10994
J. Phys. Chem. B 1997, 101, 10994-10999
Optical Kerr Effect Spectroscopy of Liquid Water: Role of Fluctuating Electronic Polarizability Badry D. Bursulaya and Hyung J. Kim* Department of Chemistry, Carnegie Mellon UniVersity, 4400 Fifth AVenue, Pittsburgh, PennsylVania 15213-2683 ReceiVed: August 20, 1997; In Final Form: October 17, 1997X
The third-order nonlinear nuclear Kerr response of liquid water is studied via molecular dynamics computer simulations. By employing the TAB/10 potential model recently developed in our group, the instantaneous adjustment of molecular polarizability to the fluctuating local electric field is properly accounted for. It is found that the fluctuating nonlinear aspect of electronic polarizability has dramatic effects on Kerr response. It strongly enhances the librational character of, and introduces short-time oscillatory structures into, the water response function, compared to a linear description with fixed polarizability. Thus despite its small anisotropy, TAB/10 water yields good overall agreement with recent experimental findings of optical Kerr effect spectroscopy.
I. Introduction The low-frequency intermolecular motions (j1000 cm-1) of liquid water have been extensively studied both theoretically1-8 and experimentally.9-14 The frequency-domain spectroscopy such as depolarized Raman scattering (DRS)13,14 and far-infrared (far-IR) absorption9-12 shows structures near 60 and 180 cm-1, which are attributed to restricted translations perpendicular and parallel to an O-H‚‚‚O bond, respectively.15 By contrast, a broad band extending from 300 to 1000 cm-1 is sensitive to the H/D isotope substitution and is thus believed to arise from the hindered rotations of water. Recent advance in time-domain techniques, e.g., optical Kerr effect (OKE) spectroscopy and impulsive stimulated Raman scattering,16,17 allows direct access to ultrafast short-time dynamics of liquids. The nuclear response of water determined from OKE measurements exhibits a rapid initial rise, followed by a fast oscillatory decay and an ensuing slow relaxation for t > 200 fs.18-20 The intermolecular dynamics responsible for the lowfrequency features in DRS spectra and OKE response of liquid water have also been studied via MD. It was found that while the contributions from hindered translations are reasonably accounted for with the inclusion of the interaction-induced effects,1b,4-7 nearly isotropic polarizability based on gas-phase experiments21 fails to reproduce the librational structure of DRS spectra.5 However, this situation improves considerably if polarizability anisotropy is enhanced in analyzing MD results.5 A similar improvement was found also in recent simulation study of Kerr response of water.8 These findings have an interesting implication that molecular polarizability in liquid water would be much more anisotropic than that in vacuum.5,8 In this Letter, we study OKE spectroscopy of liquid water by employing a truncated adiabatic basis-set (TAB) description for solvent electronic structure22 and present new insight into the roles of polarizability anisotropy and fluctuations. We will consider here only the third-order nuclear Kerr response; the details, including frequency-domain far-IR and DRS, will be published elsewhere.23
variation. In a point-dipole approximation for the charge ˆ of the distributions,24 the effective electronic Hamiltonian H solvent system consisting of N molecules is
H ˆ )
∑i hˆ i + ∑j ∑ [pˆ i‚Tij‚pˆ j + VijLJ]; i(>j)
1 (i * j) (1) Tij ) ∇i∇j |b ri - b r j|3 where i and j label the molecules (i, j ) 1, 2, ..., N), hˆ i and pˆ i are the vacuum electronic Hamiltonian and dipole operators for i, b ri denotes its position vector, Tij and VijLJ are the dipole tensor and Lennard-Jones interactions between i and j. In the TAB description, the electronic wave function ψ(i) of i in solution is represented as a linear combination of a few basis functions φ(i) ˆi µ (µ ) 1, .., r) that are the eigenstates of h r
ψn(i) )
(i) c(i) ∑ nµ φµ µ)1
where n labels the single-molecule (1M) energy states in solution (see below). Due to the intermolecular Coulombic interactions, 22 the state coefficients c(i) nµ vary with the solvent configurations. Thus the dipole moment fluctuations and associated polarizability are described by a solvation-dependent mixing of φ(i) µ in the current approach. In the self-consistent field (SCF) approximation, the ground-state wave function ΨSCF and electronic potential energy ESCF of the total solvent system are N
|ΨSCF〉 ) ˆ |ΨSCF〉 ) ESCF ) 〈ΨSCF|H
|ψ(i) ∏ 0 〉 i)1
X
Abstract published in AdVance ACS Abstracts, December 15, 1997.
