Nonlinear Spectroscopic Theory of Displaced Harmonic Oscillators

Article pubs.acs.org/JPCA

Nonlinear Spectroscopic Theory of Displaced Harmonic Oscillators with Differing Curvatures: A Correlation Function Approach Andrew F. Fidler and Gregory S. Engel* Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, United States S Supporting Information *

ABSTRACT: We present a theory for a bath model in which we approximate the adiabatic nuclear potential surfaces on the ground and excited electronic states by displaced harmonic oscillators that differ in curvature. Calculations of the linear and third-order optical response functions employ an effective shorttime approximation coupled with the cumulant expansion. In general, all orders of correlation contribute to the optical response, indicating that the solvation process cannot be described as Gaussian within the model. Calculations of the linear absorption and fluorescence spectra resulting from the theory reveal a stronger temperature dependence of the Stokes shift along with a general asymmetry between absorption and fluorescence line shapes, resulting purely from the difference in the phonon side band. We discuss strategies for controlling spectral tuning and energy-transfer dynamics through the manipulation of the excited-state and ground-state curvature. Calculations of the nonlinear response also provide insights into the dynamics of the system−bath interactions and reveal that multidimensional spectroscopies are sensitive to a difference in curvature between the ground- and excited-state adiabatic surfaces. This extension allows for the elucidation of short-time dynamics of dephasing that are accessible in nonlinear spectroscopic methods.

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INTRODUCTION Solvation processes control many chemical reactions; solvation tunes the potential energy surfaces to stabilize transition states, acts as a source of energy to surmount potential energy barriers, and provides an energy sink to stabilize products.1−3 Chemical reactions can depend critically on the solvent’s composition and concentration, and the choice of solvent often changes the outcome of a reaction.4 Optical spectroscopy probes the solvent−solute interactions by interrogating the local electronic environment of chemical species. The nuclear motions of the solvent and chromophore intrinsically lead to a broad absorption spectrum; this broadening can frustrate attempts to extract the underlying dynamics controlling the solvation process.1 Precise knowledge of the collective motions of the nuclei in the solvation process could provide insights into the chemical control of solvation. Solvation processes occur on time scales ranging from a few femtoseconds to several picoseconds.5 The excited electronic state of small molecules combined with ultrafast time-resolved spectroscopy can provide direct measurements of solvation processes.6−9 In these experiments, the chromophore is dissolved in a solvent and resonantly excited with a short laser pulse that promotes the chromophore to an electronic excited state. Because the excitation process is faster than the motion of the nuclei, the solvent is prepared in a nonequilibrium nuclear configuration. This nonequilibrium nuclear configuration then thermalizes back to an equilibrium configuration, which is measured by subsequent light−matter interactions. If the curvature of the electronic excited-state potential energy surface © XXXX American Chemical Society

is identical to that of the ground state, the solvation process can be treated as a linear perturbation away from equilibrium. In this scenario, the process of solvation in the excited electronic state is directly analogous to the solvation process in the ground electronic state. Furthermore, the equilibrium energy gap correlation function completely describes the dynamics of the process within this limit.1 Many spectroscopic techniques have been developed to measure the energy gap correlation function, in both the time and frequency domains. Fluorescence line narrowing and the closely related hole burning spectroscopies rely on clear separation of time scales in the correlation function, whereby slower modes are effectively static on the time scale of the measurement.10,11 In these experiments, a narrow band pulse resonantly excites a subensemble within an inhomogeneously broadened spectral line. A subsequent pulse interrogates the resulting distribution of frequencies present. This distribution of frequencies directly reports on the spectrum of modes to which the electronic state is coupled. These modes are typically referred to as the spectral density, which is related to the Fourier transform of the correlation function. Hence, these experiments can be thought of as frequency domain measurements. Due to the need for a separation of time scales, these experiments Special Issue: Oka Festschrift: Celebrating 45 Years of Astrochemistry Received: November 28, 2012 Revised: January 30, 2013

A

dx.doi.org/10.1021/jp311713x | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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ence of the nuclear coordinate can arise when the excited statecurvature differs from the ground-state curvature. The linear polarization for a general system with linear and quadratic coupling has been solved exactly.33 These results have been shown to be consistent with observations of the temperature dependence of the width and shift of the absorbance maximum in several systems.33,34 Similar extensions to electron-transfer theory, where the initial and final states were coupled to the bath with differing frequencies, have also been explored.35 Certain aspects of linear spectroscopic measurements are also at odds with the MBO model. The MBO theory predicts a weak shift of the peak in the absorption spectrum with temperature, while the addition of a quadratic coordinate dependence has been shown to produce a more pronounced temperature dependence of the shift.33,36 The MBO theory also predicts that the absorption and fluorescence spectra will be mirror images, assuming that the nuclear coordinates have fully equilibrated prior to fluorescence. Further, the mean absorbance frequency and mean emission frequency are independent of changes in temperature. While these results are consistent with many chromophores, several notable exceptions exist, including cyanine dyes,37 oligo(para-phenylene)s,38 and fluorescent proteins.39 In this article, we show that the introduction of a difference in curvature between the ground and excited states is consistent with these observations. Moreover, within this model, the dynamics are no longer Gaussian, and higher-order correlation functions do not necessarily vanish or factor, as has been realized in previous studies.31,33 In the first section, we define the Hamiltonian for a single chromophore, and the calculations for the second- and third-order correlation functions are presented. The next section presents the derivation of the linear and third-order optical response functions expanded up to third-order. Finally, we present calculations of a chromophore strongly coupled to a single high-frequency oscillator as well as a continuum of low-frequency modes to demonstrate the general results of the theory.

