Thermal Population Fluctuations in Two-Dimensional Infrared

Article pubs.acs.org/JPCB

Cite This: J. Phys. Chem. B 2018, 122, 3647−3654

Thermal Population Fluctuations in Two-Dimensional Infrared Spectroscopy Captured with Semiclassical Mechanics Prashanth Ramesh and Roger F. Loring* Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, United States S Supporting Information *

ABSTRACT: Time-resolved two-dimensional (2D) infrared spectra of the asymmetric stretch mode of solvated CO 2 show distinct features corresponding to ground- and excited-state thermal populations of the bend modes. The time-dependence of these peaks arises in part from solvent-driven thermal fluctuations in populations of the lower-frequency bend modes through their coupling to the higher-frequency asymmetric stretch. This observation illustrates the capacity of multidimensional vibrational spectroscopy to reveal details of the interactions among vibrational modes in condensed phases. The optimized mean-trajectory (OMT) method is a trajectory-based semiclassical approach to computing the vibrational response functions of multidimensional spectroscopy from a classical Hamiltonian. We perform an OMT calculation of the 2D vibrational spectrum for two coupled anharmonic modes, with the lower-frequency mode undergoing stochastic transitions in energy to mimic solvent-induced fluctuations in quantum populations. The semiclassical calculation reproduces the influence of thermal fluctuations in the low-frequency mode on the 2D spectrum of the high-frequency mode, as in measured spectra of solvated CO2.

I. INTRODUCTION As with their analogs in magnetic resonance, two-dimensional (2D) infrared spectra reflect not only the frequencies present in a mechanical system, but also the interactions among degrees of freedom characterized by those frequencies. Diagonal peaks in the 2D spectrum record frequencies, while cross peaks can reveal dynamics induced by interactions. The dependence of the shape and amplitude of cross peaks on the delay time in a four-wave-mixing measurement can reveal the dynamics of processes such as incoherent and coherent energy transfer among vibrational modes1,2 and the vibrational analog of chemical exchange in magnetic resonance.3−14 Infrared spectroscopy has been extensively applied15−22 to carbon dioxide as a solute in room temperature ionic liquids, because of the potential application of these solvents in carbon capture. Two-dimensional spectroscopy in particular probes structure and dynamics of solvation20,21 in these systems. Brinzer et al.15,16 and Giammanco et al.17 have measured 2D vibrational spectra of CO2 in various ionic liquids. The chromophore in these studies is the infrared-active asymmetric stretch, which undergoes an absorption red shift upon excitation of the bend mode. The 2D spectrum of the asymmetric stretch therefore shows distinct diagonal peaks corresponding to ground-state and first-excited-state populations of the bend modes. With increasing evolution time, diagonal peaks diminish in amplitude and cross peaks appear, reflecting thermal fluctuations in population of the bend modes, driven by the solvent. This process is well-described by a twostate kinetic model15−17 of incoherent population transfer for the bend modes. Such studies exemplify that 2D spectroscopy of a high-frequency mode can report on the detailed dynamics © 2018 American Chemical Society

of a multimode system to which it is coupled, here the bend modes and the solvent. Calculating 2D infrared spectra23,24 requires propagating nuclear degrees of freedom on a single (ground-state) electronic surface. Fully classical mechanical calculations25−31 of spectroscopic response functions can deviate qualitatively from quantum mechanical results at long times, although relaxation processes in condensed phases can limit accessible time scales to a regime in which classical dynamics is approximately applicable. Numerically correct quantum mechanical treatments32 of nonlinear infrared spectroscopy are not generally feasible for many degrees of freedom. Several approximation strategies have been employed to incorporate quantum dynamics into the calculation of nonlinear vibrational spectra, including the partition of a large system into a quantum subsystem interacting with a classical or semiclassical environment,33−38 the application of a tight-binding Hamiltonian with couplings dependent on structures generated in a molecular dynamics simulation,33,39−41 and mapping protocols connecting vibrational frequencies to collective coordinates, determined from quantum studies of subsystems.20,21,42−48 Another approach is to adopt semiclassical approximations49−53 to quantum dynamics that, in principle, can be applied uniformly to all degrees of freedom. The optimized mean-trajectory approximation (OMT)54−58 is a semiclassical procedure for computing nonlinear vibrational Special Issue: Benjamin Widom Festschrift Received: December 8, 2017 Revised: January 30, 2018 Published: January 31, 2018 3647

