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Dynamics of hydrogen atom and proton transfer reactions. Simplified analytic two-state formulae
CHEMICAL
Volume 101, number 6
4 November 1983
PHYSICS LETTERS
DYNAMICS OF HYDROGEN ATOM AND PROTON TRANSFER SIMPLIFIED ANALYTIC TWO-STATE FORMULAE*
REACTIONS_
Vasil K. BABAMOV * Cents de Graduados. Institrrto TecJ~o~ogicode T@cana. Apdo Postal II66. TQuana. B. C_.Mexico
and Vicente LOPEZ g and R-A. MARCUS XOWSLaboratory of Chemical Physics. CaIifomia Itistitl~te of TechnoIog_v. Pasadena, Califorrlia 9I IX.
USA
Received 10 August 1983
Analytical formulae for the threshold region of the reaction probability curve for a nearly degenerate H-atom transfer XC @en and shown to yield good agreement with e.\act results for a t\\o-state model system. Application to H and H+ transfer in complcl systems is discussed.
1. Introduction The exchange dynamics of hydrogen atont and proton transfer reactions between two heavy particles has been studied recently [l-13] *_ For the case of exact degeneracy of the reactant and product vibrational states which occurs in syntntetric exchange reactions a new kind of vibrationally adiabatic approximation in reactive collisions was introduced [I] and used to derive analytical formulae for the reaction probability [I] _Subsequent accurate numerical calculations [6,7] on symmetric H-atom exchange dentonstrated both the high degree of vibrational adiabaticity of the symmetric H-transfer reactions postulated in ref. [l] and the high accuracy*” of the analytical fomtulae. The vibrationally
* Contribution No. 6884. * Also at California Institute of Technolog. * Permanent address: Departamento de Qu?inica Fisica y Quiiniea Cuintin, Universidad AutBnoma de hladrid. Cantoblanco, Madrid-34, Spain. * The analogy of the coordinates used in electron transfer in atomic collisions with those for H-transfer between heavy atoms was discussed in ref. 121. The same coordinates were used in ref. [3] to study the tunneling in H + Hz_ ** The shift of the results of eq. (4.10) in ref. [ 11 relative to the accurate coupled channel ones [6] was misprinted in ref. [ 1 l] as 0.5 kcal/mol. It should read 0.5 x 10m3 kcal/mol.
0 009-2614/83/0000-0000/S
03.00 0 1983 North-Holland
adiabatic analysis in ref. [I] has been extensively applied in exploring various aspects of the dynamics of H-atom transfer reactions [6 --IO] as well as to indicate the possibility of forming molecules on repulsive potential energy surfaces [ 14-16]_ The symmetric exchauge treatment [I] was recently extended [ 1 l] to the closely related problem of the dynamics of H-atom exchange between two nearly degenerate vibrational states in an asymmetric reaction AH +B+A+HB*. It was shown [l l] that for this case the reaction dynamics can be represented in terms of two suitably distorted nearly degenerate vibrational states between which the eschange occurs. The resulting two coupled differential equations [ 11, which sintulate the dynamics, were solved numerically [ 121 for a model BrH + Cl + Br + HCl reaction and were shown to yield results in good agreement with accurate multichannel calculations [ 131 for the saute problem. A perturbative (DWBA) approximate treatment of the two coupled equations valid in the reaction threshold region was also developed [l 1] _ For reactions with some (at least few kcal/mol) activation energy the threshold region determines the kinetics of low and moderate temperature reactions_ In section 2 we calculate a l
The Sate parameters, listed as 0.01,0.02,
and 0.00 in ref. [ 111, refer to HBr, HCI, and BrCl, respectively.
507
Boltzmann-weighted
4 November 1983
CHEMICAL PHYSKS LETTERS
Vo’olume 101._number 6
reaction probability
PC(T) ob-
tained from the DWBA results in ref. [ 1 I] and com-
pare it with the one obtained from accurate numerical solution of the two-state approximation for the BrH + Ci model system_ A P,,-(T) is useful for testing these DWBA results since it is more representative than I’&?) of the accuracy which they can be espected to yield when used for calculations of reaction rates in actual physical processes. An outline of a possible application of the formulae developed here and in ref. [ 1 I] to reactions of H and Hf transfer between two complex molecules and to reactions in condensed phases is given in section3. Vibrational adiabaticity is discussed in section 4. An approximate anaiy~ical expression for the reaction probability P..(E). accurate over the whole energy range B where the two-state approsinlation is valid. is given in a later paper [ 13]_
(Jr-$ = Vii since the vibrational basis set is real. The method used in the present study to obtain the matrix elements F:,, Yip and Yii was an improved version [ 131 of the methods in ref. [ 1 l] ; however, use of the latter would have made little difference in the threshold region). The coupled equations [eq. (2)] were solved in ref. [I I] in the tunneling region by means of a perturbative (DWBA) treatment to give the following expression for the reactive transition probability (cf. eqs. (5.5) and (3.9) of ref. [ If] ):
Jpii= (kT$
= [exp(-
A2/2aF)]z
P S,m = [(2ir/ff~)1~2 Gj exp (ar3/24F)J2.