necessitate cryogenic temperatures. For multichromophoric systems, only the lowest energy state can be effectively interrogated due to energy-transfer processes. These issues can be circumvented with the photon echo technique, which interrogates the system in the time domain to directly measure the equilibrium time correlation function.12,13 Photon echo experiments rephase inhomogeneity to recover the homogeneous line width.14,15 Various aspects of the detected signal have been shown to directly report on the frequency−frequency correlation function. For example, the maximum of the emitted signal in the coherence time as a function of waiting time (peak shift) reveals the frequency−frequency correlation function.12,16 This method is applicable at all temperatures and for any number of chromophores. A particularly successful theoretical solvation model developed in parallel to these experiments is the multimode Brownian oscillator model (MBO).17 Fundamentally, this theory can be conceptualized as a continuum of harmonic oscillators that are displaced relative to each other on the ground- and excited-state potential energy surfaces. The difference between the excited state and ground state is then linearly related to the nuclear coordinates, and the system can be treated in terms of linear response theory. Conveniently, within this model, the frequency−frequency correlation function completely describes the dynamics of the reduced density matrix. More formally, all odd orders of the energy gap correlation function vanish, and the even orders factor into sums of products of the second-order correlation function, as shown using Wick’s theorem.18 Specifying the spectrum of modes and their coupling strength then completely defines the system−bath interaction. The introduction of new nonlinear spectroscopic methods that are more sensitive to the dynamics of the bath have begun to show the shortcomings of this model. Transient two-dimensional (2D) vibrational spectroscopy has revealed that the spectral diffusion process is frequency-dependent in some systems, meaning that underneath of the broad absorption spectrum, different members of the ensemble undergo different dynamical fluctuations.19 2D Raman measurements directly reveal the anharmonic nature of the solvation process20,21 and have provided insight into dephasing dynamics22 and collective solvation motion in hydrogen bonding environments.23 More recent experiments have been designed to monitor the lowfrequency dependence of the solvent response induced following excitation.24,25 These experiments have allowed for a more detailed view of the solvent response far from equilibrium, revealing that the spectrum of modes changes as it approaches equilibrium. Higher-order three-dimensional (3D) resonant experiments in the infrared26,27 and visible28 have also revealed line shape metrics that are indicative of nonlinear solvation processes. Line shape effects in some 2D experiments have also been attributed to higher-order solvation effects.29 Several theoretical studies have looked to generalizations of the MBO theory by the inclusion of anharmonicity or a change of the excited-state curvature. The effects of anharmonicity can be included through perturbative methods by expressing the anharmonic oscillator as a sum of harmonic oscillators.30 These results show that the introduction of anharmonicity yields little difference in the linear optical properties but that nonlinear response could provide a more detailed knowledge of the underlying dynamics. There have also been numerous studies of the temperature dependence of the absorption line shape with the addition of a quadratic dependence on the nuclear coordinates for impurities in crystals.31,32 A quadratic depend-

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SOLVATION THEORY We begin by defining the Hamiltonian for a chromophore with only two electronic states coupled to a collection of harmonic oscillators within the adiabatic approximation. H = |g⟩Hg⟨g| + |e⟩He⟨e|

(1)

Here, we assume that the nuclear potential surfaces are well described as harmonic and that the excited energy surface is allowed to differ in curvature, that is, the frequency of the oscillator may be different on the excited state compared to that on the ground state. Hg =

∑ j

pj2 2μgj

He = ℏωeg° +

+

∑ j

1 μ ωg2jqj2 2 gj pj2 2μej

+

1 μ ωeg2 (qj + dj)2 2 ej

(2)

(3)

Here, for the jth mode, pj is the momentum operator, qj is the position operator, μgj (μej) is the reduced mass of the ground (excited) state, ωgj (ωej) is the frequency of the oscillator on the ground (excited) state, dj is the displacement of the origin on the excited-state potential, and ωeg ° is the energy difference between B



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is given by ρg = e−Hg/kbT/⟨e−Hg/kbT⟩, and the thermalized energy gap ℏωeg = ⟨(He − Hg)ρg⟩ can be calculated using operator techniques41 detailed in the Supporting Information.

the minima of the two electronic states. The salient features of the model are visualized in Figure 1 for a single mode.

