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Rotational surprisal plots for H + H2, a simple theory
Volume 47, number 3
ROTATIONAL
CHEMICAL PHYSICS LElTERS
1 May 1977
SURE’RISAL PLOTS FOR H + H,, A SIMPLE THEORY
E. POLLAK Department of Chemistry. Columbia University. New York City, New York 10027, USA Received i0 January 1977 The linear surprisal plots for rotational cross sections of the H + Hz exchange reaction are shown to be a manifestation of the transfer of momentum constraint. As a result, it is shown that the complete reactive cross section matrix at a @en total energy is a function of the variable Ej + Ej*_ The surprisal of the matrix elements as a function of this variable is Linear as predicted from the constraint.
1. Introduction Surprisal analysis of rotational
can well approximate oi’,i(E)
= oy,i(E)
the inelastic exp(-
energy transfer cross sections
cross sections
in atom diatom by an exponential gap law
f0 - 6 R !+!$- Ej’ I/E) .
collisions
[l, 21 has shown that one -01
E is the total energy of reactants, j,j’ the initial and final rotational states, oy,@) is the prior rate, 0, the rotational temperature parameter and IO an energy dependent constant. Clearly, if Ei = 0 i.e. one starts a reaction with the reactant in thej = 0 state, one would expect to find a simple linear surprisal in the variable E,*. Wyatt [31 has demonstrated that for quantum mechanical calculations of H + Hz (i = 0) + HZ 0”) f H one does indeed obtain the linear plot. Recently, Schatz and Kuppermann [4] have investigated the atom exchange reaction for several different irMaI as a function of j’ (j’ = 04) for each irtitkdj, j values. They plotted the surprisal I(/“, j) = - ln[o~~,i(E)/o~,i(E)] j = 0,4. Their results show that for each initial j one obtains a linear plot in the variable E,-_ This fact is in obvious conflict with the gap law which predicts a change of slope of the surprisal around the point Ej = ET _ Even if one assumes that eq. (1) is invalid for reactive systems, one remains with the puzzling fact that the quanturn mechanical calculations give linear surprisal plots in the variable Ej*_ Obviously the system is governed by a dynamical constraint and one would like to find a theory for it. The object of this letter is to resolve the above two problems. We intend to show that the transfer of momentum constraint [5-8) alone governs the behaviour of the H + H, exchange reaction. Since all transitions are limited to energy disposal suffices for a description of the the ground vibrational state of Hz, knowledge of the translational rotational energy distribution. In section 2 we review briefly the theory of the transfer of momentum constraint. In section 3 we compare the theory to the calculations of Schatz and Kuppermann. Specifically, we show that the complete rotational cross section matrix can be well characterized by the variable Ei + Ei*, as shown in fig. I - i3dly in section 4 we discuss briefly the scope and limitations of the theory.
2. The transfer of momentum
constraint
Originally, the transfer of momentum constraint was applied only to the normalized products (charm& b) translational energy distribution P(ET*) [5,7] - It was found that P(ET-) could be well represented as
CHEMICAL
Volume 47, number 3 P(ETp) = Po(ET.)
exp [-A,
1 May 1977
PHYSICS LETTERS
iEg2 - sEy2 - R 1/2)2 - A,] .
(2)
XT. an energy dependent parameter, s, RLi2 are constants and h ensures normalP”(+) is the p rior distribution, ization. In principle, a reaction is governed by the constraint of least momentum transfer in both directions. Therefore for the opposite direction one can write P(ET) = P” (ET) exp {-XT’ [_/Zg2 - s’E$~ - (R’)‘fl J 2 - A;) .
(3)
In eqs. (2) and (3) primed values refer to channel b while unprimed values refer to channel a (the original reactants). P(ET) and P(ET.) are not totally independent - they are connected because of microscopic reversibility_ . It has been shown theoreticaliy and verified experimentally that eqs. (2) and (3) coupled with the principIe of microscopic reversibility give an explicit expression for the yield matrix elements [7-lo]. Specifically, at a given reactants energy E the reactive part cf the symmetric yield matrix [I I J Y(ETn, ET; E) is given by the expression 17.81
Y(E+.ET;0 = Y”(ET4’T;E) X exp {--XTETp - AT, ET f (ETET,)1/2 (AT + s’+) Here Y”(ET,,
+ AT, (_R’ET)1/2] ) .
