Detection of dark states in two-dimensional electronic photon-echo signals via ground-state coherence.

Detection of dark states in two-dimensional electronic photon-echo signals via ground-state coherence Dassia Egorova Citation: The Journal of Chemical Physics 142, 212452 (2015); doi: 10.1063/1.4921636 View online: http://dx.doi.org/10.1063/1.4921636 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Oscillations in two-dimensional photon-echo signals of excitonic and vibronic systems: Stick-spectrum analysis and its computational verification J. Chem. Phys. 140, 034314 (2014); 10.1063/1.4861634 Towards microscopic assignment of oscillative signatures in two-dimensional electronic photon-echo signals of vibronic oligomers: A vibronic dimer model J. Chem. Phys. 139, 144304 (2013); 10.1063/1.4822425 Analysis of cross peaks in two-dimensional electronic photon-echo spectroscopy for simple models with vibrations and dissipation J. Chem. Phys. 126, 074314 (2007); 10.1063/1.2435353 Stimulated emission three-pulse photo-echo peakshift: A mixed pump–probe and photon-echo technique for studying excited-state dynamics J. Chem. Phys. 121, 5039 (2004); 10.1063/1.1794694 Spectrally resolved femtosecond two-color three-pulse photon echoes: Study of ground and excited state dynamics in molecules J. Chem. Phys. 120, 8434 (2004); 10.1063/1.1651057

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THE JOURNAL OF CHEMICAL PHYSICS 142, 212452 (2015)

Detection of dark states in two-dimensional electronic photon-echo signals via ground-state coherence Dassia Egorova Institut für Physikalische Chemie, Christian-Albrechts-Universität zu Kiel, Olshausenstr. 40, D-24098 Kiel, Germany

(Received 10 February 2015; accepted 13 May 2015; published online 5 June 2015) Several recent experiments report on possibility of dark-state detection by means of so called beating maps of two-dimensional photon-echo spectroscopy [Ostroumov et al., Science 340, 52 (2013); Bakulin et al., Ultrafast Phenomena XIX (Springer International Publishing, 2015)]. The main idea of this detection scheme is to use coherence induced upon the laser excitation as a very sensitive probe. In this study, we investigate the performance of ground-state coherence in the detection of dark electronic states. For this purpose, we simulate beating maps of several models where the excited-state coherence can be hardly detected and is assumed not to contribute to the beating maps. The models represent strongly coupled electron-nuclear dynamics involving avoided crossings and conical intersections. In all the models, the initially populated optically accessible excited state decays to a lower-lying dark state within few hundreds femtoseconds. We address the role of Raman modes and of interstate-coupling nature. Our findings suggest that the presence of low-frequency Raman active modes significantly increases the chances for detection of dark states populated via avoided crossings, whereas conical intersections represent a more challenging task. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4921636]

I. INTRODUCTION

Dark electronic states in molecular systems have been proven responsible for such fundamental processes as photoprotection and light-harvesting (DNA bases and carotenoids being most well known examples) but remain extremely difficult to access experimentally. Their presence is usually detected indirectly, e.g., as a very low fluorescence yield, or an ultrafast fluorescence (or stimulated-emission) decay in timeresolved experiments: after light absorption, a molecule does not remain in the absorbing state but undergoes a fast (much faster than the excited-state lifetime) radiationless transition into a close-lying dark state. From the microscopic point of view, radiationless transitions to dark states occur, in molecular systems, via avoided crossings (ACs) or conical intersections (CIs) of electronic states, i.e., the associated dynamics involves the breakdown of Born-Oppenheimer approximation and is governed by strongly coupled electron-nuclear motion. This strong coupling leads to formation of the so called vibronic manifold, a dense manifold of states of mixed electronic and vibrational character. In time-resolved spectroscopy, ultrashort broadband laser pulses can excite a superposition of vibronic states and initiate excited-state coherent motion. If excited-state electronic and vibrational motions are strongly coupled (i.e., nonseparable), one can hardly distinguish electronic and vibrational contributions in the created coherence. On the other hand, nonlinear time-resolved techniques involve light-matter interactions leading to the excitation of coherent motion in the ground electronic state. In contrast to the excited states, the arising motion is, in the most cases, of purely vibrational origin. 0021-9606/2015/142(21)/212452/11/$30.00

The generated coherences can be resolved in the recorded signals as oscillatory transients, provided the employed laser pulses are short enough at the timescale of the considered motion. Already the pioneering pump-probe experiments have monitored vibrational wave-packet motion, but most recently, two-dimensional (2D) electronic photon-echo spectroscopy1 also often referred to as 2D electronic spectroscopy (2DES)2 has proven as the most selective and sensitive experimental toll to detect coherent oscillations and the reported experimental applications of this technique range from quantum nanostructures3–6 to biosystems.7–15 While the origin of the detected oscillations remains in focus of intensive discussions,16–22 most recent advance is the idea to exploit the induced coherence for a more detailed analysis and characterization of the excited states and ultrafast excited-state dynamics.23–25 Historically, photon-echo (PE) technique has been developed and employed for the elimination of inhomogeneous broadening. The 2D setup has been inspired by similar nuclear magnetic resonance (NMR) implementations and first realised in the infrared domain, where it has been successfully employed to measure couplings between the vibrational modes and anharmonicities. The subsequent extension to the optical domain immediately faced the interpretation difficulties due to electron-vibrational coupling. The vibronic interactions lead to very broaden signals and may cause significant modifications of the lineshapes. If underdamped vibrational motion is initiated, it will not only influence the signal lineshapes but will also contribute to the signal time-dependent oscillatory component. The analysis of the 2D signal oscillatory signatures is considerably facilitated if the so called beating maps20 are employed. The beating maps represent amplitudes of each oscillatory component detected. The term “map” is due to the

