A Direct Comparison of Equilibrium and Nonequilibrium Methodologi

Article Cite This: J. Phys. Chem. B 2018, 122, 6823−6828

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Decompositions of Solvent Response Functions in Ionic Liquids: A Direct Comparison of Equilibrium and Nonequilibrium Methodologies Z. L. Terranova and S. A. Corcelli*

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Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, Indiana 46556, United States ABSTRACT: Time-dependent Stokes shift (TDSS) measurements provide crucial insights into the dynamics of liquids. The interpretation of TDSS measurements is often aided by molecular dynamics simulations, where solvent response functions are computed either with an equilibrium or nonequilibrium approach. In the nonequilibrium approach, the solvent is at equilibrium with the ground electronic state of the solute and its charge distribution is instantaneously changed to that of the first excited state. The solvation response function is then calculated as a nonequilibrium average of the subsequent evolution of the solvent influence on the electronic energy gap. In the equilibrium approach, the normalized time correlation function of the fluctuations of the solvent-perturbed electronic energy gap is calculated. If the linear response approximation is valid, then the nonequilibrium solvation response function is identical to the equilibrium time correlation function. The nonequilibrium methodology conceptually mimics the experiment, but it is significantly more computationally expensive than the equilibrium approach. In multicomponent systems such as ionic liquids, it is natural to inquire how the various components affect the observed relaxation dynamics. When utilizing the nonequilibrium methodology, the solvation response naturally decomposes into a sum of responses for each component present in the system. However, the equilibrium time correlation function does not decompose unambiguously. Here, we have evaluated a decomposition strategy that is consistent with the linear response approximation for the study of solvation dynamics of coumarin 153 (C153) in the 1-ethyl-3-methyl imidazolium tetrafluoroborate, [emim][BF4], ionic liquid. The agreement of the equilibrium and nonequilibrium solvation response functions demonstrates the validity of the linear response approximation for the C153/[emim][BF4] system. Moreover, decompositions of the equilibrium time correlation function into contributions of the translational and rovibrational motions of the anions and cations are essentially identical to the same decompositions of the nonlinear solvation response. evolution of the fluorescence is typically characterized by the time-dependent solvent response function1−4

1. INTRODUCTION Observing the evolution of solvent reorganization in the vicinity of an electronically excited fluorescent solute molecule provides unique insight into the molecular interactions and motions present in a liquid.1−4 In particular, discerning these molecular motions in liquids is important to understand the role of solvents in charge-transfer reactions whose kinetic rates are influenced by solvent reorganization.5 Time-dependent fluorescence Stokes shift (TDSS) experiments can reveal these dynamics. TDSS measurements begin with an ultrafast laser pulse that electronically excites a fluorescent probe molecule present in the liquid at dilute concentrations. Following the Franck−Condon principle, the molecular geometry of the probe molecule is initially unchanged by the excitation but its charge distribution is significantly altered. After the initial excitation, the frequency dependence of the subsequent fluorescence is monitored. The frequency of maximum emission, v(t), shifts to longer wavelengths over time as the solvent molecules reorganize to establish equilibrium with the excited-state charge distribution of the probe molecule. The © 2018 American Chemical Society

S( t ) =

v (t ) − v (∞ ) v(0) − v(∞)

(1)

where v(∞) is the maximum emission frequency after the environment has completely responded to the excited-state charge distribution of the probe. By construction, the solvent response function is 1 at t = 0 and 0 at t = ∞. Central to the theoretical study of solvation dynamics is the linear response approximation, which assumes that the dynamics of a liquid responding to a nonequilibrium perturbation are equivalent to fluctuations present in the thermally equilibrated system. Linear response theory equates the nonequilibrium response function, S(t), to the equilibrium time correlation function (TCF), C0(1)(t), of the fluctuations in the solvent-perturbed electronic energy gap6−8 Received: May 3, 2018 Revised: June 12, 2018 Published: June 14, 2018 6823

DOI: 10.1021/acs.jpcb.8b04235 J. Phys. Chem. B 2018, 122, 6823−6828

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The Journal of Physical Chemistry B S(t ) ≅ C0(1)(t ) =

one C153 molecule were modeled as fully flexible, except for covalent bonds containing hydrogen which were fixed using the SHAKE algorithm.49 Simulation conditions were identical for C0(t); however, for the calculation of C1(t), the cations and anions were equilibrated in the presence of a C153 molecule with a charge distribution appropriate for its excited electronic state. The partial charges for C153 in its ground and excited electronic states were calculated and validated previously by Cinacchi et al.50 Independent trajectories generated from the ground-state equilibrium simulation were used as starting configurations for the nonequilibrium calculations, whereby the partial charges of C153 were abruptly switched to reflect the excited-state charge distribution. Production runs [comprising a total of 5.015 μs for C0(t), 2.0 μs for C1(t), and 5.0 μs for S(t)] were performed using the large-scale atomic/ molecular massively parallel simulator.51