e3.s = Ffjy & F/$!,
F$ = F$ (i #j), in the adiabatic
where the notation is that of refs. [I,1 I]
omitted. It was shown in ref. [I 11 that instead of solving eq_ (1) directly (e.g. ref. [ 171) it is more appropriate to utilize the diagonal elements in eq. (I) for generating
a pair of closefy related but not exactly equivalent coupled equations in the diabatic representation: d’/dp’
+ “ji(p)
E-5 d’/dP2
+ Gus-
1-4
50s
- E] \ki(p) = t’Tij\ki(p), E] ?tj(p)=
~i\ki(~)
(2)
(4)
representa-
Av=Qii=O,
(5)
where the a refers to the plus sign and the s to the minus. F$$ in (3) is defined below. Eq. (3) was obtain-
ed by usmg the linear approximation for V$ and for In F$ (i #j) as a function of p near the turning point. i.e. by setting
d In Vu/dp = (d In I~j/dp)po (1)
(3b)
This P,,, is the DWBA appro~iillation to the reactive transitibn probability for the corresponding symmetric problem defined by the I/-matrix elements P’$:
in eq. (2), or alternatively tion (eq. (1)) by
11 was shown earlier that the treatment of the reactive exchange dynamics in a collinear AH -I- B + A + HB re.rction can. in the presence of an HB vibrational state nearly degenerate with the initial AH one, be represented approsimately by a pair of coupled ordinary differential equations f 1.1 l]
t3a)
where
F/j = F$ = + ( FTii+ Fl,)
2. Test of the relevant aspects of the “esponentiated” DWBA formula for evaluating reaction rates
P,,,
and by approximating - F:#r) by its value 4 = ~/ii&-j) - frj+Q))
= - CY.
the resonance
defect
(6) Y&J)
(71
at the classical turning point PO for the p-motion on a potential V;(p) C$ is Vij(po)_ Eq. (3) has the disadvantage typical of DWBA formulae of yieIding probabilities higher than unity at high energies. To avoid this we can, in analogy to frequent practice in using DWBA results for calculating inelastic transition probabilities, limit the probability by setting max(P~rn)=
1.
(8)
Volume 101, number 6
CHEMICAL
PHYSICS
4 November 1983
LETTERS
.
‘A
-=-
I
\r
‘h \
Log p,,(T) -15
-
*As,
l \
i t t -25
‘k
-
‘0.
I
I
0
20
3
1000/T(K) Fig. 2. Plot of the logto of the Boltzmann averaged reaction probability POT (T) versus the inverse temperature. The full
line is obtained from the accurate numericalsolution of eq.
(2). the circles are obtained using P#E)
E(kml/moll Fig. 1. Plot of Paz versus total energy of the reactants for the model system BrH + CI + Br + HCI. The full line is the result of the numerical solution of eq. (2) and the dashed line is the approximation given by eq. (9).
in eq. (9) and the triangles are obtained from eq. (8) plus eq. (3).
PO(T) = i Pii
exp(-E/kT)
dE/kT_
(10)
0
A better approximation can be obtained by “exponentiating” [11,18] the symmetric part of the Pq, namely by writing* Pii
= exp(-
A2/~)
sin?P$:.
(9)
where Pwm is defined by eq. (3). At low values of P sym eq. (9) reduces to eq. (3a). Eq. (9) gives excellent results around the reaction threshold, as seen in fig_ 1, but not at energies much higher than the threshold due to the approximations of a linear Vz (p) and ln Vii(p) (eq. (6)). Fortunately, reaction rates for most reactions that have at least some activation energy are determined at low to moderate temperatures by the probabilities near the threshold_ In order to examine the utility of eq. (9), and of eq. (3) plus (8), for calculating reaction rates for physical processes which involve H-atom or proton transfer at low and moderate temperatures it is convenient to test its accuracy by calculating a dimensionless Boltzmannaveraged reaction probability, * Typically, the entire DWBA Pq has been ‘exponentiatcd” rather than, as in the present case, only the symmetric
part.