DOI: 10.1021/acs.jpcb.7b12122 J. Phys. Chem. B 2018, 122, 3647−3654

Article

The Journal of Physical Chemistry B

ground vibrational state are shown in Figure 1. Each diagram is a schematic plot of chromophore action variable versus time.

response functions such as the third-order response function relevant to 2D spectroscopy.59 The method employs classical trajectories subject to semiclassical quantization rules, and is based on a mapping between the double-sided Feynman diagrams59−61 customarily used to represent processes contributing to a quantum spectroscopic response function and semiclassical diagrams54 representing radiation-induced discontinuous transitions between classical trajectories. The OMT circumvents62 the inherent numerical challenge of applying semiclassical propagators to nonlinear response functions: evaluating multidimensional integrals over highly oscillatory integrands.63 This significant benefit carries the cost of assuming the existence of good action and angle variables, even for anharmonic systems of many degrees of freedom for which such variables may not exist. However, in the application to ultrafast condensed phase spectroscopy in which time scales are set by population relaxation or dephasing rates, application of the OMT only requires that the system behaves on short time scales approximately as though good action and angle variables existed. The OMT has been demonstrated to incorporate quantum anharmonic effects,54 line broadening from both dissipation and pure dephasing,56 and vibrational coherence transfer between near-resonant anharmonic vibrations.57 The OMT has been extended to include electronic as well as vibrational transitions.64 In the present work we apply the OMT to a model inspired by solvated carbon dioxide.15,17 This model is based on a classical Hamiltonian for two nonlinearly coupled anharmonic modes: a higher-frequency mode qualitatively representing the asymmetric stretch and a lower-frequency mode qualitatively representing one bend mode. Hamiltonian parameters are assigned by comparison to measured spectra of solvated CO2. The solvent is not explicitly represented; rather, its effects are incorporated through a stochastic process that adds and removes energy from the bend mode. Brinzer et al.15 analyzed 2D spectra of solvated CO2 with a stochastic model in which the frequency of the asymmetric stretch mode fluctuates between two values. The present model treats the effects of the solvent on the bend mode phenomenologically but fully includes the mechanics of the interacting bend and stretch modes within the model Hamiltonian. The OMT treatment of this model reproduces the phenomena observed in the measured spectra. The OMT approach is briefly summarized in Section II, and the model Hamiltonian is defined. Calculated 2D spectra are presented and discussed in Section III, with conclusions given in Section IV. Technical details are reserved for Supporting Information.

Figure 1. Semiclassical OMT diagrams for the amplitude of the rephasing (echo) signal, R++−(t3, t2, t1) in eq 1. The chromophore vibration is assumed to be thermally unexcited.

Horizontal solid lines represent continuous classical trajectories with time durations labeled, and vertical dashed lines indicate action changes of ±ℏ/2 resulting from the radiation-matter interaction. Solid vertices indicate phase space points at which values of momenta and coordinates are collected. Labels + and − indicate phases as defined below in eq 6. For the present model of an anharmonic chromophore mode c coupled to an anharmonic dark mode d, the sum of diagrams in Figure 1 has the value56 R++ −(t3 , t 2 , t1) =

⎛ i ⎞ 3 ⎜ ⎟∑ϵ r ⎝ ℏ7 ⎠

∫ dz(1) ∫ dz(2) ∫ dz(3)

r=1

×

γ(Jc(1)

− (3)

× δ(ϕ

ℏ/2)γ(Jd(1))ΔF(z(1))δ(ϕ(2) −

− ϕt(1)) 1

(1) (1) (2) ϕt(2))Δ(1) σr Δσr′ Q −(z )Q +(z t1 ) 2

× Q +(z(3))Q −(z(3) t3 ) (1) ∞

γ(J ) = ℏ ∑ δ(J − (n + 1/2)ℏ) (2)

n=0

ΔF(z) = F(z)