ℏωeg = ℏωeg° +

⎡ D2 β 2 ω

∑ ℏ⎢⎢ j

+ ωej

(1 − βj2)2 4βj2

⎣

j j

2

ej

+

ωej − ωgj 2

⎛ (1 + β 4 ) ⎞⎤ j ⎟⎥ + n(ωgj)⎜⎜ωej − ω gj⎟ 2 β 2 ⎝ ⎠⎥⎦ j

(4)

Here, n(ωgj) = [eℏωgj/kbT − 1]−1 is the Bose−Einstein thermal distribution. If the line shape is asymmetric, the absorbance peak will be temperature-dependent in the MBO model though the mean of the absorbance will remain temperature-independent. Unlike in MBO theory, the thermalized energy gap, or, differently defined, the mean of the absorption spectra, is temperature-dependent. The explicit temperature dependence found in our generalized theory arises from changes in the Franck−Condon factors as higher vibrational states are thermally activated. We will now introduce an operator whose value is the deviation of the energy gap between electronic states from the thermalized average.

Figure 1. Model of two harmonic potential energy surfaces that differ in curvature for a particular mode qj and are displaced a distance dj. The frequency of the ground oscillator, ωgj, differs from that of the excitedstate oscillator, ωej. Absorption of a photon causes the nuclear coordinates to populate a nonequilibrium configuration on the excited-state potential energy surface, generating a nuclear wavepacket on the excited-state surface. The solvation reorganization energy upon ° for a single mode and can physically excitation is given by λaj = ωeg − ωeg be interpreted as the excess energy of the population’s vibrational motion during the excitation process.

⎡ ω − β 2ω βj2ωej − ωgj ej j gj 2 ⎢ U = ∑ℏ Pj + Q j2 + ωejDjβj2Q j 2 ⎢ 2 2 β j ⎣ j (1 + 2n(ωgj))(2βj2ωgj − βj4 ωej − ωej) ⎤ ⎥ + ⎥⎦ 4βj2

(5)

In addition to the linear dependence on the nuclear coordinates, a quadratic dependence on the nuclear coordinates and momentum is now found. If the effective mass does not change, the quadratic momentum dependence vanishes. The time dependence of this operator will be calculated with respect to the ground-state nuclear Hamiltonian.

These harmonic modes may represent normal modes and collective motion of either the solvent or intramolecular vibrations of the solute. From the reduced theoretical standpoint presented here, the effects of the solute and solvent nuclear motion are indistinguishable. For most small molecules, the electronic transition is typically associated with a conjugated bond system, and thus, these nuclear modes would need to be present close to these systems to greatly influence the optical response. The normal modes on the excited state and ground state are not necessarily identical, and such an effect could be treated by a change in the effective mass of the normal mode. A more proper treatment would take this into account through the Duschinsky matrix,40 which relates the change in the normal modes between the ground-state and excited-state potential energy surfaces. For the sake of generality, we will allow the effective mass to change between the ground- and excited-state potential. Specification of the displacement, ground-state frequency, excited-state frequency, and ratio of the reduced masses completely determines the model. Physically, the model can be conceptualized as the ground state and excited state having different spectral densities. It is convenient to move to a dimensionless coordinate system with the dimensionless momentum Pj = (ℏμgjωgj)−1/2pj, dimensionless coordinate Qj = (μgjωgj/ℏ)1/2qj, and dimensionless displacement Dj = (μgjωgj/ ℏ)1/2dj. Note that the dimensionless units are all defined in terms of the ground-state potential surface parameters. It will also be useful to introduce the following dimensionless parameter βj = (μejωej/μgjωgj)1/2, relating the difference between the groundstate and excited-state potentials. The MBO theory is recovered in the limit of βj → 1 in all of the expressions derived in this paper. Assuming that the environmental degrees of freedom are in thermal equilibrium prior to excitation, the initial density matrix

⎛ iHgt ⎞ ⎛ iHgt ⎞ U (t ) = exp⎜ ⎟U exp⎜ − ⎟ ⎝ ℏ ⎠ ⎝ ℏ ⎠

(6)

Even though the choice of the reference Hamiltonian is formally arbitrary in our calculations, the choice of the ground-state Hamiltonian simplifies the calculation of the equilibrium correlation functions. The two-point correlation function C(2) (t1) = 1/ℏ2 ⟨U(t1)U(0)ρg⟩ for the Hamiltonian given in eq 1can then be expressed as ⎡⎛ 4 ⎞2 ⎢⎜ ωej(1 + βj ) C (t ) = ∑ ⎢⎜ − ωgj⎟⎟ (n(ωgj)2 + n(ωgj)) 2 2 β j ⎢ ⎠ j ⎣⎝ (2)

⎛ β 4 − 1 ⎞2 j ⎟ ((1 + n(ω ))2 e−i2ωgjt + n(ω )2 e i2ωgjt ) + 2⎜⎜ωej gj gj 2 ⎟ 4 β ⎝ ⎠ j ⎤ ⎛ 2 4 2⎞ ⎜ Dj βj ωej ⎟ iωgjt ⎥ −iωgit ((1 + n(ωgj))e +⎜ + n(ωgj)e )⎥ ⎜ 2 ⎟⎟ ⎥⎦ ⎝ ⎠ (7)

The first term is time-independent and can be interpreted as inhomogeneity in the system as the presence of a new state, in effect, introduces inhomogeneity in the sample. The second term arises due to the quadratic displacement and momentum terms C



