Basis Set Truncation Further Clarifies Vibrational Coherence Spectra

Coherent vibrational oscillations in femtosecond transient-absorption spectra have been interpreted since the 1990s using a model based on Gaussian wavepacket dynamics. The oscillations are often studied using probewavelength dependent plots of the oscillation amplitude and phase that are known as vibrational coherence spectra. Here we show that restricting the basis of the wavepacket to a small number of eigenstates clarifies several features in vibrational coherence spectra. Improving the understanding of vibrational coherence signatures will help distinguish them from signatures of electronic coherence that arise from measurements of strongly coupled excitonic states in molecular aggregates and light-harvesting proteins.


Introduction
A femtosecond laser pulse can excite a nonstationary vibrational wavepacket on the electronic states of molecules. The coherent oscillations of such a wavepacket produce oscillations of the nonlinear optical signal as the delay time between the pump and probe pulses is varied in a time-resolved spectroscopy measurement [1]. In transient-absorption spectroscopy, the amplitude and phase of the observed oscillations vary with probe wavelength, and these features of the oscillations began to be studied intently in the late 1980s [2]. The oscillations were observed in transient-absorption measurements of conjugated laser-dye molecules and pigment-protein complexes throughout the 1990s [3][4][5][6][7][8][9][10][11][12][13], and the effects of pulse chirp on the oscillations were explored [14,15]. Subsequently, Champion and coworkers developed a theoretical framework to model coherent vibrational oscillations in systems having nonradiative decay mechanisms such as nonadiabatic transfer [16].
More recently, researchers have gained a renewed interest in using the coherent wavepackets to resolve the dynamic Stokes shift in pigment-protein complexes and molecules [17][18][19][20], to explore coherence in electron-transfer reactions [21], to quantify ultrafast protein solvation dynamics [22], to search for signatures of nonadiabatic dynamics [23,24], to study the excited-state dynamics of a molecule used for the medical application of photodynamic therapy [25], and to characterize the evolution of vibrational coherence in isomerization reactions and electronic internal conversion [26][27][28][29][30]. Other theoretical explorations included broadening the application of the doorwaywindow approach [31,32] and studying wavepacket dynamics under intense pumping [33]. Cina and coworkers recently presented an edifying theoretical description of the coherent vibrational oscillations in transient-absorption spectra [34]. In these studies, researchers often analyze the probe-wavelength dependence of the coherent wavepacket dynamics in the frequency domain, These foundational works revealed two characteristic signatures of coherent vibrational wavepacket dynamics that are readily apparent in a vibrational coherence spectrum. There is an abrupt phase shift of the oscillations as a function of probe wavelength, and the phase shift is accompanied by an amplitude node. These two characteristic features typically appear at the probe wavelength that corresponds to the maximum signal in a steady-state fluorescence spectrum. In previous works, the coherent oscillations were interpreted through a model of coherent wavepacket dynamics [2,4,7,35]. In this description-depicted in the left panel of Figure (2)-the pump pulse promotes a Gaussian wavepacket to the displaced excited electronic state.
The wavepacket then evolves in phase space, oscillating on the excited-state potential energy surface. The probe pulse returns the wavepacket to the ground electronic state resulting in the stimulated emission of a photon. For a conceptual interpretation, this transition is often treated in the classical limit where the emitted photon must have energy equal to the instantaneous gap between the excited and ground electronic states [35,36]. For displaced harmonic oscillator potentials, the energy gap is proportional to the internuclear displacement, q. As the wavepacket evolves, the probe wavelength resolved oscillation spectrum essentially follows the wavepacket: The probe signal increases when the wavepacket is at an internuclear displacement value with a peak at that probe energy and decreases when the wavepacket moves to another internuclear displacement value. Considering the wavepacket dynamics from the perspective of a single probe wavelength, as the wavepacket moves in and out of the vicinity of that internuclear displacement value, the signal will increase and decrease. On either side of the minimum of the potential, the wavepacket-because it has some width-will pass through once per oscillation period, yielding an oscillation frequency that matches the vibrational oscillation. At the minimum of the potential, the wavepacket is observed twice per cycle-once passing in each direction-and the stim-

