Instantaneous normal mode analysis of liquid waterMinhaeng Cho, Graham R. Fleming, Shinji Saito, Iwao Ohmine, and Richard M. Stratt Citation: The Journal of Chemical Physics 100, 6672 (1994); doi: 10.1063/1.467027 View online: http://dx.doi.org/10.1063/1.467027 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/100/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Local Structural Effects on Intermolecular Vibrations in Liquid Water: The InstantaneousNormalModeAnalysis AIP Conf. Proc. 982, 410 (2008); 10.1063/1.2897830 Instantaneous normal mode analysis of orientational motions in liquid water: Local structural effects J. Chem. Phys. 121, 3605 (2004); 10.1063/1.1772759 Instantaneous normal mode analysis of Morse liquids J. Chem. Phys. 116, 10825 (2002); 10.1063/1.1479714 Instantaneous normal mode analysis of liquid methanol J. Chem. Phys. 115, 395 (2001); 10.1063/1.1376164 Instantaneous normal mode analysis of liquid Na J. Chem. Phys. 105, 9281 (1996); 10.1063/1.472758

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Instantaneous normal mode analysis of liquid water Minhaeng Cho and Graham R. Fleming Department of Chemistry and the James Franck Institute, The University of Chicago, Chicago, Illinois 60637

Shinji Saito and Iwao Ohmine Institute for Molecular Science, Myodaiji, Okazaki 444, Japan

Richard M. Stratt Department of Chemistry, Brown University, Providence, Rhode Island 02912

(Received 6 December 1993; accepted 18 January 1994)

We present an instantaneous-normal-mode analysis of liquid water at room temperature based on a computer simulated set of liquid configurations and we compare the results to analogous inherent-structure calculations. The separate translational and rotational contributions to each instantaneous normal mode are first obtained by computing the appropriate projectors from the eigenvectors. The extent of localization of the different kinds of modes is then quantified with the aid of the inverse participation ratio-roughly the reciprocal of the number of degrees of freedom involved in each mode. The instantaneous normal modes also carry along with them an implicit picture of how the topography of the potential surface changes as one moves from point to point in the very-high dimensional configuration space of a liquid. To help us understand this topography, we use the instantaneous normal modes to compute the predicted heights and locations of the nearest extrema of the potential. The net result is that in liquid water, at least, it is the low frequency modes that seem to reflect the largest-scale structural transitions. The detailed dynamics of such transitions are probably outside of the instantaneous-normal-mode formalism, but we do find that short-time dynamical quantities, such as the angular velocity autocorrelation functions, are described extraordinarily well by the instantaneous modes.

I. INTRODUCTION

Understanding just how the atoms move in a liquid is a critical step in describing various chemical and physical pro-cesses in liquids. Recent theoretical! and experimentaf stud-ies of electron transfer processes in condensed phases, e.g., have shown the vital role of solvent dynamics. When a prod-uct is produced through either electronically adiabatic or nonadiabatic transition, the excess energy of the nonequilib-rium product will be transferred from the reacting system to the bath, or more specifically, to the nuclear or electronic degrees of freedom of the surrounding molecules. The ki-netic energies of the solvent molecules will increase as a result of this dissipation process. Moreover, the changes in local liquid configuration that stabilize the product state will result in changes in the potential energy of the bath. An interesting question then is just how localized the changes in local structure are and approximately how many molecules are involved in the dissipation process within the relevant time scale.

In order to learn how to answer a question of this sort, what we shall pursue in this paper is the application of a set of microscopically defined modes of the liquid, the instanta-neous normal modes? The usefulness of working with some set of well-defined modes can be appreciated by following the example of a chemical reaction to its logical conclusion. Any local excitation, any local shift in the kinetic and poten-tial energies, will eventually relax toward a state of thermo-dynamic eqUilibrium. Though the effective size of the bath is a rather ambiguous concept, the notion of the degree of lo-calization of an effective heat bath coupled to a reacting

system can be formulated somewhat more precisely-and indeed, one might find that it depends strongly on the nature of the solvent. In the case of an atomic exchange reaction in a simple liquid (liquid Ar), Wilson and co-workers4 found that collisional processes are most effective in relaxing the energy of transition state, corresponding to an increase in the kinetic temperature of the solvent. However, the same reac-tion in liquid water'(a) showed a very different dissipation mechanism, with the long-ranged electrostatic interaction playing the dominant role. We may thus expect the effective size of the water heat bath to be larger than it is with liquid Ar.5(b) If there are useful collective variables capable of de-scribing the solvent dynamics, it is the degree of localization of these variables that could be the desired measure of the size of the bath. To be more literal, we might note that the frictional force itself, the first derivative of the intermolecu-lar potential between the reacting system and the solvent, is a sum of pairwise terms involving every solvent molecule's nuclear coordinates. Thus the collectiveness of the dissipa-tion is manifest, but the degree of cooperativity is not obvi-ous until one resolves the friction into its component modes, discriminating between minor molecular rearrangements and large-scale structural transitions of the heat bath.

The modes we focus on in our study of liquid water [the instantaneous normal modes (INMs)] are not the only pos-sible choice. Both INMs and quenched normal modes (QNMs) can be computed from an eqUilibrium set of liquid configurations without any accompanying dynamical infor-mation. The former though are defined by diagonalizing the Hessian matrix of the potential at every instantaneous liquid configuration, whereas the latter arise by performing this

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Cho at al.: Normal mode analysis of liquid water 6673

same diagonalization, but only at the local minima of the potential (which are obtained by quenching the thermal configurations).6.7 Clearly the QNMs provide a natural way of thinking about the inherent (underlying) structures of a liquid, but one of the advantages of the INMs is that they can be shown to contain a great deal of dynamical information pertinent to the finite-temperature fluid.3,8-12 With atomic liquids, the density of INM states can even be computed analytically by using liquid-theory methods. IO- 12 However, the complexities of both the intermolecular interactions and the structure of water present severe obstacles to analytical predictions. We therefore use a conventional (N, V,E) mo-lecular dynamics simulation technique here to generate a representative set of equilibrium liquid configurations.

