A Hybrid Memory Kernel Approach for Condensed Phase Non-Adiabatic Dynamics

A hybrid memory kernel approach for condensed phase non-adiabatic dynamics

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As Published http://dx.doi.org/10.1063/1.4990739

Publisher American Institute of Physics (AIP)

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Citable link http://hdl.handle.net/1721.1/113586

Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ A hybrid memory kernel approach for condensed phase non-adiabatic dynamics Diptarka Hait,1, 2 Michael G. Mavros,1 and Troy Van Voorhis1, a) 1)Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA 2)Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California, Berkeley, California 94720, USA (Dated: 13 June 2017) The spin-boson model is a simplified Hamiltonian often used to study non-adiabatic dynamics in large condensed phase systems, even though it has not been solved in a fully analytic fashion. Herein, we present an exact analytic expression for the dynamics of the spin-boson model in the infinitely slow bath limit and generalize it to approximate dynamics for faster baths. We achieve the latter by developing a hybrid approach that combines the exact slow-bath result with the popular NIBA method to generate a memory kernel that is formally exact to second order in the diabatic coupling but also contains higher-order contributions approxi- mated from the second order term alone. This kernel has the same computational complexity as NIBA, but is found to yield dramatically superior dynamics in regimes where NIBA breaks down—such as systems with large diabatic coupling or energy bias. This indicates that this hybrid approach could be used to cheaply incorporate higher order effects into second order methods, and could potentially be generalized to develop alternate kernel resummation schemes.

I. INTRODUCTION Equations of Motion (HEOM)27,28 and Quantum Monte Carlo (QMC)29,30 have therefore been used to study the Condensed phase chemical processes are strongly in- dynamics of the spin-boson model. Many of these ap- fluenced by the large number of solvent degrees of free- proaches are formally numerically exact, but often incur dom that interact with the reacting molecules1–4. Accu- significant computational cost that makes exploration of rate modeling of condensed phase dynamics thus requires parameter space difficult. Analytic approximations like 15,31,32 incorporation of solvent effects, but this is difficult to the Non-Interacting Blip Approximation (NIBA) 33–35 achieve via atomistic ab initio techniques as the compu- and generalized quantum master equations are also tational cost scales steeply with the number of solvent co- often used to study the dynamics of the spin-boson ordinates treated quantum mechanically. The generality model, although they may also be quite computation- of such atomistic approaches5–10 makes them extremely ally expensive or be effective only in certain parameter useful nonetheless11–14, but they may not represent the regimes. best tools for many problems. 1–3 In this paper, we present an exact analytic solution to The success of Marcus theory , on the other hand, the spin-boson model in the limit of an infinitely slow suggests that simple model systems containing a few pa- bath, and combine this result with NIBA to generate a rameters (obtained from experiment or calculations on hybrid kernel for the dynamics of general baths with arbi- single potential energy surfaces) could be an alternate trary spectral densities. This hybrid kernel approxima- route to study condensed phase non-adiabatic dynamics. tion has the same computational complexity as NIBA, A well known example of such a model is the spin-boson 15 but is found to yield much superior results in regimes model , which is used quite often to study the dynamics where the latter fails, being approximately as effective of electron transfer4,16–18 as it reduces to the highly suc- 17 as a fourth order resummed memory kernel approach cessful Marcus theory under appropriate limits . This for all cases tested. In particular, it is found to give suggests that it can be viewed as a dynamical generaliza- quite good performance in problematic cases involving tion of Marcus theory that could be useful for studying low temperatures, large diabatic coupling and large en- dynamics of electron transfer and non-adiabatic molecu- ergy biases, where NIBA is known to give qualitatively arXiv:1704.07477v2 [physics.chem-ph] 9 Jun 2017 lar dynamics in general. incorrect behavior. This indicates that the hybrid ap- However, an exact analytic solution for the population proach is a promising alternative to naive NIBA, as it has dynamics of the spin-boson model has not yet been found, the same computational complexity but yields superior despite the relative simplicity of the Hamiltonian. Vari- results in all regimes tested. Consequently, it can be used ous numerical approaches like Quasi-Adiabatic Propaga- to cheaply explore a large parameter space for screening tor Path Integral (QUAPI)19–24, Multi-Configurational 25,26 purposes over a more expensive method like MC-TDH. Time-Dependent Hartree (MC-TDH) , Hierarchical The developed approach is also able to approximate dynamics from second order information alone, suggesting that it can potentially be generalized to develop a suite of resummation methods that can be applied to problems a)[email protected] more general than the spin-boson model. 2

