- Email: [email protected]
The evolution of entropy along the reaction path in an atom-diatom collision
-\ToIume 33, number
3
CHEMICAL PHYSICS LETTERS
15 June1975
THE EVQLUTIQN OF ENTROPY ALONG THE REACTION PATH W AN ATOM-DIATOM
COLLISION*
G.L. HOFACKER i.ci~rsrzrhl fiir Tlreoreriscl!e Chenlie, Tecfrnische LhiversitGt.
Munich,
Gerrnany
and
Received 8 April 1975
The distribution of vibrational states at any point along the reaction path is obtained, in the ciassical path limir, for a model reactive collision problem. For reactants in the ground vibrationa state the distribution is af a Poisson-type throughout the reaction, and hence the entropy can be anrdytically computed. It is shown that there may be more extensive vibrational population inversion at an intermediate point as compared to the final pioducts.
1. Introduction
into reactants and products region. The present model, while simplistic, can account for all essential effects
It is of interest to develop explicitly soluble models for the dynamics of molecular collisions [I ] even at the e,% ense of simplifying zrsumptions. Here we show how a previously proposed model [2-51 can be solved in the classical path limit. The solution enables us to examine the different aspects of the dynamics. We pay particular attention to the distribution of the vibrational states and show ITOWthe entropy [6--I I] of this distribution evolves aIong the reaction path. The essence of the model we use [2-S] is the coupling between the motion along the reaction path and the vibration perpendicular to it [12--141. The most distinct feature of this soupling is that it depends on the local kinetic energy. In the natural coordinate system [I 2,131 it is induced by the internal centrifugal :force as the system-point moves around the bend in the reaction coordinate. There are a number of other
models for treating the vibrational-translational
cou-
pling in a chemical reaction [15-271. Some of these mode!s also cen,ter attention on the motion Aong the reaction path Chile others divide the reactive event .
and yet allow for an explicit solution (in the classical path limit). For obvious reasons simple models usually consider collinear collisions. There is then the need to account for the modifications ir going from one- to three-dimensional collisions. An approximate procedure for allowing for this, so-called, “dimension bias” has recently been introduced 1281 and validated for some concrete cases [29,30]. In this way our onedimensional (collinear) distributions can be converted to three-dimensional ones. Entropy and other information-theoretic measures have been shown [7-I l] to offer a compact and convenient description of the products distribution in a chemical reaction. It is therefore worthwhile to examine the “evolution” of the entropy in the course of the collision._ Explicitly we examine the entropy of the vibrational motion all the way from the reactants to the products region and explore the dynamical factors involved.
. We consider a collinear atom-diatom
collision on
Voll;me 3 3, number 3
a single notential energy surface. Introducing a reaction coordinate < and a vibrational coordinate n the hamiltonian can be written in the form [2,12-141
3. Solution
H= TECLr7) f P(17,E) -i-WO)
H vjj =fiw(t)&
+tW)?
.
(1)
Here Ts(E,n) =p$2p(l tKnjZ is the kinetic energy for the motion along the curvilinear coordinate e and K is the local curvature. For small displacements from the reaction path one can expand Tt(E, q) as TQ, n) = TQ,O)
+ [arE(E!0)/a77j 17+ .. . .
(2)
A quanta1 discussion of this model hamiltonian was given previously [2]. At this point we now make
a classical path approximation. We assume that the motion along the reaction coordinate (Q is to be treated in a classical fashion so that { can be determined as a function of time, ,$= g(5). The (classical) equation of motion for < need be determined so as to include the (adiabatic) centrifugal shift [3,13] of the reaction coordinate. In other words, we wish to define .$ as the dynamic reaction path [2,13] so that the potentia1 perpendicular to it is harmonic. With this definition of a reaction path one can neglect (to first order) the influencs of the vibrational motion on the motion along <_ Hence, a simple closed equation for .$ results. For a given E(f) the hamiltonian for the vibrationai motion along r) can be written as &ii
= p;/2P
+ ak(t) V2 - [P;(r)/2P]
2K77 .
(3)
Here k(t) = k(t(t)) is the 10~1 harmonic force constant. We then introduce dimensionless oscillator coordinates x = (;C/~~LJ)“~ 77and or?, = (jJzo)““p, such that
H mb = l-m(t) (:p~+$x“) Here g(f) is defined
- Rc&)g(r)x@-.
to employ
the model hamiltonian
+ti[J(t)s
+f*(t)si]
to the form
)
‘(7)
where
At> = -4Mfj
(8)
-
As is well known [3 l-341 this hamiltonisn can be solved exactly. For our purpose here it is sufrficient to
consider the solution with the boundary condition that at t G -= (i.e., before the collision) the reactants are in the ground vibrationat state. Then, an eigenstate of H.& at any subsequent time t can be written as lo> = exp [A(r) + C(t)st
] IO) .
