Hanning Chen, Mark A. Ratner and George C. Schatz- Time-Dependent Theory of the Rate of Photo-Induced Electron Transfer

8/3/2019 Hanning Chen, Mark A. Ratner and George C. Schatz- Time-Dependent Theory of the Rate of Photo-Induced Electr

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Published: August 12, 2011

r 2011 American Chemical Society 18810 dx.doi.org/10.1021/jp205262u|J. Phys. Chem. C2011, 115, 1881018821

ARTICLE

pubs.acs.org/JPCC

Time-Dependent Theory of the Rate of Photo-induced ElectronTransfer

Hanning Chen, Mark A. Ratner, and George C. Schatz*

Argonne-Northwestern Solar Energy Research Center, Department of Chemistry, Northwestern University, 2145 Sheridan Road,Evanston, Illinois 60208, United States

1. INTRODUCTION

Electrons are such mobile elementary particles1 that theirspatial movement in condensed phases from one host species toanother can either be driven by internal thermal fluctuations2 orby external electromagnetic excitation.3 Often the efficiency andcontrollability of electron transfer (ET) is of vital importance toessential biological functions of living organisms.4 One exampleis the oxidation of the Cytochrome c heme protein5 that helps toestablish a transmembrane proton concentration gradient foradenosine triphosphate (ATP) synthase.6 In this system, elec-tron release from the heme protein to molecular oxygen iscatalyzed by Cytochrome c oxidase (CcO), a terminal enzymein the respiratory chain of mitochondria.7 Interestingly, anautocatalytic cycle is formed in this redox reaction through a

conformational change in the CcO upon reduction of its boundheme group,8 suggesting a strong interplay between nuclearmotions of the reaction centers and electron transmissionthrough molecular interfaces.9,10 Another instance of pro-nounced vibration-electronic coupling arises in inelastic electrontunneling across single-molecule transistors at very lowtemperature,11,12 where the transferred electron can lose onequantum of vibrational energy at the molecular junction. In thiscase, the onset of a small conductance increase is only observedwhen the biased chemical potential between the two electrodes islarger than the junctions vibrational quanta, leading to adiscontinuity in the first derivative of the currentvoltage(IV) curve, better represented as a peak in the second

derivative of an I

Vgraph.13 In fact, even at zero temperature,

the ET is not completely suppressed thanks to the zero-pointfluctuations.14

Another typically more efficientwaytopromoteETisthroughlight irradiation,15 wherein the ET donor is excited to a higherelectronic state followed by transfer of an electron to the ETacceptor through vibronic coupling.16 The improved efficiencyofphotoinduced electron transfer (PIET) over thermally activatedET has been largely ascribed to stronger electronic couplingbetween the donor excited state and the acceptor ground statethan between the ground states,17,18 because the excited statewave function is generally much more diffuse than its groundstate counterpart.19 It is also found that ET can even occurwithout direct orbital overlap between a donor and an acceptor,in contrast with the Dexter exchange quenching mechanism.20 Inorder to explain the long-distance charge transfer between twophenyl groups linked by a polymethylene chain,21 the so-calledsuperexchange mechanism22was introduced, based upon wavefunction mixing with virtual states of the spacer molecules. Theperspective of superexchange has gained extensive supportfrom studies of donor-bridge-acceptor (DBA) systems,23,24 inwhich the PIET rate is usually observed to decay exponentially with donoracceptor separation.25,26 Although a larger free

Received: June 4, 2011Revised: August 2, 2011

ABSTRACT:A novel approachbased on constrained real-time time-dependent density functional theory (C-RT-TDDFT) is introducedto accurately evaluate the electronic Hamiltonian coupling associatedwith photoinduced electron transfer (PIET) using diabatic states thatare defined using constrained DFT (C-DFT). In combination withthe semiclassical Marcus theory, the photoexcited ET rate forcoherently coupled photoexcitation and electron transfer is deter-mined for a given incident wavelength by combining this Hamilto-

nian coupling with free energy changes and ground statereorganization energies that are obtained using an implicit solvationmodel. As an application of this method, we consider PIET for the(Ag20Ag)+ complex as a model of a plasmon-enhanced electrontransfer process. Using solar radiation intensity, the fastest PIET rate is found to be induced by an incident wavelength thatis distinct(blue-shifted) from the wavelength of strongest plasmon-like excitation associated with the Ag20

+ cluster, particularly for largedonoracceptor separations, which suggests a much more efficient coupling Hamiltonian for higher-energy molecular orbitals.Through a comparison between the PIET and absorption cross sections, the quantum efficiency for PIET is found to be a fewpercent at most at short donoracceptor distances, and it decays exponentially as they separate.


