Semiclassical initial value representation for the Boltzmann operator in thermal rate constants

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 21 1 DECEMBER 22 Semiclassical initial value representation for the Boltzmann operator in thermal rate constants Yi Zhao and William H. Miller a) Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 9472 Received 11 June 22; accepted 5 September 22 The thermal rate constant for a chemical reaction, k(t), can be expressed as the long time limit of the flux-side correlation C fs (t)tre Ĥ/2 Fˆ e Ĥ/2 e iĥt/ ĥe iĥt/. Previous work has focused on semiclassical SC approximations implemented via an initial value representation IVR for the time evolution operators exp(iĥt/) in the correlation function, and this paper shows how an SC-IVR can also be used to approximate the Boltzmann operators exp(ĥ/2). Test calculations show that over a wide temperature range little error is introduced in the rate constant by this SC approximation for the Boltzmann operator. 22 American Institute of Physics. DOI: 1.163/ I. INTRODUCTION This paper is a continuation of our efforts to develop the semiclassical SC initial value representation IVR Refs. 1 3 into a practical way of adding quantum effects to classical molecular dynamics simulations, here focused on calculating thermal rate constants for chemical reactions. A formally exact quantum mechanical expression for such rate constants is the long time limit of the flux-side correlation function, 4,5 ktq r T 1 lim C fs t, t which is given by C fs ttrfˆ ĥt, where Fˆ () is the Boltzmannized flux operator Fˆ e Ĥ/2 i Ĥ,hsqˆ e Ĥ/2, and ĥ(t) is the time-evolved projection operator onto products, ĥte iĥt/ hsqˆ e iĥt/. 1.4 s(q above is some function of the coordinates of the system that defines a dividing surface via the equation s(q), and Q r is the reactant partition function per unit volume. There is also a Kubo version 6,7 of the Boltzmannized flux operator that is sometimes used, 8 but we have found the above symmetrically split version to behave better in the numerical calculations presented in this paper. Evaluation of rate constants via Eqs thus involves two essential operators, ĥ(t) and Fˆ (), the former involving real time propagators, exp(iĥt/), and the latter the imaginary time propagator or Boltzmann operator, exp(ĥ/2). Previous work in our group and also by others on applying SC-IVR methods to evaluate rate constants via Eqs has concentrated on various ways of representing the real time propagators in ĥ(t), e.g., a linearization 9 of the difference between the two propagators in Eq. 1.4 leads to the classical Wigner model 1 15 and also what Pollak et al refer to as mixed quantum classical rate theory; the forward backward IVR Refs is a more accurate way for combining the two propagators in ĥ(t) into one effective propagator from to t and from t back to ; and the most complete semiclassical treatment is to use the full SC-IVR for both propagators independently. 27 With these various ways of treating the real time propagators in ĥ(t), the imaginary time propagators in Fˆ (), Eq. 1.3, have typically been dealt with by a harmonic approximation 23 about the transition state or essentially exactly by a Feynman path integral representation 28 evaluated by Monte Carlo methods 27,29. This paper is concerned with evaluating the Boltzmann operators in Eq. 1.3 also via SC-IVR. This will be more accurate than a harmonic approximation which has been noted to break down at low temperature and presumably more efficient than a path integral treatment. Also, it seems more esthetically pleasing and consistent to use the same kind of approximation for the operator exp(ĥ/2) as for exp(iĥt/), since they are essentially the same operator. We note that Makri and Miller 3 have recently explored use of a coherent state i.e., Herman Kluk-type 31 IVR for the Boltzmann operator, but in this paper we employ the coordinate space or Van Vleck 32 IVR. As one of us has recently shown, 33 a Gaussian filtering specifically, a modified Filinov transformation of the coordinate space IVR converts it into the coherent state IVR, so the two IVR s are effectively equivalent. We find it simpler, at least for present purposes, to formulate things in terms of the coordinate space IVR, and then to carry out any necessary filtering at the end. We also note that Pollak and Eckhardt 37 have used the SC approxia Electronic mail: miller@cchem.berkeley.edu /22/117(21)/965/6/$ American Institute of Physics

