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Cite this: DOI: 10.1039/c7cp05423h

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Dynamics theory for molecular liquids based on an interaction site model Kento Kasahara

*a and Hirofumi Sato

*b

Dynamics theories for molecular liquids based on an interaction site model have been developed over Received 9th August 2017, Accepted 29th September 2017 DOI: 10.1039/c7cp05423h

the past few decades and proved to be powerful tools to investigate various dynamical phenomena. In many of these theories, equations of time correlation functions are formulated by using the Zwanzig–Mori projection operator. Since algebriac equations are directly treated in these statistical mechanical approaches, the obtained dynamical properties are essentially free from statistical error. This perspective presents the theoretical framework of such theories and their applications, including visualization of diffusion processes,

rsc.li/pccp

collective excitations, solvation dynamics, transport properties, and diffusion controlled reactions.

1 Introduction Understanding the dynamical behavior of liquids is a main subject of solution chemistry. Diffusion is a typical example of dynamics, which plays an important role in various chemical phenomena. A diffusion-controlled reaction is a key process in many biological events, such as enzyme–substrate association, ligand binding, and so on.1,2 Traditional stochastic approaches such as the Langevin equation and its generalization are very useful for investigating liquid dynamics, and have been extensively utilized.3–5 However, most of them are almost limited to simple liquids composed of spherical particles. Commonly used

a

Department of Molecular Engineering, Kyoto University, 615-8510, Japan. E-mail: [email protected] b Department of Molecular Engineering and Elements Strategy for Catalysts and Batteries (ESICB), Kyoto University, 615-8520, Japan. E-mail: [email protected]

solvents consist of polyatomic molecules, and hence considering the rotational degree of freedom is required in addition to the translational one. Undoubtedly, molecular dynamics (MD) simulations are the most widely used to investigate liquid systems, treating the solute and solvent molecules explicitly.6,7 An alternative approach is a theory of statistical mechanics for molecular liquids based on an integral equation (IE), which has been developed over the past few decades.8 Water is a commonly used solvent. It is well known that molecules form the characteristic liquid structure which stems from hydrogen bonding. This hydrogen bonding causes an anomalous pressure dependence of both translational and rotational motions.9–11 The self-diffusion coefficient passes through the maximum as a function of pressure, and the minimum in the spin–lattice relaxation time is also shown at low temperature. This is because of the balance between two mechanisms due to the applied pressure and distortion of the hydrogen bond causing the increase of unbounded water molecules which

Kento Kasahara was born in Sakai, Japan in 1989. He received his Master’s degree (2014) in chemistry under the supervision of Prof. Hirofumi Sato from Kyoto University. His interests are mainly focused on chemical reactions and dynamics in solution.

Kento Kasahara

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Hirofumi Sato is a Professor at Kyoto University, with research interests in theoretical chemistry. After obtaining his Master’s degree and PhD (1996) from Kyoto University under the guidance of Prof. Shigeki Kato, he worked at the Institute for Molecular Science (IMS). Currently, he is also ViceDirector of the Fukui Institute for Fundamental Chemistry, Kyoto University. Hirofumi Sato

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participate in the diffusion, and compressing the unbounded molecules. This example clearly indicates that the atomistic description of liquids is essential for a complete understanding of the dynamics. The above example also indicates that the static structure is closely related to the dynamics. The structure is often evaluated as a radial distribution function (RDF) g(r), a probability of finding the atom belonging to another molecule at the distance r between two molecules. The Fourier transform of the RDF is called the static structure factor, which is observable by means of neutron and X-ray scattering measurements. From MD simulations, the RDF is calculated by averaging the many configurations of molecules in the system generated from time development based on Newton’s equation. In contrast to MD simulations, IE theory provides the function directly by solving a set of equations formulated in terms of statistical mechanics. This theory represents the static information about liquids as the distribution function.12 Evaluating various time correlation functions and transport properties is a standard approach to understand liquid dynamics. For example, the velocity autocorrelation function (VACF) Z(t) describes the correlation of the motion for a certain molecule between different two times, 0 and t. By using linear response theory, time integration of correlation functions gives the corresponding transport properties. Similar to the computation of the RDF, these quantities can be calculated with trajectories obtained from MD simulations, although the computational task is quite demanding compared with static properties because a very long trajectory is required to obtain statistically converged results. As for IE based dynamics theory, equations about time correlation functions are represented in terms of various static properties such as static structure factors. By solving the equation numerically or analytically, the time correlation function is obtained. Due to the efforts made by many researchers, methodologies to compute various types of time correlation functions are available. IE-based dynamics theory has the inherent nature of static IE theory. Since IE performs configurational integrals analytically under some approximations, an infinite number of solvent molecules is treated and the complicated composition of a solution such as electrolyte solutions and ionic liquids—which is sometimes difficult to be treated using MD simulations—can be easily treated without statistical error. Due to the analytical treatment, the computational cost is much smaller than that of MD simulations. In the present perspective, we will explain the essentials of IE-based dynamics theory and its applications.

2 Description of static molecular liquid structure: IE theory Dynamics theories require information on the static liquid structure, namely the distribution function. In this section, we briefly explain the reference interaction site model (RISM) theory as the methodology to describe the RDFs for molecular liquids.8,12–15 The theory is an extension of the Ornstein–Zernike (OZ) equation to molecular liquids. The OZ equation describes the liquid structure of simple liquids in terms of the distance r between two particles.