Simulations of water have, in fact, played a vital role in facilitating our understanding ever since the seminal molecu-lar dynamics studies of Rahman and Stillinger. 13 Indeed, the "vibrational spectrum" of liquid water has been the subject of a variety of simulation-based investigations ranging from predictions of the Raman and infrared spectra themseIves14 to calculations of the power spectra of assorted dynamical variables. 15.16 More pertinent for our purposes, there have already been a number of normal-mode investigations from the quenched-mode perspective, starting with Pohorille et al. 17 and continuing with an extensive series of studies by Tanaka and Ohmine. ls-22 In this latter work, the authors dis-covered that the reaction coordinates connecting one inherent structure in water to another seem to overlap strongly with the low frequency QNMs-i.e., they coincide with the largely translational modes. Conversely, excitations of these same modes seem to generate inherent-structure transitions. 18.20.21

Both the INM and QNM approaches have fundamentally harmonic outlooks, so it is probably worth emphasizing that we do not intend to minimize the importance of the intrinsic anharmonicity in pursuing these modes. Not only are molecule-molecule pair interactions not particularly har-monic, the low frequency collective modes are likely to be strongly anharmonic, a point emphasized by Ohmine and co-workers. 19 The fact that the lowest frequency quenched normal modes almost always lead to structural transitions from one inherent structure to another is a real indication of the complexity of the global potential surface. Nonetheless, the modes we use, especially the INMs, are of some real value. Besides illustrating the overall range of time scales pertinent to a liquid, the INMs can generate accurate predic-tions for ultrafast dynamics.3 Moreover, unlike the QNMs, which are restricted to local potential minima, each INM will have associated with it a nonzero values of the force fa acting on the liquid configuration. In conjunction with the real frequency cu"" this information suffices to predict the separation, in both energy and distance, of the nearest poten-tial surface minimum in the a direction. In addition, since some of the INM frequencies will be imaginary and the pre-diction for selected modes will be for the nearest potential barrier along the mode direction.9,23 Use has already been made of imaginary modes in quantifying the comparative fluidities of liquids, solids, and c1usters.3,8,9,24,25

Another intriguing, though somewhat indirect, connec-

tion with the instantaneous normal modes is with a number of other kinds of liquid modes that have been studied recently.26-30 As with the INMs, the modes are defined by diagonalizing some matrix that is on the order of N X N at a collection of equilibrium liquid configurations. The specifics of the matrix vary though depending on the physical property being investigated. For example, if the matrix is formulated with dipole interaction tensors for elements, the resulting po-larization modes reflect the polarization fluctuations of the liquid-which can be used to compute electronic spectral shifts of solutes28,29 and the dielectric behavior of a liquid as a whole.28,30,31 In fact, comparing the spectral shifts pre-dicted for bulk liquids and for clusters allowed some fairly definitive statements about the usefulness of a collective per-spective on solvation, at least for one system,29

In this paper, we consider a liquid's instantaneous nor-mal modes as a kind of basis set of dynamical variables that can be used to describe the time evolution of observables in terms of a linear combination of small molecular displace-ments. When we end up computing it, the actual dynamics will then simply be a superposition of sinusoidal functions of time. We shall proceed as follows: in Sec. II, we summarize the instantaneous normal mode formalism as it applies to a liquid composed of rigid molecules of arbitrary shape. Nu-merical results for the INM spectra of water, along with vari-ous dynamical quantities and measures of the potential sur-face, are then presented in Sec. IlL The results and prospects are discussed in Sec. IV.

II. INSTANTANEOUS NORMAL MODE FORMALISM

Defining for a given water molecule j (j = 1 , ... ,N) the center-of-mass position to be rj' the total Hamiltonian of liquid water can be written as

(1)

where m denotes the mass of a water molecule. Hereafter we shall use i, j, k, ... to label molecules and use p.., P, '1/, ... for intramolecular coordinate indices. In Eq. (1), Ijp. is the mo-ment of inertia along the molecular ft axis of the jth mol-ecule. The molecular-fixed coordinate frame {Xj 'Yj ,Zj} is defined by a set of Euler angles fl j = { <P j , OJ , 1/1). V (R) is the potential energy at liquid configuration R,

(2)

with rj={Xj,Yj,Zj}, and CUjx, CUjy, and CUjz are the angular velocities along the instantaneous molecular fixed axes Xj' Y j , and Z j , respectively, Using the Euler angles and direction cosine matrix,32 the angular velocities can be rewritten as

( = ( ::: ( t). (3) cujZ cos OJ 0 1 1/Ij

By expanding the potential to second order with respect to the displacement between the configuration at time t and the instantaneous (t = 0) configuration (Rt - Ro), and evalu-ating the kinetic energy at Ro, we Can now perform an in-

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6674 Cho et al.: Normal mode analysis of liquid water

stantaneous normal mode analysis.3 Without loss of general-ity, we define the 6N-dimensional mass-weighted coordinates

with

and

ZjlL ={ ,In;Xj' ,In;Yj' ,In;Zj' ffxtjx> J!;tjy , ffztjz},

for JL = 1 ,2, ... , 6

( ) (

. 0.0· ".0 r sm Uj sm 'l'j

= sin eJ cos if!J cos eO

J

cos if!J - sin if!J

o

(4)

(5)

Here eJ and if!J are the Euler angles at the instantaneous configuration Ro. In terms of these new coordinates, the instantaneous-normal-mode Hamiltonian can be recast in the form

H=!Z.Z+ V(Ro)- F(Ro)(Zt-Zo)

+!(Zt- Zo)D(Ro)(Zt- Zo), (6)

where F(Ro) and D(Ro) are the 6N-dimensional force vector and the 6NX 6N mass-weighted dynamical matrix, respec-tively, whose elements are

FjlL(Ro) = - (V jlL V)Ro'

DjlL,kv(Ro) = (V jlL V kv V)Ro

with

a VjlL=-a- .

ZjlL

(7)

Let U(Ro) be the 6NX 6N orthogonal transformation matrix that diagonalizes the dynamical (Hessian) matrix D(Ro). The transformed forces can then be written as

f,,(Ro)=2, Ua.jlL(Ro)FjlL(Ro). (8) jlL

The eigenvalues of the dynamical matrix are given as

2, Ua,jlL(Ro)DjlL,kv(Ro)Ua,kv(Ro), (9) jlL,kv

and the instantaneous normal coordinates themselves {q a} can be defined as

qa(t,Ro)= 2, Ua,jlL(Ro)[ZjlL(t)-ZjlL(O)]. (10) jlL

Using these instantaneous normal coordinates, we can re-write the instantaneous-normal-mode Hamiltonian at any given liquid configuration Ro,

(ll)

Equivalently, by defining the shifted normal coordinates

Stable Mode Unstable Mode

FIG. 1. The harmonic potential associated with an instantaneous-normal-mode coordinate xa . For a real (stable) mode, the curvature is positive, and for an imaginary (unstable) mode, the curvature is negative. In both cases, the bold X represents the instantaneous liquid configuration from which the potential is derived and the mass-weighted displacement from the loc?l minimum or maximum is given by f ,; With stable modes, the energy of the instantaneous configuration above the minimum is given by whereas for unstable modes, the instantaneous configuration is below the maximum.

(12)

the Hamiltonian can be expressed in the form

6N ( 2 ) 1'2 1 2 2 fa H=V(Ro)+2, iXa+iwaXa--2'

a=\ 2wa (13)

Here the magnitude of f al can be thought of as a predic-tion for the t = 0 displacement from the nearest extremum of the potential in the a direction-the local potential minimum for a stable modes and the local maximum for an unstable mode In the same spirit, the last term - represents the depth of the nearest potential mini-mum along the ath instantaneous normal mode, if the mode is stable. Should the eigenvalue be negative -would be positive and would represent the barrier height of this unstable mode (Fig. 1).

From Eq. (13), it is straightforward to see that the kine-matics of this coordinate and the velocity are given simply by

(14) Va(t)=Va(O)cos wat-WaXa(O)sin waf.

