insitute quantum modeling of transient infrared spectra ... · a photoinduced electron transfer...

THE JOURNAL OF CHEMICAL PHYSICS 124, 114105 �2006�

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Quantum modeling of transient infrared spectra reflecting photoinducedelectron-transfer dynamics

Birgit Strodel and Gerhard Stocka�

Institute of Physical and Theoretical Chemistry, J. W. Goethe University, Marie-Curie-Strasse 11,D-60439 Frankfurt, Germany

�Received 7 December 2005; accepted 21 December 2005; published online 20 March 2006�

A theoretical description of transient vibrational spectra following the impulsive optical excitationof a molecular system is presented. The approach combines the nonsecular evaluation of theRedfield equations to describe the dissipative dynamics of the system with an efficientimplementation of the doorway-window formalism to calculate optical pump/infrared probe �vis/IR�spectra. Both parts of the calculation scale with N2, thus facilitating the treatment of systems witha dimension up to 104. The formulation is applied to a simple model of photoinduced electrontransfer, which takes into account two coupled electronic states and a single anharmonic vibrationalmode. Despite its simplicity, the model is found to exhibit quite complex electronic and vibrationalrelaxation dynamics, which in turn give rise to rather complex time- and frequency-resolved vis/IRspectra. Interestingly, the calculated IR spectra of the electron-transfer system predict theappearance of novel vibronically induced sidebands, which may even dominate the spectrum atearly times. © 2006 American Institute of Physics. �DOI: 10.1063/1.2166629�

I. INTRODUCTION

With the advent of femtosecond lasers in the infrared�IR� range, it has become possible to perform time-resolvedvibrational studies of ultrafast chemical dynamics.1–3 Fol-lowing the preparation of the sample by an impulsive pumppulse �which may be in the IR or in the optical range�, theprobing by a time-delayed IR laser pulse allows us to moni-tor the structural evolution of the molecular system. Recentapplications of transient IR spectroscopy include the investi-gation of photoinduced charge transfer,4–6 the dynamics ofhydrogen bonds in the condensed phase,7 the structuralrearrangement and cooling following cis-trans photo-isomerization,8,9 and the conformational dynamics of photo-switchable peptides.10,11 Furthermore, the structural sensitiv-ity of time-resolved IR spectroscopy can significantly be en-hanced by employing two-dimensional detectiontechniques.12 In particular, there has been a considerableprogress in the multidimensional IR spectroscopy of theamide I band of peptides and proteins, which allows us tostudy the conformational structure and dynamics of thesesystems with a subpicosecond time resolution.13–18

In this work, we are concerned with the theoretical de-scription of transient IR spectra following the impulsive op-tical �vis� excitation of a molecular system. This vis/IRscheme appears to be closely related to optical �vis/vis� orvibrational �IR/IR� pump-probe spectroscopy, for both ofwhich well-established theoretical formulations exist.19–22

Nevertheless, apart from simple density-matrix formulationswith phenomenological damping terms23 and from thermallypopulated oscillator models to explain the time-dependentlevel shift due to vibrational cooling,8 we are not aware of acomprehensive quantum-mechanical description of a time-

a�
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resolved vis/IR pump-probe experiment. This somewhat sur-prising finding might be a consequence of the fact that amicroscopic modeling of such an experiment has to take intoaccount several issues. First, in order to facilitate the spec-troscopic identification of reactant and photoproduct states,the model should contain some kind of photophysical or pho-tochemical reaction that defines these states. Simple ex-amples are photoinduced charge-transfer and photoisomer-ization reactions. Next, the anharmonicity of the potential-energy surfaces is to be considered, since transient IR spectrareflect the excited-state absorption and stimulated emissionprocesses, which largely cancel in the harmonic approxima-tion. Furthermore, one needs to account for the subsequentvibrational cooling of the photoproducts, i.e., the vibrationalenergy redistribution between the initially excited modes andthe remaining degrees of freedom. Finally, the interaction ofthe molecular system with laser pulses of finite duration is tobe taken into account either nonperturbatively24 or, morecommonly, via time-dependent perturbation theory.19

