Letter pubs.acs.org/JPCL
Black-Box, Real-Time Simulations of Transient Absorption Spectroscopy Triet S. Nguyen, Joong Hoon Koh, Susan Lefelhocz, and John Parkhill* Department of Chemistry and Biochemistry, The University of Notre Dame, 251 Nieuwland Science Hall, Notre Dame, Indiana 46556, United States S Supporting Information *
ABSTRACT: We introduce an atomistic, all-electron, black-box electronic structure code to simulate transient absorption (TA) spectra and apply it to simulate pyrazole and a GFP-chromophore derivative. The method is an application of OSCF2, our dissipative extension of time-dependent density functional theory. We compare our simulated spectra directly with recent ultrafast spectroscopic experiments. We identify features in the TA spectra to Pauli-blocking, which may be missed without a first-principles model. An important ingredient in this method is the stationary-TDDFT correction scheme recently put forward by Fischer, Govind, and Cramer that allows us to overcome a limitation of adiabatic TDDFT. We demonstrate that OSCF2 is able to reproduce the energies of bleaches and induced absorptions as well as the decay of the transient spectrum with only the molecular structure as input.
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of electronic population changes are the most important features in TA experiments, but they are completely absent in ordinary RT-TDDFT. Moreover, it suffers from artifacts when multiple or resonant continuous fields are applied to a system of interest.35−37 Adiabatic functionals predict that the energies of excited states change following an electronic excitation. When the positions of excited states change, they can cause false bleaches and false absorptions. Recent advancements in theoretical methods and computing resources have made black-box RT-TDDFT simulations with dissipation possible.38−42 This paper extends our dissipative RT-TDDFT method, OSCF2, to treat dissipation in a TA spectrum. We combine OSCF2 with a simple correction scheme developed by Fischer et al.43 to treat the initial pump and calculate TA spectra in real time. Fischer’s reference correction simply subtracts unphysical oscillations of an initially excited density, allowing us to use a golden-rule mixture of excited densities as the starting point. We will highlight features of TA spectra caused by Pauli blocking,44−47 which ab initio theory is well suited to capture. The results also assess our two main approximations: the reference correction and OSCF2 as a model of dissipation. We examine a small molecule to demonstrate that the bleach signals correspond to the state prepared by the pump, that residual “false bleaches” are small, and that there is an orbital-specific structure in an ab initio model. Subsequently, we will employ a detailed system-bath model to reproduce the experimental TA spectra of a GFPchromophore derivative to assess OSCF2 as a model of
ransient absorption (TA) spectroscopy is the standard technique for investigating dynamics of visible excited states.1−5 Interpretation of TA spectra is indirect because the energies and intensities of excited-state absorptions are usually uncertain.6 A method that could predict observed TA spectra would offer insight and assist with experimental interpretation. Besides corroborating the assignments of peaks and relaxation channels, an ab initio simulation can calculate any timedependent property of the system, including those inaccessible by experiments.7−10 For these reasons, simulations of TA spectra are an area of active development.11−18 Numerous papers reported useful simulations of TA spectra using tight-binding (TB) and ab initio models.19−26 TB models are constructed from calculated or experimental energies and properties; the basis states are treated distinguishably. Approximations must be used for energies and couplings, and their accuracy relies on the practitioner. Hundreds of electrons are often represented by a dozen states in a TB model.27 Ab initio models lie in the opposite extreme:28 Every electron has hundreds of degrees of freedom, and they are treated as indistinguishable Fermions.29 No basis of states or couplings are chosen besides the atomic basis; all of the energies and other properties are predicted by quantum mechanics. Ab initio simulations of transient spectra are significantly more demanding than a TB approximation but provide a predictive picture of electronic dynamics. Using a coarse model of electrons, TB models can afford greater accuracy for other degrees of freedom.24,27,30−32 Real-time time-dependent density functional theory (RTTDDFT) is an ab initio method often used for simulating absorption spectra.33,34 Unfortunately, RT-TDDFT is not a satisfying model for TA without improvements.11 Time scales © XXXX American Chemical Society
Received: February 23, 2016 Accepted: April 8, 2016
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DOI: 10.1021/acs.jpclett.6b00421 J. Phys. Chem. Lett. 2016, 7, 1590−1595
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Figure 1. Zero-delay TA spectra of pyrazole pumped into two different excited states: (a) When pumped into the lowest bright state, the spectra show two prominent bleaches. They are two excited states with the same electron distribution but different hole distributions. (b) When pumped with a much higher energy, none of the same bleaches appear.