Though we are representing the dynamics of water as a har-monic bath, this normal mode description provides a picture of the liquid dynamics very different from other oscillator representations. The stochasticity that is present here arises only from the initial coordinates and velocities of the 6N normal modes-the eqUilibrium distribution functions. In particular, it is completely unnecessary for us to invoke any artificial stochastic properties to mimic the complexity of liquid dynamics.

III. NUMERICAL RESULTS

We carried out a 60 ps molecular dynamics simulation of the TIPS2 model for liquid water,33 including a single neutral spherical solute of diameter 6 A along with 264 water mol-ecules in a periodic box. The solute interacts with the oxygen

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Cho et al.: Normal mode analysis of liquid water 6675

,..... 2.5 'b -'-'

2.0 !i C/) '0 1.5

1.0

0.5

o 200 400 600 800 1000 1200 Frequency (cm'l)

FIG. 2. The probability distributions of instantaneous normal modes (solid line) and quenched normal modes (dotted line) for liquid water. As is con-ventional. imaginary modes are displayed on the negative portion of the frequency scale.

atom of the water molecules through a Lennard-Jones poten-tial. The parameters for the interaction between the solute and a water are 0"=4.30 A and €= 132 K, respectively, and the mass of the solute is 180 amu. The system volume is incremented due to the presence of the solute, starting with the volume appropriate for just 264 waters at 1 g/cm3• The solute is always located at the center of the cubic celL The temperature and density of the water were 298 K and 1 g/cm3, other details of the computational methods were given in Ref. 20. The principle reason for considering this aqueous solution, rather than neat liquid water, was to lay the ground work for a subsequent study of the neutral solute on the liquid dynamics, Nonetheless, one should keep the presence of a solute in mind, especially if the solute is to be changed to something more strongly interacting in any future work.

A. Density of states

We first calculate the quenched normal modes (QNM) spectrum of liquid water using the steepest descent method to locate the local potential minima and then performing an ordinary normal mode analysis.2o The final QNM density of states (Fig. 2) is then computed by averaging the results over 13 different configurations. Note that unlike the INM spec-trum, all of the frequencies are real because of the restriction to configurations that are potential energy minima. It is also worth pointing out that the shape of the density of states is precisely the same as that found by Tananka and Ohmine,2o despite the addition of a neutral solute here. It is this obser-vation that suggests that any modifications of the dynamics by our solute would be barely noticeable.

In examining the QNM spectrum, we find that there is a dip em-I separating two distinct groups of vibrations. Ohmine and co-workers I8 -21 presented a detailed discussion of the microscopic origin of these QNM modes. In particular, the low frequency part (0<w<350 em-I) is related to the translational degrees of freedom, whereas the higher fre-quency modes (400<w<1000 cm- I) involve mostly rota-tional, indeed librational, degrees of freedom. By carefully examining the molecular motions corresponding to quenched

normal modes of frequency (350-400 cm- I), they found that the vibrations in this range consisted of both translational and rotational motions which were very localized in space. This pronounced separation of the translational and rota-tional contributions to the QNM density of states undoubt-edly has its origins in the difference between the overall (H20) mass participating in the translation and the much lighter (H) mass defining the moments of inertia. Indeed, the distinctive separation between the translational and rotational peaks in the QNM spectrum disappears in the QNM spec-trum of liquid D20 as shown in Ref. 21.

We now tum to the instantaneous normal mode (INM) calculations. These calculations are carried out by diagonal-izing the dynamical matrix at 30 different liquid configura-tions generated at 2 ps intervals during 60 ps simulation. The resulting INM density of states in Fig. 2 shows some notice-able differences from the QNM spectrum. For one thing, the clear distinction between translational and librational modes in the QNM results has been smeared out in the INM spec-trum. Moreover, the two QNM peaks and 250 em-I have been combined into a single peak that is blue shifted by about 50 cm -I relative to the lowest QNM peak. On the balance, it is not surprising that differences exist between INM and QNM spectra, inasmuch as the QNMs define a zero-temperature density of states. Our findings in particular are quite consistent with the tendency for low frequency modes to blue shift as temperature increases, a result ob-served by Keyes and co-workers in their INM study of simple liquids.9

The other principle difference between the two kinds of modes is the existence of imaginary modes in the INM spec-trum (shown, for ease of presentation, in the negative fre-quency region). We find the stable (real) modes outnumber the unstable (imaginary) modes by about 16 to 1. This rela-tively tiny fraction of unstable modes stands in sharp contrast to the 10%-30% figure seen in simple liquids,9.11.12 suggest-ing that most of the water molecules spend an inordinately large amount of time near the minima of the potential energy hypersurface.

B. Translational and rotational components

In order to separate the translational and rotational de-grees of freedom in the INM spectrum, we define the corre-sponding projectors3

N

pTa= (U )2 .L.J.L.J a.j p. , j=1 p.=1.2.3

N (15)

(Ua.j p.)2. j= I p.=4.5.6

Because of the orthogonality of the transformation matrices U, the projectors obey the normalization condition p + = 1. These translational and rotational projectors [Fig. 3(a)] represent the relative weights of the translational and rotational components for each mode a. Hence the total density of states pew) can be partitioned into translational and rotational densities of states [Fig. 3(b)]

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6676 Cho et al.: Normal mode analysis of liquid water

1.0 1.0

0.8 0.8 of Rot

f! 0.6 .9 § 0.6 <.) Q)

<.) Q)

l 0.4 l 0.4

of Tran. 0.2 0.2

0.0 0.0 -600 -400 -200 0 200 400 600 800 lOOO 1200 0 200 400 600 800 1000

(a) Frequency (cm· l ) (e) Frequency (cm- I )

3.0 3.0

2.5 2.5

{; ...... 2.0 '-'

El '-' 2.0

<.) 1.5 8- 8- 1.5 <I.l

<I.l

1.0 Tran. ::E z 1.0 CY

0.5 0.5

" 0.0 0.0

-600 -400 -200 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200

(b) Frequency (cm· l ) (d) Frequency (cm- I )

FIG. 3. The partitioning of the normal modes of water into translational and rotational contributions. (a) The translational and rotational INM projectors defined by Eq. (15). (b) The translational (dotted), rotational (dashed-dotted), and total (solid) instantaneous-normal-mode densities of states. (c) The translational and rotational QNM projectors. (d) The translational (dashed), rotational (dotted), and total (solid) quenched-normal-mode densities of states.

(16)

With this division, the average (real) translational and rota-tional frequencies are 168 and 536 cm-I, respectively, and the corresponding root-mean-square frequencies are 206 and 577 cm- I .

Clearly the rotational degrees of freedom govern the in-stantaneous normal modes whose frequency is greater than 400 cm -I, whereas the INM frequencies which are less than 200 cm- I or are imaginary involve mostly translational mo-tion. This division is similar to, though not quite as sharp as, that seen with the quenched normal mode spectrum [see Figs. 3(c) and 3 (d)]. Actually it is revealing to press this comparison a bit more. The low frequency INMs are clearly going to be closely related to large amplitude motions, since

the potential surface associated with such modes is shallow and the displacement from the local minimum is expected to be large (vide infra).