To satisfy the above requirements, we consider the fol-lowing formulation. �i� We adopt a standard model of photo-induced electron transfer,25 which takes into account twocoupled electronic states and a single harmonic vibrationalmode. To introduce anharmonicity, the harmonic oscillatorsare replaced by Morse potentials. Furthermore, we assumedifferent vibrational frequencies for the initially excited elec-tronic state and the charge-transfer state. �ii� To describequantum-mechanical dissipation of vibrational energy, weemploy reduced density-matrix theory within the Redfieldapproximation.26,27 We use the recently proposed “nonsecu-lar” algorithm,28 which systematically includes the most im-portant nonresonant terms while retaining the advantageousN2 scaling of the standard secular approximation �N being
the dimension of the reduced density matrix of the system�.
© 2006 American Institute of Physics05-1

ense or copyright; see http://jcp.aip.org/about/rights_and_permissions

http://dx.doi.org/10.1063/1.2166629

http://dx.doi.org/10.1063/1.2166629

http://dx.doi.org/10.1063/1.2166629

114105-2 B. Strodel and G. Stock J. Chem. Phys. 124, 114105 �2006�

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�iii� To calculate transient vis/IR pump-probe spectra, adoorway-window-type formulation is employed,29 which is aperturbative approach that assumes that the pump and probelaser pulses do not overlap in time and that dissipation can beneglected during the interaction with the radiation fields. Us-ing the eigenstate representation and assuming Gaussian la-ser pulses, explicit expressions for the doorway and windowoperators are derived that also scale with N2. Since the nu-merical propagation of the Redfield equation is performed inthe eigenstate representation anyway, there is no major addi-tional effort when combining the doorway-window and non-secular Redfield formulations.

Employing this formulation, we present a detailed nu-merical study of the transient IR spectral features of an ul-trafast photoinduced electron-transfer reaction. In particular,we discuss the spectroscopic signatures of the electronic andvibrational relaxation dynamics as well as of the decayinginitial state and the arising photoproduct.

II. THEORETICAL FORMULATION

A. Model system

As a well-studied model of photoinduced electron trans-fer �ET�,30 we consider an electronic three-state system,comprising an energetically well separated electronic groundstate ��0� and two nonadiabatically coupled excited states��1� and ��2�. It is assumed that the ET process takes placebetween the two excited electronic states after excitation ofthe system at time t=0 by an ultrashort laser pulse from ��0�to the optically bright state ��2�. Adopting a diabatic elec-tronic representation, the system Hamiltonian,

HS = �n=0,1,2

��n��T + Vnn��n� + ��1�V12��2� + H.c. , �1�

is given as the sum of the kinetic energy T and the diabaticpotential matrix Vnm. In this paper, we restrict ourselves toa constant diabatic coupling V12=V21 and a single vibrationaldegree of freedom q, described by shifted Morse potentials,

Vnn = Dn�1 − e−��q−qn��2 + En. �2�

Here, the Dn are the dissociation energies, q=qn defines thelocation of the energy minimum of the potential Vnn, and theconstant � is connected to the force constant and the anhar-monicity of the Morse potential. Following Ref. 25, the pa-rameters of the model are chosen to represent the situation ofa photoinduced electron transfer promoted by a high-frequency vibrational mode.31 The fundamental frequencies�n of the Morse potentials are defined by

�n =�2Dn�2

m, �3�

which leads to �0=�1=1097 cm−1 and �2=1032 cm−1. Thepotentials curves V11 and V22 together with the eigenstates ofthe system are shown in Fig. 1�a�. The vertical displacementsE1 and E2 of the potentials are chosen to give near degenera-cies for eigenstates belonging to different electronic states,which guarantees an effective electron transfer between ��1�and ��2�.25 In the remainder of the paper, we change to di-
mensionless coordinates �i.e., �q→q� and set ��1.

B. Dissipative dynamics

To describe quantum-mechanical dissipation of vibra-tional energy of the system HS, the usual system-bathapproach26,27 is employed in which the molecular Hamil-tonian,

HM = HS + HB + HSB, �4�

is partitioned into the system Hamiltonian HS, the bathHamiltonian,

HB = ��

��

2�x�

2 + p�2� , �5�

comprising the vibrational modes x� of the environmentwithin the harmonic approximation, and the bilinear system-bath coupling,

HSB = q��

c�x�. �6�

The coupling constants c� are chosen according to an Ohmicspectral density,

J�� =�

2 ��

c�2�� − �� = ��e−�/�c, �7�

where the overall strength of the system-bath coupling isgiven by the dimensionless parameter �, while the cutofffrequency �c describes the time-scale distribution of thebath. Throughout this work, the parameters are chosen as�=0.1 and �c=1050 cm−1.

