Resonant Femtosecond Stimulated Raman Spectra: Theory and

Article pubs.acs.org/JPCA

Resonant Femtosecond Stimulated Raman Spectra: Theory and Simulations B. Jayachander Rao* Departamento de Química, and Centro de Química, Universidade de Coimbra, 3004-535 Coimbra, Portugal

Maxim F. Gelin and Wolfgang Domcke Department of Chemistry, Technische Universität München, D-85747 Garching, Germany ABSTRACT: We present a description of resonant femtosecond stimulated Raman spectra, which is based on the solution of the nonperturbative equation of motion of the chromophore in the laser fields. The theory is applicable for arbitrary shapes and durations of the Raman pulses, accounts for excited-state absorption, and describes nonstationary preparation of the system by an actinic pulse. The method is illustrated by the calculation of femtosecond stimulated Raman spectra of a model system with a conical intersection.

I. INTRODUCTION Femtosecond stimulated Raman scattering (FSRS) is a robust method for the detection of vibrational dynamics with high temporal (∼25 fs) and better than 10 cm−1 frequency resolution.1 FSRS has matured into a versatile spectroscopic technique, and there exists a rich literature on the application of FSRS to various photophysical and photochemical processes. FSRS has been used to investigate photoinduced isomerization,2,3 excited−state proton transfer,3 dynamics of hydrogen bonding in proteins,4 reactive and nonreactive pathways in photochemical ring opening reactions,5 as well as to reveal metastable states, relaxation, and structural changes.6−11 Recently, FSRS has been combined with surface-enhanced Raman spectroscopy (SERS), resulting in a new experimental technique called SE-FSRS,12−14 which is suitable to probe chromophores at low concentrations or with small Raman cross sections. In FSRS, a short actinic pulse triggers the process under study, which is subsequently probed by two pulses driving the Raman transition. The latter are a picosecond narrowbandwidth pump pulse and a femtosecond broad-bandwidth probe pulse that stimulates the scattering of vibrational modes with frequencies between 600 and 2000 cm−1. The use of an additional probe pulse to induce the Raman scattering offers an improved spectral and temporal resolution compared to traditional spontaneous Raman spectroscopy. Theoretical support is indispensable for the interpretation and quantitative simulation of spectroscopic signals. Two conceptually different theoretical methods are traditionally employed in femtosecond nonlinear spectroscopy. The © 2016 American Chemical Society

perturbative methods rely upon perturbation theory for the evaluation of the material system response to external laser fields.15 The nonperturbative methods are based on the numerical evaluation of the material system dynamics driven by external laser fields. If the system−field interactions are weak, the perturbative and nonperturbative methods yield identical results.16 If the system−field interactions cannot be considered as weak, then the nonperturbative methods allow the simulation of strong-field effects in spectroscopic responses.17 However, even if the system−field interactions are weak compared to the intramolecular interactions, it can be of advantage to solve the equation of motion (the timedependent Schrödinger equation for the wave function or the Liouville−von Neumann equation for the density matrix) nonperturbatively in the laser fields. The available theory of FSRS spectroscopy is based on the perturbative description. In the majority of FSRS experiments performed so far, the Raman pump and probe pulses are electronically off-resonant. A comprehensive theoretical description of this class of experiments has been given in refs 18−20. The description is based on a perturbative treatment of the system−field interactions, while the material system dynamics can be treated at various levels of sophistication. Special Issue: Ronnie Kosloff Festschrift Received: December 16, 2015 Revised: February 23, 2016 Published: February 24, 2016 3286