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dependence of the amplitude on the two frequencies of the conventional 2DES signal, the so called excitation and probe frequencies. The dispersed pump-probe signal can be viewed as a 2DES with the reduced resolution: it depends on the probe frequency only. An analogue of the beating maps can be also constructed for it,24 but since one of the two frequencies is missing, 2DES maps “collapse” in 1D pump-probe maps and the analysis becomes more difficult and less efficient. Each 2DES beating map is supposed to represent one particular coherence: it is recorded as the amplitude of a specific oscillation with a preselected frequency. In practice, several oscillations with close frequencies contribute to the same map. Still, the beating-map representation can provide information on the energies of the states involved in each coherence and allows one to distinguish ground-state and excited-state coherent motions.24 Further, beating maps can serve as a very sensitive detection tool and reveal very weak transitions associated with optically forbidden electronic states. Experimental demonstrations of the latter capability include an application to carotenoids12 and to singlet fission in pentacene crystals.25 In Ref. 25, the detected oscillations have been assigned to excited-state coherent motion, and the detection of the dark multiexciton state turned possible due to efficient excited-state absorption. In Ref. 12, the detected oscillations have been interpreted as ground-state coherence. Excited-state coherence carries a lot of information on the populated states and their nature. In general, it can be of mixed electron-vibrational character and may require much effort to interpret. Also, the excited-state coherent motion is expected to experience fast dephasing and can be difficult to capture experimentally. In contrast, ground-state coherence, if created, is robust and long-lived, and, in the most cases, it has a very clear origin, namely, excitation of vibrational modes in the electronic ground state. Further, ground-state motion is decoupled from ultrafast coherent dynamics in the excitedstate (conical intersections involving the ground electronic state represent an exception and are out of scope of the present work). In this study, we explore the idea to use ground-state coherence for detection of dark excited states. For this purpose, we consider several model systems that exhibit ultrafast dynamics governed by avoided crossings and conical intersections between optically bright and optically dark electronic states. Note that although several extensive theoretical studies of 2DES signals of this class of systems have been reported,16,26,27 the related beating-maps analysis is very limited.15,20,25 In this study, the respective geometry of the optically bright and optically dark excited states and their energetic location ensures ultrafast non-reversible depopulation of the bright state so that the stimulated-emission contribution to the signal is negligible. We investigate if the populated dark state can be resolved in the beating maps of 2DES provided ground-state coherent motion is created in the experiment. The presented analysis implies that excited-state contribution to the oscillatory part of the signal is negligibly weak or absent. This is the case whenever no excited-state coherence is created or if there is no efficient channel for excited-state absorption from the dark state.

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Note that the ground-state beatings themselves do not carry any information on the excited-state dynamics in general and on the dark states in particular. However, due to multiple interactions with the laser pulses, the amplitude of the beatings in the maps depends on a product of four dipoles, where each dipole represents a transition strength between a state of the electronic ground-state manifold (this can be either the ground vibrational state or a vibrationally excited state) and a state of the electronic excited-state manifold (in general, a vibronic state of mixed electronic and vibrational character). In this way, an “enhancement” of weak transitions can be achieved if stronger dipoles contribute to the product as well. In Sec. II, a more detailed explanation of the detection capabilities of 2DES beating maps is given. In Sec. III, the considered models are introduced. The results of the simulations are presented and discussed in Sec. IV, and Sec. V concludes. II. 2DES BEATING MAPS

The sensitivity of 2DES to coherent oscillations can be exploited to obtain microscopic information on the excited states populated during the experiment. For this purpose, it is most convenient to represent the signal in terms of the so called beating maps.20 The beating maps partition the oscillatory contribution to the 2D signal into several components, each component corresponds to one particular mode detected in the signal. The conventional 2DES signal depends on two frequencies: excitation frequency, ωex, and probe frequency, ωpr, as well as on the waiting time. The excitation frequency ωex reveals the transitions also accessible (but not necessarily resolved) via linear absorption. The probe frequency ωpr depends on the detection pathway: it reveals the transitions between the levels of the excited-state manifold and the levels of either the ground-state manifold (stimulated emission (SE) and ground-state bleach (GSB)) or the higher-lying excited states (excited-state absorption (ESA)). A beating map represents the amplitude of a particular oscillation detected in the signal during the waiting time as a function of ωex and ωpr. The number of possible beating maps is determined by the number of different frequencies in the oscillatory component of the signal. The larger is the number of the detected modes, the more detailed information can be obtained. The representation in terms of the beating maps is very well suited for the resolution of the individual transitions, since, as a number of experimental realizations have been demonstrated,12,14,15,23,25 broadening and overlap effects are significantly reduced as compared to the conventional 2D signal. Further, in the weak-field limit (i.e., within the applicability of perturbative description of the system-field interaction), the signal and the beating maps can be thought as a superposition of three contributions:28 GSB, SE, and ESA. The oscillatory signal component contains ground-state coherence as GSB and excited-state coherence as SE and ESA. GSB, SE, and ESA signatures in the beating maps are different and can be told apart.24 The beating maps may also provide a sensitive tool for the detection of very weak transitions associated with the essentially dark electronic states. This can be achieved due


