Coherent control of long-range photoinduced electrontransfer by stimulated X-ray Raman processesKonstantin E. Dorfmana,1, Yu Zhangb,1, and Shaul Mukamelb,c,2

aSingapore Institute of Manufacturing Technology, Singapore 638075; bDepartment of Chemistry, University of California, Irvine, CA 92697;and cDepartment of Physics and Astronomy, University of California, Irvine, CA 92697

Contributed by Shaul Mukamel, June 30, 2016 (sent for review May 3, 2016; reviewed by Alán Aspuru-Guzik and Villy Sundstrom)

We show that X-ray pulses resonant with selected core transitionscan manipulate electron transfer (ET) in molecules with ultrafast andatomic selectivity. We present possible protocols for coherently control-ling ET dynamics in donor–bridge–acceptor (DBA) systems by stimulatedX-ray resonant Raman processes involving various transitions betweenthe D, B, and A sites. Simulations presented for a Ru(II)–Co(III) modelcomplex demonstrate how the shapes, phases and amplitudes of theX-ray pulses can be optimized to create charge on demand at selectedatoms, by opening up otherwise blocked ET pathways.

electron transfer | coherent control | ultrafast X-ray spectroscopy |stimulated Raman

Long-range electron transfer (ET) over tens of angstroms inmolecular assemblies plays an essential role in many biolog-

ical processes, artificial light-harvesting schemes, and sensorapplications (1–7). Using lasers to precisely control ET pathwaysand rates has been a long-term goal of chemists (8). The mannerin which infrared light can excite molecular vibrations to affectET in donor–bridge–acceptor (DBA) systems has been studiedtheoretically (9, 10) and experimentally (11–14).The rapid development of bright X-ray lasers and high harmonic

sources has opened up new opportunities for X-ray spectroscopy(15). We have recently demonstrated that stimulated X-ray Ramanspectroscopy (16, 17) with broadband X-ray pulses can reveal thetime-evolving oxidation states of various species in the long-rangeET process of the protein azurin (18). Here we show that by cou-pling to core-excited states, resonant X-ray pulses can preciselytarget either the donor, bridge, or the acceptor site in an ET processby altering the valence electronic states in its vicinity by triggering thebridge-to-acceptor (BA), the donor-to-bridge (DB), or the bridge-to-bridge (BB) ET transfer. We show how the ET pathways in amodel DBA system ([(CN)4Ru

II(tpphz)CoIII(CN)4]3−) can be co-

herently manipulated by X-ray pulses resonant with the acceptor.Application is made to a Ru–Co light-harvesting complex (shown

in Fig. 1 A and B) where an electron is transferred from the donorRuII to the acceptor CoIII to create RuIII/CoII. X-ray pulses cancreate valence excitations via a Raman process (19), thus altering theoccupied molecular orbitals (MOs). We shall focus on the BA ETcoherent control scheme illustrated in Fig. 2A. In a stimulatedRaman process a core hole created by the X-ray pulse on the ac-ceptor is instantaneously filled by a valence electron on the bridge,resulting in a B→A ET. Such an ET process is analogous to thevalence-to-core X-ray spontaneous emission observed in transitionmetal complexes with ligand-to-metal charge transfer (20).

Effective Model Hamiltonian for the Ru–Co ComplexLong-range ET in the bimetallic Ru–Co complex [(bpy)2

1RuII

(tpphz)1CoIII(bpy)2]5+ (low spin in the ground state) was investigated

recently by transient optical absorption, X-ray absorption, X-ray diffusescattering, and X-ray emission (21–23). This complex has been pro-posed for artificial light-harvesting applications. The DB ET step isvery fast (<50 fs) but the BA ET is much slower (picoseconds).The system ends up in a [(bpy)2

2RuIII(tpphz)4CoII(bpy)2]5+ high-spin

charge-separated state. Our goal is to accelerate the BA step by astimulated X-ray Raman process. To reduce the computational cost

while keeping the essential ET physics of the original complex, we havestudied the simplified model complex [(CN)4 Ru

II(tpphz)CoIII(CN)4]3−

(Fig. 1 A and B) where the bpy ligands are replaced with (CN)−. Thiseliminates the complicated spin crossover transition at the Co center(23), because the strong ligands (CN)− favor the low-spin state.The relevant ET parameters were obtained from electronic

structure calculations. Because the Ru and Co centers are far apart,electronic structure calculations can be carried out for smaller frag-ments [(CN)4 Ru

II(tpphz)]2− and [−(tpphz)CoIII(CN)4]2− (the tpphz

ligand is negatively charged). Computational details are given inMaterials and Methods.We model the ET in this complex using two frontier orbitals

on each of the donor, the bridge, and the acceptor site; these arethe highest occupied molecular orbital (HOMO) and the lowestunoccupied molecular orbital (LUMO). Neglecting spin, eachsite has four possible states: the ground state j0i, where theHOMO is occupied and LUMO is not; the negatively chargedstate c†mj0i (both orbitals are occupied); the positive charged holestate v†mj0i (both orbitals are unoccupied); and a single electron–hole pair (Frenkel exciton) state c†mv

