Fast, Accurate Simulation of Polaron Dynamics and ... · Davydov ansätze exhibit great promises in the investigation of the polaron energy band and other static properties of the

Fast, Accurate Simulation of Polaron Dynamics and MultidimensionalSpectroscopy by Multiple Davydov Trial StatesNengji Zhou,†,‡ Lipeng Chen,‡ Zhongkai Huang,‡ Kewei Sun,¶ Yoshitaka Tanimura,§ and Yang Zhao*,‡

†Department of Physics, Hangzhou Normal University, Hangzhou 310046, China‡Division of Materials Science, Nanyang Technological University, Singapore 639798, Singapore¶School of Science, Hangzhou Dianzi University, Hangzhou 310046, China§Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

ABSTRACT: By employing the Dirac−Frenkel time-depend-ent variational principle, we study the dynamical properties ofthe Holstein molecular crystal model with diagonal and off-diagonal exciton−phonon coupling. A linear combination ofthe Davydov D1 (D2) ansatz, referred to as the “multi-D1ansatz” (“multi-D2 ansatz”), is used as the trial state withenhanced accuracy but without sacrificing efficiency. The timeevolution of the exciton probability is found to be in perfectagreement with that of the hierarchy equations of motion,demonstrating the promise the multiple Davydov trial stateshold as an efficient, robust description of dynamics of complexquantum systems. In addition to the linear absorption spectra computed for both diagonal and off-diagonal cases, for the firsttime, 2D spectra have been calculated for systems with off-diagonal exciton−phonon coupling by employing the multiple D2ansatz to compute the nonlinear response function, testifying to the great potential of the multiple D2 ansatz for fast, accurateimplementation of multidimensional spectroscopy. It is found that the signal exhibits a single peak for weak off-diagonal coupling,while a vibronic multipeak structure appears for strong off-diagonal coupling.

1. INTRODUCTION

Thanks to recent advances in ultrafast spectroscopy, femto-second photoexcitation has became a major technique inprobing elementary excitations, which brought about numerousstudies on relaxation dynamics of photoexcited entities, forexample, polarons in inorganic liquids and solids,1−3 chargecarriers in topological insulators,4,5 trapped electrons and holesin the semiconductor nanoparticles,6−8 and electron−hole pairsin light-harvesting complexes of photosynthesis.9−13 Emergingtechnological capabilities to control femtosecond pulsedurations and down-to-1-Hz bandwidth resolutions offerunpreceded windows on vibrational dynamics and excitationrelaxation. For example, progress in femtosecond spectroscopyhas enabled the observation of a coherent phonon wave packetoscillating along an adiabatic potential surface associated with aself-trapped exciton in a crystal with strong exciton−phononinteractions.14 Taking advantage of ultrashort pulse widths ofrecent lasers, the femtosecond dynamics of polaron formationand exciton−phonon dressing have been observed in pump−probe experiments.15−17 These experiments have revealed acomplex interplay between a single exciton and its surroundingphonons under nonequilibrium conditions, while theoreticaldevelopments have not been kept in parallel. In particular,modeling of polaron dynamics have not received much-deserved attention over the last six decades.18,19

From a theoretical point of view, capturing time-dependentpolaron formation requires an in-depth understanding of the

combined dynamics of the particle and the phonons in itsenvironment.20 A simple Hamiltonian is that of the extendedHolstein molecular crystal model21,22 with simultaneousdiagonal and off-diagonal exciton−phonon coupling, as shownin Figure 1a, where the diagonal coupling represents anontrivial dependence of the exciton site energies on thelattice coordinates, and the off-diagonal coupling, a nontrivialdependence of the exciton transfer integral on the latticecoordinates.23−27 A large body of literature exists on the studyof the conventional form of the Holstein Hamiltonian with thediagonal coupling only.28,29 It seems fundamental to take intoaccount simultaneously diagonal and off-diagonal coupling tocharacterize solid-state excimers24,25 as a variety of experimentaland theoretical studies imply a strong dependence of electronictunneling upon certain coordinated distortions of neighboringmolecules in the formation of bound excited states. However,complete understanding of the off-diagonal coupling and out-of-equilibrium phenomena remains elusive. Early treatments ofthe off-diagonal coupling include the Munn−Silbeytheory,26,27,30 which is based upon a perturbative approachwith additional constraints on canonical transformationcoefficients determined by a self-consistency equation. Theglobal-local (GL) ansatz,31,32 formulated by Zhao and co-

Received: December 21, 2015Revised: February 8, 2016

Article

pubs.acs.org/JPCA

© XXXX American Chemical Society A DOI: 10.1021/acs.jpca.5b12483J. Phys. Chem. A XXXX, XXX, XXX−XXX

pubs.acs.org/JPCA

http://dx.doi.org/10.1021/acs.jpca.5b12483

workers in the early 1990s, was subsequently employed incombination with the dynamic coherent potential approxima-tion (with the Hartree approximation) to arrive at a state-of-the-art ground-state wave function as well as highereigenstates.33

Because an exact solution to the polaron dynamics still eludesus, several numerical approaches have been developed. Forexample, the time-dependent Schrodinger equation can benumerically integrated in real space for a few phonon timeperiods to probe the time evolution of electron and phonondensities and electron−phonon correlation functions.34 How-ever, the method is time-consuming and impractical when thesize of the system is large. Fortunately, time-dependentvariational approaches are still valid to treat the polarondynamics in such cases as long as a proper trial wave function isadopted. Previously, static properties of the Holstein polaronand the spin-boson model were studied by Zhao and his co-workers with a set of trial wave functions based upon phononcoherent states, including the Toyozawa ansatz,31,35,36 the GLansatz,31,32,36,37 a delocalized form of the Davydov D1 ansatz,

38

and the multi-D1 ansatz.39 The results of these extendedDavydov ansa tze exhibit great promises in the investigation ofthe polaron energy band and other static properties of theHolstein polaron. However, difficulties surround accuratesimulations of the polaron dynamics from an arbitrary initialstate, such as a localized state for which the aforementionedBloch states are not well suited. Thus, the question of what typeof the variational trial state is suitable for the polaron dynamicsof the Holstein model is still open.By using the Dirac−Frenkel time-dependent variational

principle, a powerful apparatus to reveal accurate dynamics ofquantum many-body systems,40 one can study the polarondynamics of the Holstein model with simultaneous diagonaland off-diagonal exciton−phonon coupling. Time-dependentvariational parameters, which specify the trial state, are obtainedby solving a set of coupled differential equations generated

from the Lagrangian formalism of the Dirac−Frenkel variation.Validity of the trial states is carefully examined by quantifyinghow faithfully they follow the Schrodinger equation.28,29,41 Thehierarchy of the Davydov ansa tze includes two trial states ofvarying sophistication, referred to as the D1 and D2ansa tze,42−46 with the latter being a simplified version of theformer. The D1 ansatz is sufficient to describe the Holsteinpolaron dynamics with the diagonal coupling, but fails in thepresence of the off-diagonal coupling. In comparison, the D2ansatz exhibits a nice dynamical performance with the off-diagonal coupling, though the deviation from the exact solutionto the Schrodinger dynamics is not negligible.41 Instead,superposition of the D1 or the D2 ansa tze will be adopted in ourwork, which offers significant improvements in the flexibility ofthe trial state,47 thus yielding accurate polaron dynamics of theHolstein model with simultaneous diagonal and off-diagonalcoupling.Recently, two-dimensional (2D) electronic spectroscopy has

been widely used to probe ultrafast energy transfer and chargeseparation processes in photosynthetic light harvestingcomplexes.48−54 Compared to linear spectroscopy techniquesin which the spectral lines are often congested, ultrafastnonlinear spectroscopies can resolve dynamical processes withvarious time scales. In a 2D electronic spectroscopy experimentand apparatus, for example, three ultrashort laser pulses,separated by two time delays, namely, the coherence time andthe waiting time, are incident on the sample, and the resultantsignal field is spectrally resolved in a given phase-matcheddirection. The 2D contour plots of the signals provide directinformation about excitonic relaxation and dephasing in avariety of molecular systems. Simulation of 2D electronicspectra of molecular aggregates was previously carried out forthe Holstein model with the diagonal exciton−phononcoupling. However, the effect of off-diagonal coupling on the2D spectra is yet to be addressed.In this paper, the multiple Davydov trial states, called the

multi-D1 and multi-D2 ansa tze, will be adopted to simulate thepolaron dynamics of an extended Holstein Hamiltonian thatincludes the off-diagonal exciton−phonon coupling. Validity ofthese trial states is carefully examined with the linear absorptionspectra compared closely with the ground-state energy band. Inaddition, 2D spectra for systems with off-diagonal exciton−-phonon coupling will be calculated by employing the multipleD2 ansatz. The remainder of the paper is organized as follows.In section 2, we introduce the Holstein Hamiltonian and twonovel variational wave functions on the basis of the multipleDavydov trial states, together with a criterion that quantifies thedeviation of our trial states from the exact solution to theSchrodinger equation. In section 3, results are analyzedincluding the time evolution of the exciton amplitudes andthe phonon displacements, the quantitative measurement forthe trial state validity, and the linear absorption and 2D spectra.Finally, conclusions are drawn in section 4.