2 966 J. Chem. Phys., Vol. 117, No. 21, 1 December 22 Y. Zhao and W. H. Miller mation for the Boltzmann operator in Eq. 1.3 but incorporated further SC approximations e.g., steepest descent in evaluating the correlation function. Section II first describes the SC-IVR of the Boltzmann operator which is based on earlier work 38 of one of us and how it is used in Eqs Some numerical tests are then presented in Sec. III to show how well the SC approximation for the Boltzmann operator performs, and Sec. IV concludes. II. SEMICLASSICAL BOLTZMANNIZED FLUX OPERATOR A. The SC-IVR of the Boltzmann operator The standard Van Vleck SC approximation 32 for matrix elements of the time evolution operator i.e., real time propagator is q 2 e iĥt/ q 1 2i q F 2 p 1 e is t (q 2,q 1 )/, 2.1a where the sum is over all classical trajectories that go from q 1 at time to q 2 at time t, and S t (q 2,q 1 ) is the classical action integral, S t t dtpt 2 /2mVqt, 2.1b and where we have for simplicity assumed a Cartesian Hamiltonian of F degrees of freedom with all masses scaled to a common value, i.e., Hp,qp 2 /2mVq. This can be written in equivalent IVR form F as follows: 2i e iĥt/ dp dq q t p e is t (q,p )/ q t q, where q t q t (q,p ) is the coordinate at time t that has evolved from initial conditions (q,p ). It was shown some time ago 38 that Eq. 2.1 can be converted into a SC approximation for matrix elements of the Boltzmann operator by the often used replacement ti. The result involves classical trajectories in the pure imaginary time variable it, but in light of Newton s equation of motion, m d2 Vqt qt 2 dt q which thus becomes m d2 Vq q 2 d q, 2.4a. 2.4b This can be viewed as classical motion in the real time variable on the inverted potential energy surface, V(q). By introducing the real momentumlike variable p, which is related to the actual momentum p which is imaginary by p ip, the equations of motion read d d qp /m, d d p Vq, 2.5 q which we note involve only real-valued quantities. The IVR for the Boltzmann operator is then obtained from Eq. 2.3, e dp dq q Ĥ 2 p F e S (q,p )/ q q, 2.6a where q (q,p ) is the value of q() at which evolves from initial conditions (q,p ) via the equations of motion Eq. 2.5, and the action integral S (q,p ) is given by S dp 2 /2mVq. B. The flux-side correlation function. 2.6b For a Cartesian Hamiltonian Eq. 2.2, the flux operator Fˆ, Fˆ i Ĥ,hsqˆ, becomes Fˆ sq sq q q q sq q sq. 2.7a 2.7b Therefore, if one uses the IVR of Eq. 2.6 for the two Boltzmann operators in Eq. 1.3 and makes use of the derivative relations of the imaginary time action, q 2 S q 2,q 1 p 2, q 1 S q 2,q 1 p 1, the symmetry of the Boltzmann operator, q 2 e Ĥ q 1 q 1 e Ĥ q 2 2.8a 2.8b 2.9 and a judicious integration by parts then the flux-side correlation function of Eq. 1.2 becomes C fs t2 dq dp dp p q q p F sq sq q p p exps /2 q,p /S /2 q,p /qĥtq, 2.1 where qq /2 (q,p ) and qq /2 (q,p ) are the values of q() at /2 that evolve via the imaginary time equations of motion Eq. 2.5 with initial conditions (q,p ) and (q,p ), respectively. Equation 2.1 thus requires that one generates two imaginary time trajectories or real time trajectories on the inverted potential energy surface, both

