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J. Phys. Chem. B 2009, 113, 2800–2804
Solvation Dynamics in Polar Solvents Studied by Means of RISM/Mode-Coupling Theory Katsura Nishiyama,*,† Tsuyoshi Yamaguchi,‡ and Fumio Hirata§ Faculty of Education, Shimane UniVersity, Matsue, Shimane 690-8504, Japan, Department of Molecular Design and Engineering, Graduate School of Engineering, Nagoya UniVersity, Chikusa, Nagoya, Aichi 464-8603, Japan, and Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan ReceiVed: NoVember 11, 2008; ReVised Manuscript ReceiVed: December 25, 2008
The extended reference interaction site model (RISM) theory coupled with the generalized Langevin/modecoupling theory (MCT) is applied to the investigation of solvation dynamics in polar solvents. The RISM/ MCT framework used in this paper significantly upgrades the previous report by Nishiyama and co-workers [Nishiyama, K.; et al. J. Chem. Phys. 2003, 118, 2279.] for the calculation of the solvation response function, SS(t). This function is experimentally observable from dynamic Stokes shift measurements, for example. SS(t) obtained by RISM/MCT relaxes with an initial Gaussian decay followed by damped oscillation, which is in accordance with experimental results or molecular dynamics simulations published elsewhere. SS(t) is then decoupled into the acoustic and optical modes of solvent, which indicate the translational and rotational motions of solvent, respectively. The majority (>90%) of SS(t) is explained by the optical mode, whereas the slower acoustic mode also plays an important role. Resultingly, RISM/MCT is shown to be an appropriate theoretical methodology to capture a molecular view of solvation dynamics, without assuming any empirical parameters. 1. Introduction Solvent dynamic response associated with an abrupt change of the solute electronic structure is referred to as solvation dynamics, which has been intensively studied by means of ultrafast laser spectroscopy,1-11 theory,12-29 and simulation30-37 due to the close connection with chemical reaction dynamics in solution. From the experimental viewpoint, the dynamic Stokes shift has been typically observed because of the intuitive and simple interpretation of experimental data.1,2,4,5 With the development of laser sources and spectroscopic technology, transient hole-burning spectroscopy has appeared as another methodology to detect solvent fluctuation. We define the solvation response function Se(t) for the solvent relaxation process as follows
Se(t) )
V˜ (t) - V˜ (∞) V˜ (0) - V˜ (∞)
(1.1)
where V˜ (t) stands for the specific frequency of the spectra observed at time t, for instance. In general, the simple spectral peak in geometry or the first moment of the spectra has been chosen as V˜ (t). An interpretation of Se(t) is that this function measures average energy relaxation of the solute-solvent system. According to their milestone experiments, Rosenthal and coworkers1 and Cho and co-workers2 have used under -50 fs technology to detect the dynamic Stokes shift. In the former paper, Se(t) is shown to relax with the initial rapid decay of a Gaussian shape, which they call the “inertial motion” of solvent, followed by ultrafast oscillation.1 The subsequent oscillation is * To whom correspondence should be addressed. E-mail:
[email protected]. Fax: +81-852-32-9832. † Shimane University. ‡ Nagoya University. § Institute for Molecular Science.
explained by relaxation of the neat solvent they used (acetonitrile), in connection with optical Kerr effect measurements.2 On the other hand, under suitable experimental conditions, the spectral bandwidth of the ground-state hole broadens with time. Another version of the response function, SW(t), which is less popular compared with Se(t), is defined as
SW(t) )
σ˜ (t)2 - σ˜ (∞)2 σ˜ (0)2 - σ˜ (∞)2
(1.2)
where σ˜ (t) denotes the spectral bandwidth. Murakami and coworkers3 have performed ps transient hole-burning spectroscopy of laser dyes in an alcoholic solvent at lower temperature (-160 K). The conclusion that they have drawn on the basis of their experiments is that a time constant governs both for Se(t) and SW(t), leading to Se(t) ) SW(t) decaying with an exponential curve. A single exponential decay of Se(t) observed by Murakami and co-workers3 does not correspond to the relaxation feature reported by Rosenthal1 or Cho2 and co-workers, even though the time range and chemical property of solvents investigated in their works are very different. In their papers regarding dynamic Stokes shift measurements,1,2 they do not mention SW(t). However, Se(t) shows a manifold feature and not a simple exponential decay any more. A question is therefore asked of how SW(t) could relax at room temperature in less viscous solvents. One of the present author and co-worker(s) have performed sub-ps transient hole-burning spectroscopy and 10 ps dynamic Stokes shift measurements in polar solvents at room or lower temperature7,8 or in binary solvent mixtures.11 They have found that at room temperature, Se(t) and SW(t) relax with the time constants which differ from each other by an order of magnitude. They have assumed that Se(t) and SW(t) are characterized by different solvent modes; Se(t) originates in the solvent rotational diffusion, whereas SW(t) has a root from the translational