There are also things one can say about the INM rota-tional modes as well, provided we further separate the rota-tional degrees of freedom into their Cartesian components in the molecular-fixed frame, the motions around the three mo-lecular rotational axes. One can convert the (laboratory-fixed) Euler angle representation of the molecular orientation employed in the molecular dynamics to the required molecular-fixed frame by using the direction cosine matrix. The result is that [with molecular rotational axes defined in the inset to Fig. 4(a)], we obtain the corresponding rotational projectors shown in Fig. 4(a). We find that the projectors along the molecular x and z axes peak -500 and 600 cm- I ,

respectively, while the y-axis projector continues to increase as the mode frequency increases. The corresponding rota-tional densities of state computed from these projectors are displayed in Fig. 4(b). As might be expected from the behav-ior of the projectors, the spectrum corresponding to rotation about the y axis is peaked at a significantly higher frequency than that around the other two axes. The moments of inertia are Ix=2.916, ly=1.013, and lz= 1.903X 10-40 gcm2

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Cho et al.: Normal mode analysis of liquid water 6677

1.0 Total ROIational Projector

0.8 z A-, '0 u

l 0.6

0.4 'J:!

0.2

0.0 .6()() 400 -200 0 200 400 600 800 1000 1200

(a) Frequency (cm- l )

..60() 400 -200 (b)

o 200 400 600 800 1000 1200 Frequency (cm- I )

FIG. 4. The partitioning of the rotational instantaneous normal modes of water into the contributions around each of the three molecular axes [defined in the inset to (a)]. (a) The three rotational INM projectors and the total INM rotational projectors [the same as is shown in Fig. 3(a)]. (b) The rotational portion of the instantaneous-normal-mode density of states subdivided into components.

for a TIPS2 water molecule, so the rotational spectra roughly, but not entirely, track the differences in moments of inertia. It should be noted that the dipole moment of a water molecule does not change as it rotates about the molecular z axis, whereas the rotation about the x axis induces a change of the dipole moment direction. Because the x-axis rotation involves stronger intermolecular interactions with surround-ing water molecules, the cage potential is likely to be stiffer than that for the z-axis rotational motion. These two effects-the differences in moments of inertia and in strengths of intermolecular interactions-may compensate each other, making the gap between the two peak positions small.

C. Localization and delocallzation of the modes

Both the instantaneous and quenched normal modes are excitations calculated at a set of instantaneously frozen liq-uid configurations. Since these configurations are disordered, a very real possibility is that some of the excitations are localized in space.34 - 37 In order to investigate this possibility,

we compute a standard measure of localization, the inverse participation ratios,35,36 defined for the alh mode in either of two ways

L (U a ,jjl.)4, jjl.

(17)

If an instantaneous normal mode is so localized that it is effectively confined to a single degree of freedom, only one of the eigenvector elements will be nonzero. The orthogonal-ity of the U matrices would then guarantee that the both inverse participation ratios would be equal to 1. On the other hand, for a completely delocalized mode, the first inverse participation ratio would be 1/6N, since all eigenvector ele-ments would be equal to (6N) - 1/2 (6N being the total num-ber of degrees of freedom). Thus the reciprocal of is commonly taken to represent the number of degrees of free-dom involved in a given mode. The second definition of inverse participation ratio by contrast, is approximately equal to the number of molecules involved in a given mode. For example, if an INM is localized to two degrees of free-dom belonging to the same molecule, e.g., rotational motions about the x and y axes of jth molecule, then U a,jjl. (,u=4,5) = 1/.J2 and we have 1/ R and 1/ R equal to 2 and 1, respec-tively. On the other hand, if the two degrees of freedom belong to two different molecules, e.g., U a,j4 = 1/.;2 and U a,k5 = 1/ .;2, then both 1/ R and 1/ R are equal to 2. Thus, 1 / in reflecting the number of molecules participating in a given mode measures its physical localization. In Figs. 5(a) and 5(b), we show the distributions of INM and

respectively, where we have averaged over liquid configurations to calculate and and divided by the density of states.

Inspection of the outcome quickly reveals that the INMs whose frequencies are larger than 900 cm -I are indeed very localized (1 / However, most of the INMs are delo-calized in space. It is interesting that there is a dip between 200 and 400 cm -I in both the 1 / and 1 / distributions. As observed by Ohmine and co-workers,18-21 quenched nor-mal modes in this frequency region are also localized in space. This region is where the translational and rotational degrees of freedom are most strongly mixed, as evidenced by the comparable values of the projectors illustrated in Figs. 3(a) and 3(c). Also notice that the stable INMs that are largely translational (w<100 cm- I ) are especially delocal-ized, with 1 / values on the order of 80, and the unstable modes are similarly delocalized. Our findings thus provide some quantitative evidence that the low-frequency modes might be associated with large-scale motions, and imaginary frequency modes might be related to structural transitions. We note also that the size of the effective bath we mentioned in the Introduction can be roughly estimated to be a sphere of radius 8.3 A in liquid water at room temperature if we esti-mate the number of water molecules involved in a given mode to be 80.

These results for the degree of delocalization of instan-taneous normal modes can be compared with the participa-tion numbers 1 / of the quenched normal modes. The over-all shape of the distribution is almost identical to that of

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6678 Cho et al.: Normal mode analysis of liquid water

250

200

150 ......

100

50

o __ __ ______ __ __ __

·600 -400 -200 o 200 400 600 800 1000 1200 (a) Frequency (cm· l )

140

120

100

80

60

40

20

0 -400 -200 o 200 400 600 800 1000 1200

(b) Frequency (cm-I)

QNM density of states given in Fig. 2. Here again we find that the low-frequency translational modes and high-frequency librational modes are mostly delocalized (11 110), while the very high-frequency modes (w>950 cm- I ) and intermediate frequency modes (350<w<450 cm- I ) are highly localized in space. Noting that the quenched liquid configurations correspond to the presumably more highly ordered low-temperature configurations, it is not very surprising that the QNMs are generally more delocal-ized than the INMs. We should mention, as a final comment, that there could be a simulation size dependence to the par-ticipation numbers. In general, one should always perform a finite-size-scaling analysis to distinguish genuinely localized from genuinely delocalized modes. The general tendencies we have been discussing though should not change very much. We should nontheless investigate the simulation box size-dependent changes in the participation number in order to be confident of our estimate of the size of the effective bath in bulk liquid water. We should further examine the dependence of the participation number on the potential pa-rameters used in the simulation.

D. The nature of the potential energy hypersurface

In order to get some information on the structure of the potential energy hypersurface, we begin by looking at the instantaneous first derivatives of the potential, the trans-

140

120

100

40

20

200 400 600 800 1000 (c) Frequency (cm-I)

FIG. 5. The degree of localization of the nonnal modes of water as de-scribed by the inverse participation ratios of the modes. (a) The number of degrees of freedom (RI) - I participating in each instantaneous nonnal modes (six per molecule). (b) The number of molecules (RiI)-1 participating in a instantaneous nonnal mode. (c) The number of molecules (R lIt l participat-ing in a quenched nonnal mode.

formed forces I a [Eq. (8)]. Since each I a includes contribu-tions from both translational and rotational degrees of free-dom, it is of some interest to separate the translational and rotational components of each transformed force

I a= 2: U a,jp.Fjp. = + jp.