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to the fact that the amplitude of an oscillation that arises in 2DES at ωex corresponding to a very weak transition (hardly resolved in linear signals) also depends on transition dipoles corresponding to other transitions which may happen to be strong enough to enable the spectroscopic resolution. The mechanism and manifestation of this so called intensity sharing (or intensity borrowing) differ for ground-state and excited-state coherences.24 For the ground-state contribution considered in this work, the transition dipoles between the electronically excited states and the vibrationally excited states of the ground electronic state matter, this relates the GSB contribution to the Raman spectroscopy. If the system undergoes an ultrafast relaxation to the optically dark state, the SE contribution (which is essentially fluorescence) is very short lived. GSB and ESA contributions remain at longer times and provide an opportunity to characterize the dark states. Therefore, the detection by means of the beating maps requires either excited-state coherent motion and an efficient ESA channel from the dark state or Raman type excitation of coherent motion in the ground-state and a detectable GSB contribution. ESA represents an efficient channel if the dark state is optically coupled to some higherlying states. In the recent study of singlet fission in pentacene crystals, for example, such channel is provided due to very efficient triplet-triplet ESA process from the dark multiexciton state.25 If no efficient ESA channel exists, GSB is the only remaining process once the relaxation to the dark state has occurred. This specific situation is addressed in this study of the GSB potential and capabilities in the dark-state detection. Since vibrational decoherence in the ground state is slow, GSB contribution to the beating maps is expected to be most robust. To compute the beating maps, we make use of stickspectrum expressions.22 We consider the so called nonrephasing part of the GSB contribution, since its rephasing counterpart does not provide any additional information. The explicit expressions that highlight the intensity-sharing mechanism are given in Sec. III.

III. MODELS

The considered models are constructed in a way that assures an ultrafast nonradiative decay to a dark state, via an avoided crossing or conical intersection. We limit the discussion to the simplest case of two excited electronic states: an optically bright state (|B⟩) and an optically dark state (|D⟩). Several vibrational modes are coupled to the electronic excitations. The model Hamiltonian can be represented as the sum H = Hg + He , where the ground-state Hamiltonian is Hg = |G⟩hG⟨G|

harmonic normal modes in all electronic states )  ( Nm  ) ∆i ( 1 − ωk √ k ak + ak† , hi = ωk ak† ak + 2 2 k ∈R k=1

(3)

where Nm denotes the number of vibrational modes. The second sum in Eq. (3) is over totally symmetric Raman (R) active modes only, and ∆ik denote displacements of the excited-state potentials from the equilibrium geometry of the electronic ground state; hG is determined by Eq. (3) with ∆Gk ≡ 0. To describe the system-field interaction, the linear-dipole approximation is employed, with the dipole-moment operator µ = |B⟩⟨G| + H.c.

(4)

In the stick-spectrum limit (i.e., if all relaxation and broadening mechanisms are neglected), the GSB beating maps are determined by the dipole strengths µg α between the eigenstates |g⟩ of the Hamiltonian Hg (vibrational states of the state |G⟩) and the eigenstates |α⟩ of the excitedstate Hamiltonian He , He |α⟩ = Eα |α⟩ (α and β are used hereafter to denote the eigenstates of He ). Here, we consider the nonrephasing maps and assume that only the lowest vibrational state at energy E0 is initially populated (we neglect the Boltzmann distribution of vibrational excitations at finite temperature). The amplitude of the peaks in a nonrephasing map of a ground-state mode with frequency ωg = Eg − E0 is given by a product of four dipoles, A ωex = Eα − E0,ωpr = E β − E0 = µ0α µ0β µg α µg β . (5) The corresponding rephasing map can be obtained simply by reducing the probe frequency ωpr by the mode frequency ωg . Fig. 1 shows Feynman diagrams for the rephasing and nonrephasing GSB contributions. We will consider beating maps corresponding to single vibrational excitations in the ground electronic state, i.e., the state denoted as |g⟩ corresponds to a vibrationally excited state in |G⟩, with one vibrational quantum in the considered mode. For convenience, we set E0 ≡ 0. Let us briefly discuss the intensity sharing mechanism using Eq. (5). The amplitude of diagonal peaks, i.e., of peaks at ωex = ωpr = Eα , scales as µ20α µ2g α . As compared to the linear absorption at the same energy (scales as µ20α ), a chance to improve the resolution depends on the dipole µg α . Even if this dipole is small but nonzero, the diagonal peaks will be weak,

(1)

(|G⟩ denotes the electronic ground state) and excited-state counterpart reads  He = |i⟩hi + ϵ i ⟨i| + {|B⟩VBD⟨D| + H.c.} , (2) i=B,D

where ϵ i denote the vertical excitation energies of the two excited electronic states, VBD represents the interstate coupling, and the vibrational Hamiltonians hG and hi imply the same

FIG. 1. GSB pathways of 2DES leading to a formation of a beating mode with frequency ω g ;|α⟩ and |β⟩ denote eigenstates of the He ; |0⟩ and |g ⟩ correspond to the initial state and a vibrationally excited state in the electronic ground state.


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but the mechanism of the intensity sharing can work for offdiagonal peaks, ωex , ωpr. For example, if for a state |α⟩ the product µ0α µg α is weak, but the product µ0β µg β for a state | β⟩ is strong, the off-diagonal peak at ωex = Eα and ωpr = E β can be resolved. In the linear signal only the state | β⟩ is detectable in such situation. As the supplementary material,31 we provide figures contrasting the dipole products µ20α and µ0α µ0 ′α for all the models considered below. For a class of models considered and in the absence of the interstate coupling, the transition dipoles µg α are determined by Franck-Condon overlaps of the vibrational wave functions (note that we have no Duschinsky rotation and integration is separable). Otherwise, the expansion coefficients C α B of the {v k }

excited eigenstate wavefunctions |α⟩ = |B⟩

 {v kB}

α C{v B} k

Nm 

|vkB⟩ + |D⟩

 {v kD}

k=1

α C{v D} k

Nm 

|vkD⟩ (6)

k=1

enter as well and the dipoles read µg α =

 {v kB}

α C{v B} k

Nm  ⟨vkG|vkB⟩,

peaks in the considered nonrephasing GSB maps is determined by the energies of the excited eigenstates, Eα , i.e., the peaks arise at the same energies as the transitions in the linear absorption signal. In the plots below, the transitions to the lowest 20-40 eigenstates are shown. According to Eq. (5), the beating maps are symmetric with respect to the diagonal so that the two frequency axes are equivalent. A. Two-mode models

In this section, two models are considered. The first one represents an AC model with two vibrational modes, both of them may be Raman active. The second model represents the simplest possible model of a CI between two electronic states of different symmetries.29 It comprises one totally symmetric Raman active mode as well as a non-totally symmetric (Raman inactive) coupling mode. For the two modes, we chose a relatively high frequency ω1 = 0.1 eV and a lower frequency ω2 = 0.03 eV. In the CI model, the low-frequency mode is chosen as the coupling mode.