Theory
Here we seek to understand a transient-absorption spectroscopy measurement conducted with broadband laser pulses, in which the pump pulse creates a wavepacket on the excited-state potential energy surface that is a superposition of vibrational eigenfunctions. We choose to use the eigenfunctions of the conventional quantum harmonic oscillator, and, as shown in Figure (2), we focus on the wavepacket that is on the excited electronic state, |ψ = n c n |e, n , plotted as a function of the displacement, q. Following the Franck-Condon principle, for small displacements, the most significant contributions to the wavepacket will be the two lowest-energy eigenfunctions, n ∈ {0, 1}. The two eigenfunctions will interfere, and the superposition will evolve at the fundamental frequency of the mode, ω 0 . The two eigenfunctions cannot interfere where one has zero amplitude, and therefore there will be a node in the interference where the q|e, 1 function has zero amplitude.
The abrupt π phase change in the oscillations is a consequence of the sign change in q|e, 1 . Both the node and phase shift occur at the minimum of the excited-state potential, q min . Below we expand on this swift visual interpretation of the wavepacket in a truncated basis and show how it is consistent with additional features observed in the measured spectra and provides physical insight.
We consider the doorway-window approach [37] for vibrationally abrupt pulses and focus on stimulated emission because the excitation pulse in many measurements has sufficient bandwidth to select primarily for oscillations on the excited electronic state [4,17,20]. This approximation is appropriate when the transition-dipole moment is independent of internuclear displacement value and the pump pulse has a spectrum that encompasses the absorption spectrum [34,35]. We treat the doorway function as the state prepared on the excited electronic state by the pump pulse [35,36] and follow a Franck-Condon approach to calculate a distribution of vibrational eigenfunctions in the electronic excited state due to the instantaneous promotion of a portion of the ground-state eigenfunction to the excited electronic state. We then propagate the wavepacket in the field-free excited-state potential. For convenience and to emphasize the shape, we assume that (1) the system can be represented as a pure state, (2) use a temperature of 0 K, and (3) use a classical window function. We address the quantized window function in Appendix A.
We identify the two electronic states as |g and |e , each with a progression of vibrational eigenfunctions. We represent the ground state-the lowestenergy vibrational eigenfunction of the ground electronic state-at time zero as |g, 0 and the n th vibrational level of the excited state as |e, n . The density matrix of the ground state in the basis of the displaced excited-state potential at time t = 0 when the pump pulse excites the molecule is given by where the coefficients c ab = e, a|g, b give the amplitude of each vibrational eigenstate in the excited electronic state. We reserve indices n (m) for the excited (ground) state. The system then evolves on the excited electronic state for time τ , and the density matrix in the q basis is then given by whereĤ e is the nuclear Hamiltonian on the excited electronic state, E n is the energy of the n th vibrational eigenstate above the minimum of the excited state potential, and the ψ(q) are the vibrational eigenfunctions. In practice, when analyzing coherent vibrational dynamics in a measured transientabsorption spectrum, researchers typically subtract the static and slowly varying signals and then fast Fourier transform the residual oscillations to produce the vibrational coherence spectrum. Hence we focus on the Fourier transform of the density matrix over τ and find This predicts a progression of frequencies for each vibrational mode where the oscillation amplitude and phase are defined at each internuclear displacement value and are determined by sums of products of the time-independent eigenfunctions. To clarify the discussion, we initially treat the ground and excited electronic states as displaced harmonic oscillators, where the energy gap between eigenstates, E n − E n = (n − n ) ω 0 , is constant, where ω 0 is the fundamental frequency of the vibrational mode and = 1. Second harmonic oscillations have been observed in gas-phase transient-absorption measurements of molecular I 2 [35], and were observed to contribute very weakly in a measurement of a condensed-phase sample [25]. Therefore we focus on positive frequency terms that oscillate at the fundamental frequency-meaning only the n − n = 1 terms-and identify the oscillation amplitude for the fundamental frequency, M HO (q), as Vibrational coherence spectra are displayed as a function of probe wavelength, λ, and therefore we transform q to λ using the potential-energy difference between the two states, V e (q) − V g (q). For the harmonic oscillator, the probe frequency, ω, becomes where the classical turning point of the n = 0 state is m eff is the effective mass of the mode, and we have substituted the peak of the fluorescence spectrum as the frequency at the minimum of the excited- In the simplest case in which only the two lowest-energy vibrational levels are excited on the electronic excited state, we can write the amplitude of the oscillations as a function of probe wavelength, The amplitude and phase profiles of the vibrational coherence spectrum as depicted in Fig. (1) are then written as