2. METHODOLOGY

2.1. Model. The Hamiltonian of the one-dimensionalHolstein polaron is composed of

= + + + − −H H H H Hex ph ex phdiag

ex pho.d.

(1)

where Hex, Hph, Hex−phdiag and Hex−ph

o.d. represent the excitonHamiltonian, the bath (phonon) Hamiltonian, the diagonal

Figure 1. (a) Schematic of the Holstein ring. A simplified molecularcrystal is treated as a ring where each point represent a two levelmolecule and the coils denote the phonons. (b and c) Phonon wavefunctions of the Davydov D1 and D2 ansa tze, respectively. The phononpart of the D1 ansatz depends on both sites and momentum, while thatof the D2 ansatz is site independent.

The Journal of Physical Chemistry A Article

DOI: 10.1021/acs.jpca.5b12483J. Phys. Chem. A XXXX, XXX, XXX−XXX

B


exciton−phonon coupling Hamiltonian and the off-diagonalcoupling Hamiltonian, respectively, which are defined as

∑

∑

∑

∑

ω

ω

ϕ ω

= − +

=

= − +

= − +

+ − +

†+ −

†

−† − †

−†

+

†−

−

H J a a a

H b b

H g a a b b

H a a b

a a b

( ),

,

(e e ),

12

{ [e (e 1) H. c. ]

[e (1 e ) H. c. ]}

nn n n

qq q q

n qq n n

iqnq

iqnq

n qq n n

iqn iqq

n niqn iq

q

ex 1 1

ph

ex phdiag

,

ex pho.d.

,1

1

(2)

where H.c. denotes the Hermitian conjugate, ωq is the phononfrequency with momentum q, a n† (an) is the exciton creation(annihilation) operator for the nth molecule, and bq

† (bq) is thecreation (annihilation) operator of a phonon with themomentum q,

∑ ∑ = = † − † † − − †b N b b N be , eq

n

iqnn n

q

iqnq

1/2 1/2

(3)

The parameters J, g and ϕ represent the transfer integral,diagonal and off-diagonal coupling strengthes, respectively, andN is the number of sites in the Holstein ring. In this paper, alinear phonon dispersion is assumed

ω ωπ

= +| |

−⎡⎣⎢

⎛⎝⎜

⎞⎠⎟⎤⎦⎥W

q1

21q 0

(4)

where ω0 denotes a central phonon frequency, W is thebandwidth falling between 0 and 1, and q = 2πl/N represents

the momentum index with = − +l 1, ...,N N2 2

.

2.2. Multiple Davydov Trial States. In the past, twotypical Davydov trial states, i.e., the D1 and D2 ansa tze, wereused to obtain the time evolution of the Holstein polaronfollowing the Dirac−Frenkel variation scheme. The D2 ansatz isa simplified version of the D1 ansatz, since the phonondisplacements of the D1 (D2) trial state is site-dependent (site-independent), as illustrated in Figure 1b,c. Multiple Davydovtrial states with the multiplicity M are then introduced in thispaper, which can be constructed as follows

∑ ∑

∑ ∑ ∑

ψ λ

ψ λ λ

| ⟩ = | ⟩| ⟩

= | ⟩ − * | ⟩† †

t n

a b b

D ( ) ,

0 exp{ [ ]} 0

M

i

M

n

N

i n i n

i

M

n

N

i n nq

inq q inq q

1 , ,

, ex ph

(5)

and

∑ ∑

∑ ∑ ∑

ψ λ

ψ λ λ

| ⟩ = | ⟩| ⟩

= | ⟩ − * | ⟩† †

t n

a b b

D ( ) ,

0 exp{ [ ]} 0

M

i

M

n

N

i n i

i

M

n

N

i n nq

iq q iq q

2 ,

, ex ph

(6)

where ψi,n and λinq are related to the exciton probability and the

phonon displacement, respectively, n represents the site index

in the molecular ring, and i labels the coherent superposition

state. If M = 1, both the |D1M(t)⟩ and |D2

M(t)⟩ ansa tze are

reduced to the usual Davydov D1 and D2 trial states,

respectively. The equations of motion for the variational

parameters ψi,n and λinq are then derived by adopting the

Dirac−Frenkel variational principle,

ψ ψ

λ λ

∂∂ *

− ∂∂ * =

∂∂ *

− ∂∂ * =

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

tL L

tL L

dd

0,

dd

0

i n i n

inq inq

, ,

(7)

For the multi-D1 ansatz defined in eq 5, the Lagrangian L1 is

given as

= ⟨ | ℏ ∂∂

− | ⟩

= ℏ ⟨ | ∂∂

| ⟩ − ⟨ | ∂∂

| ⟩

−⟨ | | ⟩

⎡⎣⎢

⎤⎦⎥

L ti

tH t

it

tt t

tt

t H t

D ( )2

D ( )

2D ( ) D ( ) D ( ) D ( )

D ( ) D ( ) ,

M M

M M M

M M

1 1 1

1 1 1 1

1 1 (8)

where the first term yields

∑ ∑

∑ ∑ ∑

ψ ψ ψ ψ

ψ ψλ λ λ λ

λ λ λ λλ λ λ λ

⟨ | ∂∂

| ⟩ − ⟨ | ∂∂

| ⟩

= * − *

+ ** + *

− * + *

+ * − *

⎡

⎣⎢⎢

⎤

⎦⎥⎥

tt

t tt

t

S

S

D ( ) D ( ) D ( ) D ( )

( )

2

2

M M M M

i j

M

njn in jn in ji

i j

M

njn in ji

q

jnq jnq jnq jnq

inq inq inq inqjnq inq inq jnq

1 1 1 1

,

,

(9)

and the second term is

⟨ | | ⟩

= ⟨ | | ⟩ + ⟨ | | ⟩

+ ⟨ | | ⟩ + ⟨ | | ⟩− −

t H t

t H t t H t

t H t t H t

D ( ) D ( )

D ( ) D ( ) D ( ) D ( )

D ( ) D ( ) D ( ) D ( )

M M

M M M M

M M M M

1 1

1 ex 1 1 ph 1

1 ex phdiag

1 1 ex pho.d.