To describe the dissipative dynamics of the system, we

FIG. 1. �Color� Potential-energy curves V11 �red� and V22 �blue� of the �a�Morse and the �b� harmonic electron-transfer models. The red and blue linesindicate the eigenstates pertaining predominantly to the electronic states ��1�and ��2�, respectively. The arrows between the lines illustrate the allowedvibrational transitions of the system. Here, the black arrows refer to thevibrational transitions in a single diabatic electronic state, while magentaand green arrows refer to the vibrational transitions between the two coupledelectronic states ��1� and ��2�.

employ the Redfield approach, that is, a second-order pertur-


114105-3 Modeling of transient infrared spectra J. Chem. Phys. 124, 114105 �2006�

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bation theory with respect to the system-bath interactioncombined with a Markov approximation for the relaxationoperators.26,27 Adopting the system eigenstate representation,HS�l�=El � l�, the Redfield equation is given as

�

�t�ij = − i�ij�ij + �

k,lRijkl�kl, �8�

where � denotes the reduced density matrix, �ij =Ei−Ej, andRijkl represents the Redfield relaxation tensor, which de-scribes the interaction of the molecular system with the bath.

For the numerical propagation of the Redfield equation�8� we employ the recently proposed nonsecular algorithm,28

which systematically includes the most important nonreso-nant terms while retaining the advantageous N2 scaling of thestandard secular approximation �N being the dimension ofthe reduced density matrix of the system�. As shown below�see Figs. 2 and 5� as well as in the previous work,25,28 thesecular approximation �that is, including only the resonantterms of the relaxation tensor26,27� may lead to a significantoverestimation of the relaxation dynamics.

C. Definition of spectroscopic signals

Within the dipole approximation, the interaction Hamil-tonian Hint describing a vis/IR pump-probe experiment canbe written as

Hint = − elE1�t� − vibE2�t� , �9�

where E1�t� and E2�t� are the vis-pump and the IR-probeel vib

FIG. 2. �Color� Electronic and vibrational relaxation dynamics of the anhar-monic electron-transfer model as exhibited by �a� the population probabilityP2�t� of the initially excited electronic state and �b� the mean position q�t� ofthe Morse oscillator. Compared are results obtained for the undamped sys-tem �red� as well as for the damped system calculated from the Redfieldformulation, with �green� and without �blue� using the secular approxima-tion. The vibrational energy relaxation of the damped system is character-ized by the Redfield results of the mean vibrational level number �n��t� andits standard deviation ��n��t� �dashed line� shown in panel �c�.

pulse, respectively, and and denote the correspond-


ing transition dipole operators. Assuming Gaussian pulseshapes, the electric fields are given by

Ei =Ei

�4��i

e−�t − ti�2/�4�i

2�e−i�i�t−ti� + c.c., �10�

with �=1/ �16 ln 2�. The laser pulses are characterized bytheir carrier frequencies �1 and �2 and their durations 1 and2, respectively. They are centered at times t= ti, wheret1=0 and t2=�t, representing the delay time of the probepulse.

Employing the Franck-Condon approximation and as-suming that the excited electronic state ��1� is dark in ab-sorption and emission, the electronic transition dipole opera-tor el is defined as

el = ��0�02��2� + ��2�20��0� . �11�

The vibrational transition dipole operator vib can be ex-panded as a power series in the coordinate q around its equi-librium position qn in the elctronic state ��n� �see Appendix�,

vib = �n=1,2

��n��n��nn�qn� + ��nn/�q�qn�q − qn�

+ higher orders� . �12�

As is usual in time-resolved vis/IR spectroscopy, weconsider the transient absorption of the IR probe pulse,which is given as a function of the dispersed frequency �and the pulse delay time �t,19

S��,�t� = 2� Im E2��P*�� , �13�

where E2�� and P�� denote the Fourier transforms of theprobe pulse and the nonlinear polarization, respectively. It isnoted that the absorption spectrum is often normalized withrespect to the incident probe intensity �E2��2. To eliminateproblems associated with the fact that only a part of themolecules is excited by the pump pulse, in experiment usu-ally difference spectra are considered, e.g., �S=Spump-probe

−Sprobe-only. Alternatively, one may define the IR differencespectra as

�S��,�t� = S��,�t� − S��,�� , �14�

which, by construction, vanish at long times.

