Photoinduced Charge Transfer from Titania to Surface Doping Site

Photoinduced Charge Transfer from Titania to Surface Doping SiteTalgat M. Inerbaev,† James D. Hoefelmeyer,‡ and Dmitri S. Kilin*,‡

†L. N. Gumilyov Eurasian National University, Astana, Munaitpasov st. 5, 010008, Kazakhstan‡Department of Chemistry, University of South Dakota, 414 East Clark Street, Vermillion, South Dakota 57069, United States

ABSTRACT: We evaluate a theoretical model in which Ru issubstituting for Ti at the (100) surface of anatase TiO2. Chargetransfer from the photoexcited TiO2 substrate to the catalytic sitetriggers the photocatalytic event (such as water oxidation orreduction half-reaction). We perform ab initio computationalmodeling of the charge transfer dynamics on the interface of theTiO2 nanorod and catalytic site. A slab of TiO2 represents afragment of the TiO2 nanorod in the anatase phase. Titanium toruthenium replacement is performed in a way to match thesymmetry of the TiO2 substrate. One molecular layer of adsorbed water is taken into consideration to mimic the experimentalconditions. It is found that these adsorbed water molecules saturate dangling surface bonds and affect the electronic properties ofsystems investigated. The modeling is performed by the reduced density matrix method in the basis of Kohn−Sham orbitals. Ananocatalyst modeled through replacement defect contributes energy levels near the bottom of the conduction band of the TiO2nanostructure. An exciton in the nanorod is dissipating due to interaction with lattice vibrations, treated through nonadiabaticcoupling. The electron relaxes to the conduction band edge and then to the Ru site with a faster rate than the hole relaxes to theRu site. These results are of importance for an optimal design of nanomaterials for photocatalytic water splitting and solar energyharvesting.

1. INTRODUCTION

Titanium dioxide (TiO2) has received considerable attentiondue to its numerous applications in many important fields, suchas catalysis, sensors, corrosion, coatings, and solar cells.1−7

Among these applications, energy harvesting directly fromsunlight is a desirable approach toward fulfilling the need forclean energy with minimal environmental impact. Using TiO2-based materialsfunctionalized or dopedsunlight energy canbe transformed and stored into two different forms: (i) electricenergy derived by light-induced charge separation with dye-sensitized solar cells2,6,8 or (ii) chemical energy as fuel andoxidizer, e.g., molecular hydrogen and oxygen which areproducts of the water splitting reaction.1,4,9 Light-inducedcharge separation is essential for both of these applications.We focus on solar fuel catalysis that depends on at least three

key steps. The first is the light captureabsorption of thesunlight by the active material, resulting in an electron−holepair. The second is the electron transfersunlight-producedelectron and hole diffuse to the catalytic sites, separated inspace.10 This process is accompanied by electron and holerelaxation toward the conduction band (CB) minimum andvalence band (VB) maximum, respectively.11−13 During thisstep electronic excitation energy is partially lost to thermalenergy of lattice vibrations via electron−phonon interactions.The last step is catalysisthe efficient making and breaking ofthe chemical bonds using the harvested electron−hole pairs.Among the various oxide photocatalysts, TiO2 (titania) still

remains the most promising photocatalyst owing to its low cost,chemical inertness, low toxicity, and high photostability.4 Thechemical and physical properties of materials used for solar-

energy conversion strongly depend on their size, shape, andcrystalline structure.6 The variation of surface morphologyallows selection of the optimal electronic structure.14 Theorigin of such a difference was recently revealed by anexperiment where it was shown that the exciton lifetime in theanatase phase is much larger than the corresponding value forthe exciton in the rutile phase.15,16 This facilitates thetranslocation of the photon-excited electrons and holes fromthe bulk to the surface, where the photochemical reactions takeplace.17 For small TiO2 nanoparticles (<50 nm), anataseseemed more stable and transformed to rutile at temperature>973 K.18,19 Thus, at room temperature small enough titaniananocrystallites are expected to have anatase structure and bemuch more effective for sunlight-harvesting applications.Recently, there were proposed new approaches to synthesize

high-purity anatase TiO2 single crystals with a large percentageof (001) facets.20,21 These facets are much more reactive forwater-dissociation reactions than the (101) surface, whichnormally forms the majority of the surface area. However, theclean anatase (001) surface reconstructs under ultrahighvacuum conditions.22 The reconstruction strongly stabilizesthe surface23 and at the same time largely reduces its reactivity:calculations indicate that the fraction of active sites for waterdissociation is reduced ∼25% with respect to the unrecon-structed surface.24

Received: November 8, 2012Revised: February 9, 2013Published: April 9, 2013

Article

pubs.acs.org/JPCC

© 2013 American Chemical Society 9673 dx.doi.org/10.1021/jp311076w | J. Phys. Chem. C 2013, 117, 9673−9692

pubs.acs.org/JPCC

The optical band gap of titania in the anatase phase is 3.2 eV,much larger than the activation energy of 1.23 eV required forwater splitting.25,26 Therefore, use of the bare titania for solarenergy harvesting is not efficient. Band gap excitation requiresultraviolet irradiation; however, UV light accounts for only 4%of the solar spectrum compared to 45% visible. So, any shift inoptical response to the visible range will have a profoundpositive effect on photocatalytic efficiencies of the TiO2materials. Various compounds ranging from anions27,28 tocations, including a wide range of transition metals29−31 such asFe,32,33 Pt,34 Co,35 Nb,36 and others,37−39 were used for thispurpose. A remarkable catalytic effect of RuO2 on powderedTiO2 for hydrogen evolution from an ethanol−water mixturehas been observed by Sakata et al.40 To investigate thesupporting effect of RuO2 on TiO2, experiments varying theamount of RuO2 on a fixed amount of TiO2 were carried out.40

The hydrogen evolution rate for RuO2/TiO2 was found to be30 times higher than that for TiO2 alone and sensitive toincreasing amounts of RuO2, reaching a maximum at 0.4 × 10−3

molar ratio of RuO2/TiO2. Ruthenia nanoparticles here act aselectron pools (reducing sites),41 while titania is an oxidizingelectrode; i.e., hydrogen and oxygen evolution takes place onruthenia nanoparticles and titania, respectively.As it was found experimentally, at monolayer coverage of

water on the rutile TiO2 (110) surface, the ratio OH:H2O ≈0.5, while heating yields an increased OH:H2O ratio.42

Theoretical analysis also reveals that molecular and dissociatedwater can coexist on rutile (110)43 and anatase (100) and(001) surfaces.44,45

The computational modeling of the hot carrier relaxation inmolecules and nanostructures rests on partitioning the totalenergy between the electronic part and nuclear part andallowing for energy flow between them, going beyond theBorn−Oppenheimer approximation.46 Several successful com-putational strategies for electronic relaxation are based on theconcept of surface hopping between potential energysurfaces.47,48 Numerical implementations of this strategy canbe based on various methods ranging from density functionaltheory (DFT)49 to high-precision nonadiabatic excited statemolecular dynamics approach.50 It has been proved efficient touse molecular dynamics trajectory for determining the electron-to-lattice coupling in semiconductors.51 A computationalprocedure integrating relaxation of electrons as an open systemwith time-dependent density functional theory (TDDFT)treatment of excited states was recently offered.52 Historically,in the limit of long time dynamics, low couplings, and multipleelectronic states, one considers multilevel Redfield theory as acommon approach for electronic relaxation.53−62 There wererecent approaches to combine Redfield theory of electronrelaxation with on-the-fly coupling of electrons-to-latticethrough a molecular dynamics trajectory in the basis ofDFT.63 Energy and charge transfer processes may occur in theultrafast and/or coherent regime at times shorter thandephasing time. Such regimes are expected, for example, insystems with sparse energy states. In contrast, systems withdense energy states such as semiconductor surfaces, canmanifest coherent behavior only at ultrashort time scales dueto faster dephasing. However, in present studies of chargetransfer for photocatalysis, one primarily focuses on mecha-nisms of population relaxation.Recently we have used computational models to explore

charge transfer states introduced to wet anatase (100) and

(001) surfaces upon transition metal doping64−66 and protonreduction reaction on the surface of the Pt nanocatalyst.67

In the present paper we investigate relaxation of electronicexcitations on the titania anatase (100) crystallographic surfacewith Ru(IV) sites as a nanocatalyst that dopes electrons intotitania. The details of electron−hole relaxation in aqueousconditions are discussed. These results are used to predict thetime-resolved spectra and frequency-dependent responsefunctions collected from ultrafast nonlinear time-resolvedexperiments. They are important for optimal design ofphotocatalytic water splitting and solar energy harvesting. Wefound that optically enabled excitations form an electron−holepair in which the electron relaxes to the CB edge and then tothe Ru site, while the hole relaxes to the Ru site at a slowerrate,25 explaining the significant increase of water decom-position effectiveness due to the ruthenia doping. At the sametime ruthenia doping leads to an increase of the lightadsorption effectiveness by the anatase phase of titania in thevisible range.

2. METHODS: COMPUTATION DETAILSWhile studying photocatalysis in energy materials, one canrepresent the overall catalytic process as a sequence ofelementary steps responsible for the total effect. Specifically,interaction of electrons with light corresponds to elementaryprocesses of photon absorption and emission. Interaction ofelectrons with lattice vibrations corresponds to relaxation ofelectronic band populations and reorganization of ionicconfiguration. The mixing of electronic orbitals is associatedwith elementary processes of surface charge transfer and theappearance of photovoltage. The common feature of all theseelementary processes is that electronic degrees of freedom(DOF) cannot be considered as isolated.To model such processes in photocatalytically active energy

materials we employ a combination of two approaches. First isthe computational modeling of electronic structure for thesystem investigated. The density functional theory (DFT)seems to be the best suited for a class of systems with reducedsymmetry, where standard analytical methods of solid statetheory approaches developed for periodic structures are notwell applicable. Among such systems are inhomogeneoussystems, surfaces with defects, and interfaces between differentsubstances where chemical reactions take place. Our secondapproach is a modeling of quantum relaxation dynamics. Herewe simulate interaction of nonequilibrium electronic states withlight and lattice vibrations. This method has a developmentpotential as it goes beyond a basic quantum description ofisolated systems. A combination of both DFT electronicstructure calculations with further time evolution analysis of theelectronic subsystem is employed in the present investigation.