Results and Discussion
A key insight of this work is that the node and phase shift observed in the vibrational coherence spectrum can be understood swiftly using a visual q/x 0 = 0 and the sinusoidal character of the time trace at q/x 0 = 1, as example. This difference gives rise to the π phase shift in the vibrational coherence spectrum. Finally, the node and change in phase occur both at q = ∆, which is q min in Fig. (2) the minimum of the excited-state potential and will be the maximum of the steady-state fluorescence spectra in this model.
Next we quantify the projection of the wavepacket into various bases.
For the full basis, we confirmed that the projection is unity for all values of ∆/x 0 . Fig. (4) shows that even for a relatively large displacement value of ∆/x 0 = 1, the projection of the wavepacket into the basis of the two lowest-energy eigenfunctions is greater than 0.9. For displacement values of These basic evaluations demonstrate that the key features of the full wavepacket dynamics are reproduced effectively using the lowest two-eigenstate basis, and therefore we next produce and evaluate the vibrational coherence spectra. Specifically we will compare the overall amplitude and width of the oscillations between the two models.  We normalize the peak value in each case by the maximum reached in the full eigenstate case. The interference terms (ψ n ψ n+1 ) require a significant amplitude in each eigenstate, and therefore the vibrational coherence spectrum peak value grows for small displacement as the n = 1 state increases in am-   plitude. Near ∆/x 0 = 1 the peak amplitude reaches a maximum as the amplitude is spread over many eigenstates. We emphasize that for ∆/x 0 < 0.8 the two models make nearly identical quantitative predictions. Our metric of ∆/x 0 is closely related to the Huang-Rhys factor, S, S = (1/2)(∆/x 0 ) 2 .
Hence the cut-off value of ∆/x 0 ∼ 1 corresponds to S ∼ 0.5. Using that relation, we compared the results of a joint theory/measurement study [38] and found that of the 25 modes analyzed in their Table S1, most modes had S ∼ 0.02 and the largest was S ∼ 0.28. This supports the notion that the Next we quantify the width of this unusually shaped amplitude profile.
We define the width as σ 2 = ω 2 − ω 2 and calculate where N = ∞ −∞ A (ω) dω is a normalization. This expression is analytic in the two-eigenstate harmonic oscillator basis, and we find σ HO 0,1 = ω 0 ∆/x 0 . In Fig. (5) we observe the anticipated behavior for the width of the two-  [34]. In Appendix B we analyze the case of an electronic excited state with two Frack-Condon active vibrational modes. We find that the node and phase shift are unchanged, however we find that additional modes increase the width of the amplitude profiles of the vibrational coherence spectra.
In many measurements the amplitude profile of a vibrational coherence spectrum is asymmetric, meaning that the peak on one side of the node has more amplitude than the peak on the other side of the node. Often it is the case that the higher-energy side has more amplitude than the lower-energy side [16-18, 21, 22, 26]. This result was not reproduced by any simulations using the harmonic oscillator eigenfunctions. Therefore we hypothesized that an anharmonic oscillator may reproduce the measured asymmetry, as well as, perhaps, the gradual slopes that are oftentimes observed in the phase profiles. We chose to use the Morse oscillator because there are analytic expressions for its eigenfunctions [39]    from the frequency variable to the wavelength variable-as well as in heightwhich is far more interesting and results from the anharmonic nature of the potential. To explore the relation between the peak height ratio and the anharmonicity further, we ran a series of simulations at varying amounts of anharmonicity. The data revealed two key results. First, as anharmonicity increased, the relative peak-height difference increased, supporting our conclusion that that anharmonicity leads to the difference in peak heights.
Second, the vibrational coherence spectra that result from the full and truncated bases began to diverge very slightly as anharmonicity increased. For example, in the case of 66 bound states, the agreement was nearly perfect, whereas for the case of only 7 bound states, minor quantitative differences were visible.  [40,41]. Second, we have not addressed more complicated potential energy surfaces, such as those with barriers to photoproduct intermediates [22] or those that induce nonadiabatic transitions resulting from avoided crossings or conical intersections in photochemical reactions [16,42]. Third, the simulations excluded damping arising from a dissipative system-bath interaction [1]. Damping would primarily serve to broaden the peaks along the frequency axis that arise from Fourier transformation of the time-delay variable. For any particular mode, the influence on a vibrational coherence spectrum-which arises typically from a slice at the peak of the amplitude along this frequency axisis likely to be negligible. However, if a sample had two or more modes of very similar frequencies, damping may cause overlap and interference among the features.

