Dimerization-assisted energy transport in light-harvesting complexesS. Yang, D. Z. Xu, Z. Song, and C. P. Sun Citation: J. Chem. Phys. 132, 234501 (2010); doi: 10.1063/1.3435213 View online: http://dx.doi.org/10.1063/1.3435213 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v132/i23 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Dimerization-assisted energy transport in light-harvesting complexesS. Yang,1 D. Z. Xu,1 Z. Song,2 and C. P. Sun1,a�

1Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China2School of Physics, Nankai University, Tianjin 300071, China

�Received 20 February 2010; accepted 5 May 2010; published online 15 June 2010�

We study the role of the dimer structure of light-harvesting complex II �LH2� in excitation transferfrom the LH2 �without a reaction center �RC�� to the LH1 �surrounding the RC� or from the LH2to another LH2. The excited and unexcited states of a bacteriochlorophyll �BChl� are modeled by aquasispin. In the framework of quantum open system theory, we represent the excitation transfer asthe total leakage of the LH2 system and then calculate the transfer efficiency and average transfertime. For different initial states with various quantum superposition properties, we study how thedimerization of the B850 BChl ring can enhance the transfer efficiency and shorten the averagetransfer time. © 2010 American Institute of Physics. �doi:10.1063/1.3435213�

I. INTRODUCTION

To face the present and forthcoming global energy crisis,human should search for clean and effective energy source.Recently the investigations on the basic energy science forthis purpose has received great attention and experienced im-pressive progress based on the fundamental physics.1,2 Inphotosynthetic process, the structural elegance and chemicalhigh efficiency of the natural system based on pigment mol-ecules in transferring the energy of sunlight have stimulateda purpose driven investigation,3–13 finding artificial analogsof porphyrin-based chromophores. These artificial systemsreplicate the natural process of photosynthesis2 so that themuch higher efficiencies could be gained than that obtainedin the conventional solid systems.2 It is because one of themost attractive features of photosynthesis is that the lightenergy can be captured and transported to the reaction center�RC� within about 100 ps and with more than 95%efficiency.4,14

Actually, in most of the plants and bacterium, the pri-mary processes of photosynthesis are almost incommon.3,14,15 Light is harvested by antenna proteins con-taining many chromophores; then the electronic excitationsare transferred to the RC sequentially, where photochemicalreactions take place to convert the excitation energy intochemical energy. Most recent experiments have been able toexactly determine the time scales of various transfer pro-cesses by the ultrafast laser technology.16–18 These greatprogresses obviously offer us a chance to quantitativelymake clear the underlying physical mechanism of the photo-synthesis so that people can construct the artificial photosyn-thesis devices in the future to reach the photon-energy andphoton-electricity conversions with higher efficiency. For ex-ample, quantum interference effects in energy transferdynamics12 have been studied for the Fenna–Matthews–Olson protein complex, and it was found6 that for such mo-lecular arrays, the spatial correlations in the phonon bath and

its induced decoherence could affect on the efficiency of theprimary photosynthetic event. The present paper will simi-larly study the influences of spatial structure on the primaryprocesses of photosynthesis for the light-harvesting com-plexes II �LH2�.

In the past, by making use of the x-ray crystallographictechniques, the structure of light-harvesting system has beenelucidated.3,19 In the purple photosynthetic bacteria, there ex-ist roughly two types of light-harvesting complexes, referredto as light-harvesting complex I �LH1� and LH2. In LH1, theRC is surrounded by a B875 bacteriochlorophyll �BChl� ringwith maximum absorption peak at 875 nm. The LH2 com-plex, however, does not contain the RC but can transfer en-ergy excitation to the RC indirectly through LH1. In thepurple bacteria, LH2 is a ring-shaped aggregate built up byeight �or nine� minimal units, where each unit consists of an��-heterodimer, three BChls, and one carotenoid. The��-heterodimers, i.e., �-apoproteins and �-apoproteins, con-stitute the skeleton of LH2, while the BChls are embedded inthe scaffold to form a double-layered ring structure. The topring including 16 �or 18� BChl molecules is named as B850since it has the lowest-energy absorption maximum at 850nm. The bottom ring with eight BChls is called B800 be-cause it mainly absorbs light at 800 nm. In every minimalunit, the carotenoid connects B800 BChl with one of the twoB850 BChls. Excitation is transferred from one pigment tothe neighbor one through the Föster mechanism,4 while theelectron is spatially transferred via the Marcus mechanism.20

Generally, it is independent of the global geometry configu-ration of the system.

In the present paper, we will study the energy transferprocedure in LH2 by considering the structure dimerizationof the B850 ring. It has been conjectured that the dimerizedinterpigment couplings can cause the energy gap to protectthe collective excitations.15 Indeed, like the Su–Schrieffer–Heeger model for the flexible polyacetylene chain,21 thedimerization of the spatial configuration with the Peierls dis-torted ground state will minimize the total energy for thephonon plus electron. As it is well known, this model exhib-its a rich variety of nonlinear phenomena and topologicala�Electronic mail: suncp@itp.ac.cn. URL: http://www.itp.ac.cn/suncp.

THE JOURNAL OF CHEMICAL PHYSICS 132, 234501 �2010�

0021-9606/2010/132�23�/234501/10/$30.00 © 2010 American Institute of Physics132, 234501-1

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excitations including the topological protection of the quan-tum state transfer.22 Similarly, we will show that when theB850 ring in LH2 is dimerized, the excitation transfer effi-ciency may be enhanced to some extent.