II. METHOD AND MODEL In the OMT approximation, a vibrational response function is calculated from classical trajectories subject to semiclassical quantization conditions.54 Discontinuous transitions between trajectories represent radiation-matter interactions. For a single nuclear degree of freedom, the OMT method approximates the dynamics of a quantum operator proportional to nl nr with classical trajectories of action ℏ(nl + nr + 1)/2. Dynamics of a quantum population map onto classical trajectories with action equal to a half-odd-integral multiple of ℏ, while single quantum coherences are represented by trajectories with action equal to an integral multiple of ℏ. The three OMT diagrams representing the rephasing (echo) component of a thirdorder response function for a chromophore initially in its

− F (z ) Jc → Jc −ℏ /2

F (z ) =

Jc → Jc +ℏ /2

(3)

exp( −βH(z))

∫ dz′exp(−βH(z′))γ(Jd′ )γ(Jc′)

(4)

Δ(±s) = ℏ2δ(Jc(s + 1) − (Jc(s) ± ℏ/2))δ(Jd(s + 1) − Jd(s))

(5)

Q ±(z) = (qc ∓ ipc /(mcωc))/2

(6)

The index r in eq 1 labels the diagrams in Figure 1. The factor ϵr takes the value −1 for the first two diagrams in Figure 1 and is 1 for the third diagram. This sign originates from quantum commutators within the response function. A point in the phase space of both degrees of freedom is denoted z. The 3648


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ν̃c = 2370 cm−1, χ̃cc = −12 cm−1, χ̃cd = −12 cm−1, ν̃d/2 + χ̃dd = 337 cm−1. Using a measured value65−68 of χ̃dd = 1.6 cm−1 for the anharmonicity of the bend mode of gas phase CO2, we take ν̃d = 670 cm−1. Next, expanding the chromophore potential in eq 8 to fourth-order in qc and applying quantum mechanical perturbation theory65 permits the expression of potential energy anharmonicity parameters from eqs 8 and 9 in terms of the empirical parameters in eq 10

factors γ(J) in eq 1, defined in eq 2, constrain initial action (1) values J(1) c and Jd with semiclassical quantization conditions. The initial action of the dark mode J(1) d is quantized to a halfodd-integral value of ℏ representing an initial quantum population, while the initial chromophore action Jc(1) is quantized to an integral value of ℏ to represent the singlequantum coherence produced by the earliest radiation-matter interaction.54 Initial phase space points are further weighted58 by the finite difference ΔF in eq 3, with F(z) a modified58 classical thermal phase space distribution in eq 4. The finitedifference distribution arises from the effect on the thermal density operator of the first radiation-matter interaction. The factors Δ(s) σ in eq 1 with σ → ± , defined in eq 5, depend on the identity of the diagram and constrain the chromophore action to change by ±ℏ/2 with each radiation-matter interaction. Each vertex in a diagram in Figure 1 labeled + or − corresponds to a phase space point at which Q± in eq 6 is determined. In eq 6, the effective mass and harmonic angular frequency of the chromophore are denoted mc and ωc. These factors give a semiclassical representation of the transition dipole, under the assumption that this operator is proportional to the chromophore coordinate, with the coefficient of proportionality suppressed. The amplitude of the absorptive 2D spectrum is defined by59

(7)

R̂ ++−(ω3, ω1; t2) denotes the two-dimensional one-sided Fourier transform of R++−(t3, t2, t1) in eq 1 with respect to t1 and t3. The particular nonrephasing response function relevant to the absorptive spectrum R+−+(t3, t2, t1) is obtained from eq 1 (1) (1) with the replacement Q−(z(1))Q+(z(1) t1 ) → Q+(z )Q−(zt1 ). The time-evolution in eq 1 is carried out with a classical Hamiltonian for two anharmonically coupled normal modes with potential energy V (qc , qd) = VM(qc) + Vd(qd) + ccdqc2qd2

(8)

Vd(qd) = md ωd2qd2 /2 + cdqd4

(9)

(11)

ccd = χcd (mcmd ωcωd /ℏ2)