†mj0i, where the HOMO

electron is moved to the LUMO. The operators c†m (cm) and v†n(vn) create (annihilate) an electron on site m and a hole on siten, respectively. They satisfy the Fermi commutation relationsfcm, c†ng= δmn, fvm, v†ng= δmn. Out of all of the orbitals only threestates D, B, and A are relevant to the ET process and are de-scribed below. Whereas the DB coupling is strong and the BAcoupling is weak, we neglect the direct donor–acceptor coupling.Initial photoexcitation creates an electron–hole pair on the donor

jDi= c†dv†dj0i. We assume that there are no electronic coherences

and only donor population is generated. This can be achieved by, e.g.,resonant optical or X-ray Raman pulse that is tuned to the vicinity ofthe Ru atom in the donor molecule. The electron then hops fromorbital d to orbital b due to the strongD=B coupling, thereby creating

Significance

Electron transfer is ubiquitous in chemical reactions and manybiophysical processes such as respiration and photosynthesis.In this work, we show how broadband X-ray pulses can be usedto change the local electronic structures of donor–bridge–acceptor(DBA) molecular systems, thus enabling and controlling electrontransfer between designated sites in the molecule. An X-ray–stimulated Raman process can control electron transfer withatomic specificity and ultrafast timescale. With an appropriateX-ray pulse setting, electron pathways can be selectively en-hanced. The control can be implemented at the donor, bridge,or acceptor sites in a DBA system.

Author contributions: K.E.D., Y.Z., and S.M. designed research; K.E.D and Y.Z. performedresearch; K.E.D. and Y.Z. analyzed data; and K.E.D. and Y.Z. wrote the paper.

Reviewers: A.A.-G., Harvard University; and V.S., Lund University.

The authors declare no conflict of interest.1K.E.D. and Y.Z. contributed equally to this work.2To whom correspondence should be addressed. Email: smukamel@uci.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1610729113/-/DCSupplemental.

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state jBi= cdc†bjDi≡ c†bv

†dj0i. The ET is completed when an electron

hops from the bridge to acceptor due to the weak B=A couplingcreating the final state: jAi= cbc†ajBi≡ c†av

†dj0i. We had calculated

the matrix elements of the Hamiltonian in SI Appendix, Eq. S1 in thisdiabatic basis jDi,jBi,jAi.Enabling Electron Transfer Pathways by an X-Ray RamanProcessIn the absence of the X-ray pulse the ET process is described bya generalized Redfield master equation:

_ρ=−L½ρ�, [1]

where the Liouvillian L½ρ�= i=Z½Hs, ρ�−Kρ is a rank 9 tensor inthe space of D, B, A states (three populations and six coher-ences). Here Hs =HDBA +Hhop is a 3× 3 Hamiltonian matrixwhich includes the DBA diagonal part HDBA (SI Appendix, Eq.S3) plus the electron hopping off-diagonal part Hhop (SI Appen-dix, Eq. S4), and K denotes 9× 9 Redfield ET rate matrix (SIAppendix, Eqs. S6–S8). The ET includes several pathways thatinvolve both population hopping (sequential) as well as elec-tronic coherences between adiabatic states (superexchange) (24).The time evolution of the acceptor population can be ob-

served by various spectroscopic measurements including fluo-rescence and transient absorption. Using Eq. 1 we had calculatedacceptor population. After photoexcitation of the excited statepopulation of the donor state D at time t= 0 in the diabatic basis,the electron wave packet goes through various states includingthe population and coherences between states D, B, A, and fi-nally populates the acceptor state A at time t given by

Pð0ÞA ðtÞ= iZGAA,DDðtÞ, [2]

where the zero superscript indicates that this is a referencecalculation in the absence of the X-ray. GAA,DDðtÞ= ½e−Lt�AA,DD isthe Liouville space Green’s function matrix element that rep-resents electron transfer dynamics between donor and acceptorpopulations and is given by SI Appendix, Eq. S10.

We now introduce an X-ray pulse at time t1 after the photoex-citation. This pulse can promote an electron from the core orbitalto the valence orbital of the acceptor, and the core hole is simul-taneously filled by an electron from the bridge valence orbital. Inthis X-ray Raman process shown in Fig. 2A, the electron is trans-ferred from orbital b to a via an inelastic process or from a to a andb to b via an elastic process. The ET can be controlled by an X-rayRaman excitation as long as the lifetime of the valence excitedstate created by the X-ray Raman process is comparable to orlonger than the ET timescale. Interaction of the molecule with lightis described by a dipole matter–field coupling in SI Appendix, Eq.S13. By placing a core electron on a valence orbital of the acceptor(orbital a) and filling this core hole with the valence electron fromthe bridge (orbital b), the pulse enables electrons to tunnel throughthe bridge and reach the acceptor. We had calculated the acceptorpopulation induced by the X-ray pulse and its variation with t2 (thetime delay between the X-ray pulse arrival and observation time)and t1 (the X-ray pulse delay relative to the initial photoexcitation).The ET dynamics consists of three steps (see the diagram in SIAppendix, Fig. S1). During t1 (before the X-ray pulse) the systemevolves according to the Redfield master equation (Eq. 1) and ispromoted from population ρDD to some superposition of bridgeand acceptor states ρmn, m, n=A,B,D. The second step is theevolution during the X-ray pulse. We use perturbation theory inthe X-ray field–matter interaction with the diagram in SI Appendix,Fig. S1. In the first-order perturbation theory with respect to X-ray