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The simplest extension in the framework of OZ theory is to treat the six-dimensional space composed of the relative position between the center of mass and the relative orientation of the two molecules. This extended equation is called the molecular Ornstein– Zernike (MOZ) equation.12 However, its applicability is limited to small molecules because of the slow convergence of the spherical harmonics expansion. In RISM theory, on the other hand, an alternative description is proposed. Molecules are decomposed into atomic sites (interaction sites) and the liquid structure is represented in terms of a set of RDFs between the interaction sites. This enables us to avoid an explicit treatment of the orientation, and the computational cost is drastically reduced. The RISM equations for solute–solvent and solvent systems are written in reciprocal space (k) as, respectively, huv(k) = wu(k)cuv(k)wv(k) + wu(k)cuv(k)qvhvv(k),

(1)

hvv(k) = wv(k)cvv(k)wv(k) + wv(k)cvv(k)qvhvv(k),

(2)

where superscripts u and v denote solute and solvent, respectively. h is the matrix of the total correlation function defined with the RDF g(r) in real space (r) as h(r) = g(r)  1. c is the matrix of the direct correlation function. q is the diagonal matrix of the number density of the solvent, and w is the matrix of the intramolecular correlation functions, describing the molecular geometries. Since the RISM equation involves two unknown functions, h and c, an additional equation, namely the closure equation, is required to be solved. Hypernetted chain (HNC) and Kovalenko–Hirata (KH) closures are often utilized. These closure relations are written as,8,16 (HNC) hlm(r) = exp[dlm(r)]  1,   ( exp dlm ðrÞ  1 if dlm ðrÞ  0 ðKHÞ hlm ðrÞ ¼ ; if dlm ðrÞ 4 0 dlm ðrÞ dlm(r) = bulm(r) + hlm(r)  clm(r).

(3) (4)

(5)

Here, b is the inverse temperature, b = 1/kBT. ulm(r) is the interaction potential between sites l and m, including van der Waals interaction such as Lennard-Jones type potential and electrostatic interaction. One of the important features for dynamics theory is that RISM theory gives the analytical expression of a static structure and its inverse for a solvent system, respectively, as vvv(k) = wv(k) + qvhvv(k),

(6)

Uvv(k) = v1(k) = [qvwv(k)]1  cvv(k).

(7)

The original RISM theory was extended to treat the electrostatic interaction in the system by Hirata et al., referred to as XRISM.14,15 The versatility of RISM theory has been expanded by several researchers. For example, treating a multi-component system such as electrolyte solutions17,18 and improvement of the description of the dielectric properties19–23 have been achieved. One of the recent steps forward is the development of theories to compute the three-dimensional (3D) solvation structure around a solute molecule, also known as spatial distribution function (SDF).24–27 An SDF is more intuitive compared with a radial distribution function. Furthermore, the effect of an anisotropic solvent

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distribution is fully taken into account. The 3D-RISM theory proposed by Kovalenko et al. is one of the theories to compute an SDF with a reasonable computational cost28 and succeeds in revealing the role of solvent water in a variety of chemical and biological systems.29–32 The 3D-RISM equation is written as follows: h i X v v vv huv cuv (8) l ðkÞ ¼ m ðkÞ wml ðkÞ þ rm hml ðkÞ : Published on 11 October 2017. Downloaded by University of Newcastle on 11/10/2017 18:45:34.

m

Similar to the 1D-theory, the equation is solved by coupling with closure relations. Kiyota et al. proposed the 3D-RISM theory between the two different solutes (u and u 0 ) as33 X 0 X 0 u0 u0 v vu0 huu cuu cuv (9) a ðkÞra hal ðjkjÞ: l ðkÞ ¼ m ðkÞwml ðjkjÞ þ m

a2v

Recently, Jeanmairet et al. proposed another approach called molecular density functional theory (MDFT) to compute an SDF effectively by using some bulk liquid properties obtained from MD simulations.34–36

3 Dynamics theory based on an interaction site model 3.1 Treatment of rotational motion: coupled translation of interaction sites A difficult problem to describe the dynamics in molecular liquids lies in the treatment of rotational motion. Theoretical developments in which the rotational motion is explicitly treated have been achieved by many researchers.37–40 However, since most of these theories are based on spherical harmonics expansion to treat the rotational motion, the slow convergence of the expansion restricts the applications to the comparatively simple liquids such as dipolar liquids. Alternatively, the representation of rotational motion in terms of an interaction site model was proposed by Hirata.41 The rotational motion is expressed with the coupled translations of the interaction sites composed in a molecule (see Fig. 1(b)). In this description, time correlation functions (TCFs) are described with the correlation between two interaction sites. For example, the site–site van Hove correlation function between sites l and m is defined as E 1D Glm ðr; tÞ ¼ drl ðr; tÞdrm ðr0 ¼ 0; t ¼ 0Þ (10) rl

Here, dri(r,t) is the local density fluctuation of site i at time t and h  i means the statistical average. Since eqn (10) does not include the orientational variable but only position r, an explicit treatment of the rotational motion is not required. This feature makes the computation of TCFs easier compared with the explicit treatment of the rotational motion. 3.2

Diffusion equations for molecular liquids

The site–site Smoluchowski–Vlasov (SSSV) equation is the first report to describe the van Hove correlation function on the basis of the interaction site description.41 This equation was derived by using the Zwanzig–Mori projection operator method5,12,42 with the phenomenological approximation to characterize the diffusive motion. In order to apply this method, appropriate slow variables are chosen. A set of the density fluctuation and current density of interaction sites, {drl(r,t),jl(r,t)}, is chosen as the dynamical variables for deriving the SSSV equation. The SSSV equation in reciprocal space (k) is shown as @ Fðk; tÞ ¼ k2 DqUðkÞFðk; tÞ: @t

(11)