(7)

k=1

where vki ⟩ denote harmonic basis states pertaining to an electronic state |i⟩,i = G,B,D (direct product of the states of each mode) . The overlap integrals ⟨vkG|vkB⟩ depend on the displacements ∆Bk and the vibrational quantum numbers, while the coefficients C α B are determined by the nature of interstate {v k }

coupling VBD as well as by the respective geometry of the two excited states. In the following, we study the role of these two factors by (i) varying the number of Raman active modes (i.e., modes with a nonzero displacements ∆Bk ) and (ii) considering two types of interstate coupling. For this purpose, we construct two-mode and three-mode models and address the dynamics involving ACs and CIs of electronic states. For AC, a constant interstate coupling VBD is assumed, while a linear dependence in the coupling mode is chosen for VBD in the CI models.29

IV. SIMULATIONS

For each model considered, we display the time-dependent population of the optically bright electronic state |B⟩ (instantaneous excitation limit has been assumed), the stick linear absorption spectrum, and the nonrephasing GSB beating maps of the modes which experience vibrational excitation in the electronic ground state. While the beating maps are calculated in the stick-spectrum limit, population relaxation and dephasing in the manifold of excited eigenstates |α⟩ are taken into account to simulate the decay of the |B⟩state population. Multi-level Redfield description with an Ohmic harmonic bath is employed for this purpose. The stickspectrum approximation for GSB maps should be appropriate since the dephasing and population relaxation in the ground electronic state are slow. The peaks are drawn as circles, their diameter does not represent any realistic width due to dephasing/broadening mechanisms, but simply scales with the peak intensity for illustrative purposes. The location of the

1. Two-mode AC

In the AC model, the interstate coupling is constant and set to VBD = 0.02 eV. The Raman activity of the modes depends on the value of the coordinate displacement ∆Bk . First, we consider a case when only the high-frequency mode is Raman active, ∆B1 = 0.7, while the displacement along the second mode is negligibly small, ∆B2 = 0. For the dark state, we chose B ∆D 1 = −0.5 and ∆2 = 0. The vertical excitation energies of the two states are related as ε B − ε D = 0.1 eV. The population of the |B⟩ state is shown in Fig. 2, and the parameters ensure the decay to the dark state within 100 fs. Strictly speaking, the chosen parameters make the model one-dimensional, since only the high-frequency mode participates in the dynamics. However, we keep and even discuss the states corresponding to the vibrational levels of the low-frequency mode in order to facilitate a comparison with the “truly” two-mode models considered below. The stick linear absorption signal and the beating map of the high-frequency mode are shown in Fig. 4, and the same

FIG. 2. Population of the optically bright (|B⟩) state of the considered models.


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FIG. 3. Stick linear absorption spectrum (upper graph) and the beating map of the single Raman mode (0.1 eV) of the model with no interstate coupling, VBD = 0. Blue lines correspond to the energies E α , eigenstates of He . In the linear absorption graph, the splittings corresponding to the vibrational frequencies of the model are indicated. The map is diagonally symmetric and the two axes corresponding to the excitation and probe frequencies are equivalent. The arrows point in the direction in which the energy increases; the lowest state has the energy E 1, also E 2 and E 3 are marked.

observables for the uncoupled case (VBD = 0) are depicted in Fig. 3. To help the eye, blue dotted lines are drawn at the energies of the eigenstates Eα . In the absence of the interstate coupling (Fig. 3), the model represents a harmonic oscillator with one mode displaced in the excited state and the eigenstates correspond to the vibrational levels in the two electronic states. The three lowest states are the excitations of the low-frequency mode (0.03 eV) in the |D⟩ state; at higher energies, the states density increases due to the overlap of the vibrational manifolds of the two electronic states (with two modes in each manifold). The nonzero transition dipoles are observed, as expected, for the ground vibrational state and the state with one vibrational quantum in the high-frequency mode of the optically bright electronic state. Vibrationally excited states of the lowfrequency mode are not detected due to ∆B2 = 0. The beating map of the uncoupled system (displaced oscillator) reflects the Raman activity of the 0.1 eV mode.

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FIG. 4. Same as Fig. 3 but for VBD = 0.02 eV (AC model with one Raman mode).

Note that the off-diagonal peaks are negative. The sign is not determined in the experimental beating maps (since only the amplitude of the oscillations is usually tracked), but we keep it hereafter for illustrative purposes. In contrast to the linear absorption and to the diagonal peaks which both depend on the dipoles squared, the off-diagonal peaks depend on the product of four different dipoles. The product is negative if one or three of these dipoles are negative. For the considered uncoupled model, this happens if the corresponding overlap integrals are negative. In the AC and Cl case, see Eqs. (8)–(10) below, also the expansion coefficients contribute to the final sign of the dipoles and of the dipole product. We drive the attention to the fact that the same dipole products contribute to the GSB component of the conventional 2D PE spectroscopy and may lead to negative features in the signals.30 Therefore, the often implied assumption that only ESA contribution leads to negative signals fails for vibronic 2DES. Further, similar coupling mechanisms and dipole relations may make ESA contribution positive. The argument is based on the harmonic approximation and the assumption that the normal modes are not rotated in different electronic states (Duschinsky matrix is unity). However, it has best chances to hold in a more realistic case where anharmonicity and Duschinsky rotation are present.


