The Journal of
Physical Chemistry
0 Copyright, 1992, by the American Chemical Society
VOLUME 96, NUMBER 16 AUGUST 6,1992
LETTERS Resonance State Approach to Quantum Mechanical Variational Transition State Theory Donald G.Truhlar* Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431
and Bruce C. Garrett Molecular Science Research Center, Pacific Northwest Laboratory, MS K2- 18, Richland, Washington 99352 (Received: April 13, 1992; In Final Form: June 10, 1992)
Variational transition state theory is reformulated in terms of compound state resonances associated with the transition state region. The use of the formalism is illustrated for two-dimensional model problems, which are treated by the adiabatic theory of reactions and by perturbation theory. The limitations of the latter are delineated for the case of a biquadratic (quartic) coupling of the reaction coordinate to a transverse vibration.
1. Introduction
There has been much discussion of how to put quantum effects into transition state theory and variational transition state theory for bimolecular chemical These approaches all start out, in one way or another, with the identification of the transition state theory rate constant as proportional to the flux through a hypersurface dividing reactants from products. Systems within the hypersurface must be quantized, as must be the flux through the hypersurface. The great variety of formulations of quantized transition state theory may be viewed as alternative ways to get around a fundamental computational and conceptual difficulty, namely, in quantum mechanics, if we localize a system in a hypersurface, we have complete uncertainty about the momentum normal to that hypersurface (Le., in the direction of the reaction coordinate) and hence about the flux through that hypersurface. The difficulty is resolved by introducing physical models, prescriptions for ordering noncommuting operators, and/or semiclassical limits.'-23 In the present Letter we suggest another alternative, namely, reformulation of transition state theory using quantum mechanical resonance theory.
The reason that resonance theory provides a natural way around the above-mentioned difficulty may be seen by comparing the classical and quantum mechanical descriptionsof onedimensional metastable states governed by a potential V(x). In classical mechanics, such a state is completely localized at the point where dV/dx = 0 and d2V/dx2 < 0; i.e., the system must be localized precisely at the barrier top. In quantum mechanics, the metastable state associated with crossing a barrier is most naturally described as a r e s o n a n ~ e , 2which ~ ~ ~ ~is a delocalized state with a complex energy. Although resonances are delocalized as compared to classical metastable states Gust as bound systems are delocalized by zero-point motion as compared to classical equilibrium systems at the bottom of a potential well), they are more localized than typical quantum mechanical continuum states, and it is this localization that accounts for their metastability. An advantage of reformulating transition state theory in terms of resonances is that the quantum mechanical theory of resonances is highly A motivation for reformulating bimolecular transition state theory in terms of resonances was provided by recent accurate
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6516 The Journal of Physical Chemistry, Vol. 96, No. 16, 199‘2
quantum mechanical calculations on several reactions: H + H2;27328 0 H2;28*29 C1 + HCI,I HI, and I + DI;30*3’He H2+;32and Ne + H2+.33 In all these cases, the quantum mechanical cumulative reaction probability shows evidence that its structure is governed by a discrete spectrum of broad features that can be associated with quantum mechanical transition state resonances. For example, the lifetimes of the metastable transition states were predicted from the imaginary parts of their energies,28 and these lifetimes agree well with direct calculation^^^*^^ of collisional time delays from quantum mechanical scattering matrices. Resonances associated with transition states are observable in principle in reaction probabilities. One possible case where quantum mechanical resonanm associated with transition states appear to have been observed experimentally as resolved structure is the photodissociation of triplet ketene.36
+
2.