Fig. 1. (A) Three-dimensional molecular structure of the Ru–Co complex[(CN)4 RuII(tpphz)CoIII(CN)4]

3−, which is a simplified model of that studied inrefs. 21–23. The D→B and B→A ET steps are represented by the blue andgreen arrows, respectively. The B→A ET step is aided by an X-ray pulse shotat the Co center. The coordinate axes are also shown. Color code: Ru, lightblue; Co, pink; N, deep blue; C, black; H, light gray. (B) Chemical structure ofthe Ru–Co complex. The donor, bridge, and acceptor fragments are labeledand shaded with different colors.

Fig. 2. (A) The acceptor control stimulated X-ray Raman process invoked bytwo X-ray pulses, of which the carrier frequencies ω0

1 are resonant with thecobalt (acceptor) local core excitation and ω0

2 is resonant with the bridge tocobalt emission transition, respectively. (B) The two pulses E1 (blue) and E2 (red)in the time domain with their phases ϕ1 and ϕ2, respectively. See SI Appendix,Eq. S14 for detailed expressions. (C) The two pulses in the frequency domainwith the carrier frequencies ω0

1 = 7,720.65 eV and ω02 = 7,721.53 eV, bandwidths

σ1 = σ2 = 3 eV, and the maximum field intensities E1 = E2 = 5×1011 V/cm. Theamplitude ratio η is defined as E2=E1. See SI Appendix, Eq. S15 for expressions.

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intensity the density matrix of the system is changed from ρmn to ρqn,q=A,B,D. We consider two X-ray pulses (Fig. 2 B and C): E1 withphase ϕ1 and central frequency ω0

1 resonant with adiabatic transitionA′→ x, and E2 with phase ϕ2 and central frequency ω0

2 resonant withtransition B′→ x, where A′, B′, and x correspond to the acceptor,bridge adiabatic states, and the intermediate core excited state onthe acceptor, respectively. The adiabatic delocalized basis is used forcalculating the relevant dipole moments because the Hamiltonian inthis basis is diagonal (SI Appendix). In the localized (diabatic basis)one can therefore select pulses for the elastic m→ x→m and in-elastic m→ x→ q≠m, m, q=A,B,D Raman processes. Therefore,when q=m the X-ray interacts twice with the same state (ifq=m=A via E1 and if q=m=B via E2). Similarly, if q=A, m=B,or q=B, m=A interaction with X-ray occurs with both E1 and E2pulses. Finally, during the third ET step which occurs during theremaining time interval t2 electron again evolves according to Eq. 1from the state ρqn to its final state ρAA. Assuming well-separatedpulses, the incremental acceptor population linear in X-ray intensityis given by (see SI Appendix, Eqs. S6–S18 and SI Appendix, Fig. S1for details)

ΔPAðt1, t2Þ= 2RXi, j=1,2

 X

m, n, q=A,B

  GAA,qnðt2ÞαðijÞqmGmn,DDðt1Þ, [3]

where R denotes the real part, and generalized polarizabilitytensor αðijÞqm is defined by

αðijÞmn =−iZ

dω2π

ðei · μxmÞ�ej · μ*nx

�

×E*i ðωÞE j

�ω+ω0

i −ω0j +ωm −ωn

�eiðϕi−ϕjÞ

ω+ω0i −ωxm + iΓx

.

[4]

Here i= 1 if q=A, i= 2 if q=B, j= 1 ifm=A, and j= 2 ifm=B. SIAppendix, Eq. S3 contains both elastic and inelastic componentsof the ET pathways. The elastic components involve q=m=A,Band consequently i= j, which are independent of the phase of thefield, whereas inelastic components i≠ j depend upon phase dif-ference between the field E1 and E2: ϕ=ϕ1 −ϕ2. Note that αðijÞqmalso depends on the amplitude ratio η=E2=E1. Zero-, first-, andsecond-order terms in η correspond to elastic process q=m=A,inelastic process q≠m, and elastic q=m=B, respectively. There-fore, the effects of the amplitude ratio η and phase ϕ on ETcontrol will be demonstrated below.

Results and DiscussionThe following parameters were used in our simulations: donor en-ergy ED = 0 eV, bridge EB = 2.8 eV, acceptor energy EA = 5.4 eV,and hopping couplings are tDB = 0.6 eV, tBA = 0.008 eV. Thecorresponding electron transfer times are k−1DB = 20 fs and k−1BA = 2 ps.The acceptor has a single core state with energy ωx = 7,725.3 eV (toresemble the Co K-edge excitation energy) and linewidth Γx = 1.5 eV.The dipole moments in the adiabatic basis (SI Appendix) are takento be μA′x =−0.00086 a.u. and μB′x = 0.00033 a.u.. These values arerationalized in SI Appendix. We further assume Gaussian pulseenvelopes E jðωÞ=