The Pii obtained using eq. (9) for P&?) and the one obtained using eq. (3) plus (8), are presented in fig. 2, together with the PO(T) obtained from the numerical solution of the coupled equations (2). The results are in excellent agreement. Pii above is related to a “one-dimensional rate constant” k;(T) by Pii = k;(T) hQ/kT, where Q is the partition function of the reactants. The linear restrictions (es_ (6)) which limit the validity of eq. (9) to the threshold region can be removed by using the exact two-state analog of Pwm, given by eq. (4.10) ofref. [I]. We then have Pii = exp(-
AZ/&) sin’ (Es - Ea),
(11)
where $jsa are the elastic phase shifts of eq. (1) with the matrix elements defmed by eq. (5). However, obtaining accurate transition probabilities at higher energies also requires extending the procedures for evaluating the matrix elements Vii(p) in ref. [l l] to smaller p_ Such extensions, as well as including the effect of variation of the resonance defect Vii - Vii with p, are discussed in detail elsewhere [13]. 509
Volume 101. number 6 3. Extension
to more complex systems
The question is frequently raised as to whether s0me of the “reorganization” of coordinates prior to an electron transfer in solution also occurs for H and for II+ transfers. (An early review of electron transfer theor! is +ZXI in ref. [ 19]_) We consider this point in the following discussion. IVe first note that the factor multiplying P,,,, in eq. (3) can be sho\vn 1131 to have the form exp[l(T4)2]. 11hsre 7 is roughly the -‘duration” of the reactive collision. (The units are such thst fi = l_) This factor top [ - (TA)~] serves as a counterpart to the well-known DiT3c delta iunction 6 (A) which appears in the Golden Rule formula for electronic transitions. e-g_ ref. [20]. including electron transfer reactions. e-g_ refs. [ 19.7 l] _ Beci~usc11112 ratio of the heavy to the light particles in 11 or 11* transfers is much closer lo unity than it is in electron transfers_ this esp[(i4)?] is a much less pc.krd
function
of the is than IS 6 (A)_
Nerertlie1ess.a
suitable fluctwlion of other coordinatesq. prior to the II rrxnsfcr. \\ill reduce exp[4:(47)!aF]. (The 4 in ~(1. (7) can IION depend on Q.) For cxanlple, the state-to-state contribution X-,(7‘) to the rate constant is givrn 1~) eq. ( 12). using ar_runirnts analogous to rlmsc
used in ref. [X]
_
(121
when the q coordinates are treated classical1y.j is the qu.uituni state of the H motion in the product that is closest in energy to the initial (ith) quantum state of the11 motion in the reactants_ In eq_ (13)Pii(E_ q) is $ven by rq. (9) and depends on 4 mainly via A(q)_ E
include5 nor only the energy along the reaction coordinate hU1 also I’ji(p. q). which replaces the I’&) present previously and is the effective potential energy of the system when the resctants are in the ith quan1un1 state for the 11 motion. Q is the partition function of the re.warus. and Q[, is the momentum part of the
classical partition function of the q degrees of freedom_ in the q coordinates which serve to rewill increase Pij(E) and hence be favored for enhancing the reaction rate provided the?, do not increase IT&, q) too much. Typically_ Fluctuations
duce 111e value of 4(q)
one expects that eq. (I?) for the reaction rate will be dominated by the Pwm factor in eq. (3), which is -_ 510
4 November 1983
CHEMICAL PHYSICS LETTERS
smaller for D than for H transfers_ The expectation that the exp [- (TA)?] term is reduced in importance by a suitable and relatively minor fluctuation in coordinates r~ is indicated by the apparent absence of unusual isotope effects (D transfer faster than H transfer)
in the literature_ If this were not the case it might be possible to find a system in which the D-transfer rate esceeded the H-transfer rate because. accidentally, A was small for the deuterium case.
4. Discussion The high degree of vibrational adiabaticity has been one of the most striking features in accumte calculations [6 71 of symmetric heavy-light-heavy (HLH) collinear reactive collisions. From an experimentalist’s point of view vibrational adiabaticity may simply mean that the vibrational quantum number of the products equals that of the reactants. It is manifested in the results of the accurate numerical calculations by the absence of vibrational esciration
when there are higher vibrational
states open. When there are no excited vibrational states open vibrational adiabaticity is manifested by the close validity of the approximate formulae in ref. [I] derived on the basis of vibrational adiabaticity. The vibrational adiabaticity is also evident in the preservation of the vibrational action in classical trajectory calculations_ As noted in ref. [I]. however. the dynamical nature adiabaticity in symmetric HLH reactions is radically different from that in the adiabatic model used by a number of investigators [23,24] and which was termed [24] vibrationally adiabatic. In that vibrationally adiabatic model [23.X4] the system moves on a single effective potential energy surface. This motion was discussed in detail using natural collision coordinates [Xl. In the new type of vibrational adiabaticity presented in ref. [ 11, the reaction is adiabatic in the sense that the quantum state of reactants is coupled only to a quantum state of the products with the same quantum number. However, two distinct quantum states and two effective adiabatic potentials are needed* of the vibrational
to discuss the reaction. Analogously, in the present
case of a non-_symmetric
* In the case of ref. [ 11. symmetric and antisymmetric states were used for the reaction AH + A - A f- HA and would also be necessary for an exact resonance AH + B + A + HB reaction.