(12)

cd = (2χdd /3)(md ωd /ℏ)2

(13)

with χrs = hcχ̃rs. Equations 11−13 together with the spectroscopic data specify the potential energy in eqs 8 and 9, giving a two-mode model whose 2D infrared spectrum resembles that of CO2 in the region of the frequency of the asymmetric stretch mode. The form of the model potential in eqs 8 and 9 was chosen to minimize the number of parameters while mimicking the relevant portion of the 2D spectrum of CO2; other choices could also have met these goals. The quartic form of the dark mode potential was chosen because it is even in coordinate and generates a nonzero value of χ̃dd in the perturbation expansion in eq 10. The biquadratic coupling between the modes in eq 8 was chosen because it yields a nonzero value of χ̃cd in eq 10; cubic terms proportional to qcq2d or q2c qd would also have contributed65 to this parameter. Our focus is the effect on the chromophore mode of thermal population fluctuations in the dark mode. Rather than extending the potential of eq 8 to include explicit solvent degrees of freedom coupled to the dark mode, we introduce a stochastic process that induces transitions in Jd, the action of the dark mode. In the OMT approximation, the evolution of the population of state nd maps to classical trajectories with action Jd = (nd + 1/2)ℏ. Quantum transitions between nd = 0 and nd = 1 are then represented by transitions in classical action of magnitude ℏ. In contrast to the transitions induced by radiation in which the value of the angle is held constant,54 for these incoherent thermal transitions, the angle value is randomized after the event, as described in Supporting Information. We superpose onto time-evolution of the dark mode according to the potential energy in eq 8, a stochastic process in which Jd → Jd + ℏ with rate constant ku and Jd → Jd − ℏ with rate constant kd. The statistical averaging in the response function is generalized from the explicit thermal average over two degrees of freedom in eq 1 to include an average over these stochastic trajectories, which are generated by a Kinetic Monte Carlo procedure69 described in Supporting Information. The values of the rate constants ku and kd must be determined externally to this calculation. We take ku = 0.015 ps−1, with kd set by detailed balance. In CO2 at 298 K, the relative thermal populations of the ground and excited states of one bend mode are approximately 0.96 and 0.04, so that the populations of the ground and first excited states of the degenerate pair of modes are approximately 0.93 and 0.07. To mimic these populations in our two-mode model, we adopt a higher temperature of T = 383 K to double the excited-state thermal population relative to room temperature, giving kd = 0.184 ps−1. Brinzer et al.15,16 fit their measured 2D spectra of

̂ −(ω3 , −ω1; t 2) R abs(ω3 , ω1; t 2) = Im[R++ ̂ (ω3 , ω1; t 2)] + R+−+

D = −(ℏωc)2 /(4χcc )

The isolated chromophore mode has a Morse potential VM characterized by harmonic frequency ωc and well depth D. The dark mode has the quartic potential in eq 9, and the two modes are coupled with the biquadratic term in eq 8. Parameters in this model potential are assigned values to mimic the 2D infrared spectra of the asymmetric stretch mode of CO2 in an ionic liquid solvent measured by Brinzer et al.15,16 These parameters are chosen with a two-step procedure, starting with an empirical truncated expansion in anharmonicity15,65 for the vibrational energy levels of the coupled chromophore and dark modes Eñ dnc = νd̃ (nd + 1/2) + νc̃ (nc + 1/2) + χdd ̃ (nd + 1/2)2 + χcc̃ (nc + 1/2)2 + χcd̃ (nc + 1/2)(nd + 1/2) (10)

This expression contains two harmonic frequencies, two anharmonicity parameters, and one anharmonic coupling strength. These parameters are evaluated by comparison to measured 15 spectroscopic transitions in solvated CO 2 . The transition frequencies shown in Figure 5 of ref 15 allow the evaluation of all but two of the parameters in eq 10: 3649