II. THE SPIN-BOSON MODEL the memory kernels K11/22(t, t1) control the dynamics of the population pi(t) in the state |ii in the following The spin-boson model15 is one of the simplest mod- manner: els for open quantum systems, describing the interaction t between a two level system and a bath of harmonic os- dp Z 1 = (K (t, t )p (t ) − K (t, t )p (t )) dt (5) cillators. The model assumes that the diabatic coupling dt 22 1 2 1 11 1 1 1 1 V between the two system states is constant and the 0 t system-bath coupling is linear in the bath coordinates. Z dp2 There may however be a non-zero intrinsic energy bias = (K11(t, t1)p1(t1) − K22(t, t1)p2(t1)) dt1 (6) between the two system states. dt 0 Therefore, the Hamiltonian HSB, in the basis of or- 1 0 thogonal diabatic states |1i = and |2i = , is However, the only independent quantity here is the 0 1 difference p1(t) − p2(t) as p1(t) + p2(t) = 1 for all times. given by: This difference is termed as hσzi (t) since p1(t) − p2(t) = Tr [ρ(t)(|1i h1| − |2i h2|)] = Tr [ρ(t)σ ], where ρ(t) is the h V z H = 1 (1) time-dependent density matrix for the whole system. We SB V h 2 thus actually just have a single equation: 2 X pi 1 2 2 h1 = − + + miωi xi + cixi (2) t 2 2mi 2 Z i d hσzi 2 = − (K−(t, t1) + K+(t, t1) hσzi (t1)) dt1 (7) X pi 1 2 2 dt h = + + m ω x − c x (3) 0 2 2 2m 2 i i i i i i i where K (t, t ) = K (t, t ) ± K (t, t ). The harmonic bath has a set of modes {x } with fre- ± 1 11 1 22 1 i Time-dependent perturbation theory allows us to ex- quencies {ω } and masses {m }. These modes interact i i pand the population difference hσ i (t) and the kernels linearly with the system states |1i and |2i via the cou- z K±(t, t1) into: pling constants {ci}. Consequently, h{1,2} are compact descriptions for the bath Hamiltonians associated with ∞ X m (m) states |1i and |2i. hσzi (t) = V hσzi (t) (8) There are typically a large number of bath modes m=0 present but we are chiefly interested in the diabatic pop- ∞ X m (m) ulations and not the minutiae of individual bath coordi- K±(t, t1) = V K± (t, t1) (9) nates. It is therefore reasonable to treat the bath oscil- m=0 lators as a continuum via a spectral density J(ω) of the 36 (m) (m) form : where hσzi (t) and K± (t, t1) can be found via the N 2 procedure described in Appendix A. X πci J(ω) = δ(ω − ωi) (4) The popular non-interacting blip approximation 2m ω 31,32,37 i=1 i i (NIBA) approximates K by only using the second- order term (i.e. K ≈ V 2K(2)), yielding quite accurate It is known that the dynamics of the spin-boson model dynamics in the small V (‘outer sphere’) regime and re- is completely specified by V, and J(ω)31,36,37-though an ducing to Marcus theory for slow bath frequencies at high actual closed form expression remains elusive. temperatures17. It is expected that the dynamics ob- We also note that the spin-boson model is extremely tained from kernels incorporating higher order terms will simplified and neglects many physical effects (non- be even more accurate, but would be much more com- Condon effects38,39, Duschinski rotations40 and bath an- putationally expensive to obtain as kernels accurate to harmonicity, to name a few) that are relevant in many 2nth order in V require evaluation of 2n − 1 dimensional condensed phase processes of interest. However, the rela- oscillatory integrals. Therefore, alternative approaches tive difficulty in solving even this simplified Hamiltonian that can obtain reasonable approximations of higher or- makes it reasonable to use the spin-boson model as a (2) starting point for method development and then gener- der terms from only second-order terms like hσzi are alize to more complex models. desirable as they would incur a much lower computational cost.

III. MEMORY KERNELS A. Initial conditions Memory kernels41–44 represent one possible approach for studying the spin-boson and related models. We em- The rest of the paper will assume that the initial den- ploy a generalized version of the memory kernel formal- sity matrix ρ(0) = p1(0) |1i h1| ⊗ ρB + p2(0) |2i h2| ⊗ ρB0 , 33 (2k+1) ism of Sparpaglione and Mukamel in this paper, where which causes odd order power series terms hσzi (t) 3

(2k+1) (4) and K± (t) to be 0. We will perform relevant deriva- Similarly, hσzi (t) is expressed in terms of integrals of (6) tions with the even simpler initial condition of ρ(0) = f4, hσzi (t) by integrals of f6 and so on. |1i h1| ⊗ ρB =⇒ hσzi (0) = 1 for mathematical simplicity, as the expressions derived from this form can be triv- ially generalized to any initial separable diabatic density C. Non-Interacting Blip Approximation matrix due to linearity of time evolution. In particular, we would like to emphasize that Eqn 27 and beyond are Going up to second order alone, we discover that true for all ρ(0) = p1(0) |1i h1| ⊗ ρB + p2(0) |2i h2| ⊗ ρB0 . (2) K11 (t, t1) = 2Re [f2(t, t1)] is consistent with Eqn. 12. The initial bath density matrix ρB is determined by This expression is similar to the non-equilibrium golden the nature of the dynamical process of interest. Ground rule in Ref [45], although the formalism employed therein state electron transfer processes often use equilibrium is convolution free unlike Eqns 5-6. For the case of e−βh1 1 e−βh1 ρ = (β being the inverse temperature ) B −βh ρB = −βh (thermal equilibrium initial conditions), Tr [e 1 ] kBT Tr [e 1 ] as this describes a scenario where the bath modes are this reduces to the well-known Non-Interacting Blip Ap- 15,31,32 in thermal equilibrium with the reduced Hamiltonian h1 proximation (NIBA) . The second-order kernels associated with the state |1i (which has all the popu- (2) 34 K11/22 are then given by : lation). On the other hand, non-equilibrium ρB of the 0 e−βh2 K(2)(t, t ) = 2e−Q (t−t1) cos Q00(t − t ) + (t − t ) (13) type can be useful for photochemistry, as 11 1 1 1 −βh 0 Tr [e 2 ] (2) −Q (t−t1) 00 K22 (t, t1) = 2e cos Q (t − t1) − (t − t1) (14) this corresponds to a situation where the population was ∞ purely in |2i with the bath in thermal equilibrium, be- 4 Z J(ω) βω Q0(t) = (1 − cos ωt) coth dω (15) fore an excitation at t = 0− led transfer of all the dia- π ω2 2 batic population to |1i without giving the bath modes 0 ∞ an opportunity to relax accordingly, due to difference 4 Z J(ω) Q00(t) = sin ωt dω (16) in timescales. We call this initial condition the “photo- π ω2 chemical” non-equilibrium condition throughout the rest 0 of the paper for convenience, although it need not cor- where J(ω) is the spectral density defined earlier in Eqn. respond to experimentally observable photochemistry for 4. It can quickly be noted that these kernels only de- all systems. Other non-equilibrium initial conditions may pend on the time difference t − t1 and, as such, we can also make physical sense for different processes. Unless (2) simply replace the function of two variables K11/22(t, t1) specified otherwise, all our expressions are valid for any (2) −βh e 3 X with the univariate K11/22(t − t1). The resulting kernel ρB = where h3 = h0 + dixi ({di} can be (2) Tr [e−βh3 ] K (t) ≈ V 2K (t) is termed as the NIBA kernel. i 11/22 11/22 any arbitrary real number) is a bath Hamiltonian with This approach is expected to yield decent dynamics for the same frequencies and modes as h1/2, but with any small V , as it is accurate to the lowest non-zero order for arbitrary set of equilibrium positions. this model. For non-equilibrium initial conditions however, the traces involved are not time-translationally invariant as B. Power series terms they depend on both (t, t1) and not just the difference t − t1, unlike the equilibrium case. As an example, this generalization for the case of non-equilibrium initial Evaluation of the non-zero even order terms in the −βh2 e 39 power series of hσzi (t) involves tracing over bath ρB = results in kernels : Tr [e−βh2 ] modes. Time-dependent perturbation theory reveals (2n) (2) 0 that the power series coefficients hσzi (t) for ρ(0) = −Q (t−t1) K11 (t, t1) = 2e cos (φ(t, t1) + (t − t1)) (17) |1i h1|⊗ρB are completely specified by integrals of traces 0 (2) −Q (t−t1) f2n(t1, t2, t3 . . . t2n) where: K22 (t, t1) = 2e cos (φ(t, t1) − (t − t1)) (18) 00 00 00 φ(t, t1) = Q (t − t1) + 2Q (t1) − 2Q (t) (19) † † f2n(t1, t2, t3 . . . t2n) = Tr O(t1)O (t2)O(t3) ...O (t2n)ρB (10) which are very similar to the equilibrium kernels, aside from the subtle difference stemming from the phase ih1t −ih2t O(t) = e e (11) φ(t, t1) not being time-translationally invariant. The 2 (2) resulting kernels K (t, t1) ≈ V K (t, t1) cannot For instance, we have: 11/22 11/22 strictly be called NIBA kernels, but this is effectively a