Based on the open quantum system theory, we simplymodel the excited and unexcited states of a BChl pigment asa quasispin. The excitation transfer is represented by the totalleakage from a LH2. Using the master equation, we calculatethe efficiency of excitation transfer and the average transfertime in low temperature for various initial states with differ-ent superposition properties. The results explicitly indicatethat the dimerization of couplings indeed enhances the quan-tum transport efficiency and shortens the average transfertime.

This paper is organized as follows. In Sec. II, a double-ring XY model with N unit cells is presented to simulate theLH2 system. In Sec. III, the energy transfer process is de-scribed by the quantum master equation. The transfer effi-ciency ��t� and the average transfer time � are introduced tocharacterize the dynamics of the system. In Sec. IV, we rep-resent the master equation in the momentum space and showthat only the �k ,k�-blocks of the density matrix are relevantto energy transfer. In Sec. V, it is found that the transferefficiency ��t� and the average transfer time � of an arbitraryinitial state can be obtained through the channel decomposi-tion. Some numerical analysis of ��A,k��t0� and ��A,k� for allthe k-channels are presented in Sec. VI. They show that asuitable dimerization of the B850 BChl ring can enhance thetransfer efficiency and shorten the average transfer time.Conclusions are summarized at the end of the paper. In Ap-pendix A, we provide an alternative way to deal with theenergy leakage problem. In Appendix B, detail derivations oftransforming the master equation from the real space to thek-space are given. The approximate solution of ��A,k� fork=0 and k= �� channel is shown in Appendix C.

II. MODEL SETUP

The simplified model of LH2 is shown in Fig. 1. All theBChls �big and small green squares� are modeled by thetwo-level systems with excited state �ej

�c��, ground state �gj�c��,

and energy level spacing �c. The raising and lowering qua-sispin operators of the jth two-level system on the �c� ringare expressed as

� j+�c� = �ej

�c���gj�c��,� j

−�c� = �gj�c���ej

�c�� , �1�

where �c�= �a���b�� denotes the B800 �B850� BChl ring. Ap-proximately, all the couplings are supposed to be of XYtype.5 This simplification enjoys the main feature of excita-tion transfer. The Hamiltonians,

Ha =�a

2 �j=1

N

� jz�a� + J1�

j=1

N

�� j+�a�� j+1

−�a� + H.c.� �2�

and

Hb =�b

2 �j=1

2N

� jz�b� + J2�

j=1

N

��1 + ��2j−1+�b� �2j

−�b�

+ �1 − ��2j+�b��2j+1

−�b� + H.c.� , �3�

with N=8, describe the excitations of the B800 and B850BChl rings, respectively. In the B850 BChl ring, the param-eter �0 characterizes the dimerization due to the spatialdeformation of the flexible B850 BChl ring in LH2. Thecoupling constants of Hb are dimerized as J2�1+� andJ2�1−� since the intraunit and interunit Mg–Mg distancebetween neighboring B850 BChls may be different. The non-local XY type interaction

Hab = g1�j=1

N

�� j+�a���2j−1

−�b� + �2j−�b�� + H.c.� + g2�

j=1

N

�� j+�a�

��2j−3−�b� + �2j−2

−�b� + �2j+1−�b� + �2j+2

−�b� � + H.c.� �4�

is used to describe the interaction between the B800 andB850 BChl rings.

In the single excitation case, the quasispin can be repre-sented with a spinless fermion with the mapping

� j+�a� ↔ Aj

†,�2j−1+�b� ↔ Bj

†,�2j+�b� ↔ Cj

† �5�

from the spin space Vs=C2�3N to the subspace VF of the fer-

mion Fock space spanned by

�O, j� = Oj†�0��O = A,B,C; j = 1,2, . . . ,N . �6�

Hereafter, let us represent the site index as �O , j�, where jrefers to a unit cell shown in Fig. 1, and O=A ,B ,C to aposition type inside the unit cell. In the subscripts, the siteindex �O , j� is written as Oj for simplicity. The vacuum stateof the fermion system �0� corresponds to the state that all thequasispins are in their ground states,

�0� ↔ �j=1

N

�gj�a�� � �

j=1

2N

�gj�b�� . �7�

Then the total Hamiltonian HS=Ha+Hb+Hab of LH2 ismapped into

FIG. 1. The model setup of the LH2 constructed by eight unit cells. Thecouplings between the neighboring quasispins in the B850 ring are dimer-ized as J2�1+� and J2�1−�. g1 denote the nearest couplings between B850BChls and B800 BChls, while g2 denote the next nearest couplings betweenB850 BChls and B800 BChls. �a� Illustration of the whole system withg2=0. �b� Detailed drawing of three unit cells and their nonlocal couplings.�c� Legends.

234501-2 Yang et al. J. Chem. Phys. 132, 234501 �2010�

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HS = �j=1

N

��aAj†Aj + �b�Bj

†Bj + Cj†Cj�� + �

j=1

N

J1Aj†Aj+1

+ g1Aj†�Bj + Cj� + J2��1 + �Bj

†Cj + �1 − �Cj†Bj+1�

+ g2Aj†�Bj+1 + Bj−1 + Cj+1 + Cj−1� + H.c. . �8�

In the present work, no multifermion interactions are consid-ered for simplicity.

On the other hand, we use the Holstein–Primakofftransformation23 to map the quasispin into bosons. The exci-tations of the BChls can be described by quasispins with thetotal angular momentum S. Then D=A ,B ,C can be regardedas the annihilation operators of bosons for the Fock spacespanned by

�Dj†�nD,j�0��D = A,B,C; j = 1, . . . ,N;nD,j = 0,1, . . . ,2S .