Volume 101, number 6
CHEhlICAL PHYSICS LETTERS
reaction the vibrational state of the reactants is nearly degenerate with, and is coupled almost exclusively to, a vibrational state of the products which has a different quantum number_ As a result the reaction is almost completely vibrationally nonadiabatic, i.e. it leads to a vibrational population inversion in the products. Both the dynamics of H-atom transfer and the present treatment are much more akin to those of the charge transfer processes in atomic collisions than to the vibrationally adiabatic treatment of chemical reactions in terms of natural collision coordinates.
181 V. Aquikmti, S. Cavalli and A. Lagana, Chem. Pbys. Let-
1161
ters 93 (1982) 179; V. Aquilanti. G. Grossi and A. Lagana. Chem. Phys. Letlets 93 (1982) 174; J. hfol. Struct. 93 (1983) 319. D.K. Bondi. J.N.L. Connor, B.C. Garrett and D-G. Truhlar, J. Chem. Phys. 78 (1983) 5981: B.C. Garrett, D.G. Truhlar. A-F. Wagner and T-H. Dunning Jr., J. Cbem. Phys. 78 (1963) 4400. J.M. Launay and hl. Le Doumeuf, J. Phys. B15 (1962) L455. V-K. Babamov, V. Lopez and R-A. Marcus, J. Chem. Phys. 78 (1983) 5671. N. Abu Salbi. V. Lopez, V.K. Babamov, D. Kouri and R.A. hlarcus, J. Chem. Phys., to be submitted. VK Babamov. V. Lopez and R.A. Wucus. to be submitted_ E. Pollak. in: 15th Jerusalem Symposium on Quantum Chemistry and Biochemistry: Intramolecular Dynamics. ed$. B. Pullman and J. Jortner (Reidel, Dordrecbt, 1982) p. 1: J. Chem. Phys. 78 (19S3) 112s; Cbem. Phys. Letters 94 (1983) 85. J. Manz. R. hleyer. E. Polla% and J. R8melt, Chem. Phys. Letters 93 (1982) 184. J. hlanz. R. hieyer and J. Remelt, Chem. Phys. Letters 96 (1983) 607. DC Clxy zmd J.N.L. Connor, Chem. Phys. Letters 94
[I71
V-K. Babamov.
I91
[lOI IllI 1121 1131
Acknowledgement We are pleased to acknowledge
r 141
the support of this
research by a grant from the National Science Foundation_ and (to VL) a Fulbright/Spanish Ministry of Education and Science Fellowship. We thank Dr. J.N.L.
rw
Connor for his helpful comments.
(19S3)
References
[ 1] V.K. Babamov and R.A. Xl.mzus,J. Cilem. I’hys. 74 (1951) 1790. [Z] C. Nyeland and T-A. Bak, Trans. Faraday Sot. 61 (1965) 1293. 131 H.S. Johnston and D. Rapp, J. Am. Chem. Sot. 83 (1961) 1: S.G. Christov and Z.L. Ceorgiev, J. Phys. Chem. 75 (1971) 1748. [4] EJ. Shipsey. J. Chem. Phys. 58 (1973) 232: 11. Bax, J. Chem. Phys. 62 (1975)
305.
[5] [6]
Al. Ya. Ovchinnikova, Chem. Phys. 36 (1979) 65. J-A. Kaye and A. Kuppermann, Chem. Phys. Letters 77 (1981) 573; 92 (1981) 574.
[7]
J. Manz and J. RSmelt, Chem. Phys. Letters 81 (19Sl) 179; 76 (1980) 337; J. hlanz, E. Pollak and J. RBmelt, Chem. Phys. Letters 86 (1982) 26; C. Hiller. J. Manz,W.H. Miller and J. RBmelt, J. Chem. Phys. 78 (1983) 3850; D.K. Bondi, J.N.L. Connor, J. hlanz and J. Rbmelt, Mol. Phys., to be published; J. R8melt. Chem. Phys. 79 (1983) 197.