D. Doorway-window representation

Within the assumption of nonoverlapping pump andprobe pulses, the pump-probe signal can be written in theso-called doorway-window representation,19,29

S��,�t� = 2� TrW��,�2�e−iLM�tD��1� , �15�

where D��1� denotes the doorway operator describing thepreparation of the system at time t=0 by the optical pumppulse, LM¯ = �HM , ¯ � is the molecular Liouvillian, ac-counting for the dissipative time evolution of the molecularsystem during 0 t �t, and W�� ,�2� represents the win-dow operator, describing the interaction of the system withthe IR probe laser at t=�t. Assuming that the system-bathcoupling can be neglected during the interaction of the sys-
tem with the laser fields, explicit expressions for the door-


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way and window operator can be obtained for exponential32

and Gaussian-shaped33 laser pulses.Assuming a Gaussian optical pump pulse, we obtain for

the doorway operator,33

D��1� = �a,b,c

�b��d��aaeqba

el adel e−�1

2��1 − �ba�2+��1 − �da�2�,

�16�

where the sums run over the eigenstates of the system,�eq=��−�� describes the initial system in thermal equilib-rium, and ba

el denotes the Franck-Condon factor for the elec-tronic transition between the eigenstates �a� and �b� belong-ing to the electronic states ��0� and ��2�, respectively. Forsimplicity, we assume throughout the paper impulsive exci-tation of the system by the pump pulse, i.e., 1=0.

This doorway density is then propagated according tothe Redfield equation �8�, until the delay time �t is reached.At that time, the overlap between the system and the windowoperator is calculated. For a Gaussian IR probe pulse, thewindow operator reads33

W��,�2� = �c,b,d

�b��d�bcvibcd

vibe−��2��2

� e−�22��bd − ��2

�� + �bc� − �� − �dc��

+ e−�22��bd + ��2

�� + �dc� − �� − �bc�� ,

�17�

where ��=�2−�. The calculation of the matrix elementsof the vibrational dipole operator vib is described in theAppendix.

III. COMPUTATIONAL RESULTS

A. Numerical details

The system Hamiltonian �1� has been expanded in twodiabatic electronic states ��1� and ��2� and 30 bound eigen-states of each Morse oscillator. Due to the assumption ofimpulsive excitation of the system by the pump pulse, theelectronic ground state ��0� needs not explicitly be consid-ered in the calculation, since it merely defines the nonequi-librium initial state for the coupled ��1�− ��2� system. Toachieve numerical convergence of the Redfield propagation,it was sufficient to take into account the lowest 40 eigen-states of the ET system. In the nonsecular algorithm of theRedfield equation,28 only those relaxation tensor elementsRijkl are considered, which satisfy the conditions

��ij − �kl� a , �18�

��kl�t�� b . �19�

In the present work, the choice of a=160 cm−1 and b=10−4

guaranteed numerical convergence of the results with respectto calculations using the full Redfield tensor.

In all calculations, we have assumed that the system isinitially in its electronic and vibrational ground states, corre-sponding to zero temperature of the bath. Test calculationsfor a temperature T=300 K showed that the results change
only in details but not qualitatively. For the simulation of the

pump probe spectra we assumed for the durations of the laserpulses 1=0 and 2=100 fs, i.e., impulsive excitation by thevis-pump pulse and a typical IR pulse length for the probeprocess. As carrier frequency of the probe pulse we havechosen �2=1089 cm−1, which is in the range of the vibra-tional transitions for both Morse potentials V11 and V22. The� functions in Eq. �17� were replaced by a Lorentzian lineshape function with width 1/T2=1 ps−1. The spectra wereevaluated every 5 cm−1, thus mimicking a typical experimen-tal frequency resolution.

B. Electronic and vibrational relaxation dynamics

Before discussing time-dependent IR spectra, it is help-ful to characterize the photoinduced electronic and vibra-tional dynamics of our ET model. To this end, we considervarious time-dependent intramolecular observables of thesystem describing these processes. A key quantity in the dis-cussion of photoinduced ET between the electronic states��1� and ��2� is the population probability P2�t� of the ini-tially prepared electronic state.34 It is calculated as

P2�t� = Tr��2��2��t� , �20�

and shown in Fig. 2�a�. In the absence of dissipation �HSB

=0, red line�, the electronic population exhibits Rabi-typeoscillations between 0.3� P2�t� 1 and with a period time of 200 fs. The period reflects the electronic couplingV12=161.3 cm−1 of the system Hamiltonian. Taking into ac-count the coupling to the environment, the Redfield result�blue line� reveals that the system relaxes within a picosec-ond into the charge transfer state ��1�. A comparison with thesecular approximation to Redfield theory �green line� showsthat this assumption significantly exaggerates the damping ofthe coherent ET dynamics. This failure can be ascribed tonear degeneracies of the eigenlevel spectrum of the systemHamiltonian, which considerably contribute to the relaxationtensor but are neglected in the secular approximation.25,28

The simplest observable describing the vibrational dy-namics of the ET model is the expectation value of the dis-placement coordinate q�t�=Trq��t� shown in Fig. 2�b�. Forthe undamped system, the time evolution of q�t� is domi-nated by coherent oscillations with the vibrational frequen-cies of 1032 and 1097 cm−1, which correspond to a periodtime of 30 fs. Furthermore, the electronic coupling V12

shows up as a 200 fs beating of q�t�. Due to the electroniccoupling and the anharmonicity of the Morse potentials thecoherent wave-packet motion dephases on a time scale of500 fs. The Redfield results for the damped system showoscillations until t 600 fs and then relax to the position ofthe minimum of the Morse potential V11, i.e., q�t→��=q1.The results obtained for the secular approximation againshow a significant overestimation of the overall relaxation.

