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Okita K, Ito N, Morishita-Watanabe N, Umakoshi H, Kasahara K, Matubayasi N. Solvation dynamics on the diffusion timescale elucidated using energy-represented dynamics theory. Phys Chem Chem Phys 2024; 26:12852-12861. [PMID: 38623745 DOI: 10.1039/d4cp00235k] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/17/2024]
Abstract
Photoexcitation of a solute alters the solute-solvent interaction, resulting in the nonequilibrium relaxation of the solvation structure, often called a dynamic Stokes shift or solvation dynamics. Thanks to the local nature of the solute-solvent interaction, the characteristics of the local solvent environment dissolving the solute can be captured by the observation of this process. Recently, we derived the energy-represented Smoluchowski-Vlasov (ERSV) equation, a diffusion equation for molecular liquids, which can be used to analyze the solvation dynamics on the diffusion timescale. This equation expresses the time development for the solvent distribution on the solute-solvent pair interaction energy (energy coordinate). Since the energy coordinate can effectively treat the solvent flexibility in addition to the position and orientation, the ERSV equation can be utilized in various solvent systems. Here, we apply the ERSV equation to the solvation dynamics of 6-propionyl-2-dimethylamino naphthalene (Prodan) in water and different alcohol solvents (methanol, ethanol, and 1-propanol) for clarifying the differences of the relaxation processes among these solvents. Prodan is a solvent-sensitive fluorescent probe and is thus widely utilized for investigating heterogeneous environments. On the long timescale, the ERSV equation satisfactorily reproduces the relaxation time correlation functions obtained from the molecular dynamics (MD) simulations for these solvents. We reveal that the relaxation time coefficient on the diffusion timescale linearly correlates with the inverse of the translational diffusion coefficients for the alcohol solvents because of the Prodan-solvent energy distributions among the alcohols. In the case of water, the time coefficient deviates from the linear relationship for the alcohols due to the difference in the extent of importance of the collective motion between the water and alcohol solvents.
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Affiliation(s)
- Kazuya Okita
- Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan.
| | - Natsuumi Ito
- Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan.
| | - Nozomi Morishita-Watanabe
- Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan.
| | - Hiroshi Umakoshi
- Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan.
| | - Kento Kasahara
- Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan.
| | - Nobuyuki Matubayasi
- Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan.
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Okita K, Kasahara K, Matubayasi N. Diffusion theory of molecular liquids in the energy representation and application to solvation dynamics. J Chem Phys 2022; 157:244505. [PMID: 36586971 DOI: 10.1063/5.0125432] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022] Open
Abstract
The generalized Langevin equation (GLE) formalism is a useful theoretical fundament for analyzing dynamical phenomena rigorously. Despite the systematic formulation of dynamics theories with practical approximations, however, the applicability of GLE-based methods is still limited to simple polyatomic liquids due to the approximate treatment of molecular orientations involved in the static molecular liquid theory. Here, we propose an exact framework of dynamics based on the GLE formalism incorporating the energy representation theory of solution, an alternative static molecular liquid theory. A fundamental idea is the projection of the relative positions and orientations of solvents around a solute onto the solute-solvent interaction, namely the energy coordinate, enabling us to describe the dynamics on a one-dimensional coordinate. Introducing systematic approximations, such as the overdamped limit, leads to the molecular diffusion equation in the energy representation that is described in terms of the distribution function of solvents on the energy coordinate and the diffusion coefficients. The present theory is applied to the solvation dynamics triggered by the photoexcitation of benzonitrile. The long-time behavior of the solvation time correlation function is in good agreement with that obtained by the molecular dynamics simulation.
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Affiliation(s)
- Kazuya Okita
- Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
| | - Kento Kasahara
- Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
| | - Nobuyuki Matubayasi
- Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
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Yamaguchi T, Yoshida N. Solvation dynamics in electronically polarizable solvents: Theoretical treatment using solvent-polarizable three-dimensional reference interaction-site model theory combined with time-dependent density functional theory. J Chem Phys 2021; 154:044504. [PMID: 33514097 DOI: 10.1063/5.0036289] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The theory of solvation structure in an electronically polarizable solvent recently proposed by us, referred to as the "solvent-polarizable three-dimensional reference interaction-site model theory," is extended to dynamics in this study through the combination with time-dependent density functional theory. Test calculations are performed on model charge-transfer systems in water, and the effects of electronic polarizability on solvation dynamics are examined. The electronic polarizability slightly retards the solvation dynamics. This is ascribed to the decrease in the curvature of the nonequilibrium free energy profile along the solvation coordinate. The solvent relaxation is bimodal, and the faster and the slower modes are assigned to the reorientational and the translational modes, respectively, as was already reported by the surrogate theory combined with the site-site Smoluchowski-Vlasov equation. The relaxation path along the solvation coordinate is a little higher than the minimum free energy path because the translational mode is fixed in the time scale of the reorientational relaxation.