4 November 1983
Sl.
J. Chem. Phys. 69 (1978)
I181 R.D. Levine, Chem. Phys. Letters 2 (196s) 22 (1971)
3414. 76; Mol. Pbys.
497.
1191 R.A. llarcus, Ann. Rev. Phys. Chem. 15 (1964) 155. 1701 L.D. Landau and E.M. Lifshitz. Quantum mechanics (Addison Wesley. Reading. 1958) p_ 147. an advanced treatise. VoL9B,eds. II. Eyring. D. Henderson and W_ Jost (Academic Press, New York, 1970) p_ 985. R-4. hlarcus, J. Chem. Phys. 45 (1966) 213s. 123 1 J-0. Hirschfelder and E. Wigner, 1. Chem_ Phys. 7 (1939) 616; WA. Elk-on and J-O_ Hirschfelder, J. Chem. Phys. 30 (1959) 1426; L. Hofacker, 2. Naturforsch. 18a (1963) 607. [14 I R.A. hlarcus, J. Chem. Phys. 43 (1965) 1598. [25 1 R.A. hlarcus, J. Chem. Pbqs. 45 (1966) 4493,450O; 49 (1968) 2610.
1311 V-G. Levich, in: Physical chemistry:
511
Volume 101, number 6
4 November 1983
PHYSICS LETTERS
DYNAMICS OF HYDROGEN ATOM AND PROTON TRANSFER SIMPLIFIED ANALYTIC TWO-STATE FORMULAE*
REACTIONS_
Vasil K. BABAMOV * Cents de Graduados. Institrrto TecJ~o~ogicode T@cana. Apdo Postal II66. TQuana. B. C_.Mexico
and Vicente LOPEZ g and R-A. MARCUS XOWSLaboratory of Chemical Physics. CaIifomia Itistitl~te of TechnoIog_v. Pasadena, Califorrlia 9I IX.
USA
Received 10 August 1983
Analytical formulae for the threshold region of the reaction probability curve for a nearly degenerate H-atom transfer XC @en and shown to yield good agreement with e.\act results for a t\\o-state model system. Application to H and H+ transfer in complcl systems is discussed.
1. Introduction The exchange dynamics of hydrogen atont and proton transfer reactions between two heavy particles has been studied recently [l-13] *_ For the case of exact degeneracy of the reactant and product vibrational states which occurs in syntntetric exchange reactions a new kind of vibrationally adiabatic approximation in reactive collisions was introduced [I] and used to derive analytical formulae for the reaction probability [I] _Subsequent accurate numerical calculations [6,7] on symmetric H-atom exchange dentonstrated both the high degree of vibrational adiabaticity of the symmetric H-transfer reactions postulated in ref. [l] and the high accuracy*” of the analytical fomtulae. The vibrationally
* Contribution No. 6884. * Also at California Institute of Technolog. * Permanent address: Departamento de Qu?inica Fisica y Quiiniea Cuintin, Universidad AutBnoma de hladrid. Cantoblanco, Madrid-34, Spain. * The analogy of the coordinates used in electron transfer in atomic collisions with those for H-transfer between heavy atoms was discussed in ref. 121. The same coordinates were used in ref. [3] to study the tunneling in H + Hz_ ** The shift of the results of eq. (4.10) in ref. [ 11 relative to the accurate coupled channel ones [6] was misprinted in ref. [ 1 l] as 0.5 kcal/mol. It should read 0.5 x 10m3 kcal/mol.
0 009-2614/83/0000-0000/S
03.00 0 1983 North-Holland
adiabatic analysis in ref. [I] has been extensively applied in exploring various aspects of the dynamics of H-atom transfer reactions [6 --IO] as well as to indicate the possibility of forming molecules on repulsive potential energy surfaces [ 14-16]_ The symmetric exchauge treatment [I] was recently extended [ 1 l] to the closely related problem of the dynamics of H-atom exchange between two nearly degenerate vibrational states in an asymmetric reaction AH +B+A+HB*. It was shown [l l] that for this case the reaction dynamics can be represented in terms of two suitably distorted nearly degenerate vibrational states between which the eschange occurs. The resulting two coupled differential equations [ 11, which sintulate the dynamics, were solved numerically [ 121 for a model BrH + Cl + Br + HCl reaction and were shown to yield results in good agreement with accurate multichannel calculations [ 131 for the saute problem. A perturbative (DWBA) approximate treatment of the two coupled equations valid in the reaction threshold region was also developed [l 1] _ For reactions with some (at least few kcal/mol) activation energy the threshold region determines the kinetics of low and moderate temperature reactions_ In section 2 we calculate a l
The Sate parameters, listed as 0.01,0.02,
and 0.00 in ref. [ 111, refer to HBr, HCI, and BrCl, respectively.