Apart from the vibrational phase relaxation monitoredby q�t�, we are particularly interested in the vibrational en-ergy relaxation of the system. The latter is monitored by thetime-dependent energy content of the vibrational mode or,equivalently, by its occupation number. To characterize thetime evolution of the vibrational energy, Fig. 2�c� shows the
time-dependent mean vibrational level number,


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�n��t� = Tr �k=1,2

��k��k�bk†bk��t� , �21�

where bk† and bk are the creation and annihilation operators of

the Morse oscillator in electronic state ��k�. At time t=0, theinstantaneous excitation from the electronic ground state ��0�with equilibrium position q0=0 to the excited state ��2� withequilibrium position q2=2 Å results in an initial level num-ber �n��0�=3.5. This value is also expected from the acces-sible vibrational levels of the potential V22 shown in Fig.1�a�. Following the excitation, �n��t� is seen to decay on thesame time scale and in a similar pattern as the electronicpopulation P2�t� shown in Fig. 2�a�. Furthermore, Fig. 2�c�shows the standard deviation ��n��t�=��n2�− �n�2, which isa measure of the width of the vibrational level number dis-tribution. Initially, ��n��t� exhibits an ultrafast rise up to 4.5,which reflects the initial decay of the electronic population.Then, the width decays on the same overall time scale as themean level number.

For interpretational purposes, it is often instructive toassociate the vibrational energy relaxation shown in Fig. 2�c�with a cooling process of the photoexcited hot molecule inthe surrounding cool solvent. Assuming a harmonic vibra-tional mode at thermal equilibrium, we obtain the relations�n�=1/ �ex−1� and ��n�=e−x/2 / �1−e−x�, where x=�� /kBT.Inverting the relation for �n��t�, we obtain temperatures 5500, 3000, and 1500 K at times t 0, 200, and 500 fs.Inverting the relation for ��n��t�, on the other hand, we ob-tain at the above times temperatures of 7000, 3800, and1600 K. That is, at short times, the width of the vibrationalenergy distribution is larger than the one observed for a cool-ing process at thermal equilibrium. This is because the vibra-tional energy relaxation is too fast to be at thermal equilib-rium and also because two coupled vibrational potentialscontribute to the process.

Finally, we note that the strong initial vibrational excita-
tion of the system is a consequence of the fact that the model

contains a single vibrational mode which, by construction,receives all the energy. In a polyatomic molecule, typically afew modes are actively involved in the photoreaction, whilethe majority of vibrations are the so-called spectator modes,that receive photoinduced energy only through anharmoniccouplings. Hence, our study describes the vibrational relax-ation and spectroscopy of “photoactive” vibrational modes,but not of spectator modes.

C. Transient IR spectra

Let us now study to what extent the above discussedelectronic and vibrational dynamics can be monitored in atime-resolved vis/IR experiment. Evaluating Eq. �17� for100 fs probe pulses, Fig. 3 presents various transient IRspectra for delay times ranging from 0 to 1 ps. Shown are inpanel �c� the transient absorption spectrum S�� ,�t� of Eq.�13� and in panel �d� the corresponding difference spectrum�S�� ,�t� as defined in Eq. �14�. Furthermore, the total spec-trum in �c� has been decomposed in its �a� absorption�S�0� and �b� emission �S�0� contributions, respectively.All spectra consist of two bands at 900–1100 cm−1 and at1150–1250 cm−1. With increasing time, both bands exhibit ablueshift, whereby the width of the bands decreases signifi-cantly. The spectral evolution reflects two interrelated phe-nomena. �i� As a consequence of the ultrafast ��2�→ ��1� ETprocess, the fundamental vibrational frequency changes from1032 cm−1 �for potential V22� to 1097 cm−1 �for potentialV11�. �ii� As a consequence of the vibrational energy relax-ation, the vibrational transition frequency increases due tothe anharmonicity of the potentials. Furthermore the width ofthe vibrational level number distribution ��n��t� decreasesfor longer times, as shown in Fig. 2�c�.