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Affiliation(s)
- Tsuyoshi Yamaguchi
- Graduate School of Engineering, Nagoya University, Chikusa, Nagoya 464-8603, Japan
| | - Norio Yoshida
- Department of Chemistry, Graduate School of Science, Kyushu University, 744 Motooka, Nishiku, Fukuoka 819-0395, Japan
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Kasahara K, Sato H. Dynamics theory for molecular liquids based on an interaction site model. Phys Chem Chem Phys 2017; 19:27917-27929. [DOI: 10.1039/c7cp05423h] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Abstract
Dynamics theories for molecular liquids based on an interaction site model have been developed over the past few decades and proved to be powerful tools to investigate various dynamical phenomena.
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Affiliation(s)
- Kento Kasahara
- Department of Molecular Engineering
- Kyoto University
- Japan
| | - Hirofumi Sato
- Department of Molecular Engineering and Elements Strategy for Catalysts and Batteries (ESICB)
- Kyoto University
- Japan
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Hirata F. A molecular theory of the structural dynamics of protein induced by a perturbation. J Chem Phys 2016; 145:234106. [DOI: 10.1063/1.4971799] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023] Open
Affiliation(s)
- Fumio Hirata
- Toyota Physical and Chemical Research Institute (Toyota Riken), Nagakute, Aichi 480-1192, Japan
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Kasahara K, Sato H. Development of three-dimensional site-site Smoluchowski-Vlasov equation and application to electrolyte solutions. J Chem Phys 2014; 140:244110. [PMID: 24985621 DOI: 10.1063/1.4884386] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Site-site Smoluchowski-Vlasov (SSSV) equation enables us to directly calculate van Hove time correlation function, which describes diffusion process in molecular liquids. Recently, the theory had been extended to treat solute-solvent system by Iida and Sato [J. Chem. Phys. 137, 034506 (2012)]. Because the original framework of SSSV equation is based on conventional pair correlation function, time evolution of system is expressed in terms of one-dimensional solvation structure. Here, we propose a new SSSV equation to calculate time evolution of solvation structure in three-dimensional space. The proposed theory was applied to analyze diffusion processes in 1M NaCl aqueous solution and in lithium ion battery electrolyte solution. The results demonstrate that these processes are properly described with the theory, and the computed van Hove functions are in good agreement with those in previous works.
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Affiliation(s)
- Kento Kasahara
- Department of Molecular Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan
| | - Hirofumi Sato
- Department of Molecular Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan
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Nishiyama K, Hirata F, Okada T. Importance of Acoustic Solvent Mode and Solute-Solvent Radial Distribution Functions in Solvation Dynamics: Studied by RISM Theory. J CHIN CHEM SOC-TAIP 2013. [DOI: 10.1002/jccs.200000111] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Kim B, Hirata F. Structural fluctuation of protein in water around its native state: a new statistical mechanics formulation. J Chem Phys 2013; 138:054108. [PMID: 23406099 DOI: 10.1063/1.4776655] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A new statistical mechanics formulation of characterizing the structural fluctuation of protein correlated with that of water is presented based on the generalized Langevin equation and the 3D-reference interaction site model (RISM)/RISM theory of molecular liquids. The displacement vector of atom positions, and their conjugated momentum, are chosen for the dynamic variables for protein, while the density fields of atoms and their momentum fields are chosen for water. Projection of other degrees of freedom onto those dynamic variables using the standard projection operator method produces essentially two equations, which describe the time evolution of fluctuation concerning the density field of solvent and the conformation of protein around an equilibrium state, which are coupled with each other. The equation concerning the protein dynamics is formally akin to that of the coupled Langevin oscillators, and is a generalization of the latter, to atomic level. The most intriguing feature of the new equation is that it contains the variance-covariance matrix as the "Hessian" term describing the "force" restoring an equilibrium conformation, which is the second moment of the fluctuation of atom positions. The "Hessian" matrix is naturally identified as the second derivative of the free energy surface around the equilibrium. A method to evaluate the Hessian matrix based on the 3D-RISM/RISM theory is proposed. Proposed also is an application of the present formulation to the molecular recognition, in which the conformational fluctuation of protein around its native state becomes an important factor as exemplified by so called "induced fitting."