3 J. Chem. Phys., Vol. 117, No. 21, 1 December 22 Boltzmann operator representation 967 starting from the same initial position q, which lies on the dividing surface s(q ) because of the delta function in the integrand. The above SC-IVR approximation for the Boltzmann operators can now be combined with whatever treatment of the real time dynamics in Eq. 2.1 i.e., the matrix elements of ĥ(t)] that is desired. For example, if the forward backward IVR Refs. 21,22 is used for the real time dynamics, the matrix element qĥ(t)q is given by qĥtq dps 2ip s 1 dq dp 2 F C q,p e is (q,p )/ qp q p q q, 2.11a where qp q and qp q are coherent state wave functions, e.g., qp q F/4 exp 2 qq 2 ip qq /, 2.11b (p,q ) are the initial conditions for real time trajectories that evolve from time to t via the normal molecular Hamiltonian, have a momentum jump at time t, p t p t p s sq t q t, 2.11c and then evolve backward in time from t to ; S is the forward backward action S t dtlps sq t t dtl, 2.11d Lp 2 /2mV(q) being the usual Lagrangian; and C is the Herman Kluk prefactor that involves derivatives of the final variables (q,p ) with respect to the initial ones (q,p ). A more approximate but much simpler treatment of the real time dynamics is the linearization approximation, 9 which assumes that the forward ( t) and backward (t ) trajectories are infinitesimally close to one another and which is also equivalent to the classical Wigner model. Though this approximation is not able to describe interference effects between different classical trajectories 26 because the forward and backward trajectories are assumed to be infinitesimally close, it has been seen in earlier work 9,27 to be reasonably accurate for these flux correlation functions. In this approximation the matrix elements of ĥ(t) are given by qĥtq2 F dp e ip (qq)/ hsq t, 2.12 where q t q t (q,p ) is the coordinate at time t that results from a real time trajectory with initial conditions (q,p ) with q (qq)/2. When this is used in Eq. 2.1, the explicit expression for the flux-side correlation function within the linearized approximation for the real time dynamics and the SC-IVR for the Boltzmann operators is FIG. 1. A schematic depiction of the various trajectories involved in Eq. 2.13: two imaginary time trajectories originate at q, with initial momenta p and p, respectively, and are propagated for time /2, arriving at positions q and q. A real time trajectory is then involved with initial point q (qq)/2 and initial momentum p. C fs t2 dq dp dp 2F dp p p q q sq sq q p p hsq t exps /2 q,p /S /2 q,p / ip qq/, 2.13 where qq /2 (q,p ),qq /2 (q,p ),q (qq)/2, and q t q t (q,p ). Thus to evaluate the integrand above, two imaginary time trajectories are begun at position q on the dividing surface with initial momentum variables p and p, respectively, and integrated for time /2; their final values are q and q. A real time trajectory, with initial coordinate q (qq)/2 and initial momentum p, is integrated until it is clear whether it will emerge in the reactant or product region in the former case the integrand is zero. Figure 1 depicts this situation schematically. III. NUMERICAL TESTS In this section we present some numerical results to test the error involved in using the SC Boltzmann operator in the flux-side correlation function discussed above. The example treated is a standard one, the 1d Eckart barrier, for which the potential function is VqV sech 2 aq, 3.1 with the parameters chosen to correspond approximately to the HH 2 reaction: V.425eV,a1.36 a.u. and m 16 a.u. The imaginary frequency is b 2V a 2 /m a.u.1619 cm 1, and the two Eckart parameters are 2V / b 13.29, u b , respectively; the cross over temperature, corresponding to u2, is 37 K. One complicating aspect of the SC Boltzmann operator should be noted: at the lowest temperatures considered below, the monodromy matrix elements, q /2 /p, appearing in Eq. 2.1 are negative for some values of the integration variables. Since the square root of this quantity is involved, this leads to an imaginary contribution to the matrix element of the Boltzmann operator but one knows that