DOI: 10.1021/acs.jpca.5b12316 J. Phys. Chem. A 2016, 120, 3286−3295

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The Journal of Physical Chemistry A Lee and co-workers21,22 developed a theory of FSRS that is valid for both resonant and off-resonant Raman pulses (see also the work of Cina and Kovac23). This theory is based on a perturbative solution of the time-dependent Schrödinger equation in terms of the system−field interaction. The actinic pulse is not considered explicitly, but it is assumed to instantaneously create a moving wavepacket, which is subsequently interrogated by the Raman pump−probe pair. According to this theory, the FSRS signal is a sum of four components, with two stimulated Raman scattering terms and two inverse Raman scattering terms. The theory reproduces the features of FSRS spectra in both Stokes and anti-Stokes regions, but it does not account for excited-state absorption (ESA). The latter is unavoidable if the low-lying excited electronic states are coupled to higher-lying electronic states by the laser pulses. For example, the FSRS experiments on rhodamine 6G reported in ref 24 were interpreted in ref 25 within a two-electronic-state model, although the S0−S1 transition in rhodamine 6G is close in energy to the S1−S4 transition,26,27 and ESA may therefore contribute to the FSRS signal. The theory of refs 21 and 22 was also extended to evaluate cascading contributions,28 which frequently contaminate (nonresonant) FSRS signals.1,29 Very recently, Moran and co-workers gave a perturbative description of their resonant FSRS experiments performed with a multibeam laser setup, which was designed to minimize the cascading.30 In the present work, we develop a computationally efficient method for the simulation of resonant FSRS signals, which is based on a numerical solution of the time-dependent Schrö dinger equation for the total (system plus field) Hamiltonian.31−34 The motivation for this development is as follows. (i) FSRS is a set of spectroscopic techniques, which differ in the manner of preparation (with or without actinic pulse, resonant or nonresonant actinic pulse) and probing (resonant or nonresonant) of the molecular system. The developed method allows a unified description of all variants of FSRS. (ii) The time-dependent Schrödinger equation can efficiently be solved for realistic molecular Hamiltonians that may be parametrized on the basis of ab initio electronicstructure calculations. The developed method of the simulation of FSRS signals can be combined, for example, with the multiconfiguration time-dependent Hartree (MCTDH) method35,36 (which is one of the most powerful solvers of multimode nuclear quantum dynamics) or with a recently developed hierarchy approach to stochastic Schrödinger equations.37 (iii) With the developed method, FSRS signals induced by pulses of arbitrary shape, chirp, and duration are simulated at no extra computational cost. ESA is straightforwardly taken into account. Actinic pulses of any strength can be described, provided the contribution due to the Raman pump−probe pair can be neglected during the action of the actinic pulse. However, the averaging over molecular orientations and the inclusion of inhomogeneous broadening effects are in general more costly than in the perturbation approach. Simultaneous time and frequency resolution makes FSRS a promising tool for the monitoring of vibrational wavepackets near conical intersections (CIs). For example, time-resolved two-dimensional (2D) FSRS experiments revealed the

dominant coupling and tuning modes in charge transfer dimers.38,39 Several off-resonant40,41 and resonant42 FSRS signals of systems with CIs have been simulated. Herein, we apply the nonperturbative method of ref 31 for the simulation of FSRS signals for a well-understood two-state two-mode model of the S1(nπ*)−S2(ππ*) CI in pyrazine.43,44 The model is augmented with a higher-lying excited electronic state giving rise to ESA. The paper is organized as follows. Section II briefly introduces the Hamiltonians and relevant notation. Section III derives the nonperturbative equation of motion method of the simulation of FSRS signals. The calculated FSRS signals are presented and discussed in Section IV. Our main findings are outlined in Section V.

II. HAMILTONIAN, SCHRÖ DINGER EQUATION, AND INITIAL CONDITIONS We consider a chromophore with electronic ground state |e0⟩, two closely spaced low-lying vibronically coupled excited electronic states |e1⟩, |e2⟩, and a third higher-lying electronic state |e3⟩. The molecular Hamiltonian is written as H = Hg + Hle + Hhe (1) (the subscripts g, le, and he stand for ground, low-excited, and higher-excited) where Hg = |e0⟩h0⟨e0|

(2)

Hle = |e1⟩(h1 + ϵ1 − iγle)⟨e1| + |e 2⟩(h2 + ϵ2 − iγle)⟨e 2| + V (|e1⟩⟨e 2| + |e 2⟩⟨e1|)

(3)

Hhe = |e3⟩(h3 + ϵ3 − iγhe)⟨e3|

(4)