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Affiliation(s)
- Bongsoo Kim
- Department of Physics and Institute for Soft and Bio Matter Science, Changwon National University, Changwon 641-773, South Korea
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Iida K, Sato H. An extended formula of site-site Smoluchowski-Vlasov equation for electrolyte solution and infinitely dilute solution. J Chem Phys 2012; 137:034506. [DOI: 10.1063/1.4732760] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/15/2023] Open
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Iida K, Sato H. A theory for time-dependent solvation structure near solid-liquid interface. J Chem Phys 2012; 136:244502. [DOI: 10.1063/1.4729750] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023] Open
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Ishizuka R, Hirata F. Dynamics of the solvent around a solute: generalized Langevin theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:011202. [PMID: 20365359 DOI: 10.1103/physreve.81.011202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2009] [Indexed: 05/29/2023]
Abstract
The generalized Langevin theory for a solution has been derived as the infinite dilution limit of the theory for a two component mixture. Following a similar formalism, the mode coupling approximations of the memory kernel have been also obtained. We have applied this method for one component bulk liquid of Lennard-Jones spheres and proved this approximation theoretically. The analysis of the space and time pair correlation proposed by Van Hove has been carried out as a function of solute particle sizes. It is found that the size of the solute particle is deeply related to the relaxation process of the solvation structure characterized around a solute particle at equilibrium. We have also investigated the relation between the different thermodynamic environments and relaxation process. From these studies, we have obtained the useful information about the rapidity of the relaxation of the solvation structure around a solute at equilibrium.
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Affiliation(s)
- R Ishizuka
- Department of Chemistry and Biochemistry, University of California at San Diego, La Jolla, California 92039-0365, USA
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Ishida T. Solvent Motions and Solvation Processes in a Short-Time Regime: Effects on Excited-State Intramolecular Processes in Solution. J Phys Chem B 2009; 113:9255-64. [DOI: 10.1021/jp901964n] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022]
Affiliation(s)
- Tateki Ishida
- Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan
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Nishiyama K, Yamaguchi T, Hirata F. Solvation Dynamics in Polar Solvents Studied by Means of RISM/Mode-Coupling Theory. J Phys Chem B 2009; 113:2800-4. [DOI: 10.1021/jp809926g] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Katsura Nishiyama
- 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
| | - Tsuyoshi Yamaguchi
- 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
| | - 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
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Raineri FO, Friedman HL. Solvent Control of Electron Transfer Reactions. ADVANCES IN CHEMICAL PHYSICS 2007. [DOI: 10.1002/9780470141663.ch2] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/30/2023]
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Kobryn AE, Yamaguchi T, Hirata F. Site-site memory equation approach in study of density/pressure dependence of translational diffusion coefficient and rotational relaxation time of polar molecular solutions: acetonitrile in water, methanol in water, and methanol in acetonitrile. J Chem Phys 2007; 122:184511. [PMID: 15918733 DOI: 10.1063/1.1884998] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We present results of the theoretical study and numerical calculation of the dynamics of molecular liquids based on the combination of the memory equation formalism and the reference interaction site model (RISM). Memory equations for the site-site intermediate scattering functions are studied in the mode-coupling approximation for the first-order memory kernels, while equilibrium properties such as site-site static structure factors are deduced from RISM. The results include the temperature-density (pressure) dependence of translational diffusion coefficients D and orientational relaxation times tau for acetonitrile in water, methanol in water, and methanol in acetonitrile--all in the limit of infinite dilution. Calculations are performed over the range of temperatures and densities employing the extended simple point charge model for water and optimized site-site potentials for acetonitrile and methanol. The theory is able to reproduce qualitatively all main features of temperature and density dependences of D and tau observed in real and computer experiments. In particular, anomalous behavior, i.e, the increase in mobility with density, is observed for D and tau of methanol in water, while acetonitrile in water and methanol in acetonitrile do not show deviations from the ordinary behavior. The variety exhibited by the different solute-solvent systems in the density dependence of the mobility is interpreted in terms of the two competing origins of friction, which interplay with each other as density increases: the collisional and dielectric frictions which, respectively, increase and decrease with increasing density.