1 (10)

Detailed derivations of the equations of motion for the

variational parameters are given in Appendix A.Similarly, the equations of motion for the multi-D2 ansatz

can be derived using the Dirac−Frenkel variational principle ineq 7 with the Lagrangian L2 defined as



C


= ⟨ | ℏ ∂∂

− | ⟩

= ℏ ⟨ | ∂∂

| ⟩ − ⟨ | ∂∂

| ⟩

− ⟨ | | ⟩

⎡⎣⎢

⎤⎦⎥

L ti

tH t

it

tt t

tt

t H t

D ( )2

D ( )

2D ( ) D ( ) D ( ) D ( )

D ( ) D ( )

M M

M M M M

M M

2 2 2

2 2 2 2

2 2(11)

Assuming the trial wave function |D1,2M (t)⟩ = |Ψ(t)⟩ at time t,

we introduce a deviation vector δ (t) to quantify the accuracy ofthe variational dynamics based on the multiple Davydov trialstates

δ χ γ = −

= ∂∂

|Ψ ⟩ − ∂∂

| ⟩

t t t

tt

tt

( ) ( ) ( )

( ) D ( )M1,2 (12)

where the vectors χ (t) and γ(t) obey the Schrodinger equationχ = ∂|Ψ ⟩ ∂ = |Ψ ⟩ℏt t t H t( ) ( ) / ( )

i1 and the Dirac−Frenkel varia-

tional dynamics γ(t) = ∂|D1,2M ⟩/∂t in eq 7, respectively. Using

the Schrodinger equation and the relationship |Ψ(t)⟩ = |D1,2

M (t)⟩, the deviation vector δ (t) can be calculated as

δ =ℏ

| ⟩ − ∂∂

| ⟩ti

H tt

t( )1

D ( ) D ( )M M1,2 1,2 (13)

Thus, deviation from the exact Schrodinger dynamics can beindicated by the amplitude of the deviation vector Δ(t) =∥δ(t)∥. In order to view the deviation in the parameter space(W,J,g,ϕ), a dimensionless relative deviation σ is calculated as

σ = Δ ∈tN t

t tmax{ ( )}

mean{ ( )}, [0, ]

errmax

(14)

where Nerr(t) = ∥χ (t)∥ is the amplitude of the time derivative ofthe wave function,

= ⟨ ∂∂

Ψ | ∂∂

Ψ ⟩

= ⟨ | | ⟩

≈ Δ

N tt

tt

t

t H t

E

( ) ( ) ( )

D ( ) D ( )M M

err

1,22

1,2

(15)

since ⟨E⟩= ⟨D2M(t)|H(t)|D2

M(t)⟩ ≈ 0 in this paper.Two types of initial states are considered, i.e., the exciton

either sits on a single site for diagonal coupling cases or on twonearest-neighboring sites for off-diagonal coupling cases. Otherinitial states, such as Gaussian and uniform distributions for theexciton occupation, have also been investigated, leading tosimilar results but with larger relative errors. To avoidsingularity, noise satisfying the uniform distribution [−10−5,10−5] is added to the variational parameters ψi,n and λiq (λinq) ofthe initial states. With the wave functions |D1

M(t)⟩ and |D2M(t)⟩

at hand, the energy of the Holstein polaron Etotal = Eex+ Eph+Ediag+ Eoff is calculated, where Eex = ⟨D1,2

M |Hex|D1,2M ⟩, Eph = ⟨D1,2

M |Hph|D1,2

M ⟩, Ediag = ⟨D1,2M |Hex−ph

diag |D1,2M ⟩ and Eoff = ⟨D1,2

M |Hex−pho.d. |D1,2

M

⟩. In addition, the exciton probability Pex(t, n) and the phonondisplacement Xph(t, n) are defined as follows

= ⟨ | | ⟩

= ⟨ | + | ⟩

†

†

P t n a a

X t n b b

( , ) D D ,

( , ) D D

Mn n

M

Mn n

M

ex 1,2 1,2

ph 1,2 1,2 (16)

Optical spectroscopy is another important aspect for theinvestigation of the polaron dynamics, as it provides valuableinformation on various correlation functions. First of all, thelinear absorption spectra F(ω) calculated from the polarondynamics on the basis of different ansa tze will becomprehensively studied. The autocorrelation function F(t)of the exciton−phonon system is introduced

= ⟨⟨ | | | ⟩ | ⟩

= ⟨⟨ | ⟨ | | ⟩ | ⟩

− †

− †

F t P P

P P

( ) 0 0 e e 0 0

0 0 e 0 0

iHt iHt

iHt

ph ex ex ph

ph ex ex ph (17)

with the polarization operator

∑μ = | ⟩ ⟨ | + | ⟩ ⟨ | †P a a( 0 0 0 0 )n

n nex ex ex ex(18)

The linear absorption F(ω) is then calculated by means of theFourier transformation,

∫ωπ

= ω∞

F F t t( )1

Re ( )e di t

0 (19)

In addition to the information provided by the linearabsorption spectra, 2D electronic spectra provide directknowledge on coupling between different exciton states anddephasing and relaxation processes that are elusive in theoutput from the traditional 1D spectroscopy. Theoreticalsimulation of 2D spectra involves the calculation of thirdorder polarization P(t), which can be expressed in terms of thenonlinear response functions Ri, where i goes from 1 to 4.55−57

The 2D electronic spectra are measured in two configurationsthat correspond to the rephasing (subscript R) and non-rephasing (subscript NR) contribution to the third orderpolarization P(t), which, in the impulsive approximation, can bewritten as

τ τ τ

τ τ τ

∼ − +

∼ − +

P t T i R t T R t T

P t T i R t T R t T

( , , ) [ ( , , ) ( , , )],

( , , ) [ ( , , ) ( , , )]

R

NR

(3)2 3

(3)1 4 (20)

where τ (the so-called coherence time) is the delay timebetween the first and second pulses, T (the so-called populationtime) is the delay time between the second and third pulses,and t is the delay time between the third pulse and measuredsignal. The rephasing and nonrephasing 2D spectra can be thenobtained by performing 2D Fourier−Laplace transformation ofeq 20 as follows

∫ ∫

∫ ∫

ω ω τ τ

ω ω τ τ

=

=

τω τ ω

τω τ ω

∞ ∞− +

∞ ∞+

τ

τ

S T t iP t T

S T t iP t T

( , , ) Re d d ( , , )e ,

( , , ) Re d d ( , , )e

R t Ri i t

NR t NRi i t

0 0

(3)

0 0

(3)

t

t

(21)

The total 2D signal is defined as the sum of the nonrephasingand the rephasing part

ω ω ω ω ω ω= +τ τ τS T S T S T( , , ) ( , , ) ( , , )t R t NR t (22)

In this work, we will apply the multiple D2 states to calculatethe nonlinear response functions Ri with special attention paidto the role of the off-diagonal exciton−phonon coupling on the2D spectra. The reader is referred to the Appendix D for moredetails on the applications of the multiple D2 ansa tze to thesimulation of 2D spectra.



D


3. NUMERICAL RESULTS3.1. Validity of Variational Dynamics. Figure 2 illustrates

the time evolution of the system energies, including the exciton

energy Eex, the phonon energy Eph and the exciton−phononinteraction energy Ediag for a diagonal coupling only case withtransfer integral J = 0.1, bandwidth W = 0.5 and couplingstrength g = 0.1. For a molecular ring of 16 sites, the energiesobtained with three different ansa tze are compared (the opencircles, the solid triangles and the solid line correspond to thesingle D2, D2

M−32, D1M−5 ansatz, respectively ). Results obtained

with the multi-D2 ansatz with M = 32 display obviousdeviations from those by the single D2 ansatz, demonstratingthe improvement produced by the multiple Davydov trial statesover its single ansatz counterpart. In addition, the dynamicsgenerated on the D1 trial state can be made more accurate bythe D1

M=5 ansatz, and results of Eex, Eph and Ediag by the D1M=5

ansatz are in perfect agreement with those obtained with theD2

M=32 ansatz, which indicates the robustness of the polarondynamics based on the multiple Davydov trial states when themultiplicity M is sufficiently large.A comprehensive test of the validity for our new trial states

consisting of the multiple Davydov ansa tze is performed forvarious parameters sets (J,W,g,ϕ). In Figure 3a, the relativedeviation σ, given by eq 14, is displayed as a function of 1/M,for the diagonal coupling case of J = 0.1, W = 0.5, g = 0.1 and ϕ= 0. AsM increases, the relative error σ monotonically deceases,and the value σ = 0.067 obtained at 1/M = 0.2 is very small,which indicates the length of the deviation vector δ (t), asdefined in eq 12, is negligibly small with respect to those of thevectors χ (t) and γ(t). Moreover, the result that the deviationobtained by the D1

M=5 ansatz is comparable with σ = 0.033obtained with the D2

M=32 ansatz demonstrates the accuracy ofthe multiple Davydov trial states when M is sufficiently large.