t natural extension of the idea behind NIBA, and is used d Z in its place for non-equilibrium initial conditions. Simi- hσ i(2) (t) = −4 Re [f (t, t )] dt (12) dt z 2 1 1 lar expressions had been derived earlier for both memory 0 kernel26 and non-equilibrium golden rule studies38. 4

D. Resummed Kernels random {di} over a large number of randomly selected time indices {ti}, indicating that it is correct to 60th or- NIBA is only correct to second order, and performs der at least, and very likely beyond as well (though as of quite poorly for systems with large diabatic coupling, now, we do not possess a proof for this conjecture). This large energy bias or low temperature. The natural step result is remarkable because it implies that for any spec- forward seems to be adding fourth or sixth order terms to tral density, the dynamics is a function of f2 alone. That the approximate kernel. However, simply stopping there is to say that, in principle, one can predict all of the dy- might be problematic as ﬁnite truncations of perturba- namics using only f2(t1, t2) as an input. In analogy with 51 tion theory series are often not convergent46. Resum- time density functional theory (in which the dynamics mation techniques help ameliorate this complication by of a many particle system is written as a functional of the approximating the unknown higher order terms from the one particle density alone), one can envision a f2(t1, t2) known low order terms. The recipe for memory kernel functional theory - in which the dynamics are directly resummation has been outlined in Refs [34,47] and other predicted by some functional of f2 alone. NIBA is one places, and we will not discuss the details here. We will such theory, but the result above suggests that the exact merely note that this approach yields signiﬁcantly supe- result can in principle be constructed from f2 alone. rior results compared to NIBA34,47,48 but this increased accuracy comes with the much higher computational cost of evaluating K(4) or other higher order kernels exactly. V. CASE OF THE SLOW BATH

Though Eqn. 20 is quite useful in finding all the per- IV. TRACE RELATIONS turbation theory traces, it does not eliminate the need to numerically integrate f2n over the 2n time indices in or- (2n) Closed-form expressions for the traces f2n for n ≥ 1 der to find hσzi (t). This quickly becomes intractable (2n) (2n) due to the grid size growing exponentially with n, creat- are necessary for calculating hσzi or K . We dis- ing a practical upper limit for applying naive perturba- covered that all traces f2n can be expressed in terms of tion theory even though we can find traces to arbitrary the second order trace f2 alone for any integer n and any choice of the spectral density; if the initial bath density order 2n rather simply via Eqn. 20. e−βh3 X However at short times, we may expand ln f2(t1, t2) as matrix ρB equals for h3 = h1 + dixi ({di} a Taylor series about t = t = 0 to obtain: Tr [e−βh3 ] 1 2 i 2 can be any arbitrary set of real numbers of cardinality ln f2(t1, t2) = ib(t1 − t2) − a(t1 − t2) + ... (22) N). Specifically, we have: where 2n−1 2n X X i+j+1 b = Tr [(h1 − h2)ρB] (23) ln f2n(t1, t2, t3 . . . t2n) = (−1) ln f2(ti, tj) 2 i=1 j=i+1 Tr (h − h )2ρ − (Tr [(h − h )ρ ]) a = 1 2 B 1 2 B (24) (20) 2

For example: Higher order terms can be neglected when ωt1,2 1, where ω is the characteristic response frequency of the f (t , t )f (t , t )f (t , t )f (t , t ) f (t , . . . t ) = 2 1 2 2 2 3 2 3 4 2 1 4 (21) system. Taking this short-time/slow-bath approximation 4 1 4 2 f2(t1, t3)f2(t2, t4) and defining a Gaussian function g(t) = e−at +ibt, we obtain: and similarly for f6, f8 etc. Essentially, an initial bath density matrix correspond- f2(t1, t2) ≈ g(t1 − t2) (25) ing to a thermal state of any harmonic bath h3 with f2n(t1, t2 . . . t2n) ≈ g(t1 − t2 ... + t2n−1 − t2n) (26) same frequencies and modes as h1/2 is sufficient to sat- isfy the trace factorization relationship Eqn. 20 (irre- which enables us to approximate all these traces as Gaus- spective of the equilibrium positions of the bath modes sian functions that can be analytically integrated. Car- in h3) for any bath spectral density. Similar relationships rying out the integrations over the time indices ti and had been presented earlier for equilibrium initial condi- summing over all orders in V (in the manner described 37,49 in Appendix C), we find that: tions of h3 = h1 but we believe that the case of general h3 has not previously been reported. This rela- t (2) Z (2) q tionship has been analytically verified to only 12th order ˙ 2 ˙ 3 ˙ 2 2 hσzi(t) = V hσzi (t) − 2V J1(2V t1)hσzi t − t1 dt1 for h = h (equilibrium initial conditions) and 6th or- 3 1 0 der for h3 = h2 (“photochemical” non-equilibrium initial (27) conditions) due to the rather steep memory cost for the  2  √ − b 4a 2at√+ib evaluation of the integrals needed to find the analytic ex- (2) 2 πe erf 2 a 50 hσ˙ i (t) = −Re  √  (28) pressions with Mathematica . It has however been veri- z  a  fied numerically up to f60 for multiple h3 resulting from 5