relaxation rates. OSCF2 is an open-systems theory in the Markov approximation. It decomposes relaxation rates into environmental modes with different coupling strengths but does not propagate explicit nuclear trajectories. Other methods, such as surface hopping, provide a more accurate treatment of nuclear motion at additional expense. Our implementation of OSCF2 in Q-Chem quantum chemistry package48 was used with B3LYP/ 6-31G*, unless some other method is specified. Details of our calculations are given in the Supporting Information (SI). Pauli-Blocking in Pyrazole. Many bright TDDFT states occur in pyrazole (C3H4N2) below 15 eV, which we can use to assess the reference correction. In particular, we would like to confirm that pumping a specific state causes a bleach of that state in the absorption spectrum and that any other bleaches can be explained by states whose electron donor or acceptor states overlap significantly with the pumped transition. Figure 1 shows the ground state, excited state, and TA spectra of pyrazole in two different scenarios. In Figure 1a, the molecule is pumped into its lowest singly excited state at 6.32 eV. More than 90% of this transition is of HOMO-2 → LUMO orbital character. A prominent “ground-state bleach” appears in the transient spectrum at this energy, and the absorption of nearby states is undisturbed. Another bleach appears just above 13 eV at the energy of a bright state with >90% HOMO-6 → LUMO character. The appearance of this bleach is not erroneous but rather a physical consequence of the orbital occupations. A simplified model of the TDDFT Liouville equation explains the origin of the effect. An absorption (or emission) signal at ωh,l = (ϵh − ϵl) is produced when the norm of an associated coherence density matrix element changes d γ = −iω h,lγh,l + iFh,l(t )(γh,h − γl,l) dt h,l
Figure 2. TA spectra of pyrazole pumped into its lowest bright state with a strong and simplified bath model and the 3D profile of the dynamics. Although all bath couplings are identical, the rates of the linked bleaches still vary because of single-orbital differences between the states.
Time evolution of the TA spectra for pyrazole is depicted in Figure 2. Here we employed a simplified bath model with identical bath couplings for all orbitals. Although the bath coupling is constant in this simulation, the decay rates still vary. We see additional evidence of the value of a singleelectron basis. Both bleaches correspond to excited states that fill the same orbital, but the higher energy one relaxes at twice the rate. If we did not have detailed ab initio knowledge of the nature of each state, we would probably infer a different dynamical picture from the transient spectrum than from this simulation. GFP-Chromophore Derivative. In 2010, Petkova and coworkers published transient spectra of the free GFP chromophore and several derivatives in a variety of solvents.52 We are interested in the transient kinetics of one derivative, hereafter GFP-3, in acetonitrile at room temperature (inset of Figure 3). We aimed
(1)
If Fh,l(t) oscillates at ωh,l, signal may result if (γh,h − γl,l) ≠ 0. If a system begins in the density of HOMO-2 → LUMO excitation, absorption of HOMO-6 → LUMO cannot occur because (γh‑6,h‑6 − γl,l) = 1 − 1 = 0. This blocking-bleach effect is observed indirectly in intrinsic semiconductors through the broadening of the apparent band gap, known as the Burstein− Moss effect.44−47,49−51 Tight binding models are usually used to study coupled single excitations of weakly interacting chromophores; however, a TB approximation that uses a few states for each chromophore and a distinguishable Coulomb coupling model may miss this blocking effect. Figure 1b shows the scenario in which pyrazole is pumped to a much higher energy (12.6 eV), which has no blocking partner. The weak signals near the lowest bright state are small artifacts of TDDFT, discussed further in the SI.
Figure 3. Absorption spectrum of GFP-3 in acetonitrile computed with OSCF2 (6-31G*/B3LYP, solid line) compared with the linearresponse stick spectrum (dashed line). Inset: optimized structure of GFP-3 in acetonitrile (6-31G*/B3LYP). 1591
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550 nm band because this band disappears quickly after zero delay. The 2.5 ps component reproduces the blue shifts of the bleach and the appearance of the two positive bands. The 21.2 ps component includes a further blue shift of the bleach and a red shift of the 600 nm band. Furthermore, the intensity of the two ESA bands is essentially identical after zero delay; this is clearly seen in the experimental spectra. We have presented a black-box, real-time ab initio method to simulate experimental TA spectra. The time scales of TA experiments are millions of times longer than the attosecond time scales of electronic motion, and thus several approximations have been introduced to produce a spectrum without prohibitive expense. Our dissipation is Markovian, so vibronic feedback effects that appear in TA spectra such as coherent phonon oscillations in solids will not be reproduced by our method. The bath model captures the differences between excited and ground states; however, it is derived from a weak-bath perturbation theory and inappropriate for bond-reorganization dynamics. Multiphonon relaxation is ignored, and we explicitly neglect coherent portions of the TA signal. We have demonstrated that several features of experimental spectra are successfully reproduced, and this method can provide information about electronic dynamics that augments experiments. We emphasized that TA spectra are sensitive to the time-dependent orbital populations of the states they probe.46 Shortcomings of available theories for correlated electronic dynamics remain the greatest obstacle to quantitatively reproducing transient spectra. Because our method employs adiabatic TDDFT as its basis for electronic dynamics, multiple excitations will not be treated properly, and charge-transfer states may cause issues.56 Our group is working on densitymatrix correlated dynamics methods to tackle these problems.