10.1021/jp809926g CCC: $40.75 2009 American Chemical Society Published on Web 02/04/2009
Solvation Dynamics Studied by RISM/Mode-Coupling Theory diffusion of solvent. At lower temperature, slower SW(t) catches up with Se(t).8 Their results make a bridge between those from Rosenthal1 or Cho2 and Murakami,3 published with their coworkers. Given the experimental situation, theoretical or simulation studies have focused attention to capture the property of Se(t) detected by the dynamic Stokes shift measurements. Raineri and co-workers14 have presented a new theory on the basis of the generalized Langevin equation, implemented with the extended reference interaction site model (RISM)38 theory for molecular description. We parenthetically note that in this paper, we simply call this theory developed by Hirata and Rossky38 “RISM”, having in mind that the theory is an upgraded version of “conventional” RISM theory by Chandler and Andersen.39 As a check of theoretical and numerical efficiency, they have chosen solute models varying from dipoles or three-site quadrupoles to water-like structures. Actually, they have reproduced the general aspect of the relaxation, represented by an initial Gaussian decay followed by oscillation. From experimentalists, however, the solute models that they have employed are somewhat simple compared with realistic molecules used in spectroscopy. In the previous publication by two of the present authors and co-worker,27 hereafter, called Paper I, we have provided solutedependent features of solvation dynamics in depth, within a framework of the RISM/SSSV (site-site Smoluchowski-Vlasov) equation theory. Although the RISM/SSSV theory does not reproduce Gaussian decay curves followed by oscillation in its appearance, they have represented general properties of response functions in the diffusive limit. In this contribution, we apply the RISM theory to solvation dynamics, with aid of the generalized Langevin/mode-coupling theory (MCT). Here is an absolutely upgraded version of Paper I, targeted for similar solute-solvent systems. RISM/MCT helps us make the first-principle calculation on solvation dynamics, while an empirical parameter is used in Paper I. In our preliminary reports,28,29 we have applied RISM/MCT to basic solute-solvent systems to check the applicability of the theoretical framework, and this theory has turned out to be a powerful tool to obtain an accurate molecular view of solutedependent dynamics. The remainder of this paper is organized as follows. Section 2 overviews RISM/MCT employed in the present work. Section 3 presents how the multiplicity of the electronic pole and the geometrical structure of solute alter the solvation response. Discussion in relation with SW(t) is given in section 4. Section 5 summarizes the entire results. 2. Overview of Theoretical Calculations 2.1. Theoretical Framework. In this section, we follow the theoretical framework of RISM/MCT. For the details of the RISM expression for solvation dynamics and the application of MCT, the reader is referred to Paper I and the reports by Nishiyama and co-workers,28,29 and references therein. In theory or simulation, it is a common practice to calculate SS(t), which has the same physical meaning as Se(t)
δε(t) - δε(∞) Ss(t) ) δε(0) - δε(∞)
(2.1)
where δε(t) stands for the solvation energy change at time t due to the transition of the electronic structure of the solute. The overbar is an ensemble average regarding solutes under
J. Phys. Chem. B, Vol. 113, No. 9, 2009 2801 the experimental observation. In eq 2.1, we substitute the subscript for S(t), notifying the reader that the quantity is obtained experimentally (e) or theoretically (s). Under the linear response theory, eq 2.1 is given as
Ss(t) )
〈δε(t)δε(0)〉 〈δε(0)δε(0)〉
(2.2)
where 〈 · · · 〉 expresses an ensemble average over the solvent configuration around the solute. With a help of the surrogate approximation, a site-site description of eq 2.2 is given as14
Ss(t) )
∫0∞ dkk2 ∑ jj′ Fjj′(k, t)Bjj′(k) ∫0∞ dkk2 ∑ jj′ Fjj′(k, t ) 0)Bjj′(k)
(2.3)
where Fjj′(k,t) is the intermediate scattering function of neat solvent. The subscript j represents the solvent site, and k is the wave vector. In this paper, we employ a combination of a generalized Langevin equation and MCT to calculate F(k,t), which characterizes the entire dynamics part of SS(t). As the details of derivations of F(k,t) and the solute-solvent interaction part of Bjj′(k) are provided by the papers of Yamaguchi and co-workers40 and Paper I, respectively, we omit the description here for the sake of brevity. 2.2. System Description. For the site-site intermolecular potential, we employ an additive pairwise potential function of the Lennard-Jones and Coulomb terms, uRβ(r)
[( ) ( ) ]
uRβ(r) ) 4εRβ
σRβ r
12
-
σRβ r
6
+
qRqβ r
(2.4)