⟨δ ΔE(0)δ ΔE(t )⟩0(1) ⟨(δΔE)2 ⟩0(1)

(2)

where δΔE(t) = ΔE(t) − ⟨ΔE⟩0(1) and ⟨···⟩ is an equilibrium ensemble average with the probe molecule modeled in the ground, ⟨···⟩0, or excited electronic state, ⟨···⟩1. Here, ΔE(t) = ΔEe(t) − ΔEg(t) is the difference in the interaction energy of the probe molecule with its environment for the excited Ee and ground Eg electronic states. An important criteria for the validity of linear response theory is that the time-dependent distribution of ΔE(t) be adequately described by a Gaussian distribution.7−9 Deviations from linear response occur when the electronic transition of the probe molecule undergoes significant change in magnitude, size, or when the excitation alters the solvation structure in such a way that it is no longer similar to ground-state fluctuations, for example, altering the hydrogen-bonding capabilities.10−13 There have been extensive experimental14−29 and theoretical29−47 studies of solvation dynamics in ionic liquids (ILs), which reveal a broad range of timescales ranging from 50 fs to 10 ns. Moreover, the solvation response exhibits a complex, multiexponential kinetic profile. In a previous study utilizing an equilibrium approach to calculate and decompose C0(t), we found that the solvation mechanism in imidazolium-based ILs is dominated by translational movements of anions into and out of the first solvation shell surrounding the fluorescent probe molecule.40 Although our calculations compared favorably with experiment, they rely on the validity of the linear response approximation. A number of investigations have examined the validity of the linear response theory in ILs, and the general consensus is that the approximation is reliable.27,38,45,46,48 For example, a recent study directly compared equilibrium and nonequilibrium solvation response functions over the full range of experimentally accessible timescales for solutes in ILs that were simulated with coarsegrained models.45 There was good agreement between the equilibrium and nonequilibrium calculations, both with each other and with experiment, over the full range of the solvation response, which is compelling evidence that the linear response approximation is valid in ILs. In contrast, another study does call into question the validity of the linear response approximation,43 so the issue is not fully resolved. Here, we will compare equilibrium and nonequilibrium calculations of the solvation response of coumarin 153 (C153) in the 1-ethyl3-methyl imidazolium tetrafluoroborate, [emim][BF4], IL. Both the solute and IL are modeled in full atomistic detail and the solvation responses are computed until fully decayed at ∼1 ns. Much like the previous studies involving coarse-grained models, the results support the conclusion that the linear response approximation is robust for the C153/[emim][BF4] system. We then go a step further to validate a rigorous protocol for decomposing the equilibrium response into contributions from the components present in the system, including the C153 solute, [BF4] anions, and [emim] cations. The results will show that the equilibrium decompositions are virtually identical to the benchmark nonequilibrium decompositions.

3. RESULTS AND DISCUSSION Figure 1 shows the ground- and excited-state equilibrium TCFs, C0(t) and C1(t), respectively, along with the non-

Figure 1. Normalized solvation response without the contribution of the dye for the nonequilibrium S(t) (black), equilibrium C1(t) (red), and C0(t) (green) from 50 fs to 1 ns. The inset is the identical response including intramolecular interactions of the dye molecule responsible for the significant oscillations below 10 ps.

equilibrium response function, S(t), for the C153 solute in the [emim][BF4] IL. The equilibrium TCFs are in reasonable agreement with each other and with the nonequilibrium response function over the full range of solvation dynamics extending to 1 ns. The errors of the equilibrium results throughout the paper are less than the thickness of the lines used to present the data. The nonequilibrium results are computed with 5000 trajectories and are converged to better than 3% accuracy. Thus, overall, the results support the notion that the linear response approximation is sensible for investigating solvent fluctuations in the IL. The inset of Figure 1 contains the total solvation response, including the oscillations due to internal motions of the C153 probe molecule. These oscillations in the solvation response have been observed in high temporal resolution measurements in solution52 and for a dye molecule (Hoechst 33258) bound to DNA.53 To facilitate comparison, these oscillations were subtracted from the total response, which is shown in the main panel of Figure 1. The calculated response functions all

2. METHODS The simulation box was constructed and equilibrated using an identical protocol as in our previous study of [emim][BF4].40 In brief, a total of 256 [emim] cations, 256 [BF4] anions, and 6824