(9)
where !O) is the initial, ground vibrational state. To verify the solution (9) one need merely substitute in the time-dependent Schrodinger equation iEd] a)/dt = Hm%(r) I a). Using the identity thaL for some (weU defined) function F,
exp {xs) rTs, st) = F(s, st +x) exp (xs) ,
(10)
one obtains idA/dr
= f((r) C(f)
(11)
and
For any given $(I) one obtainsf(r) as in (8) and hence can solve the ordinary differential equation for C(f). Such a solution is possible even if w(f) is t-dependent. The simplest way of haridling any r (i.e., t) dependence of w is to introduce an “interaction-like” change of variable
C(t) = c(t) exp (--i(‘w(L’)
such tha: (3) and (4) agree,
it is convenient
We have brought
(4)
,e
Finally,
.15June1975
CHEhlICAL PfIYSICS LETTERS
creation
(51 and an-.
nihilatjon operators for the oscillator. As usual, x = 2-“2(sT+~). Then,
df’) ,
03)
when the secular term will be eliminated, To sirnpliCy the notation we shall take w tc be constant and write C(r) = c(t) exp (-ior). In those cases when there are large changes in the vibrational frequency along the reaction path one should defmed in (I 3). From (12) idcjdt
‘E
exp (iwt)J*(r)
replace czt by the integral
as
Volume 33.‘numbcr
CHIZMICAL PHYSICS LETTERS
3
1.5 June 1975
. Using (18) one readily
E(C) = -i
_/ exp (iwt’)j’*(t’)di’ _m
.
(21)
For the present purpose a!1 we need is ~(‘IE).However, from (11) we can readily determine A(f). Gnce the state lo) is determined one can compute the mean vibrhticnal energy. In sopsing E(t) this ener,T needs to be.subtracted from the total energy in order to deterrnine the energy available for the classical motion along-the reaction path. .’ To bring the explicit solution to a more familiar form we consider p,l(t). P,(t)=
shows that
09
I~nlo>12 ,
(16)
the probabiiity of exciting the ,Ith vibrational state at the time r. Starting with the definition lni = (~r+l)“~ In+11 where In> is the 12th vibrzional level, it follows that (ft)mln)= ~(~~?t~z)!/~1!J1’*I~z+~~2)and hence, usings power series expansion uf expx.. that
Since D(t) is not necessarily a monotonically increasing function of time P,I(l) need not steadily increase to its asymptotic value. Some vibrational states can, in principle, be more heavily populated during the collision then at the asymptotic region. A computational illustration of this phenpmenon for reactive collisions has been noted previously [35] and has been verified for many different systems since*. The average value of the deviance of p,,(t) from P,r(m) is the entropy
st
esp [C(r) s’] (0) = F
[C(i)]”
lrz)/(f2!)“2
.
(17)
Thus, with the result (9) for lo), U,(r)
= [r-1(1)]” exp [--l.r(tIr]/~!
The factor exp -p ensures rnalised a: all times and H(f) = l&)1” We bution is well that it
.
that P,,(t)
-
(18) is indeed
nor-
09)
have thus shown that for any time t the distriof vibmtional state; is Poisson-like. This result kndwn for t f =_ M’hat we have shown here is holds for all intermediate times*.
This, strictly non-positive, (18) and (21) to be
measure
is found,
s(t) = At> Un [~(+~(~)I
+ 1 - M-)/At)1
from
1.
(23)
Using the inequality In x 4x1, one sees that S(r) is indeed non-positive and vanishes only when p(t) = ,u(-). The initial value of S(f) is S(-0~) = -p(m) and the rise ofs(t) from its initial value to the final value of zero elso need not be monotonic. If, during the collision, p(r) > u(w) the entropy production will oscillate. Physically this is very obvious. If p(f) > p(m) then the intermediate distribution of states is “wider” than the final one. The essence of the problem is the time dependence of y(r). We thus need a better understanding of the factors that govern this dependence.
5. Summary 4. The evolution of entropy We wish to compare the distribution of stxes a1 any‘time t [{Pn(t)j, cf. (18)] with the final (t + -) distribution of states $
We have shown that at any point along the reaction coordinate the dis!ribution* of vibrational states is Poisson-like (when starting from the reactants ground vibrational
state).