In this equation, all functions take the form of N  N matrices, where N is the number of interaction sites in the solvent. F(k,t) is an intermediate scattering function equivalent to the Fourier transform of the van Hove correlation function G(r,t). D and q are diagonal matrices defined as Dlm = Dldlm and rlm = rldlm, where Dl is the diffusion coefficient of site l. U(k,t) is the inverse of the density–density static correlation function v(k,t). The analytical form of the correlation function is provided by the RISM equation. In the original paper,41 the theory was applied to a pure water system, showing good agreement with MD simulation results in the long time region. This theory has been utilized to describe the response of the solvent distribution to a sudden change in the system (solvation dynamics)43–54 by combining with the surrogate Hamiltonian method as reported by Raineri et al.55–57 The original theory is limited to neat solvents such as pure water, but the extension to the multi-component systems including the infinitely diluted system was achieved by Iida et al.58 Recently, further extension to compute the van Hove correlation function in 3D-space was achieved by Kasahara et al.59 This theory enables us to understand the molecular diffusion more intuitively. The distinct part of the 3D intermediate scattering function Fuv(k,t) is expressed as Fuv(k,t) = k2P(k)Bvv(k,t)P1(k)Dvqvcuv(k) + P(k)exp[hv(k)t]P1(k)qvhuv(k),

(12)

where 8 1   v  > exp yl ðkÞt k2 Dul  yvl ðkÞ > > <   ½Bvv ðk; tÞlm ¼  exp k2 Dul t dlm if yvl ðkÞak2 Du ; > > >   : if yvl ðkÞ ¼ k2 Du : t exp yvl ðkÞt dlm (13) Fig. 1 The description of rotational motion. (a) Rotation is explicitly treated, and (b) represented in terms of coupled translation of each interaction site.

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P(k,t) and hv(k) are respectively the unitary matrix and eigenvalues obtained by diagonalizing k2DvqvUvv(k). Since the SSSV equation is

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derived through Markov approximation and overdamped limit, the theory is valid only in the diffusion regime. To overcome this problem, another theory based on the generalized Langevin equation (GLE) has been developed as shown in Section 3.3. However, it should be noted that SSSV theory and its extensions are still useful for analyzing the dynamics in the long time region, because the analytical solution of the equation is available and the van Hove correlation function at an arbitrary point of time can be computed. Another type of diffusion equation for molecular liquids was proposed by Kasahara et al. to describe the diffusion process of a polyatomic molecule in 3D-space under an external field produced by another tagged molecule.60 From the Zwanzig–Mori projection operator method in which a set of single particle density fields in the presence of an external field, {rsl(r,t)}, is chosen as the dynamical variables, the following equation is obtained. " Xð @ s rl ðr; tÞ ¼ Dr  dr0 gl ðrÞrwlm 1 ðjr  r0 jÞrsm ðr0 Þ @t m (14)  1 s s rUl ðrÞrl ðr; tÞ : hl ðrÞrrl ðr; tÞ þ kB T Here, Ul(r) is the potential of the mean force acting on site l defined as Ul(r) = kBT log gl(r). Eqn (14) can be regarded as a molecular version of the Smoluchowski equation. The equation was successfully applied to the diffusion-controlled reaction in a diatomic molecule liquid. In the derivation, the approximated three-body density correlation function was employed, which is obtained from the functional derivative of the HNC closure and the 3D-RISM equation between two tagged solutes (uu-3D-RISM).33 The prescription to obtain approximated many-body density correlation functions from HNC closure was proposed by Yamaguchi et al. for simple liquids.61 3.3

Site–site generalized Langevin equation (SSGLE)

The site–site GLE (SSGLE) for the intermediate scattering function F(k,t) was derived for both the collective and single-particle cases by Chong et al.62,63 The SSGLE is briefly outlined only for the collective case here, because that for the single particle case is almost the same. In the derivation, a set of density fields and longitudinal current density fields of interaction sites represented in k-space, {drl(k,t),jL,l(k,t)}, are chosen as dynamical variables. The resultant expression is shown as ðt  2 @2 @ Fðk; tÞ þ x dtKL ðk; t  tÞ Fðk; tÞ ¼ 0; Fðk; tÞ þ k 2 @t @t 0 (15) Eqn (15) is mathematically the same as the equation for simple liquids except that the above equation takes a matrix form. KL(k,t) is the memory kernel. hxk2i is the normalized second frequency moment of the dynamic structure factor S(k,o) and the n-th frequency moment is defined as ð 1 1 doon Sðk; oÞUðkÞ: (16) hxkn i ¼ 2p 0 S(k,o) is defined as the Fourier transform of F(k,t) about time t. From a well-known relationship between the dynamic structure

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factor and the static longitudinal current field correlation function, J(k) = hjL(k)jL(k)i is given as5,12 ð 1 1 JðkÞ ¼ doo2 Sðk; oÞ; (17) 2p 0 hxk2i can be computed with the static properties. Since the general expression of J(k) for rigid molecules was derived by Yamaguchi et al.,64 the versatility of the SSGLE theory has been greatly expanded. The initial value of KL(k,t) is also expressed with frequency moments as KL(k,0) = hxk4ihxk2i1  hxk2i  D(k)

(18)