+ 2 [AT (RETs)~;~
ET; E) is the prior value of the yield matrix element
(4)
[ 1,l l] (5)
Y”(ET’.ET;E)=hQpCET)~(ET,)p(x) is the transIationaI density of states and Q is an energy dependent scopic reversibility implies the folIowing relationship [7,8]
constant.
tn addition
to eq_ (4), rnicro-
XTS =X7’s’ -
(6)
The reaction we will be dealing with is the symmetric atom exchange reaction. Since channels a and b are physically ind.istinguishabIe it follows that s = s’ and X, = XT*. For a symmetric exchange reaction one also expects [7,8] that R1j2 = (R’)lj2 = 0 (i.e. the net distortion on a symmetric potential energy surface is to a fust approximation expected to equal zero). As a result, eqs. (2)-(4) become much more simplified.
3. Application
of the theory
to H + Hz
Eq. (2) suffices for explaining the linearity of the rotational surprisal plots. To see this, one must only remember that the quantum mechanical calculations have only the lowest vyorational channel open. As a result, by energy conservation, the translational energy distribution is the rotational energy distribution_ Energy conservation means ET+Ej
=E=ET.+Ej,_
(7)
Here,j,i’ refer to reactants and products last section) into eq. (2) gives
rotational
states respectiveIy.
P(q)
= P” (ET) exp {-XT [(E - Eip)lj2 - s(E - Ei>li2 12 - b}
I&*)
= -In[P(Ef)/P”(Ei*)l
Introducing
eq. (7) (and the results of the
,
(8)
or = AT [E(l + s2) - Ei* - s2Ej-
2s(E - Ej*)1’2(.
If EjfE and Ej*/E are smaller than 0.4 then we can use the following (1 -EjlE)“‘(l
514
-Ej*/E)‘12
~ 1 -EjlzE-~/2E-
excellent
- Ej)1’2] + h -
(9)
approximation (10)
Volume 41, number 3 Inserting
CHEMICAL PHYSICS LETTERS
this approximation
I(Ej*)=X-r(l
1 May I977
in (9) one finds
-s)[(l
-s)E+(+Ej)]
+hu.
00
Since AT is dependent in principle on E, the slope of the surprisal XT(s - 1) can be dependent on E asfoundby _ the quantum mechanical calculation [4] _On the other hand, for a given E and different initial j’s the slope is constant. This is also consistent with the calculation 141. In obtaining eq. (11) we have utilized only normalized product distributions. A stronger result is obtained if one uses eq. (4) One can compare cross sections for different initial i states. The ctoss section is proportional [I ] to the yield [with a proportionality factor Irl$ (2j + l)] . For H f Hz eq. (4) thus gives Uj’_jQ
UT,j(E)
=
exp [ - hT (2E - Ej - Ej*- 2s(E- Ej)"'(E- E,,)"'] -
cry,i is the prior cross section o~,j(~ S is a constant ri_,i(~
and is given by
= S[(E- Ei~)"'/(E-Ej)"'](2i'+ 1) (S = 4n2R/h3). inserting = - In [UI’,j
(121
Q/U?,jQI
(E3),
eq. (11) into (12) one finds for the surprisal
= A=(1 - S)[x - Cq+ Ej.11 -
(14
We have thus found, that the transfer of momentum constraint as applied to H + Hz predicts a linear surprisal for the complete reactive cross section matrix as a function of the variable (Ej+ Ejr)To check this prediction we have taken all cross sections given in fig. 22 of ref. [4] and plotted their surprisal as a function of Ej f Ef-This is shown in fig. 1. The most notable aspect of this figure is its disagreement with eq. (I), the exponential gap law. Eq_ (14) also predicts that the slope of the surprisal should have opposite signs if plotted once as a function of El f Ef with E constant and once as a function of E with Ej + Ejvconstant. Fig. 2 verifies this prediction. Note that the su~risal is not linear as a function of E,XT depends quite strongly on E [4].