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in the energy gap operator and causes two-phonon transitions to become directly allowed. The final term is solely due to the linear displacement term and involves only single-phonon transitions. Because the current theory is beyond the limit of linear response, higher-order correlation functions need not be zero or be decomposed into a sum of products of the two-point correlation function. The three-point correlation function, C(3)(t1,t2) = (1/ ℏ3)⟨U(t2)U(t1)U(0)ρg⟩, is found to be C(3)(t1 , t 2) =

∑ [αj3nj(1 + nj)(1 + 2nj) j

+ 4αjδj2(nj2(nj + 1)(e i2ωgjt1 + e i2ωgj(t2 − t1)) + nj(nj + 1)2 (e−i2ωgjt1 + e−i2ωgj(t2 − t1)) + (nj + 1)3 e−i2ωgjt2 + nj3e i2ωgjt2) + αjγj2(2nj(nj + 1)(cos(ωgjt1) + cos(ωgj(t 2 − t1))) + (nj + 1)2 e−iωgjt2 + nj2e iωgjt2) + 2γj2δj (nj2(e−iωgj(t1− 2t2) + e iωgj(t1+ t2)) + (nj + 1)2 (e iωgj(t1− 2t2) + e−iωgj(t1+ t2)) + 2nj(nj + 1) cos(ωgj(2t1 − t 2)))]

(8)

Figure 2. Real (left) and imaginary (right) contour plots of the threepoint correlation function of the current model. For (A) and (B), ωg = 500 cm−1 and ωe = 370 cm−1, and for (C) and (D), ωg =370 cm−1 and ωe = 500 cm−1.

Here, nj = n(ωgj) is shorthand for the Bose−Einstein thermal distribution and the auxiliary constants αj = [(ωej(1 + β4j )/2β2j ) − ωgj], δj = ωej[(β4j − 1)/4β2j ], and γj = (Djβ2j ωej)/√2 are related to the material properties. The first two terms arise solely from the quadratic displacement and momentum, while the next two terms are mixed terms, involving both the linear and quadratic. It is somewhat counterintuitive that three-body correlations could arise in a purely harmonic system, but this effect can most easily be understood as arising simply due to counterpropagating wavepackets moving at different velocities. The overlap between these wavepackets will then show some three-body correlations as they initially oscillate in phase and then drift relative to each other due to the difference in the propagation velocities. We note that the presence of odd-order correlation has been noted in previous studies and will never be zero when the deviations from equilibrium involve a quadratic displacement of a harmonic oscillator.31 Figure 2 shows the real and imaginary portions of the correlation function (eq 8). The initial value of the correlation function can be either positive or negative, depending on the relative difference in curvature. These correlation functions will be used in the calculation of the absorption line shape and nonlinear response detailed in the next section. In order to calculate the fluorescence line shape, we will assume that the environmental degrees of freedom have reequilibrated in the excited-state potential surface because the fluorescence lifetime is typically on the order of a few nanoseconds and solvation processes occur on time scales of femtoseconds to hundreds of picoseconds. The initial density matrix is then given by ρe = e−He/kbT/⟨e−He/kbT⟩, and the thermalized energy gap ℏω′eg = (He − Hg)ρe can again be calculated using operator techniques. ℏωeg′ = ℏωeg° +

⎡ D 2ω j gj

∑ ℏ⎢⎢− j

− ωgj

(1 − βj 2)2 4βj 2

⎣

2

+

We now define the excited-state deviations away from equilibrium. U ′ = He − Hg − ℏωeg′

The time dependence of this operator will be propagated with respect to the excited-state nuclear Hamiltonian. ⎛ iH t ⎞ ⎛ iH t ⎞ U ′(t ) = exp⎜ e ⎟U ′ exp⎜ − e ⎟ ⎝ ℏ ⎠ ⎝ ℏ ⎠

(11)

The second-order and third-order fluorescence correlation functions are obtained by interchanging the material properties of the ground- and excited-state manifolds of eqs 7 and 8. Note that Dj has a hidden factor of μgjωgj in its definition. During the solvation process, vibrational energy is transferred from the chromophore into the surrounding environment. The loss of energy is reflected in the shift in wavelength between the absorbance and fluorescence spectra. The mean shift yields an average energy that is given to the bath in the solvation process and is referred to as the solvent reorganization energy λ. 2λ ≡ ωeg − ωeg′ =

⎡ D2

∑ ⎢⎢ j

j

⎣ 2

(βj 2ωej + ωgj)

⎛ 1 + β4 ⎞ ⎛ 1 + β4 j j ⎟ + n(ω )⎜ω + n(ωej)⎜⎜ωgj − ω ej ⎟ gj ⎜ ej 2 2 2 2 β β ⎝ ⎠ ⎝ j j ⎤ ⎞ (1 − β 2)2 j ⎥ ( ) − ωgj⎟⎟ + ω + ω ej gj ⎥ 2 4 β ⎠ j ⎦

ωej − ωgj

(12)

The temperature dependence of this expression can again be understood as the difference in the Franck−Condon factors of higher-lying vibrations. The shift in the mean absorbance and fluorescence is sometimes improperly referred to as the Stokes shift in the literature. The Stokes shift is related to the difference in the maxima of the absorption and fluorescence spectra, not the