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The intensity of the lines in the linear signal of the coupled system (Fig. 4) is determined by the dipoles µ0α =

 v1B

α G B C{v B,0B}⟨01 |v1 ⟩ 1

2

(8)

with the sum running over vibrational excitations of the Raman G mode only; the initial state |g⟩ = |0G 1 ⟩|02 ⟩ has been assumed G B (the integral ⟨02 |02 ⟩ is unity; this indicates the redundancy of the low-frequency mode in the AC model in case ∆B2 = ∆D 2 = 0). In the low-energy region, the eigenstates remain almost unperturbed by the coupling VBD. Nevertheless, the lowest eigenstate (E1) gets a weak dipole moment due to admixture of the |B⟩ state (but it is hardly detectable in the linear signal). At higher energies, the simple vibronic progression of the Raman mode in the uncoupled case splits for nonzero VBD into pairs of states of mixed electronic and vibrational character. The detected mixing occurs between the ground vibrational state of the |B⟩ state and the vibrationally excited state (with one vibrational quantum in the high-frequency mode) of the |D⟩ state as well as between the vibrationally excited state of the |B⟩ state (one vibrational quantum in the high-frequency mode) and one of the close-lying states of the |D⟩ character (two vibrational quanta in the high-frequency mode). The diagonal peaks in the beating map of the AC model scale as µ20α µ2g α , where µ2g α is determined by Eq. (8) with G ⟨0G 1 | replaced by ⟨11 | (the vibrationally excited state is |g⟩ G G = |11 ⟩|02 ⟩). The possibility to improve the resolution using the diagonal peaks is therefore determined by the relation G between the zero-zero overlap integral ⟨0G 1 |01 ⟩ and the 1-0 G G overlap integral ⟨11 |01 ⟩. On the other hand, the off-diagonal peaks at ωex = Eα and ωpr = E β represent an “average” of the two diagonal peaks at ωex = ωpr = Eα and ωex = ωpr = E β . This is what makes the intensity borrowing possible and the off-diagonal peaks can become very good sensors of weak transitions (provided that the other involved transitions are strong). The beating map in Fig. 4 highlights the splittings in the higher-energy region but does not improve the resolution of the lowest eigenstate (E1) the system relaxes to. Interestingly, although the lowest eigenstate and the highest detectable eigenstate considered here have very similar transition dipoles in the linear signal, the higher state is much better detected in the beating map. The reason is a better overlap with the corresponding vibrationally excited state in the electronic ground state, leading to | µg α | > | µ0α | for the considered transition (see the supplementary material31). This observation proves the idea that the resolution of weak transitions may be considerably improved in the GSB beating maps. However, only the low-lying states with very weak transition dipoles can be unambiguously identified experimentally as dark states, and these are also the states the system relaxes to and one searches to characterize. Therefore, in the following, we focus on the resolution of the low-lying dark states only. If we make the low-frequency mode Raman active as well and assume equal displacements for the both modes, D ∆B1 = ∆B2 = 0.7 and ∆D 1 = ∆2 = −0.5, in the absence of the interstate coupling, linear absorption signal shows vibronic progressions for the both modes, Fig. 5. The beating maps

FIG. 5. Stick linear absorption spectrum (upper graph, the splittings corresponding to the modes’ frequencies are indicated), and the beating maps of the model with two Raman modes and no interstate coupling, VBD = 0. The map of the mode of 0.03 eV is shown in the middle graph, and the counterpart for the mode of 0.1 eV is shown in the bottom graph.

in the same figure reflect the Raman activity of the modes according to Eq. (5) with the dipoles determined by the corresponding overlap integrals of vibrational wave functions. The constant coupling VBD, when turned on (Fig. 6), makes all the low-lying transitions weakly allowed, including the essentially dark states that involve vibrational excitations in the low-frequency mode (E2,E3). Due to nonzero ∆B2, Eq. (8) for the transition dipoles from the ground state must be modified


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vibrationally excited states of essentially |D⟩ character, have chances to be resolved in the beating map of the low-frequency mode (Fig. 6). The transitions to these states (the second and the third lowest eigenstates) attain very weak intensity in the linear signal, but the dipoles between them and the vibrationally excited ground state µeg (with one quantum in the 0.03 eV mode) are strong enough to enable their detection via cross peaks in the beating maps (the expression for the G dipoles is as Eq. (9) with ⟨0G 2 | replaced by ⟨12 |). We observe, therefore, weakly allowed (due to vibronic interactions) electronic transitions via excitation of low-frequency vibrational motion. As for the beating map of the high-frequency mode shown Fig. 6, its strongest peaks are similar to those shown in Fig. 4 and simply indicate the mode Raman activity. At higher energies, a considerable resolution improvement is observed for weak transitions as compared to the linear signal, but the relation of these transitions to the dark state would not be straightforward experimentally. A comparison of beating maps in Fig. 5 (no coupling) and Fig. 6 supports the idea that these are the peaks in the low-energy region, which provide an experimentally detectable proof of dark-state existence. 2. Two-mode CI

