Kinetic Rate Kernels via Hierarchical Liouville–Space Projection

Article pubs.acs.org/JPCA

Kinetic Rate Kernels via Hierarchical Liouville−Space Projection Operator Approach Hou-Dao Zhang† and YiJing Yan*,†,‡ †

Department of Chemistry, Hong Kong University of Science and Technology, Hong Kong, China Hefei National Laboratory for Physical Sciences at the Microscale, iChEM (Collaborative Innovation Center of Chemistry for Energy Materials), University of Science and Technology of China, Hefei 230026, China

‡

ABSTRACT: Kinetic rate kernels in general multisite systems are formulated on the basis of a nonperturbative quantum dissipation theory, the hierarchical equations of motion (HEOM) formalism, together with the Nakajima-Zwanzig projection operator technique. The present approach exploits the HEOM−space linear algebra. The quantum non-Markovian site-to-site transfer rate can be faithfully evaluated via projected HEOM dynamics. The developed method is exact, as evident by the comparison to the direct HEOM evaluation results on the population evolution.

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INTRODUCTION Quantum network kinetics focuses on quantum transfer or reaction rate properties of multisite (or multichromophoric) systems, coupled nonperturbatively with a quantum bath environment. It concerns various quantum transport phenomena,1,2 including the coherent energy transfer in photosynthesis light-harvesting antenna complexes,3−9 and the subsequent charge transfers and separations in the reaction center.10−12 These nanostructured functional molecular complexes have comparable intrasystem coupling and system−environment coupling strengths. The time scales of transfer kinetics and bath fluctuations are also comparable. The consequent finite quantum coherence in space and in time, including correlated system-and-bath interferences, are believed to support the optimal functional performance.13−16 The kinetic rate problem deals with the evolution of populations in the form of Pȧ (t ) =

∑∫ b

t

0

dτ 2ab(t − τ )Pb(τ )

Various methods have been developed in the evaluation of non-Markovian rate kernels and Markovian rate constants. In the weak interchromophoric coupling regime, the rate can be expressed in terms of linear transfer coupling correlation functions. This results in the Marcus electron transfer rate17−19 and also the Förster resonance energy transfer theory.20 These linear correlation function-based theories cannot account for the multisite quantum coherence. Recently, Wu and coworkers9,21 developed a continued fraction method to systematically include high-order nonlinear corrections to the multichromophoric Föster theory. A similar block matrix inversion approach in the Laplace space for the time-integrated rate matrix has also been proposed.22,23 This advanced approach has been applied to excitation energy transfer kinetics in biological clusters.23,24 The significance of quantum coherence among several chromophores is verified.2−4,23,24

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THEORETICAL METHODS In this work, we revisit multisite quantum rate kernels on the basis of a nonperturbative quantum dissipation theory, the hierarchical equations of motion (HEOM) formalism,25−34 together with the Nakajima-Zwanzig projection operator technique.35,36 The basic idea in this development is as follows. First of all, the HEOM is linear and can be recast as (see eqs 13 and 14)

(1)

The populations, {Pa(t) = ⟨a|ρS(t)|a⟩}, are the diagonal elements of the reduced system density operator, ρS(t) = trBρT(t), in a given representation. The underlying particle number conservation leads to

ρ̇ (t ) = −i 3ρ(t )

∑ 2ab(t ) = 0

The involving generator 3 and dynamical variables,

(2)

a

ρ(t ) = {ρ(0)(t ) ≡ ρS (t ); ρn(n > 0)(t )}

Non-Markovian rate kernel matrix elements, {2ab(t )}, are the key quantities of the study. In the long-time or Markovian kinetics regime, eq 1 reduces to Ṗa(t)= ∑b KabPb(t), with the rate constants, Kab ≡

∫0

∞

dt 2ab(t )

(5)

Special Issue: Ronnie Kosloff Festschrift Received: December 1, 2015 Revised: January 11, 2016 Published: January 12, 2016

(3) © 2016 American Chemical Society

(4)

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DOI: 10.1021/acs.jpca.5b11731 J. Phys. Chem. A 2016, 120, 3241−3245