V is the vibronic coupling of the states |e1⟩ and |e2⟩, while we ignore possible vibronic couplings of |e1⟩ and |e2⟩ with |e0⟩ and |e3⟩, which are assumed to be energetically well-separated. The ϵi are the vertical electronic excitation energies. Employing the linear vibronic-coupling model,45 we take into account two vibrational modes with dimensionless normal coordinates Qt, Qc, dimensionless momenta Pt, Pc, and frequencies ωt, ωc. The vibrational Hamiltonians h0, h1, h2, and h3 are written as

h0 = TN + V0 TN =

1 (ωcPc2 + ωtPt2) 2

V0 =

1 (ωcQ c2 + ωt Q t2) 2

h1 = h0 + κ1Q t h2 = h0 + κ2Q t h3 = h0 + κ3Q t

Qt is a totally symmetric “tuning mode” with excited-state gradients κ1, κ2, and κ3. Qc is a nontotally symmetric “coupling mode” that couples the states |e1⟩ and |e2⟩ in first order V = λcQ c

where λc is the linear vibronic coupling constant.45 The chromophore may be perturbed by an environment. In FSRS, the chromophore is monitored within a short time 3287


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The Journal of Physical Chemistry A interval determined by the duration of the femtosecond probe pulse. On this time scale, vibrational energy relaxation can be neglected, while electronic dephasing must be taken into account. Electronic dephasing can be described by the inclusion of intermolecular vibrational modes in the system Hamiltonian.46,47 In the density matrix formalism, electronic dephasing can be treated either phenomenologically48 or microscopically49 in the system−bath framework. In the present work, electronic dephasing is accounted for phenomenologically by including effective electronic lifetime parameters γle and γhe in the Hamiltonians of eqs (3) and (4).21,22 The Hamiltonian describing the interaction of the chromophore with external fields consists of two contributions HF(t ) = Hac(t ) + Hpp(t )

(5)

Figure 1. (top) Schematic view of the pulses in a FSRS experiment: actinic pulse (red), Raman pump pulse (blue), Raman probe pulse (magenta). (bottom) The level scheme and allowed electronic transitions.

The first term is responsible for the interaction of the chromophore with the actinic pulse Hac(t ) = −X† ,ac(t ) − X ,*ac(t )

(6)

while the second term describes interaction of the chromophore with the Raman pump and probe pulses

assuming that the molecule is initially in its ground electronic state |e0⟩ and its ground vibrational state |0⟩. δpr̅ is a technical parameter that is set to δpr̅ = 3δpr in the calculations. If the actinic pulse is applied, it is convenient to set its arrival time to zero, τac = 0. Then τ is the time delay between the actinic pump pulse and the Raman pump−probe pair. In this case, one must solve the time-dependent Schrödinger equation

Hpp(t ) = −X†(,pu(t ) + ,pr(t )) − X(,*pu(t ) + ,*pr(t )) (7)

The Hamiltonians (6) and (7) are written in the rotating wave approximation (RWA). The transition dipole moment operator is specified as X = v02|e0⟩⟨e 2| + v13|e1⟩⟨e3| + v23|e 2⟩⟨e3|

∂ |Ψ(t )⟩ = −i(H + HF(t ))|Ψ(t )⟩ ∂t

(8)

The electronic states |e1⟩ and |e2⟩ are assumed to be optically dark (v01 = 0) and bright (v02 ≠ 0), respectively, from the electronic ground state. The parameters v13 and v23 represent optical couplings of the low-lying electronic states |e1⟩, |e2⟩ with the higher-lying state |e3⟩. The laser fields are written as ,α(t ) = λαEα(t − τα)exp{i(k αr − ωαt )}

with the initial condition |Ψ(t = −δac̅ )⟩ = |e0⟩|0⟩

(9)

III. EVALUATION OF FEMTOSECOND STIMULATED RAMAN SCATTERING SIGNALS The numerical solution of the Schrödinger eq 12 yields the total (complex-valued) polarization P(t ) = ⟨Ψ(t )|X |Ψ(t )⟩

⎧ (t − τα)2 ⎫ ⎬ Eα(t ) = exp⎨−4ln2 δα2 ⎭ ⎩

(14)