Section IV A 1 suggests that the presence of the lowfrequency Raman modes can potentially improve the resolution of very weak transitions associated with the dark electronic state in the AC model. Before a further exploration of this idea (Sec. IV B below), we address the role of the second factor which can influence the maps: the nature of electronic coupling. For this purpose, we consider the CI model and turn the low-frequency mode into the so called coupling mode. It means that this mode cannot be Raman active and we put ∆B2 = ∆D 2 = 0. The high-frequency mode keeps its excitedstate geometry as above, ∆B1 = 0.7 and ∆D 1 = −0.5. We are, therefore, back to the first example considered (AC model with one Raman mode), but the interstate coupling has now a linear dependence on the low-frequency vibrational coordinate; we ( ) † √ eV a2 + a . chose VBD = 0.02 2 2 The linear absorption and the beating map of the highfrequency mode are shown in Fig. 7. The transition intensities can be rationalized using Eq. (8); the coefficients C α B differ, {v k }

FIG. 6. Same as Fig. 5 but for VBD = 0.02 eV (AC with two Raman modes).

to include an additional sum over the excitations in the lowfrequency mode, v2B,  α G B G B C{v (9) µ0α = B, v B}⟨01 |v1 ⟩⟨02 |v2 ⟩. v1B, v2B

1

2

As a consequence of the Raman activity of the second mode, the intensity redistribution in the spectral region of the strongest transition involves more states as compared to Fig. 4, since the excitations in the low-frequency mode contribute to the mixing as well. The relaxation to the dark state occurs within about 200 fs (Fig. 2). The so populated lowest eigenstates, in particular

of course, from the AC case. Namely, the interstate coupling provides only two very weak additional transitions in the linear signal as compared to the uncoupled case. In contrast to the AC model, the lowest eigenstate (E1) remains dark and is not detected. The interstate coupling is such that only the dark states with one vibrational quantum in the lowfrequency coupling mode acquire some dipole strength due to the mixing with the |B⟩ state. This is quite different to the AC models considered above and suggests that the nature of the interstate coupling in polyatomic molecules is, in principle, spectroscopically distinguishable. The relaxation to the dark state occurs at a longer time scale as compared to the AC model but is still quite fast (300 fs, see Fig. 2). The beating map shows extremely weak features associated with the dark state and does not provide any additional information as compared to the linear signal. Therefore, the dark states populated via CI nature of the


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have the same Raman activity ∆B1 = ∆B2 = ∆B3 = 0.7 and equal D D displacements in the dark state ∆D 1 = ∆2 = ∆3 = −0.5. The interstate coupling and the vertical excitation energies are kept as in Sec. IV A 1. The resulting eigenstates produce population dynamics shown in Fig. 2. The |B⟩ state decays within 200 fs; the dynamics is very similar to that of the AC model with two Raman modes. The linear absorption signal and the beating maps are shown in Fig. 8. The additional vibrational mode per state increases further the state density, and the mixing and the intensity redistribution are very efficient: many states of predominantly |D⟩ character gain some dipole strengths. The line intensities in the linear signal are determined by µ0α =

 v1B, v2B, v3B

α G B G B G B C{v B, v B, v B}⟨01 |v1 ⟩⟨02 |v2 ⟩⟨03 |v3 ⟩. 1

2

3

(10)

nonradiative coupling have worse (as compared to AC type of coupling) chances to be detected in the beating maps if only ground-state coherence contributes to the oscillatory component of the signal. The low-frequency coupling mode is hardly excited upon electronic transitions. However, a close look at the dipoles µg α of the model suggests that a weak excitation of a mode with a twice vibrational frequency of the coupling mode is possible in the ground electronic state. This issue deserves a more profound exploration; here, we only provide a short explanation of the effect in the supplementary material.31

The peaks in the beating map of the kth mode depend also on µg α , which are determined by Eq. (10) with ⟨0Gk | replaced by ⟨1Gk | (see the supplementary material31 for graphical representation of µ0α µg α products). The resolution of the low-lying states of mostly |D⟩ character in the beating maps improves further as compared to the AC model with one and two Raman modes (Figs. 4 and 6), especially in the maps of the low-frequency modes. Each map is most sensitive to the eigenstates involving vibrational excitations in the considered mode: the second eigenstate (E2) in the map of the 0.03 eV mode and the third eigenstate (E3) in the map of the 0.05 eV mode. The diagonal peaks corresponding to these transitions are quite weak, but the off diagonal peaks gain much more intensity than the lines in the linear absorption signal. In this way, coherent vibrational motion induced in the ground state helps to detect the dark-state levels corresponding to vibrational excitation in the corresponding mode, i.e., the Raman induced vibrational excitation enables the detection of very weak electronic transitions. In the spectral region where strong mixing occurs (in the vicinity of AC), the peak structure of the low-frequency beating maps becomes very dense and their experimental counterparts would be difficult to interpret. The map of the high-frequency mode is less sensitive for this mixing and keeps its four strong peaks also found in Figs. 4 and 6, although the relative intensities are slightly different. The resolution of the lowest eigenstates remains poor in this map. We conclude, therefore, that for the AC type of interstate coupling, these are the low-frequency Raman modes (if present in the considered system) that can serve as a probe of very weak transitions via ground-state coherence.

B. Three-mode models

2. CI with two Raman modes

In order to further explore the role of low-frequency Raman modes in the detection of the dark states, we extend the considered models and include a third mode with vibrational frequency ω3 = 0.05 eV.

Let us now check if an additional Raman mode can improve the resolution of the dark state populated via a CI. We assign the mode 2 (ω2 = 0.03 eV) to the coupling mode, and modes 1 (ω1 = 0.1 eV) and 3 (ω3 = 0.05 eV) are Raman modes D with ∆B1 = ∆B3 = 0.7 and ∆D 1 = ∆3 = −0.5. Fig. 9 shows the signals obtained, and the population of the |B⟩ state is depicted in Fig. 2. The |B⟩-state population decays faster as compared to the two-mode CI model, but its timescale is a bit longer than for the considered AC models. The state density in Fig. 9 is

FIG. 7. Stick linear absorption spectrum (upper graph) and the beating map of the Raman mode (0.1 eV) for the two-mode CI model.