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18811 dx.doi.org/10.1021/jp205262u |J. Phys. Chem. C2011, 115, 1881018821

The Journal of Physical Chemistry C ARTICLE

energy difference between donor and acceptor does not alwaysensure a faster charge transfer,27,28 it is nevertheless an essentialfactor as demonstrated by the strong dependence of the PIETrate on the size-modulated injection potential in a quantum dot-metal oxide junction.29 In addition, the ease of solvent reorienta-tion upon charge transfer relative to the transient lifetime of theexcited species is also widely believed to be critical to PIET.30,31

The standard Marcus expression for the electron transfer rateis32

kET 2p

jHDAj2 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4kBT

p eG0 2=4kBT 1

where HDA is the electronic coupling strength, is the reorga-nization energy,G0 is the free energy difference between donorand acceptor, and Tis the absolute temperature. This expressionassumes that the vibrational states are governed by a classicalBoltzmann distribution (i.e., pvib , kBT). In theBornOppenheimer approximation, the vibronic wave function,(R,r), is separable into a nuclear part,jN(R), and an electronicpart, e(r):

R, r jNRer 2So, eq 1 includes a density-weighted FranckCondon

(DWFC) factor, (1/(4kBT)1/2)e(G0 + )

2/(4kBT) , whicharises from the vibrational orbital overlap, jN(D)|jN(A),between the donor (D) and the acceptor (A), and an electroniccoupling Hamiltonian term HDA given by

HDA eDjH0jeA 3For ET assisted by high-frequency molecular vibrations with

pvib . kBT , the more generally applicable MarcusDogonadze Jortner equation accounts for nuclear tunnelingeffects:33

kET jHDAj2

p

ffiffiffiffiffiffiffiffiffiffikBT

r

j 0 e

D

j! Dj

!eG0 jpvib

2

=4kBT

4where D is the vibrational-electronic coupling strength, a simplefunction of the reduced bond strength change. Because thepresent study is restricted to the study of PIET induced by theslow solvent reorientation, our methodology development andthe discussions thereafter will be exclusively based on eq 1, thereduced semiclassical form of Marcus theory. Thus, the non-adiabatic ET rate is readily determined once three key factors,namelyHDA, , and G0, are known.

For decades, a challenging task with the theoretical modelingof ET has been the reliable construction of the diabatic ET

states34,35 in which charge is localized on one of the two reactioncenters.36 Recently, constrained density functional theory(C-DFT) has been developed, which provides this capabilitythrough the imposition of a desired charge constraint onto theposition-dependent Hartree potential.37 Although C-DFT stillsuffers from the self-interaction error (SIE) within each con-strained region (which is a well-known deficiency of DFT38

particularly for strongly correlated electronic systems39), it wasfairly successful in applications to charge recombination rates foranthraquinone dye molecules36 and to the exchange-couplingconstants of transition-metal complexes.40 As an added benefitfor our calculation of long-range ET, C-DFT directly builds thecharge-transfer (CT) state, in which the energy of the lowest

unoccupied molecular orbital (LUMO) can be regarded as theelectron affinity.41 By contrast, only the highest occupied mo-lecular orbital (HOMO) energy is meaningful in conventionalDFT through the use of the ground state as the reference state, asthere is a energy discontinuity with respect to electron number.42

It is also known that when the donor and the acceptor arespatially well separated, the soundness of time-dependent density

functional theory (TDDFT)43

is questionable due to the negli-gible orbital overlap if the popular local exchange-correlationfunctionals are used.44 Consequently, C-DFT is seemingly apromising solution for tackling the longstanding issue of theunderestimated excitation energy in TDDFT45which can resultin incorrect values for the ET driving force.

Real-time time-dependent density functional theory (RT-TDDFT), a time-domain variant of TDDFT, has receivedincreasing attention since its first application in calculating thedipole response of large atomic clusters in an external perturbingelectric field.46 In concert with rapid developments in lasertechnology using ultrashort pulses and intense power,47 RT-TDDFT enables the direct study of electron dynamics in acomputationally feasible manner.48 The commonpractice of RT-

TDDFT is to numerically solve the time-dependent KohnSham equation49 subject to an electric dipole interaction thatincludes a time-dependent external electromagnetic field. Be-cause RT-TDDFT allows for an arbitrary number of externalperturbations, both linear50 and nonlinear51 optical propertiescan handily be incorporated within same theoretical frameworkby solving coupled dipole-field response equations. A hybridquantum mechanics/classical electrodynamics (QM/ED) ap-proach was recently formulated to account for the polarizationeffect of a nanoparticle on the optical properties of a nearby dyemolecule.52 Besides flexibility and extensibility, RT-TDDFT isalso noted for its smaller basis set requirements as it avoidsexplicit construction of the unoccupied (virtual) molecularorbitals.53 In most cases, a basis set sufficient for a ground state

calculation can be adopted for RT-TDDFT, and in the presenceof a modest light source (such as solar radiation) there is only asmall fluctuation around the optimized ground state wavefunction. Another attractive feature of RT-TDDFT is its cap-ability to determine any linear physical property over a broadrange of incident frequencies from a single simulation.54 Thisallows for systematic design of dye molecules with desired opticalproperties, including PIET.

Despite considerable efforts devoted to the description of ETin both closed53,55 and open56,57 systems using RT-TDDFT, thesimulation duration is still severely limited by the computationalcost associated with the explicit electron dynamics. Therefore,only ET phenomena associated with a few hundred femto-seconds of simulation can be simulated by RT-TDDFT. How-

ever for PIET processes involving plasmonic nanoparticles,which comprise the application of interest in this paper, thistime limitation is not a problem as plasmon relaxation occurs onthis same time scale.