(a) (b)

(c) (d) −4 FIG. 1. Comparison between GaussSB, NIBA and HEOM at V = 1, ωc = 10 and = 0. where J1 is a Bessel function of the first kind. We note approximate form given in Eqn 28. This is done in the that Eqn 27 holds for any initial density matrix of the di- hope of generalizing Eqn 27 to baths with non-Gaussian 2 agonal form p1(0) |1i h1|⊗ρB +p2(0) |2i h2|⊗ρB0 , courtesy f2 as the substitution makes Eqn 27 exact to order V the linearity of time evolution. for any set of parameters. We call this version of Eqn For equilibrium initial conditions we have b = + λ 27 GaussSB as it is exact for spin-boson problems with ∞ ∞ Gaussian f2. 2 Z βω 4 Z J(ω) and a = J(ω) cosh dω; where λ = dω π 2 π ω 0 0 λ is the Marcusian reorganization energy. a ≈ in the VI. BENCHMARKING ACCURACY OF GAUSSSB β limit of frequencies {ω } → 0. The expression for hσ i (t) i z The lack of a rigorous proof for Eqn. 20 raises some obtained with these values for a, b appears to be consis- well warranted questions about the accuracy of GaussSB tent with the results obtained for a slow bath by Horn- even in the Gaussian limit, leading us to compare it bach and Dakhnovski52, which makes us more confident against HEOM27,28—a method known to be numerically in overall accuracy of our result and the applicability to λ ωωc general initial conditions beyond the equilibrium case. exact for J(ω) = 2 2 if a sufficiently large The “photochemical” non-equilibrium initial condition, 2 ω + ωc number of Matsubara frequencies are employed and a on the other hand, yields the same a, but b = − λ in- high hierarchy depth used for calculations. This particu- stead, indicating that for slow-baths at least, equilibrium lar form of J(ω) appears to be suboptimal for GaussSB and “photochemical” conditions lead to same dynamics on account of the long, slowly decaying high-frequency if = 0. tail, but we nonetheless stick with it for convenience in ˙ (2) We can also substitute the exact hσzi (t) = benchmarking. We also compared GaussSB to NIBA in t Z order to determine the differences between the two, since −4 Re [f2(t, t1)] dt1 in Eqn 27 instead of the Gaussian they both attempt to approximate the dynamics with the (2) 0 second order term hσzi (t) alone. Comparisons were 6

(a) (b)

−4 FIG. 2. Comparison between GaussSB, NIBA and HEOM at ωc = 10 ,V = 1, λ = 1, β = 5 and = −1 for both equilibrium and “photochemical” non-equilibrium initial conditions.

done in both the slow bath limit (where f2 is Gaussian cellent agreement between GaussSB and HEOM, despite and GaussSB is expected to be exact) and the fast bath GaussSB and NIBA both only employing second order (2) limit (where f2 is decidedly non-Gaussian and GaussSB information hσzi (t) alone. is likely to fail). HEOM calculations were done with the PHI code53 while both GaussSB and NIBA were implemented in C++ employing the error function implementation by Johnson54 and the GSL implementation of Quadpack integration routines55,56.

A. Slow Baths

GaussSB is expected to be valid in the ωc → 0 limit −4 (slow bath limit) and so ωc was set to 10 for all calculations in this subsection. We also set V = 1 throughout in order to fix the timescale of oscillations in diabatic populations. We first checked for cases without bias ( = 0) over a wide range of temperatures and reorganization energies (0.1 ≤ λ, β ≤ 10). Only equilibrium initial conditions were tested, as “photochemical” initial conditions (a) give the same Gaussian parameters a, b for cases where = 0, and thus should yield same dynamics. Four different behaviors were observed in this regime and these are depicted in Fig 1. There is visual agreement between GaussSB and HEOM in each case presented (as well as in many other cases tested that resulted in qualitatively similar dynamics), while NIBA often even fails to reproduce the qualitative behavior. These tests are consistent with our claim that GaussSB is exact in the slow bath limit. We next consider cases with bias, where equilibrium and “photochemical” dynamics are expected to give different results. Two selected cases are depicted in Fig 2 (b) and we observed similar behavior over other ranges of FIG. 3. Behavior of GaussSB in fast baths (ωc = 1): (a) The parameters as well. NIBA is known to be problematic in coherent regime, and (b) The dissipative regime. cases with large |β|, often yielding absurd |hσzi (t)| > 1. We observe the same behavior here, but also find ex- 7