1. AC with three Raman modes

The model of this section is as in Sec. IV A 1 but with the third mode added. For simplicity, all the modes are assumed to


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FIG. 9. Three-mode CI model: stick linear absorption spectrum (upper graph) and the beating maps of the two Raman modes (0.05 eV, middle, and 0.1 eV, bottom).

FIG. 8. AC model with three Raman modes: stick linear absorption spectrum (upper graph) and the beating maps of the three Raman modes (0.03 eV, 0.05 eV, and 0.1 eV, from top to bottom).

similar to the case of the three-mode AC (Fig. 8), but interstate mixing of the intensity redistribution is a way less efficient. As compared to Eq. (10), apart from the different coefficients, the expression for the transition dipoles now misses the sum

over the vibrational excitations in the coupling mode. A closer look at the linear signal reveals that only the states with one vibrational quantum in the coupling mode (and any number of vibrational quanta in the Raman modes) experience “dipole borrowing” from the |B⟩ state. The only detectable state in the low-energy region is, as in the case of CI with one Raman mode (Sec. IV A 2, Fig. 7), the dark state with one vibrational quantum in the coupling mode and a weak admixture from the |B⟩ state. It is not resolved in


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the map of the low-frequency mode (0.05 eV) and only weakly resolved in map of the high-frequency mode, hardly better as compared to Fig. 7. We conclude that the resolution of the dark states in case of CI cannot be achieved in the same manner as for AC type of coupling (the low-frequency beating maps are insensitive to dark states in this case) and alternative strategies must be explored. Very probably, several coupling modes can improve the situation since more states in the excitedstate manifold are expected to acquire a nonzero transition strength form the ground state. Exploration of these effects is out of scope of the present study and a subject of ongoing research.

V. CONCLUSIONS

We have considered a class of systems where the dark electronic state is located lower than the optically accessible state and a fast nonradiative transition to the dark state via AC or CI is energetically favorable. We have explored 2DES beating maps as the dark-state detection tool. Since the fluorescence is very short-lived and the SE contribution to the 2DES signal is weak in the considered systems, ESA and GSB are the two remaining contributions to the signal and the maps. ESA can be quite strong if the dark state is optically coupled to higher-lying electronic states; it can be resolved in the beating maps if long-lived excited-state coherences are created upon the laser excitation. Otherwise, the only chance to resolve dark states is to make use of ground-state coherence (GSB contribution), and this specific situation has been studied here. We have employed stick-spectrum approximation. It involves neglect of relaxation and dephasing processes as well as the limit of infinitely short pulses. The approximation is expected to be justified for the ground-state coherence since it is robust at the considered timescales and shall not be significantly distorted by dephasing and pulse overlap effects. As for the finite pulse durations, these would, of course, influence the spectral resolution. We have chosen the nonrephasing GSB maps as the observable, since the rephasing counterpart does not provide any relevant addition information. The peaks in the nonrephasing GSB beating maps are located at the system absorption frequencies and their location does not depend on the beating-mode frequency.24 This makes the identification of the energies of the dark-state transitions especially reliable. For the considered systems, all the transitions detected in the nonrephasing GSB beating maps at energies below the absorption maximum can be unambiguously identified as essentially dark states. Therefore, we have concentrated on the resolution of the transitions to the lowest eigenstates with very weak dipoles. GSB in beating maps reflects the most robust coherence (ground-state vibrational motion), but its performance for the dark-state resolution has been found satisfactory only in case of AC and in the presence of several Raman active mode. In particular, our findings suggest that the coherent motion in the electronic ground state can serve as a probe of the dark excited states if low-frequency Raman modes are present in the

J. Chem. Phys. 142, 212452 (2015)

system. The Raman activity is necessary to excite the modes in the ground state and to detect the corresponding coherence. The low-frequency modes are the best candidates since they also constitute the low-lying eigenstates of the essentially dark electronic character the system relaxes to and we aim to detect. These states are getting only a weak admixture from the optically bright state and lie well below the spectral region where the two electronic states are very close and the interstate coupling forms a very dense manifold of states of strongly mixed electronic and vibrational character. The resolution of dark states accessible via CI appears very challenging. The reason is that the admixture of the bright-state character occurs only for the dark states involving excitations (specifically, one vibrational quantum for the considered form of the interstate coupling) in the coupling mode. However, the coupling mode is not Raman active (due to symmetry reasons) and cannot be directly excited in the ground state (see supplementary material31 for the discussion of the possible excitation of the coupling mode in the considered models). Therefore, the only way to detect the dark state via GSB beating map is to use other modes (with a sufficient Raman activity) as probes. We have not observed any evidence that a larger number of Raman modes can improve the dark-state resolution in case of CI. Eventually, some improvement can be achieved if several coupling modes are present. Exploration of this idea is out of scope of the present study and a subject of ongoing research. At the current stage, we conclude that for CI of electronic states of different symmetries, ESA (if present) provides a more reliable channel for the detection of dark states in 2DES beating maps. We note that the Duschinsky mixing was neglected in all models considered. If taken into account, it can provide additional possibilities for “dipole borrowing” and eventually improve the resolution of weak transitions. The experimental realization of the beating map analysis for molecular systems with very short-lived fluorescence (low fluorescence yield) due to decay to lower-lying dark states can be facilitated if the following suggestions are taken into account (the necessary condition is, of course, the existence of oscillatory signatures in 2DES signal). The rephasing and nonrephasing beating maps should be recorded separately; this can help to tell the difference between ground-state and excited-state coherent motions. The excited-state contribution, if revealed, is dominated by ESA and identification of ESA channels helps the analysis. If no excited-state coherence is created or no efficient ESA pathway exists, Raman active vibrational modes (which are usually well known) can serve, at least in the case of AC type of interstate coupling, as darkstate probes, and the low-frequency modes are expected to be most sensitive. The GSB dominated beating maps (we believe that the nonrephasing maps are best suited for this purpose) can be superimposed with the linear absorption signal in order to identify transitions not resolved in the linear signal and, therefore, attributed to dark states. Significant differences in the patterns of beating maps corresponding to AC and CI models suggest that the beatingmaps analysis can be also potentially used for the experimental characterization of the nature of the interstate electronic coupling.