Another theoretical difficulty with using RT-TDDFT tomodel ET arises in the construction of the initial and finalelectronic states, which typically are not eigenstates of the fullelectronic Hamiltonian.41,58 In earlier work, a RT-TDDFT study was carried out to track the molecular conductance across apolyacetylene wire after a chemical potential bias had beenestablished through an electron density gradient that was gener-ated by a charge constraint.59 Although steady state electrontransmission was never reached in that study due to the


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persistently decreasing bias voltage, the idea of incorporatingC-DFT into RT-TDDFT calculations to define diabatic ET is animportant suggestion. Another encouraging attempt to treat thephoton-phonon coupling was made in the description of excitedcarrier dynamics in a carbon nanotube. Here C-DFT wasemployed to build an initial excited state to mimic photoexcita-tion and then an Ehrenfest molecular dynamics (EMD) calcula-

tion was used to describe nonradiative decay leading to chargedensity variation.60 However, an issue with the combinedC-DFT/EMD approach is that it is not always easy to identifythe appropriate charge constrained states as the initial conditionin EMD leads to considerable uncertainty about the excitationenergy generated by DFT.44 For example, the energy-leveldegeneracy of the excited states can impose ambiguity.

In light of the growing interest in modeling PIET, a reliableand feasible first-principle approach is needed for evaluatingelectron transfer rates that are important in numerous applica-tions, including high-efficiency solar cells,61 photoconductivemolecular electronics62 and photocatalytic water splitting.63

We herein present a time-dependent formalism to evaluate thePIET rate within the framework of Marcus ET theory.64A novel

component of our work is the rigorous determination of theHamiltonian coupling matrix element between the excitedcharge transfer state and the diabatic ground state by trackingtheir time-dependent wave function overlap along a RT-TDDFTtrajectory. During a RT-TDDFT simulation, a constrainedposition-dependent Hartree potential is applied not only toconstruct the initial diabatic wave function, but also to guidethe electron dynamics with the diabatic Hamiltonian. Theemphasis in this work is on plasmonic PIET processes, so weonly consider short-time coherent dynamics that can occurbefore plasmon relaxation occurs on the 10100 fs time scale.This relaxation is included using empirical damping factors, andthe nuclei are assumed fixed. Generalizations to include nuclearmotions are possible, but will not be considered.

The remainder of this paper is organized as follows. In section2, the protocol to calculate the ground state electron transfer rateis introduced based on the C-DFT, an implicit solvation modeland the Marcus formalism. In section 3, the constrained real-timetime-dependent density functional theory (C-RT-TDDFT) isformulated to determine the frequency-dependent PIET matrixelement that will be used to evaluate the PIET rate and theexcited state ET rate. In section 4, we report numerical results fora model plasmonic system, the complex (Ag20Ag)+, consistingof a Ag20 cluster and an isolated Ag atom, which we use to studythe correlation between photoabsorption and photoinducedelectron transfer. Finally, we summarize our results and brieflydiscuss possible applications in section 5.

2. GROUND STATE ELECTRON TRANSFER RATE, kg

As illustrated in Figure 1, PIET within a closed system can besimplified to a cyclic three-state process (here with spin-forbid-den intersystem crossing65 and excited state decay66 beingneglected). In this picture, the three states are the ground donor(D), the ground acceptor (A) and the excited acceptor (A*), andthe three transition paths between these diabatic states arevertical excitation, excited state ET and ground state ET. Pleasenote that we have assumed that the ground state acceptor isbelow the ground state donor, while excited state acceptor isabove ground state donor. Excitation of the ground donor is notconsidered due to its relatively low population in comparison to

the ground acceptor. Also, the thermal acceptor to donor rate isassumed to be very slow and is thus neglected.

In ET theory, the diabatic states are defined by imposing

charge localization on the reaction centers.58

In this sense, theatomic charge distribution, QC , of a given diabatic state mustsatisfy the definition of that diabatic state:

QC QeR, r 5where Q is an operator to transform the delocalized electronicwave function into localized atomiccharges. Upon ET, there is anassociated free energy change, G0, (Figure 2) which receivescontributions from both the reactive system and from thesurrounding environment, e.g., solvent:

G0 G0,intra G0, sol 6

In this paper, the entropy component of eq 6 is ignored due tothe use of an implicit solvent model. Accordingly, G0 can berewritten as

G0 EA ED N

i1UA, iQA, i

N

i 1UD, iQD, i 7

where ED and EAare the energies of the donor and acceptor andUD,i and UA,i are the solvation potentials acting on the ith atom.The latter can be determined by solving the Poisson equation

3 rUr 4Qr 8where is the position-dependent dielectric constant and Q(r)are the atomic charges derived from the C-DFT calculations.

Figure 1. Energy level diagram which shows the relationship of photo-induced electron transfer to ground state ET. pn0 is the verticalexcitation energy.

Figure 2. Energy level diagram forground state ET and excited state ET.


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Similarly, another essential parameter of ground state ET, 0, isevaluated as

0 N

i 1UD, iQA, i

N

i 1UA, iQA, i 9

to reflect the energy penalty of solvent reorientation.FollowingtheC-DFTprotocolpresentedbyWuandVanVoorhis,67

the procedure to calculate the electronic Hamiltonian matrixelement, HDA, is briefly summarized as follows:

(i) A constrained energy minimization with respect to elec-tron density is performed for each of the desired QD andQA to determine the charge-constrained states ofCD

0 andCA

0 , respectively:

H0 VCwC0C FC0C 10where both the constraining strength, VC, and the con-straining operator, wC are unique for a given QC. Addi-tionally, H0 is the standard DFT Hamiltonian thatincludes the kinetic energy, Coulomb repulsion, ion-electron attraction, and exchange-correlation term.68

(ii) Since CD0

and CA0

are not guaranteed to be orthogonal,a diagonalization of the overlap matrix

S 1 0CDj0CA

0CAj0CD 1

!

is needed to determine the diabatic transformationmatrixC

wCC SCN 11

where

wC QD 0CD

wC0CA

0CA

wC0CD QA

0BBBBB@

1CCCCCA

is the constraining matrix and Nis a diagonal matrix.