In Figure 3b, the relative deviation σ is displayed as afunction of the transfer integral J with circles and triangles

Figure 2. Energies of the exciton, the phonons, and the exciton−phonon interaction, i.e., Eex(t), Eph(t), and Ediag(t), are displayed as afunction of the time t for the weak coupling case of J = 0.1, g = 0.1, W= 0.5, and ϕ = 0. The open circles, the solid triangles and the solid linecorrespond to the results obtained with the single D2, D2

M=32, and D1M=5

ansa tze, respectively.

Figure 3. (a) Relative deviation σ of the multi-D1 ansatz in a 16-sitemolecular ring is displayed as a function of 1/M. The set of parametersJ = 0.1, g = 1, W = 0.5, and ϕ = 0 is used. Moreover, the relativedeviation σ for the diagonal coupling case is also plotted as a functionof the transfer integral J in part b and diagonal coupling strength g inpart c. For both cases, the lines with circles and triangles correspond tothe results obtained with the multiplicity M = 1 and M = 4,respectively.



E


corresponding to results obtained by the multi-D1 ansa tze withM = 1 and 4, respectively. Other parameters used in thesimulation are g = 0.1,W = 0.5 and ϕ = 0. An obvious reductionin the relative error σ has been found when the multiplicityM isincreased for the entire J regime. Similarly, the relative error σagainst the diagonal coupling strength g is displayed in Figure3c for M = 1 and 4, respectively. The relative error σ isobviously reduced for the multiplicity M = 4 in comparisonwith that for M = 1 when g < 0.3. However, these two curvesoverlap for g > 0.3 as the exciton is self-trapped in one of thesites. The above results indicate that the multiple Davydov trialstates will significantly improve the accuracy of the delocalizedstate, while in the localized state the single D1 ansatz issufficient. In addition, the multiple Davydov trial states in theoff-diagonal coupling case are also investigated with thenonzero value of ϕ. Taking the set of parameters ϕ = 0.4and g = J = W = 0 as an example, the relative error σ isdisplayed as a function of 1/M in Figure 4. As M increases, the

relative error σ decreases, similar to the diagonal coupling caseas shown in Figure 3a, although the value of σ for M = 6 (σ =0.54) remains relatively large. For off-diagonal coupling,considerable improvements in accuracy can be achieved byutilizing multi-D2 with the increase of multiplicity M (seediscussions in ref 47).3.2. Exciton Probabilities and Phonon Displacements.

Dynamical properties of the Holstein polaron, including theexciton probabilities and phonon displacements, are inves-tigated by using the multiple Davydov trial states, and incomparison with those obtained with the single Davydov ansatzand the numerically exact HEOM method58−61(see AppendixB). Figure 5 illustrates the time evolution of the excitonprobability Pex(t, n) for the case of J = 0.5, W = 0.5, g = 0.1 andϕ = 0. For simplicity, a small ring with N = 10 sites is used inthe simulations. As depicted in Figure 5a,b, distinguishabledeviation in Pex(t, n) can be found between the variationalresults from the D1

M=1 and D1M=8 ansa tze. Interestingly, the

exciton probability Pex(t, n) obtained from the HEOM methodalmost overlaps with that obtained by the D1

M−8 ansatz (seeFigure 5b,c). Furthermore, the exciton probability differencebetween the variational method and the HEOM method,ΔPex(t, n), as depicted in Figure 5d, is two orders of magnitudesmaller than the value of Pex(t, n). It indicates that the

variational dynamics of the Holstein polaron can be numericallyexact if the multiplicity M of the D1 ansatz is sufficiently large.In Figure 6, the exciton probabilities Pex(t, n) at the site n = 5

and 10 are plotted in the top and the bottom panels with thesolid line, the dashed line and the circles, corresponding to thevariational results obtained with the single D1 and D1

M=8 ansa tzeand the HEOM results, respectively. The near overlap of thedashed line and the circles further reconfirms the validity of themulti-D1 ansatz.

Figure 4. Relative deviation σ from the multi-D1 ansatz is displayed asa function of 1/M for the off-diagonal coupling case with the couplingstrength ϕ = 0.4, and other parameters J = W = g = 0 are set.

Figure 5. Time evolution of the exciton probability Pex(t, n) for thecase of J = 0.5, W = 0.5, g = 0.1 and ϕ = 0 is displayed in parts a−c,corresponding to the results obtained with the single D1 ansatz, theD1

M=8 ansatz and the HEOM method, respectively. The differenceΔPex(t, n) between the HEOM and the D1

M=8 variational method is alsodisplayed in part d. We set the size of the molecular ring N = 10 insimulations.

Figure 6. Time evolution of the exciton probability Pex(t, n) at n = 5and 10 are displayed in the top and bottom panel for the case of J =0.5,W = 0.5, g = 0.1 and ϕ = 0. In each panel, the solid line, the dashedlines and the circles correspond to the variational results obtained withthe single D1 and D1

M=8 ansa tze and the HEOM results, respectively.The size of the molecular ring is set to N = 10.



F


Displayed in parts a and c of Figure 7 are the excitonprobability Pex(t, n) and the phonon displacement Xph(t, n)

obtained with the single D1 ansatz, respectively, for the case ofW = 0.5, g = 0.1, J = 0.5, and ϕ = 0. For comparsion,corresponding results of Pex(t, n) and Xph(t, n) obtained by themulti-D1 ansatz with M = 4 are presented in parts b and d ofFigure 7, respectively. Quite obvious difference is found in theexcitonic behavior for the two cases when t/(2π/ω0) > 3.Specifically, the exciton probability calculated by the single D1ansatz staggers around two sites in the ring before beingeventually trapped near site 8 accompanied by a thickenedphonon cloud (cf. Figure 7c), while that obtained by the multi-D1 ansatz with M = 4 continues to propagate in two oppositedirections. The former behavior is apparently an artifact as thecombination of J = 0.5 and g = 0.1 places the system firmly inthe large polaron regime, incompatible with any form of self-trapping at long times. This shows that the single D1 ansatz istoo simplistic to capture accurate polaron dynamics at longtimes, especially in the weak coupling regime.Next, we investigate the improvement on the polaron

dynamics by the multi-D2 trial state for the off-diagonalcoupling case. The exciton probability Pex(t, n) calculated bythe multi-D2 ansatz with M = 16 for two different sets of theparameters, (J = 0.1, g = 0, ϕ = 0, W = 0.5) and (J = 0.1, g = 0,ϕ = 0.1, W = 0.5), are displayed in Figures 8a,b, respectively.Corresponding Pex(t, n) obtained by the single D2 ansatz withthe same two sets of parameters are shown in Figure 8c,d,which reveals a similar pattern of the exciton motion with thesame speed of the exciton packet, v = ω0/2π, despite theincrease of the off-diagonal coupling strength from 0 to 0.1. Incontrast, the exciton probability obtained with the multi-D2ansatz shows localization signatures for off-diagonal couplingstrength ϕ = 0.1, which is absent if ϕ = 0. It indicates that thecombined effect of the transfer integral and the off-diagonalcoupling will confine the exciton to the sites of the initialcreation, despite that acting alone, either the transfer integral orthe off-diagonal coupling may propagate the exciton wavepackets. This phenomenon can be better understood after

analyzing the energy band near the zone center where a discreteself-trapping transition occurs.36 Our calculations show thateffective mass in the case of ϕ = 0.1 is larger than that of ϕ = 0,resulting in a less mobile polaron. It demonstrates again thatthe polaron dynamics obtained with the multiple Davydov trialstates is more accurate than that by the single Davydov trialstate.

3.3. Absorption Spectra. In this subsection, we employthe multiple Davydov trial states to study the linear absorptionspectra F(ω) defined in eq 19. To facilitate comparisons,spectral maxima are normalized to unity, and a damping factorof 0.08 ω0 is used.