2: 2: U a,jp.Fjp., j p.= 1,2,3

2: 2: U a,jp.Fjp.' j p.=4,5,6

We show the absolute values of the transformed forces and their components in Fig. 6. It should be noted that the density of states has been divided out of the distributions of these forces in plotting the figure. Note also that since I a can be negative, + *' II al in general.

The rotational force terms appear to make up the vast majority of the total forces. In particular, the total force for the high-frequency modes seems to be dominated by the ro-tational forces. This behavior is, of course, consistent with that of the translational and rotational projectors [Fig. 3(a)]; the rotational degrees of freedom become progressively more important as the INM frequency increases, but notice also that while the translational degrees of freedom dominate

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Cho et al.: Normal mode analysis of liquid water 6679

-400 -200 o 200 400 600 800 !ooo Frequency (cm!)

FIG. 6. The magnitude of the forces If al for the various instantaneous normal modes, along with the translational and rotational components.

both the low real and imaginary frequency modes (see Figs. 3), even the forces for these regions are largely determined by the rotational terms. Thus, for liquid water, it always seems to be the rotational torques that act as the driving forces propagating the instantaneous liquid configurations along an instantaneous-normal-mode direction.

Were an instantaneous liquid configuration Ro actually situated in one of the local minima of the potential hypersur-face (one of the inherent structures), the forces F(Ro) (and therefore our transformed forces) would all vanish. More generally though, the quantities f j that appear in Eq. (12) can be used as measures of how much any instantaneous configuration Ro is displaced from the nearest potential ex-tremum along each of the normal coordinate directions a. If taken literally, Eq. (12) suggests that for real frequencies, the extremum should be a minimum and for imaginary frequen-cies, a maximum. We plot the distribution of these displace-ments in Figs. 7 on a log-log scale with the density of states normalized out. That is, we plot

10g"l ( 8(W-W.») )

vs 10glO(w),

The sizes of these predicted displacements make for an interesting study in themselves. A 200 cm -I mode, e.g., has a (mass-weighted) displacement of about 1 (glmol)112 A, a rather tiny amount for a molecule with a mass of 18 glmo!. However, a 10 cm -I INM, whether real or imaginary, has a displacement 40 times larger. These figures do make it seem likely that the lowest-frequency real and imaginary modes are the ones most closely tied to structural rearrangements. Certainly, if we were to take the position of the potential maximum along the ath mode to correspond to the location of a transition state for a barrier-crossing process in the a direction, we would naturally expect at least some of the imaginary modes to be associated with large-scale structural transitions.

Equally important, perhaps, the form of the dependence of these displacements on the frequency of the modes has

2

1.5 Stable Modes

] 00 0.5 o

0.0

'8" ......, -0.5 !±::!

.s-1.0 -1.5 __

(a)

2

1.5 0< C! -,..... '0

0.5 o

0.0

'8" ......, -0.5 !±::!

.3 -1.0

1.4 1.8 2.2 2.6 3 LogIJoo) (00: cm-I)

'-'- Unstable Modes '-

slope = -1 -

-1.5 L-__ __ __ __ __ __ __

1.4 1.8 2.2 2.6 (b) LogIJlool) (00: cml )

FIG. 7. The magnitude of the mass-weighted displacements If j for the (a) real and (b) imaginary instantaneous normal modes, plotted on a 10gujlogIO scale. The dashed lines correspond to Eq. (18) with '7= -1 in (a) and -I and -2 in (b).

implications for the degree of anharmonicity inherent in a liquid's potential surface. If the true potential surface really were harmonic with a (mass-weighted) curvature in the a direction of the equilibrium-average root-mean-square displacement in that would be (kBT)II2/wa' Hence, a genuinely harmonic topography would imply that the log-log plot shown in Fig. 7(a) should be linear with a slope of - 1. As one can see, the curve is indeed reasonably linear for the real modes

(18)

but with 7]= 1.44 instead of 1. The imaginary modes [Fig. 7(b)] of course permit no such equilibrium estimation; it is not possible for the potential in any given direction to remain an inverted parabola forever (and still be bounded from be-low, as a physical potential must be). Nonetheless, Eq. (18) still seems to hold, albeit with 7]= 1.9.

In much the same sense that these represent displacements, Eq. (13) tells us that the magnitude of

represents the depth of the harmonic potential sur-face along the ath normal mode, measured from the instan-taneous position--or at least it should for a stable mode (Fig. 1). Since the instantaneous configuration is almost always

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6680 Cho et al.: Normal mode analysis of liquid water

(a)

(b) Unstable Modes

FIG. 8. How the potential energy hypersurface of a liquid looks from the perspective of instantaneous and quenched normal modes. (a) At any instan-taneous liquid configuration X, the configuration space can be divided into orthogonal (INM) directions, some of which are stable, others of which are unstable. The expectation would be then that the potential surface, if ex-trapolated in each direction, would resemble the parabolas shown. (b) From the inherent structure (QNM) point of view, the true potential surface has locally stable configurations (here, SI and S2) separated by barriers, along with a reaction path connecting them (here, the thick solid line). If one were to carry out an INM analysis at every point along this special path through configuration space, one would indeed find that the reaction coordinate would be a stable mode in the vicinity of the minima and an unstable mode near the barrier (shown here as parabolas). Hence the energies and f7J2 might be used to estimate the well depths and barrier heights. From an INM perspective, however, only a vanishingly small fraction of configu-ration space is in the intermediate vicinity of a literal potential well. Almost all the extrema of the potential hypersurface are saddle points-they have a macroscopic number of both upward and downward curvatures.

well removed from a local potential minimum, we may ex-pect a stable mode to evolve on its positive-curvature poten-tial surface for some period of time; for just how long de-pends on the time scale of the structural transitions and mode-coupling processes induced by the anharmonicity. By way of contrast to the stable mode situation, the magnitude of for an unstable modes might be considered as the height of the barrier separating one bound state from another along the ath normal coordinate (again, with respect to the instantaneous position). We draw a schematic picture describing these alternatives in Fig. 8, with the thick line in Fig. 8(b) meant to convey a slice through the true potential energy hypersurface along the direction of a "reaction coor-dinate. "

If we so desire, we can calculate the actual magnitude of these "well depths" and "barrier heights." What we show in Fig. 9 is the distribution of these objects with, as usual, the density of states divided out. Just as we expected to see a frequency dependence of w;;1 for the displacements of the stable INMs, for truly harmonic stable modes, the equilib-rium average potential depth should be kBTI2, regardless of the mode frequency. Our distribution of well depths and bar-

rier heights though is markedly peaked around Iwl=o. The INM-predicted well depths are significantly greater than kBTI2 at low and imaginary frequencies and somewhat smaller than k B T /2 at frequencies larger than 600 cm -I. Ac-tually, these observations are quite consistent with the asso-ciation of low-magnitude frequencies with structural changes. A barrier height of 2kBT, say, could simply mean that major rearrangements are e 2 = 7.4 times slower than some characteristic attempt frequency. We should emphasize, however, that both Figs. 7 and 9 still leave us with clear indications of the anharmonicity of the potential surface.