(iii) The true diabatic ground states with strict orthogonality,i.e., A

0 |D0 = 0 = D

0 |A0 , are constructed from

CD0 and CA

0 with the aid of the Hermitian transposeof the matrixC

0

D0A

! C 0CD0CA ! 12Finally, the ground state diabatic Hamiltonian couplingmatrix element, HDA

0 is the off-diagonal term of theresultant matrix

CED FA0CDj0CA VAC 0CDjwACj0CA

FD0CAj0CD VDC 0CAjwDCj0CD EA

!C 13

With G0, 0 , and HDA0 on our hands, we are in a good

position to calculate kg using eq 1 if the temperature is highenough for a sufficient population in the vibrational states of thediabatic potential well (Figure 2).

3. PHOTOINDUCED ELECTRON TRANSFER RATE, kPIET()

Assuming the absence of structural relaxation upon electronicexcitation, the PIET driving force and reorganization energy aregiven by

G0 G0 p 14and

1 N

i 1UA, iQD, i

N

i 1UD, iQD, i 15

where is the incident light angular frequency.

For a system under a time-dependent Hamiltonian,^

H1(t), itstime evolution operator, U(t0,t), follows the time-dependentSchrodinger equation

ipUt0, t

t H0 H1tUt0, t 16

and it can be separated into a stationary part, U0(t0,t), and aperturbative part, U1(t0,t)

Ut0, t U0t0, tU1t0, t 17where

U0t0, t eiH0t t0=p 18

With the boundary condition

^

Ut0, t0 1 19it is easy to recognize that

U1t0, t 1 ip

Ztt0

dt0 eiH0t t0=pH1t0eiH0t t0=pU1t0, t0

20Therefore, U1(t0,t)canbeexpandedtoanyorderinarecursive

manner

U1t0, t 1 ip

Ztt0

dt0 eiH0t t0=pH1t0eiH0t t0=p

1

i

pZ

t

t0

dt0 eiH0t t0=pH1t0

eiH0t t0=pU1

t0, t

0

::: ! 21

Ifasystems wave function is expressed as a linear combinationof the eigenstates of the constrained Hamiltonian, H0 + Vcwc

t n

cntneiEnt=p 22

The coefficient of a given diabatic state,n, follows the Dysonseries

cnt c0n c1n c2n ::: 23


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where

c0n 0n

c1n i

p

Zt0t0

dt0 njH1t0j0ein0t0 t0

c2n i

p2

m

Ztt0

dt0 njH1t0jmeinmt0 t0

Zt0

t0

dt00 mjH1t00j0eim0t00 t0

::: 24

In the context of PIET (Figure 1), 0 is the ground acceptorstate, m refers to all FranckCondon allowed excited stateswithin the acceptor manifold, and n represents the grounddonor state,D. The coupling Hamiltonian H1(t) is the sum ofthe full Hamiltonian H0 (which leads to diabatic coupling) andthe dipole interaction EBx(t) 3Bx.

In the small perturbation limit, D can be added to the basisset used to represent (t) due to the assumptions

0jD 0 and cnt 0 n 6 0 25

In addition, cn(1)(t), which refers to the direct transition from

0 toD can be taken to be zero, as there is no diabatic couplingbetween these states due to eq 25 and their optical coupling isneglected as discussed previously. As a result, the first nonzeroterm in eq 24 involves cn

(2)(t), corresponding to the indirecttransition between 0 and D via the intermediate state, m.In this evaluation, the coupling matrix element between 0 and

m is within the acceptor manifold is therefore based onconstrained states. This matrix element arises from the dipoleinteraction, while the couplingm andD is between acceptorand donor manifolds and therefore involves the diabaticcoupling.

Using the same notation as in section 2, 0 and Dare jA(R)A

0 (R,r) and jD(R)D0 (R,r) respectively. To

simplify the FranckCondon factor evalution, we assumethat the only excited states, m, that contribute to the PIETprocess have the same initial vibrational wave function as0, i.e.

mt0 jAR, t0mR, r 26

If a continuous wave, Ex(eit+ eit), is turned on at t0 = 0

to vertically excite the system, then the matrix element ofH1(t) between ground and excited states within the acceptormanifold is

mjH1t0j0 mj EBx 3~xj0Aeit eit 27

Similarly the matrix element of H1(t) between m and Dinvolves the diabatic coupling

DjH0jm 0DjH0jmjDjjA 28

By inserting eqs 27 and 28 into eq 24, the second-orderDyson coefficient of the donor state is given by

c2D

i

p

2m

Zt0

dt0 DjH0jmeimdt0Zt0

t0

dt00 eit00

eit00 mj EBx 3~xj0eim0t00

ip

2m

Zt

0dt0 jDjjA0DjH0jmeimdt0

Zt0

t0

dt00 eit00 eit00 mj EBx 3~xj0Aeim0t00

ip

2m

0DjH0jmmj EBx 3~xj0AjDjjA

Zt

0dt0 eimdt

0 eim0 t0 1

im0 eim0 t

0 1im0

!

i

p 2

m 0Dj^H0jmmj EBx 3~xj

0AjDjjA

eid0 t 1

m0 d0 eidmt 1

m0 dm

eid0 t 1

m0 d0 eidmt 1

m0 dm

29

Note that the sum over m in this expression is a restrictedsum thatonlyreferstothosestatesthathaveaFranckCondonfactorofunity.