507
Boltzmann-weighted
4 November 1983
CHEMICAL PHYSKS LETTERS
Vo’olume 101._number 6
reaction probability
PC(T) ob-
tained from the DWBA results in ref. [ 1 I] and com-
pare it with the one obtained from accurate numerical solution of the two-state approximation for the BrH + Ci model system_ A P,,-(T) is useful for testing these DWBA results since it is more representative than I’&?) of the accuracy which they can be espected to yield when used for calculations of reaction rates in actual physical processes. An outline of a possible application of the formulae developed here and in ref. [ 1 I] to reactions of H and Hf transfer between two complex molecules and to reactions in condensed phases is given in section3. Vibrational adiabaticity is discussed in section 4. An approximate anaiy~ical expression for the reaction probability P..(E). accurate over the whole energy range B where the two-state approsinlation is valid. is given in a later paper [ 13]_
(Jr-$ = Vii since the vibrational basis set is real. The method used in the present study to obtain the matrix elements F:,, Yip and Yii was an improved version [ 131 of the methods in ref. [ 1 l] ; however, use of the latter would have made little difference in the threshold region). The coupled equations [eq. (2)] were solved in ref. [I I] in the tunneling region by means of a perturbative (DWBA) treatment to give the following expression for the reactive transition probability (cf. eqs. (5.5) and (3.9) of ref. [ If] ):
Jpii= (kT$
= [exp(-
A2/2aF)]z
P S,m = [(2ir/ff~)1~2 Gj exp (ar3/24F)J2.
e3.s = Ffjy & F/$!,
F$ = F$ (i #j), in the adiabatic
where the notation is that of refs. [I,1 I]
omitted. It was shown in ref. [I 11 that instead of solving eq_ (1) directly (e.g. ref. [ 171) it is more appropriate to utilize the diagonal elements in eq. (I) for generating
a pair of closefy related but not exactly equivalent coupled equations in the diabatic representation: d’/dp’
+ “ji(p)
E-5 d’/dP2
+ Gus-
1-4
50s
- E] \ki(p) = t’Tij\ki(p), E] ?tj(p)=
~i\ki(~)
(2)
(4)
representa-
Av=Qii=O,
(5)
where the a refers to the plus sign and the s to the minus. F$$ in (3) is defined below. Eq. (3) was obtain-
ed by usmg the linear approximation for V$ and for In F$ (i #j) as a function of p near the turning point. i.e. by setting
d In Vu/dp = (d In I~j/dp)po (1)
(3b)
This P,,, is the DWBA appro~iillation to the reactive transitibn probability for the corresponding symmetric problem defined by the I/-matrix elements P’$:
in eq. (2), or alternatively tion (eq. (1)) by
11 was shown earlier that the treatment of the reactive exchange dynamics in a collinear AH -I- B + A + HB re.rction can. in the presence of an HB vibrational state nearly degenerate with the initial AH one, be represented approsimately by a pair of coupled ordinary differential equations f 1.1 l]
t3a)
where
F/j = F$ = + ( FTii+ Fl,)
2. Test of the relevant aspects of the “esponentiated” DWBA formula for evaluating reaction rates
P,,,
and by approximating - F:#r) by its value 4 = ~/ii&-j) - frj+Q))
= - CY.
the resonance
defect
(6) Y&J)
(71
at the classical turning point PO for the p-motion on a potential V;(p) C$ is Vij(po)_ Eq. (3) has the disadvantage typical of DWBA formulae of yieIding probabilities higher than unity at high energies. To avoid this we can, in analogy to frequent practice in using DWBA results for calculating inelastic transition probabilities, limit the probability by setting max(P~rn)=
1.
(8)
Volume 101, number 6
CHEMICAL
PHYSICS
4 November 1983
LETTERS
.
‘A
-=-
I
\r
‘h \
Log p,,(T) -15
-
*As,
l \
i t t -25
‘k
-
‘0.
I
I
0
20
3
1000/T(K) Fig. 2. Plot of the logto of the Boltzmann averaged reaction probability POT (T) versus the inverse temperature. The full
line is obtained from the accurate numericalsolution of eq.
(2). the circles are obtained using P#E)
E(kml/moll Fig. 1. Plot of Paz versus total energy of the reactants for the model system BrH + CI + Br + HCI. The full line is the result of the numerical solution of eq. (2) and the dashed line is the approximation given by eq. (9).
in eq. (9) and the triangles are obtained from eq. (8) plus eq. (3).