Perhaps in view of these signatures of anharmonicity, a word is in order about the appropriateness of applying instantaneous-normal-mode ideas. The existence of anhar-monicity in the equilibrium properties of a liquid comes as no great surprise.8,19,38 Near their triple points, the heat ca-pacities of solid Ar and solid Na are C vlk8=2.89 and 3.1, reasonably close to the harmonic equipartition values of 3.00. Yet, under largely the same thermodynamic conditions, liquid Ar and liquid Na have CvlkB values of 2.32 and 3.4, respectively.39 Stillinger and Weber showed how the latent heat of melting ice is distributed among the following ele-ments; 85%-90% of the latent heat is attributed to upward shift in potential energy of inherent structures across the transition; the remainder is attributed to changing anharmo-nicity of potential well curvature.7 On a more microscopic level, by propagating quenched mormal modes, Tanaka and Ohminel9 found that both cubic and quartic anharmonicities could be pertinent. Still, the optimum domain of the instan-taneous normal modes is the short-time dynamics of the liq-uid. The full potential surface, in all its anharmonic splendor, is always included in INM dynamics through the initial con-ditions of the INM trajectories.

How far then can one push our harmonic fortune telling for the liquid's potential surface? If an instantaneous liquid configuration Ro is located at one of the X's in Fig. 8, the

\-.. , .... o ____ __ __ ____ __

-400' -200 o 200 400 600 800 1000

Frequency (cm!)

FIG. 9. The magnitudes of the "well depths" and "barrier heights" for the instantaneous normal modes. The translational and rota-

tional components of these energies are indicated by dotted and dashed lines, respectively. The value of the thermal energies kBT and kBTI2 are shown for reference.

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Cho et al.: Normal mode analysis of liquid water 6681

eigenvalues of the dynamical matrix yield the 6N curvatures of the potential in the 6N orthogonal directions. Along any one of these directions, we might well try to identify a struc-tural transition from one stable structure SI to another S2 as a literal barrier-crossing process. A rough estimate of the well and barrier frequencies, e.g., might be generated by comput-ing the average stable and unstable frequencies

(iw>u=Io'" do' ap(a) (a=iw),

respectively, with pew) the normalized INM density of states (Fig. 2). These averages tum out to be (w>s=365 cm- I and (iw>u= 142 cm- I. Unfortunately, there are a number of dif-ficulties with this naive view that have their roots in the intrinsically multidimensional character of the potential sur-face. Seductive as it may be to think of the real harmonic frequencies as reactant modes and the imaginary harmonic frequencies as barrier modes, the picture of there being tran-sitions from one stable well to another is just too simplistic. For one thing, the initial state is not necessarily a thermally equilibrated state of the hypothetical harmonic well, so the well-known barrier-crossing theories (such as Kramer's rate expression and its subsequent generalizations)4o may not be correct. More fundamentally, one never has purely stable motion. The fact that 6% of the instantaneous normal modes in liquid water are unstable means that at any arbitrary point in configuration space, 6% of the directions are unstable. For the concept of genuinely stable wells to apply, 6% of the thermally accessible configurations would have to be barrier-like-a rather different scenario. We actually know that it is the former picture that is correct because INM spectra are self-averaging. That is, one knows from the study of disor-dered systems in other contexts that the eigenvalues for any one configuration are representative of all the configurations, except for a set of configurations of measure zero.41 Water molecules apparently do not hop between stable regions of the potential surface; they probably try to move in one direc-tion after another until they find a suitably unstable direction-and only then does significant center-of-mass mo-tion occur.

Of course, in the quenched-normal-mode picture, it is a bit more straightforward to begin a formulation of the large-scale dynamics and diffusive motion of liquids. The inherent-structure language6.8 makes it natural to describe a structural transition as a motion from a liquid configuration within the "drainage basin" of one inherent structure SI to a configu-ration within the basin of another S2' The barrier height is then the energy gap between SI and the top of the barrier along some unknown reactive pathway leading to S2' One can even perform a normal mode analysis at structure SI (a QNM study) and attempt to use the resulting modes to char-acterize the SI-+S2 transition, 18,20,21 but what is not clear, and what remains to be investigated, is the relationship be-tween this kind of approach and the thermal-liquid-configuration (INM) view of liquid dynamics.

We might also note that besides posing this conundrum, the presence of an anharmonic potential surface invites us to rethink the issue of mode localization. What is clearly the next step here is to investigate how the instantaneous anhar-monicity leads to interactions between the otherwise inde-pendent instantaneous normal modes. These interactions could very well lead to a type of localization closely allied to the normal-mode-to-Iocal-mode transition that is common-place in highly vibrationally excited small molecules,42 but that would not be seen in measures such as our inverse-participation ratio.35,42

E. Translational and angular velocity correlation functions

We close this section, in a less speculative vein, with an explicit calculation of liquid dynamics from the INMs. Since we know the translational and rotational spectra, we can compute the corresponding normalized translational and an-gular velocity autocorrelation functions3

C:(t)= I dw p:(w)cos wt (/-L=X,y,z),

respectively. Here the subscript /-L refers to the molecular rotational axis defined in Fig. 4(a). As discussed by Buchner et al} inclusion of the unstable modes in these formulas leads to an unphysical divergence of the correlation functions (owing to the fact that cos wt grows exponentially with time when w is imaginary). Fortunately, because the imaginary modes contribute only a minuscule fraction of the net density of states, we can confine ourselves to the stable parts of the spectra (with an appropriately modified normalization). The net results are the autocorrelation functions shown in Figs. 10 and 11.

The physical origins of the oscillatory behavior in the correlation functions follow immediately from their con-struction out of sinusoidal functions of time. Note that while there is no damping in the INM formalism, the dephasing induced by the broad distribution of INM frequencies suf-fices to lead to correctly decaying time correlation functions. Moreover, the predictions for the angular velocities, in par-ticular, are in striking agreement with the exact time depen-dencies. The dynamics described by the INMs thus provides a certain amount of physical insight into the microscopic nature of the collective molecular motions.

It may be worth noting that although the correlation functions computed here were fundamentally single-body quantities, it still proved to be useful to use the collective language of the instantaneous normal modes. As we re-marked in the Introduction, the INMs really find their opti-mum utility as a kind of dynamical basis set for the liquid. Here, we have literally resolved the translational and angular velocities of the individual molecules into linear combina-tions of the INMs with weighting factors equal to the trans-lational and rotational projectors.

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6682 Cho et al.: Normal mode analysis of liquid water

0.8 (a) 0.4 Molecular x-axis 0.0

-0.4

0.00 0.05 0.10 0.15 0.20 0.25

0.8 (b) 0.4 Molecular y-axis 0.0

-0.4

0.00 0.05 0.10 0.15 0.20 0.25 TIME (ps)

0.8 (c) 0.4 Molecular z-axis 0.0

-0.4

0.00 0.05 0.10 0.15 0.20 0.25

FIG. 10. The angular velocity autocorrelation functions for liquid water about the three molecular axes [defined in the inset to Fig. 4(a)]. The solid lines are the outcome of an exact 60 ps molecular dynamics run and the dotted lines are the instantaneous-normal-mode predictions derived from the real part of the density of states shown in Fig. 4(b).