28,29 In Figure 9, the linear absorption spectraF(ω) of a 16-site ring is displayed for the case of g = 0.2, J = 0.1,W = 0.1, and ϕ = 0. In Figure 9a, we compare results obtainedby the single D1 (solid) and single D2 (dashed) ansa tze. Largedifferences are found between these two curves, and negativevalues in the spectrum obtained by the single D2 ansatz point toits apparent invalidity. The multiple D1 trial states are capableto correct such inaccuracies in its single-D1 counterpart, asdemonstrated in Figure 9b for the multiple D1 trial state withmultiplicity M = 4. Similar corrections are also achieved by amulti-D2 ansatz with a multiplicity of 16, as shown in the samepanel. Moreover, the position of the zero-phonon line, denotedby ωm (in unit of ω0), is marked by the vertical dash-dotted lineat −0.75(1).The zero-phonon line can be also determined by the ground-

state polaron energy band Ek, where k is the crystal momentum.In order to identify the relationship, the transition moment Pkquantifying the transition probability between the vacuum stateand the exciton state is introduced and defined as Pk= ex⟨0 |ph⟨0|P†|Ψk⟩, where P = μ∑n(a n†|n⟩ex ex⟨n| + |n⟩ex ex⟨n|a n) is thepolarization operator, and Ψk is the ground-state trial wavefunction with the crystal momentum k. By employing thevariational method with the Toyozawa and delocalized D1ansa tze (details are shown in Appendix C), the ground-statewave function Ψk can be obtained, and corresponding polaronenergy band Ek = ⟨Ψk|H|Ψk⟩ can be calculated accordingly.

Figure 7. Time evolution of the exciton probability Pex(t, n) and thephonon displacement Xph(t, n) obtained with the single D1 ansatz (leftpanel) and the D1

M=4 ansatz (right panel) are displayed in parts a−d forthe case of W = 0.5, g = 0.1, J = 0.5, and ϕ = 0.

Figure 8. Time evolution of the exciton probability Pex(t, n) isdisplayed for the case of J = 0.1 and ϕ = 0 in the left column and thecase of J = 0.1 and ϕ = 0.1 in the right column. Other parameters usedare g = 0,W = 0.5, and N = 16 for both cases. Two different trial states,the D2

M=16 and D2M = 1 ansa tze, are used in the parts a and b and parts c

and d, respectively.



G


Variational calculations carried out for different k values areindependent of each other, and the set of Ek constitutes avariational estimate (an upper bound) for the polaron energyband. In Figure 10, polaron energy bands Ek/ω0 calculatedvariationally for the case of g = 0.2, ϕ = 0, J = 0.1, and W = 0.1,are plotted as a function of the crystal momentum k/π with thesolid and open circles corresponding to the delocalized D1 andthe Toyozawa ansa tze, respectively. For simplicity, we set μ =ω0 = 1. Interestingly, the normalized position of the zero-phonon line, ωm/ω0 in Figure 9b, is consistent with the value ofEk=0/ω0. It indicates that Ψk=0 is the bright state responsible forthe zero-phonon line, in perfect agreement with the obtainedtransition probability Pk, which is nonzero only at the crystalmomentum k = 0 as depicted in the inset.Moreover, absorption spectra in the presence of off-diagonal

coupling (ϕ ≠ 0) are investigated with the aid of a multi-D2ansatz with M = 16 (we set J = g = W = 0 for simplicity). Asshown in Figure 11, with an increase in the off-diagonalcoupling strength ϕ, phonon sidebands of the linear absorptionspectra become broadened and the intensity of the zero-phonon line is reduced. Vertical dashed lines shown in the 4panels of Figure 11 denote the positions of the zero-phononlines (ωm/ω0 = −0.08, −0.369, −0.956, and −1.93). For strong

off-diagonal coupling, such as the case of ϕ = 1, the linearabsorption spectra, shown in Figure 12, behave quite differentlyfrom those in weak off-diagonal coupling cases, such as ϕ = 0.1and 0.2 (cf. Figure 11). All of the sharp peaks are smeared out,and the zero-phonon line almost disappears. In order to betterunderstand the line shape, we plot the absorption spectrum in alog−log scale in the inset. A power-law fitting (dashed line)yields a slope of 2.1(1) indicating that the phonon sidebanddeviates from the Gaussian line shape. A Lorentzian line-shapefunction (dotted line) is then introduced for the fitting,

Figure 9. Linear absorption spectra F(ω) for a 16-site, one-dimensional ring of a coupled exciton−phonon system are displayedin (a) for the single D1 and D2 ansa tze and in (b) for the single D1,D1

M=4 and D2M=16 ansa tze. The set of parameters J = 0.1, g = 0.2, W =

0.1, and ϕ = 0 are used. A rescaled factor is adopted to normalize thespectral maxima to facilitate comparisons. The vertical dash-dotted lineindicates the location of the zero-phonon line ωm/ω0 = −0.75(1).

Figure 10. Polaron energy bands Ek/ω0 are calculated variationallyusing the delocalized D1 ansatz (solid line) and the Toyozawa ansatz(open circles) for the case of g = 0.2, J = 0.1, W = 0.1, and ϕ = 0. Theposition of the zero-phonon line ωm/ω0 is marked by the dashed line,consistent with the values of Ek=0/ω0. A lattice of N = 16 sites is usedin calculations. In the inset, the transition moment Pk is plotted as afunction of the crystal momentum k.

Figure 11. Linear absorption spectra F(ω) obtained with the D2M=16

ansatz are displayed in parts a−d for the off-diagonal coupling caseswith the nonzero coupling strengths ϕ = 0.1, 0.2, 0.3, and 0.4,respectively. Other parameters g = W = J = 0 and N = 16 are set. Thevertical dash-dotted lines indicate locations of zero-phonon lines.



H


consistent with the absorption spectrum obtained from thevariational method.3.4. 2D Spectra. In addition to the linear absorption

spectra, fast and accurate implementation of the multidimen-sional spectroscopy is possible via the time-dependentvariational method developed here. As an example, we presentin this subsection 2D spectra calculated for a molecular ring of10 sites using the multiple D2 ansatz. For the secondary bathwhose spectral density is defined by eq 53, we adopt theoverdamped Brownian oscillator model with the Drude-Lorentz type spectral density

ω η γωω γ

=+

D( ) 2 2 2 (23)

The resulting line shape function [cf. eq 58] can be evaluatedanalytically,55

∑

ηγ

γβ γ ηγ

γ

ηγβ

νν ν γ

= + − − + −

++ −

−

γ γ

ν

− −

=

∞ −

g t t i t

t

( ) cot2

[e 1] [e 1]

4 e 1( )

t t

n

tn

n n12 2

n

(24)

where νn = 2πn/β is the Matsubara frequency. In ourcalculations, we set η = 0.1, β = 5, and γ = 0.02.In Figure 13, 2D spectra of the 10-site ring are displayed for

the case of ϕ = 0.1 (left panel) and ϕ = 0.4 (right panel). Forsimplicity, we set J = g = W = 0, and adopt the toy model of J-aggregates with the tangential (head-to-tail) orientations of thetransition dipoles. We first consider weak off-diagonal coupling(ϕ = 0.1). The 2D spectra are shown in parts a−c of Figure 13,corresponding to the population times T = 0, 20, and 40,respectively. At T = 0, the signal exhibits a single peak locatedat (ωτ,ωt) = (−0.08,−0.08), which is elongated along thediagonal line. As the population time increases, the elongationbecomes less pronounced, and the peak appears more rounded.We then study the case of strong off-diagonal coupling with ϕ =0.4, as depicted in the right column of Figure 13 for several

values of the population time (see Figure 13, parts d−f for T =0, 20, and 40, respectively). Overall, it is found that strongexciton phonon coupling induces a pronounced vibronicmultipeak structure in the 2D spectra. With increasingpopulation time, the shapes as well as the strengths for thepeaks change, and we also find population cascades from highto low energy regions with lower ωt for larger values of T, asdemonstrated in Figure 13d−f.