In the experiment, usually only the difference spectrum�S�� ,�t� can be measured. A comparison to the spectrumS�� ,�t� reveals, though, that both spectra essentially carry

FIG. 3. �Color� Cuts of the vis/IRpump-probe spectrum for the anhar-monic electron-transfer model, ob-tained for delay times �t=0 �black�,100 fs �red�, 250 fs �green�, 500 fs�blue�, and 1 ps �cyan�. Shown are the�a� absorption, �b� emission, �c� totalspectrum, and �d� difference spectrum.

the same information. In the calculation, moreover, one may



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distinguish between absorption and emission processes. Asthe two contributions largely cancel each other at early times,this decomposition does add new information. For example,Fig. 3 demonstrates that—given sufficient spectralresolution—the decomposed contributions allow us to esti-mate the number of excited vibrational levels. For longertimes, the cancellation of the two contributions vanishes, be-cause the emission dies out as the system relaxes into thevibrational ground state.

The above results demonstrate that a time-resolvedvis/IR experiment is able to reveal numerous aspects of theelectronic and vibrational dynamics of an ultrafast ET pro-cess. So far, however, two questions remain unsolved. Thatis, the origin of the unusual spectral band at 1200 cm−1,and to what extent the electronic and vibrational relaxationscontribute to the observed spectral evolution.

To clearly distinguish the effects of anharmonicity andvibronic coupling on the transient IR spectrum, we study twolimiting cases, that is, �i� an adiabatic model without elec-tronic coupling �V12=0� and �ii� a harmonic version of theET system. For the adiabatic model, the corresponding tran-sient IR spectra are shown in Fig. 4. In this case, the time-dependent blueshift of the absorption and emission bands issolely driven by the anharmonicity of the potential V22. As aconsequence, the blueshift as well as the overall width of theband is significantly smaller than in the IR spectrum of thecoupled system. Furthermore, the spectral band at 1200 cm−1 is missing. Reflecting the vibrational energy re-laxation of a single anharmonic mode, the interpretation ofthe transient IR spectrum in Fig. 4 is obviously quitestraightforward.

To focus on the effects introduced by the nonadiabaticcoupling, we consider a harmonic version of the ET model
�1� with diabatic potentials,

Vnn =�n

2q2 + �nq + En, �22�

where the frequencies match the fundamentals of the Morsemodel, �0=�1=1097 cm−1 and �2=1032 cm−1, and thevibronic coupling constants are given by �n=−�nqn. Asin the anharmonic ET model, the vertical displacement

of the ground electronic state is E0=−16000 cm−1, and

E1=−242.0 cm−1 and E2=3742.4 cm−1 eV have been chosento give near degeneracies for eigenstates belonging to differ-ent electronic states. The potential-energy curves and eigen-states of the harmonic and the Morse ET models are com-pared in Fig. 1. As expected, the Morse potentials are steeperfor distances q shorter than the equilibrium geometry qn andmore shallow for q�qn.

Let us briefly compare the electronic and vibrational re-laxation dynamics of the harmonic model shown in Fig. 5 tothe dynamics of the Morse model shown in Fig. 2. As aconsequence of the choice of the vertical displacements �seeabove�, the ET dynamics as represented by the electronicpopulation P2�t� is quite similar in both models. In the ab-sence of anharmonicity, the vibrational phase relaxation ex-hibited by q�t� is somewhat slower than for the Morse model.This effect is only significant for the undamped systems,while there are surprisingly small differences in the timeevolution of q�t� in the presence of dissipation. Finally, wecompare the vibrational energy relaxation of the two models,exhibited by mean vibrational level number �n��t� and itsstandard deviation ��n��t�. Although the overall appearanceof these quantities is quite similar for both models, theinitial level number �n��0� 2 of the harmonic model issignificantly lower than in the case of the Morse model�n��0� 3.5. A similar situation is found for the initial width��n��0� of the vibrational level number distribution. Consult-

FIG. 4. �Color� As in Fig. 3, but forthe adiabatic electron-transfer model,i.e., in the absence of vibronic cou-pling �V12=0�.

ing the potential-energy curves of both models shown in



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Fig. 1, this effect is clearly a consequence of the steeperpotential well for q�q2 of the Morse potential compared tothe harmonic potential.

The transient IR spectra corresponding to the harmonicET model are shown in Fig. 6. We first discuss the timeevolution of the main spectral band at 1000–1100 cm−1. Inthe absence of anharmonicity, the time-dependent blueshiftof the absorption and emission bands is mainly caused by theET-driven change of the fundamental vibrational frequencyfrom 1032 cm−1 �for potential V22� to 1097 cm−1 �for poten-tial V11�. As a consequence, the blueshift as well as the over-all width of the band are significantly smaller than in the IRspectrum of the anharmonic ET system. Thus, we again havefound that the evolution of the transient IR spectrum mainlyreflects the time-dependent vibrational level number distribu-tion of the system.