PO(T) = i Pii
exp(-E/kT)
dE/kT_
(10)
0
A better approximation can be obtained by “exponentiating” [11,18] the symmetric part of the Pq, namely by writing* Pii
= exp(-
A2/~)
sin?P$:.
(9)
where Pwm is defined by eq. (3). At low values of P sym eq. (9) reduces to eq. (3a). Eq. (9) gives excellent results around the reaction threshold, as seen in fig_ 1, but not at energies much higher than the threshold due to the approximations of a linear Vz (p) and ln Vii(p) (eq. (6)). Fortunately, reaction rates for most reactions that have at least some activation energy are determined at low to moderate temperatures by the probabilities near the threshold_ In order to examine the utility of eq. (9), and of eq. (3) plus (8), for calculating reaction rates for physical processes which involve H-atom or proton transfer at low and moderate temperatures it is convenient to test its accuracy by calculating a dimensionless Boltzmannaveraged reaction probability, * Typically, the entire DWBA Pq has been ‘exponentiatcd” rather than, as in the present case, only the symmetric
part.
The Pii obtained using eq. (9) for P&?) and the one obtained using eq. (3) plus (8), are presented in fig. 2, together with the PO(T) obtained from the numerical solution of the coupled equations (2). The results are in excellent agreement. Pii above is related to a “one-dimensional rate constant” k;(T) by Pii = k;(T) hQ/kT, where Q is the partition function of the reactants. The linear restrictions (es_ (6)) which limit the validity of eq. (9) to the threshold region can be removed by using the exact two-state analog of Pwm, given by eq. (4.10) ofref. [I]. We then have Pii = exp(-
AZ/&) sin’ (Es - Ea),
(11)
where $jsa are the elastic phase shifts of eq. (1) with the matrix elements defmed by eq. (5). However, obtaining accurate transition probabilities at higher energies also requires extending the procedures for evaluating the matrix elements Vii(p) in ref. [l l] to smaller p_ Such extensions, as well as including the effect of variation of the resonance defect Vii - Vii with p, are discussed in detail elsewhere [13]. 509
Volume 101. number 6 3. Extension
to more complex systems
The question is frequently raised as to whether s0me of the “reorganization” of coordinates prior to an electron transfer in solution also occurs for H and for II+ transfers. (An early review of electron transfer theor! is +ZXI in ref. [ 19]_) We consider this point in the following discussion. IVe first note that the factor multiplying P,,,, in eq. (3) can be sho\vn 1131 to have the form exp[l(T4)2]. 11hsre 7 is roughly the -‘duration” of the reactive collision. (The units are such thst fi = l_) This factor top [ - (TA)~] serves as a counterpart to the well-known DiT3c delta iunction 6 (A) which appears in the Golden Rule formula for electronic transitions. e-g_ ref. [20]. including electron transfer reactions. e-g_ refs. [ 19.7 l] _ Beci~usc11112 ratio of the heavy to the light particles in 11 or 11* transfers is much closer lo unity than it is in electron transfers_ this esp[(i4)?] is a much less pc.krd
function
of the is than IS 6 (A)_
Nerertlie1ess.a
suitable fluctwlion of other coordinatesq. prior to the II rrxnsfcr. \\ill reduce exp[4:(47)!aF]. (The 4 in ~(1. (7) can IION depend on Q.) For cxanlple, the state-to-state contribution X-,(7‘) to the rate constant is givrn 1~) eq. ( 12). using ar_runirnts analogous to rlmsc
used in ref. [X]
_
(121
when the q coordinates are treated classical1y.j is the qu.uituni state of the H motion in the product that is closest in energy to the initial (ith) quantum state of the11 motion in the reactants_ In eq_ (13)Pii(E_ q) is $ven by rq. (9) and depends on 4 mainly via A(q)_ E
include5 nor only the energy along the reaction coordinate hU1 also I’ji(p. q). which replaces the I’&) present previously and is the effective potential energy of the system when the resctants are in the ith quan1un1 state for the 11 motion. Q is the partition function of the re.warus. and Q[, is the momentum part of the
classical partition function of the q degrees of freedom_ in the q coordinates which serve to rewill increase Pij(E) and hence be favored for enhancing the reaction rate provided the?, do not increase IT&, q) too much. Typically_ Fluctuations
duce 111e value of 4(q)
one expects that eq. (I?) for the reaction rate will be dominated by the Pwm factor in eq. (3), which is -_ 510
4 November 1983
CHEMICAL PHYSICS LETTERS
smaller for D than for H transfers_ The expectation that the exp [- (TA)?] term is reduced in importance by a suitable and relatively minor fluctuation in coordinates r~ is indicated by the apparent absence of unusual isotope effects (D transfer faster than H transfer)
in the literature_ If this were not the case it might be possible to find a system in which the D-transfer rate esceeded the H-transfer rate because. accidentally, A was small for the deuterium case.