B. Behavior for fast baths kernel typically decays much more rapidly than the population transfer rate35, and thus it seemed more appro- GaussSB cannot be expected to work exactly for fast priate to connect the memory kernels of GaussSB and baths since we explicitly used a small-time approxima- NIBA together in order to better utilize the short-time exactness of GaussSB. This led us to numerically in- tion taylor series to generate Gaussian traces and sum GSB the perturbation theory series. Numerical tests indicate vert GaussSB hσzi to obtain kernels K11/22(t) and then that this is indeed the case, with GaussSB being qualita- smoothly interpolate between these kernels and the exact (2) tively incorrect in the dissipative regime. We believe that second order kernels K11/22(t, t1) to get hybrid kernels the failure of GaussSB stems from slow decay of coher- that tend to pure second-order at long times and pure ent oscillations courtesy of the Gaussian approximation GaussSB at short times. Mathematically, this implies: (which implicitly assumes slow bath response), as it is I GSB the bath response that ultimately causes the transition K11/22(t, t1) = u(t − t1)K11/22(t − t1) from oscillatory behavior to dissipation. A related conse- (2) quence of this is that the formal rate constant associated + (1 − u(t − t1)) K11/22(t, t1) (29) with GaussSB (obtained from integrating the memory kernel from t = 0 to t → ∞) is always zero, indicating where KI is the interpolated kernel and u(t) is an inter- that GaussSB cannot reproduce correct equilibrium pop- polating function with the following properties: ulations or replicate long time exponential decay of the type predicted by Marcus theory and experimentally ob- 1. u(t = 0) = 1. served in chemical processes at high temperatures. This 2. lim u(t) = 0. represents quite a major obstacle to using pure GaussSB t→∞ for realistic problems where the dissipative regime is involved. 3. The rate of decay of u(t) (or equivalently, the On the other hand, we do obtain qualitatively correct growth rate of 1 − u(t)) is inversely related to the behavior with GaussSB in regimes where the dynamics is bath response time. This ensures that GaussSB be- controlled by coherent oscillations as opposed to dissipa- havior dies on the timescale of the bath response, tion (namely cases with large β, V, ), but the agreement since it becomes increasingly inaccurate at long is not quantitative. We do however note that GaussSB times. does much better than NIBA in this regime, again despite 4. u(t) is well-behaved (i.e., non divergent and contin- both of them using only second order information alone. uous, preferably differentiable). Examples for behavior in both regimes are presented in Fig. 3. I The kernel K11/22(t, t1) is thus almost pure GaussSB when the two time indices t, t1 are close, almost pure second order kernel when t, t1 are far and intermediate VII. INTERPOLATED KERNELS between the two extremes at intermediate separations. (2) For the equilibrium case, K11/22(t, t1) is the NIBA ker- Despite the issues related to applicability to fast baths, nel, and NIBA behavior is recovered at long times. we know GaussSB is accurate for time-scales shorter than There exist many possibilities for u(t), depending on bath response, as evidenced by its accuracy in the slow what choices are made regarding bath response times. bath limit. NIBA is simultaneously known to be accu- λ ωωc For the previously employed J(ω) = 2 2 form, it rate at times significantly larger than the bath response 2 ω + ωc as it can reproduce dissipative behavior, indicating that is at least possible to unambiguously define a timescale an approach that smoothly ties these two regimes to- by means of ωc, but such an option is not open for all gether could potentially be more effective than either. spectral densities. This leads us to obtain decay rate ∞ This interpolation scheme should recover pure GaussSB Z βω type behavior as t → 0, NIBA type behavior as t → ∞ J(ω) cos ωt coth dω 2 and behavior intermediate between the two at all times 0 in between.This is distinct from hybrid quantum-classical via θ(t) = ∞ as θ(t) corresponds 25,26 Z βω methods like MC-TDH , as it combines exact quan- J(ω) coth dω tum mechanical results at two limiting cases, as opposed 2 0 to discriminating between different bath modes by treat- to the normalized energy gap fluctuation autocorrelation ing only a fraction quantum mechanically. Re [hδ∆E(t)δ∆E(0)i] function , where the numerator is One issue in constructing an interpolating scheme is hδ∆E(0)δ∆E(0)i that GaussSB and NIBA approach population dynam- a quantity often computed in MD simulations to obtain ics in completely different ways. NIBA employs a non- J(ω) and is thus readily available in most cases. This de- ˙ Markovian memory kernel to connect hσzi and hσzi while cays smoothly from 1 → 0 as t runs from 0 → ∞. Conse- ˙ GaussSB directly obtains hσzi via Eqn 27. The memory quently, θ(t) is a measure of the dissipation rate of initial 8

(a) (b)

FIG. 4. Comparison of Interpolated kernel populations against HEOM, NIBA and Padéresummed kernel correct to fourth order (with ωc = 1, β = 0.125, λ = 4 and = 0). The interpolated kernel approach performs better as V increases. energy fluctuations and thus gives a timescale for switch- creases in magnitude57, indicating that p can be a decent ing to the dissipative regime. Furthermore, θ(t) = e−ωct metric for controlling the GaussSB to NIBA switching for Debye spectral densities at small β (high tempera- rate. The absolute value in u(t) = |θ(t)|p is taken since tures, i.e. the classical limit), indicating that it in fact though θ(t) ≥ 0 in most cases, a few pathological param- decays with the same timescale as the bath frequencies. eter sets might lead to θ(t) < 0, which would make taking non-integer powers problematic. Taking |θ(t)| solves θ(t) thus is a good metric for the rate of decay. That this issue, though it can lead to non-differentiable u(t) at is however not the only factor we should consider, as al- points where θ(t) = 0 changes sign. This is unfortunate, though θ(t) contains information about bath frequencies but we feel it is an acceptable compromise due to its rar- as well as some temperature effects, it does not at all ity and because it does not lead to non-differentiable hσzi account for the energy gap which is known to affect as it only affects the kernel. population coherence time-scales as well. Furthermore, NIBA is known to be especially terrible for large |β| (as For determining the accuracy of this interpolated ap- multiple figures in the preceding section highlight) and proach, we considered a few Hamiltonian parameters this weakness also ought to be considered for building studied earlier via memory kernel resummation in Ref u(t). We therefore propose u(t) = |θ(t)|p to be a general [34] (all using equilibrium initial conditions) and com- form for the interpolating function u(t), where p is very pared the performance of HEOM, NIBA, Interpolated small for β 1 (allowing GaussSB to dominate) but GaussSB and Padé resummed memory kernels58 (ob- reduces to just 1 for = 0. p = sechβ was chosen by us tained from data generated for Ref [34]) correct to fourth as it satsifies the above requirements and corresponds to order in V . The results for cases with = 0 are given 1 in Fig 4 and it appears that the interpolated kernel ap- the eβA(), where A () = − ln cosh β is the component β proach is comparable to (and for large V is superior to) of the Gibbs-Boguliubov free energy dealing with bias57. dynamics obtained from the much more computation- NIBA performance is known to deteriorate as A() in- ally expensive fourth-order resummed kernel approach. 9