FIG. 5. �Color� As in Fig. 2, but for the harmonic electron-transfer model.


Let us now turn our attention to the unusual spectralbands at 900 and 1220 cm−1 of the harmonic ET model.These features can readily be explained by considering thematrix elements ij

vib of the vibrational dipole operator vib

with respect to the eigenstates �i� and �j�. With this end inmind, all allowed transitions with a sufficiently high oscilla-tor strength �ij

vib�2 are indicated in Fig. 1 and listed togetherwith their transition frequency �ij in Table I. The first twosets of transitions in Table I take place in a single diabaticelectronic state, ��1� or ��2�. Consequently, these transitionsoccur close to one of the fundamental frequencies 1032 cm−1

�for potential V22� and 1097 cm−1 �for potential V11�, andgive rise to the main band of the IR spectrum.

However, there are also allowed vibrational transitionsij

vib between the two coupled electronic states ��1� and ��2�.Because we have chosen our ET model to give near degen-eracies of the uncoupled vibrational levels �see above�, thecorresponding eigenstates are shifted from the uncoupled vi-brational levels by approximately ±V12. Hence, the two un-usual spectral bands at 900 and 1220 cm−1 reflect vibroni-cally induced transitions at frequencies which are shiftedfrom the fundamental frequency by approximately 160 cm−1

towards higher or lower frequencies. The lower intensityof the side bands compared to the main band reflect theirsmaller oscillator strength and the fact that their frequencyis somewhat off-resonant to the carrier frequency�2=1089 cm−1 of the probe pulse.

The latter effect also explains the different intensities ofthe side bands in the transient IR spectra of the Morse modelin Fig. 3 and the harmonic model in Fig. 6. Because bothside bands of the Morse model are shifted to lower frequen-cies, the high-frequency band is enhanced and the low-frequency band is suppressed compared to the harmoniccase. Furthermore, we notice that the vibrational energy re-laxation leads to a blueshift of the high-frequency band forthe anharmonic system and to a �minor� redshift for the har-

FIG. 6. �Color� As in Fig. 3, but forthe harmonic electron-transfer model.



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monic system. As shown in Table I, this reflects the fact thatthe corresponding vibrational transitions exhibit increasingand decreasing frequencies, respectively.

IV. CONCLUDING REMARKS

We have outlined a theoretical description of transientvibrational spectra following the impulsive optical excitationof a molecular system. The approach combines the nonsecu-lar evaluation28 of the Redfield equation to describe the dis-sipative dynamics of the system with an efficient implemen-tation of the doorway-window formalism33 to calculate time-and frequency-resolved vis/IR spectra. Both parts of the cal-culation scale with N2, thus facilitating the treatment of sys-tems with a dimension up to 104.

As a first application of the formulation, we have con-sidered a simple model of photoinduced ET, which takes intoaccount two coupled electronic states and a single anhar-monic vibrational mode. Despite its simplicity, the modelwas found to exhibit quite complex electronic and vibra-tional relaxation dynamics, which in turn is reflected in arather complex transient vis/IR spectrum. The spectrum ofthe single anharmonic mode consists of three vibrationalbands, the standard main band and two vibronically inducedside bands. With increasing time, the spectrum exhibits ablueshift, whereby the width of the bands decreases signifi-cantly. The spectral evolution reflects two interrelated phe-nomena. �i� As a consequence of the ultrafast ET process

TABLE I. Frequencies �ij �in cm−1� of the allowed vibrational transitionsbetween eigenstates �i� and �j�, as obtained for the harmonic and Morseelectron-transfer models, respectively. The first two sets of transitions takeplace in a single diabatic electronic state, ��1� or ��2�. The other two setstake place between the two coupled electronic states ��1�± ��2�, where the �

indicates that the corresponding transitions are shifted by approximately±V12.