Fig. 2. The rotational surprisaI for j = j’ = 0 as a function of the available energy E (defined as in fig. 1). The sotid points are taken from fig. 23 of ref. [4]_
Fig- I. The rotational surprisal [eq. (I4)] is plotted as a func-
tion of fj + fj* for H + Hz 62 * Hz (i’) f H. fj is defied as Ej/E and similarly fi*- Here, E isthe avaiIabIeenerg rebtive to the bottom of the potential well. The solid points are taken
0
I
I
I
I
I
01
02
a3
04
05
f, ‘r,.
from fig. 22 of ref. 141. (To obtain the cross sectionaI srrrpri.saIone milst multipty by (1 - Glr/2 .) E = 0.6 eV; The Iine is _. a least square fit of eq. (14), AT(1 - s) = 33 eV-’ _ Since me . pnor cross section is detimcd up to an arbitrary constant K only, we were not able to check the ratio of hT(t - s) to Io,e to see whether it conforms to eq. (14).
Volume 47, number 3
CHEMICALPHYSICSLETTERS
1 May
1977
4. Discussion
The exponential gap law [eq. (I)] h as b een empirically documented for diverse systems [I, 2,121. Even so, an unconstrained reaction would not (by defmition) show an exponential gap behatiour. This seems to be the case for the H + HZ reaction. The transfer of momentum constraint suffices for a description of the reactive cross sections and no additional constraint need be invoked. For a general reaction, eq. (7) is not valid. Vibrational energy plays a role. Furthermore, as a result of the large barrier in the H3 potential surface one needs a considerable amount of initial translational energy to overcome the barrier. Therefore the fraction of available energy in initial and final rotation is small and so one can use eq. (IO). This is not so in the more general case, and surely not so for inelastic collisions. _ It should be stressed that eq. (4) describes the yield only in the poor persons limit [7]. In deriving eq. (4) one explicitly ‘assumes a microcanonical distribution of ah other independent variables. For the H3 reaction there is only one other variable Ep and it is dependent via the total energy E. Therefore (4) gives also the rotational yield matrix. Since in general this is not the case one should not generalize the results of section 3. On the other hand, fig. 1 is an additional piece of experimental evidence that the transfer of momentum constraint is quite general. It has again been verified that eq. (4) is a good representation of the yield matrix. Although we have found that at a given reactants total energy E, the H f HZ atom exchange reaction cannot be characterized by an exponential gap law, one would expect that thermal averaging of the reactants would give an exponential gap law. This has been shown to be the case for vibrational energy transfer in reactive and non reactive collisions. Using the methods of ref. [S] it is easily seen that Boltzmann averaging of the initia1 translational energy in eq. (12) will yield an approximate exponential gap law.
References R.D. Levine, R.B. Bernstein, P. Kahana, I. Procaccia and E.T. Upchurch, J. Chem. Phys. 64 (1976) 796. I. Procacch and R.D. Levine, J. Chem. Phys. 64 (1976) 808. R.E. Wyatt,Chem. Phys. Letters 34 (1975) 167. G.C. Schatz and A. Kuppermann, 3. Chem. Phys. 65 (1976) 4668. A. Wri, E. Pollak, R. Kosioff and R.D. Levine, Chem. Phys. Letters 33 (1975) 201. A. Kafri, Y. Shimoni, R.D. Levine and S. Alexander, Chem. Phys. 13 (1976) 323. E. PoUak and R.D. Levine, Chem. Phys. Zl(1977) 61. E. PoUak. Chem. Phys. 22 (19?7) 151. H. Kaplan and R.D. Levine. Chem. Phys. 18 (1976) 103. Bf. Tamir and R.D. Levine, Chem. Phys. 18 (1976) 125. R.B. Bernstein and R.D. Levine, Advan. At. Mol. Phys. 11 (1975) 215. M.D. PattengiU and R.B. Bernstein, J. Chem. Phys. 65 (1976) 4007.