2.1. Computational Modeling of Nonadiabatic Tran-sitions. Due to the fundamental role of radiationless processesin chemical reactivity, there is abundant literature devoted toextracting nonadiabatic coupling from computer simulations.Theoretical approaches utilized for this purpose fall into twogeneral categories: trajectory-based methods and techniquesbased on direct propagation of the reduced density operator.The latter set of methods includes the mean-field

Ehrenfest,68 surface-hopping,47,69 semiclassical initial valuerepresentation,70 linearization,71,72 and classical mapping73,74

approaches. The former techniques are based on evolution ofthe reduced density matrix (RDM) and consider Redfieldequations75 and the noninteracting blip approximation,76 which

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp311076w | J. Phys. Chem. C 2013, 117, 9673−96929674

serves as a time-dependent generalization of Forster theory.77

In the present paper the reduced density operator formalism isemployed for description of the nonadiabatic charge dynamicsin bare and ruthenia doped anatase (001) titania of the anatasephase for practical design material for sunlight harvesting.2.1.1. Reduced Density Operator (RDO) Theory. The basic

idea in microscopic theory of nonadiabatic charge dynamics is adecomposition of a total system into a system and bath parts. Inthe particular case of the present research, the former onecorresponds to electronic, while the latter one is an ionic part.Complete Hamiltonian of a problem under consideration iswritten as a sum of electronic Hamiltonian Hel, an ionicHamiltonian Hion, and their interaction Hint

= + + H H H HR r R r R R r( , ) ( , ) ( ) ( , )el ion int(1)

where R and r are ionic and electronic coordinates. Thedynamics of the electronic subsystem is described by theRDO,78 which is obtained by tracing the density operator of thewhole system over the ionic bath DOF. Dependence of Hel onthe ionic DOF emphasizes the nonadiabatic character of chargedynamics in the investigated system.Hereafter we focus our attention on charge dynamics

initiated by incident light beam. Let us consider the formationof an excited state at the surface of interest continuouslyirradiated by light of frequency Ω, consistent with anexperimental setup. Light promotes electrons up in energyfrom the VB to the CB, while relaxation processes lead to de-excitation back toward the ground electronic state. Theseprocesses are described within a reduced density operator ρ =∑ij|i⟩ρij⟨j| and RDM ρij treatment within Born and Markovapproximations.78 Born approximation corresponds to the weakelectron−ion coupling, so that the back-action of the systemonto the bath can be neglected. Within the Markovapproximation it is assumed that system correlation time isshort.Either way, one can pose a formal problem of time-

dependent evolution in terms of density operator ρ which, innumerical representations, is composed of N2 elements,compared to N elements for wave function. The densityoperator follows the von-Neumann equation of motion(EOM). The density operator is a powerful theoretical toolcombining quantum mechanics and thermodynamics. It allowsto describe an open system interacting with the environment:solvent, substrate, lattice vibrations, or quantized radiation.55−61

The density operator provides a quantum picture of energydissipation, photoemission, and dephasing. We focus on EOMfor electronic DOF and take into account the influence ofthermalized ionic motion in an approximate fashion, similar tothe Redfield approach.53 The density operator for electronicand ionic DOF ρtot is projected on electronic DOF byintegrating out the ionic DOF ρel = Trion(ρ

tot). Here we assumethat the total density operator can be factorized on electronicρel and ionic ρion components and that the ionic densityoperator obeys thermal equilibrium distribution. In thisapproach the EOM for the reduced density operator forelectronic DOF gains an additional term representing theapproximate infuence of lattice vibrations on the electonicDOF. This term, often referred to as the Redfield term, isobtained79 as second-order perturbation with respect to theelectron−phonon interaction Hamiltonian as described below.There one applies several approximations: perturbative in

electron-to-lattice ion interaction, Markov, convolutionless

Equation 2 provides a time increment of the electronic densityoperator ρel and linearly depends on its value at specific timeρel(t). The first term of the equation is trivial, contains phaseaccumulation due to the electronic Hamiltonian, plus aninfluence of a driving laser field, if any, and corresponds to theelastic, reversible dynamics.In contrast, the second term is very specific. It uses

Hamiltonian of interaction of electrons and lattice vibrationsHint. This Hamiltonian is small compared to Hel. The secondterm appears as a perturbative correction of the second ordersince the first-order correction vanishes upon average. Thesecond term has prefactor of ℏ−2. The dimension of the secondterm is the same as the dimension of the first term: time−1. Theintegration limit runs from the instant the interaction wasswitched on until the time we focus on. The symbol Trioncorresponds to an average over states of the ionic subsystem, inthermal state, which is characterized by the ion density operatorρion. The double commutator appears from the second-orderperturbation. The Hamiltonian of interaction Hint enters theretwice: at the initial instant of time t = 0 and at the delayedinstant of time t = τ. This corresponds to two elementaryevents of interaction, separated by the time interval τ. Duringthe interval between these events, there appears anaccumulation of phase, corresponding to noninteracting ionsand electrons. The density operator of the electronic systementers there as an argument, at time t. The interactionHamiltonian is often factorized as a bilinear product of relevantoperators affecting the electronic subsystem and ionic latticesubsystem. The second-order perturbation term, upon averageover lattice motion, contains an autocorrelation function forlattice operators.In case a specific basis {|i⟩} is chosen, the density operator

converts into density matrix ρij = ⟨i|ρ|j⟩ for electonicDOF.53,55−61 RDM obeys Redfield EOM

Elements of the Redfield tensor R are expressed in terms ofpartial rates Γ±

∑ ∑δ δ= Γ + Γ − Γ − Γ+ − + −Rijkl ljik ljik jlm

immk ikm

jmml(4)

Here, the partial rates Γ± are expressed in terms ofautocorrelation functions M of the interaction Hamiltonian

∫ τ τ ω τΓ = −+ M id ( )exp{ }ljik

t

ljik ik0 (5)

∫ τ τ ω τΓ = − −− M id ( )exp{ }ljik

t

ljik lj0 (6)

The phase increment factors exp{−iωljτ} appear from the timeevolution of the electronic DOF and use energy difference ℏωij= εj − εi. The autocorrelation function of the interactionHamiltonian reads

τ τ τ= ′ =M M H H( ) ( , 0) Tr { ( ) (0)}ljik ljik lj ikionint int

(7)

with an assumption M′(τ,0) = M′(τ + τ′,τ′). Here we define



τ τ τ= ⟨ | − | ⟩H l iH H iH j( ) exp{ } (0)exp{ }ljint ion int ion

(8)

Note that approximations which we have committed lead to thetime-independent coefficients Rijkl for this system of lineardifferential equations. Later, for the practical implementation ofthe Redfield eq 3, we replace the general Hamiltonian with theFock operator or one-electron Kohn−Sham operator, inagreement with used computational basis. For the focus onenergy relaxation and charge transfer, it is sufficient to includeonly single excitations of electronic states in the explicittreatment of Hel. In what follows the interaction Hamiltonian isexclusively based upon nonadiabatic coupling elementsactivated by motion of ions.In numerical form, based on the derivative of the

Hamiltonian (Fock operator), on the basis |k⟩, |l⟩ with respectto normal mode displacements or to the position of the Ith ionRI = {RI

(x), RI(y), RI

(z)}, the interaction Hamiltonian is composedof the terms including products of gradient matrix elementsVkl(I,α) = MI

−1⟨k|((∂)/(∂RI(α)))|l⟩Δt, α = x, y, z, and momentum

as follows Hint = ∑k,l,I,α|k⟩⟨l|Vkl(I,α)PI

(α)(t). Here, for each timestep, the difference between the αth Cartesian coordinate of theIth atom at time moments of t and t + Δt can be expressed interm of momentum ΔRI

(α) = ((PI(α)(t))/(MI))Δt, where MI is

mass of the Ith atom. The autocorrelator 7 is composed of theterms including products of non-adiabatic coupling matrixelements Vkl

(I,α) and momentum−momentum correlationfunction

∑τ τ= ⟨ + ⟩α β

α β α β−M N V V P t P t( ) { } ( ) ( )klmnt I J

klI

mnJ

I J1

, , , ,

( , ) ( , ) ( ) ( )

(9)

In numerical form, based on “on-the-fly” calculation, theautocorrelator 7 is composed of the terms including products ofmatrix elements of the interaction Hamiltonian at differentinstants of time

∑τ τ= +−M N H t H t( ) { ( ) ( )}klmnt

kl mn1 int int

(10)

= ⟨ | ∂∂

| ⟩H t m tt

n t( ) ( ) ( )mnint

plus permutations. Here, the average over both states Trion isrepresented by the summation of snapshots of multipletrajectories or one long trajectory. Note that in the last twoequations the autocorrelation function includes only theinteraction Hamiltonian and is expected to decay rapidly insystems with broad and dense spectra of lattice vibrations. Anautocorrelation of the total Hamiltonian include components,which might decay at longer time scale and contribute to sucheffects as pure dephasing.80−82

2.2. Practical Application of Methodology to Func-tionalized Titania Slabs. The equations below representimplementation of the Redfield equation on the basis of Kohn−Sham orbitals including dipole interaction with optical field offrequency Ω under rotating wave approximation as explained inAppendices I−III.83,84 Below, a tilde accent denotes matrixelement in rotating frame.

ρ ρ ε ε ρ ρ = Ω > =( ) ( ) ( ) ( ) ( )t t i t t texp , ,ij ij i j ii ii

∑ρ ρ ρ γ ρ ρ∂∂

= Ω − − − t

i2

( ) ( )jjk

jk kj jk jj jj jjeq

(11)

ρ ρ ρ ρ γ ρ ρ∂∂

= − Δ − Ω − − −

≠t

i i

j k

( ) ( ),jk jk jk jk kk jj jk jk jkeq

(12)

where γjj = ∑k≠jκjk and γjk = ∑l(κjl + κlk)/2 + γjk0 are population

and coherence relaxation rates and κjk = 1/ℏ2|Vjk|2J(ωjk)|

f BE(ωjk,T)| is a state-to-state transition rate in the Markoff limit.Here Ωjk = −Djk·E0/ℏ and Δjk = Ω − ℏωjk stand for Rabi anddetuning frequencies, respectively. Also, ℏωjk = εj − εk, J, f BE =(exp{(ℏω)/(kBT) − 1})−1, and T are described in refs 83, 85,and 86 and stand for an energy difference, spectral density ofbosons, their thermal distribution for vibrational frequencies ω,and temperature, respectively, while γjk

0 stands for a puredephasing component, obtained through an autocorrelation oftime-dependent KSO energy ⟨εi(t)εj(τ)⟩.

80,82 The initial valuesfor the RDM correspond to the system initially at thermalequilibrium and then excited by light, during a very long time.The thermal equilibrium state in our electron system isspecified by the Fermi−Dirac distribution ρjk

eq = δjk f FD(εj;T),

εε ε

=−

+−

⎪ ⎪

⎪ ⎪⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎧⎨⎩

⎫⎬⎭

⎞⎠⎟⎟f T

k T, exp

( )1j

jFD

Fermi

B

1

as applies to our system at equilibrium at temperature T beforeexcitation.

2.2.1. Solution in the Limit (i): Steady State. As soon as wehave established formal EOM 11 for the electronic DOFperturbed by light and lattice vibrations, we have to solve it. Weconsider the solution of eq 11 in two practically importantlimits, first in the limit of continuous photoexcitation at a givenfrequency. This corresponds to experiment with monochro-matic CW excitation. In this limit, at a long time after theexcitation was switched on, the excitation of the model by lightand deexcitation due to interaction with lattice vibrations, inaverage, compensate each other.In such a case, the average of any element of density matrix

experiences no change in time. The model comes into so-calledsteady state ρss. In the steady state, the differential EOM fordensity matrix 11 transforms into a linear algebraic equationwhich establishes a connection between the equilibrium densitymatrix ρeq and the steady state density matrix ρss.A system driven by steady light of a given frequency shows

two competing tendencies that establish a steady state: lightpromotes electronic population from the valence band to theCB, and interaction with the medium returns population backto the valence band. Furthermore, interaction with the mediumover long times damps rapidly oscillating coherences. In such acase, as long as the intensity of irradiation is kept constant, it ispossible to search for solutions of the EOM with steady statevalues ρij

ss = limt→∞ρij(t) for times t ≫ γij−1, which can be

obtained imposing the condition ∂ρij/∂t = 0. In the limit ofweak optical field intensity for which |gjk| ≪ γjj, a Taylor seriesexpansion of matrices leads to

∑ρ ρ γ ρ ρ = + − − g ( )jj jj jjk

jk kk jjss eq 1 eq eq

(13)

ρ γ γ ρ ρ = Ω + Δ − ≠−i k j( ) ( ),kj jk jk jk jk kk jjss 2 2 1 ss ss

(14)

with gjk(Ω) = γjkΩjk2 [γjk

2 + Δjk(Ω)2]−1 and details given in refs 84and 87. Interestingly, ρjk

ss of the off-diagonal elements, for k andj in different bands, corresponds to transition density matrix.