Conclusions
We have evaluated the amplitude and phase profiles of vibrational coherence spectra observed in transient absorption spectroscopy, focusing on insights gleaned by truncating the basis of vibrational eigenstates. This description yielded an intuitive picture of the source of the shape of the amplitude and phase profiles, particularly when focusing on only the two lowestenergy vibrational eigenstates. Using anharmonic potentials produced vibrational coherence spectra that were asymmetric in relative peak amplitude, which is a common occurrence in measured spectra. In addition, the models led to predictions about the shape of the vibrational coherence spectrum that could be compared with experimental measurements. These details are a key method by which one can distinguish vibrational coherences from electronic coherences arising from strongly coupled excitonic states in molecular aggregates. This distinction is critical for studies of electronic energy transfer mechanisms in molecular aggregates and photosynthetic light-harvesting proteins.
An ideal molecular sample to which these predictions can be compared would have a high fluorescence quantum yield with minimal photactivity and a large Stokes shift to distinguish clearly the ground-state wavepackets from those on the excited state. Finally, it would be convenient for the molecule to have a limited number of dominant vibrational modes so that excessive post-processing of the dataset is not necessary. Creating a significant signal from excited-state wavepackets and suppressing wavepacket oscillations on the ground state requires a laser pulse shorter than the vibrational period [35], hence an ideal molecule would have an absorption peak and a fluores-cence peak in a wavelength range where few-cycle laser pulses are readily produced, typically 500 nm to 800 nm.

Appendix A. Window in a Vibrational Eigenstate Basis
In posing this model, we have used the doorway-window approach in which we have treated the doorway and time evolution in a vibrational eigenstate basis, but we treated the window classically. We assumed that projecting the wavepacket back into the ground electronic state requires only that the laser energy match the instantaneous energy gap between the ground and excited state potential at the internuclear displacement value of the wavepacket. In other words, we chose a convenient, analytic route for the transformation to Eqns. (7b) because we aimed to study not the full nonlinear optical spectroscopy measurement but rather a key conceptual interpretation.
To provide a more complete model of a transient-absorption spectroscopy measurement, in this Appenix we consider projecting the excited-state wavepacket to the ground electronic state using a window operator represented in a vibrational eigenstate basis. Without line broadening, this leaves only transitions at discrete energies ω n,m = E n −E m . The q dependence becomes obscured because the Franck-Condon factor for the transition between any two states is a constant, independent of time. The laser frequency dependence then arises from the sum of, potentially, many contributing |e, n → |g, m transitions with distinct amplitudes and phases.
We follow prior work-specifically Eqn. (3.8a) in Ref. 37-and assume that the laser pulse duration is short compared to the nuclear oscillations but long compared to the dephasing of the electronic transition; we calculate a bare window operator of the form where γ is the dephasing rate of the electronic transition. Up to overall constants, the spectrally resolved signal is given by where Tr[. . . ] is the trace evaluated here in a basis of the vibrational eigenstates on the excited electronic state. We find We choose to focus on the terms oscillating at the fundamental vibrational frequency (n = n + 1). In this case, the vibrational coherence spectrum is given by This is a sum of Lorentzian lines centered on the |e, n → |g, m transition energies with various amplitudes and phases. The amplitudes are dominated by the overlap integrals c nm .
We compared the calculated vibrational coherence spectrum for the purely vibronic model with the classical window model and found that at large displacements the two spectra overlap very well and at small displacements the classical model predicts a much narrower width. This highlights one of the limitations of any classical interpretation of wavepacket dynamics. For small displacements, the relative variation in ω = V e (q)−V g (q) over the full motion of the wavepacket is less than ω 0 , however, from the vibrational eigenstates perspective, the transitions are discrete and, for the harmonic oscillator, are spaced by ω 0 .

Appendix B. Two-Mode Analysis
We consider two vibrational modes on the excited electronic state which lead to a two-dimensional wavepacket on the excited electronic state. In addition to oscillations at the fundamental frequency and overtones of each mode, the wavepacket now includes combination bands composed of sums and differences of the two frequencies, as has been observed in prior multi-mode analyses [25,43]. While the combination bands merit further investigation, we focus the discussion here on the fundamental frequencies to be consistent with the remainder of this work.
The wavefunction is composed of a product of the basis functions of the eigenstates ψ (s) n (q s ), where we label the different vibrational eigenfunctions and modes with s = 1 or 2. While the wavefunctions are independent in an internuclear displacement basis, the difference potential V e − V g depends on both modes. For a given value of the laser frequency ω there are many combinations of q 1 and q 2 where V e − V g = ω that must be accounted for to calculate the full contribution to the vibrational coherence spectrum. For the harmonic oscillator case, these points make a straight line through the two-dimensional space so for this case we perform a Radon transform [44] to project the q dependent amplitude profiles along lines of constant ω. As an example we calculate the mode amplitude profile at the fundamental frequency ω 1 of the s = 1 mode as a function of laser frequency assuming that only the lowest energy vibrational eigenstates needed to create the coherence contribute to the signal and find where ω s and x s are the fundamental frequency and classical turning point of the s vibrational mode and the normalized displacement ζ s = ∆ s /x s is defined for convenience. We highlight that the general form of the equation is unchanged, but the width of the vibrational coherence spectrum is increased due to the additional mode. A more thorough treatment could consider the details of the shape as additional eigenstates are included.