�9�

For S�1 /2, one local BChl has more than one excitedstates. In this case, higher order coherence could be includedfor further generalization.

In the following, we focus on the single excitation case.Then the temperature should be suitable to ensure there is nohigher order excited state.

III. TRANSFER EFFICIENCY AND AVERAGETRANSFER TIME VIA THE MASTER EQUATION

Next we consider the energy transfer from an initial state

�̂�0� = �j,l

�Aj,Al�0��A, j��A,l� , �10�

which is a coherent superposition or a mixture of those localstates �A , j� on the B800 ring. As time goes by, the initialstate will evolves a state distributing around both the B800and the B850 rings. Since there exists a difference in chemi-cal potential. �=�a−�b, energy is transferred between thetwo rings during the time evolution. For an isolated LH2system, such energy transfer is coherent, namely, the systemoscillates between the B800 and the B850 rings. However,when a LH2 is coupled to a heat reservoir with infinite de-grees of freedom, irreversible energy transfer occurs. As il-lustrated in Fig. 2, in the real photosynthetic system, energyis transferred from one LH2 to another LH2 or LH1 throughthe B850 ring.15 Therefore, we regard the first excited LH2as an open system and the sum of others as the a heat reser-voir. The energy transfer now can be manipulated as theenergy leakage from the B850 ring to the environment.

In order to describe such a procedure that the excitationsare finally transferred from the B850 ring to the heat reser-voir, the Markovian master equation

d�̂

dt= − i�HS, �̂� + L��̂� �11�

in the Lindblad form is employed for determining the timeevolution of the density matrix. Here two kinds of loss pro-cesses, dissipation and dephasing, are considered as Lindbladterms,

L��̂� = �j=1

N

�Ldiss,j��̂� + Ldeph,j��̂�� . �12�

We suppose that each quasispin on the B850 ring is coupledto an independent heat reservoir,6 which reflects the localmodes of phonons and other local fluctuations. Then the dis-sipation from the jth unit cell is described as

Ldiss,j��̂� = � j �O=B,C

�Oj�̂Oj† −

1

2Oj

†Oj, �̂ , �13�

where · , · denotes the anticommutator. Here, the sink rate� j at the jth point may be site dependent. For the dynamicsconstrained on the subsystem described by O operators, thelast term of Ldiss,j��̂� gives contribution −� j�Oj,Oj tod�Oj,Oj /dt; thus dissipation results in the reduction in thetotal population. Therefore, the dissipation term Eq. �13� rep-resents the incoherent transfer of energy into the environ-ment.

On the other hand, the dephasing term reads

Ldeph,j��̂� = � j� �O=B,C

�Oj†Oj�̂Oj

†Oj −1

2Oj

†Oj, �̂ . �14�

Compared with the dissipation term, the dephasing oneLdeph,j��̂� does not contribute to any time local change of theprobability distribution, i.e., the derivative of the diagonalelements of the density matrix is irrelevant to this term. Thusthe total population �O=A,B,C� j�Oj,Oj would be conserved ifonly the dephasing terms were present. However, the dephas-ing process is also incoherent since it makes the nondiagonalelements of the density matrix tend to zero.

The above two contributions force the LH2 system toultimately reach a steady state �̂steady= �0��0�= �̂v,v, namely, inthe long-time limit, all excitations are sinked away. The samesteady state is obtained from Eq. �11� in the superoperatorform

FIG. 2. The first excited LH2 is treated as an open system, while the otherLHs are regarded as heat reservoirs. The energy transfer process is equiva-lent to the excitation leakage from the B850 BChl ring of the LH2 system tothe environment.

234501-3 Transport with dimerization in photosynthesis J. Chem. Phys. 132, 234501 �2010�

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d

dt��� = M��� , �15�

where ��� denotes the column vector defined by all matrixelements in some order and the super-operator M is deter-mined by

M��� = �− i�HS, �̂� + L��̂�� . �16�

In this sense the steady state is just the nontrivial eigenstateof M with vanishing eigenenergy. Usually, from det M =0,the steady state can be found.

However, we are interested in the system dynamics on ashort time scale, i.e., how soon the excitations can be trans-ferred from one LH2 to the other light-harvesting complexes.To this end, the transfer efficiency ��t� is defined as thepopulation �v,v�t� of the vacuum state �0� at time t,

��t� = �v,v�t� . �17�

The corresponding master equation �11�

d�v,v

dt= �

j=1

N

� j�0��Bj�̂Bj† + Cj�̂Cj

†��0� = �j=1

N

� j �O=B,C

�Oj,Oj

�18�

means that only the first term of Ldiss,j��� contributes to thetime derivatives of �v,v�t�. The transfer efficiency is given bythe integral of the above formula,5–7

��t� = �0

t

�j=1

N

� j �O=B,C

�Oj,Oj�t��dt�. �19�

The average transfer time � is further defined as5,6

� = limt→�

1

��t��0

t

t��j=1

N

� j �O=B,C

�Oj,Oj�t��dt�

=1

�̄�

0

�

t��j=1

N

� j �O=B,C

�Oj,Oj�t��dt�, �20�

where usually

�̄ = limt→�

��t� = 1. �21�

Therefore, an efficient energy transfer requires not only aperfect transmission efficiency � but also a short averagetime �. In Appendix A, we present an equivalent non-Hermitian Hamiltonian method, which can also be utilized tostudy the dynamics of the open system.