2.2.2. Solution in the Limit (ii): Evolution of Non-equilibrium Excitation. Another limit, when eq 11 is importantto solve, is the limit of time evolution of the nonequilibriumexcited state created by an instantaneous ultrafast pulse or byabrupt switching off of the steady excitation. In this limit, EOMkeeps only terms for phase increment and for lattice-inducedevolution and does not include interaction with light. TheEOM is solved by numerical diagonalization of the matrixequation87,88

ρ ρ + = Ω ( ) JJ

J( ) ( )(15)

where elastic and inelastic terms in the Redfield equation are

transformed into and supermatrices, providing theelectron density matrix ρ(t) = ∑jAJρ

(J) exp{−iΩjt} at eachtime. Here, ρ(j) stands for an eigenmatrix of the RDM EOM.The coefficient AJ = Tr[ρ(0)†ρ(J)] is obtained from the initialconditions, and J = (i,j) stands for a diadic superindex.63,88

2.2.3. Applying RDO Solutions to Explore Properties ofMaterials. As soon as one has solved the EOM for the densitymatrix one may want to connect this knowledge aboutelectronic system details to the real world, to observableswhich reflect steady and time-dependent system properties. Weselect such observables which are either easy to measure for thisclass of systems or reflect the most important features of themodels (see Appendix IV).For models of functionalized surfaces for energy applications,

one is interested in the ability of the material to absorb lightand to convert the relevant energy into the form of chargetransfer and, later, into chemical form. One can quantify theability of material to participate in the charge transfer event inan average way. A surface of a material can be approximatelyconsidered as a double layer capacitor with one layer Arepresenting the inner area of material well below the surfaceand the second layer B representing the outer area of thematerial, facing the vacuum or other type of environment. Aphotoexcitation may induce average charge imbalance betweensuch layers. This creates a homogeneous electric field across thecapacitor plane and a voltage between layers of the double-layercapacitor. The corresponding electric potential differencebetween charged layers is proportional to their average chargeand to the distance between oppositely charged layers and canbe cast for a continuous charge distribution as in ref 89 anddescribed in Appendix IV.Other quantities we follow are very helpful for understanding

the mechanism of energy relaxation and charge transfer in themodels of titania surfaces, energy distribution of photoexcitedcarriers as a function of energy and time, and space distributionof photoexcited carriers as a function of position and time. Weconsider the average energy of a carrier as a function of time toquantify average features of energy relaxation. We quantify theamount of charge transfer by considering the averagerepartitioning of charge between the titania slab and rutheniananocatalyst.A population distribution in the energy domain reads n(ε,t)

= ∑iρii(t)δ(εi − ε), while a relative change of population withrespect to the equilibrium distribution reads

ε ε εΔ = −n t n t n( , ) ( , ) ( )eq(16)

and represents an increase of population (electrons, Δn > 0), adecrease (holes, Δn < 0), or unchanged population for Δn = 0.The ruthenia nanocatalyst and Ru doping of titania surfaces

have the potential to promote half reaction of water splitting.

Therefore, one focuses on the details of the hole transfer andon behavior of carriers in the occupied valence band.The charge density distribution in the VB/CB as a function

of z reads, with r = (x,y,z)

∫∑ ρ ϕ ϕ= *≤

P z t t x y r r( , ) ( ) d d ( ) ( )a b

i jij

a bi jVB

( , )

( , ) HO

( , )

(17)

∫∑ ρ ϕ ϕ= *≥

P z t t x y r r( , ) ( ) d d ( ) ( )a b

i jij

a bi jCB

( , )

( , ) LU

( , )

(18)

for the initial nonequilibrium (a, b) photoexcitation andcharacterizes the electron density localized at an adsorbate attime t. Below we solve for the RDM and obtain the abovedistributions to perform a case study of photoinduced evolutionin our models of titania slabs, with and without Ru adsorbate.The expectation value of the hot carrier (hole) energy Δεh is

given by

∫ε ε ε ε ε⟨Δ ⟩ = − Δ −ε

−∞t n t( ) d ( , )( )h Fermi

Fermi

(19)

The expectation value of the hot carrier (electron) energy Δεeis given by

∫ε ε ε ε ε⟨Δ ⟩ = + Δ −ε

∞t n t( ) d ( , )( )e Fermi

Fermi (20)

To compute the average rate of energy dissipation, weconvert expectation values ⟨Δεe⟩(t) and ⟨Δεh⟩(t) into theirdimensionless counterparts

ε εε ε

⟨ ⟩ =⟨Δ ⟩ − ⟨Δ ⟩ ∞⟨Δ ⟩ − ⟨Δ ⟩ ∞

E tt

( )( ) ( )

(0) ( )e,he,h e,h

e,h e,h (21)

Both of these functions change between 0 and 1. We try to fitthe dissipation of energy in the CB and VB into amonoexponential decay ⟨Ee⟩(t) = exp{−ket} and ⟨Eh⟩(t) =exp{−kht}. Here the decay rate is approximated as inverse ofthe time integral

∫= ⟨ ⟩∞

−k E t t{ ( )d }e,h0

e,h1

(22)

Average partitioning of the charge can be quantified asfollows

∫=t zP z tCT ( ) d ( , )Z

a bVB

0VB( , )max

(23)

so that the charge presence above the titania surface z > 0 yieldsincrement proportional to the amount of charge transfer.

2.3. Electronic Structure Calculations. A set of Kohn−Sham (KS) orbitals is computed using density functional theory(DFT) with the Perdew−Burke−Ernzerhof (PBE) para-metrization90 as is implemented in VASP code.91,92 Anobtained set of KS orbitals is used to calculate matrix elementsof the relevant density operators.92−94 The Ti 3d states intitania were explicitly treated as valence states through theprojector augmented wave method95 and cutoff energy valueequal to 400 eV. The slab replicated along the (100)crystallographic surface contains eight layers of thicknesscontaining totally 96 atoms and was cleaved from the anatasephase structure of titania downloaded from the NRLdatabase.96 The z axis of expected charge transfer is chosento be orthogonal to the exposed (100) surface.



The periodic boundary conditions a = 7.570 Å and b = 9.908Å are imposed along the x ⟨010⟩ and y ⟨001⟩ directions. Avacuum layer of 8 Å was added to the simulation cell along thez ⟨100⟩ direction to prevent spurious interactions betweenperiodic replicas. The above-described model of the titaniasurface was modified to approximately represent surfacenanocatalyst. To do so, one Ti atom at top and bottomsurfaces covered by water was replaced by a Ru atom. Theserepresent “doped” and “undoped” models, which are studied asfollows.Here we restrict ourselves by consideration of water

monolayer adsorption on investigated surfaces. Recenttheoretical results indicate that the presence of a second layerof water molecules causes those in the first water molecule layerto be reoriented, and a monolayer model was found at bestlimited applicability to understand the water−TiO2 interface.

97

The resulting structure was saturated by H2O molecules tomimic a realistic situation of the titania nanorod immersed inwater. The resulting geometry is shown in Figure 1(left). Theeffect of explicit thermal motion of the surface layer of watermolecules is included in ab initio molecular dynamicstrajectories involved in the computation of correlationfunctions, Redfield tensor elements, and relaxation rates. Thiscan be best visualized by refs 98 and 99.The energies of KS orbitals {εi(t)}, transition dipoles {Dij},

spatial distributions of KS orbitals partial charge density ρi(r),normal modes, ab initio molecular dynamics trajectories{RI(t)}, average nonadiabatic transition coefficients Hij

int(t),and autocorrelation functions Mijkl(τ) were extracted bypostprocessing of the output of VASP DFT calculation foreach model.63 All coefficients for the Redfield equation arecomputed based on the ab initio data.

3. RESULTS AND DISCUSSION3.1. Effect of Doping on Ground State Electronic

Properties. Geometry Modification and Binding Energies.As the Ti−O bond length is shorter than Ru−O, replacementof Ti(IV) with Ru(IV) leads to slight distortion of the lattice.The distortion affects the in-surface bonds differently than M−O bonds perpendicular to the surface as shown in Table 1.

Calculated by eq 24, binding energy of Ru to the titania surfaceis also shown in Table 1.

μ μ= − + −E E Edopingbing

doped clean host atom doping atom (24)

where the energies of clean and doped surfaces are Eclean =E(Ti32O64 ·8H2O) = −973.5859 eV and Edoped =E(Ru2Ti30O64·8H2O) = −962.5596 eV. The chemical potentialsof Ru and Ti atoms were calculated for hexagonal Ru and α-Ticrystals, μdoping atom = E(Ru) = −8.568 eV/atom and μhost atom =E(Ti) = −7.658 eV/atom, respectively. Plugging these valuesinto eq 24 one obtains Edoping

bind = 6.424 eV per Ru atom.Upon optimization, water molecules are arranged to optimize

hydrogen bonding. H atoms of each H2O molecule protrudeparallel to the surface plane to the nearest O atom from the

Figure 1. Atomistic model (left) and schematic representation of electronic structure of the anatase (100) surface, undoped (center) and doped(right) by the Ru ion. Left: Atomic model of the doped surface. Atoms are color coded as yellow, blue, red, and white spheres standing for Ti, Ru, O,and H atoms. The figure shows only one-half of the structure. Periodic boundary conditions are implied along the x⟨010⟩ and y⟨001⟩ directions.Dangling surface bonds are compensated by water molecules. Center, right: Upon replacement doping, ten of the Ti 3d unfilled orbitals are removedfrom the conduction band (A) and replaced by ten Ru 4d orbitals (B)−(C), with splitting due to the nearly octahedral crystal field. Six of theseorbitals are unfilled (B, red dashes) and partitioned between the CB and bandgap region, while four of those orbitals are filled (C, solid red lines) andcontribute to the bandgap energy region. Basic electron−hole excitations in the undoped (1) and doped (1′)−(7′) models are symbolized by verticalarrows and are explained as follows. Notation is chosen in consistency with Table 2. (1) Transition in the undoped model is neutral, it oxidizes Oand reduces Ti, in a delocalized way. (2′) Transition in the doped model is neutral, it oxidizes O and reduces Ti, in delocalized manner. (7′) is aneutral transition and it promotes an electron from the filled to unfilled Ru 4d orbital; this excitation easily recombines into the ground state. (6′)Charged transition, it oxidizes O and reduces the Ru ion as follows: Ru(4+) + e− → Ru(3+); negative charge migrates to the surface. (1′) Chargedtransition, it oxidizes the Ru ion and reduces the Ti ion. The Ru reduction reaction follows as: Ru(4+) → Ru(5+) + e−; positive charge migrates to thesurface.