2

⎛ 1 + βj 4 ⎞⎤ ⎜ ⎟⎥ + n(ωej)⎜ωej − ωgj 2βj 2 ⎟⎠⎥⎦ ⎝

(10)

(9) D



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approximation should not be problematic in cases where only a single mode exhibits a difference of curvature. In this paper, we explore the effects of the above theory on 2D electronic spectroscopy.14,15 In these experiments, a sequence of three short pulses of light interact with a dilute sample. The first pulse resonantly excites a coherence between the ground state and first resonant excited state, which will evolve for a time period known as the coherence time (τ). The next pulse interacts with the system and brings it into either the excited electronic state or the ground electronic-state manifolds; the system then evolves for a time period referred to as the waiting time (T). The final pulse brings the system back to a coherence between the ground and excited states that evolves during the rephasing time (t). On the basis of the time ordering and wave vectors of the incoming pulses, as well as the direction of the detected signal, the phase evolution during the rephasing time can be controlled. If the phase evolution during the rephasing time is in the same direction as the phase evolution in the coherence time, the signal is referred to as a nonrephasing signal. Otherwise, the phase evolution is in the opposite direction, and the signal is referred to as a rephasing signal. In the case of rephasing pathways, if the system has retained a phase memory of its initial state, a photon echo will be produced, which is emitted after the final pulse arrives. The 2D signal is then typically visualized after a 2D Fourier transform is taken along the coherence and rephasing time axes.

difference in the means. Hence, the MBO theory does predict a small change in the Stokes shift with temperature, even though the means of the absorption and emission line shapes will remain temperature-independent. In the current theory, both the means and maxima will shift with temperature, and the latter will shift by a greater amount that that in the MBO model. Finally, in the MBO model, the absorption and fluorescence mean transitions, eqs 4 and 9, are shifted by the same amount from the 0−0 electronic energy difference, ωeg = ω°eg + λ and ω′eg = ω°eg − λ. Due to differing Franck−Condon factors during the excitation and fluorescence process, this is no longer the case in our generalized model, ωeg = ωeg ° + λa, λ and ωeg ′ = ωeg ° − λf, so that the total reorganization energy is given by 2λ = λa + λf.

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OPTICAL RESPONSE FUNCTIONS AND 2D SPECTROSCOPY To describe the optical properties of a dilute solution of chromophores, we will utilize a semiclassical response function formalism, wherein the electric field will be treated classically and the material response treated quantum mechanically.17 The linear absorption and nonlinear spectra can then be described in terms of the above-derived correlation functions. Within the Condon approximation, the transition dipole moment is assumed to be weakly dependent on nuclear coordinates and is treated as constant. The linear absorbance spectrum is expressed as the Fourier transform of the linear response function. The linear response function can be calculated through a cumulant expansion method, which we have truncated at third-order. A(ω) = |μeg⃗ |2

∫0

S2D(ωτ , T , ωt ) ∝

(17)

∞

exp[i(ω − ωeg )t ]

(2)

(3)

(3)

exp[−g (t ) + ig (t )] dt

Here, P (τ,T,t) is the third-order polarization as a function of the relative timings of the three pulses. We only consider purely absorptive spectra in this article, which results from the sum of the rephasing and nonrephasing contributions to the response. Additional pathways involving coherence between the doubly excited states and the ground exist as well but are not considered in this work. The resultant spectrum can then be interpreted as a correlation diagram between the absorption frequencies ωτ and the emitted frequencies ωt as a function of the waiting time T. The nonlinear polarization can also be calculated utilizing the cumulant expansion technique, which we have expanded to third-order. In general, any resonant third-order nonlinear experiment can be expressed as a function of a four-point dipole correlation function,17 expressed below for a two-electronic state system.

(13)

Here, μ⃗eg is the transition dipole moment and should not be confused with the reduced mass. The transition dipole moment can be distinguished by the vector sign in the present work. The second-order and third-order line shape functions, g(2) (t) and g(3) (t), are time-ordered integrals of the two-point and threepoint correlation functions. g(2)(t ) =

∫0

g(3)(t ) =

∫0

∬ iP(3)(τ , T , t ) exp[iωττ − iωt t ] dt dτ

t

dτ2

∫0

dτ3

∫0

t

τ2

τ3

dτ1 C(2)(τ2 − τ1) dτ2

∫0

τ2

(14)

dτ1 C(3)(τ2 − τ1 , τ3 − τ1) (15)

⟨μge⃗ (t1)μeg⃗ (t 2)μge⃗ (t3)μeg⃗ (t4)ρg ⟩

Similarly, the fluorescence line shape is expressed in terms of the linear response function, assuming that the initial density matrix is thermalized in the excited-state potential energy surface. 2

F(ω) = |μeg⃗ |

∫0

= ⟨μge⃗ μeg⃗ μge⃗ μeg⃗ ⟩ exp[− iωeg (t1 − t 2 + t3 − t4)]L(2) (t1 , t 2 , t3 , t4)L(3)(t1 , t 2 , t3 , t4)