IV. k-SPACE REPRESENTATION OF THE MASTEREQUATION

In this section we present the k-space representation ofthe master equation above so that we can reduce the dynam-ics of time evolution in some invariant subspace. If all thedissipation and dephasing rates are homogeneous on theB850 BChl ring, i.e., � j =� and � j�=��, the whole systemhas translational symmetry. For each unit cell containingthree BChls shown in Fig. 1, we introduce the Fourier trans-formation,

Ok† =

1�N

�j=1

N

eikjOj†, �22�

for O=A ,B ,C. Then in the k-space the Hamiltonian �8� isrepresented as HS=�kHk with

Hk = 2J1 cos kAk†Ak + �g1 + 2g2 cos k��Ak

†Bk + Ak†Ck�

+ J2��1 + � + �1 − �e−ik�Bk†Ck + H.c. . �23�

Here k are chosen as discrete values

k =2�l

N, for l = 1,2, . . . ,N . �24�

In the subspace of the single excitation plus the vacuum withthe basis

��0�, �O,k� � Ok†�0��k =

2�l

N;l = 1,2, ¯ N;O = A,B,C� ,

�25�

the general density matrix is decomposed into

�̂ = �̂v,v + �k1,k2

�̂k1,k2+ �

k

��̂v,k + �̂k,v� , �26�

where

�̂v,v = �v,v�0��0� �27�

is the vacuum block, while

�̂k1,k2= �

O,O�=A,B,C

�Ok1,O�k2�O,k1��O�,k2� �28�

is called the �k1 ,k2�-block. For fixed k1 and k2, �Ok1,O�k2form

a matrix

��Ak1,Ak2�Ak1,Bk2

�Ak1,Ck2

�Bk1,Ak2�Bk1,Bk2

�Bk1,Ck2

�Ck1,Ak2�Ck1,Bk2

�Ck1,Ck2

� . �29�

The k-space representation of the density matrix is illustratedin Fig. 3 for the N=8 system.

In the k-space, the master equation �11� is reduced to

d�̂k1,k2

dt= − i�Hk1

�̂k1,k2− �̂k1,k2

Hk2�

+ �O=B,C

���

N�

k

Ok†Ok1

�̂k1,k2Ok2

† Ok2+k−k1

−1

2�� + ����Ok1

† Ok1�̂k1,k2

+ �̂k1,k2Ok2

† Ok2�� �30�

for all the k1, k2,

d�̂k,v

dt= − iHk�̂k,v −

1

2�� + ��� �

O=B,COk

†Ok�̂k,v,

d�̂v,k

dt= i�̂v,kHk −

1

2�� + ��� �

O=B,C�̂v,kOk

†Ok, �31�

for all the k, and

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d�̂v,v

dt= �

k�

O=B,C�Ok�̂k,kOk

†. �32�

The details of the calculation are shown in Appendix B.We notice that the equations about �̂k,v and �̂v,k are com-

pletely decoupled from �̂k1,k2and �̂v,v. It follows from Eq.

�30� that when no dephasing exists, i.e., ��=0, the�k1 ,k2�-block �̂k1,k2

is decoupled with other �̂k1�,k2�for

�k1� ,k2��� �k1 ,k2�. Thus �̂k1,k2only evolves in the

�k1 ,k2�-block. However, when the dephasing is present����0�, the term

�O=B,C

��

N �k

Ok†Ok1

�̂k1,k2Ok2

† Ok2+k−k1�33�

actually induces the coupling between the �k1 ,k2�-block andthe �k ,k2+k−k1�-block. The initial �̂k1,k2

may evolve into�̂k,k2+k−k1

as time goes by. A typical example of �k ,k2+k−k1�-blocks are shown by the black hollow dot-dashedsquares in Fig. 3. The momentum difference k1−k2 is con-served during the evolution since

k2 − k1 = �k2 + k − k1� − k . �34�

In addition, Eq. �32� means that only the �k ,k�-blocks of thedensity matrix result in energy transfer, which are marked bythe eight green solid squares in Fig. 3. All the other k1�k2

blocks do not affect the transfer efficiency ��t� and averagetransfer time � at all. Especially, the initial component �̂k1,k2with k1�k2 will not influence ��t� or � at any time t after-ward since it cannot evolve to the blocks with k1=k2. There-fore, only considering the dynamics of the �k ,k�-blocks isenough for the present purpose.

V. TRANSFER EFFICIENCY AND AVERAGETRANSFER TIME WITH CHANNEL DECOMPOSITION

In this section we use the k-space representation of themaster equation to calculate the average transfer time andtransfer efficiency by the standard open quantum system

method. As a highly organized array of chlorophyll mol-ecules, the LH2 acts cooperatively to shuttle the energy ofphotons to elsewhere when sunlight shines on it. In thissense, we use the density matrix

�̂�0� = �k1,k2

�O,O�=A,B,C

�Ok1,O�k2�0��O,k1��O�,k2� �35�

to describe the excitations in the initial state. From the dis-cussions in the last section, only the k1=k2=k blocks arerelevant to energy transfer. Therefore, there exists an equiva-lence class of initial states,

��̂��0�� = �̂��O,k��̂�O�,k� = �Ok,O�k�0� , �36�

that results in the same transfer efficiency and average trans-fer time as that for �̂�0�. For further use, a special densitymatrix is chosen from the equivalence class �̂�0�� ��̂��0��,

�̂�0� = �k

�O,O�=A,B,C

�Ok,O�k�0��O,k��O�,k� = �k

�̂�k��0� ,

�37�

which satisfies �O ,k1��̂�0��O� ,k2�=�Ok,O�k�0� for k1=k2=kand �O ,k1��̂�0��O� ,k2�=0 for k1�k2. �̂�0� plays an equiva-lent role for determining the transfer efficiency and averagetransfer time. Here,