Table 1. Geometry Modification of the Titania Surface byRuthenium Ion Functionalization, Doping ReplacementEnergy, and Water-to-Surface Binding Energy

Me = Ti Me = Ru

dMe−O,∥ Å 1.977 ± 0.1 1.982 ± 0.1

dMe−O,⊥ Å 1.842 ± 0.005 1.844 ± 0.003

EMebind, eV 0 6.4237

EMe−H2Obind , eV −0.2612 −0.3365



neighboring H2O molecule. The average binding energy ofH2O molecule to the doped and undoped surface equals to

= − −− +EN

E E E1

( )NTi H Obind

adsslab ads slab2 ads (25)

where Nads stands for number of H2O ligands covering thesurface, Eslab+ads total energy of a model covered with water, Eslabtotal energy of a model with bare surface, and ENads total energyof all water adsorbants. Numerical values ETi−H2O

bind = −0.2612 eVand ERu−H2O

bind = −0.3365 eV are also summarized in Table 1.Density of States. The electronic configuration of titanium,

oxygen, and ruthenium are Ti = [Ar]3d24s2, O = [He]2s2p4,and Ru = [Kr]4d75s1. The electronic structure of doped andundoped models is schematically summarized in Figure 1 andsupported by computational results as follows. Symbolically, theK−S effective Hamiltonian operator of pure TiO2 can belabeled as F0

KS. Introduction of Ru doping created severalchanges of the electronic structure. The model Hamiltonian forthe doped slab includes a term Fdop

KS for the electronic changesdue to a dopant in the host lattice and a term Fdist

KS for theinduced distortion of the lattice, so that

= + + F F F FKS0KS

dopKS

distKS

(26)

Features of electronic structure and K−S eigen energies ofdoped and undoped slabs are analyzed using density of statesD(ε) = ∑jδ(εj − ε). For better visualization, the DOS for bothmodels is shifted in energy with respect to εFermi

undoped for theundoped model. Comparison of the DOSs D0 and D for pureand doped slabs helps to identify changes due to electroniccontributions from dopants and due to lattice distortion, with

− = Δ = Δ + ΔD D D D D0 dop dist (27)

Figure 2(a),(b) illustrates the effect of doping on theelectronic properties of the ruthenia-doped TiO2 slab andshows calculated DOS, total and projected on atomic orbitals.The DOSs D0(ε) versus ε shows a null region typical of the gapbetween VB and CB of the titania crystal, as seen in Figure2(a). On the basis of calculated projected DOS, one noticesthat for the undoped model all orbitals in the valence bands arecomposed as a superposition of O 2p orbitals of oxygen, and allorbitals in the CB are composed as superposition of Ti 3dorbitals of titanium. The density of states for the doped model,in Figure 2(b), approximately reproduces the DOS for theundoped model, in Figure 2(a), in the energy range below −1.5eV (VB) and above 0.5 eV (CB). Slight change of peak positionis attributed to electronic states being perturbed by latticedistortion Fdist

KS . The unique changes intoduced by doping FdopKS

are manifested as states within the bandgap.The equilibrium population of the undoped model is trivial:

all states in the VB (below 0 eV) are occupied, and all states inthe CB (above 0 eV) are unoccupied. In the doped model,pseudooctahedral Ru ion contributes t2g orbitals to thebandgap. Their filling is consistent with the d4 configurationof Ru(IV). This feature plays a role in understanding the chargetransfer mechanism. These results may potentially be used as amodel for more complex Ru-based sites such as clusters ornanoparticles. We expect that the Ru band in the bandgaprange will gain more states and will broaden for the largerruthenia nanocatalyst, providing similar qualitative trends in theelectronic structure.

Electronic absorption spectra of doped and undopedmodels are presented in Figure 2(c). Examples of electronictransitions contributing to the spectra are given in Table 2. Forboth models, bright transitions appear above the opticalbandgap of the undoped model. The absolute values of theoptical band gap energies can be systematically improved byemploying GW corrections and the Bethe−Salpeter equationapproach.26

The Ru-doped surface exhibits a significant red-shift inelectronic absorption spectrum relative to the undoped titania.Ru-doping and relevant lattice distortion effect perturb theposition of the main absorption peaks above 2.3 eV incomparison to the undoped model. Ru-doping also adds severalsemibright transitions below the optical bandgap. Thesetransitions are involved in the formation of charge separatedstates and are discussed in what follows. Individual transitionswith maximal oscillator strength are shown in Figure 2(c), forthe doped model in a form of vertical stems with heightsproportional to the oscillator strength. As expected, KS orbitalslocalized on adsorbed water do not contribute to the electronic

Figure 2. Basic electronic structure features for the doped andundoped TiO2 anatase (100) surface. Panels a, b show comparison oftotal density of states (lines) and partial density of states componentson O 2p (long dashes), Ti 3d (short dashes), and Ru 4d (dot-dashes)for undoped (a) and doped (b) models. Undoped surface has abandgap of over 2 eV. Doped surface shows states contributed bydoping in the bandgap area. Part of Ru-doping-related states arepopulated. Panel c shows absorption spectra of undoped (solid) anddoped (dashes) models. Electronic transitions contributing at most tothe doped model are represented by vertical stems with height equal totheir oscillator strength. Labels of individual excitations are consistentwith the scheme in Figure 1 and notations in Tables 2 and 3.



absorption spectrum. Interestingly, the two most intensetransitions in both doped and unroped models have the sameorientations of transition dipoles, along the ⟨100⟩ and ⟨010⟩directions.Ab initio molecular dynamics trajectories at ambient

temperatures were used to compute couplings and Redfieldtensor elements.98,99 Due to contact to the thermostat, all ionsand fragments included in the model experience quasichaoticmotion. Ti, Ru, and O ions of the slab perform motion ofoscillatory character, with different phases, while solvent H2Omolecules saturating surface dangling bonds perform a morecomplicated motion. These motions include rotation, forming,

and breaking hydrogen bonds and desorption and readsorptionback of water from/to the surface. This approach approximatelytakes solvent effects into account.One sees two ways to address dynamics solvent reorganiza-

tion at higher precision: (i) compute multidimensionalpotential energy surfaces by including vibrational degrees offreedom, all or most important ones, into explicit consid-eration100−102 and (ii) update the ionic trajectory according toelectron dynamics.103,104 Both approaches will demandsubstantial computational resources, compared to the presentapproach. Nonadiabatic molecular dynamics implies generationof the vibrational forces and trajectorirs for each of the possible

Table 2. Featured Electronic Transitionsa

Cartesian projections

orbital pairs main data ⟨010⟩ ⟨001⟩ ⟨100⟩

index description hole index electron index oscillator strength transition energy, eV Dx Dy Dz

Undoped (100) TiO2 anatase + H2O, HO = 2881 Bright, first HO-7 LU 3.4018 2.4003 −4.3875 0.1813 0.00252 Bright, second HO-8 LU+3 3.0160 2.7111 −0.0317 −0.0648 3.8899

Ru-doped (100) TiO2 anatase + H2O, HO′ = 2921′ Charged, Ru → slab HO′-2 LU′+4 0.8932 1.3504 0.0887 −0.7205 −2.91092′ Bright, first analog HO′-10 LU′+2 2.1672 2.5076 −3.4274 −0.0128 0.10783′ Bright, distortion-due HO′-7 LU′+3 2.7428 2.5915 3.7948 −0.0099 −0.00444′ Bright, third analog HO′-21 LU′+2 1.9881 2.8562 −0.0275 −0.3825 3.05355′ Bright, second analog HO′-11 LU′+4 1.4667 2.8725 0.0729 −0.0262 2.63476′ Charged, slab → Ru HO′-13 LU′+1 0.1820 2.1087 −1.0814 −0.0715 −0.01697′ Neutral, Ru → Ru HO′-2 LU′+1 0.1839 0.5293 0.1680 −0.3840 2.1339

aThe featured electronic transitions are presented for undoped and doped models. The criterion for selecting the states is the vicinity of thetransition energy to the bandgap, large value oscillator strength, and potential to contribute to the charge transfer. Indices of doped orbitals areindicated with a prime sign. Bright excitations are enlabeled in accordance with Figure 1(c). Cartesian projections are included to identify thetransitions of similar symmetry.

Figure 3. Representative examples of correlation functions Mljik, eq 7, (a), (c) and selected elements of Redfield tensor (b), (d) for studied atomicmodels without doping (a), (b) and with Ru doping (c), (d). The units for Redfield tensor elements are inverse femtoseconds. Here we focus onanalysis of those elements of the autocorrelation function and Redfield tensor, which correspond to population transfer. Interestingly, the maximalabsolute values appear for Rj,j,j±1,j±1. Note that the elements of Redfield tensor connecting states near the bandgap |j − HO| < 2, |k − HO| < 2 arequalitatively different for doped and undoped models. For the undoped model, the values of Rjjkk are vanishing. For the Ru-doped model, the valuesof Rjjkk are of the order of 0.01 fs −1. The Ru ion contributes states which facilitate the relaxation near the bandgap.



electronic states or even for each possible sequence ofelectronic states taken by the system. This approach worksefficiently for lighter elements and smaller systems, such asdendrimers.104 It is an interesting idea to try such an approachfor the class of models considered here. Specifically, some basisKS orbitals have a portion of charge localized on water.Energies of such orbitals may get modified in a treatment withupdated trajectories.Nonradiative Transitions. Figure 3 shows representative

examples of correlation function components Mljik(t), eq 7, andselected elements of Redfield tensor, eq 10, for studied atomicmodels with and without Ru doping. Here we focus on analysisof those elements of the autocorrelation function and Redfieldtensor, which correspond to population transfer. It is veryinteresting and important that the autocorrelation functionelement Mljik(t) decays rapidly as a function of time. Thisnumerical observation provides validation for applying Markoffapproximation. The computed Redfield tensor elementsresponsible for population transfer exhibit a feature: specifically,the maximal absolute values appear for those elementsdescribing relaxation between states with close energiesRj,j,j±1,j±1. Note that the elements of Redfield tensor connectingstates near the bandgap |j − HO| < 2, |k − HO| < 2 arequalitatively different for doped and undoped models. For theundoped model, the values of Rjjkk are vanishing andcorrespond to long nonradiative lifetime. For the Ru-dopedmodel, the values of Rjjkk are of the order of 0.01 fs −1. Ru ioncontributes states which facilitate the relaxation near thebandgap and may contribute to nonradiative recombination.The computed atomistic data are used for parametrizing EOMfor RDM, solved in the following sections.3.2. Surface Photovoltage Created by cw Photo-

excitation. We analyze the interaction of electromagneticradiation with doped and undoped titania surfaces. We startanalysis by considering continuous wave (cw), low intensityirradiation leading to a steady state excitation of titania surfacemodels. This analysis is helpful for interpreting absorptionspectra in Figure 2(c) and photovoltage spectra in Figure 4(a).Optical perturbation of a given frequency creates coherencebetween occupied and unoccupied orbitals, referred to astransition density matrix. At the same time, optical excitationperturbs the equilibrium populations of states in valence andconduction bands. In average, originally occupied orbitals ofvalence band yield decrement in population, and originallyunoccupied orbitals of conduction band yield increment ofpopulation.105 This corresponds to optical excitation promotingan electron from the VB to the CB. Steady state solutions of theEOM for the density matrix for titania surface models provide atransition density matrix ρjj

ss(Ω) and steady state populations oforbitals ρjj

ss(Ω) as a function of incident light frequency. Theabsolute value of the relative change of such populations withrespect to their equilibrium values Δρjjss(Ω) = |ρjj

ss(Ω) − ρjjeq| is

shown in Figure 4(b),(c) for undoped and doped models. Anonequilibrium excited state created by light of a givenfrequency Ω can be considered as a superposition of multipleelectron−hole pairs and perturbs both VB and CB orbitals.Light of lower frequency perturbs only few orbitals in thebandgap area HO − Δ < i < HO + Δ, Δ < 5. Light of higherfrequency perturbs orbitals farther away from the bandgap. Athigher frequency, more orbitals become involved into theformation of a nonequilibrium excited state, Δ > 5.For the undoped model, the excitation with frequency