In the vicinity of m0,

c2D

i

p

20DjH0jmmj EBx 3~xj0AjDjjA

eid0 t 1

m0 d0 eidmt 1

m0 dm

!30

Accordingly, the time evolution of the electron population atD subject to an incident photon energypm0 is

jcDj2 ip

40DjH0jm2mj EBx 3~xj0A2jDjjA2

eid0 t 1

m0 d0 eidm t 1

m0 dm

!

eid0 t 1

m0 d0 eidmt 1

m0 dm

!

1p

40DjH0jm2mj EBx 3~xj0A2jDjjA2

sin2

d0 t2

m0 2 d0 2 2 sin

2 dmt

2

n0 2 dm2 2

0BBBB@

2cosd0 t cos dmtm0 2d0 dm sin2

m0 t2

m0 2

2d0 dm

1CCA

31


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For a continuous band characterized by a density of states, F(d)

jcDj2 1p

40DjH0jm2mj EBx 3~xj0A2jDjjA 2

Z

dd0 Fd

sin2

d0 t2

d0 2

2m0 2

sin2

dmt

2

m0 2 dm2 2

0BBBB@ 2cosd0 t cos dmtm0 2d0 dm

sin2

m0 t2

m0 2

2d0 dm

1CCCA

1p 4

0DjH0jm2mj EBx 3~xj0A2jDjjA2 2tF

dm0 2

2t

p

0DjH0jm2mj EBx 3~xj0A2

2

jDjjA2Fdp

32

where an empirical damping factor of is added to the resonanceenergy, i.e., pm0 f pm0 + i, to reflect the broadening of theresonancepeak dueto thequantum dephasingandvibronic coupling.In combination with the density-weighted FranckCondon factorbetween the potential energy surfaces ofD

0 and m as derived byMarcus,32 the PIET rate equals

kPIET djcDj2

dt 2

p

jDjjm2Fdp

0DjH0jm2mj EBx 3~xj0A2

2

2p

DWFCdm0DjH0jm2mj~xj0A2

2

Ic0

33

where c is the speed of light,0 is the vacuumpermittivity,I() is theradiation intensity, and D

0 |H0|mm|Bx|A0 is the absorption-

weighted electronic coupling strength.As mightbe expected, the PIET rate (eq 33) is proportional to

the square of the diabatic coupling matrix element multiplied bythe square of the dipole matrix element, and divided by thesquare of the excited state width. This is consistent with theassumed coherent mechanism in which photoexcitation isdirectly coupled to electron transfer, and it is only processes which take place during the excited state lifetime that can

contribute to the rate.Inspired by the RT-TDDFT,54 a numerically convenient way toevaluate D

0 |H0|m2m|Bx|A

02 is to track the time evolutionof the electronic coupling strength between D

0 and A(t) after ashort electric pulse ofEx is applied to the acceptor for a time oft

HDAt 0DjH0jAt 0DjH0j0A m

cmt0DjH0jm

H0DA m

ip

0DjH0jmt0Aj EBx 3~xjm

eim0t eim0t 34

Subsequently, a Fourier transform of the perturbative part ofHDA(t) at m0 normalized byExtyields

DA, x

Zdt HDAtei t

Zdt Ex

0DjH0jmt0Aj EBx 3~xjmF

pExt

0DjH0jmmj~xj0A

35

where DA,x is defined as the electron transfer matrix elementthat has the same unit as electric dipole moment. In general, thespatial averaging has to be taken account of for randomlypolarized incident light

DA

1

3DA, x

DA,y

DA, z

36

Alternatively, HDA(t) can be written as

HDAt DtjH0j0A H0DA Dtj0A0AjH0j0A

Dtj0D0DjH0j0A H0DA Dtj0AEA

Dtj0DH0DA H0DA 37In the limit of a small perturbation

Dtj0D 1 38

we end up with a concise form for HDA(t),

HDAt 0DjAtEA 39Now, the computational task of determining the excited state

ET rate has been simplified to an evaluation of the timedependence of the diabatic wave function overlap.

For each of the two charge-constrained states, CD0 and

CA0 that are determined by C-DFT in eq 10, the perturbation

field,Ex, is added to their Hamiltonian. Then, their wave functionevolution can be described by solving the time-dependentKohnSham equation

ijCt

t

H0

VCwC

EBx 3~x

jC

t

40

Following a diabatic transformation using the matrixC

0DjDt 0DjAt0AjDt 0AjAt

!

C 0CDjCDt 0CDjCAt

0CAjCDt 0CAjCAt

!C 41

D0 |A(t) is the off-diagonal term of the resultant matrix. Note

that this same calculation can also be used to determine theoverlap needed to calculate the PIET rate for the reverse process.


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Finally, the PIET rate is written as

kPIET 2p

eG0 p 12=41kBTffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

41kBTp 2DA

Ic0

42

and the linear relation between kPIET() and I() allows us todefine a PIET cross section, PIET()

PIET pkPIETI

2c0

eG0 p 12=41kBTffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

41kBTp 2DA 43

This cross section is useful for quantifying the efficiency ofelectron transfer. In particular, the conversion rate from photonsto electrons can be also represented by the quantum yield (QY)

Q Y PIETabs PIET 44

where abs() is the absorption cross section defined as the ratiobetween the stimulated transition rate and the incident photoflux density. The detailed protocol to evaluate abs() is given inref 52.