�̂�k��0� = �O,O�=A,B,C

�Ok,O�k�0��O,k��O�,k� �38�

is called as the k-channel component of the density matrix.According to the above observation, we first choose every�̂�k��0� as the initial state to obtain the final state �̂�k��t�,which gives the k-channel transfer efficiency at time t,

��k��t� = ��0

t

�k�

�O=B,C

�Ok�,Ok��k� �t��dt�, �39�

and the k-channel average transfer time

��k� =�

�̄�

0

�

t��k�

�O=B,C

�Ok�,Ok��k� �t��dt�. �40�

Then we prove a general proposition: for an arbitrary initialstate �̂�0� (Eq. �35�) of the LH2 complex, the transfer effi-ciency at time t and the average transfer time are the sum of��k��t� and ��k� over all k-channels, respectively,

��t� = �k

��k��t�

� = �k

��k�. �41�

In order to prove the above proposition, we notice that theeffective initial state �̂�0� evolves into

�̂�t� = �k

�̂�k��t� . �42�

Since the corresponding transfer efficiency and averagetransfer time of �̂�0� are

FIG. 3. Configuration of the density matrix of the N=8 system in the sub-space expanded by �0� , �O ,k� with O=A ,B ,C and k= �2� /8�1,2 , . . . ,8.An initial state localized in the �k1 ,k2�-block can be evolved to other�k ,k2+k−k1�-blocks �black hollow dot-dashed squares�. Only the diagonal�k ,k�-blocks �green solid squares� are related to the average transfer time.

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��t� = ��0

t

�k�

�O=B,C

�Ok�,Ok��t��dt�

� =�

�̄�

0

�

t��k�

�O=B,C

�Ok�,Ok��t��dt�, �43�

Eq. �41� is obtained from Eqs. �39�, �40�, �42�, and �43�.Namely, ��t� and � are the sum of ��k��t� and ��k� for differ-ent momentum k channels.

The present experimental observations15 have providedsome potential pathways for light-harvesting. One of themoriginates from the excitations on the B800 BChl ring.It shows that the excitations are transferred to the RCthrough B800�LH2�→B850�LH2�→B850�another LH2�→¯→B875�LH1�→RC. As to our model, the initial stateis specialized as

�̂�0� = �k1,k2

�Ak1,Ak2�0��A,k1��A,k2� . �44�

Accordingly, the k-channel component of the effective initialstate �̂�0� becomes

�̂�k��0� = �Ak,Ak�0��A,k��A,k� = �Ak,Ak�0��̂�A,k��0� . �45�

Taking

�̂�A,k��0� = �A,k��A,k� �46�

as the initial state, we obtain the transfer efficiency and theaverage transfer time

��A,k��t� = ��0

t

�k�

�O=B,C

�Ok�,Ok��A,k� �t��dt�,

��A,k� =�

�̄�

0

�

t��k�

�O=B,C

�Ok�,Ok��A,k� �t��dt�. �47�

Hereafter, the superscript �A ,k� denotes that the initial stateis Eq. �46�. Similar to the above analysis about the proposi-tion, we present a corollary: the transfer efficiency ��t� andaverage transfer time � of the initial state in Eq. �44� are theweighted average of ��A,.k��t� and ��A,k�, respectively,

��t� = �k

�Ak,Ak�0���A,.k��t� ,

� = �k

�Ak,Ak�0���A,k�. �48�

In the following, we will show the analytical and numericalresults of ��A,.k��t� and ��A,k�.

First we consider the case without dephasing, i.e.,��=0. The time evolution from initial state �̂�A,k��0� onlytakes place in the �k ,k�-block. According to Eq. �30�, themaster equation of �̂k,k,

d�̂k,k

dt= − i�Hk�̂k,k − �̂k,kHk� −

�

2

�O=B,C

�Ok†Ok�̂k,k + �̂k,kOk

†Ok� , �49�

gives the average transfer time

��A,k� =�

�̄�

0

�

t� �O=B,C

�Ok,Ok�A,k� �t��dt�. �50�

When k=0, Eq. �49� about �Ok,O�k is rearranged as a systemof differential equations about v j�t� �j=1, ¯ ,4� andv5�t�= �v4�t���,

v1 = �Ak,Ak,

v2 = �Bk,Bk + �Ck,Ck,v3 = �Bk,Ck + �Ck,Bk,

v4 = �Ak,Bk + �Ak,Ck,v5 = �Bk,Ak + �Ck,Ak. �51�

It is

d

dtv1�t� = ig+�v4�t� − v5�t�� ,

d

dtv2�t� = − ig+�v4�t� − v5�t�� − �v2�t� ,

d

dtv3�t� = − ig+�v4�t� − v5�t�� − �v3�t� ,

d

dtv4�t� = 2ig+v1�t� − ig+�v2�t� + v3�t��

− �2i�J1 − J2� + i � +�

2�v4�t� , �52�

with initial conditions

v1�0� = 1,v2�0� = v3�0� = v4�0� = v5�0� = 0. �53�

Here g+= �g1+2g2�. Solving the above differential equations,we obtain

��A,k=0� =�

�̄�

0

�

t�v2�t��dt�

=g+

2 + �J1 − J2 + �/2�2 + �02/4

g+2�0

, �54�

with �0=� /2, which is independent of the dimerization pa-rameter . Similarly, when k= ��, the average transfer timeof �̂�A,k��t0� is

��A,k=��� =g−

2 + �J1 + J2 − �/2�2 + �02/4

g−2�0

, �55�

where g−= �g1−2g2�. It is a quadratic function with respect to. The optimal parameter with the shortest transfer timesatisfies

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opt�A,k=��� =

�/2 − J1

J2.