4. CONCLUSIONSIn this work, we have studied the dynamical properties of theHolstein polaron in a one-dimensional molecular ring using theDirac−Frenkel time-dependent variational principle and anextended form of the Davydov trial states, also known as the“multi-D1 ansatz” (“multi-D2 ansatz”), which is a linearcombination of the single Davydov D1 (D2) trial states. Forboth diagonal and off-diagonal exciton−phonon coupling, therelative error quantifying how closely the trial state follows theSchrodinger equation is found to decrease with the multiplicityM, reflecting the improvement in accuracy of the multipleDavydov trial states. Moreover, exciton probabilities calculatedby the multiple Davydov trial states are obtained, in perfectagreement with those from a numerically exact approachemploying the hierarchy equations of motion, demonstratingthe great promise the multiple Davydov trial states hold as anefficient, robust description of dynamics of the complexquantum systems.An abnormal self-trapping phenomenon is uncovered in the

dynamical behavior of polaron with the increase of the off-diagonal coupling. Besides, the optical spectrum is also studied

Figure 12. Linear absorption spectrum F(ω) obtained by the D2M=16

ansatz is displayed for the off-diagonal coupling case with ϕ = 1. In theinset, the power-law and the Lorentzian fittings are given in the log−log scale with the dashed and dotted lines, respectively.

Figure 13. 2D spectra of the molecular ring for off-diagonal couplingstrengths ϕ = 0.1 (left column) and ϕ = 0.4 (right column). Upper,middle and lower panels correspond to the population time T = 0, 20,and 40, respectively. Other parameters g = W = J = 0 and N = 10 areset.



I


as a sensitive indicator of the accuracy of the variational polaron

dynamics. Among our findings, linear absorption spectra from

the multi-D1 ansatz with a multiplicity of 4 can be reproduced

by the multi-D2 ansatz with a multiplicity of 16, and the

positions of the zero-phonon lines are in good agreement with

ground-state energy bands calculated by the Toyozawa and the

delocalized D1 ansa tze in the weak electronic coupling (transfer

integral) regime. Moreover, for the first time, 2D spectra have

been calculated for systems with off-diagonal exciton−phononcoupling by employing the multiple D2 ansatz to compute the

nonlinear response function, testifying to the great potential of

the multiple D2 ansatz for fast, accurate implementation of

multidimensional spectroscopy. It is also found that the signal

exhibits a single peak for weak off-diagonal coupling, while a

vibronic multipeak structure appears for strong off-diagonal

coupling.

■ APPENDIX A. THE MULTI-D1 TRIAL STATE

The individual energy terms can be respectively calculated as

follows:

∑ ∑ ψ ψ

ψ

⟨ | | ⟩ = − *

+

+ +

− −

t H t J t t S

t S

D ( ) D ( ) ( )[ ( )

( ) ]

Mx

M

i j

M

nj n i n j n i n

i n j n i n

1 e 1,

, , 1 , ; , 1

, 1 , ; , 1 (25)

∑ ∑ ∑ψ ψ ω λ λ⟨ | | ⟩ = * *t H t SD ( ) D ( )Mph

M

i j

M

njn in

qq jnq inq ji1 1

, (26)

∑ ∑

∑

ψ ψ

ω λ λ

⟨ | | ⟩ = − *

× + *

−

−

t H t g

S

D ( ) D ( )

(e e )

Mex phdiag M

i j

M

njn in

qq

iqninq

iqnjnq ji

1 1,

(27)

∑ ∑

∑ ∑

ϕ ω ψ

ψ λ

λ ϕ ω ψ

ψ λ λ

⟨ | | ⟩ = *

× − + −

× * + *

− + − *

− +

+ +− −

−

−−

−−

t H t S t

t t

t S t

t t

D ( ) D ( )12

( )

( )[e (e 1) ( ) e (e 1)

( )]12

( )

( )[e (1 e ) ( ) e (1 e ) ]

Mex pho d M

n q i j

M

q j n i n j n

i niqn iq

i n qiqn iq

jnqn q i j

M

q j n i n j n

i niqn iq

i n qiqn iq

jnq

1. .

1, ,

, ; , 1 ,

, 1 , 1,

, ,, ; , 1 ,

, 1 , 1,

(28)

where the Debye−Waller factor is formulated as

λ λ

λ λ

= ⟨ | ⟩

= ⟨ | ⟩+ +

S

S

,ij i j

j n i n j n i n, ; , 1 , , 1 (29)

The Dirac−Frenkel variational principle leads to equations of

motion:

∑ ∑ ∑

∑

∑ ∑ ∑ ∑

∑ ∑

∑ ∑

ψ ψ λ λ λ λ λ λ

ψ ψ

ψ ω λ λ ψ ω

λ λ ϕ ω ψ

λ λ

ϕ ω ψ

λ λ

− − * − * − *

× = +

− * +

× + * −

× − + − *

× − −

× + − *

+ + − −

−+

+− −

+ −−

−−

−

i Si

S J S S

S g

S t

t t

S t

t S

2(2 )

( )

(e e )12

( )

[e (e 1) ( ) e (e 1) ( )]12

( )[e (1 e )

( ) e (1 e ) ]

i

M

in kii

M

inq

knq inq inq inq inq inq

k ii

M

i n k n i n i n k n i n

i

M

inq

q knq inq kii

M

inq

q

iqninq

iqnknq ki

i

M

qq i n

iqn iqi n q

iqn iqknq

k n i ni

M

qq i n

iqn iq

i n qiqn iq

knq k n i n

, , 1 , ; , 1 , 1 , ; , 1

, 1

, 1,

, ; , 1 , 1

, 1, , ; , 1 (30)

∑ ∑ ∑

∑

∑

∑ ∑

∑ ∑ ∑

∑

∑ ∑

∑ ∑

ψ ψ λ ψ ψ λ ψ ψ λ

λ λ λ λ λ λ

ψ ψ λ ψ λ

ψ ψ λ ω ω λ λ

ψ ψ ω ψ ψ λ ω

λ λ ϕ ω ψ ψ

ψ

ϕ ω ψ ψ λ

λ λ

ϕ ω ψ ψ λ

λ λ

− * − * − *

× * − * − *

= * +

× − * + *

+ * + *

× + * − *

× − + −

− * −

+ − *

− * −

+ − *

+ + + − −

−

−

−+

−

−+ −

−−

+ +

− −+ +

−−

−

−− −

i S i Si

S

J S

S S

g S g

S

S S

S

S

2(2 )

(

) ( )

e

(e e )12

[ e

(e 1) e (1 e ) ]

12

[e (e 1)

e (e 1) ]12

[e (1 e )

e (1 e ) ]

i

M

kn in inq kii

M

kn in inq kii

M

kn in inq

k ip

knp inp inp inp inp inp

i

M

k n i n i n q k n i n i n i n q

k n i ni

M

kn in inq ki qp

p knp inp

i

M

kn in qiqn

kii

M

kn in inqp

p

ipninp

ipnknp k i

i

M

q kn i niqn

iqk n i n i n

iqn iqk n i n

p i

M

p k n i nipn ip

i n p

ipn ipknp i n q k n i n

p i

M

p k n i nipn ip

i n p

ipn ipknp i n q k n i n

,

, , 1 , 1, , ; , 1 , 1 , 1,

, ; , 1

, , 1

, ; , 1 , 1 , ; , 1

, , 1 , 1,

, 1, , ; , 1

, , 1 , 1,

, 1, , ; , 1 (31)

■ APPENDIX B. HIERARCHY EQUATION OF MOTION

For the Holstein model [eq 1], let us denote |n⟩ = a n†|0⟩ex where

|0⟩ex stands for the exciton vacuum. Then the reduced density

matrix element for the exciton system is expressed in the path

integral form with the factorized initial condition as62

∫ ∫ρ ρ′ = ′ ′

× ′ ′−

n n t n n n n t

F n n t

( , ; ) ( , ; )

e ( , ; )eiS n t iS n t

0 0 0

[ ; ] [ ; ](32)

where S[n] is an action of the exciton system and F[n,n′] is the

Feynman−Vernon influence functional



J


∫ ∫∑ ω

βω ω

ω

′ = − ′ *

× ′ − ′

− ° ′ − ′

×

×

F n n s s V s

V s s s

iV s s s

[ , ] exp( d d ( )

[ ( ) coth( /2) cos( ( ))

( ) sin( ( ))]).

qq

t

t

t

s

q

q q q

q q

2

0 0

(33)

In the above equation, β is the inverse of temperature (β = 1/kBT), and the abbreviations

= − ′ ° = + ′×V V n V n V V n V n( ) ( ), ( ) ( )q q q q q q (34)

are introduced with Vq† = g∑n an

† aneiqn.