to reproduce the experimental TA spectra and the decayassociated spectra (DAS) of this system. Our model consists of one Z-isomer of GFP-3 solvated in a box of acetonitrile molecules. An equilibrium dynamics simulation was performed to construct an orbital-specific bath model. The solvent was explicitly represented using the built-in QM/MM feature of Q-Chem and the CHARMM27 force field.53,54 Further details, including geometries and bath parameters, are presented in the SI. The computed linear-absorption spectrum of GFP-3 in acetonitrile is shown in Figure 3. The OSCF2 spectrum is homogeneously broadened due to relaxation and is in good agreement with the reported experimental maximum (393 nm). The system was vertically pumped to the lowest excited state from the ground-state optimized geometry to simulate the TA spectra and the DAS of GFP-3. The computed TA spectra shown in the top panel of Figure 4b consist of a negative band around 375 nm that represents the bleach of S0−S1 and a broad positive band centered around 550 nm that is the S1 excitedstate absorption (ESA). The intensity of both the bleach and the ESA bands decays to essentially zero as the simulation progresses. The ESA band broadens and exhibits two peaks around 450 and 600 nm that shift slightly at longer time delays. The shift is a TDDFT artifact further discussed in the SI. Besides the spectral change exhibited at early time, the shape of the computed spectra largely reproduces its experimental counterpart. TA spectra are often interpreted by producing DAS that best fit the transient spectrum with a simple kinetic model. We obtained the DAS of our computed TA spectra in the program Glotaran55 (bottom panel of Figure 4b) to produce an “applesto-apples” comparison of OSCF2 with the experimentally observed kinetics. The resulting DAS are associated with three exponential decays with 0.8, 2.5, and 21.2 ps time constants. All three time scales are a factor of 2 above their experimental counterparts (Figure 4a). This overestimation is consistent with our previous experiences with OSCF2; lifetimes tend to be overestimated within an order of magnitude. In addition to the S0−S1 bleach feature, the 0.8 ps component also includes the
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METHODS Transient absorption spectroscopy is one of the most common experimental techniques for studying ultrafast electronic dynamics.57−60 In this technique, a laser pulse is split into two components: a pump used to excite and a delayed probe whose
Figure 4. Transient absorption spectra of GFP-3 in atomistic acetonitrile simulated in the present work and their decay-associated spectra (b) compared with the experimental counterpart52 (a). We include the maximum and minimum of the experimental TA spectra as vertical dashed lines in panel b to guide the eye. 1592
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which experiences a field, the other without, and the resulting dipoles are subtracted. To first-order approximation, this eliminates unphysical signal from the oscillations of the initial state. Combined with OSCF2, this leads to a straightforward yet efficient real-time approach for calculating TA spectra. The various phases of propagation are represented in Figure 5. We propagate ρe, allowing the system to relax during the delay
absorption is measured. By varying the time delay between the pump and the probe, the dynamics of electronic excitations are studied indirectly. The method developed in this paper is applicable to delay times longer than the pulse length. Specializing on longer time scales allows us to disregard short-time coherent contributions61−64 to the TA signal. The electronic dephasing time scale is fast relative to most TA delays,65 so this trade-off is desirable. Because decoherence can be ignored during the delay time, the applicability of our method improves. Assuming the pump pulse prepares the initial excited density of the system-of-interest at t = −τ so that ρe = ρ(−τ), the perturbative expression for the absorption of the probe pulse, centered at t = 0, is26,66 A (ω , τ ) =
4πω Im(-{P(t , τ )/E(t )}) c
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where P(t) is the polarization of the system induced by a weak probe pulse, E(t). A relatively short P(t) sampling time (tabs ≥ 10 fs) is required to resolve an absorption spectrum following the probe. The ground-state absorption is subtracted out of the probe absorption: ΔA = A − Ags. A propagator, ρ(t ) =