where R or β means the interaction site of either solute or solvent. The qγ (γ ) R, β) is the charge put on the site, and ε and σ have the usual meanings. The Lorentz-Berthelot combination rule is used for interaction between the different species of the sites. T ) 298 K is chosen for acetonitrile (MeCN) and water, and 249 K is chosen for methyl chloride (MeCl). Details of the solute and solvent models are provided in Paper I, and we follow the identical nomenclatures concerning solutes also in this work. At the very end, in the case of a solute nXy, n () 2 or 8) represents the number of the solute sites corresponding to dumbbell or cubic, respectively. X ) I, D, Q, or O denotes the ion (I) or the multiplicity of the pole, the dipole (D), quadrupole (Q), or octapole (O), respectively. The y ) a, b, or c indicates the distance of the charges for the dipole (0.6 nm, 0.6(2)1/2 nm, or 0.6(3)1/2 nm, respectively). We also calculate simple ions, chloride (Cl), or model ions. Except for model simple ions, the parameters εRR and σRR are set as identical to those of Cl for the sake of simple comparison. At t < 0, all of the site(s) are neutral. We then put charge(s) at t ) 0 so as to produce nXy or an ion, with varying of the transition of charge δq ) 0.2-1.0 e where necessary. 3. Results 3.1. Applicability of RISM/MCT for Solvation Dynamics. In Figure 1, we present how RISM/MCT is applicable for the investigation of solvation dynamics. Figure 1a and c compares the present results (drawn in bold lines) with those from the RISM/SSSV theory (dotted lines) given in Paper I. When we use RISM/MCT, SS(t) relaxes with an initial Gaussian decay followed by the damped oscillation. Note that we have observed
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Nishiyama et al.
Figure 3. The dynamic response function SS(t) of 2Da obtained in MeCN. The solid, dash-dotted, dotted, and dashed lines indicate δq ) 1.0, 0.75, 0.5, and 0.2 e, respectively.
Figure 1. Comparisons among the results obtained from RISM/MCT and the RISM/SSSV theory or molecular dynamics (MD) simulations. The solvents and solutes investigated are labeled within the figure. The MD results are taken from the literature reported by Maroncelli and co-workers, (b)30 and (d),31 respectively. The properties of the model solute “S” and “L” are provided in the literature.30,31 Calculated with the transition of charge; δq ) 1.0 e. Figure 4. Dependence on the solute-charge distribution for SS(t) in (a) MeCN and (b) water with the solute 8Xy (X ) D, Q, O, and I; y ) a, b, c, or without a suffix). The solutes studied are labeled inside of the figure, with the same order of decay curves indicated by the arrow inside of the panel. Calculated with the transition of charge; δq ) 0.2 e.
Figure 2. Plots of Fourier transforms of SS(t), denoted as SS(ω), calculated with Cl. The solvents investigated are (a) MeCN and (b) water. Calculated with the transition of charge; δq ) 1.0 e.
the larger amplitude of the underdamped oscillation with a shorter period in water compared with that in MeCN. The curves from the RISM/SSSV theory do not reproduce the Gaussian and oscillating characters. In this sense, RISM/MCT is a significantly upgraded version compared with the RISM/SSSV theory applied in the diffusive limit. However, we point out that at the earliest stage of the relaxation (90% of the overall relaxation of SS(t) is superimposed by the optical mode of solvent. For Cl, for example, fNN ) 0.037, fNZ ) 0.010, and fZZ ) 0.943. Similar trends are obtained for other solutes. These results are in agreement with those from Paper I. Furthermore, Nishiyama and co-workers have suggested that most parts of Se(t) can be achieved by solvent rotation at room temperature based on their experiments.7,8,11 However, even though the weighting factor is small, the acoustic mode takes part in the relaxation processes. In their paper,19 Nishiyama and co-workers have calculated the time-dependent radial distribution function, g(r,t), followed by Cl f Cl- in MeCN, where r represents the distance between solute and solvent sites. They have found that the first peak at around r ) 0.4 nm, which is assigned to the first solvation shell, relaxes within 0.4 ps. On the contrary, the relaxation of the second peak at around r ) 0.8 nm, assigned to the second shell, has a relaxation time of 2 ps. They suggest that the reorganization of the second or further shell, mainly achieved by the translation mode of the solvent, can be responsible for SW(t).19 With a closer look at Figure 5, regarding the acoustic mode, we find a decay after the sudden falling down (>300 fs), with the time constant at around 2 ps. Although the solvents employed are different, MeCN and MeCl are typical rod-like polar solvents with similar chemical properties. We conjecture that the slower decay observed for the acoustic mode has a close relation with SW(t). At the present stage of our investigation, a direct estimation of SW(t) by theory is not available due to difficult calculations of the correlation function of the fourth order. After we tackle this issue, SW(t) is directly accessible by theory and comparable with experiments.