516
ROTATIONAL
CHEMICAL PHYSICS LElTERS
1 May 1977
SURE’RISAL PLOTS FOR H + H,, A SIMPLE THEORY
E. POLLAK Department of Chemistry. Columbia University. New York City, New York 10027, USA Received i0 January 1977 The linear surprisal plots for rotational cross sections of the H + Hz exchange reaction are shown to be a manifestation of the transfer of momentum constraint. As a result, it is shown that the complete reactive cross section matrix at a @en total energy is a function of the variable Ej + Ej*_ The surprisal of the matrix elements as a function of this variable is Linear as predicted from the constraint.
1. Introduction Surprisal analysis of rotational
can well approximate oi’,i(E)
= oy,i(E)
the inelastic exp(-
energy transfer cross sections
cross sections
in atom diatom by an exponential gap law
f0 - 6 R !+!$- Ej’ I/E) .
collisions
[l, 21 has shown that one -01
E is the total energy of reactants, j,j’ the initial and final rotational states, oy,@) is the prior rate, 0, the rotational temperature parameter and IO an energy dependent constant. Clearly, if Ei = 0 i.e. one starts a reaction with the reactant in thej = 0 state, one would expect to find a simple linear surprisal in the variable E,*. Wyatt [31 has demonstrated that for quantum mechanical calculations of H + Hz (i = 0) + HZ 0”) f H one does indeed obtain the linear plot. Recently, Schatz and Kuppermann [4] have investigated the atom exchange reaction for several different irMaI as a function of j’ (j’ = 04) for each irtitkdj, j values. They plotted the surprisal I(/“, j) = - ln[o~~,i(E)/o~,i(E)] j = 0,4. Their results show that for each initial j one obtains a linear plot in the variable E,-_ This fact is in obvious conflict with the gap law which predicts a change of slope of the surprisal around the point Ej = ET _ Even if one assumes that eq. (1) is invalid for reactive systems, one remains with the puzzling fact that the quanturn mechanical calculations give linear surprisal plots in the variable Ej*_ Obviously the system is governed by a dynamical constraint and one would like to find a theory for it. The object of this letter is to resolve the above two problems. We intend to show that the transfer of momentum constraint [5-8) alone governs the behaviour of the H + H, exchange reaction. Since all transitions are limited to energy disposal suffices for a description of the the ground vibrational state of Hz, knowledge of the translational rotational energy distribution. In section 2 we review briefly the theory of the transfer of momentum constraint. In section 3 we compare the theory to the calculations of Schatz and Kuppermann. Specifically, we show that the complete rotational cross section matrix can be well characterized by the variable Ei + Ei*, as shown in fig. I - i3dly in section 4 we discuss briefly the scope and limitations of the theory.
2. The transfer of momentum
constraint
Originally, the transfer of momentum constraint was applied only to the normalized products (charm& b) translational energy distribution P(ET*) [5,7] - It was found that P(ET-) could be well represented as
CHEMICAL
Volume 47, number 3 P(ETp) = Po(ET.)
exp [-A,
1 May 1977
PHYSICS LETTERS
iEg2 - sEy2 - R 1/2)2 - A,] .
(2)
XT. an energy dependent parameter, s, RLi2 are constants and h ensures normalP”(+) is the p rior distribution, ization. In principle, a reaction is governed by the constraint of least momentum transfer in both directions. Therefore for the opposite direction one can write P(ET) = P” (ET) exp {-XT’ [_/Zg2 - s’E$~ - (R’)‘fl J 2 - A;) .
(3)
In eqs. (2) and (3) primed values refer to channel b while unprimed values refer to channel a (the original reactants). P(ET) and P(ET.) are not totally independent - they are connected because of microscopic reversibility_ . It has been shown theoreticaliy and verified experimentally that eqs. (2) and (3) coupled with the principIe of microscopic reversibility give an explicit expression for the yield matrix elements [7-lo]. Specifically, at a given reactants energy E the reactive part cf the symmetric yield matrix [I I J Y(ETn, ET; E) is given by the expression 17.81
Y(E+.ET;0 = Y”(ET4’T;E) X exp {--XTETp - AT, ET f (ETET,)1/2 (AT + s’+) Here Y”(ET,,
+ AT, (_R’ET)1/2] ) .