∞

exp[i(ω − ωeg′ )t ]

exp[−g f(2) *(t )

−

ig f(3) *(t )]

(18)

Here ωeg is the thermalized energy gap, L (t1,t2,t3,t4) and L(3) (t1,t2,t3,t4) are the second- and third-order cumulant contributions to the line shape, respectively, and ⟨μ⃗ geμ⃗ eqμ⃗ geμ⃗eg⟩ is an isotropic average over the dipole orientation. Both line shape contributions are composed of a sum of integrals of the equilibrium energy gap correlation functions, and the detailed form of these functions is given in the Supporting Information. We note that the above solvation model combined with the expression for the four-point dipole correlation function could also easily be used to describe other related nonlinear techniques, such as transient absorption, transient grating, and three-pulse photon echo peak shift.17 The transient absorption is in fact (2)

dt

(16)

(3) A * indicates a complex conjugate, and g(2) f (t) and f (t) are the second-order and third-order fluorescence line shape functions, defined as in eqs 13 and 14, with the fluorescence correlation function used in place of the ground-state correlation function. Rigorously, all orders of correlation will contribute to the response functions. These formulas should be considered as a short-time approximation valid for small t and βj. Because the optical dephasing time at room temperature for most condensedphase systems is on the order of tens of femtoseconds, this

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The vibronic progression is more clearly visible in the fluorescence line shape, and the main transition is narrower in the fluorescence spectrum than the absorption spectrum. When both modes have the same frequency, the line shapes show mirror image symmetry about their respective means (C and D). Finally, we consider the case when the ground-state vibrational mode is more weakly bound than the excited-state mode (E and F), which results in the same line shapes of A and B with the role of the absorption and fluorescence interchanged. This result simply illustrates that the projection of the wavepacket onto the new potential surface is driving the change in line shape. Furthermore, this result is symmetric with respect to the interchange of the ground and excited states, as illustrated by the mirror image symmetry of A with F and B with E. The third moment of the line shape, a value that describes whether the asymmetry of the line shape is to the red or blue of the mean, changes for the different cases, being determined by the sign of C(3) (0) (see Figure 2). Now, we consider generally the results of tuning the curvature of the excited- and ground-state surfaces on the peak and width of the absorbance and fluorescence spectra. Here, we will use the second moment of the energy gap, C(2) (0), as a measure of the width. The results of these calculations are shown in Figure 4, with the maxima plotted in the top row (A−C) and widths in the bottom row (D−F). In the first column, we consider the case in which the groundand excited-state curvatures are identical, and we tune the vibrational frequency (A and D). The difference between the absorption and fluorescence maxima increases very weakly with increasing vibrational frequency, while the widths, which remain identical, increase dramatically with increasing frequency. In the middle column, we consider cases where the ground-state potential changes with the excited potential fixed at 1500 cm−1 (B and E). When the ground-state potential is tuned to higher frequencies than the excited potential, there is little dependence of the absorption spectrum width or maximum, but the fluorescence peak shifts and broadens. In this regime, the fluorescence line width is always greater than the absorbance line width. Tuning the ground state to frequencies less than the excited state reverses this trend, causing the absorption peak to shift and broaden and the fluorescence peak to narrow and shift only slightly. Finally, we consider the case where the excited-state surface is tuned while the ground-state potential is fixed at 1500 cm−1 (C and F). These results are complementary to the previous ones with the role of absorption and fluorescence interchanged. For example, when the excited-state potential is tuned to higher frequencies, the fluorescence maximum and width do not change appreciably, while the absorption peak shifts and broadens. These results are consistent with experiments on yellow fluorescent protein, where alterations of the protein matrix were found to shift the width and maximum of the absorption spectrum while leaving the fluorescent spectrum relatively unchanged in width and maximum.43 Interestingly, our calculations indicate that the particular normal mode responsible for this behavior stiffens upon optical excitation. These results suggest a direct route to control of optical properties of a chromophore if one could reliably predict the effects of a chemical modification on the adiabatic surfaces of the ground and excited states. For example, to design a broad-band absorbing chromophore with a narrow-band emission, the excited-state potential should be tuned to a higher frequency than the ground-state vibration. One could potentially realize this through a π* → π transition, wherein the bond order increases

equivalent to a projection of the 2D signal onto the rephasing frequency axis.