The task left to numerically solve eqn (15) is to give the explicit form of the memory kernel KL(k,t). Since obtaining an exact expression of KL(k,t) is actually impossible except for its initial value KL(k,0), some approximations are required. Chong et al. utilized the two approaches proposed for simple liquids, a simple exponential model (SEM)62,63,65,66 and a mode-coupling theory (MCT) based approach.67–69 In an SEM, the memory kernel is assumed as a simple exponential function with the k-dependent time constant s(k). It should be noted that s(k) is not an adjustable parameter but analytically calculated by requiring that S(k,o) reproduces the free-particle behavior at the limit k - N, o = 0. This treatment is a natural extension of Lovesey’s prescription proposed for monoatomic liquids.70,71 This theory has been applied to the collective excitation in diatomic molecular liquids,62,63 pure water,65,66 and reorientational relaxation of water molecules.72 In an MCT based approach, the memory kernel is decomposed ¨gren’s theory.5,73–75 into fast and slow portions, according to Sjo The fast portion is related to the binary collision process in the short time regime and written as Kfast(k,t) = g[ts1(k)]D(k) where the shape function g(x) satisfies g(x) E 1  x2 for small x such as ¨ L(k,0) g(x) = exp(x2) and g(x) = sech2(x). s(k) is determined from K and KL(k,0) represented in terms of hxkni (n = 2, 4, and 6). As for the slow portion Kslow(k,t) related to the correlated collision process, the form is constructed based on the MCT so as to satisfy that Kslow(k,t) evolves at the order of t4 in the short time regime. For convenience, this MCT based theory is referred to as RISM/MCT-1 in this perspective. The strategy to extract the information about the translational and rotational motions from the interaction site description was proposed.68 Chong et al. derived another formulation of the MCT based on the Mori–Fujisaka method (RISM/MCT-2)76 to investigate the glass transition for a dumbbell molecule system.69,77 Yamaguchi et al. revealed that Chong’s theory underestimates the friction on the reorientational motions and modified the theory by including interaxial coupling (RISM/MCT-3).78 3.4

Time correlation functions and transport properties

From the site–site intermediate scattering functions F(k,t), various transport properties can be evaluated. The site–site velocity correlation function Z(t) is described as5,64   3 @2 s Zlm ðtÞ ¼ vl ð0Þ  vm ðtÞ ¼  lim 2 2 Flm ðk; tÞ; k!0 k @t

(19)

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where vl(t) is the velocity of site l at time t, and the superscript s means the self-part. The diffusion coefficient D is obtained by time integration of Zlm(t) divided by 3, and the choice of sites l and m can be arbitrary. The first-rank reorientational correlation Cl(t) is defined as64

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Cl(t) = hl(0)l(t)ih|l|2i1,

(20)

where the vector l is defined as the linear combination of the P position of the sites in a molecule, lðtÞ ¼ al rl ðtÞ, and a set of l P al should satisfy al ¼ 0. The second derivative of Cl(t) is l

expressed with Z(t) as D E X d2 Cl ðtÞ ¼  al am Zlm ðtÞ jlj2 1 : 2 dt lm

(21)

The integration of Cl(t) about time gives the first-rank reorientational relaxation time tl. The relationship between the wavevector- and frequencydependent dielectric function, e(o) = e 0 (k,o) + ie00 (k,o) and F(k,t), is represented as79,80 ð 1 1 4pbior 1 iot X  ¼ dte zl zm Flm ðk; tÞ; eðk; oÞ eðk; o ¼ 0Þ k2 0 lm (22) and at the limit k - 0, 1 1 4pbio  ¼ eðoÞ e0 V

ð1

dteiot hMz ð0ÞMz ðtÞi:

(23)

0

Here, V is the volume of the system, zl is the point charge on site l, e0 is the static dielectric permittivity (e0 = e(o = 0)), and Mz(t) is the z-component of the electric dipole moment of the system. Atomic units are used in eqn (22) and (23). The dielectric relaxation time tD is defined as ð1   tD ¼ e0 dthMz ð0ÞMz ðtÞi Mz2 ð0Þ 1 : (24) 0

The site–site velocity correlation, reorientational, and frequencydependent dielectric functions described as eqn (19), (21) and (23), respectively, are defined in the limit k - 0 of F(k,t). In the calculations of these functions, the limit k - 0 is taken at the stage of solving the SSSV equation or the GLE. Afterwards, the time propagations of these functions are computed. The time-dependent shear viscosity Z(t) was derived by Yamaguchi et al. with an MCT approximation as81 

ð kB T 1 4 @ fvðkÞ  wðkÞg ZðtÞ ’ k dkTr UðkÞ  60p2 0 @k @wðkÞ  fvðkÞ  wðkÞg @k @wðkÞ  fvðkÞ  wðkÞg  @k i  UðkÞ  Fðk; tÞÞ2



(25)

This formula was checked by applying it to neat water under ambient conditions.81 In their calculations, F(k,t) obtained

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from the RISM/SEM was used as the input of eqn (25). The theoretical result shows that Z(t) decays within a range of around 1 ps, which is qualitatively in good agreement with the MD simulation results by Smith et al.82 The ordinary shear viscosity obtained by integrating Z(t) is 0.62 cP, which is comparable to the experimental value, 0.82 cP.83 The viscosity was also reported from MD simulations, but depends on the simulation methods, 0.43–0.91 cP.82,84–86 3.5

Other dynamics theories for molecular liquids

Raineri, Friedman and Resat developed reference memory function approximation (RMFA) theory, in which the normalized l-th memory function for the chosen dynamical variable was replaced with that for a reference dynamical variable.79,87–89 Yamaguchi et al. derived a Smoluchowski-like equation for ion pairs in solution to formulate the theories for computing ultrasonic absorption spectra90,91 and frequency-dependent electrical conductivity.92,93 Kobryn et al. proposed a statistical–mechanical theory to treat ultrasonic processes such as the propagation and absorption of ultrasound in molecular liquids.94 Kim et al. proposed an SSGLE which is different from eqn (15) to treat the structural fluctuation of a solute molecule such as protein based on 3D-RISM theory.95,96 Yoshimori derived a time-dependent density functional theory (TD-DFT) represented in terms of the interaction site description.97 Bertolini et al. proposed a theory of computing generalized thermodynamic and transport properties for molecular liquids by using the various time correlation functions obtained from MD simulations.98,99 Schilling and his coworkers derived a molecular mode-coupling theory (MMCT) in which both the translation and rotation are explicitly treated and proved that their theory is applicable to molecular liquids such as water.100–103