eq 33 can be rewritten as

∫ ∫∑ ω

βω

βω

′ = − ′ *

×′

′ − ° ′

+′

′ + ° ′

ω

ω

×

−×

− −×

F n n s s V s

V s V s

V s V s

[ , ] exp( d d ( )

[e

2( ( ) coth( /2) ( ))

e2

( ( ) coth( /2) ( ))])

qq

t

t

t

s

q

i s s

q q q

i s s

q q q

2

( )

( )

q

q

0 0

(35)

Taking derivative of eq 32, one has

∫ ∫

∫

∑

ρ ρ

ω ρ

βω

βω

∂∂

′ = − ′

− * ′ ′

× ′′

′ − ° ′

+′

′ + ° ′

× ′ ′

ω

ω

×

−×

− −×

−

⎡⎣⎢

⎤⎦⎥

tn n t i n n t

V t n n n n t

s V s V s

V s V s

F n n t

( , ; ) ( , ; )

( ) ( , ; )

de

2( ( ) coth( /2) ( ))

e2

( ( ) coth( /2) ( ))

e ( , ; )e

qq q

t i t s

q q q

i t s

q q q

iS n t iS n t

20 0 0

0

( )

( )

[ , ] [ , ]

q

q

(36)

If we use the following superoperator

ω

βω

Φ = *

Θ = ∓ °

×

±×

t V t

t V t V t

( ) ( )/2,

( ) ( ) coth( /2) ( )

q q q

q q q q

2

(37)

Equations 35 and 36 then can be simplified as

∫ ∫∑′ = − ′ Φ

× ′ Θ ′ + ′ Θ ′ω ω−+

− −−

F n n s s s

s s

[ , ] exp( d d ( )

[e ( ) e ( )])q t

t

t

s

q

i s sq

i s sq

( ) ( )q q

0 0

(38)

∫ ∫

∫

∑

ρ ρ

ρ

∂∂

′ = − ′

− Φ ′ ′

× ′ ′ Θ ′

+ ′ Θ ′ × ′ ′

ω

ω

−+

− −−

−

tn n t i n n t

t n n n n t

s s

s F n n t

( , ; ) ( , ; )

( ) ( , ; )

d [e ( )

e ( )] e ( , ; )e

qq

ti t s

q

i t sq

iS n t iS n t

0 0 0

0

( )

( ) [ , ] [ , ]

q

q(39)

In order to derive the equations of motion, we introduce theauxiliary operator ρm1±,m2±,···,mN±

(n,n′;t) by its matrix element as

∫ ∫ ∫

∫

∏

ρ

ρ

′

= ′ ′ Θ

× Θ × ′ ′

ω

ω

⋯

=

−+

− −−

−

± ± ±

+

−

n n t

n n n n t s s

s s F n n

( , ; )

( , ; ) ( d e ( ))

( d e ( )) e ( , )e

m m m

q

N

t

ti t s

qm

t

ti t s

qm iS n t iS n t

, , ,

0 0 01

( )

( ) [ , ] [ , ]

N

q q

q q

1 2

0

0

(40)

for non-negative integers m1±,m2±,...,mN±. Note that onlyρ0···..0(t) = ρ(t) has a physical meaning and the others areintroduced for computational purposes only. Differentiatingρm1±,m2±,...,mN±

(n,n′;t) with respect to t, we can obtain thefollowing hierarchy of equations in the operator form:

∑

∑

∑

ρ ρ

ω ρ

ρ

ρ

ρ

ρ

∂∂

= −

− −

− Φ

+

+ Θ

+ Θ

⋯ ⋯

− + ⋯

⋯ + ⋯

⋯ + ⋯

+ + ⋯ − ⋯

− − ⋯ − ⋯

± ± ± ±

± ±

± + − ±

± + − ±

± + − ±

± + − ±

tt i t

i m m t

t

t

m t

m t

( ) ( )

( ) ( )

( ( )

( ))

( ( )

( ))

m m m m

qq q q m m

qq m m m m

m m m m

qq q m m m m

q q m m m m

, , , ,

, ,

, , 1, , ,

, , , 1, ,

, , 1, , ,

, , , 1, ,

N N

N

q q N

q q N

q q N

q q N

1 1

1

1

1

1

1 (41)

The HEOM consists of an infinite number of equations, butthey can be truncated using a number of hierarchy elements.The infinite hierarchy of eq 41 can be truncated by theterminator as

∑ρ ω

ρ

∂∂

= − + −

×

− +± ±

± ±

tt i i m m

t

( ) ( ( ))

( )

m mq

q q q

m m

,...,

,...,

N

N

1

1 (42)

The total number of hierarchy elements can be evaluated as Ltot= (Ntrun + 2N)!/Ntrun!(2N)!, while the total number oftermination elements is Lterm = (Ntrun + 2N − 1)!/(2N − 1)!Ntrun!, where Ntrun is the depth of the hierarchy for mq±(q = 1,..., N). In practice, we can set the termination elements to zeroand thus the number of hierarchy elements for the calculationcan be reduced as Lcalc = Ltot − Lterm.

■ APPENDIX C. DELOCALIZED D1 ANSATZ AND THETOYOZAWA ANSATZ

Our interest in this work includes the polaron ground-stateenergy band, computed as

κ κ κ= ⟨Ψ | |Ψ ⟩E H( ) ( ) ( ) (43)

where |Ψ(κ)⟩ is an appropriately normalized, delocalized trialstate, and H is the system Hamiltonian. The joint crystalmomentum is indicated by the Greek κ. It should be noted thatthe crystal momentum operator commutes with the systemHamiltonian, and energy eigenstates are also eigenfunctions ofthe crystal momentum. Therefore, variations for distinct κ areindependent. The set of E(κ) constitutes a variational estimate



K


(an upper bound) for the polaron energy band. The relaxationiteration technique, viewed as an efficient method foridentifying energy minima of a complex variational system, isadopted in this work to obtain numerical solutions to a set ofself-consistency equations derived from the variationalprinciple. To achieve efficient and stable iterations toward thevariational ground state, one may take advantage of thecontinuity of the ground state with respect to small changes insystem parameters over most of the phase diagram and mayinitialize the iteration using a reliable ground state alreadydetermined at some nearby points in parameter space. Startingfrom those limits where exact solutions can be obtainedanalytically and executing a sequence of variations along well-chosen paths through the parameter space using solutions fromone step to initialize the next, the whole parameter space can beexplored.The D1 and D2 ansa tze are localized states from the soliton

theory, but without considering a form factor of a delocalizedstate. The polaron state have been analyzed with thedelocalized D1 and Toyozawa ansa tze, both of which areBloch states with the designated crystal momentum. The D1and D2 ansa tze can be transformed to the delocalized D1 andToyozawa ansa tze via a projection operator Pκ

∑ δ κ = = − κκ− − P N Pe ( )

n

i P n1 ( )

(44)

where

∑ ∑ = +† †P ka a qb bk

k kq

q q(45)

The delocalized D1 ansatz are then obtained after thedelocalization onto the usual D1 ansatz

κ κ κ κ|Ψ ⟩ = | ⟩⟨ | ⟩−( )11/2

(46)

∑ ∑

∑

κ α

β

| ⟩ =

× − − | ⟩

κ κ

κ

−†

− −†

a

b

e

exp[ ( H. c. )] 0

n

i n

nn n n

nn n n n n

1

,

1 1

21 2 2

(47)

where H.c. stands for the Hermitian conjugate, |0⟩ is theproduct of the exciton and phonon vacuum states, αn1−n

κ is the

exciton amplitude, and the phonon displacement βn1−n,n2−nκ

depends on n1 and n2, respectively, the sites at which anelectronic excitation and a phonon are generated.After the delocalization onto the usual D2 ansatz, the

Toyozawa ansatz is given by

κ κ κ κ|Ψ ′ ⟩ = | ′⟩⟨ ′| ′⟩−( )21/2

(48)

∑ ∑

∑

κ ψ

λ

| ′⟩ =

× − − | ⟩

κ κ

κ

′−′ †

−′ †

a

b

e

exp[ ( H. c. )] 0

n

i n

nn n n

nn n n

1

2

1 1

2 2(49)

where ψn1−nκ′ is the exciton amplitude analogous to αn1−n

κ in the

delocalized D1 ansatz, and λn2−nκ′ is the phonon displacement.