When g1=2g2, Eq. �55� shows that ��A,k=���=� correspondsto �̄=0; the energy transfer is prevented at this time.

If the dephasing is present, i.e., ���0, we can provideapproximate solutions for ��A,k=0� and ��A,k=���,

��A,k=0� =g+

2�4�s − ���/� + �2J1 − 2J2 + ��2 + �s2/4

2g+2�s

,

��A,k=��� =g−

2�4�s − ���/� + �2J1 + 2J2 − ��2 + �s2/4

2g−2�s

,

�56�

where �s=�+��. They almost exactly agree with the nu-merical calculation below and can also be confirmed by Eqs.�54� and �55� when ��=0. The details are shown in Appen-dix C.

VI. ENERGY TRANSFER EFFICIENCY AND AVERAGETRANSFER TIME IN NUMERICAL CALCULATION

For a general k, the analytical solution of ��A,k��t� and��A,k� is not easy to get. Nevertheless, the numerical results of��A,k� as a function of are plotted as blue scatter lines in Fig.4. Here we have chosen

N = 8,J1

�= 0.3,

J2

�= 1,

g1

�= 0.5,

�

�= 0.1,

��

�= 1, �57�

g2 /�=0 for the upper panel, g2 /�=0.125 for the lowerpanel, and t is in the unit of �1 /�� and is long enough toensure �̄=1. It shows that when k�0 and varies from �1to 1, there always exist optimum cases opt

�A,k��0 with andshorter average transfer time. This fact reflects the enhancedeffect of dimerization.

We then take the mixed initial density matrix �̂�0�= �̂mix

A �0� as an example,

�̂mixA �0� = �

j=1

N

�Aj,Aj�0��A, j��A, j�

= �k1,k2

�Ak1,Ak2�0��A,k1��A,k2� . �58�

The weight �Ak,Ak always satisfies

�Ak,Ak�0� =1

N�j=1

N

�Aj,Aj�0� =1

N. �59�

From Eq. �48�, the transfer efficiency and the average trans-fer time of �̂mix

A are

�mix�t� =1

N�

k

��A,k��t� , �60�

�mix =1

N�

k

��A,k�, �61�

where �mix is also verified numerically and shown in Fig. 4 asthe red solid lines.

In order to see the dynamics of the transfer processclearly, we plot �mix with respect to the dimerization degree and time t in Fig. 5�a�, i.e., �mix=�mix� , t�. At a certaininstant t0=12, �mix� , t0� as a function of is plotted in Fig.

-1.0 -0.5 0.0 0.5 1.00

5

10

15

202

4

6

8

10

τ

δ

τ mix

k = + π/2k = + 3π/4k = + π

g2 / Γ = 0.125

τk = 0k = + π/4

g2 / Γ = 0

τ [A,k]

FIG. 4. The average transfer time ��A,k� of �̂�A,k��0� �blue scatter lines� and�mix of the initial mixed state �̂mix

A �0� �red solid lines� with respect to thedimerization degree of the B850 BChl ring. Here N=8, J1 /�=0.3,J2 /�=1, g1 /�=0.5, � /�=0.1, �� /�=1, g2 /�=0 �upper panel�, andg2 /�=0.125 �lower panel�. � is in the unit of �1 /�� and �̄=1. It shows thateach ��A,k� �k�0� curve has a minimum at opt

�A,k��0. �mix is the equal-weighted average of a complete set of ��A,k�.

0

1

0 1

η mix(δ,t 0)

(b)

ηmix(δ0,t)

(c)

-1.0 -0.5 0.0 0.5 1.00

5

10

15

20

δ

t

00.20.40.60.81

(a)ηmix(δ,t)

FIG. 5. �a� The contour map of the transfer efficiency of the initial mixedstate �mix� , t� as a function of dimerization degree and time t for the samesetup as that in the lower panel of Fig. 4. �b� The profile of �mix alongt0=12 �black dashed line in �a��. �c� The profile of �mix along 0=−0.5 �reddotted line in �a��. It shows that �mix� , t� increases over time, and an opti-mum can enhance the transfer efficiency.

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5�b�, while for a certain dimerization degree 0=−0.5,�mix�0 , t� as a function of t is plotted in Fig. 5�c�. Here theparameters are chosen as same as the ones in Fig. 4, exceptthat g2 /�=0.125. The contour map in Fig. 5�a� and the pro-files of �mix� , t� in Figs. 5�b� and 5�c� show that �1��mix� , t� increases monotonously as time goes by. In thelarge t limit, �mix� , t� equals to 1. �2� At any certain shortinstant, an optimum can enhance the transfer efficiency.

Similar to �mix and �mix, in general, there exists an opti-mal opt�0 for an arbitrary initial �̂�0�, which means that asuitable distortion of the B850 ring is helpful for the excita-tion transfer. This result agrees with the x-ray observationthat the Mg–Mg distance between neighboring B850 BChlsis 9.2 Å within the ��-heterodimer and 8.9 Å between theheterodimers reported in Ref. 19. The B850 ring is indeeddimerized in nature.