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RESULTS AND DISCUSSION To illustrate the results of our new model, we consider the effect of a single high-frequency mode that shows a difference of curvature between the ground and excited manifolds. This mode represents a coherent intramolecular vibration. In addition, we add a continuum of low-frequency MBO modes that represent the coupling between the chromophore and surrounding solvent. Formal arguments have been made that the solvation process for an electronic transition will generally be a Gaussian process due to the large number of modes coupling to the transition and hence be properly modeled within the MBO theory.42 We assume a Gaussian form for this spectrum of modes29 with a characteristic decay of 300 fs and coupling strength of 300 cm−1. For the coherent mode, we chose a displacement of (μgj/ℏ)1/2dj = 2.06 fs−1/2. For a 500 cm−1 mode in the MBO model, this would correspond to a Huang−Rhys factor of D2j /2 = 0.2. The displacement is kept constant throughout all of the calculations below. We further assume that the excited-state and ground-state reduced masses are identical. These parameters were used previously to describe the effects of a single high-frequency vibrational mode to recreate some features observed in 2D measurements of a long-chain cyanine dye 1,1-diethyl-4,4dicarbocyanine iodide (DDCI-4).29 The values for the excitedstate and ground-state curvatures are varied to ascertain their influence on the optical properties of the chromophore. The temperature for the calculations is set to 300 K. As a starting point, we consider the linear absorption and florescence line shapes for different values of the excited-state and ground-state curvature. Shown in Figure 3 are the absorption and fluorescence line shapes for three different pairs of values of the excited- and ground-state curvature. First, we consider the case when the ground-state vibrational mode has a higher frequency than the excited state (A and B).

Figure 3. Calculations of absorption (blue) and fluorescence (red) line shapes for the current model with different values of the excited-and ground-state potential frequencies. For (A) and (B), ωg = 500 cm−1 and ωe = 370 cm−1; for (C) and (D), ωg = ωe = 500 cm−1; and for (E) and (F), ωg = 370 cm−1 and ωe = 500 cm−1. Only in the case of both potential energy surfaces having the same curvature do the absorption and fluorescence line shapes show mirror symmetry. F



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Figure 4. Calculations of the absorbance (blue) and fluorescence (red). The top plots display the peak maximum (A−C) and the bottom plot the second moment or width (D−F) as a function of the curvature of the harmonic mode. For (A) and (D), both the ground and excited modes have the same curvature, and the vibrational frequency is varied; for (B) and (E), ωe = 1500 cm−1 and ωg is varied; and for (C) and (F), ωg = 1500 cm−1 and ωe is varied. Note the different scales and units used in the vertical axis between the top and bottom rows.

Figure 5. Comparison of the effects of a difference of curvature between the ground and excited states on the absorptive 2D spectrum. (A−F) The evolution of the 2D spectrum with waiting time T from 0 to 270 fs. For the left column (A, C, E), the excited-state and ground-state curvatures are both set to ωg = ωe = 500 cm−1, while in the right column (B, D, F), the excited-state curvature is set to ωe = 370 cm−1 and the ground-state curvature is set to ωg = 500 cm−1. Each spectrum is individually normalized, and the linear colormap extends from −0.1 to 1. Traces of the waiting time dependence for the three points marked in the 2D spectra are shown in panels (G−I), where the sold line corresponds to the case when the two potentials are equal and the dashed line to the case when the excited potential is changed. The traces are taken from the points ωτ = −12650 cm−1, ωt = 12150 cm−1 (G); ωτ = −12500 cm−1, ωt = 12500 cm−1 (H); and ωτ = −12150 cm−1, ωt = 12650 cm−1 (I).

Guzik group has shown the feasibility of calculating the normal modes and the ground and excited states of a small molecule for determining the optical proprieties from first principles.44 Calculations of this sort could be used to inform the proper chemical route to reach the desired goal.

upon optical excitation. These results also show a direct strategy for altering fluorescent properties while leaving the absorption properties unchanged and vice versa by stiffening the groundstate potential for the former case and stiffening the excited-state potential for the latter case. A recent study from the AspuruG



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curvatures are the same, the main diagonal feature is symmetric in width along the diagonal and rounds with waiting time, reflecting spectral diffusion of the lower-frequency modes. When the curvatures differ, the diagonal peak is initially symmetric and then skews toward the higher frequencies, showing a slightly broader antidiagonal line width at the blue edge than that at the red edge. A similar effect of the line width has been seen in theoretical studies of 2D infrared spectroscopy.51 The diagonal line shape also remains diagonally elongated at longer waiting times when the curvature differs, reflecting the increase in inhomogeneity discussed earlier. The most pronounced differences arise when comparing the waiting time dependence of the signal, shown in the right-hand side of Figure 5. Here, we see that when the curvature differs (dashed lines) that higher frequencies are present. A Fourier transform of the signal (data not shown) reveals that the harmonics of the vibrational frequencies become much more pronounced, which arises from the quadratic coupling that allows the two-phonon transition to become directly accessible. 2D electronic spectroscopy experiments have observed many of the common features of the MBO model and our generalized result. The effect of lower-frequency vibrations has been shown to induce peak shape and amplitude oscillations at the frequency of the vibrational mode.52 Higher-frequency vibrations can lead to vibrational cross peaks and typically generate a characteristic four-peak53−55 or six-peak pattern,29 dependent on the molecular details and experimental bandwidth available. Determining unambiguous signatures of the differing excited-state and ground-state curvatures from these previous studies is difficult due to the rather subtle differences discussed above, which are mostly prevalent by examining the waiting time dependence. Studies of N,N′-bis(2,6-dimethylphenyl)perylene-3,4,9,10-tetracarboxylicdiimide (PERY) has shown multiple frequencies present in the transient grating and 2D signal,52 two of which appear to be close to the fundamental and first harmonic of a 143 cm−1 mode. This may be attributable to two-phonon transitions that are present in the current model. The ability to unambiguously separate excited-state and ground-state wavepackets in 2D spectroscopy29,56 may be the best experimental test for this model, where the differing frequencies of the excited state and ground state should be directly observable as different frequencies present for different time traces. This effect has been observed in pump−probe studies57 and is present in our calculations (data not shown), though the excited-state frequency does not exactly match the excited-state curvature. The lack of agreement in the excited-state frequency is presumably due to the effective short-time approximation employed here.