So far, our discussions have been focused on the coherenttransition rate from one diabatic ground state to another via theexcited states. However, the so-called excited state ET rate,ke(), can also be evaluated using this formalism. This is thequantity that is often measured using the transient absorptionspectroscopy,69 and in contrast to the PIET process it imagines

that the excited state lives long enough that the population ofexcited states is well-defined. In our notation, ke() is thetransition rate from m to D

ke 2p

DWFC0DjH0jm2 2p

PIETabs 45

Therefore, by calculating the ratio ofPIET() and abs(),we can determine the value ofke(). Note that in our real-timeformalism, this rate constant refers to the average over all excitedstates within the approximate energy width of. Except for that,this rate constant does not explicitly depend on the excited statelifetime.

4. APPLICATION TO PIET FOR THE (Ag20Ag)+COMPLEX

a. Simulation Details. Our model system is a water-solvated(Ag20Ag)+ complex consisting of a tetrahedral Ag20 cluster andan isolated Ag atom or ion (Figure 3). This system provides asimple model of a plasmonic silver nanoparticle (modeled asAg20based on past work

70,71which shows this has plasmonic-likeproperties) interacting with a silver atom or ion in solution.Motivated by experimental work with much larger silver particles which shows plasmon-assisted reduction of silver ions insolution,72,73 we will use this system to study plasmon-assistedelectron transfer, with emphasis on whether the wavelengths thatoptimize PIET are similar to those which optimize absorption.Note that the thermodynamics of our model system looks likethatinFigure2inwhichthedonorisAg20Ag+ and the acceptoris Ag20

+Ag. The PIET process in this case involves oxidation ofthe Ag atom to Ag+.

With this model system and the optical excitation, we will

study the PIET rate between Ag20+

and Ag driven by thereorientation of the solvent as described by the Pois-sonBoltzmann implicit solvation model.74 A charge distribu-tion function, QC, is defined to facilitate numerical definition ofthe diabatic states

QC QAg20 QAg 46

where QAg20 and QAgare the charges of Ag20 and Ag, respectively.According to their relative energies, Ag20Ag+ is the donor withQD = 1, whereas Ag20+Ag is the acceptor with QA = +1. Toexamine the effect of donoracceptor distance on the PIET rate,the separation from Ag to one of the triangular surfaces of Ag20 isvaried from 6 to 10 equally spaced by 1 . The nuclei are

otherwise fixed as we imagine that there are negligible structuralchanges associated with the electron transfer.

The APBS package74 was utilized to solve the PoissonBoltzmann equation to obtain the solvation potential (eq 8), acrucial component used in evaluating G0, 0 , and 1. Adielectric constant of 78.54 and a molecular radius of 1.4 werechosen for the implicit water model, while the Blochl scheme75

was employed to determine the atomic charges of the explicitsolute based on ground state electronic wave functions from theC-DFT calculations. The basic idea of the Blochl scheme is touse localized atom-centered Gaussian packets to fit the multi-pole moments and the charge widths of a given electron den-sity distribution, making it an excellent choice for charge fitting

Figure 3. Model system consisting of a (Ag20Ag)+ complex solvatedin water. The density of transferring electron is indicated by the

blue shade.Figure 4. Ground state driving force and reorganization energy as afunction of Ag20Ag separation.


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of optical properties that are subject to the multipole-fieldinteractions.

The DFT, C-DFT, and C-RT-TDDFT calculations were all

performed using the CP2K software76with the PerdewwBurkeErnzerhof(PBE) exchange-correlation functional,77 the GoedeckerTeterHutter (GTH) pseudopotential78 and the polarized-valence-double- (PVDZ) basis set.79 The geometry of Ag20was first optimized by DFT with zero net charge, and then theionized Agwas added along the x axis. In the CDFT calculation(eq 10), the constrained wave function optimization continueduntil a deviation from the target QC is less than 10

6. Afterward,the optimized wave functions were used as the initial states in theC-RT-TDDFT simulations (eq 40), in which a short electricpulseExwithadurationof0.0121fsanda fieldstrength of 1.47 106 V/m was applied along the x axis to drive the electrondynamics. No polarization averaging is used in the present study.The time-dependent Schrodinger equation was integrated by the

enforced-time-reversible-symmetry (ETRS) algorithm80 with atime step of 0.0121 fs. A RT-TDDFT trajectory was propagatedfor a total of 4000 steps to converge all optical and ET propertiesusing a value of 0.1 eV for the damping factor. (This choice hasoften been used to empirically represent damping in a plasmonicsystem with frozen nuclei.70,81,82) The average solar radiationintensity of 342 Watt/m283 was chosen to evaluate kPIET. Thisvalue is in the range where linear optical response should occur.

b. GroundState ET Rate,kg, for the Reaction Ag20 + Ag+

f

Ag20+ + Ag. As shown in Figure 4, the magnitude of G0

decreases slightly from 1.2 to 1.0 eV when R is increasedfrom 6 to 10 . This small change reflects the size of electro-static interactions between the ion and polarizable atom for the

distances considered. By contrast, 0 was found to increasesubstantially by 0.5 eV over the same range of R , as theseparated Ag20

+ and Ag+ are more strongly solvated than themerged complex. The optimum condition of Marcus theory, i.e.,

G0 =

0, almostoccurs at R= 6, leading to a sizable DWFC.The calculated HDA0 is presented in Figure 5. This exhibits arapid decay with increasing Rdue to the exponential variation ofthe overlap of electronic wave functions.84 Byfitting it to a shiftedexponential function of the form of

H0AD H0eR R0 47the value 1.38 1 is obtained for the decay coefficient, .Compared to the DBA systems which typically have < 0.61,25 the more rapid variation of HDA

0 in our (Ag20Ag)+complex is likely caused by the absence of a -conjugated bridgemolecule.

