Analytic continuation of real-time correlation functions to obtain thermal rate constants for chemical reaction

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1 9 July 1999 Chemical Physics Letters Analytic continuation of real-time correlation functions to otain thermal rate constants for chemical reaction Haoin Wang, William H. Miller ) a Department of Chemistry, UniÕersity of California, Berkeley, CA 9472, USA Chemical Sciences DiÕision, Lawrence Berkeley National Laoratory, Berkeley, CA 9472, USA Received 2 March 1999; in final form 4 May 1999 Astract It is shown how a particular correlation function, computed quantum mechanically or semiclassically as a function of real-time, can e analytically continued to otain thermal rate constants Ž or other thermally averaged quantities. over a wide range of temperature. The correlation function is first evaluated at a reference temperature, and then analytically continued to otain information at other desired temperatures. The method is applied to the calculation of the thermal rate constant for the ClqH HClqH Ž Js. 2 reaction using the flux correlation function formalism, and it is seen to e capale of otaining accurate rate constants over a road temperature range. q 1999 Elsevier Science B.V. All rights reserved. 1. Introduction There has een a great deal of progress wx 1 in recent years in the development of rigorous quantum mechanical methods for calculating thermal rate constants of chemical reactions; cf. work in our group w2 4 x, that of Light et al. w5 7 x, that of Manthe et al. w8 1 x, and that of other w11 x. These approaches are ased on formally exact expressions w12 14x for the rate constant kt either in terms of a flux time correlation function, e.g., H ` y1 r k T sq T dtc t, 1a ) Corresponding author. Fax: q ; miller@neon.cchem.erkeley.edu where C t str e Fe e Fe, y Hr2 y Hr2 i Htr " yi Htr " Ž 1. Hand F eing the Hamiltonian and flux operators, respectively, or in terms of the cumulative reaction proaility NŽ E., 1 ` y E kž T. s H d E e NŽ E., Ž 2. 2p"Q Ž T. y` r where Q Ž T. r is the reactants partition function per unit volume and sž k T. y1. When using Eq. Ž. B 1, one typically calculates C Ž. t, and thus kt, for a single value of temperature T, while if one calculates NŽ E. for a suiciently wide range of energy E, then Eq. Ž. 2 can e used to otain kt for a range of values of T r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S

2 464 ( ) H. Wang, W.H. MillerrChemical Physics Letters In this Letter, we consider an alternate way of otaining kt over a wide temperature range in this case solely from the time correlation function Ž and its analytic continuation.. Section 2 presents the theoretical development and discusses how the resulting procedure relates to earlier work, Section 3 illustrates the approach y application to the ClqH 2 HClqH reaction Ž in 3d for Js., and Section 4 concludes. 2. Theory We consider the time integral of the flux correlation function of Eq. Ž 1. as a function of, R 'kž T. Q Ž T. H ` r y Hr2 y Hr2 ihtr " yi Htr " s dt tr e F e e F e, Ž 3. which we note can also e expressed w13x as the long time limit of the flux-side correlation function, y Hr2 y Hr2 ihtr " yi Htr " R slim tr e F e e h e, t ` Ž 4. where hsh Ž q. is the Heaviside function which is i related to the flux operator Fs wh,h x ". Taking to e complex, s y2it r", Ž 5. where Ž s 1rk T. B corresponds to a reference temperature and t is a real-time variale, the function R Žt. is defined as R Ž t.'r, Ž 6a. y2it r " and is given explicitly y ` y Hr2 y Hr2 R Ž t. sh dt tr e F e i H Ž tqt. r" yi H Ž tyt. =e F e r", Ž 6. The procedure, therefore, is to calculate R Žt. from Eq. Ž 6. for a range of Ž real. values of t and then use these to otain R Žand thus kt. y Ž numerically. analytically continuing it to t s i"dr2, i.e., RsR Ž i"dr2., Ž 7. where Dsy. The calculation of R Žt. is carried out y the same