It can be Fourier-decomposed as P(t ) =

δα being the pulse durations (full-width at half maximum). In FSRS, the actinic pulse comes first, while the Raman pump and probe pulses come together with a time delay.1 Hence: τpu = τpr ≡ τ

∑ Pk(t )eikr k

where the wave vector takes the values of k = nackac + npukpu + nprkpr (nac, npu, and npr are arbitrary integers). We need to extract the FSRS polarization, which obeys the phase matching condition

The corresponding pulse sequence as well as the scheme of the electronic levels and allowed transitions are sketched in Figure 1. If the actinic pulse is not applied (λac = 0), we can put τ = 0. The time-dependent Schrödinger equation describing the evolution of the chromophore under the influence of the Raman pump−probe pair reads (ℏ = 1)

k = k pr

(15)

In the simplest version of FSRS, the actinic pulse is not used (λac = 0). The Raman pump and probe pulses are applied to the system in its ground state, and the FSRS polarization can be evaluated as the third-order pump−probe polarization (3) 2 PFSRS (t ) ∼ λpu λpr

(10)

(16)

If the system is pre-excited by a weak actinic pulse, the FSRS signal is determined by the fifth-order polarization

In the absence of the actinic pulse, it is solved with the initial condition |Ψ(t = −δpr̅ )⟩ = |e0⟩|0⟩

(13)

where δac̅ = 3δac .

where λα, kα, ωα, and τα denote the effective amplitude, wave vector, frequency, and the arrival time of the pulse, where α = ac, pu, pr. The dimensionless pulse envelopes are assumed to be Gaussian

∂ |Ψ(t )⟩ = −i(H + Hpp(t ))|Ψ(t )⟩ ∂t

(12)

(5) 2 PFSRS (t ) ∼ λac2λpu λpr

(11) 3288

(17) DOI: 10.1021/acs.jpca.5b12316 J. Phys. Chem. A 2016, 120, 3286−3295

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The Journal of Physical Chemistry A

pump−probe pair couples (resonantly or off-resonantly) the low-excited and higher-excited electronic states.1,7 This experiment yields the polarization

Below we consider the two cases separately. The evaluation of the FSRS signal induced by a strong actinic pulse (beyond second order in the system−field coupling) is considered in the Appendix. The FSRS polarization PFSRS(t) is then a function of λ2ac, with the leading term in the perturbation expansion given by eq 17. A. Third-Order Polarization without Actinic Pulse. Within the RWA, any two-pulse-induced third-order polarization can be Fourier-decomposed into four different directions: k = kpu, kpr, 2kpu − kpr, and 2kpr − kpu (the contributions along 3kpr, 2kpu + kpr, etc. are forbidden within the RWA). This result is independent of the number of electronic states in the system Hamiltonian and of the initial state of the system. As shown in ref 31, the separation of these four contributions requires the evaluation of the total polarization (14) at four different phase angles. The FSRS polarization for the phase-matching condition (15) can be calculated as31 (3) PFSRS (t ) =

⎛ π⎞ 1 ⎛ Re⎜P(t , 0) + P ⎜t , ⎟ + P(t , π ) ⎝ 2⎠ 2 ⎝ ⎛ 3π ⎞⎞ ⎟⎟ + P ⎜t , ⎝ 2 ⎠⎠

(5) PFSRS (t ) = Ple(t )

C. The Femtosecond Stimulated Raman Scattering Signal. Once P(m) FSRS(t) (m = 3, 5) has been evaluated, the intensity of the FSRS signal is computed as follows:21,22 IFSRS(ω) =

Epr(ω) =

(22)

1 2π

∞

∫−∞ eiωtEpr(t )dt

(23)

and (̃ m) PFSRS (ω) =

1 2π

∞

(̃ m) (t )dt ∫−∞ eiωtPFSRS

(24)

Here

(18)