The parameter
of the distribution,
.’ P,(m) = .[p(h)]” *l-i-@ di&ibutIon
exp ‘[+(-)I
/,I! .
(20)
is Poisson&ke.only if the initial tibrational state is the gound slate. Par more general initial states the ;, _ciistriiution is.a generalized Poisson one, kg. ref, [t].
z ‘Ihe probabilities
of the different vibrntional states are the diagonal elements of the density matrix in &he,n-representation. f Private communications by J. Manz, M. TarGr and LT..Halavee. * Fodtnote see next page.
Volww
33,number 3
CHEMICAL
PHYSICS
values when f(tj, eq. (8), varies exp (iwt). p(r) can however decrease due to destructive interference between f(r) and exp (iot). The entropy analysis also verifies that if p(t) > P(W) there is more population inversion during than after the collision. The temperature parameter [&I I] of the Prl(rjdistribution as compared to the asymptotic one is [cf. eq. (2 1 )J , &t), attains
significant
On 2 time scalecomparable‘to
Ah
=
=
d{-In
[P,(rj/Pf] + In
[P,(===)/P,s)] }/d!z
-In ld~>/PWI .
Again we see that it-p(r) > p(m) the vibrational bution is “hotter” during the collision.
(24) distri-
LETTERS
1s Jurre
1975
[S] R.D. Levine. Chem. Phys: Letters IO (1971) 510. [Cl E.T. Jaynes, Phys- Rev. 206 (1957) 620. [7] R.B. Bernstein and R.D. Levine. J. Chem. Phys. 57 (1972) 434. [8 ] A. Ben-Shaul, R.D. Levine and R.B. Bcrnstcin, J. Chem. Phys. 57 (1972) 5427. [9] G.L. Hofackcr and R.D. Levine, Chcm. Phys. Letters 15 (1972) 163. U. Dinur and R.D. Levine. Ckem. Phys. 9 (1975)
17. R.D. Levine and R.B. Bernstein, Accounts Chem. Res. 7 (1974) 393. [12] C.L. Hofacker, 2. Naturforsch. 18~1(1963) 607. [13] R.A. Marcus, J. Chem. Phys. 45 (1966) 4500. (141 S.F. F&her, G.L. Hofacker and R. Seiler, J. Chem. Phys. 51 (1964) 3951. 1151S.1:. Fischer and h!. Ratncr, I. Chem. Phys. 57 (1972)
[la]
[ll]
2769.
1161 h1.V. Basilcvsky, Mol. Phys. 26 (1973) 765. [171 N.H. Hijazi and K.J. Laidler, J. Chem. Phys. 58 (1973) 349.
Acknowledgement We would like to thenk Dr. J. Manz and Mr. U. Halavee for useful comments on the manuscript.
1181M. Baer, J. Chem. Phys. 60 (1974) 1057. (191 hid.Berry, Chem. Phys. Letters 29 (1974) 323, 329. i201 J.P. Simons and P.\v’. Tasker, Chem. Sot. Faraday Trans. II 70 (1974) 1496.
[?I1 Y. Band and R.F. Freed, Chcm. Phys. Letters 28 (1974) 328.
* For the offiIiagona1 elements of the density matris one can show that they are small whenever j-(rj varies slowly compared to esp (iwf). In particular this includes the asymptotic (post collision) region where/(r) = Cland the density matrix is thus necessarily diagonal. In particular since C(r) = c(i) cup (--iwf) it also follows that the loss of phase
memory of the oscillator occurs over a sirlgle vibrational period.
References [l J R.D. Levine and R.B. Bernstein, XIolecular rcacrion dynamics (Chrendon Press, Oxford, i974). [ 21 G.L. Hofacker and R.D. Levine, Chem. Phys. Letters 9 (1971) 617. [3] G.1,. Hofxker snd N. Rijsch, Ber. Bunsenges. Physik. Chem. 77 (1973) 661. [4] G.L. Hofacker and W. Michel, Ber. Bunsenges. Physik. Chem. 78 (1974) 174.
[22] S. Mukamcl and J. Jurrner,
J. Chenx Phys. 60 (1974) 4760. [23l M.D. Pattcngill and J.C. Polanyi, Chcm. Phys. 3 (1974) 1. Chem. Phys. [241 J.C. Light and A. Altenbcrgcr-Siczck, Lerters 30 (1973) 19.5. (25 1 U. Halavee and hl. Shapiro, to be published.