It thus appears that the interpolated kernel approach could be a cheap and effective way to incorporate V 4 and higher order terms to NIBA at a significantly lower computational cost than numerical evaluation of higher order kernels for resummation. We also note that in ad- dition to our chosen form of u(t), we experimented with some different forms of u(t) (decaying exponentials and Gaussians in ωct) for these cases and obtained roughly similar behavior. This aspect however was not explored in greater detail as other u(t) explored did not easily generalize to non Drude-Lorentz spectral densities where a characteristic ωc is not apparent. Cases with a non-zero pose a greater challenge, as we know that neither GaussSB nor NIBA can accurately determine the equilibrium populations in such cases and (a) thus the combined approach is unlikely to succeed in this task either. It has been shown34 that it is possible to adjust the resummation scheme to yield kernels correct to fourth order that recover equilibrium populations in a least-squares sense. This indicates that the interpolated kernel in these cases should be expected to perform worse than the optimal fourth order resummed kernel as the latter has more information than the former. Fig 5 depicts the comparison of HEOM, NIBA, intepolated kernel and optimized Landau–Zener resummation59 (found to be superior to Padéfor cases with bias34) for some systems studied in Ref [34]. This comparison indicates that the interpolated kernel approach gives dynamics of comparable quality to fourth-order resummed kernels, although it tends to perform worse at long times as it does (b) not contain information about hσzi (t → ∞) that the resummation technique employs. Nonetheless, it recovers quite reasonable short-time dynamics, with less information and at a fraction of the cost relative to the fourth order methods. Performance for the difficult cases where exact hσzi (t → ∞) is close to ±1 (such as Fig VII(a)) can potentially be further improved by altering u(t) to ensure recovery of correct equilibrium populations (perhaps via tuning the power parameter p).

VIII. CONCLUSION AND FUTURE DIRECTIONS

We have found a relationship between perturbation theory traces of the spin-boson model that allowed us to (c) solve for population dynamics analytically for the case of a Gaussian trace f2(t1, t2). We call this method GaussSB FIG. 5. Comparison of interpolated kernel populations and validate its accuracy by benchmarking it against against HEOM, NIBA and optimized Landau–Zener re- the highly accurate HEOM method for infinitely slow summed kernel correct to fourth order for cases with 6= 0. baths. GaussSB is unfortunately not particularly useful for many problems of chemical interest since f2(t1, t2) is on the timescale of the bath. This decay is controlled by only Gaussian for baths with small frequencies, though an interpolating function u(t), in order to prevent higher it appears to recover the non-dissipative behavior at order GaussSB terms (which become increasingly non- low temperatures with qualitative accuracy for even fast exact at long times) from becoming too influential on baths. timescales larger than bath response. This hybrid ap- We remedy this by formulating a hybrid kernel that proach yields qualitatively accurate dynamics for even is exact to second order in V and includes higher order fast baths that is comparable in quality to results ob- correction terms incorporated via GaussSB that decay tained from using more computationally expensive re- 10 summed memory kernels accurate to fourth order. The of Eqn 27 of the form hybrid kernel method is especially effective in regimes where NIBA normally breaks down (large β, V, ), indi- ˙ 2 ˙ (2) hσzi(t) =V hσzi (t) cating that the interpolation approach was successful in t incorporating critical amounts of higher order behavior. Z (2) q 3 ˙ 2 2 − 2V J1(2V f(t1)t1)hσzi t − t1 dt1 0 (30)

˙ where the function f(t1) is chosen to ensure hσzi(t) is ex- We have thus devised a hybrid method that interpo- act to fourth power in V . The kernel resulting from this lates between NIBA and GaussSB to give quite accurate expression will be exact to fourth order as it contains dynamics for the spin-boson model for the same compu- both second and fourth order information. Generalizing tational complexity as NIBA (since it does not need eval- even further, it appears that it might possible to generate uation of expensive triple or higher order integrals). This other resummation methods employing the general form hybrid approach could thus be used in place of NIBA to of Eqn. 27 that are exact in the limit of Gaussian f2. explore dynamics across a large parameter space since it They would still incur significant computational expense has superior performance to NIBA in traditionally prob- in order to have exact fourth (or higher) order behav- lematic regimes and is not observed to ever perform any ior, but could potentially prove more effective than pure worse that NIBA, on account of it incorporating higher Padéor Landau–Zener resummation and might not even order effects that NIBA misses. need interpolation on account of the extra information supplied. It may also be possible to recover the correct equilibrium populations with such kernels, and we hope to study the accuracy of such kernels (both with interpolation and without) in the future to determine whether this is indeed the case. It should in principle be possible to generalize this approach to other model Hamiltonians like the linear vi- bronic coupling (LVC) model39, as long as equivalent ACKNOWLEDGMENTS slow-bath solutions like GaussSB exist. In practice, there are problems stemming from the fact that simple trace This work was supported by NSF Grant No. CHE- factorization relations like Eqn 20 have not been found 1058219. D.H. would like to thank the MIT UROP office (and may simply not exist) for more sophisticated sys- for support. M.G.M. would also like to thank the NSF tems. Directly employing the GaussSB kernel (using GRFP program for funding. (2) the exact hσ˙zi (t) for the model) might be an acceptable compromise for such cases, even though it would sacrifice exactness in the slow-bath limit. The result- Appendix A: Time-Dependent Perturbation Theory ing method would resemble a resummation scheme as it would employ the exact second order term to approxi- Time-dependent perturbation theory is the necessary mate higher order terms, and would furthermore be much first step for any perturbative approaches to quantum cheaper than actually computing fourth order kernels for dynamics. We treat the diabatic coupling V as the the problem. We hope to explore the effectiveness of this perturbation, which allows us to define the zeroth or- approach in the future to determine whether it has any der Hamiltonian , consequently enabling us to express promise. operator A(t) in the interaction picture as AI (t) = eiH0tA(t)e−iH0t. The perturbation in the interaction 0 V picture is thus V (t) = eiH0t e−iH0t = V σx(t) I V 0 I (σx being the Pauli matrix in the x direction) and the interaction picture density matrix can be expressed as iH0t −iH0t In the future, we also seek to explore routes to general- ρI (t) = e ρ(t)e (where ρ(t) is the actual time ize GaussSB to have kernels correct to fourth (or higher) dependent density matrix). The von-Neumann equation order at all times and see if that yields improved dynam- for the time-evolution of the interaction-picture density ics. The easiest way to achieve this is via a generalization matrix allows us to state that: 11