4 968 J. Chem. Phys., Vol. 117, No. 21, 1 December 22 Y. Zhao and W. H. Miller FIG. 2. The flux-side correlation function for the 1d Eckart potential, using the SC-IVR for the Boltzmann operator and the exact quantum matrix element of ĥ(t); upper figure is for T1 K, and the lower one for T2 K. these matrix elements are real-valued! The most consistent way to deal with this is simply to set the integrand to zero for such values, and this is what has been done in the calculations presented below. Appendix A gives a discussion of the stationary phase approximation for the case of real exponents which justifies this procedure. To provide an unambiguous test of the accuracy of the SC Boltzmann operator for this example, we first carried out calculations using the exact quantum matrix elements of ĥ(t); i.e., we solved the time-dependent Schrödinger equation in a discrete variable representation in order to obtain qĥ(t)q exactly. The SC-IVR for the Boltzmann operator is thus the only approximation in this case. Figure 2 shows C fs (t) for a high temperature T1 K, upper figure and a low temperature T2 K, lower figure compared to the exact quantum results i.e., using exact quantum results for the Boltzmann operator and ĥ(t)]; the corresponding values of the rate constant are given in Table I and shown in Fig. 3. The conclusion is that the SC approximation for the Boltzmann operator is quite good for a wide range of temperature. We also carried out calculations within the linearized approximation for the real time dynamics, i.e., using Eqs In the present 1d example it is possible to carry out the integral over p in Eq analytically as well as that over q by virtue of the delta function, to give the following explicit expression: C fs 2 2 dp dp q p q p p p 2m qq exp S /2,p /S /2,p / iqq2m V b V qq 2 /. 3.2 A Filinov filtering of the integrand was necessary to tame the oscillatory character of the above integral particularly for the lowest temperatures, and Appendix B describes this procedure which is a slight generalization of what we have used before 39 in more detail. The Filinov parameter was taken sufficiently large 2 to be essentially in the limit i.e., larger values gave no change in the results; in this limit the Filinov procedure yields the value of the original integral. The integrand in Eq. 3.2 is quite localized in the two-dimensional space (p,p ), so that only a modest number of these values are needed to calculate the integral here by grid methods. For systems with more degrees of freedom, one would expect that Monte Carlo methods would make the multidimensional integrals efficient. The results of this linearized approximation for the real time dynamics are also given in Table I and Fig. 3, and one sees that this approximation degrades the quality of the rate constant, more so the lower the temperature, e.g., at T2 K, where the tunneling correction factor is 2, the result of the linearization approximation is about a factor of 2 too small. This is consistent with our earlier results 8,9 using the linearization approximation in the tunneling regime where the Boltzmann operator was treated exactly, by Monte Carlo path integration. Thus use of the SC Boltzmann operator with the linearized approximation for the real time dynamics is essentially just as good as that obtained with the exact treatment of the Boltzmann operator. IV. CONCLUDING REMARKS It has been shown how a SC-IVR can be utilized for the Boltzmann operator in reactive flux correlation functions. This can be combined with whatever level of treatment of the real time dynamics one chooses. Tests on a simple but illustrative example suggests that little error is introduced by treating the Boltzmann operator semiclassically, even at low temperatures where tunneling corrections are quite large. ACKNOWLEDGMENTS This work has been supported by the Director, Office of Science, Office of Basic Energy Sciences, Chemical Sci- TABLE I. Comparison of tunneling correction factors 2e V k(t) calculated by different methods. T K Semi. F Quan. ĥ(t) a Semi. F Semi. ĥ(t) b Quantum mechanical c a Calculated with the SC-IVR for the Boltzmann operator e Ĥ and the exact quantum result for the projection operator ĥ(t) see Eq b Calculated with the SC-IVR for e Ĥ and the linearized semiclassical approximation for ĥ(t) see Eq c Calculated by the exact quantum mechanical method.