Figure 5. The dynamic response function SS(t) and the decomposed terms of (a) Cl, (b) 2Da, (c) 8Da, and (d) 8O in MeCl. The thin solid, bold solid, dotted, and dashed lines indicate SS(t), SNN(t), SNZ(t), and SZZ(t), respectively. The data sets decomposed are scaled by the weighting factors defined in eq 3.1. Calculated with the transition of charge; δq ) 1.0 e.
In the present report, SS(t) depends on the multiplicity of the pole and the value of the solute-charge transition, even though the quantitative difference is small. With the latest experimental technology, the solute dependence reported in this work is surely detectable. In addition, if we handle more realistic solutes with more complex molecular and electronic structures, the dependence from the calculation might become larger. Our present work may encourage experimentalists to extract the solventdriven dynamics from Se(t) and SW(t) and to evaluate the contribution from the solute properties. As a development of the present study, we can use solute models with more realistic formulas such as laser dyes, for example. In this case, we should take the solvent-induced electronic structure change regarding solute into account. A RISM/SCF theory44 may help us tackle this topic. Another extension of the present framework may be achieved by the three-dimensional (3D) RISM theory, which has been presented recently.45 By utilizing this theory, we can describe solvent site distribution around the solute site in a 3D format. A combination of the 3D RISM theory and time-dependent description may become a powerful methodology for solvation dynamics. 5. Summary We have employed the extended reference interaction site model (RISM) theory along with the generalized Langevin equation/mode-coupling theory (MCT) in order to describe solvation dynamics in polar solvents. This work is an upgraded version of the previous work by Nishiyama and co-workers.27 We have varied the solute from simple ions or dipoles to octapoles. Acetonitrile (MeCN), methyl chloride (MeCl), and water have been chosen as polar solvents. The solvation response function, SS(t), has been calculated with RISM/MCT, the function of which corresponds to Se(t) detected by dynamic Stokes shift measurements. RISM/MCT has turned out to be very efficient to describe molecular aspects of solvation dynamics. By virtue of RISM/MCT, we have calculated the dynamic response functions without assuming any empirical parameters.
2804 J. Phys. Chem. B, Vol. 113, No. 9, 2009 SS(t) depends on the solute-charge distribution or multiplicity of the pole. This is a clue for experimentalists when they choose a solute for solvation dynamics. We have then decoupled SS(t) into SNN(t), SZZ(t), and their cross term SNZ(t). The vast majority of SS(t) is explained by SZZ(t), the optical mode, or the solvent rotation. On the other hand, SNN(t), the acoustic mode, or solvent translation, decays in the slower relaxation domain. We suggest this slower relaxation may have a close tie with SW(t), the relaxation of the spectral bandwidth observed by the transient hole-burning or time-resolved fluorescence spectroscopy. Acknowledgment. K.N. thanks Professor Emeritus T. Okada for discussions and encouragements. K.N. and F.H. thank the financial support of the Joint Studies Program from the Institute for Molecular Science, and K.N. thanks the Grant-in-Aid as Encouragement for Young Scientists (B) (15750010) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. References and Notes (1) Rosenthal, S. J.; Xie, X.; Du, M.; Fleming, G. R. J. Chem. Phys. 1991, 95, 4715. (2) Cho, M.; Rosenthal, S. J.; Scherner, N. F.; Ziegler, L. D.; Fleming, G. R. J. Chem. Phys. 1992, 96, 5033. (3) Murakami, H.; Kinoshita, S.; Hirata, Y.; Okada, T.; Mataga, N. J. Chem. Phys. 1992, 97, 7881. (4) Chapman, C. F.; Fee, R. S.; Maroncelli, M. J. Phys. Chem. 1995, 99, 4811. (5) Horng, M. L.; Gardecki, J. A.; Papazyan, A.; Maroncelli, M. J. Phys. Chem. 1995, 99, 17311. (6) Ma, J.; Bout, D. V.; Berg, M. J. Chem. Phys. 1995, 103, 9146. (7) Nishiyama, K.; Asano, Y.; Hashimoto, N.; Okada, T. J. Mol. Liq. 1995, 65/66, 41. (8) Nishiyama, K.; Okada, T. J. Phys. Chem. A 1997, 101, 5729. (9) Passino, S. A.; Nagasawa, Y.; Joo, T.; Fleming, G. R. J. Phys. Chem. 1997, 101, 725. (10) Chudoba, C.; Nibbering, E. T. J.; Elsaesser, T. Phys. ReV. Lett. 1998, 81, 3010. (11) Nishiyama, K.; Okada, T. J. Phys. Chem. A 1998, 102, 9729. (12) Fried, L. E.; Mukamel, S. J. Chem. Phys. 1990, 93, 932. (13) Bagchi, B. J. Chem. Phys. 1994, 100, 6658.
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