IV. CONCLUDING REMARKS

In this paper, we have examined the implications of one from of the vibrational density of states of liquid water-its instantaneous-normal-mode spectrum. The usefulness of this analysis, as with the analogous quenched-normal-mode stud-ies, stems from the ability it gives us to define the collective modes of water in a genuinely microscopic and unambiguous way. We have therefore deliberately gone beyond the previ-ous studies which concerned themselves solely with spectra and endeavored here to learn what kinds of lessons about the molecular dynamics of water were available from the modes themselves, the eigenvectors.

This comment is not to say that the spectra themselves were uninteresting. In contrast to the situation in liquid Ar, where 20%-30% of the modes are imaginary,l1 we found that only 6% of the instantaneous normal modes in water are imaginary. We also found that the translational and rotational densities of states occupied entirely different portions of the spectrum, unlike the situation with homonuclear diatomics,3 where the two components largely overlapped. This separa-tion was even more pronounced with the quenched normal modes. Yet, except for the relatively small fraction of imagi-nary modes, the translational spectrum of water bears a strik-ing resemblance to the translational INM spectrum of every other liquid; it exhibits the same right-triangle shape and even the same range of frequencies.3.9,1I,12 What apparently sets water apart is its high frequency librational modes.

We set out to investigate the modes themselves in sev-eral ways. We first attempted to quantify the degree of local-ization of the modes by computing the inverse participation ratios. To our surprise, both the translational and rotational modes ended up being largely delocalized by this measure. Only the highest frequency rotational modes and the mixed rotation-translation modes were clearly localized (and even the latter were only definitively localized in the quenched-mode version). Somewhat more detailed information in this vein was available from the instantaneous forces acting along each normal mode direction. Although it is a concept logically separate from using the INMs to predict time evo-lution, one could imagine assuming that the locally harmonic character of the potential surface extends beyond local re-gions. In that event, the forces would carry information about the statistics of the nearest potential extrema. Our results for these minima/maxima statistics actually betrayed the signifi-cant anharmonicity of water's potential hypersurface, but to the extent that we can trust a harmonic extrapolation, our results were clear in attributing the largest scale motion to the modes with the lowest magnitude frequencies.

These comments on the geometry of the modes aside, the essence of the instantaneous normal modes lies in their ability to help us understand short-time dynamics. Here the INMs meet their true test. In this context, we should empha-size how remarkably faithful to the real molecular dynamics of water the INM angular-velocity autocorrelation functions proved to be. Indeed, the quantitative agreement is all the more impressive in view of the pronounced oscillations of the simulated correlation functions. The fact that one can interpret these oscillations as resulting from a superposition of instantaneous normal modes led real credence to the no-tion that the collective motions responsible for ultrafast liq-uid dynamics are accurately captured by the INMs.

This kind of agreement, moreover, gives us confidence to pursue the next step. What is now a fairly sizeable body of

1.0

0.8

0.6

0.4

0.2

0.0

0.0 0.2 0.4 0.6 0.8

Time (ps)

FIG. 11. The center-of-mass (translational) velocity autocorrelation function for liquid water. The solid line is the exact molecular dynamics result and the dotted line is the instantaneous-normal-mode prediction. Note the differ-ence in time scale between this figure and Fig. 10.

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Cho et al.: Normal mode analysis of liquid water 6683

evidence has accumulated to the effect that solvation dynam-ics largely runs its course, at least in common solvents, in an extremely short period of time.43- 45 It therefore seems natu-ral to try to express real solvation correlation functions as a sum over instantaneous normal modes-thereby resolving the elementary steps leading to solvation into discrete and identifiable molecular motions. This effort is, in fact, underway.46.47

ACKNOWLEDGMENTS

This work was supported by the National Science Foun-dation and U.S.-Japan science cooperation programs. RMS acknowledges the support of NSF grants CHE-9119926 and 9203498 and GRF NSF CHE-8819678. I. O. and S. S. are partially supported by the grant in aid for scientific research on the priority area of "chemical reaction theory" and the Japan-U.S. joint research program on "ultrafast reactions." M.e. thanks colleagues in Professor Ohmine's group and Professor Yoshihara at the Institute for Molecular Science in Japan for their hospitality during his stay at the Institute and thanks colleagues in Professor Stratt's group at Brown Uni-versity for stimulating discussions during his visit to Brown.

I R. A. Marcus and N. Sutin, Biochirn. Biophys. Acta. 811, 265 (1985); 1. Rips and J. Jortner, J. Chern. Phys. 87, 2090 (1976); 87, 6513 (1976); L. D. Zusrnan, Chern. Phys. 49, 295 (1980); 1. T. Hynes, J. Phys. Chern. 90, 3701 (1986); B. B. Smith, A. Staib, and J. T. Hynes, Chern. Phys. 176,521 (1993); J. S. Bader, R. A. Kuharski, and D. Chandler, 1. Chern. Phys. 93, 230 (1990); M. Marchi, J. N. Gehlen, D. Chandler, and M. Newton, J. Am. Chern. Soc. 115,4178 (1993); X. Song and R. A. Marcus, J. Chern. Phys. 99.7768 (1993).

2p. F. Barbara and W. Jarzeba, Adv. Photochern. 15, I (1990); P. F. Barbara, T. 1. Kang, W. Jarzeba, and T. Fonseca, in Perspectives in Photosynthesis, edited by J. Jortner and B. Pullman (K1uwer, Dordrecht, 1990); T. 1. Kang, W. Jarzeba, P. F. Barbara, and T. Fonseca, Chern. Phys. 149,81 (1990); K. Torninaga, G. C. Walker, T. 1. Kang, P. F. Barbara, and T. Fonseca, 1. Phys. Chern. 95, 10485 (1991); M. J. Weaver, Chern. Rev. 92, 463 (1992); M. J. Weaver and G. E. McManis, Ace. Chern. Res. 23, 294 (1990).

3M. Buchner, B. M. Ladanyi, and R. M. Stratt, J. Chern. Phys. 97, 8522 (1992).

41. Benjamin, B. J. Gertner, N. J. Tang, and K. R. Wilson, J. Am. Chern. Soc. 112. 524 (1990).

sea) B. J. Gertner, R. M. Whitnell, K. R. Wilson, and J. T. Hynes, J. Am. Chern. Soc. 113. 74 (1991); (b) 1. Ohmine, J. Chern. Phys. 85, 3342 (1986).

6The concept of inherent structures in liquids has been explored in a series of papers by Stillinger, Weber, and co-workers: F. H. Stillinger and T. A. Weber, Phys. Rev. A 25,978 (1984); T. A. Weber and F. H. Stillinger, J. Chern. Phys. SO. 2742 (1984); SO, 4434 (1984); F. H. Stillinger and R. A. LaViolette, ibid. 83, 6413 (1985).

7The inherent structure of water was first described by F. H. Stillinger and T. A. Weber. J. Phys. Chern. 87, 2833 (1983).

8R. A. LaViolette and F. H. Stillinger, J. Chern. Phys. 83, 4079 (1985); 85, flJ27 (1986).