PI
D.G. Truhlar,
to bc published.
[271 B.C. ELI, Chcm. Phys. 5 (1974) 95. [2Sl R.B. Bernstein and R.D. Lcvinc, Chem. Phys. Letters 29 (1974) 314. 1291 J.C. Polanyi and J.L. Schreibcr, Chem. Phys. Letters 29 (1974) 319. 1301 A. Ksfri, E. Pollack, R. Kodoff and R.D. Levine, Cnem. Phys. Letters :3 (1975) 201. R.J. Glaubcr, Phys. Rev. 84 (1951) 395. f:‘;2 E. Kerner, Can. J. Phys. 36 (1958) 371. (331 V.S. Popov and &hi. Peorlomov, J. Expt. Thcor. Phys 30 (1970) 910. (341 X. Chapuisrt and J. hlanz, hiol. Phys. 26 (1973) 1577. 135 J R.D. Levine and S.F. Wu, Chcm. Phys. Letters l! (1971) 557.
,407
3
CHEMICAL PHYSICS LETTERS
15 June1975
THE EVQLUTIQN OF ENTROPY ALONG THE REACTION PATH W AN ATOM-DIATOM
COLLISION*
G.L. HOFACKER i.ci~rsrzrhl fiir Tlreoreriscl!e Chenlie, Tecfrnische LhiversitGt.
Munich,
Gerrnany
and
Received 8 April 1975
The distribution of vibrational states at any point along the reaction path is obtained, in the ciassical path limir, for a model reactive collision problem. For reactants in the ground vibrationa state the distribution is af a Poisson-type throughout the reaction, and hence the entropy can be anrdytically computed. It is shown that there may be more extensive vibrational population inversion at an intermediate point as compared to the final pioducts.
1. Introduction
into reactants and products region. The present model, while simplistic, can account for all essential effects
It is of interest to develop explicitly soluble models for the dynamics of molecular collisions [I ] even at the e,% ense of simplifying zrsumptions. Here we show how a previously proposed model [2-51 can be solved in the classical path limit. The solution enables us to examine the different aspects of the dynamics. We pay particular attention to the distribution of the vibrational states and show ITOWthe entropy [6--I I] of this distribution evolves aIong the reaction path. The essence of the model we use [2-S] is the coupling between the motion along the reaction path and the vibration perpendicular to it [12--141. The most distinct feature of this soupling is that it depends on the local kinetic energy. In the natural coordinate system [I 2,131 it is induced by the internal centrifugal :force as the system-point moves around the bend in the reaction coordinate. There are a number of other
models for treating the vibrational-translational
cou-
pling in a chemical reaction [15-271. Some of these mode!s also cen,ter attention on the motion Aong the reaction path Chile others divide the reactive event .
and yet allow for an explicit solution (in the classical path limit). For obvious reasons simple models usually consider collinear collisions. There is then the need to account for the modifications ir going from one- to three-dimensional collisions. An approximate procedure for allowing for this, so-called, “dimension bias” has recently been introduced 1281 and validated for some concrete cases [29,30]. In this way our onedimensional (collinear) distributions can be converted to three-dimensional ones. Entropy and other information-theoretic measures have been shown [7-I l] to offer a compact and convenient description of the products distribution in a chemical reaction. It is therefore worthwhile to examine the “evolution” of the entropy in the course of the collision._ Explicitly we examine the entropy of the vibrational motion all the way from the reactants to the products region and explore the dynamical factors involved.
. We consider a collinear atom-diatom
collision on
Voll;me 3 3, number 3
a single notential energy surface. Introducing a reaction coordinate < and a vibrational coordinate n the hamiltonian can be written in the form [2,12-141
3. Solution
H= TECLr7) f P(17,E) -i-WO)
H vjj =fiw(t)&
+tW)?
.
(1)
Here Ts(E,n) =p$2p(l tKnjZ is the kinetic energy for the motion along the curvilinear coordinate e and K is the local curvature. For small displacements from the reaction path one can expand Tt(E, q) as TQ, n) = TQ,O)
+ [arE(E!0)/a77j 17+ .. . .