very eicient procedure we have used earlier for calculating flux correlation functions w2 4 x; here it involves a Lanczos calculation with the Boltzmann operator for the reference temperature, expž y Hr2., and then real-time propagation Ž e.g., via the split-operator algorithm. to generate the time evolution operators expwyi Ht" Ž t. r" x. The numerical analytic continuation is carw15x Ža type ried out y Schlessinger s point method of Pade Approximant w16 x., and we find that a range of t of several "D is suicient to otain stale results. ŽWe note that this approach was suggested earlier wx 1 y one of us for the case ', ut experience has shown that it works much etter with a finite reference temperature.. We now discuss several aspects of this result and procedure. First, recall that some years ago another type of analytic continuation was used for flux correw13,17 19 x. The lation functions and rate constants goal then was to use Monte Carlo path integration methods to represent the exponential operators in evaluating Eq. Ž 1., and to avoid oscillatory integrands which cause troule for such calculations one considered the pure imaginary time correlation function which involves only Boltzmann-like Ži.e., real exponential. operators, CŽ t.'c Ž yit. Ž yr2qtr". H Ž yr2ytr". H str e F e F. Ž 8a. The procedure was thus to calculate CŽ t. for a range of real t values and then use these to analytically continue Ž numerically. to imaginary t s i t to otain the flux correlation function, i.e., C Ž t. scž it.. Ž 8. A major limitation of this procedure, however, is that the real values of t for which Eq. Ž 8a. can e evaluated are restricted to the interval " " y -t-, Ž meaning that it is not possile to generate numerical analytic continuation to real-times t much longer than ;" r2. This is suiciently long to descrie direct arrier crossing dynamics Žincluding tunneling., ut not long enough to descrie non-transition

3 ( ) H. Wang, W.H. MillerrChemical Physics Letters state theory dynamics, i.e., dynamical eects related to flux that recrosses the transition state region. The present approach, ased on Eqs. Ž. 4 and Ž. 7, suers no such limitation since one can carry out calculations for as wide a region of the real variale t as necessary in order to guarantee a stale numerical continuation to t si"dr2. The present analytic continuation procedure is in a sense, therefore, the opposite of the earlier one; there the goal was to have all the exponential operators appear as Boltzmann-like operators, so that Monte Carlo path integration would e well-ehaved, while the present approach is to have most Ž all in the limit. of the exponential operators as real-time propagators. This makes the present approach especially advantageous when using semiclassical approximations for the real-time dynamics; e.g., one can choose suiciently small Ž i.e., the reference temperature T suiciently high. for the high temperature approximation to expž yhr2. to e valid, and then the semiclassical initial value representation Ž SC-IVR. w2 32x can e used to approximate the real-time propagators expwyi Ht" Ž t. r" x. It is also clear that the analytic continuation procedure of Eqs. Ž. 4 and Ž. 7 can e applied to any thermal correlation function of the form y Hr2 y Hr2 ihtr " yi Htr " C Ž t. str e A e e B e. Thus with s y2it r", one otains Ž 1. C Ž t,t.'c Ž t., Ž 11a. y2it r " which is given y C t,t str e A e y Hr2 y Hr2 = e i H tqt r" B e yi H tyt r", Ž 11. One evaluates Eq. Ž 11. for a range of real values of t and uses these to analytically continue to t s i"dr2, giving C Ž. t as C Ž t. sc Ž t,i"dr2.. Ž 12. Finally, we note that in comparison with similar approaches y Light et al. wx 6. and Manthe et al. wx 9 our current method is not restricted to the quantum mechanical calculations that employ finite asis sets, as are necessary in previous work ased on Fourier transforms involving certain wavefunctions w6,9 x. Instead, it can e applied as long as the correlation function C Žt,t. in Eq. Ž 11. can e conveniently evaluated without referring to the specific Žtime-de- pendent. wavefunctions. It is thus applicale to a wider class of quantum and semiclassical calculations of time correlation functions. ŽWe note that Matzkies and Manthe also recently proposed a wx method 9 ased on the Fourier transform of C Žt,t. in Eq. Ž 6., which can e viewed as another application of this correlation function.. 