9G. Seeley and T. Keyes, in Spectral Line Shapes, edited by J. Szudy (Ossolineurn, Wroclaw, Poland, 1989), Vol. 5, p. 649; J. Chern. Phys. 91, 5581 (1989); B. Madan. T. Keyes, and G. Seeley, ibid. 92, 7565 (1990); 94, 6762 (1991); G. Seeley, T. Keyes, and B. Madan, ibid. 95, 3847 (1991); J. Phys. Chern. 96, 4074 (1992); B. Madan and T. Keyes, J. Chern. Phys. 98. 3342 (1993).

10B._c. Xu and R. M. Stratt, J. Chern. Phys. 92, 1923 (1990). II Y. Wan and R. M. Stratt, J. Chern. Phys. (in press). 12T._M. Wu and R. F. Loring, J. Chern. Phys. 97, 8568 (1992). 13A. Rahman and F. H. Stillinger, J. Chern. Phys. 55, 3336 (1971). 14 R. W. Irnpey, P. A. Madden, and 1. R. McDonald, Mol. Phys. 46, 513

(1982); P. A. Madden and R. W. Irnpey, Chern. Phys. Lett. 123, 502

(1986); R. Bansil, T. Berger, K. Toukan, M. A. Ricci, and G. Ruocco, ibid. 141, 297 (1987); M. A. Ricci, G. Ruocco, and M. Sampoli, Mol. Phys. 67, 19 (1989); V. Mazzacurati, M. A. Ricci, G. Ruocco, and M. Sarnpoli, Chern. Phys. Lett. 159, 383 (1989); w. B. Bosma, L. E. Fried, and S. Mukarnel, J. Chern. Phys. 98,4413 (1993); S. Sastry, F. Sciortino, and H. E. Stanley (preprint).

1Sp. H. Berens, D. H. J. Mackay, G. M. White, and K. R. Wilson, J. Chern. Phys. 79, 2375 (1983).

16H. Tanaka and I. Ohrnine, 1. Chern. Phys. 87, 6128 (1987); M. Sasai, I. Ohmine, and R. Ramaswamy, ibid. 96, 3045 (1992).

11 A. Pohorille, L. R. Pratt, R. A. LaViolette, M. A. Wilson, and R. D. MacElroy, J. Chern. Phys. 87, 6070 (1987).

18I. Ohmine, H. Tanaka, and P. G. Wolynes, 1. Chern. Phys. 89, 5852 (1988). 19H. Tanaka and I. Ohrnine, J. Chern. Phys. 91, 6318 (1989). 2oH. Tanaka and I. Ohrnine, J. Chern. Phys. 93, 8138 (1990). 211. Ohmine and H. Tanaka, Chern. Rev. (in press). 22 See also, F. Sciortino and S. Sastry, 1. Chern. Phys. (to be published). 23 J. E. Straub and D. Thirurnalai, Proc. Natl. Acad. Sci. USA 90, 809

(1993). 24R. O. Rosenberg, D. Thirurnalai, and R. Mountain, J. Phys. Condensed

Matter I, 2109 (1989). 25 J. E. Adams and R. M. Stratt, J. Chern. Phys. 93, 1332 (1990); 93, 1632

(1990). 26B._C. Xu and R. M. Stratt, 1. Chern. Phys. 91, 5613 (1989); Z. Chen and

R. M. Stratt, ibid. 94,1426 (1991); K. Ganguly and R. M. Stratt, ibid. 95, 4418 (1991); 97, 1980 (1992).

27D. E. Logan and M. D. Winn, 1. Phys. C 21, 5773 (1988); M. D. Winn and D. E. Logan, Phys. Condensed Matter 1, 1753 (1989); 1, 8683 (1989).

28Y._C. Chen, J. L. Lebowitz, and P. Nielaba, J. Chern. Phys. 91, 340 (1989). 29R. M. Stratt and J. E. Adams, J. Chern. Phys. 99, 775 (1993); 1. E. Adams

and R. M. Stratt, ibid. 99, 789 (1993). 3OZ. Chen and R. M. Stratt, J. Chern. Phys. 95, 2669 (1991). 31 J. A. Leegwater and S. Mukamel, J. Chern. Phys. 99, 6062 (1993); J. Cao

and B. J. Berne, ibid. 99, 6998 (1993). 32R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chem-

istry and Physics (Wiley, New York, 1988). 33w. L. Jorgenson, J. Chern. Phys. 77, 4156 (1982). 34The general phenomenon of Anderson localization is reviewed by P. Lee

and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985). 35S. R. Nagel, A. Rahman, and G. S. Grest, Phys. Rev. Lett. 47, 1665

(1981); S. R. Nagel, G. S. Grest, and A. Rahman, ibid. 53, 368 (1984); J. Canisius and J. L. van Hemmen, 1. Phys. C 18,4873 (1985); M. L. Wil-liams and H. J. Maris, Phys. Rev. B 31, 4508 (1985); R. Biswas, A. M. Bouchard, W. A. Karnitakahara, G. S. Grest, and C. M. Soukoulis, Phys. Rev. Lett. 60, 2280 (1988).

36B. B. Laird and H. Schober, Phys. Rev. Lett. 66, 636 (1991); H. Schober and B. B. Laird, Phys. Rev. B 44, 6746 (1991); u. Buchenau, Philos. Mag. B 65, 303 (1992); H. R. Schober, C. Oligschleger, and B. B. Laird, J. Non-Cryst. Solids 156-158, 965 (1993).

31 S. John, H. Sornpolinski, and M. J. Stephen, Phys. Rev. B 27, 5592 (1983).

38H. Jonsson and H. C. Andersen, Phys. Rev. Lett. 60, 2295 (1988). 39p. A. Egelstaff, An Introduction to the Liquid State (Clarendon, Oxford,

1992), pp. 4-6. 40 J. T. Hynes, in The Theory of Chemical Reaction Dynamics, edited by M.

Baer (CRC, Boca Raton, FL, 1985), Vol. 4; P. Hiinggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990).

41 R. J. Elliott, J. A. Krumhansl, and P. L. Leath, Rev. Mod. Phys. 46, 465 (1974).

42 See, for example, M. S. Child and L. Halonen, Adv. Chern. Phys. 57, I (1984); R. Bruinsma, K. Maki, and J. Wheatley, Phys. Rev. Lett. 57,1773 (1986).

43 P. F. Barara and W. Jarzeba, Adv. Photochern. IS, I (1990); M. Maroncelli, J. Mol. Liquids 57, I (1993).

44M. Cho, S. Rosenthal, N. F. Scherer, L. D. Ziegler, and G. R. Fleming, J. Chern. Phys. 96, 5033 (1992).

45L. Perera and M. Berkowitz, J. Chern. Phys. 97, 5253 (1992); E. A. Carter and J. T. Hynes, ibid. 94, 5961 (1991); M. Maroncelli, ibid. 94, 2084 (1991).

46R. M. Stratt and M. Cho, J. Chern. Phys. (in press). 41M. Cho, G. R. Fleming, S. Saito, I. Ohrnine, and R. M. Stratt (to be

published).

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