Control of the quadratic coupling also has relevance for energy transfer in multichromophoric systems. A critical parameter in the energy-transfer dynamics is the dephasing induced by the bath.45−47 Proper tuning of the excited-state curvature could allow the bath-induced fluctuations to decrease after excitation and equilibration of the nuclear motions on the excited state, that is, the homogeneous line width of the fluorescent state can be significantly less than the absorbance line width. This is possible because energy-transfer events occur on time scales longer than solvation. In this regime, the absorption spectrum is broader than the emission spectrum. The decrease in dephasing could then allow for the simultaneous control of the excited-state dephasing rate, to allow for coherent energy-transfer mechanisms to be active, while maintaining a broad absorption spectrum. This type of coupling has previously been found to be consistent with the absorption maximum shift with temperature in a photosynthetic reaction center;36 therefore, it may be that this coupling represents a design principle in photosynthetic light-harvesting complexes. Finally, we consider the effects of tuning the excited-state curvature on 2D spectroscopy. We consider two cases, first, when the two potentials are identical (ωg = ωe = 500 cm−1) and, second, when the excited-state curvature is decreased (ωg = 500 cm−1, ωe = 370 cm−1). Because we have truncated the cumulant expansion, effectively making a short-time approximation, our solution gives nonphysical results (the response function having a value greater than 1) at waiting times > 1 ps. Apparently, all orders of equilibrium correlation functions would be required to properly describe the equilibration of the excited-state wavepacket within this model. For our calculations, we consider the real part of the absorptive 2D spectrum.48 The results of the calculations are shown in Figure 5. The thermalized transition between the ground and excited states was set to 12500 cm−1, and we have assumed delta function excitation pulses. There are several features that are present in both calculations of the 2D spectra. The early time spectra for both models show similar features, with highly diagonally elongated peak shapes, indicating that the system is inhomogeneous on the time scale of a few femtoseconds. The zero waiting time spectra, Figure 5A and B, are excessively diagonal when compared to experimental measurements and result from the δ-function excitation pulses assumed as well as the relatively slow spectral diffusion time scale of 300 fs. Small negative features are also observed below the diagonal at early waiting times. While negative features in 2D spectroscopy are typically assigned to excited-state absorption, indicating the presence of an additional electronic state, they can also be present in two-state systems. In the case of a two-state system, the negative feature is typically assigned to either a fourlevel four-wave mixing processes49 or a vibrational wavepacket.50 Cross peaks, resulting from transitions between different vibrational states, are also cleanly resolved at early waiting times but are slowly encompassed into the main diagonal peak at longer waiting times due to the spectral diffusion process. In general, the differences between the two calculations are rather subtle, and the discrepancies can be seen more in the waiting time dependence than when comparing individual 2D spectra. At early times, the cross peaks arising from the transition to the higher vibrational state are reduced when the excited-state curvature is decreased. The reduction of curvature causes the excited-state wavepacket to be projected onto more vibrational states, thereby reducing the cross peak of the first transition and redistributing the amplitude to other transitions. The line shape dynamics also show differences between the two cases. When the

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CONCLUSION We have presented a nonlinear model for solvation and determined its influence on the optical properties of a singlechromophore system. The model predicts stronger temperature dependence in both the line shape and maxima of the absorbance and fluorescence spectra and an asymmetry between the absorption and emission line shapes. Nonlinear spectroscopic measurements provide an experimental tool capable of quantifying these effects through line shape metrics and waiting time dependence of the experiments. It would be of interest to see if higher-order spectroscopies are more sensitive to these changes as previous theoretical studies have suggested that fifthorder measurements can yield more detailed information. Most surprisingly, the theory indicates that the solvation process is no H



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longer Gaussian, revealing that even in a harmonic model, nonGaussian processes can occur. Tuning the relative curvature of electronic states could allow for a chemically plausible route to simultaneous or independent tuning of absorption and fluorescent properties, providing control of the optical properties of a chromophore and even the excited-state dephasing dynamics in multichromophoric systems.

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ASSOCIATED CONTENT

S Supporting Information *

A description of the operator method for the calculation of the equilibrium correlation functions and complete expressions for the nonlinear response functions. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank NSF MRSEC (Grant No. DMR 08-02054), AFOSR (Grant No. FA9550-09-1-0117), DTRA (HDTRA1-10-1-0091 P00002), and the DARPA QuBE program (Grant No. N66001-10-1-4060) for supporting this work. A.F.F. acknowledges support from the DOE SCGF program. The authors thank Dugan Hayes, Justin Caram, and Phil Long for a critical reading of the manuscript.

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Nonlinear Spectroscopic Theory of Displaced Harmonic Oscillators

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