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M. Jonas, Annu. Rev. Phys. Chem. 54, 425 (2003). M. Branczyk, D. B. Turner, and G. D. Scholes, Annalen der Physik 526, 31 (2014). 3C. Y. Wong and G. D. Scholes, J. Lumin. 131, 366 (2011). 4S. T. Cundiff, J. Opt. Soc. Am. B 29, A69 (2012). 5J. R. Caram, H. Zheng, P. D. Dahlberg, B. S. Rolczynski, G. B. Griffin, A. F. Fidler, D. S. Dolzhnikov, D. V. Talapin, and G. S. Engel, J. Phys. Chem. Lett. 5, 196 (2014). 6E. Cadett, R. D. Pensack, B. Mahler, and G. D. Scholes, J. Opt. Soc. Am. B 29, A69 (2012). 7G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Man˘ cal, Y.-C. Cheng, R. Blankenship, and G. R. Fleming, Nature 446, 782 (2007). 8E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer, and G. D. Scholes, Nature 463, 644 (2010). 9G. Panitchayangkoon, D. Hayes, K. A. Fransted, J. R. Caram, E. Harel, J. Wen, R. E. Blankenship, and G. S. Engel, Proc. Natl. Acad. Sci. U. S. A. 107, 12766 (2010). 10E. Harel and G. S. Engel, Proc. Natl. Acad. Sci. U. S. A. 109, 706 (2012). 11S. Westenhoff, D. Palecek, P. Edlund, P. Smith, and D. Zigmantas, J. Am. Chem. Soc. 134, 16484 (2012). 12E. E. Ostroumov, R. M. Mulvaney, R. J. Cogdell, and G. D. Scholes, Science 340, 52 (2013). 13A. F. Fidler, V. P. Singh, P. D. Long, P. D. Dahlberg, and G. S. Engel, J. Phys. Chem. Lett. 4, 1404 (2013). 14M. Ferretti, V. I. Novoderezhkin, E. Romero, R. Augulis, A. Pandit, D. Zigmantas, and R. van Grondelle, Phys. Chem. Chem. Phys. 16, 9930 (2014). 15F. D. Fuller, J. Pan, A. Gelzinis, V. Butkus, S. S. Senlik, D. E. Wilcox, C. F. Yocum, L. Valkunas, D. Abramavicius, and J. P. Ogilvie, Nat. Chem. 6, 706 (2014). 16D. Egorova, Chem. Phys. 347, 166 (2008). 17J. R. Caram, A. F. Fidler, and G. S. Engel, J. Chem. Phys. 137, 024507 (2012). 2A.

J. Chem. Phys. 142, 212452 (2015) 18N. Christensson, H. F. Kauffmann, T. Pullertis, and T. Man˘ cal, J. Phys. Chem.

B 116, 7449 (2012). Butkus, D. Zigmantas, L. Valkunas, and D. Abramavicius, Chem. Phys. Lett. 545, 40 (2012). 20V. Butkus, D. Zigmantas, D. Abramavicius, and L. Valkunas, Chem. Phys. Lett. 587, 93 (2013). 21V. Tiwari, W. K. Peters, and D. M. Jonas, Proc. Natl. Acad. Sci. U. S. A. 110, 1203 (2013). 22D. Egorova, J. Chem. Phys. 140, 034314 (2014). 23E. E. Ostroumov, R. M. Mulvaney, R. J. Cogdell, and G. D. Scholes, J. Phys. Chem. B 117, 11349 (2013). 24D. Egorova, J. Phys. Chem. A 118, 10259–10267 (2014). 25A. A. Bakulin, S. E. Morgan, T. B. Kehoe, M. W. B. Wilson, A. Chin, D. Zigmantas, D. Egorova, and A. Rao, “Real-time observation of multiexcitonic states and ultrafast singlet fission using coherent 2D electronic spectroscopy,” Nature Chemistry (submitted); A. A. Bakulin et al., Ultrafast Phenomena XIX (Springer International Publishing, 2015), pp. 226–229. 26J. Krcmar, M. F. Gelin, D. Egorova, and W. Domcke, J. Phys. B: At., Mol. Opt. Phys. 47, 124019 (2014). 27K. A. Kitney-Hayes, A. A. Ferro, V. Tiwari, and D. M. Jonas, J. Chem. Phys. 140, 124312 (2014). 28S. Mukamel, Principles of Nonlinear Optical Spectroscopy (University Press, Oxford, 1995). 29H. Köppel, W. Domcke, and L. S. Cederbaum, Adv. Chem. Phys. 57, 59 (1984). 30A. Schubert, “Kohärente und dissipative wellenpaketdynamik und zeitaufgelöste spektroskopie: Von zweiatomigen molekülen zu molekularen aggregaten,” Ph.D. thesis (Technische Universitaet Muenchen, 2012). 31See supplementary material at http://dx.doi.org/10.1063/1.4921636 for the 2 µ 2 and following: (i) a graphical comparison of the dipole products µ 0α 0α µ 0α µ gα for all the systems considered; (ii) a brief description of the excitation of the coupling mode in the CI models. 19V.


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