ffiffiffiffiffi2π

pσjEje

−ð1=2Þσ2j ω2−iϕj, with the carrier frequencyω01 = 7,720.65 eV, ω0

2 = 7,721.53 eV, bandwidth σ1 = σ2 = 3 eV, andmaximum field intensity E1 =E2 = 5× 1011 V/cm. Due to theweak dipole transitions this strong electric field can still betreated within perturbation theory. Ionization cross-sectionis suppressed for Co when the pulse is resonant to a coreexcitation.We first study the variation with t1 for t2 = 3 fs. The time growth of

the acceptor population Pð0ÞA ðt1Þ in the absence of the X-ray pulse

shown in Fig. 3A is limited by the slow BA ET rate reaching 0.5% att1 = 20 fs. The incremental acceptor population ΔPAðt1Þ caused bythe X-ray pulse is governed by GAA,qnðt2Þ as shown in Fig. 3B. This

Fig. 3. Acceptor population vs. t1 at t2 = 3 fs (right after the X-ray pulse) inthe Ru–Co model complex. (A) Population without the X-ray pulse Pð0Þ

A ðt1Þ.(B) Incremental population due to X-ray ΔPA for different phases ϕ; (C) fordifferent amplitude ratio η and ϕ= 0; (D) Same as in C but for ϕ= π.

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large population increase (up to 8% at 20 fs) may be used fortransistor or gating applications (25). One can see that coherentcontrol is possible and ΔPA can change from 5% to 8.5% by varyingϕ from π=4 to π. Fig. 3C shows the variation of ΔPA with η at ϕ= 0.Note that increasing η from 0.2 to 1 decreases the acceptorpopulation from 6% down to 1%. Fig. 3D shows that for ϕ= πthe acceptor population increases from 5% to 15%. This is dueto the interference in various pathways that involve differentpowers of η from zeroth power (elastic A→ x→A) to first (in-elastic B→ x→A) to quadratic (elastic B→ x→B).We next turn to the variation of the acceptor population after the

X-ray pulse has passed. In Fig. 4A we show the unperturbedpopulation Pð0Þ

A vs. t2 at different t1. It shows the gradual increase inpopulation with increase of the pulse delay due to more populationtransferred to acceptor at later times via the ET kinetics. Thecorresponding population changes from 0% to 0.4% at t2 = 0 to0.6% to 1.1% at t2 = 20 fs. Similar increase can be observed for theincremental population ΔPA as shown in Fig. 4B. However, in thiscase population reaches 4.5%, which is much larger than the un-perturbed value. The phase control is depicted in Fig. 4C. Themaximum population change occurs at t2 = 0 from ϕ= 0 reaching8% down to 5% at ϕ= π=4 and intermediate values at ϕ= π=2 andϕ= 3π=4. At t2 = 20 fs the population changes from 9% at ϕ= 0, πdown to 6.7% at ϕ= π=4. Finally, the field amplitude ratio η canalso control the population. In this case, independently of the valueof ϕ the population always increases with increase of the strength ofinelastic component at t2 = 0 from 7% at η= 0.2 to 8.2% at η= 1and at t2 = 20 fs from 7.2% at η= 0.2 to 9.5% at η= 1.The X-ray pulse can significantly affect the acceptor population

in a limited parameter regime (e.g., appropriate dipole momentsfor the core transitions μB, μA, electron transfer rates KBA and KDB,as well as electric field intensities Ij =

��E j��2). We use a weak X-ray

pulse, so that ΔPA is less than 10%. Assuming ultrashort δ pulsesEjðt− t1Þ= I1=2j δðt− t1Þ for equal parameters of both pulses I1 = I2,σ1 = σ2, we obtain that the ratio of the inelastic component (thatinvolved B→ x→A transition) to elastic (A→ x→A) (SI Appendix,Eq. S19) should obey

jμBxjjμAxj

>KBA

KDB. [5]

Eq. 5 provides an important restriction on the pulse and matterparameters suitable for ET coherent control where inelasticcomponent of the stimulated Raman process contributes sub-stantially to the acceptor population.Other ET control schemes are possible. The present scheme

shown in Fig. 2A is based on control of the core hole on acceptor.However, if the core hole is on the bridge, the X-ray Raman processmay induce A→B ET and the D→B→A ET process is hindered.Another control scenario may arise in DB and BB coherent controlschemes. As sketched in Fig. 5 A–C, one can either use X-rays toinduce electronic transitions between donor and bridge MOs (Fig. 5A and B), or move electrons between MOs on the same or differentbridges (Fig. 5C), thus enabling or blocking ET pathways. The DBcontrol scheme is analogous to the BA discussed above. The onlydifference is the ET process manipulated by the X-ray pulse is theD→B step. In Fig. 5C strong and weak interactions between orbitalson different sites are represented by red solid and blue dashed lines,respectively. The donor orbital has strong coupling to the bridgeorbital b1 but weak coupling to b2; whereas the acceptor orbital hasstrong coupling to b2 but weak coupling to b1. The ET process is thushindered by the weak couplings. To overcome this, the bridge orbitaloccupations are altered by an X-ray Raman process (see the blackarrows in Fig. 5C), so that both D/B and B/A interactions becomestrong. The superexchange and sequential mechanisms are two lim-iting cases of ET (26). This BB control scheme can represent bothsequential ET in which the transferred electron actually populates the

bridge, and a superexchange ET mechanism in which the transferredelectron does not populate the bridge and the X-ray pulse simplypaves the pathway for the electron. The orbitals involved in the X-rayRaman process are not necessarily localized on the same bridge site ifthe DBA system contains more than one bridge site. In this case theX-ray pulse controls the electron flow between bridges.