(transition energy) below the bandgap does not perturb

populations of orbitals. Starting with frequency matching, theoptical bandgap (in this calculation 2.1438 eV) light perturbspairs of orbitals with nonzero transition dipoles, as shown inFigure 2(b).For the doped model, light of much lower frequency is able

to perturb some orbitals since there are Ru 4d t2g orbitalscontributed by surface Ru-doping. The actual bangap of thedoped model (Egap

doped = 0.3213 eV in this calculation) is smallerthan the optical bangap of titania anatase. Several band gap areatransitions have small values of transition dipole; therefore,noticeable perturbations of KS orbital populations arecalculated at the transition energies of about 1 eV, as shownin Figure 4(c). The transitions in the range of excitation energybetween 1 and 2 eV involve orbitals localized on Ru(IV) and

Figure 4. (a) Calculated surface photovoltage and correspondingrelevant population changes of Kohn−Sham orbitals b,c. Panel a showscalculated surface photovoltage as a function of photon excitationenergy for undoped (solid) and doped surfaces (dashes). Similarfeatures of the spectra are indicated by symbols a, b for undoped anda′, b′ for doped models. The feature c′ for the doped model is uniqueand is appearing due to the presence of doping. The blue squaresymbols correspond to experimental data.107 The features, similar tothe calculated ones, are labeled as a″ and b″. (b) Absolute value ofphotoinduced change of population of KS orbitals of an undopedtitania anatase (100) surface. Orbitals belonging to the CB or VB aredrawn as blue and red bars, respectively. (c) The same for the dopedtitania anatase (100) surface.



contribute to the formation of O 2p → Ru 4d and Ru 4d → Ti3d charge separated states as discussed later in this section.Earlier calculations of SPV for crystalline, amorphous, and

doped Si (c-Si)84,87,105,106 provide the correct sign of thephotovoltage and qualitatively correct positions of peaks in thedependence on photon energy in comparison to experimentaldata. This provides a partial justification of the method and abackground to expect that applying the same approach todoped TiO2 surfaces provides reasonable insight into the detailsof formation of an integrated charge transfer state on thesurface. The solution for the steady state density matrix ρjj

ss(Ω)is plugged into the equation for surface photovoltage Vs(Ω) −Vs(0) = −e[ε0εrA]−1∑ijρij

ss(Ω)⟨i|z|j⟩, with details provided in eq51, Appendix IV.The resulting Vs(Ω) dependences for both doped and

undoped models are presented in Figure 4(a) along theexperimental results of SPV for porous anatase film of 16 nm ina vacuum.107 This is the best relevant data available in theliterature. Surface photovoltage spectra of doped (calculatedand measured) and undoped models exhibit similar features, atthe approximately same frequency, sign, and intensity. Thesefeatures are labeled as (a), (b) for the undoped model and as(a′), (b′) in the doped model, and as (a″), (b″) in the undopedexperimental data. In addition to these features, the SPVspectrum for a doped model exhibits a high intensity negativepeak at about 1.5 eV, labeled as (c′). This peak establishesevidence of formation of a charge transfer state on the dopedtitania surface. A negative sign of photovoltage indicates thedirection of charge transfer: upon excitation at transition energyof ℏΩ = 1.5 eV the surface is getting depleted with electronsand enriched with holes. Note that this information isfrequency dependent, not time dependent. The detailedmechanism and nature of this event is analyzed later in thissection. SPV calculation has proved the h+ transfer from titaniaactive media to the ruthenia nanocatalyst.3.3. Electronic Dynamics after Instantaneous Photo-

excitation. Here we consider the solution of the RDM EOMin the time domain. The instantaneous optical pulse creates anonequilibrium state of the model by promoting an electronfrom occupied orbitals in the valence bands (a) to unoccupiedorbitals in the conduction band (b). This event is mostprobable in the case the oscillator strength fab is maximal andtransition energy εab = ωab/ℏ is in resonance with the centralfrequency of the exciting laser pulse Ω.In what follows, we assume that the system is optically

excited at t < 0 by steady light of frequency Ω ≈ (εb − εa)/ℏ,that promotes an electron from occupied orbital a tounoccupied orbital b, and that at t ≥ 0 its electric field iszero. The initial value of the density matrix is therefore ρij(0) =δij·νj, with orbital occupation numbers νj = 1 for occupied j andzero otherwise, except that νa = 0 and νb = 1. Its change overtime is shown as ρij

(a,b)(t).This initial state, specified by indices a,b, is evolving in time

due to the interaction of electronic DOF with lattice vibrations.Figure 5 shows representative examples of photoexciteddynamics in models with and without surface doping for thefollowing initial conditions a = HO-8, b = LU+3, created bypulse of central frequency ℏΩ = 2.7111 eV, and a = HO′-21, b= LU′+3, created by pulse of central frequency ℏΩ = 2.8562eV. We analyze the photoexcited dynamics of the non-equilibrium electron distribution as a function of time. Figures5(b),(e) show isocontours of the population Δn(a,b)(ε,t), eq 16,giving the dynamics of similar photoexcitations calculated for

the titania surface with and without a surface-adsorbed Runanocatalyst, with red indicating large gain and blue large lossof electron density with respect to equilibrium distribution.Over time, the maxima are found to relax to a lower excitationenergy. It corresponds to the electronic part of excitation (red)approaching the CB minimum and the hole part of excitation(blue) approaching the VB maximum. As the systems relax, thepopulations of some orbitals change by large amounts, goingthrough maxima before they decrease.In the undoped model, Figure 5(b), The relaxation of holes

takes place at earlier time (10−100 fs) than the electronrelaxation (1−10 ps). Both the electron and hole relax to thelowest available excitation for the undoped model. Suchexcitation, HO → LU, undergoes a longer recombinationprocess, kHO,LU = 2.85 × 10−2 ps−1, τHO,LU ≈ 35 ps. This isquicker than electron−hole recombination time in water-dissolved titania quantum dots that is reported as equal to 500ps at 12 K108 and varies with nanoparticle size from 66 to 405ps at room temperature.109 The rate of recombination isdetermined by several key parameters, which should match inthe system and in the model: (a) temperature, (b) nuclearfluctuations, (c) value of bandgap, (d) surface-to-volume ratio,(e) details of surface functionalization. The influence of eachparameter is discussed in the Discussion section. In the dopedmodel, Figure 5(e), the relaxation of electrons occurs earlier intime at about .5−1 ps. The relaxation of holes experiencesseveral subsequent steps. The hole part of excitationsubsequently visits several orbitals from the VB. In general,these steps of hole relaxation take longer than in the undopedmodel since these relaxation events involve new sparsely spacedorbitals contributed by Ru-doping. Interestingly, the electron

Figure 5. (a, d) Atomic models of thin semiconductor films with andwithout metal adsorbate. (a) Ti32O64·8H2O; (d) Ti30Ru2O64·8H2O. Allmodels contain eight monatomic layers, with 12 atoms each. Thevertical axis z is perpendicular to the surface and is positive above thesurface. Periodic boundary conditions are implied along x and ydirections. Dangling surface bonds are compensated by watermolecules. (b, e) Isocontours of the population Δn(a,b)(ε,t) givingthe dynamics of similar photoexcitations calculated for the anataseTiO2 surface with and without Ru-doping. Here, red, green, and bluecolored areas label the distribution Δn(a,b)(ε,t) in eq 16 for gain, no-change, and loss, respectively, in comparison with the equilibriumdistribution; red areas can be understood as relating to electrons andblue ones to holes. (c, f) Spatial distributions of charge for VB asfunctions of time from eq 17. Colors from blue to red symbolizecharge density value from 0 to 1. The figure presents the timeevolution of hole density distribution in the normal direction to thesurface for a titania anatase surface with and without Ru-doping.



relaxes to the CBM LU′ earlier (0.5 ps) than the hole relaxes tothe VBM HO′ (2 ps). The strongest attention is focused on thetime interval .5 ps < t < 2 ps, when one carrier (e−) arrives tothe surface earlier than another one (h+). In such a regime, thesurface is being charged and is ready to facilitate reactions.Finally, both e− and h+ localize on the same surface. However,in a realistic setup with both oxidizing and reducingnanocatalysts, the e− carrier is expected to be swept off to thereducing catalyst providing an excess of h+ charges on thesurface with oxidizng RuO2 nanocatalyst.Next, we focus on the time evolution of the valence band

(VB) charge distribution in space since evolution of the holepart of excitation plays a key role in triggering an oxidizingreaction on the RuO2 nanocatalyst. Spatial distributions asfunctions of time from eq 17 are presented in Figure 5(c), (f).The dynamics of the hole part of excitation for the undoped

titania surface, Figure 5(c), shows subsequent maximaloccupation of orbitals HO-8, HO-1, and HO. This correspondsto spatial charge evolving from being equally distributedbetween all inner layers of the slab toward semidelocalizeddistribution over two atomic layers near the surface.The dynamics of the hole part of excitation for the doped

titania surface, Figure 5(f), shows that maximal occupation oforbitals HO′-21, HO′-11, HO′-3, and HO′ corresponds tospatial charge evolving from being equally distributed betweenO 2p of all inner layers of the slab toward localization on the Ru4d surface doping. This illustrated the dynamics towardformation of the charge separated state. Possibly, this processcan be summarized by approximate redox reactions involving Oand Ru

+ →− − −eO O2 (28)

→ ++ + + −eRu Rux x( 1) (29)

It is noted that these equations are symbolic simplifications. Inreality, the electron is getting transferred between orbitals,which are delocalized over multiple oxygen ions. Interestingly,titania thin film surface charge transfer is reflected in thephotovoltage observable which slowly decays in time as excitedstates recombine to the equilibrium.110,111 The informationabout time evolution of a photoexcited carrier for a givenphotoexcitation can be, in average, consolidated into a singlefunction, such as evolution of an average energy of the carrier,eq 19. Figure 6 illustrates the dependence of such averageenergy of hole as a function of time, for different initialphotoexcitations, for doped and undoped models. Table 3summarizes the numerical measure of carrier energy dissipationfor various initial excitations as computed according to eq 22.3.4. Analysis of Active Orbitals. The models are oriented

in such way that x, y, and z coordinates correspond to b =⟨010⟩, c = ⟨001⟩, and a = ⟨100⟩ crystallographic axes,respectively. For the undoped model, the two brightesttransitions have transition dipoles oriented along ⟨001⟩ and⟨010⟩ axes. The electronic configuration of titanium andruthenium are Ti = [Ar]3d2 4s2 and Ru = [Kr]4d7 5s1. In oursimulations, the orbitals near the bandgap are contributed byvalence electrons only. While doping, we replace two Ti atoms,from the top and bottom surfaces of the slab by Ru atoms,gaining four additional valence electrons per each dopant. HereHO′ = HO + 4, HO = 288, and HO′ = 292 correspond to twicethe number of valence electrons in the model. Our calculationis not spin polarized. The dopant atoms contribute fouradditional orbitals within the bandgap. Three of them are

occupied, and one of them is unoccupied. This analysisprovides an argument that the index of VB orbitals of a givensymmetry is shifted down by 3 in the doped model comparedto the undoped one. By analogy, the index of unpopulated CBorbitals is shifted up by +1 in the doped model compared to theundoped one. This expectation is partially fulfilled for twolowest and brightest transitions, and their indices are as follows:pair of orbitals (1) = (HO-7; LU)undoped converts into a pair oforbitals (2′) = (HO′-10; LU′+2)doped. By analogy, a pair oforbitals (2) = (HO- 8; LU+3)undoped converts into a pair of

Figure 6. Dependence of hole relaxation dynamics on the initialexcitation. Representatives on individual excitations are selectedaccording to electron−hole transitions with maximal oscillatorstrength. Only the hole dynamics is represented here. The expectationvalue of the hole energy is calculated by integrating the energydistribution of charge below the Fermi energy weighted by the relevantenergy value, eq (19). Panel a shows average hole energy dynamics forthe undoped titania anatase (100) surface, for different initialexcitations. Panel b averages hole energy dynamics for the Ru-dopedtitania anatase (100) surface, for different initial excitations. The indexof the VB orbital where initial nonequilibrium population is created isexplicitly labeled with the index of the orbital.