As shown in Fig. 4, ��A,k=0� and ��A,k=��� are particularsince nearly all the other ��A,k� are within the range of���A,k=0� ,��A,k=����, so is the average transfer time � of anarbitrary �̂�0�. Besides, the absolute value of opt

�A,k=��� for thek= �� case is larger than the one of other ��0�, i.e., �opt�� �opt

�A,k=����. Hence, once we have known the properties of��A,k=0� and ��A,k=���, the behavior of a general � can be con-jectured to some extent. Compare the lower panel of Fig. 4with the upper panel; a larger g2 /� can increase ��A,k=��� butdecrease ��A,k=0�. In the g2 /�=0.125 case, the homogeneouspure state �̂�A,k=0��0� is better than the mixed state �̂mix

A �0� forenergy transport. However, the upper panel with g2 /�=0gives the contrary result.

The minimal ��A,k=��� is reachable at opt�A,k=���= � � /2

−J1� /J2,

�min�A,k=��� =

�g1 − 2g2�2�4� + 3��� + ��� + ���2/42�g1 − 2g2�2��� + ���

. �62�

In the toy model illustrated in Fig. 4, J2�J1 and g2�g1.When

g2/g1 = �g = 12 + �2 − ��1 + �2, �63�

we have �min�A,k=���=��A,k=0�, where

� =�� + ���

2�2J2 − 2J1 − ��. �64�

On the side of 0�g2 /g1��g, �min�A,k=������A,k=0�, while on

the other side, �g�g2 /g1�1, �min�A,k=������A,k=0�.

In general, the shortest average transfer time of an arbi-trary initial �̂�0� is within the range of ��min

�A,k=��� ,��A,k=0��.The mean value of �min

�A,k=��� and ��A,k=0� can roughly reflectthe influence of parameters on the transfer process,

�̄ = 12 ��min

�A,k=��� + ��A,k=0�� . �65�

In Fig. 6, we plot �̄ with respect to g1 /� for differentJ2 /�=0,1 ,2 ,3. Here, J1 /�=0.3, = � � /2−J1� /J2, g2 /g1

=0.25, � /�=0.1, �� /�=0.5, and �̄ is in the unit of �1 /��.It shows that �̄ decreases monotonously as g1 /� increases. Inthe short g1 /� limit, �̄ tends to infinity, which is reasonablesince the two BChl rings are decoupled in this case. More-over, �̄ is larger when J2 /� is larger.

VII. CONCLUSION

In summary, we have studied the craggy transfer in light-harvesting complex with dimerization. We employed theopen quantum system approach to show that the dimerizationof the B850 BChl ring can enhance the transfer efficiencyand shorten the average transfer time for different initialstates with various quantum superposition properties. Actu-ally our present investigation only focuses on a crucial stagein photosynthesis—the energy transfer, which is carried bythe coherent excitations in the typical LH2. Here the LH2 ismodeled as two coupled BChl rings. With this modeling, theordinary photosynthesis is roughly described as three basicsteps: �1� stimulate an excitation in LH2, �2� transfer it toanother LH2 or LH1, and �3� the energy causes the chemicalreaction that converts carbon dioxide into organic com-pounds. Namely, the excitations are transferred to the RCthrough B800�LH2�→B850�LH2�→B850�another LH2�→¯→B875�LH1�→RC. Obviously, the first two are ofphysics; thus our present approach can be generalized to in-vestigate these physical processes. Although photosynthesishappens in different fashions for different species, some fea-tures are always in common from the point of view of phys-ics. For example, the photosynthetic process always startsfrom the light absorbing and energy transfer.

Another important issue of the photosynthesis physicsconcerns the quantum natures of light.24,25 Since the experi-ments have illustrated the role of the quantum coherence ofcollective excitations in light-harvesting complexes, it isquite natural to believe that the excitation coherence may beinduced by the higher coherence of photon. Therefore, in aforthcoming paper we will report our systematical investiga-tion on how the statistical properties of quantum light affectthe photosynthesis.

ACKNOWLEDGMENTS

This work was supported by NSFC under Grant Nos.10474104, 60433050, 10874091, and 10704023 and byNFRPC under Grant Nos. 2006CB921205 and2005CB724508.

0.1 0.3 0.5 0.7 0.9 1.10

50

100

150

0.6 0.7 0.82

4

6

8

10

τ

g1 / Γ

τ

g1 / Γ

J2 / Γ =0123

FIG. 6. Plots of �̄ as a function of the dissipation ratio g1 /� withJ2 /�=0,1 ,2 ,3, where J1 /�=0.3, = � � /2−J1� /J2, g2 /g1=0.25, � /�=0.1, �� /�=0.5, and �̄ is in the unit of �1 /��. It shows that �̄ de-creases as g1 /� increases but increases with the increase in J2 /�.

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APPENDIX A: EQUIVALENT NON-HERMITIANHAMILTONIAN

In the case without dephasing, i.e., � j =0, an equivalentnon-Hermitian Hamiltonian is introduced to study the dy-namics of the open system,

H = HS − i�j=1

N� j

2�Bj

†Bj + Cj†Cj� . �A1�

The equivalence between Eqs. �A1� and �11� is shown asfollows. On the one hand, the Schrödinger equation,

id

dt��� = �HS − i�

j=1

N� j

2�Bj

†Bj + Cj†Cj����� , �A2�

and its Hermitian conjugate

− id

dt��� = ����HS + i�

j=1

N� j

2�Bj

†Bj + Cj†Cj�� , �A3�

give the evolution equation of the density matrix �̂= ������,

d�̂

dt= � d

dt��� ��� + ���� d

dt���

= − i�HS, �̂� − �j=1

N� j

2Bj

†Bj + Cj†Cj, �̂ . �A4�

On the other hand, when the dephasing terms are absent, themaster equation �11� becomes

d�̂

dt= − i�HS, �̂� + �

j=1

N

�O=B,C

� j�Oj�̂Oj† −

1

2Oj

†Oj, �̂� .