ConclusionsWe have demonstrated how a stimulated X-ray Raman processcan induce the B→A transition, thus enabling the D→B→A ETprocess. An X-ray pulse resonant with a core transition inter-acting with a pair of bridge and acceptor electronic states may be

Fig. 4. Incremental acceptor population without the X-ray pulse ΔPð0ÞA vs. t2

for different t1 (A), with the X-ray pulse ΔPA vs. t2 for different t1 (B), phase ϕ(C), amplitude ratio η at ϕ= 0 (D).

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used to coherently control the ET process in a DBA system.Alternatively, one can induce ET by an ionization with an off-resonant X-ray pulse. However, this may cause the system todecay through multiple channels, which complicates the process.Pulse delays, shapes, phases, and intensities can be tuned toenhance the ET. This coherent control tool for ET processes canbe used for processing of functional electronic materials (27),disease diagnostics (28), and X-ray sterilization (29).Infrared (IR) pulses have been used recently to excite selected

vibrational modes after triggering ET by UV pulses (11, 12). Inthe superexchange ET mechanism, vibrations can affect the in-terferences between various ET pathways, so that some ETpathways could be totally switched off (9). Similar experimentalobservations have been reported, but for the sequential ETmechanisms (12). IR pulses only weakly perturb the electronicstructure and the molecular geometry, which facilitates theirapplication for biomolecular ET coherent control.

The X-ray pulses used here can substantially alter the electronicstructure of the bridge, whereas vibrations generated by IR pulsesonly change phases of different ET pathways and therefore affecttheir interferences (9–12, 30, 31). UV-vis pulses may access thesame valence excited states prepared by an X-ray Raman process.However, UV-vis excitations lack the site selectivity of X-rays. Inaddition, short X-ray pulses can capture ultrafast ET dynamics thatgoes beyond the reach of UV pulses.Coherent control has been successfully used to manipulate the

relative strength of competing optical processes (32) but the majorgoal has always been the steering of chemical reactions in desireddirections by properly shaped and timed optical pulses (33–39). TheX-ray control schemes presented here offer opportunities for ma-nipulating and monitoring chemical reactions which involve anelectron transfer step. These could provide new synthetic routes formolecules and materials. X-ray pulses allow for a molecular levelcontrol with atomic selectivity and high temporal resolution whichis not possible by optical techniques (40). Our earlier studies haddemonstrated that stimulated X-ray Raman spectroscopy can beused to elucidate the catalytic reaction mechanism of the cyto-chrome P450 complex (41) by probing the oxidation state history ofreactants, products, and various intermediates. By combining nar-rowband and broadband pulses, the hybrid X-ray Raman technique(42) [previously denoted as attosecond stimulated X-ray Ramanspectroscopy (43)] offers a unique combination of spectral andtemporal resolution, making it possible to take snapshots of ultra-fast chemical reaction dynamics, which is not possible by conven-tional IR and optical techniques. Raman lineshapes are not affectedby the core lifetime and are thus much narrower than those of X-rayabsorption, which significantly enhances their resolution and se-lectivity. Going beyond the detection of reaction intermediates, withthe control schemes proposed here one can selectively enhance orsuppress ET steps, thereby facilitating or hindering selected chemicalreaction steps, and then probe the reacting system by additionalsequences of X-ray pulses (18). These offer a class of applications toX-ray free-electron laser and high harmonic light sources.Finally, we note that competing decay channels with the

resonant X-ray Raman of a core-hole state, such as Auger elec-tron emission, must be taken into account. Auger processes aredominant in the core-hole decay processes of light atoms, but areless important for heavy atoms. For Co, the atomic spontaneousK-edge X-ray fluorescence and Auger yields are 0.373 and 0.627,respectively (44). However, unlike the abovementioned sponta-neous processes, in stimulated X-ray Raman processes intensepulses can be used to enhance the Raman cross-section andsuppress the Auger decay channels. For example, it was observedthat the charge-transfer excitations in NiO were enhanced byresonant X-ray Raman scattering (45). In this study we focused onthe acceptor control scheme. However, we can also use the samestrategy to control the electron transfer via the donor, Ru atom.For Ru, the spontaneous K-edge Auger decay yield is only 0.206(44). Considering many heavy metal complexes used as photo-sensitizers in solar cell applications, it is safe to neglect the effectof Auger decay when applying our proposed X-ray electrontransfer control scheme to these systems. For example, in solid-state systems, the saturation of the stimulated emission on the Si(100) surfaces with X-ray free-electron laser was observed (46),which indicates the Auger decay channels have been suppressed.It was demonstrated that the stimulated X-ray Raman in COmolecules can compete well with the Auger processes (47, 48).