Table 3. Dynamics of Featured Electronic Transitions, forUndoped and Doped Modelsa

indexradiative

lifetime, τ, nshole relax. rate,

kh, ps−1

electron relax.rate ke, ps

−1energy relax.rate kx, ps

−1

Undoped (100) TiO2 anatase + H2O, HO = 2881 7.40 2.1394 NA 2.13942 6.54 2.1651 0.9138 3.0789

Ru-doped (100) TiO2 anatase + H2O, HO′ = 2921′ 89.07 1.3726 1.3145 2.68712′ 10.64 1.2185 2.0219 3.24043′ 7.87 1.9735 1.0590 3.03254′ 8.94 0.9708 2.0219 2.99275′ 11.98 0.9848 1.3145 2.29936′ 179.21 0.9793 48.8368 49.81617′ 2815.10 1.3726 48.8368 50.2094

aThis table sumarizes the rates of computed dynamics. The values for6′ and 7′ are high since these relaxation channels involve nearlydegenerate orbitals.



orbitals (5′) = (HO′-11; LU′+4)doped. The difference, if any, isdue to perturbation imposed on all orbitals due to the presenceof the dopant. These four examples of pairs of orbitals areillustrated in Figures 7 and 8 and can be symbolicallyrepresented by redox reactions

→ +− − −eO O2 (30)

+ →+ − +eTi Ti4 3 (31)

There is another sequence of transitions involving orbitalscontributed by doping. Such orbitals are spatially localized inthe vicinity of the doping atom. The host, TiO2 orbitals aretypically delocalized over the whole slab. We considertransitions from host to dopant, from dopant to host, and Rud−d transitions. The representative examples of suchtransitions are mentioned in the records in the rows 1', 6',

and 7' of Table 2. The mean position of electron orbitaldistribution ⟨i|r|i⟩ is localized in the surface region for dopingorbitals and localized in the inner area of the slab for hostorbitals. This fact is responsible for the specific character ofpartial contribution of a given pair of orbitals to the formationof charge transfer state and to photovoltage signal at a giventransition frequency.For example, the transition (1′) = (HO′-2 → LU′+4)

corresponds to the electron being promoted from the surfacedopant to the Ti 3d inner area of the slab as summarized by theredox equation

→ ++ + + −eRu Rux x( 1) (32)

+ →+ − +eTi Ti4 3 (33)

Here we hypothesize x = 4, but this must be supported byadditional computational data. The electron is transferred from

Figure 7. Bright excitations (1) and (2) for the undoped TiO2 surface.Pairs of occupied and unoccupied KSOs contributing to low-energyoptical transitions, for the atomic model of anatase TiO2 with the(100) face exposed and covered by a monolayer of water. Red, green,and blue stand for Ti, O, and H atoms. Gray clouds symbolizeisosurfaces of KSO. Panels a,b,d,e show partial charge density of agiven KSO. Panels c,f show partial charge density integrated over thex,y direction, as a function of z; the ⟨100⟩ direction is orthogonal tothe plane of the figure. Lowest brightest transition HO-7 to LU attransition energy 2.40 eV with oscillator strength f = 3.40 isrepresented by panels a−c and corresponds to index (1) in Table 2and the scheme in Figure 1. Panel (a) shows 3D isosurface of HO-7orbital. Panel (b) shows 3D isosurface of LU orbital. Green dashes inpanel (c) represent 1D projection of HO-7 orbital on z = <100>direction. Solid red line in panel (c) represents 1D projection of LUorbital. The second brightest transition HO-8 to LU+3 at transitionenergy 2.71 eV with oscillator strength f = 3.01 is represented bypanels d−f and compared to index (2) in Table 2. Panels (d), (e)show 3D isosurfaces of HO-8 and LU+3 orbitals. Panel (f) representsprojections of orbitals on z-direction. Green dashes in panel (f) standfor HO-8 orbital. Solid red line in panel (f) stands for LU+3 orbital.Interestingly, all occupied orbitals are composed as a superposition ofp-orbitals of oxygen and clearly exhibit p-character. All unoccupiedorbitals are composed as superposition of d-orbitals of titanium. Thevision is difficult since the atoms of O and Ti are stacking in the ⟨001⟩view directions.

Figure 8. Bright excitations (2′) and (5′) for the doped TiO2 surface.Pairs of occupied and unoccupied KSOs contributing to low-energyoptical transitions, for the atomic model of anatase TiO2 with (100),exposed, doped by Ru atom, and covered by a monolayer of water.Red, green, and blue stand for Ti, O, and H atoms. The position of theRu surface doping atom is indicated by a dashed circle. Gray cloudssymbolize isosurfaces of KSO. Panels a,b,d,e show partial chargedensity of a given KSO. Panels c, f show partial chagre densityintegrated over the x,y direction, as a function of z. The ⟨001⟩direction is orthogonal to the plane of the figure. This figure showsbright transitions which have similar orbital symmetry with those ofthe undoped titania surface. Lowest brightest transition HO′-10 toLU′+2 at transition energy 2.50 eV with oscillator strength f = 2.16 isrepresented by panels a−c and labeled as (2′) in Table 2. Panels (a),(b) show 3D isosurfaces of HO'-10 and LU'+2 orbitals. Panel (c)represents projections of orbitals on z-direction. Green dashes in panel(c) stand for HO'-10 orbital. Solid red line in panel (c) stands forLU'+2 orbital. The second bright transition similar to the one in theundoped structure is HO′-11 to LU′+4 at transition energy 2.87 eVwith oscillator strength f = 1.46 and is represented by panels d−f andlabeled as (5′) in Table 2. Panels (d), (e) show 3D isosurfaces of HO'-11 and LU'+4 orbitals. Panel (f) represents projections of orbitals onz-direction. Green dashes in panel (f) stand for HO'-11 orbital. Solidred line in panel (f) stands for LU'+4 orbital.



the surface to inside of the slab. Exciting such a transitioncreates electron depletion on the surface. In other words, thiscan be interpreted as hole transfer to the surface. At the sametime, this transition has nonzero oscillator strength. Therefore,this elementary process of photoexciting this transitioncontributes to the photovoltage negative peak shown in featurec′ in Figure 2(a). The Kohn−Sham orbitals for this transitionare visualized in Figure 9(a)−(c).

Another featured transition, (6′) = HO′-13 → LU′ + 1,corresponds to transition from the O 2p occupied hostdelocalized orbital to an unoccupied Ru 4d localized dopingorbital which corresponds to an electron being promoted fromthe inner area of the slab to the surface doping as summarizedby the redox reaction

→ +− − −eO O2 (34)

+ →+ − − +eRu Rux x( 1) (35)

The electron is transferred from the inside of the slab to thesurface. Exciting such transition creates electron enriching onthe surface. In other words, this can be interpreted as an

electron transfer to the surface. Therefore, this elementaryprocess of photoexciting this transition could contribute to thephotovoltage positive peak. However, because this transitionhas low oscillator strength, the contribution is negligible andnot seen in Figure 2(a). The Kohn−Sham orbitals for thistransition are visualized in Figure 9(d)−(f).We offer a hypothesis that the negative sign contribution by

Ru doping in the photovoltage spectrum can be rationalized asfollows: As seen in the DOS, Figure 1(a), the number ofinterband occupied orbitals contributed by Ru doping is largerthan the number of interband unoccupied states contributed byRu doping. Therefore, the relative number of transitions withh+ transfer to the surface, similar to (1′) = (HO′-2 → LU′+4),overcomes the number of transitions with e− transfer to thesurface, similar to (6′) = (HO′-13 → LU′+1). By addingtogether the partial contributions from transitions of bothtypes, the h+-transfer transitions overweight and give overall h+

transfer and a negative peak in the SPV spectra.The last type of transition, from localized doping orbital to

another localized doping orbital, such as (7′) = (HO′-2 →LU′+1) does contribute neither to optical absorption nor tocharge transfer. Such transitions, however, are the lowestavailable transitions. Any excitation is expected to relax to them.Such excitations are expected to contribute to either Ru-basedluminescence, non-radiative recombination, or for triggering achemical reaction.

4. DISCUSSION

Quantitative description of relaxation in Redfield formalism canbe useful for description of a broad range of phenomena from(a) hot carrier relaxation to (b) multiple exciton generation and(c) reaction dynamics. Here we focus on (a) energy relaxationand charge transfer. Therefore, it is sufficient to include onlysingle excitations of electronic states in the explicit treatment.For implementing such formalism to (b) and (c), one needs toconsider a Hamiltonian in a bigger basis. For (b) one needs toexplicitly include double excitations etc. For (c) one needs totreat few nuclear DOFs explicitly.Pure TiO2 is a wide gap semiconductor. For such material,

Multiple Exciton Generation (MEG) would be available at highUV excitation energy. This is probably of limited practicalinterest. MEG seems to be possible if involving dopant states.Atomistic consideration of MEG on the basis of Kohn−Shamorbitals was reported for silicon quantum dots.112,113 Adescription of multiexciton generation in the density matrixformalism was reported on the basis of KP and effective masstheory.114 A combination of both atomistic and density matrixapproaches need to add a few modules to the code. Practically,a description of MEG would need two modules: (i) includemultiple excitations into basis and (ii) explicitly includeColoumb matrix element computation.The photoexcitation induces several pathways: cooling,

trapping, recombination, and others, taking place at differenttime scales from femtoseconds to nanoseconds.108,109,115,116 Incomparing experimental observations and computationalmodeling, there is an issue of careful assigning of lifetime ofcharge trapping versus charge recombination processes. Therate of recombination is determined by several key parameters,which should match in both the experimental system andatomistic model: (a) temperature, (b) character of nuclearfluctuations, (c) value of bandgap, (d) surface-to-volume ratio,(e) details of surface functionalization.