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The Journal of Physical Chemistry A are all well-defined.30−34 Recent development of dissipaton dynamics theory37−39 renders a quasi-particle picture of the hybrid bath dynamics. ρ(n>0) (t) are now identified to be the n dissipaton density operator (DDO) of the specified n-particle configuration.37−39 Thus, HEOM (eq 4) is actually a complete theory, as the system−bath entanglement is concerned.9,24,37−41 This observation qualifies the direct application of the commonly used Nakajima−Zwanzig projection operator technique35,36 onto the hierarchical system population subspace. To proceed, let us start with the reduced system subspace projection operators, 7 and 8 ≡ 1 − 7 , for partitioning ρS(t) ≡ ρ(0)(t) into the population and coherent components, respectively: (0)

7ρ (t ) ≡

∑

ρaa(0)(t )|a⟩⟨a|

≡

a

8ρ(0)(t ) ≡

generator 3 at the new temperature. However, for most of the rate problems, the initial total density operator is assumed in the factorized form ρT(0) = ρS(0)ρeq B (T), with the thermal −hB/(kBT) equilibrium bath, ρeq /trBe−hB/(kBT). It is easy to B (T) = e prove34 that the initial factorization ansatz amounts to > 0) ρ(n (0) = 0. Together with the diagonal initial ansatz of n ρS(0) = ∑a Pa(0)|a⟩⟨a|, the last term in eq 10 vanishes. In the following, we present the detail about the HEOM evaluation of rate kernel matrix, 2(t ) of eq 11. Its elements {2ab(t )} that enter eq 1 will be specified. Let us start with the typical HEOM expression, ρn(̇ n) = −(i 3S +

∑ Pa(t )|a⟩⟨a|,

k

It specifies the HEOM-space dynamical generator the genetic form of

The corresponding HEOM-space projection operators, 7 and 8 = 0 − 7 , are then defined via

⎡ 3S (k 0 0 ⎢ ⎢ *k 3S − iγk (k 0 3=⎢ 3S − 2iγk (k 2*k ⎢0 ⎢ ⎣⋮ ⋮ ⋮ ⋮

7ρ(t ) = {7ρ(0)(t ); 0, 0, ⋯} ≡ p(t ), 8ρ(t ) = {8ρ(0)(t ); ρn(n > 0)(t )} ≡ σ (t )

(7)

We can now recast the HEOM (eq 4) in terms of

⎡ ̂ (0) ⎤ ⎢X ⎥ X = ⎢ X̂ (1)⎥ ⎢ k ⎥ ⎢⎣ ⋮ ⎥⎦

(8)

The formal solution to σ(t) reads dτ e−i 83(t − τ)83p(τ )

(9)

t

(14)

dτ 2(t − τ )p(τ ) − i 73e−i 83t σ (0)

(n)

X̂ n =

(10)

(16)

The actions of 7 and 8 on X are (cf. eq 7) (11)

(0)

7X = {7X̂ ; 0, 0, ⋯},

In writing eq 10, we have also used the identity, −i 73p(t ) = 0

(n)

∑ X̂ n;ab|a⟩⟨b| ab

with the rate matrix kernel being formally of 2(t ) = −73e−i 83t 837

(15)

It can be either ρ of eq 5, or p or σ of eq 8. Each element in eq 15 is an operator in the reduced system subspace, denoted as

which leads to the first identity of eq 8 in the expression,

∫0

⋯⎤ ⎥ ⋯⎥ ⎥ ⋯⎥ ⎥ ⋮⎦

The genetic Liouville−space vector is defined as

⎛ 737 738 ⎞⎛ p ⎞ ⎛ ṗ ⎞ ⎟⎜ ⎟ ⎜ ⎟ = − i⎜ ⎝ σ̇ ⎠ ⎝ 837 838 ⎠⎝ σ ⎠

p ̇ (t ) =

(13)

k

(6)

a≠b

t

k

−i ∑ nk *kρn(−n − 1)

∑ ρab(0)(t )|a⟩⟨b|

∫0

+ k

k

a

σ (t ) = e−i 83t σ (0) − i

∑ nkγk)ρn(n) − i ∑ (kρn(n+ 1)