+
+
Letters tionally adiabatic theory will also be relevant to understanding the calculations of ref 21. To illustrate these ideas, we consider the ground level of the transition state for the two-dimensional Hamiltonian H = Ho VI where 1 1 1 Ho = --(Ps2 2P + p,Z) + V -k,p2 2 -kIlu2 2 (5)
+
+
+
with k,, < 0 and kl I > 0, and where VI is the anharmonicity. The zero-order resonance energies are given by
E, = f l
+ hwl(ul + !I2)
(6)
hl(k~s/P)l/~1
(7)
and
r, where
Theory
According to adiabatic transition state theory,5-7,10314 the reaction rate is
where T is temperature, h is Planck’s constant, aR(T)is the reactant partition function per unit volume for bimolecular reactions and the reactant partition function for unimolecular reactions, E is total energy, kBis Boltzmann’s constant, and K,(E) is the transmission probability for transition state T at energy E. Now, following recent work,24,25,27,28,31 we identify each transition state with a resonance. The complex energy of a resonance is written26 era(^) = E,
-r,/2
(2)
where E, is the resonance energy and, for isolated narrow resonances, r, is the full width at half-maximum of the resonance feature. In the present paper we assume that the effectivie bamer in the vicinity of a transition state is parabolic. For such a transition state, it is easily shown, by combining the standard treatment of transmission by a parabolic barrier37with the resonance treatment of parabolic barrier^,^^,^^ that (3) where
rT= hlw,fl
(4)
and w,* is the imaginary frequency associated with the effective parabola. Substituting eq 3 into eq 1 and carrying out the thermal average (the thermal average of (3) is described in detail else~here’~) yields the semiclassical results of Wigner.] In the general case, one would find E, and r, by standard approximate methods for finding resonance energies.26 Or one could use a recent stationarity principle39for complex poles of the resolvent, which allows systematically converged calculations of E, and I”,. There is no simple relation between K,(E)and these parameters in the general case. One possible approximationwould be to obtain E, and r, by general methods, including anharmonicity and mode coupling, but still use (3) to relate K,(E)to the resonance energy and width. This is the approach we pursue m the present paper. In particular, we will compare two methods of calculatingthe complex resonance energies-perturbation theory and vibrationally adiabatic theory. If one uscs perturbation theory, the present theory becomes very similar to the semiciassical transition state theory of Miller,” which was not based on resonances, so this is an encouraging result. Miller, Hernandez, Handy, and co-workers21have implemented Miller’s semiclassical theory by second-order perturbation theory based on a quartic expansion about the saddle point. Since this is similar to the perturbation theory calculations presented here, except that we treat the transition states as quantum mechanical resonances, our comparison of perturbation theory to the vibra-
WI
= (kll/P)1/2
(8)
3. Calculations 3.1. Cubic Anharmonicity. In this case the anharmonic part of the potential is VI = kl12u2s (9) The ground-state energy of the transition state may be obtained more accurately than eqs 6 and 7 by treating the cubic coupling as a perturbation. Using second-order perturbation theoryIMthis yields h2kl12’ 4W12 - 3W2’
AE,=-
(10)
8p3wl2wZ24wI2 - w22
and
where wZ2 = k8Jp < 0. The change in the real part of the resonance energy is positive, which is the correct physical sign since the perturbation should shift the dynamical bottleneck to higher energy. An alternative treatment for the same Hamiltonian is to use adiabatic transition state t h e ~ r y . ~ ~ JThe O J ~real part of the ground state resonance energy is the maximum of the ground state adiabatic potential curve 1 k l l 2kIl2s VaG= V + -k,p2 (12) 2 2
+
+
)
Setting dVaG/dslm. = 0 yields a cubic equation that can be solved for small S, to yield
and hence max VaGz V
1 h2kl122 + -hwl -2
8p3012w22
(14)
The last term is pitive, and it agrees exactly with the perturbation theory treatment in the limit where w12 >> 1 ~ ~which ~ 1 ,is where the adiabatic theory is most valid. In the vibrationally adiabatic theory, we calculate the width in the local parabolic barrier top approximation from eq 4 with
where s* is the location of the maximum of VaG(s). This yields
which agrees perfectly with eq 11 when w 1 2>> Iw212. Again, in
The Journal of Physical Chemistry, Vol. 96, No. 16, 1992 6517
Letters the adiabatic limit, the standard adiabatic theory and the new resonance approach agree. 3.2. Quartic Anharmonicity. Next consider a biquadratic (quartic) perturbation, where
VI = kllZ2u2s2
-
(17) and where higher-order terms make V,, 0. The leading perturbation theorya correction for the ground state is easily calculated
which is pure imaginary. But this VIis precisely the kind of term that can cause a double maximum in the vibrationally adiabatic potential curve and, thereby, a large shift in the real part of the transition state energy. Here adiabatic transition state theory has an advantage because we can find a global solution without approximating s as small. For example, consider the ground state; the maximum of VaG Vt
+ -k,p2 1 + ;h1 2
(
kll + ;llzzs2)~/2
(19)
occurs at s = 0 when k1122 < ko, where
ko = IksslP@l/h (20) This result agrees with the first-order perturbation theory result, i.e., no effect. But in many interesting cases k l l z 2is larger, and eq 19 correctly exhibits a nonzero variational effect on the real > part of the transition state energy. In particular, when k1122 ko, the maximum of the vibrationally adiabatic curve is given by
For values of k1122 close to but greater than ko, the predicted shift in the transition state energy is
and using eq 15 at the maximum yields
[
Ar7= hlwz(
(1
- x)1’2
- 11
k11222
(23)
Although conventional second-order perturbation theory in spectrmcopp includes second order for cubic terms and first order for quartic terms, it is interesting in the present case to go to second order for the quartic term. This yields
where the first term in (25) is the fsst-order result. At this order of perturbation, hE, is positive but the perturbation theory result of (24) varies as k1122,2 whereas the result of eq 22 from the global vibrationally adiabatic formalism varies as ( k ,122 - ko)2,because for small quartic force constants the dynamical bottleneck remains at the saddle point. We conclude that perturbation theory through second order is qualitatively incorrect in such a case. However, when kl 122 < ko, the adiabatic approach yields
AFT = h1w21[ ( 1 -
2)”’ ] -1
Agaii,for small enough perturbations, the adiabatic and resonance approaches agree perfectly when lw21