5 J. Chem. Phys., Vol. 117, No. 21, 1 December 22 Boltzmann operator representation I SPA 2 Ax k k 2 e Sx k S(x k )/, 969 A7 where x k are all the roots of Eq. A5 for which Eq. A6 is also true. That is, one does not include all the stationary points of S(x) in the sum in Eq. A7, only those for which S(x k ). Note that if one forgot and included the stationary roots in Eq. A7 for which S(x k ), these would give an imaginary contribution, which is nonsensical since the original integral, Eq. A4, is manifestly real. FIG. 3. An Arrhenius plot of the thermal rate constant for the 1d Eckart barrier. The solid curve is the exact result, and the dashed line the classical result. The triangles and squares are the results of present calculations that use the SC-IVR for the Boltzmann operator, and the exact quantum result for the projection operator ĥ(t) triangles, or the linearized semiclassical approximation for ĥ(t) squares. ences Division of the U. S. Department of Energy under Contract No. DE-AC3-76SF98, and by the National Science Foundation Grant No. CHE Y.Z. gratefully thanks N. Makri for stimulating discussions. APPENDIX A: STATIONARY PHASE WITH REAL EXPONENTS The standard stationary phase approximation for a one dimensional integral of the form is I 1 dxaxe is(x)/ I 1 SPA k Ax k 2i e Sx k is(x k )/, A1 A2 where x k are all the points of stationary phase, i.e., roots of the equation Sx k. A3 In extending this result to the case of real exponents, i.e., for an integral of the form I 2 dxaxe S(x)/ A4 one notes that the dominant contributions to the integral come from regions where S(x) has a local minimum i.e., where the integrand has a local maximum with Sx k Sx k. A5 A6 Expanding S(x) quadratically about each such point and carrying out the usual Gaussian integral gives the analog of Eq. A2, APPENDIX B: A GENERALIZED FILINOV TRANSFORMATION FOR THE SC-IVR A recent paper 39 described the effectiveness of a generalized Filinov filtering procedure for evaluating multidimensional integrals with an oscillatory integrand, i.e., integrals of the form, I dz Rze i(z). B1 The procedure is to insert into the integrand of Eq. B1 the following representation of unity: 1 F e 1/4 T 1 dz e (zz )T (zz ) e it (zz ) B2 expand the exponent (z) quadratically about z, and then perform the Gaussian integral over z, giving the following result: I dz Rz 2 2iz exp 1 4 T z T 2iz 1 z. e i(z ) B3 The generalization in the procedure, compared to the original version, is inclusion of the linear term in the exponent in Eq. B2. In the treatment above, it was assumed that the preexponential factor in the integrand of Eq. B1, R(z), is sufficiently slowly varying that it was replaced by R(z ) in obtaining Eq. B3, but this may not always be the case. For example, for the following integral: I dq q e Aq 2 /2 e i pq, B4 which is in fact characteristic of some of the integrals encountered in SC-IVR applications, Eq. B3 gives the incorrect result I. It is easy to remedy this defect, however, by expanding R(z) to first order about z in the treatment leading up to Eq. B3. The resulting generalization of Eq. B3 is easily found to be