(̃ m) (m) (m) PFSRS (t ) = PFSRS (t ) − Poff (t ) (5) where P(3) FSRS(t) and PFSRS(t) are calculated via eqs (18) and (m) (19), respectively, and P(m) off (t) is calculated as PFSRS(t) but without pump pulse (λpu = 0). Hence, the computation of (3) P̃ FSRS (t) requires the numerical solution of five auxiliary Schrödinger equations: four for the evaluation of P(3) FSRS(t) via eq 18 and one for the evaluation of P(3) off (t). Alternatively, P̃(3) FSRS(t) can be evaluated via the wave function version of the equation-of-motion phase-matching approach.53−55 In this case, the numerical solution of five (without ESA) or six (with ESA) auxiliary Schrödinger equations is required. The evaluation of P̃ (5) FSRS(t) via eq 19 necessitates, in general, the numerical solution of 15 Schrödinger equations, five for Pg(t), Ple(t), and P(3) FSRS(t), respectively. If eqs (20) and (21) are applicable, the numerical effort is significantly reduced.

(19)

IV. NUMERICAL ILLUSTRATIONS A. Model and Computational Details. As an illustrative application, we simulated FSRS spectra of a two-mode model representing the CI of the S1(1B3u(nπ*)) and S2(1B2u(ππ*)) states in pyrazine.43,44 We included a higher-lying excited electronic state |e3⟩ to account for ESA. The mode ν10a of B1g symmetry is the single normal mode of pyrazine that can couple the 1B3u and 1B2u states in first order. The totally symmetric ring-bending mode ν6a is included as the dominant tuning mode of pyrazine. The numerical values of the parameters of the model are taken from refs 43 and 44. The frequencies of the coupling and tuning modes are ωc = 952 cm−1 (2π/ωc = 35 fs) and ωt = 597 cm−1 (2π/ωt = 56 fs). The intrastate electron-vibration coupling constants are κ1 = −847 cm−1 and κ2 = 1202 cm−1. The interstate vibronic coupling constant is λc = 2110 cm−1. The vertical excitation energies are ϵ1 = 31 800 cm−1 and ϵ2 = 39 000 cm−1. The parameters specifying the higher-excited state are as follows: κ3 = 1613 cm−1 and ϵ3 = 70 815 cm−1. Cuts through the potential-energy surfaces along the normal coordinates Qt and Qc are shown in Figure 2.

Here Pg(t) and Ple(t) are the contributions coming from the electronic ground state and the lower-excited electronic states, respectively. The explicit formulas for the calculation Pg(t) and Ple(t) are given by eq A22. P(3) FSRS(t) is subtracted to remove the contribution without actinic pulse. Note that eq 19 can directly be applied for the simulation of signals measured in broad-band impulsive vibrational spectroscopy (BB-IVS).52 If the actinic pulse is weak and off-resonant, only the groundstate contribution remains: (5) (3) PFSRS (t ) = Pg(t ) − PFSRS (t ) + O(λac4)

8π 2lC (̃ m) * (ω)PFSRS (ω)} ωIm{Epr 3n

Here l, C, n, and ω denote the cell length, number of molecules per unit volume, refractive index, and dispersive frequency of the probe field, respectively,

Here, the total polarization P(t, ϕpu) is computed by the Schrödinger eq 10 with the initial condition (11) for ϕpr ≡ kprr = 0 and specific value of ϕpu ≡ kpur. The procedure of the obtaining P(3) FSRS(t) via eq 18 can be interpreted in terms of phase cycling,50 since both approaches are equivalent for weak pulses. B. Fifth-Order Polarization with Actinic Pulse. Let us assume that the interaction of the system with the Raman pump−probe pair can be neglected during the action of the actinic pulse. This requirement holds if the actinic pulse and the Raman pulses are temporally well-separated or if the pulses overlap, but the amplitude of the actinic pulse is much higher than the amplitude of the Raman pulses, λac ≫ λpu, λpr.51 The detailed derivation of the fifth-order FSRS polarization in the phase-matching direction (15) is given in the Appendix. The result is (5) (3) PFSRS (t ) = Pg(t ) + Ple(t ) − PFSRS (t ) + O(λac4)

(21)

(20)

Note that (3) Pg(t ) = PFSRS (t ) + O(λac2)

P(5) FSRS(t) thus contains the net contribution due to the excitation of the ground state by the actinic pulse. In a common variant of the FSRS experiment, the carrier frequency of the actinic pulse is in resonance with the excitation energy of the low-excited electronic state(s), while the Raman 3289


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Figure 2. Cuts through the adiabatic potential-energy surfaces of the electronic ground state |e0⟩ (blue), the two low-lying excited electronic states |e1⟩ (green), |e2⟩ (red), and the higher-lying excited electronic state |e3⟩ (black) along the normal coordinates Qt (a) and Qc (b).