Materials and MethodsThe geometry of the Ru–Co model complex was optimized using the Beckethree-parameter and Lee–Yang–Parr hybrid (B3LYP) exchange-correlation en-ergy functional (49, 50). The Ru atom was decribed by the Stuttgart/Dresdenrelativistic pseudopotential and its corresponding basis set (51). The Co atomwas described by the improved default triple-zeta valence basis set with smallpolarization (def2-TZVP) (52), and the 6–31G* basis set (53) was used for other

A

B

C

Fig. 5. The DB and BB coherent control schemes. ET processes with X-raycontrol are represented by magenta dashed arrows. (A) The DB controlscheme which facilitates the D→B→A ET process (core excitation on thebridge). (B) The DB control scheme which hinders the D→B→A ET process(core excitation on the donor). (C ) The BB control scheme. The red solidlines represent the strong interactions between orbitals, and bluedashed lines represent weak interactions. Relevant molecular orbitalsare also labeled. Orbitals b1 and b2 can be on the same or differentbridges.

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light atoms. Solvation effect was considered by using the polarized continuummodel (54–56) with the solvent acetonitrile. Valence excited-state calculationswere done at the time-dependent density functional theory (TDDFT) level oftheory using the Coulomb-attenuating method version of the B3LYP functional(CAM-B3LYP) (57). All geometry optimization and valence excitation calcula-tions were done with the quantum chemistry program package Gaussian(58). To compare the transition dipoles of charge transfer and localizedcore excitation around the Co center, the Co 1s core excitation calcula-tions were done with the quantum chemistry program package NWChem(59) at the TDDFT/Tamm–Dancoff approximation (60) level of theory

using the exchange-correlation functional CAM-B3LYP. The def2-TZVP basisset was used for Co and N, and the 6–31G* basis set was used forother atoms.

ACKNOWLEDGMENTS. The support of the Chemical Sciences, Geosciences andBiosciences Division, Office of Basic Energy Sciences, Office of Science, US Depart-ment of Energy (DOE), Grant DE-FG02-04ER15571, is gratefully acknowledged. K.E.D.and Y.Z. were supported by the DOE grant. We also acknowledge the support ofthe National Science Foundation (Grant CHE- 1361516). K.E.D. is grateful forsupport from Science and Engineering Research Council at Singapore Agency forScience, Technology, and Research X-ray Photonics Program, Grant 1426500053.

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10006 | www.pnas.org/cgi/doi/10.1073/pnas.1610729113 Dorfman et al.

Supporting Information for “Coherent Control ofLong-range Photoinduced Electron Transfer byStimulated X-ray Raman Processes”Konstantin E. Dorfman ∗, Yu Zhang †, and Shaul Mukamel †

∗Singapore Institute of Manufacturing Technology, 71 Nanyang Dr 638075, Singapore, and †Dept. of Chemistry, University of California, 450 Rowland Hall, Irvine, California

92697, USA

Submitted to Proceedings of the National Academy of Sciences of the United States of America

The Effective HamiltonianThe system is described by the effective Hamiltonian

Hs =∑m,n

[t(e)mnc†mcn + t(h)mnv

†mvn −W (eh)

mn c†mv†nvncm], [S1]

where m (n) runs over the initially unoccupied (occupied) or-

bitals. t(e)mn (t

(h)mn) is the electron (hole) hoping rate, W

(eh)mn =

V(eh)(|rm − rn|) is the Coulomb attraction between the elec-tron and the hole. We neglect electron-electron, hole-holeCoulomb repulsion and the dipole-dipole interaction betweentwo excitons at different sites. One can now calculate the ma-trix elements of the Hamiltonian in Eq. S1 in the three-statebasis (|D′⟩,|B′⟩, |A′⟩) using

⟨0|cmvnHsc†kv

†l |0⟩ = δnlt

(e)mk + δmkt

(h)nl − δmkδnlW

(eh)mn . [S2]

We assume that the direct electron hoping between donor

and acceptor t(e)ba and t

(e)ab is weak compared to the donor-

bridge hoping t(e)db , t

(e)bd . This is justfied for remote orbitals

with a large energy difference - few eV. We further neglectdirect hoping of electrons between donor and acceptor such

that t(e)da = t

(e)ad = 0. The Hamiltonian (Eq. S1) (ℏ = 1) can

thus be recast as Hs = HDBA+Hhop where the diagonal partis given by

HDBA = ωD|D⟩⟨D|+ ωB |B⟩⟨B|+ ωA|A⟩⟨A|, [S3]

and

Hhop = tDB |D⟩⟨B|+ tBA|B⟩⟨A|+H.C. [S4]

is the electron hopping Hamiltonian. Here ωD = t(e)d,d + t

(h)dd −

W(eh)d,d , ωA = t

(e)a,a + t

(h)dd − W

(eh)a,d , ωB = t

(e)b,b + t

(h)dd − W

(eh)b,d

are electronic energies, and tDB = t(e)d,b, tBA = t

(e)b,a are the

hopping rates between many-electron states.The X-ray parameters are based on the adiabatic (delocal-

ized) basis. By diagonalizing the 3×3 Hamiltonian we obtainthe adiabatic exciton Hamiltonian

Hs = ω′D|D′⟩⟨D′|+ ω′

B |B′⟩⟨B′|+ ω′A|A′⟩⟨A′|, [S5]

where the exciton basis set is obtained via unitary transfor-mation U(t): |{D′, B′, A′}⟩ = U(t)|{D,B,A}⟩.