3. Flux correlation function calculations for the ClHH HClHH ( Js) reaction 2 As a test example, the current analytic continuation methodology is applied to the quantum mechanical calculation of kt for the ClqH 2 HClqH reaction Ž in 3d for J s. over a wide range of temperature. The procedure is as descried in the paragraph following Eq. Ž. 6 aove, with a slightly dierent Ž ut equivalent. expression for R Žt. H t max y Hr4 y Hr4 R t s dt tr e F e =e e F e i HŽ tqt. r" y Hr4 y Hr4 yi HŽtyt.r " =e. Ž 13. y Hr4 y Hr4 The Boltzmannized flux operator e F e is first diagonalized via a Lanczos procedure to give its eigenfunctions c 4 and eigenvalues f 4 w2 1 x i i, and the flux correlation function is otained y propagating these eigenfunctions weighted y the appropriate eigenvalues w2 1 x. Matrix elements of yi Ht the time evolution operator G Ž. t s² c < e < c : ij i j, which are also needed in calculating a single rate constant wx 4, are otained at each time step during the propagation. These G Ž. ij t s are then used to generate all the Žt,t. pairs necessary in evaluating Eq. Ž 13.. In calculations presented elow, 1 fs of real-time propagation was carried out in 5 time steps, so that

4 466 ( ) H. Wang, W.H. MillerrChemical Physics Letters Fig. 1. Arrhenius plot of the thermal rate constant etween 2 and 15 K. The solid line is from the analytic continuation procedure with reference temperature T s15 K, and the circles are from separate calculations Žas in Ref. wx. 3 for each temperature. ranges from to 5 fs. Finally, other specific details of the calculation ŽHamiltonian, asis set, etc.. are the same as in the previous work wx 3. t Fig. 1 shows the comparison of kt otained via the current analytic continuation approach with a high reference temperature T s 15 K Ž solid line. Fig. 2. Same as Fig. 1, except for the reference temperature T s5 K.

5 ( ) H. Wang, W.H. MillerrChemical Physics Letters to the rate constants otained previously wx 3 from separate calculations for each temperature Ž circles.. The agreement is excellent Ž to within a few percent. for a wide temperature range Ž 2 15 K.. This is the way we envision one will usually employ the approach, i.e., with a high reference temperature Ž small. and analytic continuation to a range of lower temperature. This is especially convenient ecause the high temperature approximation for the Boltzmann operator is so simple. Analytic continuation is also possile Žthough not as reliale. for the opposite direction, i.e., from a low reference temperature T continuing to higher temperatures. Fig. 2 shows the Arrehenius plot of kž T. otained from analytic continuation with T s 5 K. Here again, they are in very good agreement with the previous results ut for a somewhat smaller temperature range Ž 5 15 K.. This, however, gives more flexiility for the analytic continuation method, since sometimes one may wish to perform a single reference calculation at a intermediate temperature T, and then continue to oth lower and higher temperatures. Finally, we note that one can also otain an accurate representation of the flux correlation func tion C Ž. t itself via the analytic continuation proce- dure, i.e., Eqs. Ž 11. and Ž 12. with AsBsF. Fig. 3 shows C t for three temperatures, otained via analytic contiunation with T s15 K. They are in excellent agreement with the previous calculations Ž. tially no eect on the integral of C the