reporter P.G. Ashmore (The Chemical Society, London, 1975) p. 243. PI M.G. Sceats, Chem. Phys. 96 (1985) 299. PI M.G. Sceats, G.E.C. Fell and D.P. Millar, to be published. WI S. Nordholm, D.L. Jolly and B.C. Freasier, Chem. Phys. 23 (1977) 135; b.L. Jolly, B.C. Freasier and S. Nordholm, Chem. Phys. 21 (1977) 211. VI M.G. Sceats and D.P. Millar, to be published. WI M. Borkovec and B.J. Beme J. Chem. Phys. 84 (1986) 4327. iI31H.A. Kramer% Physica 7 (1940) 284.
57