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excited states of the chromophore mode. The overtone peaks that are anharmonically red-shifted in ω3 resulting from transitions between the first and second excited states of the chromophore are not shown here. With increasing t2, there is increasing likelihood of thermal transitions between ground and excited populations of the dark mode during this interval, leading to cross peaks. The red arrow indicates a transfer of peak amplitude to a cross peak resulting from thermal transitions of the dark mode from the ground to the first excited state during the t2 interval, producing a red shift in ω3. Figure 2B shows as an example the uppermost OMT diagram from Figure 1 with the t2 interval colored in red. Below it is a schematic of an example stochastic trajectory of the action of the dark mode, indicating a transition during the t2 interval from Jd = ℏ/2 to Jd = 3ℏ/2, representing a thermal transition from the ground to the first excited state. Incoherent thermal transitions between ground- and excited-state populations are represented by action changes of ±ℏ, in contrast to the spectroscopic transitions involving one application of a raising or lowering operator which are semiclassically represented by action changes of ±ℏ/2. This stochastic trajectory is one example of a process that transfers amplitude as indicated by the red arrow in the spectrum. The lower-amplitude diagonal peak in Figure 2A results from processes in which the dark mode is excited for significant portions of the t1 and t3 intervals. The blue arrow in the spectrum shows transfer of amplitude to a cross peak resulting from processes such as that shown in the blue portions of the diagrams in Figure 2C. The upper image shows an OMT diagram, and the lower image shows an example of a stochastic trajectory of the action of the dark mode. The dark mode begins the t2 interval in the excited state and finishes in the ground state. An incoherent transition from the excited to the ground state of the dark mode results in a blue shift of the chromophore frequency in the ω3 dimension. While the OMT method avoids the “sign problem” of highly oscillatory integrands arising in other semiclassical methods,63 some numerical challenges remain. Principal among these is the necessity of calculating the time-domain response function as in eq 1 at sufficiently large numbers of values of t1 and t3 to permit the double Fourier transform in eq 7.56 If time propagation is carried out naively from the instant of the first radiation-matter interaction, i.e., from left to right in Figure 1, each choice of a new value of t1 requires repeating the calculation of the response function. Alemi and Loring56 implemented a more efficient “forward−backward” approach,70 in which the state of the system is sampled after the second interaction with the radiation field at time t1. Starting at the second vertex in any diagram in Figure 1, time is propagated backward for −t1 and forward for t2 and t3. One initial condition then provides the response function at any desired number of t1 and t3 values. The validity of this approach relies on the response function computation being an equilibrium calculation, on Liouville’s theorem, and on the approximate assumption that “good” action variables can be identified, so that applying a semiclassical quantization condition at a later time is equivalent to applying the same condition at an earlier time. The assumption of the existence of effectively good action variables underlies the justification of the rest of the OMT procedure54,55 as well. Calculations presented in Section III employ the forward−backward procedure of ref 56, modified for the present model as described in Supporting Information. Evaluating OMT diagrams as shown in Figure 1 requires implementing canonical transformations between Cartesian

solvated CO2 with a model in which the chromophore frequency undergoes transitions between two values. Their best fit value16 of ku is approximately a factor of 6 smaller than the value used here. The value of this rate constant sets the evolution-time-dependence of the 2D spectra, and also contributes to line shapes of spectral peaks. Our calculated spectra presented below do not include other line-broadening mechanisms present in experimental spectra and so are given nonphysical line widths by application of window functions prior to Fourier transformation. These spectra show a time evolution approximately comparable to that measured in ref 15 for CO2. A schematic 2D spectrum for the chromophore mode in Figure 2A illustrates the formation of cross peaks from

Figure 2. (A) Schematic 2D spectrum illustrates the formation of cross peaks by spectral exchange during the t2 interval. Arrows colored red or blue indicate amplitude transfer from diagonal peaks to cross peaks that are, respectively, red- or blue-shifted in the ω3 dimension. (B) Representative OMT diagram and stochastic trajectory for the action of the dark mode producing a red shift. The t2 interval is colored red in both diagrams. (C) Representative OMT diagram and stochastic trajectory for the action of the dark mode producing a blue shift. The t2 interval is colored blue in both diagrams.

exchange processes during the t2 interval. The higher-amplitude diagonal peak reflects time evolution of the chromophore during t1 and t3 with the dark mode mostly in its ground state, while the lower-amplitude diagonal peak results from chromophore dynamics with primarily the dark mode excited. At t2 = 0, these diagonal peaks dominate the spectrum. All peaks in Figure 2A result from processes in which all spectroscopic transitions are between the ground and first 3650