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Affiliation(s)
- Alexander E Kobryn
- Department of Theoretical Study, Institute for Molecular Science, Myodaiji, Okazaki, Aichi 444-8585, Japan
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Nishiyama K, Yamaguchi T, Hirata F, Okada T. Solvation dynamics in water investigated by RISM/mode-coupling theory. J Mol Liq 2005. [DOI: 10.1016/j.molliq.2004.10.009] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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17
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Pressure dependence of diffusion coefficient and orientational relaxation time for acetonitrile and methanol in water: RISM/mode-coupling study. J Mol Liq 2005. [DOI: 10.1016/j.molliq.2004.10.003] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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Nishiyama K, Hirata F, Okada T. Solute-structure dependence of solvation dynamics studied by reference interaction-site model theory. J Chem Phys 2003. [DOI: 10.1063/1.1532345] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Nishiyama K, Hirata F, Okada T. Nonlinear response of solvent molecules induced by instantaneous change of solute electronic structure: studied by RISM theory. J Mol Struct 2001. [DOI: 10.1016/s0022-2860(00)00778-x] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Average energy relaxation and rearrangement of solute-solvent radial distribution function in solvation dynamics: A connection between spectroscopic results and RISM theory. J Mol Liq 2001. [DOI: 10.1016/s0167-7322(01)00128-3] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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22
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Relaxation of average energy and rearrangement of solvent shells in various polar solvents in connection with solvation dynamics: studied by RISM theory. Chem Phys Lett 2000. [DOI: 10.1016/s0009-2614(00)00944-1] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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Ishida T, Hirata F, Kato S. Solvation dynamics of benzonitrile excited state in polar solvents: A time-dependent reference interaction site model self-consistent field approach. J Chem Phys 1999. [DOI: 10.1063/1.479083] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Nishiyama K, Okada T. Relaxation Dynamics of Inhomogeneous Spectral Width in Binary Solvents Studied by Transient Hole-Burning Spectroscopy. J Phys Chem A 1998. [DOI: 10.1021/jp983025h] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Katsura Nishiyama
- Venture Business Laboratory, Osaka University, Suita, Osaka 565-0871, Japan
| | - Tadashi Okada
- Department of Chemistry, Graduate School of Engineering Science and KYOKUGEN, Osaka University, Toyonaka, Osaka 560-8531, Japan
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Hirata F. Chemical Processes in Solution Studied by an Integral Equation Theory of Molecular Liquids. BULLETIN OF THE CHEMICAL SOCIETY OF JAPAN 1998. [DOI: 10.1246/bcsj.71.1483] [Citation(s) in RCA: 62] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/23/2023]
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Yamaguchi T, Kimura Y, Hirota N. Solvent and Solvent Density Effects on the Spectral Shifts and the Bandwidths of the Absorption and the Resonance Raman Spectra of Phenol Blue. J Phys Chem A 1997. [DOI: 10.1021/jp971310f] [Citation(s) in RCA: 45] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- T. Yamaguchi
- Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-01, Japan
| | - Y. Kimura
- Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-01, Japan
| | - N. Hirota
- Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-01, Japan
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Nishiyama K, Okada T. Relaxation of Inhomogeneous Spectral Band Width of Dye Molecules in Polar Solvents Studied by Time-Resolved Hole and Fluorescence Spectroscopy. J Phys Chem A 1997. [DOI: 10.1021/jp961972p] [Citation(s) in RCA: 38] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Katsura Nishiyama
- Department of Chemistry, Faculty of Engineering Science, and Research Center for Extreme Materials, Osaka University, Toyonaka, Osaka 560, Japan
| | - Tadashi Okada
- Department of Chemistry, Faculty of Engineering Science, and Research Center for Extreme Materials, Osaka University, Toyonaka, Osaka 560, Japan
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Yoshida S, Hirata F, Munakata T. Shear-induced distortion of intramolecular and intermolecular correlations in liquids: Time-dependent density-functional theory. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:1763-1769. [PMID: 9965254 DOI: 10.1103/physreve.54.1763] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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