Figure 3. Intensity-scaled FSRS spectra as a function of the Raman shift for different ωpu = ωpr indicated on the right. No actinic pulse; ESA is neglected.

The durations of the Raman pump and probe pulses (fullwidth at half maximum) are set to δpu = 500 fs and δpr = 5 fs. For the pulse amplitudes, we take λpu = λpr = 0.8 cm−1 (which is in the weak-field limit). The actinic pulse is specified by δac = 5 fs and λac = 8 cm−1. This value of the actinic pulse amplitude is still in the weak-field limit but insures the fulfillment of the requirement λac ≫ λpu, λpr which guarantees the validity of the present method of the evaluation of FSRS signals for overlapping actinic and pump pulses. The carrier frequencies of the pulses are varied. The dipole coupling of the electronic ground state with the bright electronic state is set to unity (v02 = 1). The parameters v13 and v23 representing dipole couplings of the low-lying electronic states with the higher-lying state are varied. As in ref 42 the excited-state lifetime parameter is set to γle = 175 cm−1 (γ−1 le = 30 fs). This is close to the electronic dephasing 16 parameter of 152 cm−1 (γ−1 to le = 35 fs), which was shown reproduce the absorption spectrum of pyrazine measured in the gas phase at room temperature.56 γhe is set to zero. The time-dependent Schrö dinger equations are solved numerically on a grid, as explained in detail in ref 57. The field-matter interaction is treated exactly. The fourth-order Runge−Kutta method is used for the numerical propagation of the wave functions. B. Femtosecond Stimulated Raman Scattering Signals. We start with the consideration of the simplest FSRS signal, where the actinic pulse is not present and the Raman pump and probe pulses are applied to the molecule in its ground electronic state. In this case, the FSRS signal is a thirdorder signal, and the FSRS polarization is evaluated via eq 18. Figure 3 shows resonant FSRS spectra for v13 = v23 = 0 (no ESA). The signals are plotted as a function of the Raman shift ω−ωpr for different values of the carrier frequencies of the pump and probe pulses. The carrier frequencies are assumed to be the same (ωpu = ωpr) and are varied from 30 000 to 40 000 cm−1 in steps of 1000 cm−1. This range of carrier frequencies allows one to record spectral features in both Stokes and antiStokes regions. With ωpu = ωpr ≈ 38 000 cm−1 the bright electronic state is probed, while for ωpu = ωpr ≈ 31 000 cm−1 the pulses are in resonance with the dark state. The FSRS signals exhibit two types of peaks, stationary and moving, which correspond to different pathways in the perturbative formalism.42 When the first two interactions of the system are with the pump and probe pulses, stationary