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Figure 3. Two-dimensional spectra for the two-mode model of eq 8. In (A), evolution time t2 = 0, and in (B), t2 = 55 ps. Each line is labeled with peak frequencies (ω1, ω3). Labels n and n* respectively denote frequencies of one-quantum chromophore transitions from n − 1 to n, with the dark mode in the ground or excited states.

frequency dark mode mostly occupies its ground state and by n* if the dark mode is primarily singly excited. From the fit of CO2 data15 to eq 10, frequencies 1 and 2 are approximately 2340 cm−1 and 2316 cm−1 and frequencies 1* and 2* are approximately 2328 cm−1 and 2304 cm−1, respectively. The chromophore anharmonic shift is −24 cm−1, whether or not the dark mode is excited. In Figure 3A, the evolution time t2 = 0. This spectrum shows anharmonically shifted dominant diagonal peaks at (1, 1) and (1, 2), arising from processes with the dark mode in the ground state. The initial interaction with radiation in each case is the fundamental 0 → 1 transition, with the second interaction probing either the same transition or the overtone 1 → 2. The two peaks are therefore shifted along the ω3 axis by the anharmonicity of the chromophore mode. The analogous pair of diagonal peaks with the dark mode excited occurs near (1*, 1*) and (1*, 2*). For the absorptive 2D spectrum as defined in eq 7, anharmonically shifted peak amplitudes are of different sign, as indicated by red (positive amplitude) and blue (negative amplitude) in Figure 3. Figure 3B shows the spectrum at t2 = 55 ps. As the model does not contain population relaxation of the chromophore mode, this spectrum resembles the asymptotic long t2 limit for this calculation. For the solvated CO2 system of ref 17, the vibrational lifetime of the asymmetric stretch mode has been measured to be 64 ps, longer than time scales of population dynamics of the bend mode. The diagonal peaks in Figure 3B have diminished in amplitude, and four cross peaks have appeared through thermal transitions in populations of the dark mode, like those indicated by blue and red arrows in Figure 2. The diagonal peak at (1, 1) has lost amplitude to a cross peak at (1, 1*) and the anharmonically shifted diagonal peak at (1, 2) has transferred amplitude to (1, 2*). The diagonal peak at (1*, 1*) in Figure 3A is no longer visible in Figure 3B, as amplitude has transferred to the cross peak at (1*, 1). Similarly, the anharmonically shifted peak at (1*, 2*) in Figure 3A is no longer apparent in Figure 3B, because of the growth of the cross peak at (1*, 2). The 2D spectra in Figure 3 at short and long times agree qualitatively with measured 2D spectra of the asymmetric stretch mode of solvated CO2.15,17 The t2 dependence of 2D spectra as in Figure 3 arises from incoherent excitation and deexcitation of the dark mode with

coordinates and momenta and action-angle variables. Timepropagation is carried out in the former, while quantized action jumps representing interactions with radiation or incoherent thermal transitions require the latter. Previous applications of the OMT54−57,71 have demonstrated that low-order classical mechanical perturbation theory in anharmonicity72,73 suffices to carry out these transformations in this application. Transitions in chromophore action were treated to second-order in the cubic anharmonicity of the Morse potential and to first-order in quartic anharmonicity of the dark mode and in the biquadratic coupling between modes. Transitions in action of the dark mode were treated to zeroth-order in anharmonicity, as described in Supporting Information. This perturbation theory is additionally applied to evaluate the thermal weight ΔF in eq 3 as also described in Supporting Information. Classical perturbation theory is only used to perform canonical transformations; all propagation is carried out with the full anharmonic Hamiltonian.