Using the Marcus formalism of eq 1, and HDA0 from Figure 5,

the ground state rate kg at 300 K is shown in Figure 6. Notsurprisingly the exponential decay law is again satisfied, this timewith an even larger value of 3.65 1for due to the presence of|HDA

0 |2 (Figure 5) in eq 1 and the dependence ofG0 and0 onR(Figure4)intheexponentontherightsideofeq1.For R 9 , kgdrops to the nanosecond time scale that is comparablewith large-scale nuclear motions.

c. PIET Rate, kPIET(), for the ReactionAg20+ + Ag f

hAg20 +

Ag+. The excited state ET rate uses the reorganization energy1,which is plotted in Figure 7. As expected, the reorganizationenergy increases with Ag20 Ag separation much as with 0.However, 1 is much larger than 0 because of the larger

Figure 5. Exponential decay of the ground state electronic couplingstrength.

Figure 6. Computed ground state ET rates.

Figure 7. Reorganization energy of the donor.

Figure 8. Calculated absorption cross section of Ag20+Ag.


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solvation energy of the isolated Ag+ than that of Ag20+, in which

the extra positive charge is delocalized over the whole cluster.This higher reorganization energy exceeds the PIET drivingforce, G0 p, even in the visible spectrum, and as a resultthe PIET rate is relatively slow for solar incident flux (see below).

In our derivation of the PIET rate, the excited molecularorbitals are assumed to provide the essential intermediate statesfor the stepwise ET relay. To quantitatively reveal the relationbetween light absorption and PIET, the absorption cross sectionabs()oftheirradiatedacceptorAg20

+AgisshowninFigure8.For visual simplicity, only results for 6 and 10 are presented.Nevertheless, the two spectra are nearly identical with a sharpintraband peak centered at 3.33 eV in addition to a broad plateauto the blue. The peak location is in good agreement with aprevious theoretical study using frequency domain TDDFT onAg20.

71 Note that this transition can be thought of as defining aplasmon-like excitation which for larger clusters becomes theplasmon-resonance excitation. For the Ag20

+

Ag system, the

invariance of the spectrum to the location of the Ag atomindicates that electronic excitation predominantly occurs in theAg20

+ moiety, with little influence from the Ag. This can beunderstood since the lowest excitation energy of silver atom iswell above that for Ag20

+.The squared magnitude of the frequency-dependent PIET

matrix element, ||2, calculated by C-RT-TDDFT is plotted as afunction of frequency in Figure 9. For small separations Re 7 ,there exists a flat plateau centered at 3.38 eV, in addition to asharp primary peak at3.63 eV. It is very interesting to note that||2 does not maximize at the absorption maximum (3.33 eV inFigure 8), suggesting that molecular orbitals with high oscillatorstrength are not the most effective in producing electron transfer.

This hypothesis is even more apparent in the appearance of ||2

at larger separations R g 8 where the flat plateau nearly

disappears, leaving a peak at 3.60 eV as the only feature. Notealso that the primary peak height decreases significantly from1.4 1011 to 1.7 109 debye2 as R increases from 6 to 10 ,corresponding to an approximately equivalent to 5-fold reduc-tion for every angstrom of additional separation.

Assuming an incident intensity of 342 W/m2 , the calculatedkPIET is shown in Figure 10. This demonstrates a similarfrequency dependence to ||2 except for a notable increasetoward the blue. This arises because the maximum DWFCoccurs in the UV, i.e., p = 8.0 eV. By plotting the maximumkPIET (near 3.6 eV) as a function ofR, and fitting it to a shiftedexponential function (Figure 11), a smaller value of 1.38 1 for is found in comparison to 3.65 1 in kg. The slower decay in

Figure 9. Squared magnitude of the PIET matrix element, ||2.Figure 10. PIET rates, kPIET().

Figure 11. Maximum kPIET as a function of R. The PIET resonancewavelengths used to calculate kPIET are given under the circle markers.


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ET rate with R for photoexcited states is a consequence of thestronger Hamiltonian coupling as induced by spatially more

extended molecular orbitals.The PIET cross section, PIET(), is shown in Figure 12 and

exhibits a similar pattern to abs(), yet with a much smallerresonant magnitude even for small separations. For instance,when R= 6 , PIET() at p = 3.63 eV is approximately 20times smaller than abs() at p = 3.33 eV, suggesting that the(Ag20Ag)+ complex is still much more efficient in aiding vertical excitation than facilitating parallel ET when its twocomponents are narrowly separated. For the largest separationofR=10,theratiobetween PIET()andabs() significantlydecreases to 105, essentially shutting-offelectron transfer. Therather low PIET efficiency is also exhibited in Figure 13, whichpresents the quantum yield at p = 3.60 eVas a function ofR. It

turns out that at most 5% of the light energy can be transferredinto electrons R=6evenfortheoptimalwavelengthof344nm.

At the same incident wavelength, the excited state ET rateconstant, ke(), shows a dramatic decay with distance from fs

1

at R= 6 to ns1 at R= 10 (Figure 14). The reason why the

excited state rate constant is so dramatically larger than the PIETrate constant can be understood by examining eqs 33 and 45.This shows that the ratio of the PIET and excited state ratesinvolves light intensity multiplied by dipole matrix elementsquared and divided by the excited state width. In the presentapplication, the width factor corresponds to decay in 40 fs, sogiven the weak light intensity, the population of excited statescapable of undergoing electron transfer is extremely small. Tomake the rates more comparable would require intensities thatare 12 orders of magnitude higher.