Figure 9. Charged excitations (1′) and (6′) for the doped TiO2surface. Abbreviations and symbols are defined the same way as in theprevious figure. Panels a,b,d,e show partial charge density of a givenKSO. Panels c, f show partial charge density integrated over x,ydirections, as function of z. This figure shows charged transitionswhich are associated with surface Ru-doping and do not have similarorbital symmetry with those of the undoped titania surface. Lowestcharged, semibright transition HO′-2→LU′+4 at transition energy εij =1.35 eV with oscillator strength f = 0.89 is represented by panels a−c,labeled as (1′) in Table 2. Panels (a), (b) show 3D isosurfaces of HO'-2 and LU'+4 orbitals. Panel (c) represents projections of orbitals on z-direction. Green dashes in panel (c) stand for HO'-2 orbital. Solid redline in panel (c) stands for LU'+4 orbital. Second charged, semibrighttransition is HO′-13 → LU′+1 at transition energy εij = 2.1087 and2.87 eV with oscillator strength f = 0.18 is represented by panels d−f,labeled as (6′) in Table 2. Panels (d), (e) show 3D isosurfaces of HO'-13 and LU'+1 orbitals. Panel (f) represents projections of orbitals onz-direction. Green dashes in panel (f) stand for HO'-13 orbital. Solidred line in panel (f) stands for LU'+1 orbital.



(a) Temperature increase is expected to speed up therecombination according to the Bose−Einstein distribution f BE,introduced after eq 10.(b) Fluctuations of ionic positions are the key feature

promoting electronic transitions. The amplitude of fluctuationsmay depend on the size of the model. Specifically, in largermodels, energy is expected to partition as 1/2 kT over eachnuclear degree of freedom. In smaller models, away fromstatistical regime, this partition may be violated for a limitedtime, leading to higher rates of electronic transitions.(c) The value of the bandgap and, generally, the energy

difference between orbitals experiencing transitions have acritical influence on the recombination and relaxation rates.This dependence can be partitioned on three factors: coupling,availability of phonon modes to accommodate energy differ-ence, and phonon occupation function. Coupling is expected todepend on the inverse of energy difference, contributing factor|εi − εj|

−2 into the overall rate. A bandgap of 2eV instead of 3eV will accumulate a factor roughly estimated as (1/(2/3)2) =2.25, providing a computed recombination rate 2.25 timesquicker. Correction of the computed bandgap can be achievedby applying the GW approach with BSE corrections. Hybridfunctionals, e.g. HSE06, might work as a “quick-fix” solution.Other factors, availability of phonon modes and Bose−Einsteindistribution function, contain more complicated dependenceson energy difference and also affect the recombination rates.(d) Larger surface-to-volume ratio provides a higher

concentration of surface recombination sites and, in average,facilitates quicker relaxation. This dependence can be exploredby manipulating the thickness of the model slab, leaving thesame surface area, and changing the volume of the model.(e) The present work delivers results for water molecules

passivating surface active sites. Other options of experimentalimportance are hydroxyls, carboxyls, amines, etc. Each of themmay contribute additional electronic states and vibrationalmodes, making a strong effect on surface charge transfer andelectron-hole recombination.

5. CONCLUSIONSWe have presented theoretical and computational treatment ofcharge transfer at the Ru-doped wet anatase titania surface. Theapplied theoretical method covers two major challenges at thesame time: (i) atomistic details of changes in electronicstructure introduced by defects, surfaces, and interfaces and (ii)dynamics of open systems: electrons interacting with latticevibrations and light. Used method combines density matrixformalism and ab initio electronic structure calculations usingan on-the-fly method to compute electron−lattice vibrationnonadiabatic couplings. Then, a time average of theautocorrelation function of the electron-to-lattice couplingsprovides coefficients of electronic transitions that enter into theequation of motion for the electronic degrees of freedom. Thisapproach allows for the description of relaxation phenomenaand allows calculation of dynamics of electronic chargeredistribution in the model. Optimum results are expected inthe following conditions: ions are considered as point charges,lattice vibrations instantaneously equilibrate with a thermostat,coupling autocorrelation function decays abruptly leading toMarkov approximation, vibrational reorganization is neglected,and excited state potential energy surfaces are assumed to havethe same shape as for the ground state.This method provides a way for atomistic computation of

time-resolved observables, with the only input being a set of

atomic positions for the model of reasonable practical size. Awide range of time intervals, from femto- to nanoseconds, couldbe explored without any additional computational cost. This isdue to the solution of equation of the motion bydiagonalization of the Liouville operator with constantcoefficients. Applicable systems include rigid solids, whoselattice vibrations are close to harmonic law: void of bondbreaking, bond formation, and low-frequency nonlinear motionof atoms such as spatially confined semiconductor nanostruc-tures at times longer than the vibrational period.Application of the method to wet titania surface dopants

reveals additional pathways for photoinduced surface chargetransfer. Dynamics of charge separation demonstrate thesubsequent arrival of opposite charges to the catalytic site.According to computed data, the direction of charge transfercan be controlled. In the time domain, the surface dopant firstgains negative charge that is later neutralized upon arrival of ahole to the same site. In the frequency domain, with CWexcitation, the computation shows formation of a stable statewith a net positive charge in the surface. We are lookingforward to verifying this conclusion by experiment or analternative computational procedure. A more complex systemwith an additional functional group(s) may serve for capturingan electron and having oxidizing and reducing sites separated inspace.Our results suggest specific surface redox reactions in the

system. The computed results support the expectations thatoxidizing and reducing nanocatalysts deposited on a semi-conductor substrate are expected to exhibit higher catalyticactivity upon charge transfer from the substrate to thenanocatalyst.117−121

Specifically, water splitting reducing and oxidizing half-reactions facilitated by charging of the surface should besimulated by ab initio molecular dynamics of charged models.Combining together the simulation of photon absorption,64−66

charge transfer from a substrate to reducing and oxidizingnanocatalysts,63,122−124 and a charge-facilitated surface reac-

Figure 10. Schematic representation of a photoelectrochemical cellconverting light into chemical form. Light is efficiently absorbed byactive media, such as a titania nanorod with absorption enhanced by aphotosensitizer. An exciton is broken into free carriers, electrons, andholes, each of which are swept off to surface nanocatalysts, adsorbedon opposite sides of the active media. Holes move to the oxidationnanocatalyst, e.g., ruthenia nanoparticle. Electrons move to thereducing nanocatalyst, such as a platinum nanoparticle. Chargetransfer to nanocatalysts triggers the photocatalytic half reactions,leading to producing a fuel compound, e.g., H2, on the reducingnanocatalyst and oxidizer compound, e.g., O2, on the oxidizingnanocatalyst. In the case that the whole nanocatalytic cell is immersedin water, the relevant half-reactions look as follows: 2H2O = 4H+ + 4e−

+ O2 and 2H+ + 2e− = H2. These half reactions are connected by twocharge transfer channels: electron transfer across the nanocatalytic celland proton transfer in the surrounding solvent.



tion67 complete the pathway of subsequent elementaryprocesses contributing to the photocatalytic water splitting(Figure 10).The theoretical and computational modeling help in

interpreting available experimental results and predicting anoutcome of a possible experiment and in suggesting a design ofa novel material with desired optical and electronic character-istics.

■ APPENDIX I. NUMERICAL IMPLEMENTATION OFRDO IN KOHN−SHAM BASIS

Diagonal elements of the density matrix give state populations,and off-diagonal elements of the density matrix correspond toquantum coherences. The evolution of electronic states of amodel system is conveniently described by the EOM in theSchrodinger picture of quantum mechanics81,125,126

∑ρ ρ ρ ρ = −ℏ

− +iF F d dt( ) ( / )jk

ljl lk jl lk jk rel

(36)

= − ·F F tD E( )KS

Equation 36 contains populations ρii and coherences ρij forKohn−Sham orbitals (KSOs) of a system with periodicboundary conditions, given by ϕj(r) = ∑|G|<Gcutoff

Cj,G exp-{−iG·r} and used as a basis set. Here j and k denote KSOs; thevectors G are the reciprocal lattice vectors; Gcutoff is the value ofthe reciprocal lattice vector for a truncated Fourier expansion;and Cj,G stands for a KSO in the momentum representa-tion.88,93,106 These KSOs are eigenfunctions of the Kohn−Sham effective Hamiltonian FKS with energies εj, and they willbe shown as |j⟩ = |ϕj⟩.The density matrix EOM (eq 36) contains a reversible

dynamics term from the operator FKS and the potential energycoupling of the dipole vector operator D to the external electricfield E and an irreversible dynamics relaxation rate term, bothof which are needed to describe transient phenomena. Thereversible term describes phase accumulation and interactionwith light. It is constructed from KSO energies, transitiondipoles, and the explicit form of the electric field. The relevantKSO energies εj satisfying Fjl

KS = εjδjl are imported from theelectronic structure calculations at fixed atomic positions. Thetransition dipole operator D = ∑ij|i⟩Dij⟨j| is obtained fromtransition dipole matrix elements calculated according to thef o r m u l a ⟨ i | D | j ⟩ = ℏ 2 e [ m e | ε i − ε j | ]

− 1

∑|G|,|G′|<GcutoffCi,G* GδG,G′Cj,G′.

88,106 The electric field for a photo-excitation created by steady irradiation with light of frequencyΩ and electric field strength is given by E(t) = E0(t)(exp{−iΩt}+ exp{iΩt}) with E0 = const for t ≥ π/(2Ω) and E0 = 0 for t <π/(2Ω), which is effectively zero in our time scales. Anexpression for the relaxation rate can be obtained in theabsence of light to second order in the interaction energybetween the system of electrons and lattice vibrations.

■ APPENDIX II. COMPUTATIONAL PROCEDURE FORINTERACTION HAMILTONIAN AND ITSAUTOCORRELATION FUNCTION

HamiltonianLet one consider the general Hamiltonian operator for anelectronic system with coordinates r which depend on positionsof ions R as a parameter

= H H r R( , ) (37)

If the electronic DOFs are considered as a quantum systemsurrounded by bath of lattice vibrations, the Hamiltonian canbe partitioned as

= + + H H H Hr R R r R r R( , ) ( ) ( , ) ( , )ion el int(38)

where Hint denotes here the electron−ion interaction.For each specific geometry configuration, such as equilibrium

relaxed geometry R = Req one can find the electroniceigenfunctions of the Hamiltonian |n⟩eq which are orthogonalto each other.