4 Applications 4.1

Visualizing diffusion process

The van Hove correlation function G(r,t) and its Fourier transform F(k,t) are fundamental properties to understand the dynamics in liquid states and deeply related to neutron and X-ray scattering experiments.104 The function G(r,t) has information about collective motions of liquids and can be decomposed into a self part (Guu(r,t)) and a distinct part (Guv(r,t)). Both parts were often computed to analyze the dynamical behavior of various glassy and supercooled liquids based on MD simulations.105–113 The function is conventionally represented in terms of the radial coordinate (r) and possesses a limited amount of information about the structural change of liquids. Zassetsky and Svischev have computed the function in 3D-space for water and NaCl aqueous solution for the first time.114,115 The computation in 3D-space has two obvious advantages: the 3D function is able to fully take into account anisotropic environments produced by a solute molecule, and the visualization of the function makes us understand the liquid dynamics easier compared with the 1D function. However, the task of computing such a space– time correlation function is highly demanding. Alternatively,

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the 3D-SSSV theory developed by Kasahara et al.59 gives the 3D van Hove function for each solvent site at arbitrary times without numerically solving the differential equation eqn (11) under the diffusion limited approximation (see eqn (12)). Fig. 2 shows the distinct part of the normalized van Hove 0 correlation function Guv (r,t) = q1Guv(r,t) + 1 for the pure water system obtained from the 3D-SSSV theory (eqn (12)).116 The function at t = 0 is equivalent to the SDF. Peaks I and II at t = 0 correspond to the hydrogen (Hw) and oxygen (Ow) atoms in solvent water which form hydrogen bonds with solute water. Peak III represents the distribution of Ow, which is bound to the Hw of peak I. At t = 0.1 ps, the characteristic structure of hydrogen bonding is still maintained, although the areas, in which the probabilities of finding Hw and Ow are high, are decreased by the diffusive motion. The rapid decay of peak III is shown but peak I still exists at t = 0.25 ps. Finally, both peaks I and III vanish at t = 0.50 ps. On the other hand, peaks II and IV are shown, and such a behavior is also detected by the MD simulations with the same potential parameters.59 The MD simulation results of Svishchev et al.114 give a similar tendency, although the time-scale of the decay is somewhat slower because of the difference in the potential functions.117 3D-SSSV theory was also applied to 1 M LiClO4/ethylene carbonate (EC), a typical electrolyte solution of lithium-ion secondary batteries, and detected the cooperative motion of EC and Li+.59 In the case of LiCl aqueous solution, the same trend of the cooperative motion was discussed by Iida et al.58 Although 3D-SSSV theory gives reasonable results in these applications, an assumption in the original SSSV theory is also employed; the diagonal diffusion matrix is assumed in the diffusion limit. This means that the oxygen atom of the carboxyl group of EC feels a stronger interaction with the surroundings

Fig. 2 Normalized 3D van Hove correlation function around solute water 0 uv0 in H2O, Guv Hw ðr; tÞ (orange) and GOw ðr; tÞ (blue) obtained from 3D-SSSV theory. The thresholds for Hw are 1.50 (mesh surface) and 2.40 (solid surface), and those for Ow are 2.25 (mesh surface) and 4.00 (solid surface) at t = 0, 0.10 and 0.25 ps. At t = 0.50 ps, the thresholds for Hw and Ow are 1.30 and 1.80, respectively.

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than the methylene groups do, but such a difference is not included in the SSSV approximation. 4.2

Collective excitations in water

The first application of the SSGLE is to describe the collective excitations in a diatomic polar liquid with RISM/SEM as reported by Chong et al.62,63 The dispersion relations for acoustic and optical modes can be obtained from the peak frequencies in the corresponding longitudinal current spectrum. After demonstrating the validity of the theoretical treatment in diatomic polar liquid, they applied the theory to a water system.65,66 The collective excitation in water attracts a lot of attention in solution chemistry. Teixeira et al. found that the phase velocity of the acoustic mode is much faster than the ordinary sound velocity in the low-k region from the inelastic neutron scattering experiment.118 The results of the inelastic X-ray scattering experiments show that the collective excitation changes from the ordinary sound to the fast sound mode continuously.119–121 This fast sound was also detected by the MD simulation studies.122–126 The optical mode was also investigated with the simulation reported by Ricci et al.127 and Bertolini et al.128 However, since the simulation cell size of MD simulations is finite, the minimum value of the wave vector which is accessible for MD simulations is comparatively large and the extrapolation to k = 0 is somehow arbitrary, although fruitful insights about these modes have been obtained. In contrast to MD simulations, since IE based dynamics theory can cover the virtually infinite distance, such a difficulty does not appear. Resat et al. successfully applied RMFA theory to describe the optical mode.88,89 Chong et al. obtained the three eigenmodes of the system by diagonalizing hxk2i as acoustic modes (ordinary sound mode), optical modes (OMs) I and II. Since the energy density is not included as a dynamical variable, the theory is not able to describe the adiabatic nature of sound. Hence, the propagation of the ordinary sound mode occurs at the isothermal sound velocity. This treatment is reasonable for the sound dispersion because the thermal expansion coefficient of water is small enough. Chong et al. clearly assigned the motions associated with OM-I and OM-II as the pitch and roll librational motions, respectively (see Fig. 3). OM-I is the same as reported by Ricci et al.127 OM-II corresponds to a single-molecule motion, which did not appear in previous studies because of the different choice of dynamical variables from those of RISM/SEM.