(2)
A quanta1 discussion of this model hamiltonian was given previously [2]. At this point we now make
a classical path approximation. We assume that the motion along the reaction coordinate (Q is to be treated in a classical fashion so that { can be determined as a function of time, ,$= g(5). The (classical) equation of motion for < need be determined so as to include the (adiabatic) centrifugal shift [3,13] of the reaction coordinate. In other words, we wish to define .$ as the dynamic reaction path [2,13] so that the potentia1 perpendicular to it is harmonic. With this definition of a reaction path one can neglect (to first order) the influencs of the vibrational motion on the motion along <_ Hence, a simple closed equation for .$ results. For a given E(f) the hamiltonian for the vibrationai motion along r) can be written as &ii
= p;/2P
+ ak(t) V2 - [P;(r)/2P]
2K77 .
(3)
Here k(t) = k(t(t)) is the 10~1 harmonic force constant. We then introduce dimensionless oscillator coordinates x = (;C/~~LJ)“~ 77and or?, = (jJzo)““p, such that
H mb = l-m(t) (:p~+$x“) Here g(f) is defined
- Rc&)g(r)x@-.
to employ
the model hamiltonian
+ti[J(t)s
+f*(t)si]
to the form
)
‘(7)
where
At> = -4Mfj
(8)
-
As is well known [3 l-341 this hamiltonisn can be solved exactly. For our purpose here it is sufrficient to
consider the solution with the boundary condition that at t G -= (i.e., before the collision) the reactants are in the ground vibrationat state. Then, an eigenstate of H.& at any subsequent time t can be written as lo> = exp [A(r) + C(t)st
] IO) .
(9)
where !O) is the initial, ground vibrational state. To verify the solution (9) one need merely substitute in the time-dependent Schrodinger equation iEd] a)/dt = Hm%(r) I a). Using the identity thaL for some (weU defined) function F,
exp {xs) rTs, st) = F(s, st +x) exp (xs) ,
(10)
one obtains idA/dr
= f((r) C(f)
(11)
and
For any given $(I) one obtainsf(r) as in (8) and hence can solve the ordinary differential equation for C(f). Such a solution is possible even if w(f) is t-dependent. The simplest way of haridling any r (i.e., t) dependence of w is to introduce an “interaction-like” change of variable
C(t) = c(t) exp (--i(‘w(L’)
such tha: (3) and (4) agree,
it is convenient
We have brought
(4)
,e
Finally,
.15June1975
CHEhlICAL PfIYSICS LETTERS
creation
(51 and an-.
nihilatjon operators for the oscillator. As usual, x = 2-“2(sT+~). Then,
df’) ,
03)
when the secular term will be eliminated, To sirnpliCy the notation we shall take w tc be constant and write C(r) = c(t) exp (-ior). In those cases when there are large changes in the vibrational frequency along the reaction path one should defmed in (I 3). From (12) idcjdt
‘E
exp (iwt)J*(r)
replace czt by the integral
as
Volume 33.‘numbcr
CHIZMICAL PHYSICS LETTERS
3
1.5 June 1975
. Using (18) one readily
E(C) = -i
_/ exp (iwt’)j’*(t’)di’ _m
.
(21)
For the present purpose a!1 we need is ~(‘IE).However, from (11) we can readily determine A(f). Gnce the state lo) is determined one can compute the mean vibrhticnal energy. In sopsing E(t) this ener,T needs to be.subtracted from the total energy in order to deterrnine the energy available for the classical motion along-the reaction path. .’ To bring the explicit solution to a more familiar form we consider p,l(t). P,(t)=
shows that
09
I~nlo>12 ,
(16)
the probabiiity of exciting the ,Ith vibrational state at the time r. Starting with the definition lni = (~r+l)“~ In+11 where In> is the 12th vibrzional level, it follows that (ft)mln)= ~(~~?t~z)!/~1!J1’*I~z+~~2)and hence, usings power series expansion uf expx.. that
Since D(t) is not necessarily a monotonically increasing function of time P,I(l) need not steadily increase to its asymptotic value. Some vibrational states can, in principle, be more heavily populated during the collision then at the asymptotic region. A computational illustration of this phenpmenon for reactive collisions has been noted previously [35] and has been verified for many different systems since*. The average value of the deviance of p,,(t) from P,r(m) is the entropy
st
esp [C(r) s’] (0) = F
[C(i)]”
lrz)/(f2!)“2
.
(17)
Thus, with the result (9) for lo), U,(r)
= [r-1(1)]” exp [--l.r(tIr]/~!
The factor exp -p ensures rnalised a: all times and H(f) = l&)1” We bution is well that it
.
that P,,(t)
-
(18) is indeed
nor-
09)
have thus shown that for any time t the distriof vibmtional state; is Poisson-like. This result kndwn for t f =_ M’hat we have shown here is holds for all intermediate times*.