Actually, λn2−nκ′ is just one column of the phonon displacement

matrix βn1−n,n2−nκ in the delocalized D1 ansatz.

■ APPENDIX D. SIMULATION OF 2D SPECTRAUSING MULTIPLE D2 ANSATZE

In order to describe the population decays and dephasingsinduced by solvent, we add additional term HB+HSB to theHamiltonian 1

= + + + + +

= + + − −H H H H H H H

H H H

ex ph ex phdiag

ex pho d

B SB

S B SB

. .

(50)

where we have included vibrational modes with significantexciton−phonon coupling into system Hamiltonian, i.e., HS =Hex+ Hph + Hex−ph

diag + Hex−pho.d. , and treated the rest of vibrational

modes as a heat bath. We assume a harmonic bath with site-independent and diagonal system bath coupling56,57,63

∑ = ℏΩ †H c cBj

j j j(51)

∑ ∑ κ = ℏΩ +=

† †H c c a a( )SBj n

N

j j j j n n1 (52)

Here, cj(cj†) is the annihilation (creation) operator of the jth

bath mode with frequency Ωj, and κj is the correspondingexciton-bath coupling strength. The bath spectral density isspecified by

∑ω κ δ ω= Ω − ΩD( ) ( )j

j j j2 2

(53)

It is noted that system-bath Hamiltonian HSB commutes withthe system Hamiltonian HS, and as a result, the nonlinearresponse function can be represented as a product of thesystem and bath. Furthermore, by making use of the fact thatthe system-bath coupling is the same for all excitons, the effectof bath can be taken into account through line shape factors Fiin the framework of second-order cummulant expansion.Finally, we arrived at the formulas for the nonlinear responsefunction56

∑

∑

∑

∑

τ τ

τ τ

τ τ

τ τ

=

× ⟨ |⟨ | | ′⟩⟨ ″| | ‴⟩| ⟩

=

× ⟨ |⟨ | | ′⟩⟨ ″| | ‴⟩| ⟩

=

× ⟨ |⟨ | | ′⟩⟨ ″| | ‴⟩| ⟩

=

× ⟨ |⟨ | | ′⟩⟨ ″| | ‴⟩| ⟩

τ

τ

τ

τ

′ ″ ‴′ ″ ‴

− + +

′ ″ ‴′ ″ ‴

+ − +

′ ″ ‴′ ″ ‴

−

′ ″ ‴′ ″ ‴

− −

R t T F t T C

n n n n

R t T F t T C

n n n n

R t T F t T C

n n n n

R t T F t T C

n n n n

( , , ) ( , , )

0 e e 0

( , , ) ( , , )

0 e e 0

( , , ) ( , , )

0 e e 0

( , , ) ( , , )

0 e e 0

n n n nn n n n

iH T H t T

n n n nn n n n

iH T H t T

n n n nn n n n

iH H t

n n n nn n n n

iH t H

1 1, , ,

, , ,

ph( )

ph

2 2, , ,

, , ,

ph( ) ( )

ph

3 3, , ,

, , ,

ph ph

4 4, , ,

, , ,

ph ph

S S

S S

S S

S S

(54)

Here

μ μ μ μ=′ ″ ‴ ′ ″ ‴C e e e e( )( )( )( )n n n n 1 n 2 n 3 n 4 n, , , (55)



L


are the geometrical factors which must be averaged overorientations of the transition dipole moments μn. For simplicity,we can assume all laser fields have the same polarization, thenthe averaging can be done analytically, leading to

μ μ μ μ

μ μ μ μ μ μ μ μ

=

+ +

′ ″ ‴ ′ ″ ‴

″ ′ ‴ ‴ ″ ′

C1

15(( )( )

( )( ) ( )( ))

n n n n n n n n

n n n n n n n n

, , ,

(56)

The line shape factors Fi can be easily evaluated as55

τ

τ

τ

τ

=

=

=

=

τ τ τ

τ τ τ

τ τ τ

τ τ τ

− * − − * + * + + + − + +

− * − * + − + − * + + * + +

− − * + * − * + − * + + * + +

− − − + + + + − + +

F t T

F t T

F t T

F t T

( , , ) e

( , , ) e

( , , ) e

( , , ) e

g t g g T g T t g T g T t




1( ) ( ) ( ) ( ) ( ) ( )

2( ) ( ) ( ) ( ) ( ) ( )

3( ) ( ) ( ) ( ) ( ) ( )

4( ) ( ) ( ) ( ) ( ) ( )

(57)

where g(t) is the line shape function

∫ ω ωω

ωβ ω ω ω

=

× ℏ − + −

∞

⎡⎣⎢

⎤⎦⎥

g tD

t i t t

( ) d( )

coth2

(1 cos ) (sin )

0 2

(58)

The next crucial step is to approximate the propagator interms of the multiple D2 ansatz, i.e,

∑ ∑ ∑ψ λ λ| ⟩| ⟩ = | ⟩ − * | ⟩− † †n a b be 0 0 exp{ [ ]} 0iH t

phi

M

n

N

i n n exq

iq q iq q, phs

(59)

Explicitly, we have final expressions for the nonlinear responsefunction

∑ ∑

∑ ∑

∑ ∑

∑ ∑

τ τ ψ

ψ τ

τ τ ψ τ

ψ

τ τ ψ τ

ψ

τ τ ψ

ψ τ

=

× + +

×

= +

× +

×

=

×

×

= −

×

×

λ λ τ

λ λ τ

λ τ λ

λ τ λ

λ τ λ

λ τ λ

λ λ τ

λ λ τ

′ ″ ‴′ ″ ‴

=′*

″‴ − ∑ | | +| + + |

∑ + +

′ ″ ‴′ ″ ‴

=′*

″‴ − ∑ | + | +| + |

∑ + +

′ ″ ‴′ ″ ‴

=′*

″‴ − ∑ | | +| |

∑

′ ″ ‴′ ″ ‴

=′*

″‴ − ∑ | − | +| |

∑ −

ω

ω

ω

ω

‴

* ‴

‴

* ‴

‴

* ‴ +

‴

* ‴ −

R t T F t T C T

T t

R t T F t T C T

T t

R t T F t T C

t

R t T F t T C t

( , , ) ( , , ) ( )

( )e

e

( , , ) ( , , ) ( )

( )e

e

( , , ) ( , , ) ( )

( )e

e

( , , ) ( , , ) ( )

( )e

e

n n n nn n n n

i j

M

jnn

inn T T t

T T t

n n n nn n n n

i j

M

jnn

inn T T t

T T t

n n n nn n n n

i j

M

jnn

inn t

t

n n n nn n n n

i j

M

jnn

inn t

t

1 1, , ,

, , ,, 1

1/2 ( ( ) ( ) )

( ) ( )e

2 2, , ,

, , ,, 1

1/2 ( ( ) ( ) )

( ) ( )e

3 3, , ,

, , ,, 1

1/2 ( ( ) ( ) )

( ) ( )e

4 4, , ,

, , ,, 1

1/2 ( ( ) ( ) )

( ) ( )e

q jqn

iqn

q jqn

iqn i qt

q jqn

iqn

q jqn

iqn i qt

q jqn

iqn

q jqn

iqn i q t T

q jqn

iqn

q jqn

iqn i qT

2 2

2 2

2 2

( )

2 2

(60)

■ AUTHOR INFORMATIONCorresponding Author*(Y.Z.) E-mail: [email protected]. Telephone: +(65)65137990.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe authors thank Vladimir Chernyak for insightful discussionand Jiangfeng Zhu for help with numerics. Support from theSingapore National Research Foundation through the Com-petitive Research Programme (CRP) under Project No. NRF-CRP5-2009-04 is gratefully acknowledged. N.Z. and K.S. arealso supported in part by the National Natural ScienceFoundation of China under Grant Nos. 11205043, 11404084,and 11574052.

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Fast, Accurate Simulation of Polaron Dynamics and ... · Davydov ansätze exhibit great promises in the investigation of the polaron energy band and other static properties of the