Fig. 3 Schematic representation of the optical modes (OM) in water as revealed by Chong et al.65 (a) OM-I and (b) OM-II.

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The dispersion relation corresponding to the acoustic mode evaluated from the peak frequency of longitudinal current spectra exhibits the positive dispersion apparently. Furthermore, the phase velocity is E3700 m s1, which is close to that predicted by MD simulations, and much larger than isothermal sound velocity of the acoustic eigenmode (E1200 m s1). They also carried out the MD simulations for comparison. The current longitudinal fields for the acoustic and optical dynamics are in good agreement with those from MD simulations, although the peak frequency for the optical dynamics is small compared with the MD simulations. 4.3

Solvation dynamics

Solvation dynamics is defined as the response of the solvent distribution change to a sudden change in the system such as the excitation of a solute molecule.38,39,129 A fundamental property for the solvation dynamics is the solvation time correlation function (STCF) defined as SðtÞ ¼

deðtÞ  deð0Þ ; deð1Þ  deð0Þ

(26)

where de(t) = e(t)  hei is the solvation energy change after the photo-excitation. By using the fluctuation dissipation theorem, eqn (26) can be rewritten as S(t) = hde(t)de(0)ihde(0)de(0)i1. Raineri et al. proposed a useful theory called the surrogate Hamiltonian method to investigate solvation dynamics. In this theory, the solute–solvent interaction is not a bare interaction, but represented as the renormalized potential based on linear response theory. This theory provides a tractable expression of the STCF as Ð1 P 2 Flm ðk; tÞBlm ðkÞ 0 dkk lm Ð P SðtÞ ¼ 1 ; (27) 2 Flm ðkÞBlm ðkÞ 0 dkk lm

Blm ðkÞ ¼

X

Dfal ðkÞwab ðkÞDfbm ðkÞ;

(28)

ab

where Flm(k,t) is the site–site intermediate scattering function for neat solvent, and wab(k) is the intramolecular correlation function for the solute molecule. Dfal(k) is defined as Dfal(k) = E G fEal(k)  fG al(k) where fal(k) and fal(k) are the solute–solvent couplings for the ground and excited states, respectively. fal(k) is defined as fal(k) = kBTcal(k) where cal(k) is the direct correlation function between the solute and the solvent. According to eqn (27), the solvation dynamics can be described with an intermediate scattering function of bulk solvents, indicating that information on the dynamics of the solute–solvent system is not required. The effect from the solute on the dynamics is included via the direct correlation function. Raineri et al. applied the theory to the ion solvation in acetonitrile and in water, and the solvation of a diatomic solute in acetonitrile, in methanol and in water, and formaldehyde in water.55–57 The obtained STCFs are qualitatively in good agreement with the non-equilibrium MD results. Hirata et al. proposed SSSV theory combined with a surrogate Hamiltonian.43 Then, Ishida et al. developed a theory to describe the time development of the electronic structure of a

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solute associated with solvent relaxation,44 which is based on RISM-SCF theory, a combined theory of RISM and quantum mechanics.130–134 They applied it to the solvation dynamics of the excited state of C6H5CN in H2O, CH3OH and CH3CN, yielding time constants comparable to the MD simulation results. Afterwards, Ishida extended the theory to treat the inertial effect neglected in SSSV theory.53 Further investigations based on SSSV theory have been carried out by Nishiyama et al. They applied the theory to the system that a simple ion is immersed in MeCN solvent.45–47 They found that the rearrangement of the second solvation shell is much slower compared with that of the first shell. Furthermore, the decomposition of the STCF reveals that the greatest part is ascribed to the optical mode. They also investigated the solute dependence of the STCF by considering various shapes of solutes, simple ions, dumbbell-like dipoles, quadrupoles, and octapoles, etc.48,49 Their results show that the nonlinear behavior of the STCF appears for multipoles and the STCF decays faster as the distance of charges increases. The decomposition of the STCF reveals that the fast mode (optical) with a time constant of 0.2 ps is responsible for the majority of the STCF. Nishiyama et al. upgraded theoretical treatment regarding the intermediate scattering function from SSSV theory to RISM/MCT-3 theory and applied it to the systems employed in the works of SSSV theory.50–52 They succeeded in reproducing the experimentally observed Gaussian decay of STCFs in the short time regime and damped oscillation. Another promising way to effectively describe the solvation dynamics has been developed for simple liquids by Yamaguchi et al.61 In their theory, solvent fluctuation under an external field produced by a solute is treated explicitly and the solvation dynamics is described by employing the linear response approximation. 4.4

Transport properties

Evaluating transport properties is crucial for understanding the dynamics. The theories explained in Section 3.3 enable us to get such information without empirical parameters except for classical force fields. Let us start to discuss the accuracy of the transport properties predicted by IE based theory. Yamaguchi et al. computed the translational and rotational diffusion coefficients of water under ambient conditions by RISM/SEM.72 The obtained translational diffusion coefficient is 3.0  105 cm2 s1, showing good agreement with experiments (2.3  105 cm2 s1) and MD simulations (2.7  105 cm2 s1). On the other hand, the two rotational diffusion coefficients are 7.3 and 4.1 ps, which are much faster than the experimental results estimated from the second-order reorientational time. They clarify that the long time part of the memory kernel which is not considered in the SEM is dominant in these properties. Fig. 4 shows the pressure dependence of the diffusion coefficient D and the first-rank reorientational relaxation time t1R obtained from RISM/MCT-2 for neat acetonitrile at 298 K as reported by Yamaguchi et al.,64 together with the experimentally observed diffusion coefficient135 and the second-rank relaxation time t2R.136 It is seen that the quantitative agreement between the theory and experiments is not good for both the diffusion coefficient and relaxation time because of