Transitioni↔ j

Electronicstate

Harmonic modelfrequency �ij

Morse modelfrequency �ij

1↔2 1 1094 10822↔3 1 1073 10453↔5 1 1062 10366↔8 1 1072 10298↔10 1 1077 101510↔12 1 1082 1006

4↔6 2 1067 10485↔7 2 1056 10317↔9 2 1051 10249↔11 2 1047 1013

2↔4 1+2 1217 12133↔6 1+2 1211 12155↔8 1+2 1222 12107↔10 1+2 1243 11949↔12 1+2 1274 1176

4↔5 1−2 917 8686↔7 1−2 906 8528↔9 1−2 885 846

10↔11 1−2 855 843

which affects a change of the electronic potential-energy sur-


face, the fundamental vibrational frequency of the potentialchanges. �ii� As a consequence of the vibrational energy re-laxation, the vibrational transition frequency increases due tothe anharmonicity of the potentials. Furthermore the width ofthe vibrational level number distribution decreases for longertimes. Due to the high speed of the reaction, the initial vi-brational energy relaxation can only roughly be explained interms of a cooling process at thermal equilibrium. The strongvibrational excitation and the resulting large blueshift is aconsequence of the fact that the single-mode model accountsfor the relaxation and spectroscopy of a “photoactive” vibra-tional mode, not of a spectator mode.

It is interesting to note that the calculated IR spectra ofthe ET system predict the appearance of vibronically inducedside bands. In particular, at early times the transient IR spec-trum in Fig. 3 is even dominated by the high-frequency sideband. Performing similar calculations for a number of one-and two-mode ET models, this effect has proven quite gen-eral. To our knowledge, vibronically induced side bands in atransient vis/IR spectrum are a new phenomenon that stillawaits experimental verification. Since in a realistic systemwith many vibrational modes possible side bands maybe eas-ily missassigned, this requires a joint experiment and theo-retical investigation.

ACKNOWLEDGMENTS

We thank Peter Hamm, Michael Thoss, and JosefWachtveitl for inspiring and helpful discussions. This workhas been supported by the Frankfurt Center for ScientificComputing and the Deutsche Forschungsgemeinschaft.

APPENDIX: MATRIX ELEMENTS OF THE MORSEOSCILLATOR

In the numerical representation of the anharmonic ETmodel �1� the bound states �v� of the Morse potential V00 arechosen, for which the eigenvalues are given by

Ev = ��0�v + 12� − ��v + 1

2�2, v = 0,1, . . . ,vmax. �A1�

Here, �=�2� /2m represents the anharmonicity of the Morsepotential, vmax=�C− 1

2� is the number of bound states with

�x� denoting the largest integer smaller than x, and

C =�0

2�. �A2�

For both excited electronic states it is thus necessary to cal-culate the matrix elements �v�T+Vnn�v�� n=1,2�. With thedefinitions

b = 2C − 2v − 1, �A3�

A =�2m

��, �A4�

z = 2Ce−�q, �A5�

they can be written as



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�v�T + Vnn�v�� = �Dn + En −b2

A2��vv� + � C

A2 −Dne�qn

C�

��v�z�v�� + �Dne2�qn

4C2 −1

4A2��v�z2�v�� ,

�A6�

with the matrix elements,35,36

�v�z�v�� = �v��z�v� = �− 1�v+v��v�!��2C − v��bb�

v!��2C − v� �1/2

,

�A7�

�v�z2�v�� = �v��z2�v�

= v�z�v��v + 1��2C − v� − v��2C − v� − 1�� ,

�A8�

where ��x� is the Gamma function.The number of bound states for our ET model is

vmax=99 for the electronic states ��0� and ��1�, whereas it isvmax=93 for ��2� �cf. Eq. �A2��. However, in the numericaltreatment it is sufficient to take the lowest ni=30�i=0,1 ,2� bound states into account.

For the numerical representation of the relaxation tensorelements Rijkl as well as the vibrational dipole operator vib itis necessary to express the vibrational displacement q in thebasis �v� of the unshifted Morse potential V00. The matrixelements �v�q�v�� with v�v� can be written as35,37

�v�q�v�� = �v��q�v�

= −�v�z�v��

��v� − v��2C − v − v� − 1��v� � v� . �A9�

To keep numerical round off errors small, it is advantageousin the case of v=v� to employ an asymptotic power expan-sion for the displacement coordinate q, giving38

�v�q − qe�v� =1

2C��3u +

1

2C�7

2u2 +

5

24�

+1

�2C�2�5u3 +3

4u�

+1

�2C�3�31

4u4 +

17

8u2 −

23

960� + ¯ �

�A10�

with u=v+ 12 .

1 P. O. Stoutland, R. B. Dyer, and W. H. Woodruff, Science 257, 1913�1992�.

2 G. H. Atkinson, J. Phys. Chem. A 104, 4129 �2000�.


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14 M. T. Zanni, N.-H. Ge, Y. S. Kim, and R. M. Hochstrasser, Proc. Natl.Acad. Sci. U.S.A. 98, 11265 �2001�.

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insitute quantum modeling of transient infrared spectra ... · a photoinduced electron transfer...

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