�A5�

The above equation is written on the expanded Hilbert spacewith an additive vacuum basis �0�. Compared with Eq. �A4�,the additive term in Eq. �A5�, � j=1

N �O=B,C� jOj�̂Oj†, has only

contribution to d�̂v,v /dt, which does not change the dynam-ics of the system. Equations �A4� and �A5� are equivalent fordetermining the time evolution of �̂Oj,O�j�.

For the non-Hermitian Hamiltonian, the correspondingtransfer efficiency and the average transfer time are

��t� = �0

t

�j=1

N

� j �O=B,C

��0�Oj���t����2dt�,

� =1

�̄�

0

�

t��j=1

N

� j �O=B,C

��0�Oj���t����2dt�. �A6�

Due to the equivalence of the non-Hermitian Hamiltonianand the dissipative master equation, the results of Eq. �A6�are the same as the ones calculated by Eqs. �19� and �20�.The non-Hermitian Hamiltonian method has an advantageover the master equation one for saving computer time. In-stead of N2 equations, only a system of N equations isneeded to be solved in the non-Hermitian Hamiltonian case.

However, when the dephasing terms are present, there isno equivalent non-Hermitian Hamiltonian. In this case,compared with Eq. �A4�, the additional term

�O=B,C� j�Oj†Oj�̂Oj

†Oj cannot be omitted any more. It can alsoaffect the evolution of the density matrix of the LH2 system.

APPENDIX B: TRANSFORM THE MASTER EQUATIONTO THE k-space

In this section, we will transform the master equationfrom the real space �Eqs. �11�–�14�� to the k-space �Eqs.�30�–�32��. Since HS=�kHk and �̂ is expressed as Eq. �26�,we have

− i�HS, �̂� = − i��k

Hk, �k1,k2

�̂k1,k2+ �

k

��̂v,k + �̂k,v��= − i �

k1,k2

�Hk1�̂k1,k2

− �̂k1,k2Hk2

�

− i�k

�Hk�̂k,v − �̂v,kHk� . �B1�

Here �HS , �̂v,v�=0 since they are in the different subspaces.According to the Fourier transformation Eq. �22�, the

term � jOj�̂Oj† becomes

�j

Oj�̂Oj† = �

k1,k2,k3,k4

1

N�

j

ei�k3−k4�jOk3�̂k1,k2

Ok4

†

= �k1,k2

1

N�

j

ei�k1−k2�jOk1�̂k1,k2

Ok2

†

= �k1,k2

k1,k2Ok1

�̂k1,k2Ok2

†

= �k

Ok�̂k,kOk†. �B2�

The term � jOj†Oj�̂ is transformed as

�j

Oj†Oj�̂ = �

k1,k3,k4

1

N�

j

e−i�k3−k4�jOk3

† Ok4

��k2

�̂k1,k2+ �̂k1,v

= �k1,k3

1

N�

j

e−i�k3−k1�jOk3

† Ok1��k2

�̂k1,k2+ �̂k1,v

= �k1,k3

k1,k3Ok3

† Ok1��k2

�̂k1,k2+ �̂k1,v

= �k1,k2

Ok1

† Ok1�̂k1,k2

+ �k

Ok†Ok�̂k,v. �B3�

Similarly,

�j

�̂Oj†Oj = �

k1,k2

�̂k1,k2Ok2

† Ok2+ �

k

�̂v,kOk†Ok. �B4�

Finally, the term � jOj†Oj�̂Oj

†Oj is written as

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�j

Oj†Oj�̂Oj

†Oj = �k1,k2,k3,k4,k5,k6

1

N2

�j

e−i�k3−k4+k5−k6�jOk3

† Ok4�̂k1,k2

Ok5

† Ok6

= �k1,k2,k3,k6

1

N2

�j

e−i�k3−k1+k2−k6�jOk3

† Ok1�̂k1,k2

Ok2

† Ok6

=1

N�

k1,k2,k3,k6

k6,k2+k3−k1Ok3

† Ok1�̂k1,k2

Ok2

† Ok6

=1

N�

k1,k2,kOk

†Ok1�̂k1,k2

Ok2

† Ok2+k−k1. �B5�

Therefore, Eqs. �30�–�32� are obtained by summarizing Eqs.�B1�–�B5�.

APPENDIX C: APPROXIMATIVE MASTER EQUATIONSFOR SPECIAL CASES

In the cases of k=0 and k= ��, we have the approxi-mate master equation,

d�̂k,k

dt= − i�Hk�̂k,k − �̂k,kHk� + �

O=B,C���Ok

†Ok�̂k,kOk†Ok

−� + ��

2�Ok

†Ok�̂k,k + �̂k,kOk†Ok�� . �C1�

It is verified numerically that the term ��Ok†Ok�̂k,kOk

†Ok inEq. �C1� plays the same role as ��� /N��k�Ok�

† Ok�̂k,kOk†Ok� in

Eq. �30� for k=0, ��. For the �k=0,k=0�-block, Eq. �52�becomes

d

dtv1�t� = ig+�v4�t� − v5�t�� ,

d

dtv2�t� = − ig+�v4�t� − v5�t�� − �v2�t� ,

d

dtv3�t� = − ig+�v4�t� − v5�t�� − �� + ���v3�t� ,

d

dtv4�t� = 2ig+v1�t� − ig+�v2�t� + v3�t��

− �2i�J1 − J2� + i � +� + ��

2�v4�t� , �C2�

Solving the above differential equation, we have ��A,k=0�

shown in Eq. �56�. The average transfer time ��A,k=��� for thek= �� channel is also obtained similarly.

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