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Fig. 4 Pressure dependence of the transport properties for neat acetonitrile at 298 K. (a) Diffusion coefficient D. (b) First-rank reorientational relaxation time t1R (RISM/MCT-2) and second-rank relaxation time t2R (experiment) scaled by 3. t1R is expected to be three times as large as t2R in the rotational diffusion limit. The result of RISM/MCT-2 is taken from ref. 64 and the experimental results of D and t2R are taken from ref. 135 and 136, respectively.

the approximations involved in RISM theory and the memory function. Yamaguchi et al. extensively examined the validity of the MCT for solute–solvent systems, Lennard-Jones fluids and ions in water systems by using MD simulations.137 For a pure solvent system, one of the solvent molecules is treated as a solute molecule in the computations of the diffusion coefficient and the reorientational relaxation time. They clarify that the factorization approximation in the MCT becomes worse when a strong attractive interaction exists such as electrostatic interaction between small ions (solute) and water molecules (solvent). On the other hand, the theory qualitatively reproduces the pressure dependence of the transport properties: the diffusion coefficient and relaxation time monotonically increase and decrease with increasing pressure. In many cases as shown later, IE based theory captures the important feature of the dynamics in molecular liquids. Furthermore, due to analytical treatment, the theories provide new viewpoints about the dynamics in

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molecular liquids by decomposing the transport properties into physically meaningful factors. The description of the dynamics of ions in polar liquids by Chong et al. is one of such analyses. It is well known that the simple Stokes’ law based on the hydrodynamic theory predicts that the friction would increase proportionally to ionic radii, but the experimental studies revealed that the ion-size dependence is opposite to the Stokes’ law for small ions. To explain this peculiar behavior, the solventberg and dielectric friction mechanisms have been proposed. Chong et al. investigated the dynamics of ions in diatomic polar liquid138 and liquid water139 based on RISM/MCT-1. In the former system, the fast portion of the memory kernel is not included. After revealing that the minimum in the ionic friction as a function of solute size is reproduced, they theoretically decomposed the friction in terms of collective excitations of the solvent into hydrodynamic (zNN) and dielectric contributions (zZZ), corresponding to acoustic and optical modes, respectively, and their couplings (zNZ). Their results indicate that all the contributions are important for small ions and zNN becomes large, and the other two contributions decrease with increased solute size. In the latter system, the fast portion is considered. The alkali metal ions (Li+, Na+, K+, Rb+ and Cs+) and halide ions (F, Cl, Br and I) are treated as solutes. The same decomposition is successfully applied to the system. The changes in diffusion coefficients with the cation and anion sizes in which a maximum is found are qualitatively in good agreement with the MD simulations by Koneshan et al.140 The pressure (density) dependence of the mobility of water has attracted many scientists for a long time. For example, the shear viscosity of water as a function of pressure exhibits the minimum value below 30 1C, although the shear viscosity of most molecular liquids increases with increasing density. Yamaguchi et al. investigated the pressure dependence of the transport properties of neat acetonitrile with RISM/MCT-264 and neat water with RISM/MCT-3.141 For both the liquids, the theory qualitatively reproduces the changes in the properties with increasing pressure observed by the experimental measurements. The diffusion coefficient decreases and orientational relaxation time increases for acetonitrile. In the case of water, the diffusion coefficient shows the maximum with increasing pressure, and the minimum appears in the viscosity, dielectric relaxation time and reorientational relaxation time. From the comparison between the results of acetonitrile and water, they concluded that the anomalous behavior in water is attributed to the small collisional friction on the reorientation due to the spherical repulsive core, and the strong short-range Coulombic interaction caused by the formation of hydrogen bonding. Kobryn et al. investigated the pressure dependence of the transport properties for solute molecules, acetonitrile in water, methanol in water, and methanol in acetonitrile, by means of RISM/MCT-3.142–144 Their results reproduce the tendency observed from the experiment, and clarify that the density dependence of the mobility in different solute–solvent systems is ascribed to the two competing origins of friction, the collisional and dielectric frictions. Yamaguchi et al. studied the dynamics of hydrophobic hydration in a water–hydrophobic

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solute (benzene and rare gases) mixture with RISM/MCT-3, succeeding in revealing that the number density fluctuation of the solvent due to the cavity produced by the solute enhances the friction on the collective polarization through the dielectric friction mechanism.145,146 For concentrated solutions of electrolytes, various kinds of pairs, contact ion pairs (CIPs), solvent separated ion pairs (SIPs), and doubly solvent separated ion pairs (SSIPs), contribute to the bulk transport properties. Ultrasonic absorption spectra strongly reflect the formation and dissociation processes of such pairs. Yamaguchi et al. developed the theory to compute the ultrasonic absorption spectra and applied it to an aqueous solution of MgSO4.90,91 The spectrum shows the four peak frequencies at 3 THz, 100 GHz, 300 MHz and 30 kHz, which is consistent with the experimental observation. They succeed in theoretically assigning these peaks without employing conventional three-state or four-state models. By extending the theory for ultrasonic absorption spectra, Yamaguchi et al. developed the theory to compute the frequency dependent electric conductivity in electrolyte solutions.92 The application to the aqueous solution of MgSO4 suggests that the competition between the rates of the dissociation and reorientation of contact ion pairs (CIPs) is an important factor. Afterwards, they extended the theory to treat the hydrodynamic interaction with the Oseen approximation and applied it to an aqueous solution of NaCl.93 They succeed in reproducing the concentration dependence of the conductivity observed in the experimental measurement, while the agreement of the dependence of the transport number is not so good as compared to the conductivity. They suggest that the development of the theoretical expression of the hydrodynamic interaction is essential for further understanding. Yamaguchi et al. applied RISM/MCT-2 to dimethylimidazolium chloride, a typical room temperature ionic liquid (RTIL).147 The site–site structure factor obtained from MD simulations is employed as an input of RISM/MCT-2 instead of that from RISM. They reveal that the theory qualitatively reproduces the dynamics of the ionic liquid obtained from the experiment and MD simulations. They conclude that dielectric relaxation is mainly ascribed to the translational mode in this RTIL. 4.5