4. Discussion The high degree of vibrational adiabaticity has been one of the most striking features in accumte calculations [6 71 of symmetric heavy-light-heavy (HLH) collinear reactive collisions. From an experimentalist’s point of view vibrational adiabaticity may simply mean that the vibrational quantum number of the products equals that of the reactants. It is manifested in the results of the accurate numerical calculations by the absence of vibrational esciration
when there are higher vibrational
states open. When there are no excited vibrational states open vibrational adiabaticity is manifested by the close validity of the approximate formulae in ref. [I] derived on the basis of vibrational adiabaticity. The vibrational adiabaticity is also evident in the preservation of the vibrational action in classical trajectory calculations_ As noted in ref. [I]. however. the dynamical nature adiabaticity in symmetric HLH reactions is radically different from that in the adiabatic model used by a number of investigators [23,24] and which was termed [24] vibrationally adiabatic. In that vibrationally adiabatic model [23.X4] the system moves on a single effective potential energy surface. This motion was discussed in detail using natural collision coordinates [Xl. In the new type of vibrational adiabaticity presented in ref. [ 11, the reaction is adiabatic in the sense that the quantum state of reactants is coupled only to a quantum state of the products with the same quantum number. However, two distinct quantum states and two effective adiabatic potentials are needed* of the vibrational
to discuss the reaction. Analogously, in the present
case of a non-_symmetric
* In the case of ref. [ 11. symmetric and antisymmetric states were used for the reaction AH + A - A f- HA and would also be necessary for an exact resonance AH + B + A + HB reaction.
Volume 101, number 6
CHEhlICAL PHYSICS LETTERS
reaction the vibrational state of the reactants is nearly degenerate with, and is coupled almost exclusively to, a vibrational state of the products which has a different quantum number_ As a result the reaction is almost completely vibrationally nonadiabatic, i.e. it leads to a vibrational population inversion in the products. Both the dynamics of H-atom transfer and the present treatment are much more akin to those of the charge transfer processes in atomic collisions than to the vibrationally adiabatic treatment of chemical reactions in terms of natural collision coordinates.
181 V. Aquikmti, S. Cavalli and A. Lagana, Chem. Pbys. Let-
1161
ters 93 (1982) 179; V. Aquilanti. G. Grossi and A. Lagana. Chem. Phys. Letlets 93 (1982) 174; J. hfol. Struct. 93 (1983) 319. D.K. Bondi. J.N.L. Connor, B.C. Garrett and D-G. Truhlar, J. Chem. Phys. 78 (1983) 5981: B.C. Garrett, D.G. Truhlar. A-F. Wagner and T-H. Dunning Jr., J. Cbem. Phys. 78 (1963) 4400. J.M. Launay and hl. Le Doumeuf, J. Phys. B15 (1962) L455. V-K. Babamov, V. Lopez and R-A. Marcus, J. Chem. Phys. 78 (1983) 5671. N. Abu Salbi. V. Lopez, V.K. Babamov, D. Kouri and R.A. hlarcus, J. Chem. Phys., to be submitted. VK Babamov. V. Lopez and R.A. Wucus. to be submitted_ E. Pollak. in: 15th Jerusalem Symposium on Quantum Chemistry and Biochemistry: Intramolecular Dynamics. ed$. B. Pullman and J. Jortner (Reidel, Dordrecbt, 1982) p. 1: J. Chem. Phys. 78 (19S3) 112s; Cbem. Phys. Letters 94 (1983) 85. J. Manz. R. hleyer. E. Polla% and J. R8melt, Chem. Phys. Letters 93 (1982) 184. J. hlanz. R. hieyer and J. Remelt, Chem. Phys. Letters 96 (1983) 607. DC Clxy zmd J.N.L. Connor, Chem. Phys. Letters 94
[I71
V-K. Babamov.
I91
[lOI IllI 1121 1131
Acknowledgement We are pleased to acknowledge
r 141
the support of this
research by a grant from the National Science Foundation_ and (to VL) a Fulbright/Spanish Ministry of Education and Science Fellowship. We thank Dr. J.N.L.
rw
Connor for his helpful comments.
(19S3)
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