t t t Z Z Z 1 ρI (t) = ρI (0) − i [VI (t1), ρI (0)] dt1 − dt1 [VI (t1), [VI (t2), ρI (0)]] dt2 ... (A1) 0 0 0 t t t Z Z Z 1 x 2 x x = ρI (0) − iV [σI (t1), ρI (0)] dt1 − V dt1 [σI (t1), [σI (t2), ρI (0)]] dt2 ... (A2) 0 0 0 ∞ X m (m) = V ρI (t) (A3) m=0

(m) ∞ where ρI (t) are independent of V . X m (m) K11/22(t) = V K11/22(t) as a power series in V and The diabatic populations p1/2(t) can be obtained from m=0 the density matrix by tracing out the bath modes and equating terms with same order of V on both sides of taking the diagonal elements. Mathematically: Eqns 5 and 6 allows us to express power series coefficients (m) (m) K (t) in terms of hσzi (t), which we have already p1(t) = Tr [ρI (t) |1i h1|] (A4) 11/22 ∞ shown can be found from perturbation theory. Going X m h (m) i (k) up to some finite order m and finding all hσzi (t) for = V TrB ρI (t) |1i h1| (A5) m=0 k ≤ m would thus provide us with the correct kernel (k) p2(t) = Tr [ρI (t) |2i h2|] = 1 − p1(t) (A6) coefficients K11/22(t), which can be directly employed or be resummed to approximate the exact kernel correct to p1/2(t) are not independent as p1(t) + p2(t) = 1, and all orders in V . thus the population dynamics can be specified with only knowledge of the difference p1(t) − p2(t). This is termed as hσzi (t) as p1(t)−p2(t) = Tr [ρI (t)(|1i h1| − |2i h2|)] = Tr [ρI (t)σz]. Thus we can express hσzi (t) as a power series in V , Appendix B: Taylor series for ln f2(t1, t2) (m) h (m) i with the mth order term hσzi (t) = TrB ρI (t)σz for all non-negative integers m. Expressing the kernels We have:

ih1t1 −ih2t1 ih2t2 −ih1t2 f2(t1, t2) = Tr e e e e ρB (B1) ∂f (t , t ) 2 1 2 ih1t1 −ih2t1 ih2t2 −ih1t2 = iTr e (h1 − h2)e e e ρB (B2) ∂t1 ∂f (t , t ) 2 1 2 ih1t1 −ih2t1 ih2t2 −ih1t2 = −iTr e e e (h1 − h2)e ρB (B3) ∂t2 ∂2f (t , t ) 2 1 2 ih1t1 2 −ih2t1 ih2t2 −ih1t2 2 = −Tr e (h1 − h2) e e e ρB (B4) ∂t1 ∂2f (t , t ) 2 1 2 ih1t1 −ih2t1 ih2t2 2 −ih1t2 2 = −Tr e e e (h1 − h2) e ρB (B5) ∂t2 ∂2f (t , t ) 2 1 2 ih1t1 −ih2t1 ih2t2 −ih1t2 = Tr e (h1 − h2)e e (h1 − h2)e ρB (B6) ∂t2∂t1

Since 12

∂ ln f (t , t ) 1 ∂f (t , t ) 2 1 2 = 2 1 2 (B7) ∂t1 f2(t1, t2) ∂t1 ∂ ln f (t , t ) 1 ∂f (t , t ) 2 1 2 = 2 1 2 (B8) ∂t2 f2(t1, t2) ∂t2 2 2 2 2 ∂ ln f2(t1, t2) 1 ∂ f2(t1, t2) 1 ∂f2(t1, t2) 2 = 2 − (B9) ∂t1 f2(t1, t2) ∂t1 f2(t1, t2) ∂t1 2 2 2 2 ∂ ln f2(t1, t2) 1 ∂ f2(t1, t2) 1 ∂f2(t1, t2) 2 = 2 − (B10) ∂t2 f2(t1, t2) ∂t2 f2(t1, t2) ∂t2 2 2 2 ∂ ln f2(t1, t2) 1 ∂f2(t1, t2) ∂f2(t1, t2) ∂ f2(t1, t2) = − + f2(t1, t2) (B11) ∂t1∂t2 f2(t1, t2) ∂t1 ∂t2 ∂t1∂t2

Expanding ln f2(t1, t2) as a Taylor series about t1 = t2 = 0 gives:

1 ln f (t , t ) = 0 + i(t − t )Tr [(h − h )ρ ] − Tr (h − h )2ρ − (Tr [(h − h )ρ ])2 (t − t )2 ... (B12) 2 1 2 1 2 1 2 B 2 1 2 B 1 2 B 1 2 Higher order terms can be neglected if:

3 ∂ ln f2(t1, t2) 2 3 3 3 = i 3Tr (h1 − h2) ρB Tr [(h1 − h2)ρB] − Tr (h1 − h2) ρB − 2 (Tr [(h1 − h2)ρB]) (B13) ∂t1

etc are small, which is the case if the bath frequen- for all integers n > 1. This form is more convenient to cies {ωi} are small. As an example, the combined handle in Laplace space where we have: e−βh1 third order term for equilibrium ρB = Tr [e−βh1 ] ∞ 2i Z is − (t − t )3 ωJ(ω) dω, which is guaranteed to 3π 1 2 0 be small if J(ω) is only considerable for small ω. Similarly, the fourth order term for this case is: ∞ (2n) 2 1 d (2n−2) (t − t )4 Z βω L hσ˙ i (s) = L hσ˙ i (s) (C2) 1 2 ω2J(ω) coth dω, which will be small if z n − 1 s ds z 6π 2 0 4 1 d 1 d (2n−4) = L hσ˙ i (s) J(ω) is only large for small ω. (n − 1)(n − 2) s ds s ds z (C3) n−1 1 2 d (2) = L hσ˙ i (s) (n − 1)! s ds z Appendix C: Derivation of Slow Bath Solution (C4)