DOI: 10.1021/acs.jpcb.8b04235 J. Phys. Chem. B 2018, 122, 6823−6828

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The two alternative decomposition schemes are applied to C0(t) in Figure 2. The top panel shows the decomposition into

include an ultrafast inertial relaxation followed by a complex nonexponential decay, which is known to be viscositydependent.14,22,28,54 C0(t) better reproduces the short-time inertial component of S(t) than C1(t), a consequence of the similar solvent configurations explored in C0(t) and S(t) simulations at early times. At times beyond about 1 ps, C1(t) is in better agreement with S(t) than C0(t). This agrees with the theoretical results of Laird and Thompson that C1(t) represents a more faithful approximation to S(t) because the solvent dynamics occur with the solute in its excited electronic state, as is the case for the nonequilibrium trajectories used to calculate S(t). The accuracy of the response functions compared to experiment was examined in a previous paper.40 Because the solvent-perturbed energy gap, ΔE(t) = ∑αΔEα(t), can be expressed as a linear sum of contributions, ΔEα(t), from the anions, cations, and intramolecular electrostatic interactions within the probe molecule itself, the decomposition of S(t) is uncomplicated S(t ) =

∑ S α (t ) = α

ΔE α(t ) − ⟨ΔE α⟩0 ΔE(0) − ⟨ΔE⟩0

Figure 2. Comparison of two decomposition strategies, eqs 4 and 5, as applied to C0(t). (Top) Decomposition of C0(t) into auto- and cross-correlation functions, Cαβ 0 (t). (Bottom) Decomposition of C0(t) into component correlation functions, Cα0 (t).

(3)

In eq 3, the overbar represents a nonequilibrium average and the superscript α signifies the solvent component of interest. The decomposition of the equilibrium response function is more ambiguous. Substituting ΔE(t) = ∑αΔEα(t) into eq 2 C0(1)(t ) =

αβ (t ) ∑ ∑ C0(1) α

=

auto- and cross-correlation functions and the bottom panel shows the component correlation functions. Many of the same qualitative inferences can be drawn from the two different analyses. In particular, the anion−anion autocorrelation function is the dominant contributor to the response in the top panel, much as the anion component correlation function is dominant in the lower panel. The dashed lines in the top panel depict the cross-correlation functions. The crosscorrelation functions involving the dye (anion−dye and cation−dye) are negligible, but there is modest anticorrelation between the anion and cation contributions to the response, as evidenced by the negative value of the anion−cation crosscorrelation function. Figure 3 shows a direct comparison between the nonequilibrium and equilibrium component correlation functions for the anion (top panel) and cation (bottom panel). The agreement between the results is nearly quantitative, which empirically demonstrates the equivalence of the nonequili-

β

∑∑ α

⟨δΔE α(0)δΔEβ (t )⟩0(1)

β

⟨(δΔE)2 ⟩0(1)

(4)

leads to the appearance of auto- (α = β) and cross (α ≠ β)correlation functions. Equation 4 can be rearranged into a form that is formally consistent with the linear response approximation,55−58 as follows C0(1)(t ) =

∑

α C0(1) (t )

α

=

∑ α

⟨δΔE(0)δΔE α(t )⟩0(1) ⟨(δΔE)2 ⟩0(1)

(5)

A key motivation of the present study is to evaluate the accuracy of the linear response approximations, Cα0(1)(t), to the unambiguous decompositions of the nonequilibrium response, Sα(t). The correlation functions for each solvent component in eq 5, Cα0(1)(t), are related to the auto- and cross-correlation functions in eq 4, Cαβ 0(1)(t), through the sum rule α C0(1) (t ) =

αβ (t ) ∑ C0(1) β

(6)

The advantage of the solvent component correlation functions is the theoretical connection to the nonequilibrium solvent component correlation functions, that is, Sα(t) ≅ Cα0(1)(t), within the linear response approximation.6−8 The functions Sα(t) ≅ Cα0(1)(t) are useful for addressing how each of the components of a multicomponent mixture contribute to solvent response dynamics, and this formalism has been successfully applied to analyze TDSS measurements in liquids,40 as well as proteins58 and DNA56,59,60 in aqueous solution. Although the auto- and cross-correlation functions in eq 4 provide complementary information about the interrelationship of the dynamics of the components, none of the αβ C0(1) (t) functions answer the pivotal question of how component α relates to the total solvation response.