Electron Transfer DynamicsElectron transfer dynamics in diabatic basis is described by amaster equation (1) where the Liouville space components ofthe ET rate matrix are

KDD,DD = kDB ,KDD,BB = −kBD, [S6]

KBB,DD = −kDB ,KBB,BB = kBD + kBA,KBB,AA = −kAB ,[S7]

KAA,BB = −kBA,KAA,AA = kAB , [S8]

and the rest of the components are zero. The correspondingGreen’s function in Liouville space is given by

Gmn,pq(t) = − i

ℏθ(t)

[e−Lt

]mn,pq

. [S9]

Eq. (S9) can be recast as

Gmm,nn(t) = − i

ℏθ(t)

∑p

χRmp(Wpp)

−1e−λptχLpn. [S10]

Here, χL,R denote the left (right) eigenvectors of the rate ma-trix K which satisfy the following equations∑

n

KmmnnχRnp = λpχ

Rmp, [S11]

and ∑m

χLpmKmmnn = λpχ

Lpn. [S12]

W = χL ·χR and λp are the identical eigenvalues for both setsof eigenvectors.

Interaction of the Molecule with X-ray PulseThe dipole matter-field coupling is formulated in the adia-batic basis. In the interaction picture and the rotating waveapproximation (RWA) (1) it reads

H ′x(t) = V′†(t) ·E(t− T ) +V′(t) ·E†(t− T ), [S13]

where

E(t− T ) =

∫ ∞

−∞

dω

2π

∑j=1,2

ejEj(ω)e−i(ω+ω0

j )(t−T ), [S14]

and

Ej(ω) = (2π)1/2σje−σ2

jω2/2−iϕj [S15]

Reserved for Publication Footnotes

www.pnas.org/cgi/doi/xxxx/pnas.xxxxxxxx PNAS Issue Date Volume Issue Number 1–S3

is the Gaussian envelope of the component j = 1, 2 withpulse bandwidth σj and phase ϕj . Pulse E(t − T ) hastwo components tuned such that E1 nearly resonant withx − A′ transition and E2 - with x − B′. These are thepositive frequency components of the field centered at de-lay T , polarization unit vector ej , center frequency ω0

j , j =

1, 2. V′(t) =∑

n′=A′,B′ µn′xe−[iωn′x+Γx]t|n′⟩⟨x| represents

a lowering operator that promotes an electron from coreorbital x to valence orbital b′, a′. The corresponding ex-pression in diabatic basis reads V(t) = U(t)V′(t)U−1(t) =∑

n=A,B,D µnxe−[iωnx+Γx]t|n⟩⟨x|, which couples all the three

diabatic states A,B,D to the core state x. The correspondinggeneralized polarizability tensor in diabatic basis that origi-nates from X-ray interaction with molecule is given by Eq.(4).

Electron Transfer Control with X-ray Resonant Raman

ProcessThe lowest order correction to acceptor population is linear inX-ray intensity. The corresponding diagram is shown in Fig.S1 (see Ref. (2) for diagram rules). One can read the diagramand obtain

∆PA(t1, t2) = 2Riℏ∫ t1+t2

−∞ds1

∫ s1

−∞ds2

∑i,j=1,2

∑m,n,q=A,B,D

× E∗i (s1 − t1)Ej(s2 − t1)(ei · µxq)(ej · µxm)

× GAA,qn(t1 + t2 − s1)Gxn,xn(s1 − s2)Gmn,DD(s2),[S16]

where Gxn,xn = −i/ℏθ(t)e−(iωxn+Γx)t, n = A,B,D is a Liou-ville space greens function that governs core to valence elec-tron dynamics. Under assumption of well separated pulsesone can evaluate the time integrals in Eq. (S16) and obtainEq. (3) in the main text. Here i = 1 if q = A, i = 2 if q = B,j = 1 if m = A and j = 2 if m = B.

Parameters for ET ControlIn the limit of well separated pulses Eq. 3 in the main textthen contains dominant elastic component where m = n =q = A

∆PAel(t1, t2) = 2RGAA,AA(t2)|µAx|2σ21I1GAA,DD(t1),

[S17]

and inelastic correction q = n = A, m = B

∆PAin(t1, t2) = 2RGAA,AA(t2)GBA,DD(t1)

× µAxµBxσ1σ2I1/21 I

1/22 e−iϕ. [S18]

The X-ray effect is maximal at t2 = 0 and in the lowest orderapproximation in the weak B/A coupling we obtain

∆PAin(t1, 0)

∆PAel(t1, 0)≃ σ2

2I1/22

σ1I1/21

|µBx||µAx|

KDB

KBA≥ 1. [S19]

The inequality (5) in the main text therefore provides an im-portant restriction on the pulse and matter parameters suit-able for ET coherent control where the inelastic contributionoriginated from coherences play an important role. Under as-sumption of for equal parameters of the both components ofthe field I1 = I2, σ1 = σ2 Eq. (S19) becomes Eq. (5).