III. RESULTS AND DISCUSSION Two-dimensional infrared spectra from eqs 1 and 7 for the twomode model of eq 8 are shown in Figure 3. The stochastic transitions in energy of the dark mode take place throughout the calculation, including during the t1 and t3 intervals, and provide one source of line broadening of the chromophore spectrum. Besides this stochastic process, the model does not include direct solvent-induced dephasing of the chromophore mode that would broaden peaks associated with this mode. To smooth the spectra resulting from a discrete Fourier transform, we employ apodization, as described in Supporting Information, effectively broadening the lines with nonphysical dephasing. Our window function in both t1 and t3 dimensions cuts off at 7.9 ps, giving line widths approximately comparable to those of the solvated CO2 system of ref 15, for which the pure dephasing time is determined to be 2.6 ps. The spectra in Figure 3 were converged using 75 sets of initial angle variables for the chromophore and dark modes and 9950 stochastic Kinetic Monte Carlo trajectories for the action of the dark mode. To identify spectral features, we label the frequency of the n − 1 → n transition in the chromophore mode by n if the low3651


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reporting on thermally driven population fluctuations in a lowfrequency mode, as observed15−17 in solvated CO2. Both the thermal population dynamics and the radiation-induced quantum state changes are represented here by changes in values of classical action variables. The thermal dynamics are mimicked by a stochastic process. The method should be practically applicable to a completely mechanical model in which a bath of explicitly treated degrees of freedom induces classical mechanical energy fluctuations in a dark mode, thereby influencing a chromophore mode; this implementation remains for future work. The calculations presented here provide a distinct example of this classical-trajectory-based approach describing the phenomenology of multidimensional vibrational spectra of solvated anharmonic vibrations.

respective rate constants ku and kd as specified in Section II. The diagonal peaks should thus decay and cross peaks should grow in with increasing t2 with the sum of these rate constants.15,17 The decay of the diagonal peak at (1*, 1*) and the associated growth of the cross peak at (1*, 1) arise from the deexcitation of the excited dark mode during the t2 interval, resulting in a blue shift in ω3. Peak volumes calculated from 2D spectra as described in Supporting Information are shown for these features in Figure 4. Both peak volumes are

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.7b12122. Classical-mechanical action-angle perturbation theory, Kinetic Monte Carlo implementation, and apodization procedure (PDF)

■

Figure 4. Dependence of peak volumes on evolution time t2. The diagonal peak at (1*, 1*) reflects processes with spectroscopic transitions between ground and first excited states of the chromophore with the dark mode primarily in its excited state during t1 and t3 intervals, while the cross peak at (1*, 1) arises from deexcitation of the dark mode during t2, resulting in a blue shift in ω3. The volume of the diagonal peak (●) decays as the volume of the cross peak (▲) grows. Curves show fits to exponential time-dependence with decay rate equal to the sum of rate constants for excitation and deexcitation.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Roger F. Loring: 0000-0003-3954-9749 Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under CHE-1361484.

scaled by the volume at t2 = 0 of the dominant fundamental peak (1, 1). Therefore, the t2 = 0 value of the scaled (1*, 1*) peak volume is approximately equal to the ratio of the equilibrium populations of excited and ground states for the dark mode. Curves show fits to exponential time-dependence with decay rate equal to ku + kd. The dependence of peak amplitudes on t2 is shown to follow the thermal population dynamics of the dark mode. The process (1*, 2*) → (1*, 2) similarly arises from radiationless deexcitation of the dark mode, and the associated peak volume plots are very similar to those in Figure 4. The processes (1, 1) → (1, 1*) and (1, 2) → (1, 2*) result from thermal excitation of the dark mode. Peak volume plots for these features follow the same exponential kinetics as in Figure 4, but are significantly noisier for the same number of trajectories. This arises in part because the 2D spectrum for this model with initial frequency 1 is more congested than that with initial frequency 1*, as can be seen from Figure 3B, complicating the definition of peak volumes. The convergence of the calculation is also impeded by changes in line shape resulting from processes with multiple stochastic transitions in population of the dark mode during intervals t1 and t3.

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IV. CONCLUSIONS The semiclassical OMT approach has been shown to reproduce a range of phenomenology associated with 2D vibrational spectra of coupled anharmonic vibrations, including pure dephasing,56 energy relaxation,56 and coherence transfer.57 The calculations shown in Figures 3 and 4 demonstrate that the method can capture the spectrum of a high-frequency mode 3652


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