Integrating Gaussian f2n analytically yields a relation ˙ (2n) between population growth terms hσzi :

t (2n) 2 Z (2n−2) hσ˙ i (t) = − t hσ˙ i (t ) dt (C1) z n − 1 1 z 1 1 by repeated application of Eqn C2. This permits us to ˙ 0 sum the inﬁnite power series for hσzi in Laplace space, 13 as we have: Appendix E: Kernels from Populations

" ∞ # h i X (2n) GaussSB gives hσ i (t) directly and does not employ L hσ˙ i (s) = L V 2nhσ˙ i (s) (C5) z z z any memory kernels. Nonetheless, it would be prefer- n=1 able to interpolate between time non-local memory ker- ∞ n−1 X V 2n 2 d (2) nels that population growth rates to fully exploit the the = L hσ˙ i (s) (n − 1)! s ds z complete short-time information contained in GaussSB, n=1 compelling us to determine ways to invert the GaussSB (C6) populations hσzi (t) to yield kernels. 2V 2 d (2) We begin from the one time-index version of the 2 ˙ 33 = V exp L hσzi (s) (C7) Sprapaglione-Mukamel formalism as there is no way (or s ds reason) to introduce a second time index whose behav- d (2) ior we cannot solely recover from the univariate hσ i (t). = V 2 exp 4V 2 L hσ˙ i (s) (C8) z ds2 z Speciﬁcally, we use a variant of Eqn 7 where: (2) t 2 ˙ p 2 2 d hσ i Z = V L hσzi ( s + 4V ) (C9) z = − (K (t − t ) + K (t − t ) hσ i (t )) dt (E1) dt − 1 + 1 z 1 1 0 t t Returning to the time domain, we have via known inverse Z Z 60 Laplace transforms : = − K+(t1) hσzi (t − t1)dt1 − K−(t1)dt1 (E2)

0 0 t (2) Z (2) q Differentiating Eqn. E2 with respect to time, we obtain: ˙ 2 ˙ 3 ˙ 2 2 hσzi(t) = V hσzi (t) − 2V J1(2V t1)hσzi t − t1 dt1 t Z 0 ¨ ˙ hσzi(t) = − K+(t1)hσzi(t − t1)dt1 − K−(t) − K+(t) hσzi (0) (C10) 0 t (E3) (2) Z hσ˙ i (t) = −4 Re [f (t, t )] dt (C11) z 2 1 1 Let us consider two initial conditions: hσzi (0) = ±1 0 and call the corresponding hσzi (t) as hσzi± (t). Then we 2  √ b 2at+ib  have: − 4a √ 2 πe erf 2 a = −Re √ (C12)   t a Z ¨ ˙ hσzi±(t) = − K+(t1)hσzi+(t − t1)dt1 − K−(t) ∓ K+(t) 0 where a 6= 0. J1 is a Bessel function of the first kind, giving us an analytical solution to the spin-boson model (E4) t for all baths with trace f Gaussian in time. Z 2 ¨ ¨ ˙ =⇒ hσzi+(t) − hσzi−(t) = − K+(t1) hσzi+(t − t1) 0 ˙ Appendix D: GaussSB in the limit of Rabi Oscillations −hσzi−(t − t1) dt1 − 2K+(t) (E5) GaussSB gives us hσ˙ i (t−t ) from which we can obtain If we consider the trivial case of J(ω) = 0, we obtain z ± 1 i(t1−t2) ¨ f2(t1, t2) = e , which is a complex exponential and the second derivatives hσzi±(t − t1) via direct differenti- not actually a Gaussian. Thus a = 0 and we can’t use ation of Eqn 27 or finite differences (which was the route Eqn. 28 directly, but it is easy to use Eqn. C10 and find: we took). Then we can solve Eqn. E5 numerically on a grid for K+(t) (by treating the integral as a finite sum t Z via trapezoid rule and solving the resulting system of lin- ˙ (2) 4 sin t hσzi (t) = −4 Re [f2(t, t1)] dt1 = − (D1) ear equations for instance, as we have done: but other approaches like Archimedes summation or other Newton- 0 Coates approaches should be equally valid), making use 2 of the fact that K+(0) = 4V since only second order √ 2 2 terms matter then. Finally, back substitution of K+(t) ˙ 4 sin t 4V + hσzi(t) = − √ (D2) in Eqn E4 is sufficient to recover K−(t), yielding the ker- 4V 2 + 2 √ nels. 4V 2 cos t 4V 2 + 2 + 2 =⇒ hσ i (t) = (D3) z 4V 2 + 2 Appendix F: Kernels in Laplace Space which is the exact result that can be directly calculated for this trivial example, showing that it correctly recovers Eqn. E5 offers an interesting alternative approach the Rabi limit. for accessing kernels via Laplace space. Taking Laplace 14

transforms on both sides, we obtain:

h ˙ i h ˙ i h ˙ i h ˙ i s L hσzi+ (s) − L hσzi− (s) = −L [K+](s) L hσzi+ (s) − L hσzi− (s) − 2L [K+](s) (F1) h ˙ i h ˙ i s L hσzi+ (s) − L hσzi− (s) =⇒ L [K ](s) = − (F2) + h ˙ i h ˙ i 2 + L hσzi+ (s) − L hσzi− (s)

(2) √ (2) √ 2 ˙ 2 2 ˙ 2 2 sV L hσzi+ ( s + 4V ) − L hσzi− ( s + 4V ) = − (F3) (2) √ (2) √ 2 ˙ 2 2 2 ˙ 2 2 2 + V L hσzi+ ( s + 4V ) − V L hσzi− ( s + 4V )

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