5. DISCUSSION AND CONCLUSION

We have developed a real-time TDDFT approach for calculat-

ing rates of photoinduced electron transfer and have demon-strated its use in describing PIET for a plasmonic-like modelcorresponding to oxidation/reduction of Ag/Ag+ near a silvercluster. C-DFT was first employed to construct the diabaticground states for donor and acceptor, from which the drivingforce and solvent reorganization energies were obtained incombination with an implicit solvent model. To describe PIET,an external electric field is applied to the diabatic ground statesand the time-dependent KohnSham equation is solved todetermine donor and acceptor wave functions. By calculatingthe time evolution of the diabatic wave function overlap, thePIET cross section in the frequency domain is readilydeterminedthrough a Fourier transform. For reference, we have alsodetermined absorption cross sections using a related real-time

approach so that the relative efficiencies of these two processescould be compared.

In our application to the (Ag20Ag)+ complex, C-RT-TDDFTshows a plasmonic-like absorption spectrum, which peaks at 3.33eV, similar to what has been found in past studies of Ag20. For thePIET cross section, a sharp primary peak is found at3.60eVforalldonoracceptor separations, indicating that excitation which opti-mizes PIET is distinct from excitation which optimizes absorption.The PIET cross sectionPIET() is found tobe 0.15

2 for R= 6 and 3.63 eV, which is only about 5% of the absorption crosssection, indicating that PIET is a small component of the overalllight absorbed by the silver cluster. In addition, the quantumefficiency of PIET undergoes a significant 4-fold reduction at

Figure13. Quantum yield at thePIET resonancewavelength of 344 nmas a function ofR.

Figure 14. Excited state ET rate, ke, at the PIET resonance wavelengthof 344 nm as a function ofR.

Figure 12. Frequency-dependent PIET cross section, PIET().


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R= 7 due to the diminished electronic coupling strength as thedonor andacceptor move apart. Overall this shows that PIET is veryinefficient for the silver cluster system, and this is primarily due tostrong damping which occurs for plasmonic systems.

From a technical perspective, the most striking advantage ofthe C-RT-TDDFT approach is the elimination of the explicitconstruction of the excited states, which are the major sources of

CPU consumption and memory allocation in most excited statesimulations. On the theory side, the optimum scheme to imposechargeconstraints for excited states is not yet clear as C-DFT wasoriginally developed for the diabatic ground states.37 By dealing with the response of the diabatic ground states to externalelectromagnetic perturbation, C-RT-TDDFT avoids this ambi-guity and determines the electronic coupling strength at noadditional cost compared to ground state simulations. As a result,C-RT-TDDFT enables the investigation of PIET on systemswith hundreds of atoms, large enough to cover most organicelectronics of interest. Moreover, C-DFT is in principle amodified ground state approach that eliminates the necessity ofconstructing the localized ET states from the dubiously definedunoccupied Kohn

Sham orbitals. Thus, the time propagation of

the charge constrained states in C-RT-TDDFT is more likely toimitate the diabatic state evolution than the unconstrained RT-TDDFT, which usually starts from a mixture of the occupied andunoccupied KohnSham orbitals. Nevertheless, several assump-tions are made in our development that might limit its applic-ability, including neglect of structural relaxation upon electronicexcitation, and neglect of internal conversion that competes withexcited state ET.85 Several studies have already been carried outto estimate the nonradiative decay rate,8688 and their incorpora-tion into our approach is an important future direction. Anotherpossible limitation of the C-RT-TDDFT comes from the single-determinant and single-particle characteristics of the KohnSham orbitals, which usually fail to capture the static part of theelectron correlation when there are nearly degenerate electron

configurations.89 However, a configuration interaction (CI)based C-DFT study has shown that the missing static correlationof the localized diabatic states can be partially recovered througha multireference active space, suggesting a promising solution toextend C-RT-TDDFT for investigating PIET in strongly corre-lated systems.90

As a fundamental chemical process, PIET is important innumerous chemical, biological and medical applications9193

where damping effects are less important than in the presentapplication. However because the damping is less important,longer calculations will be required. Thus the theoretical model-ing of PIET remains a challenging yet rewarding task as lightradiation offers unique opportunities for control over ET.Indeed, coherent control94,95 has become a powerful tool forselectively breaking or forming chemical bonds using multiplelaser pulses shorter than the molecular vibrational period, and itsapplication to PIET could be explored using our approach. Also,since C-RT-TDDFT does not impose any limitations on thenumber of perturbed electric fields and their amplitudes, fre-quencies, shapes, and durations, it is an excellent choice for theinvestigation of nonlinear control.

AUTHOR INFORMATION

Corresponding AuthorFax: 847-491-7713. Phone: 847-491-5657. E-mail: [email protected].

ACKNOWLEDGMENT

The research was supported by Grant DE-SC0004752 fundedby theU.S. Department of Energy, OfficeofScienceandOfficeofBasic Energy Sciences. The computational resources utilized inthis research were provided by Shanghai Supercomputer Center.

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Documents

Hanning Chen, Mark A. Ratner and George C. Schatz- Time-Dependent Theory of the Rate of Photo-Induced Electron Transfer