ET Parameter for the Ru-Co Model ComplexFrom the TDDFT excited state calculation of the Ru-Comodel complex, the first valence excited state with significantRu-to-tpphz character (D→B transition) is the No. 2 excitedstate with an excitation energy of 2.8013 eV. The significantMOs involved in the transition are shown in Fig. S2. Toreduce the computational cost, in the B→A transition calcu-lations, we simply use the [(−tpphz)CoIII(CN)4]

2− molecularfragment cut from the Ru-Co complex. For this fragment, thefirst valence excited state with significant tpphz-to-Co char-acter (B→A transition) is the No. 9 excited state with anexcitation energy of 2.6070 eV. Thus we choose the followingET parameters: the donor energy 0 eV, the bridge energyEB = 2.8 eV, and the acceptor energy EA = 2.8 + 2.6 = 5.4eV. The electronic couplings for the D→B and B→A ET pro-cess are estimated by using the generalized Mulliken-Hushscheme (3–6):

tDB,BA =µ12∆E12√

(µ1 − µ2)2 + 4µ12

, [S20]

where µ12 is the transition dipole between the initial and finalstates of the ET process; ∆E12 is the energy difference of theinitial and final states; and µ1,2 are the dipole moments of theinitial and final state, respectively. Since the ET direction isalong the x axis (see Fig. 1(a) in the main text), we consideronly the x-component of the dipole in electronic coupling esti-mation. For tDB , TDDFT calculations show µ12 = 1.6137 a.u., ∆E12 = 2.7826 eV, µ1 = 27.0603 Debye and µ2 = 8.3054Debye, so tDB = 0.557512 eV; for tBA, TDDFT calculationsshow µ12 = 0.0362 a. u., ∆E12 = 2.6070 eV, µ1 = −30.1568Debye and µ2 = −60.2611 Debye, so tBA = 0.0079674 eV.Thus we choose tDB = 0.6 eV and tDB = 0.008 eV. Theelectron transfer times kDB = 20 fs and kBA = 2 ps wereestimated from the experimental data in Ref. (7). Co 1s coreexcitation calculations on the simple Co complex fragmentshow that a typical Co→CN ligand core transition has a tran-

sition dipole µ(x)

A′x = 0, µ(y)

A′x = −0.00003, µ(z)

A′x = −0.00086a. u., and a typical Co→tpphz bridge core transition has a

transition dipole µ(x)

B′x = 0.00033, µ(y)

B′y = 0, µ(z)

B′z = 0 a. u.,

thus we used these numbers in our model simulation. ParticleMOs involved in these transitions are shown in Fig. S3.

1. Louisell WH (1973) Quantum Statistical Properties of Radiation (Wiley, New York).

2. Mukamel S, Rahav S (2010) in Advances In Atomic, Molecular, and Optical Physics,

eds Arimondo E, Berman P, Lin C (Academic Press) Vol. 59, pp 223–263.

3. Cave RJ, Newton MD (1996) Generalization of the Mulliken-Hush Treatment for the

Calculation of Electron Transfer Matrix Elements. Chem. Phys. Lett. 249:15–19.

4. Mulliken RS (1952) Molecular Compounds and their Spectra. II. J. Am. Chem. Soc.

74:811–824.

5. Hush NS (1967) in Progress in Inorganic Chemistry, ed Cotton FA (Interscience Pub-

lishers) Vol. 8, pp 391–444.

6. Hush N (1968) Homogeneous and Heterogeneous Optical and Thermal Electron Trans-

fer. Electrochim. Acta 13:1005–1023.

7. Canton SE, et al. (2015) Visualizing the Non-equilibrium Dynamics of Photoinduced In-

tramolecular Electron Transfer with Femtosecond X-ray Pulses. Nat. Commun. 6:6359.

2 www.pnas.org/cgi/doi/xxxx/pnas.xxxxxxxx Footline Author

Fig. S1. Ladder diagram corresponding to ∆PA(t1, t2) (Eq. (3) in the main text). Blue

arrows represent interactions with X-ray pulse. Red arrows at t = 0 represent photo-induced

ET preparation. Red arrows at t correspond to the fluorescence detection of the acceptor

population. |x⟩ is the intermediate resonant core state that affects the polarizability α (Eq.

(4) in the main text). Diagram rules are presented in (2)

.

Footline Author PNAS Issue Date Volume Issue Number 3

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Fig. S2. Representative molecular orbitals of the D→B valence excitation of the Ru-Co

model complex. The corresponding CI coefficients are also shown. Color code: Ru (light

blue), Co (yellow), N (deep blue), C (gray), H (white). Results from TDDFT calculations with

Gaussian09.

4 www.pnas.org/cgi/doi/xxxx/pnas.xxxxxxxx Footline Author

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Fig. S3. Representative particle molecular orbitals of the B→A (a) and A→A local core

excitation (b) of the simplified Co fragment complex. The corresponding hole orbital is always

the Co 1s orbital (not shown). The corresponding CI coefficients are also shown. Color code:

Co (light brown), N (blue), C (gray), H (white). Results from restricted excitation window

TDDFT calculations with NWChem. Core excitation energies are not shifted.

Footline Author PNAS Issue Date Volume Issue Number 5