+ 2 [AT (RETs)~;~
ET; E) is the prior value of the yield matrix element
(4)
[ 1,l l] (5)
Y”(ET’.ET;E)=hQpCET)~(ET,)p(x) is the transIationaI density of states and Q is an energy dependent scopic reversibility implies the folIowing relationship [7,8]
constant.
tn addition
to eq_ (4), rnicro-
XTS =X7’s’ -
(6)
The reaction we will be dealing with is the symmetric atom exchange reaction. Since channels a and b are physically ind.istinguishabIe it follows that s = s’ and X, = XT*. For a symmetric exchange reaction one also expects [7,8] that R1j2 = (R’)lj2 = 0 (i.e. the net distortion on a symmetric potential energy surface is to a fust approximation expected to equal zero). As a result, eqs. (2)-(4) become much more simplified.
3. Application
of the theory
to H + Hz
Eq. (2) suffices for explaining the linearity of the rotational surprisal plots. To see this, one must only remember that the quantum mechanical calculations have only the lowest vyorational channel open. As a result, by energy conservation, the translational energy distribution is the rotational energy distribution_ Energy conservation means ET+Ej
=E=ET.+Ej,_
(7)
Here,j,i’ refer to reactants and products last section) into eq. (2) gives
rotational
states respectiveIy.
P(q)
= P” (ET) exp {-XT [(E - Eip)lj2 - s(E - Ei>li2 12 - b}
I&*)
= -In[P(Ef)/P”(Ei*)l
Introducing
eq. (7) (and the results of the
,
(8)
or = AT [E(l + s2) - Ei* - s2Ej-
2s(E - Ej*)1’2(.
If EjfE and Ej*/E are smaller than 0.4 then we can use the following (1 -EjlE)“‘(l
514
-Ej*/E)‘12
~ 1 -EjlzE-~/2E-
excellent
- Ej)1’2] + h -
(9)
approximation (10)
Volume 41, number 3 Inserting
CHEMICAL PHYSICS LETTERS
this approximation
I(Ej*)=X-r(l
1 May I977
in (9) one finds
-s)[(l
-s)E+(+Ej)]
+hu.
00
Since AT is dependent in principle on E, the slope of the surprisal XT(s - 1) can be dependent on E asfoundby _ the quantum mechanical calculation [4] _On the other hand, for a given E and different initial j’s the slope is constant. This is also consistent with the calculation 141. In obtaining eq. (11) we have utilized only normalized product distributions. A stronger result is obtained if one uses eq. (4) One can compare cross sections for different initial i states. The ctoss section is proportional [I ] to the yield [with a proportionality factor Irl$ (2j + l)] . For H f Hz eq. (4) thus gives Uj’_jQ
UT,j(E)
=
exp [ - hT (2E - Ej - Ej*- 2s(E- Ej)"'(E- E,,)"'] -
cry,i is the prior cross section o~,j(~ S is a constant ri_,i(~
and is given by
= S[(E- Ei~)"'/(E-Ej)"'](2i'+ 1) (S = 4n2R/h3). inserting = - In [UI’,j
(121
Q/U?,jQI
(E3),
eq. (11) into (12) one finds for the surprisal
= A=(1 - S)[x - Cq+ Ej.11 -
(14
We have thus found, that the transfer of momentum constraint as applied to H + Hz predicts a linear surprisal for the complete reactive cross section matrix as a function of the variable (Ej+ Ejr)To check this prediction we have taken all cross sections given in fig. 22 of ref. [4] and plotted their surprisal as a function of Ej f Ef-This is shown in fig. 1. The most notable aspect of this figure is its disagreement with eq. (I), the exponential gap law. Eq_ (14) also predicts that the slope of the surprisal should have opposite signs if plotted once as a function of El f Ef with E constant and once as a function of E with Ej + Ejvconstant. Fig. 2 verifies this prediction. Note that the su~risal is not linear as a function of E,XT depends quite strongly on E [4].