6 961 J. Chem. Phys., Vol. 117, No. 21, 1 December 22 Y. Zhao and W. H. Miller I dz 2 2iz Rz i Rz T 2iz 1 z e i(z ) exp 1 4 T z T 2iz 1 z. B5 It is not hard to show that Eq. B5 gives the exact result for the test case in Eq. B4. We now specialize Eq. B5 to the type of IVR integrals considered in this paper: the complex function (z) is written in terms of its real and imaginary parts, zzi z, B6 and it is assumed that the Hessian of the real part, (z), is negligible. Equation B5 then becomes I dz 2 2z Rz i Rz T 2z 1 z i z expiz z 1 4 T z i z T 2z 1 z i z. B7 The Gaussian width matrix and vector are the parameters that control the accuracy and efficiency of the procedure. The original integral is recovered in the limits and.) This above procedure, Eq. B7, was applied to the linearized SC-IVR expression, Eq. 2.13, in the variable p p p. It is necessary to carry out the Filinov filtering in only this one variable in order to obtain a well-behaved integrand for Monte Carlo integration. The explicit result of carrying out this procedure for Eq is C fs dq dp dp dp sq with 1 sq hsq t RiR2 1 q i2 1 M qp q,p M qp q,p 2 2 expi i 2 1 i], B8 Rp p, R1, S /2 q,q S /2 q,q, B9a B9b B9c 1 2 p /2 M qp q,p p /2 M qp q,p, B9d 1 4 M ppq,p M qp q,p M pp q,p M qp q,p, p qq/, p 2 M qpq,p M qp q,p, where M qp represents a monodromy matrix. B9e B9f B9g 1 W. H. Miller, J. Chem. Phys. 53, D. J. Tannor and S. Garashchuk, Annu. Rev. Phys. Chem. 51, W. H. Miller, J. Phys. Chem. A 15, W. H. Miller, J. Chem. Phys. 61, W. H. Miller, S. D. Schwartz, and J. W. Tromp, J. Chem. Phys. 79, T. Yamamoto, J. Chem. Phys. 33, G. A. Voth, D. Chandler, and W. H. Miller, J. Phys. Chem. 93, X. Sun, H. B. Wang, and W. H. Miller, J. Chem. Phys. 19, H. Wang, X. Sun, and W. H. Miller, J. Chem. Phys. 18, E. J. Heller, J. Chem. Phys. 65, R. C. Brown and E. J. Heller, J. Chem. Phys. 75, H. W. Lee and M. O. Scully, J. Chem. Phys. 73, J. S. Cao and G. A. Voth, J. Chem. Phys. 14, R. E. Cline and P. G. Wolynes, J. Chem. Phys. 88, V. Khidekel, V. Chernyak, and S. Mukamel, in Femtochemistry: Ultrafast Chemical and Physical Processes in Molecular Systems, edited by M. Chergui World Scientific, Singapore, 1996, p J. Shao, J. L. Liao, and E. Pollak, J. Chem. Phys. 18, J. L. Liao and E. Pollak, J. Chem. Phys. 111, Y. Zheng and E. Pollak, J. Chem. Phys. 114, J. L. Liao and E. Pollak, J. Chem. Phys. 116, N. Makri and K. Thompson, Chem. Phys. Lett. 291, W. H. Miller, Faraday Discuss. 11, X. Sun and W. H. Miller, J. Chem. Phys. 11, H. Wang, M. Thoss, and W. H. Miller, J. Chem. Phys. 112, R. Gelabert, X. Giménez, M. Thoss, H. Wang, and W. H. Miller, J. Phys. Chem. A 14, H. Wang, M. Thoss, K. L. Sorge, R. Gelabert, X. Giménez, and W. H. Miller, J. Chem. Phys. 114, R. Gelabert, X. Giménez, M. Thoss, H. Wang, and W. H. Miller, J. Chem. Phys. 114, T. Yamamoto, H. Wang, and W. H. Miller, J. Chem. Phys. 116, P. R. Feynman, Rev. Mod. Phys. 2, E. Jezek and N. Makri, J. Phys. Chem. 15, N. Makri and W. H. Miller, J. Chem. Phys. 116, M. F. Herman and E. Kluk, Chem. Phys. 91, J. H. V. Vleck, Proc. Natl. Acad. Sci. U.S.A. 14, W. H. Miller, Mol. Phys. 1, V. S. Filinov, Nucl. Phys. B 271, N. Makri and W. H. Miller, Chem. Phys. Lett. 139, J. D. Doll, D. L. Freeman, and T. L. Beck, Adv. Chem. Phys. 78, E. Pollak and B. Eckhardt, Phys. Rev. E 58, W. H. Miller, J. Chem. Phys. 55, H. Wang, D. E. Manolopoulos, and W. H. Miller, J. Chem. Phys. 115,