rate constant. Ž dot dashed lines. wx 3. The indication of recrossing Ž eects i.e., a negative loe in C Ž.. t, which are visile at the high reference temperature Ž Fig. 3a., disappear for lower temperatures Ž Fig. 3,c., exactly as the results otained efore wx 3. For the lowest temperature, Ts2 K, there are very small oscilla tions in the analytically continued C Ž. t due to the finite precision of the computation. As seen from Fig. 3a,c, C Ž. t changes y aout 12 orders of magnitude from 15 to 2 K, so it is not surprising for a small amount of numerical noise to appear for the farthest extrapolation; this, of course, has essen Ž. t which gives Fig. 3. Flux flux autocorrelation function. Ž. a Result of the direct calculation Žcf. Ref. wx. 3 for T s15 K. Result of the present analytic continuation procedure with a reference temperature T s 15 K, for T s8 K. Ž. c Same as Ž. except for T s2 K. 4. Concluding remarks The analytic continuation approach presented in this Letter thus provides a simple and eicient way

6 468 ( ) H. Wang, W.H. MillerrChemical Physics Letters of otaining thermal rate constants Žor other thermally averaged quantities. over a road temperature range from one reference calculation. Although it is tested for the case of a rigorous quantum mechanical calculation, the approach is designed Ž and expected. to e more useful for a calculation employing the semiclassical initial value representation Ž SC-IVR. w2 32x where rigorous evaluation of the Boltzmann operator is diicult for low temperatures. The success in the present example suggests that it is promising to y-pass this diiculty y carrying out a SC-IVR calculation for a high reference temperature, and then extract information for a road range of lower temperatures. Such applications are currently under investigation. Acknowledgements This work was supported y the Director, Oice of Science, Oice of Basic Energy Sciences, Chemical Sciences Division of the US Department of Energy under Contract No. DE-AC3-76SF98, y the Laoratory Directed Research and Development Ž LDRD. project from National Energy Research Scientific Computing Ž NERSC. Center, Lawrence Berkeley National Laoratory, and also y National Science Foundation Grant CHE References wx 1 For reviews see W.H. Miller, Faraday. Discuss. 11 Ž wx 2 W.H. Thompson, W.H. Miller, J. Chem. Phys. 16 Ž wx 3 H. Wang, W.H. Thompson, W.H. Miller, J. Chem. Phys. 17 Ž wx 4 H. Wang, W.H. Thompson, W.H. Miller, J. Phys. Chem. A 12 Ž wx 5 D. Brown, J.C. Light, J. Chem. Phys. 97 Ž wx 6 D.H. Zhang, J.C. Light, J. Chem. Phys. 14 Ž wx 7 D.H. Zhang, J.C. Light, S.-Y. Lee, J. Chem. Phys. 19 Ž wx 8 U. Manthe, J. Chem. Phys. 12 Ž wx 9 F. Matzkies, U. Manthe, J. Chem. Phys. 18 Ž w1x F. Matzkies, U. Manthe, J. Chem. Phys. 11 Ž w11x See, for example, Z.J. Zhang, R. Wyatt Ž Eds.., Dynamics of Molecules and Chemical Reactions, Marcel Dekker, New York, w12x W.H. Miller, J. Chem. Phys. 61 Ž w13x W.H. Miller, S.D. Schwartz, J.W. Tromp, J. Chem. Phys. 79 Ž w14x See also, T. Yamamoto, J. Chem. Phys. 33 Ž w15x L. Schlessinger, Phys. Rev. 167 Ž w16x G.A. Baker Jr. Essentials of Pade Approximants, Academic, New York, w17x D. Thirumalai, B.J. Berne, J. Chem. Phys. 79 Ž w18x R. Jaquet, W.H. Miller, J. Phys. Chem. 89 Ž w19x K. Yamashita, W.H. Miller, J. Chem. Phys. 82 Ž w2x W.H. Miller, J. Chem. Phys. 53 Ž w21x M.F. Herman, E. Kluk, Chem. Phys. 91 Ž w22x B.E. Guerin, M.F. Herman, Chem. Phys. Lett. 286 Ž w23x E.J. Heller, J. Chem. Phys. 94 Ž w24x F. Grossman, E.J. Heller, Chem. Phys. Lett. 241 Ž w25x K.G. Kay, J. Chem. Phys. 11 Ž w26x G. Campolieti, P. Brumer, J. Chem. Phys. 19 Ž w27x S. Garashchuk, D.J. Tannor, J. Chem. Phys. 19 Ž w28x A.R. Walton, D.E. Manolopoulos, Chem. Phys. Lett. 244 Ž w29x W.H. Miller, J. Chem. Phys. 95 Ž w3x B.W. Spath, W.H. Miller, J. Chem. Phys. 14 Ž w31x. Sun, W.H. Miller J. Chem. Phys. 18 Ž w32x D.V. Shalashilin, B. Jackson, Chem. Phys. Lett. 291 Ž