Diffusion-controlled reactions (DCRs)

Diffusion controlled reactions (DCRs) appear in various fields of chemistry and physics such as biochemistry, photochemistry, and materials science.2 In this perspective, our attention is confined to the DCR for an anisotropic system such as a polyatomic molecule system, and the DCR includes the reactions both with finite and infinite reactivity. Brownian dynamics (BD) simulation148 is a powerful tool to study the DCR for such systems. Exceptional methodologies for describing DCRs based on BD simulations were developed by McCammon and his coworkers,149,150 as well as Sangyoub Lee and his coworkers.151–154 Complicated reactions oriented towards biological systems such as enzyme–substrate association, protein–protein association, and ligand binding to protein, have been extensively studied.149,150,152,155–174 These studies have proven the importance of the hydrodynamic interaction, the electrostatic

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interaction between reactants, and the structural fluctuation of biomolecules. MD based methodologies were also proposed.175–178 Dong, Baros, and Andre (DBA) proposed the method to calculate the survival probability of reactants related to the rate constant by counting reaction events directly.175 Since the advent of the DBA model, efficient methods have been developed based on the 1st recollision probability reported by van Beijeren et al.176 and on returning probability reported by Lee et al.177,178 Recently, the latter method was applied to the dismutation of O2 catalyzed by human Cu/Zn superoxide dismutase (SOD) for the wild and mutant types, yielding the rate constants which are in good agreement with the experimental result. An alternative approach is to utilize the transport equation such as the Smoluchowski equation with reaction boundary conditions179,180 or with reaction sink functions.181,182 In this approach, since the various properties about the reaction dynamics are obtained from solving the equations numerically or analytically, the computation is free from statistical error. Along this line, attempts to treat the anisotropic systems have been carried out by using spherical particles and hemispheres with reactive patches on their surfaces.151,159,183–203 Indeed, these developments are limited to spherically symmetric systems. Recently, Kasahara et al. derived a molecular version of the Smoluchowski equation represented in terms of the interaction site description, which is applicable to the anisotropic systems originated from the molecular geometry.60 The theory was applied to the DCR in diatomic liquids by imposing the absorbing boundary condition. The time-dependent distribution function of a reactant molecule around another one and the time-dependent rate constant obtained from the theory were compared with those from MD simulations (DBA method). Similar to the conventional Smoluchowski equation with the absorbing boundary condition, the difference between the theory and MD results is large in the short time regime, but the change of the distribution function is shown in good agreement with MD simulation results. In the case of the long time regime, furthermore the theoretical result is in good agreement with the MD simulation results. There is still room for improvement in their theory. The Markov approximation used in SSSV theory is adopted in the derivation of eqn (14), and the hydrodynamic interaction is neglected. Furthermore, the absorbing boundary condition is employed in this application, and upgrading the condition is essential for further deep understanding of the reaction dynamics.

5 Concluding remarks In this perspective, dynamics theories for molecular liquids based on integral equation theory are summarized. In Section 2, the integral equation (IE) theory for polyatomic molecular systems, and reference interaction site model (RISM) theory, are briefly explained. Section 3 presents the theoretical frameworks of the dynamics theories for polyatomic systems and the applications are exhibited in Section 4.

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The outstanding advantages of the IE based dynamics theories compared with the other molecular theories and MD simulations are summarized as follows. The dynamics theories explained in Section 3 are based on the interaction site description and an explicit treatment of the orientational degrees of freedom is not required. Hence, the theories are applicable to a variety of solution systems including electrolyte solutions and ionic liquids. Furthermore, in contrast to MD simulations, as information about the dynamics is obtained by solving the equations, the results are free from the statistical errors that sometimes appear as a serious problem in MD simulations. This advantage is provided by the analytical nature of IE theory. Of course, challenging problems still lie in the molecular theory of dynamics. For example, all the theories introduced in this article are applicable only to liquids in which solvent molecules are rigid. As for the description of static liquid structures, IE theories for flexible molecules represented with realistic intramolecular potentials have been proposed.204–206 The development of a dynamics theory to treat the flexibility of solvent molecules based on these IE theories is highly desired. The refinement of the description of dynamical properties is also an important problem. Several studies reveal that explicit treatment of solvent fluctuation under an external field produced by a solute is essential for refinement.61,137 At the same time, development of hydrodynamic interactions from the microscopic point of view is important. The authors believe that further development of molecular theory will lead to deeper understanding of the dynamics of solution systems.

Conflicts of interest There are no conflicts to declare.

Acknowledgements K. K. is thankful for the Grant-in-Aid for the Japan Society for the Promotion of Science (JSPS) Fellows. The work was financially supported in part by a Grant-in-Aid for Scientific Research (B) (JP17H03009). A part of this work was performed under a management of ‘Elements Strategy Initiative for Catalysts & Batteries (ESICB)’. Theoretical computations were partly performed using the Research Center for Computational Science, Okazaki, Japan. All of the authors were supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

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