(0) (n > 0) 8X = {8X̂ ; X̂ n }

(12)

(17)

Here, 7 and 8 are defined according to eq 6. Having eq 14, we can also evaluate

This term describes the direct population-to-population transfer, which in general does not exist,42 as it directly related to the reduced system Liouvillian tensor elements of S S ⟨⟨aa|3S|bb⟩⟩ = (Hba − Hab )δab = 0. The projected 2(t ), eq 11, is defined with the HEOM-space operator notation, where a ̂ * , m ′ n ′ = Onm ̂ , n ′ m ′. Hermitian operator has the property of Omn Consequently, 2ab(t ) = 2aa , bb(t ) is real. The inhomogeneous term in eq 10 depends on the initial preparation of σ (t = 0) = {8ρ(0)(0); ρn(n > 0)(0)} (cf. eq 7).

⎡ ̂ (0)⎤ ⎢Y ⎥ Y = −i 3X = ⎢ Y ̂ (1)⎥ ⎢ k ⎥ ⎢⎣ ⋮ ⎥⎦

(18)

We now introduce Z(t ) ≡ e−i 83t X

(0)

Both the reduced system coherence 8ρ (0) (cf. eq 6) and the > 0) DDOs {ρ(n (0)} may be nonzero, for example, in a n temperature jump experiment. In this case, the initial values of ρ(t = 0), thus, σ(t = 0) and p(t = 0) via eq 7, can be determined via the steady-state solutions to the HEOM at the initial temperature.34 The temperature jump-induced kinetic rate processes are then studied on the basis of the HEOM

(19)

The propagation of Z satisfies the HEOM of Ż (t ) = −i(83)Z(t ) = −i 8[3Z(t )]

(20)

with the initial condition Z(0) = X. The above evaluation involves the calculation of −i 3Z(t ), i.e., the conventional HEOM propagation, followed by the 8 −projection. 3242

DOI: 10.1021/acs.jpca.5b11731 J. Phys. Chem. A 2016, 120, 3241−3245

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The Journal of Physical Chemistry A Equations 14−20 constitute all the ingredients for evaluating the matrix elements of reaction kernel, 2(t ) of eq 11. For the bth column elements, we start with ⎡ ̂ (0) ⎤ ⎡|b⟩⟨b|⎤ ⎢X ⎥ ⎢ ⎥ ⎢ ⎥ (1) =⎢ 0 ⎥ X b = X̂ ⎢ k ⎥ ⎢⎣ ⋮ ⎥⎦ ⎢⎣ ⋮ ⎥⎦ b

(21)

and evaluate (cf. eqs 17 and 18) Yb = 837X b = 83X b

(22)

With Zb(0) = Yb as the initial condition, we then integrate the equation of motion, Ż b(t ) = −i(83)Zb(t )

(23)

and obtain Zb(t ) = e−i 83t Yb . Finally, we have ⎡ 2 (t )|a⟩⟨a|⎤ ⎥ ⎢∑ ab a ⎥ 73Zb(t ) = −⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎦ ⎣ ⋮

Figure 1. Rate kernels for a three-site system: β(ϵ2−ϵ3) = 2β(ϵ2−ϵ1) = 2, βV12 = βV23 = 1, and V13 = 0, with β = 1/(kBT). The bath spectral density adopts the Drude model, with parameters βλ = 1 and βγ = 2, for all three sites subject to longitudinal fluctuations. (24)

The involving {2ab(t )} are the site-to-site rate kernels for eq 1 or eq 10. We would like to emphasize that all the accelerating techniques with standard HEOM propagator, such as the indexing43,44 and on-the-fly filtering45 algorithms, can be readily implemented to facilitate the integration of eq 23. In particular, the dynamical filtering algorithm helps to truncate the hierarchy automatically and numerically exact results can be anticipated.45

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RESULTS AND DISCUSSION For numerical illustrations, we consider a multisite system, HS =

∑ ϵa|a⟩⟨a| + ∑ Vab(|a⟩⟨b| + |b⟩⟨a|) a

a≠b

(25)

Each site is subject to a longitudinal coupling with a Drude bath of the spectral density J(ω) =

2λγω ω2 + γ 2

Figure 2. Frequency-resolved rates for the three-site system in Figure 1. KR(ω) and KI(ω) report the real and imaginary parts of K(ω), respectively.