Figure 3. Normalized solvation response decomposed into anion (top) and cation (bottom) contributions for nonequilibrium S(t) (black), equilibrium C1(t) (red), and C0(t) (green) from 50 fs to 1 ns. 6825

DOI: 10.1021/acs.jpcb.8b04235 J. Phys. Chem. B 2018, 122, 6823−6828

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of symmetry and therefore no permanent dipole, the contributions arising from rovibrational motions are expected to be negligible. The translational motions of cations also contribute to the total solvation response, albeit to a lesser degree than the translational movements of the anions; however, these contributions are largely negated by rotational motion of the cations. In this sense, the anticorrelation can be understood as the rotations of cation molecules as being counterproductive in stabilizing the excited-state charge distribution of the C153 solute. The component decomposition strategy unambiguously identifies the respective motion of the constituents in the IL responsible for accommodating the charge perturbation and we are able to confidently identify the solvation mechanism in [emim][BF4]. The level of agreement between the component decomposition and translational/rovibrational analysis of S(t), C1(t), and C0(t) indicates the robustness of the linear response approximation for investigating the solvation dynamics of ILs.

brium, eq 3, and equilibrium, eq 5 component decomposition strategies. This highlights the strength of the equilibrium component decomposition, eq 5, versus the decomposition into auto- and cross-correlation functions, eq 4, for unambiguously identifying the role of each species in the multicomponent mixture to the solvation dynamics. Here, the cross-correlation functions are not particularly large in magnitude. In other systems, however, the cross-correlation functions can be sufficiently large to obscure the true contribution of each component to the overall dynamics. Of course, the cross-correlation functions can be helpful for revealing correlated dynamics between the different components in the system. Another advantage of the component decomposition approach is the ability to perform further analysis to reveal the contributions of specific motions (e.g., translational, rotational, etc.) of the components to the overall solvation response. Although this would also be possible with auto- and cross-correlation functions, there would be a proliferation of response functions which could hinder the interpretation of the results. The translational contribution, ΔEαtrans, to ΔEα for the cations and anions is computed by regarding each IL molecule as a single point charge located at its respective center of mass. α The rovibrational contribution, ΔErovib , to ΔEα is then obtained simply as the difference, ΔEαrovib = ΔEα − ΔEαtrans. With these definitions, the correlation functions for the cations and anions decompose into translational and rovibrational contributions, Cα(t) = Cαtrans(t) + Cαrovib(t), where the functions Cαtrans(t) and Cαrovib(t) are computed with eq 5 using ΔEαtrans(t) and ΔEαrovib(t) in place of ΔEα(t). In Figure 4, the total equilibrium and nonequilibrium component response functions for the anions and cations are

4. CONCLUDING REMARKS In this paper, we have directly compared equilibrium and nonequilibrium approaches for computing the TDSS response of C153 in the [emim][BF4] IL. The agreement of the equilibrium and nonequilibrium response functions demonstrates the validity of the linear response approximation for describing solvation dynamics in the [emim][BF4] IL. Nonequilibrium response functions for mixtures naturally decompose into response functions for each of the components present in the solution and even specific (translational vs rovibrational) motions of the components. In contrast, the equilibrium response functions do not naturally decompose. Here, we have evaluated the accuracy of a decomposition strategy for the equilibrium response functions based on the linear response approximation. A direct comparison of the nonequilibrium decompositions, which serve as a benchmark, to the equilibrium decompositions demonstrates the equivalence of the two approaches when the linear response approximation is operative. Because the equilibrium calculations are more tractable than the nonequilibrium calculations, having a well-validated strategy for addressing how each component contributes to the overall solvation response is vital to extracting physical understanding from solvation dynamics calculations. The present study focused on a single ILfurther investigations are necessary to reveal the quality of the linear response approximation in other ILs.

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AUTHOR INFORMATION

Corresponding Author Figure 4. Normalized solvation responses of S(t) (black), equilibrium C1(t) (red), and C0(t) (green) decomposed into contributions due to translational (solid) and rovibrational (dashed) motions.

*E-mail: [email protected].

further decomposed into contributions resulting from translational and rovibrational motions. The translational and rovibrational contributions originating from the anions are displayed in the top panel and the cations in the bottom panel, where the translational component (solid) and rovibrational component (dotted) lines are presented for S(t) (black), C1(t) (red), and C0(t) (green). In agreement with our previous study,40 translational motion of the anion is the dominant contributor to the solvation response with a negligible rovibrational contribution. Because the anion possesses a lack

Notes

ORCID

S. A. Corcelli: 0000-0001-6451-4447 The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the American Chemical Society Petroleum Research Fund (52648-ND6) and the National Science Foundation (CHE-1565471). The authors are thankful for high-performance computing resources and support from the Center for Research Computing at the University of Notre Dame. 6826

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DOI: 10.1021/acs.jpcb.8b04235 J. Phys. Chem. B 2018, 122, 6823−6828

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DOI: 10.1021/acs.jpcb.8b04235 J. Phys. Chem. B 2018, 122, 6823−6828