This, strictly non-positive, (18) and (21) to be
measure
is found,
s(t) = At> Un [~(+~(~)I
+ 1 - M-)/At)1
from
1.
(23)
Using the inequality In x 4x1, one sees that S(r) is indeed non-positive and vanishes only when p(t) = ,u(-). The initial value of S(f) is S(-0~) = -p(m) and the rise ofs(t) from its initial value to the final value of zero elso need not be monotonic. If, during the collision, p(r) > u(w) the entropy production will oscillate. Physically this is very obvious. If p(f) > p(m) then the intermediate distribution of states is “wider” than the final one. The essence of the problem is the time dependence of y(r). We thus need a better understanding of the factors that govern this dependence.
5. Summary 4. The evolution of entropy We wish to compare the distribution of stxes a1 any‘time t [{Pn(t)j, cf. (18)] with the final (t + -) distribution of states $
We have shown that at any point along the reaction coordinate the dis!ribution* of vibrational states is Poisson-like (when starting from the reactants ground vibrational
state).
The parameter
of the distribution,
.’ P,(m) = .[p(h)]” *l-i-@ di&ibutIon
exp ‘[+(-)I
/,I! .
(20)
is Poisson&ke.only if the initial tibrational state is the gound slate. Par more general initial states the ;, _ciistriiution is.a generalized Poisson one, kg. ref, [t].
z ‘Ihe probabilities
of the different vibrntional states are the diagonal elements of the density matrix in &he,n-representation. f Private communications by J. Manz, M. TarGr and LT..Halavee. * Fodtnote see next page.
Volww
33,number 3
CHEMICAL
PHYSICS
values when f(tj, eq. (8), varies exp (iwt). p(r) can however decrease due to destructive interference between f(r) and exp (iot). The entropy analysis also verifies that if p(t) > P(W) there is more population inversion during than after the collision. The temperature parameter [&I I] of the Prl(rjdistribution as compared to the asymptotic one is [cf. eq. (2 1 )J , &t), attains
significant
On 2 time scalecomparable‘to
Ah
=
=
d{-In
[P,(rj/Pf] + In
[P,(===)/P,s)] }/d!z
-In ld~>/PWI .
Again we see that it-p(r) > p(m) the vibrational bution is “hotter” during the collision.
(24) distri-
LETTERS
1s Jurre
1975
[S] R.D. Levine. Chem. Phys: Letters IO (1971) 510. [Cl E.T. Jaynes, Phys- Rev. 206 (1957) 620. [7] R.B. Bernstein and R.D. Levine. J. Chem. Phys. 57 (1972) 434. [8 ] A. Ben-Shaul, R.D. Levine and R.B. Bcrnstcin, J. Chem. Phys. 57 (1972) 5427. [9] G.L. Hofackcr and R.D. Levine, Chcm. Phys. Letters 15 (1972) 163. U. Dinur and R.D. Levine. Ckem. Phys. 9 (1975)
17. R.D. Levine and R.B. Bernstein, Accounts Chem. Res. 7 (1974) 393. [12] C.L. Hofacker, 2. Naturforsch. 18~1(1963) 607. [13] R.A. Marcus, J. Chem. Phys. 45 (1966) 4500. (141 S.F. F&her, G.L. Hofacker and R. Seiler, J. Chem. Phys. 51 (1964) 3951. 1151S.1:. Fischer and h!. Ratncr, I. Chem. Phys. 57 (1972)
[la]
[ll]
2769.
1161 h1.V. Basilcvsky, Mol. Phys. 26 (1973) 765. [171 N.H. Hijazi and K.J. Laidler, J. Chem. Phys. 58 (1973) 349.
Acknowledgement We would like to thenk Dr. J. Manz and Mr. U. Halavee for useful comments on the manuscript.
1181M. Baer, J. Chem. Phys. 60 (1974) 1057. (191 hid.Berry, Chem. Phys. Letters 29 (1974) 323, 329. i201 J.P. Simons and P.\v’. Tasker, Chem. Sot. Faraday Trans. II 70 (1974) 1496.
[?I1 Y. Band and R.F. Freed, Chcm. Phys. Letters 28 (1974) 328.
* For the offiIiagona1 elements of the density matris one can show that they are small whenever j-(rj varies slowly compared to esp (iwf). In particular this includes the asymptotic (post collision) region where/(r) = Cland the density matrix is thus necessarily diagonal. In particular since C(r) = c(i) cup (--iwf) it also follows that the loss of phase
memory of the oscillator occurs over a sirlgle vibrational period.
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