peaks are produced. When the first two interactions are solely with the pump pulse, moving peaks result. The spectra in Figure 3 exhibit both types of peaks. The stationary peaks reveal the energy levels of the tuning mode at ±ωt, ±2ωt, etc. The moving peaks scale linearly with ωpu. The first overtone of the coupling mode at ±2ωc is clearly seen in Stokes and antiStokes regions for ωpu = ωpr = 31 000 and 32 000 cm−1, which are in resonance with the dark state. When ωpu = ωpr are in resonance with the bright state, the signatures of the coupling mode are missing. The peaks produced by the coupling mode of the pyrazine model in spontaneous Raman spectra behave similarly.58 Figure 4 illustrates the influence of ESA on the FSRS signals. The actinic pulse is not applied, and the FSRS polarization is calculated by eq 18. Panel (a) corresponds to the optical coupling of the low-lying dark excited state to the higher-lying state (v13 = 1, v23 = 0 in eq 8), while panel (b) corresponds to the optical coupling of the low-lying bright state to the higherlying state (v23 = 1, v13 = 0). The comparison of Figure 3 with Figure 4a,b reveals that the spectra for nonresonant pump− probe frequencies are almost unaffected by ESA: The populated vibronic levels of the |e1⟩ − |e2⟩ manifold are out of resonance with the Franck−Condon accessible vibrational levels of the |e3⟩ state. As expected, ESA manifests itself significantly at frequencies ωpu = ωpr = 34 000−36 000 cm−1, which couple the low-lying vibronic levels of the |e1⟩ state with those of the |e3⟩ state. For these ωpu = ωpr, the shapes of the ESA peaks in panels (a) and (b) differ, but both panels reveal moving peaks at multiple frequencies of the tuning mode. These moving peaks featuring the tuning mode can be considered as signatures of ESA. If the higher-lying excited state is shortlived (γhe > ωc, ωt), the ESA peaks merge and produce broad featureless structures (not shown). ESA produces additional weak peaks at ±ωc, ±2ωc, etc. and combination peaks at ωc ± ωt (recall that the fundamental ±ωc is absent without ESA). This indicates that ESA can be used for probing the coupling mode (cf. ref 59). Figure 5 exemplifies nonstationary preparation of the system by a weak actinic pulse, which is either off-resonant (ωac = 22 900 cm−1, panel (a)) or resonant (ωac = 39 000 cm−1, panel (b)) with the bright state |e2⟩. The Raman pump and probe 3290


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Figure 4. Intensity-scaled FSRS spectra as a function of the Raman shift for different ωpu = ωpr indicated between the panels. No actinic pulse; ESA is allowed. (a) ESA from the dark state. (b) ESA from the bright state.

Figure 5. FSRS signals with preparation by (a) an off-resonant (ωac = 22 900 cm−1) and (b) resonant (ωac = 39 000 cm−1) weak actinic pulse. The Raman pump and probe pulses are in electronic resonance (ωpu = ωpr = 39 000 cm−1); ESA is absent. The delay time τ is indicated between the panels.

pulses are in resonance with the |e0⟩ − |e2⟩ transition (ωpu = ωpr = 39 000 cm−1). ESA is absent (v13 = v23 = 0). The time delay τ between the actinic pulse and the Raman pump−probe pair is indicated between the panels. In this case, FSRS is a fifth-order spectroscopy. The polarizations are calculated via eq 20 (panel (a)) and eq 19 (panel (b)). The signals exhibit conspicuous progressions of stationary peaks and dips revealing fundamentals and overtones of the tuning mode. The shapes of the peaks differ considerably in the Stokes and anti-Stokes region. As in the case of stationary preparation (ωpu = ωpr = 39 000 cm−1 in Figure 3), the coupling mode does not reveal itself in the signal. The amplitudes and shapes of the peaks are delay-time dependent, reflecting wavepacket motion in the ground electronic state and in the coupled low-lying excited electronic states. Since 3ωt = 1791 cm−1 and 2ωc = 1902 cm−1, 3ωt ≈ 2ωc corresponds to an

overtone resonance, which manifests itself in the split peak shapes in both Stokes and anti-Stokes sides of the spectra in Figure 5. The amplitudes of the peaks in Figure 5b exhibit more pronounced dynamics than those in Figure 5a. This can be understood as follows. On the one hand, an off-resonant actinic pulse excites exclusively the tuning mode in the ground state, because the coupling mode is Franck−Condon inactive. A resonant actinic pulse, on the other hand, prepares vibrational wavepackets in both ground and excited electronic states. Because of the vibronic coupling, the wavepacket in the bright electronic state involves both tuning and coupling modes. The wavepacket is highly anharmonic, while its counterpart in the electronic ground state is harmonic. The wave packet evolutions in the ground and excited electronic states are therefore very different, which results in pronounced 3291


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condition (13). Let us introduce the time evolution operator corresponding to this Schrödinger equation

modulations of the peak amplitudes in Figure 5b. This suggests that the duration of the actinic pulse can be fine-tuned to enhance amplitudes of selected peaks. Strong (giving the contributions beyond ∼λ2ac) actinic pulses may also be of interest.

|Ψ(t )⟩ =

Resonant Femtosecond Stimulated Raman Spectra: Theory and

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