(26)

Note that for a two-site system, the analytical expression of rate, based on continued fraction resolution to HEOM, has been developed.22,46,47 We have verified the present HEOM +projection operator method gives the numerically identical results. In the following, we focus on a three-site system. Figure 1 depicts the evaluated rate kernels 2(t ) for the threesite system specified in the figure caption. In our simulation, [1/1]-Padé expansion of the Bose function is sufficient for the present parameter settings.32,33,48,49 The tolerance error for the numerical filtering is set to be 2 × 10−5, which leads to the automatical hierarchy level truncation45 at L = 15. We also verify the relation in eq 2. Figure 2 presents the frequencyresolved rates, defined as K (ω) =

∫0

∞

Note that the present hierarchical Liouville-space projection method is exact. Figure 3 shows the population evolutions, comparing between the direct HEOM evaluation and the indirect approach via the numerical rate kernel results reported in Figure 1. The indirect evaluation involves the Fourier− Laplace transform and its numerical inverse. The Fourier− Laplace domain correspondence to eq 1 reads −izPã (z) − Pa(t = 0) =

b

(28)

∞

izt where K (z) = ∫ dteizt 2(t ) and P̃a(z) = ∫ ∞ 0 dte Pa(t), with z 0 = ω + iζ. Apparently, the half-sided Fourier transform, which corresponds to ζ = 0, would add an oscillating term, [eiωtPa(t)]t→∞, to the left-hand-side of eq 28. To avoid this complication, we choose ζ > 0 as a finite small constant. With

iωt

dte 2(t )

∑ Kab(z)Pb̃ (z)

(27)

The zero-frequency values {Kab(ω = 0)} are the long-time Markovian rate constants (cf. eq 3]. 3243

DOI: 10.1021/acs.jpca.5b11731 J. Phys. Chem. A 2016, 120, 3241−3245

The Journal of Physical Chemistry A

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REFERENCES

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Figure 3. Kinetics (dashed) versus exact HEOM evaluation (solid) on the population dynamics for the three-site system in Figure 1. The initial factorization of total density operator is assumed, and eq 10 reduces to eq 1.

the evaluated rate kernels 2(t ), the resultant K(z) is obtained via the half-sided Fourier transform of [e−ζt 2(t )]. Then, we solve the linear eq 28 for {P̃ a(z)}, whose numerical inverse Fourier transform amounts to {e−ζtPa(t)}. Finally, by multiplying the outcomes with eζt, we obtain the population kinetics (dashed curves) in Figure 3. Evidently, the rate kernels reported in Figure 1 do faithfully reproduce the exact HEOM results. Note also that the present method, which evaluates the rate kernel matrix one column at a time, as inferred from eqs 21−24, would be rather affordable, in comparing with the nonlinear correlation function-based approach.8,9 It provides an accurate and feasible numerical means for the quantum kinetic rate analysis, such as coherent excitation energy transfer pathway clustering in biological systems.23,24 Apparently, the non-Markovian rate, as inferred from eqs 21−24 does not depend on the initial populations. However, the initial coherence and system−bath correlations do affect the population evolution, according to the last term of eq 10. This expression is generally valid and can be used in, for example, temperature jump-induced rate process analysis. Initial coherence within system and/or between system and bath may play roles in the pathway selection of the energy and charge carriers dynamics in reaction center.10−12 One might argue that HEOM is originated at the path integral influence functional formalism50 that involves the initial factorization of total density operator. However, the recent dissipaton dynamics theory that reproduces the HEOM formalism supports arbitrary initial conditions.37−39 Moreover, the dissipaton theory can be readily extended to treat non-Gaussian quadratic bath coupling. The resulting dynamical generator 3 remains linear. The present projection operator approach, including eq 10, would still be valid.

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The support from the Hong Kong UGC (AoE/P-04/08-2) and Strategic Priority Research Program (B) of CAS (XDB01020000) is gratefully acknowledged. 3244

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