ε | ⟩ = | ⟩H n nn (39)

However, small perturbation of ionic positions R = Req + ΔRperturbs the Hamiltonian

= + ∂ ∂

·Δ + ΔH HH

OR

R R( )02

(40)

For this perturbed Hamiltonian, the equilibrium electronicstates |n⟩eq are not the eigenstates anymore. The displacementof ions breaks orthogonality of the chosen set of electronicstates. This nonorthogonality may lead to electronic transitions.This effect can be quantified by estimating the increment to thematrix element of the Hamiltonian

Here the expression H obeys the eigenstate property 39, andsince eigenvectors of the nonpertubed Hamiltonian H0 areorthonormal, eq 41 can be re-written as

ε δ ε= + ⟨ | ∂∂

| ⟩·Δ + ΔH m n OR

R R( )mn n mn n2

(42)

Here one can notice the equivalence of the two ways to accountfor lattice vibrations: through increment of the Hamiltonian ineq 40 and through increment of the electronic state in eq 42,both reflecting the deviation of ionic configuration fromequilibrium.The method of overlap of displaced orbitals can be efficiently

computationally implemented based on “on-the-fly” numericadiabatic ab initio molecular dynamics.49,63,104 There, in theframework of molecular dynamics description of lattice DOF,trajectories are represented by a set of subsequent atomicconfigurations and momenta. For each timestep, the differencebetween the αth Cartesian coordinate of the Ith atom at timemoments of t and t + Δt can be expressed in terms ofmomentum ΔRI

(α) = (∂RI(α)/∂t)Δt = ((PI

(α)(t))/(MI))Δt, whereMI is the mass of the Ith atom. For each atomic configuration,there is a corresponding set of eigenstates. In eq 41 the gradientterm times displacement ΔR is replaced by the time derivativeof the electronic wavefunction. For a discrete time increment,this all yields a discretized time-dependent nondiagonalHamiltonian, which characterizes interaction of electron-to-lattice vibrations, with matrix element

∑

= ⟨ | ∂∂

| ⟩

= ⟨ | ∂∂

| ⟩ Δ

=α

α

α

α α

H t m tt

n t

m tR

n tPM

t

V P

( ) ( ) ( )

( ) ( )

mn

I

I

mn I

int

( )

( )

( ) ( )(43)



The features of the electron-to-lattice coupling can beunderstood by analysis of the autocorrelation function of theinteraction operator

τ τ = ⟨ ⟩M H H( ) Tr { (0) ( ) }ionint int

(44)

In numerical form, based on the derivative of the Hamiltonian,the autocorrelator 44 is composed of the terms includingproducts of matrix elements Vkl

(α) = MI−1⟨k|((∂)/(∂RI

(α)))|l⟩Δtand momentum−momentum correlation function

∑τ τ= ⟨ + ⟩α β

α β α β−M N V V P t P t( ) { } ( ) ( )klmnt

kl mn I J1

, ,

( ) ( ) ( ) ( )

(45)

In numerical form, based on “on-the-fly” calculation, theautocorrelator 44 is composed of the terms including productsof matrix elements

∑τ τ= +−M N H t H t( ) { ( ) ( )}klmnt

kl mn1 int int

(46)

plus permutations. Here, the average over ionic bath states Trionis represented by the summation of snapshots of multipletrajectories or one long trajectory.

■ APPENDIX III. NUMERIC IMPLEMENTATION OFRDO EOM IN KSO DFT130

This Appendix provides details on numerical implementation ofan equation of motion for electronic density matrix in case thenon-adiabatic couplings are already found.83,126 Electronicrelaxation phenomena occurring in the long-time limit aredescribed in terms of Redfield tensor elements Rijkl, obtainedwith autocorrelation of non-adiabatic coupling matrix elements.Their change in presence of low-intensity light is assumedsmall. In what follows, density matrix is cast in the Schroedingerpicture (SP), interaction picture (IP) symbolized by superscript“I”, and in the rotating frame, symbolized by tilde accent. Therelaxation part of the EOM in Schroedinger picture reads

∑ρ ρ =t R t( ) ( ) ( )ijkl

ijkl klrel(47)

The EOM for the RDM is transformed to the interactionpicture (IP) of quantum mechanics to identify terms that areoscillating slowly over time. These terms survive a long-timeaverage of the EOM. In detail, a term in the EOM containingthe phase factor exp(iΔ·t), averaged over a long time τ aroundt, gives 2τ−1∫ t−τ

t+τdt′ exp(iΔ·t′) = exp(iΔ·t)sin(Δ·τ)/Δ·τ ≅ 0, forτ ≫ Δ−1. Therefore, the equations keep only terms whichoscillate slowly over time.In the IP, ρjk(t) = exp(−iωjkt)ρjk

(I)(t), with ωjk = (εj − εk)/ℏ,and similarly for other operators.

III.A Elimination of fast oscillatory terms for opticalprocessesWe consider first oscillatory terms describing light absorptionand emission. Light of frequency Ω activates only transitionsallowed by Djk ≠ 0, and after time averaging it involvestransitions with detuning Δjk = Ω − (εj − εk)/ℏ, smaller thandephasing rates γkj defined in what follows. Here one considersonly interband optical transitions so that i and j belong todifferent bands: CB, i ≥ LUMO and VB, j ≤ HOMO. Thedipole coupling operator terms in the IP are decomposed as

∑ ∑ ω

ω

ω

ω

· = · | ⟩ ⟨ | − Ω

+ | ⟩ * − ⟨ | Ω

+ | ⟩ * − ⟨ | − Ω

+ | ⟩ ⟨ | Ω

≥ ≤

t i i t j i t

j i t i i t

j i t i i t

i i t j i t

D E E D

D

D

D

( ) { exp( ) exp( )

exp( ) exp( )

( ) exp( )

exp( ) exp( )}

I

i LUMO j HOMOij ij

ij ij

ij ij

ij ij

( )0

The first two terms describe processes where a semiconductorsurface gets excited from the VB KSO |j⟩ to the CB KSO |i⟩ byabsorbing light or gets de-excited from the CB KSO |i⟩ to theVB KSO |j⟩ by emitting light. These terms involve slow time-dependent contributions oscillating with frequencies Ω − |ωij|.We keep these slow changing terms and omit the fast ones, asin the Rotating Wave Approximation (RWA)127,128 for thecoupling potential. The resulting EOM can be expressed interms of the rotating frame density matrix elements

ρ ρ ε ε = Ω >t t i t( ) ( )exp( ),ij ij i j (48)

and ρii(t) = ρii(t).

III.B. A procedure of eliminating fast oscillating terms alsoaffects relaxation terms in the EOM for density matrix.Turning to oscillating terms in the relaxation rate (ρij)rel within36. Relaxation term in the interaction picture reads

∑ρ ε ε ε ε ρ = ℏ − − + ∼ it R( ) exp{ / ( )}ijI

kli k j l ijkl kl

Irel

(49)

In order to keep only slowly oscillating terms one inspects howwell the following condition is satisfied: εi − εk − εj + εl = 0.Keeping only the matching terms one arrives at the secularapproximation.128 Specifically, this condition is fullfilled in twospecific cases (i) εi = εj, εk = εl providing relaxation (both gainand loss) rates Riijj for populations ρii, and (ii) εi = εk, εj = εlproviding dephasing rates Rijij for coherences ρij. The otherterms describe coherence transfer, or coupling betweenpopulations and coherences. They give smaller contributionsupon averaging over long times, and they are neglected in whatfollows. Nonadiabatic coupling matrix elements Vjk, mediatedby vibrational motion and charge polarization (bosonicexcitations) in the medium, are assumed small for thephenomena of present interest. This justifies an expansion ofthe solutions to the relaxation equation in powers of thestrength of the R-coefficients. The leading population loss rateis then given by γjj = Rjjjj, and the balance of loss minus gain canbe written as −γjj(ρjj − ρjj

eq), using detailed balance at thermalequilibrium for a collection of electrons. Similarly, therelaxation rate for the quantum decoherence of state pair (jk)is written as γjk(ρjk − ρjk

eq), with γjk = Rjkjk.Hence, combining elastic (reversible) and inelastic (irrever-

sible) dynamics terms and staying within the mentionedapproximations, the EOM reads83−87,126 as presented inequations (11) and (12).

■ APPENDIX IV. OBSERVABLESThe electronic excitations induced in a slab by a lightadsorption lead to creation of a photoinduced polarizationwith a time-dependent component.129 The polarizationgenerates an electric field, and if the slab is brought intocontact with the electric circuit it leads to electrical voltagegeneration. Measured photovoltage is related to the timeaverage of surface electric dipoles, and these in turn could be



theoretically derived from a time-dependent density matrix(TDDM) ρ using the KSO basis set.Consider the situation when a surface acquires an excess or

deficit of electronic charge density with respect to ground statedistribution. The electronic charge density is assumed to beperiodically distributed in the XY plane. The dependence of theelectronic charge density along the Z coordinate perpendicularto the surface leads to inhomogeneity in total surface chargedensity across the slab ctot(z) = ∫ XYctot(x,y,z)dxdy, which

becomes a source of an electric field = ( ), ,x y z z =

ctot(z)/(εrε0), perpendicular to the surface. The correspondingelectric potential difference between charged layers is propor-tional to to their average charge and the distance between thelayers with excess and deficient planar electronic chargedistribution as

∫ε ε= − −V zc z z[ ] ( )dw

s r 01

0tot (50)

where w, z, ctot, ε0, and εr stand for the width of a photoactivelayer, the distance perpendicular to the surface, the total chargedensity per unit length across the surface, the dielectricconstant of a vacuum, and the dielectric coefficient of thephotoactive medium, respectively. Note that this universalequation is valid for ctot representing the ground or excited stateof the semiconductor surface. The total charge density containsboth ionic and electronic contributions, so that ctot(z) = cion(z)+ cel(z), with cion(z) = ∑σ,iσCσδ(Ziσ − z) and cel(z) = −en(z).Here, Cσ is the ion charge; Ziσ is the z component of theposition vector of the iσ

th atom of type σ; e is an elementarycharge; and n(z) is the electronic charge density along the z-axis.Within DFT, the electronic charge density is modeled as the

sum of partial charge densities from KS orbitals. The electroniccharge follows from the electronic density matrix of the wholesystem in the coordinate representation at each time moment tas cel(r,t) = −eρ(r,r,t). The electronic density matrix ρ(r,r′,t) =∑ijρij(t)ϕi*(r)ϕj(r) is expressed on a basis set of KS orbitals{φi(r)} and the surface photovoltage (SPV) VSPV = Vs(Ω) −Vs(0)

106 constructed from

∑ ∑ε ε ρ Ω = − Ω ⟨ | | ⟩σ

σ σ−

σ

V A C Z e i z ij( ) [ ] { ( ) }i

ij

ijs r 01

, (51)

with A being the surface area of the simulation supercell, Cσ andZiσ the atomic core charge and position in the lattice, and ⟨i|z|i⟩an average electronic position, so that VSPV(Ω) is obtained withrespect to the voltage of the nonexcited system.87 The densitymatrix elements ρij(Ω) depend on the frequency of incidentlight Ω as it is shown below.

■ AUTHOR INFORMATIONCorresponding Author*Phone: 605-677-7283. E-mail: [email protected] authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis paper is dedicated to the memory of B. I. Stepanov,04.28.1913-12.07.1987, in recognition of his contribution totheoretical spectroscopy. This research was supported by SouthDakota Governor’s Office of Economic Development, NSFaward EPS0903804, and by DOE, BES - Chemical Sciences,

NERSC Contract No. DE-AC02-05CH11231, allocationAwards 85213 and 86185 “Computational Modeling ofPhotocatalysis and Photoinduced Charge Transfer Dynamicson Surfaces”. The authors acknowledge computationalresources of USD High Performance computational facilitiescosponsored by South Dakota IDeA Networks of BiomedicalResearch Excellence NIH 2 P20 RR016479 and operated byDouglas Jennewein. DK thanks Sergei Tretiak for hosting thevisit at the Los Alamos National Lab during manuscriptpreparation. DK thanks Svetlana Kilina, David Micha, and OlegPrezhdo, for fruitful discussions. Editorial proofreading byStephanie Jensen is gratefully acknowledged.

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