Fig. 2. The rotational surprisaI for j = j’ = 0 as a function of the available energy E (defined as in fig. 1). The sotid points are taken from fig. 23 of ref. [4]_
Fig- I. The rotational surprisal [eq. (I4)] is plotted as a func-
tion of fj + fj* for H + Hz 62 * Hz (i’) f H. fj is defied as Ej/E and similarly fi*- Here, E isthe avaiIabIeenerg rebtive to the bottom of the potential well. The solid points are taken
0
I
I
I
I
I
01
02
a3
04
05
f, ‘r,.
from fig. 22 of ref. 141. (To obtain the cross sectionaI srrrpri.saIone milst multipty by (1 - Glr/2 .) E = 0.6 eV; The Iine is _. a least square fit of eq. (14), AT(1 - s) = 33 eV-’ _ Since me . pnor cross section is detimcd up to an arbitrary constant K only, we were not able to check the ratio of hT(t - s) to Io,e to see whether it conforms to eq. (14).
Volume 47, number 3
CHEMICALPHYSICSLETTERS
1 May
1977
4. Discussion
The exponential gap law [eq. (I)] h as b een empirically documented for diverse systems [I, 2,121. Even so, an unconstrained reaction would not (by defmition) show an exponential gap behatiour. This seems to be the case for the H + HZ reaction. The transfer of momentum constraint suffices for a description of the reactive cross sections and no additional constraint need be invoked. For a general reaction, eq. (7) is not valid. Vibrational energy plays a role. Furthermore, as a result of the large barrier in the H3 potential surface one needs a considerable amount of initial translational energy to overcome the barrier. Therefore the fraction of available energy in initial and final rotation is small and so one can use eq. (IO). This is not so in the more general case, and surely not so for inelastic collisions. _ It should be stressed that eq. (4) describes the yield only in the poor persons limit [7]. In deriving eq. (4) one explicitly ‘assumes a microcanonical distribution of ah other independent variables. For the H3 reaction there is only one other variable Ep and it is dependent via the total energy E. Therefore (4) gives also the rotational yield matrix. Since in general this is not the case one should not generalize the results of section 3. On the other hand, fig. 1 is an additional piece of experimental evidence that the transfer of momentum constraint is quite general. It has again been verified that eq. (4) is a good representation of the yield matrix. Although we have found that at a given reactants total energy E, the H f HZ atom exchange reaction cannot be characterized by an exponential gap law, one would expect that thermal averaging of the reactants would give an exponential gap law. This has been shown to be the case for vibrational energy transfer in reactive and non reactive collisions. Using the methods of ref. [S] it is easily seen that Boltzmann averaging of the initia1 translational energy in eq. (12) will yield an approximate exponential gap law.
References R.D. Levine, R.B. Bernstein, P. Kahana, I. Procaccia and E.T. Upchurch, J. Chem. Phys. 64 (1976) 796. I. Procacch and R.D. Levine, J. Chem. Phys. 64 (1976) 808. R.E. Wyatt,Chem. Phys. Letters 34 (1975) 167. G.C. Schatz and A. Kuppermann, 3. Chem. Phys. 65 (1976) 4668. A. Wri, E. Pollak, R. Kosioff and R.D. Levine, Chem. Phys. Letters 33 (1975) 201. A. Kafri, Y. Shimoni, R.D. Levine and S. Alexander, Chem. Phys. 13 (1976) 323. E. PoUak and R.D. Levine, Chem. Phys. Zl(1977) 61. E. PoUak. Chem. Phys. 22 (19?7) 151. H. Kaplan and R.D. Levine. Chem. Phys. 18 (1976) 103. Bf. Tamir and R.D. Levine, Chem. Phys. 18 (1976) 125. R.B. Bernstein and R.D. Levine, Advan. At. Mol. Phys. 11 (1975) 215. M.D. PattengiU and R.B. Bernstein, J. Chem. Phys. 65 (1976) 4007.
516