S1089-5647(97)02720-X CCC: $14.00
(3)
∑i (i)0 - ∑j ∑ bp (i)00‚Tij‚pb(j)00 i(>j)
II. Truncated Adiabatic Basis-Set Description We begin with a brief reprise of the theoretical formulation developed in ref 22a to describe the solvent electronic structure
(2)
+
∑j ∑ VijLJ
(4)
i(>j)
(i) where the 1M SCF wave function and energy, ψ(i) n and n (n ) 0, 1, ..., r - 1), satisfy a set of “Fock” equations
© 1997 American Chemical Society
Letters
J. Phys. Chem. B, Vol. 101, No. 51, 1997 10995
(i) (i) ˆf i|ψ(i) ˆ i + Vˆ i1M]|ψ(i) ˆ i1M ≡ n 〉 ) [h n 〉 ) n |ψn 〉; V
∑ bp (j)00‚Tji‚pˆ i
j(*i)
(5)
and b p(i) p(i) 00 is the ground-state dipole expectation value b 00 ) (i) i (i) 〈ψ0 |pˆ |ψ0 〉. In the point-dipole description, one can determine exactly at the SCF level the total electronic polarizability tensor Π (ref 22a)
Π)
πi‚Tij‚πj + ∑ πi‚Tij‚πj‚Tjk‚πk + ...; ∑i πi - ∑ i,j i,j,k πi ≡ 2
∑
n(*0)
b p (i) p (i) n0b n0 (6) (i) (i) n - 0
where b p(i) n0 is the transition dipole moment between the ground (i) i and excited states ψ(i) 0 and ψn . π is the 1M polarizability that would result if the so-called “dipole-induced” contributions25 (i.e., terms involving Tij in eq 6) are neglected.26 Since the 1M wave function and energy of each molecule vary with its local environment [cf. eqs 2 and 5], resulting πi is also modulated by the intermolecular interactions. Thus even when the dipole-induced terms are ignored, the 1M molecular polarizability varies with the solvent configuration in our theory; this is essentially nonlinear electronic response, closely related to hyperpolarizability effects.22 This contrasts with many existing theories where fixed vacuum polarizability is used in eq 6 instead of πi. In this Letter, we study the effects of polarizability fluctuations on OKE spectroscopy, whose signal is characterized by a third-order nonlinear nuclear response function R(3)(t) (ref 17)
R(3)(t) ) -
θ(t) ∂ 〈Π (t) Πab(0)〉 kBT ∂t ab
(7)
Here θ(t) is the Heaviside step function, kB is Boltzmann’s constant, T is the temperature of the system, and Πab (ab ) xy, xz, yz) is an off-diagonal component of Π in the lab frame. III. Water Potential Model and Simulation Method In the present study, we employ the TAB/10 potential for water with SPC geometry,27 developed in ref 22b. In this model,28 the electronic structure variation of water is described by a total of 10 basis functions. Both the diagonal and overlap charge distributions associated with the basis functions were represented by partial point charges centered on the interaction sites of each molecule. In addition to three real atomic sites (O and two H’s), two fictitious sites off the water molecular plane were introduced to describe out-of-plane polarizability. The parameters employed for the vacuum Hamiltonian and dipole operators as well as the LJ interactions are compiled in Tables 1 and 2. For more details, see ref 22b. The simulations were performed in the canonical ensemble of 128 water molecules at a density of 0.997 g cm-3 and temperature of 298 K, using the extended system method of Nose´.29 Periodic, truncated octahedral boundary conditions30 were employed. All solvent bonds were constrained with the SHAKE algorithm.31 The trajectories were integrated with a time step of 2 fs using the Verlet algorithm.32 At each time step the 1M wave functions were calculated by solving eq 5 iteratively; the SCF solutions were converged with a relative tolerance of 10-8 in total electrostatic energy.22 The Coulombic interactions were computed with the Ewald method33 with account of the self-consistency condition between the central
and image molecule charges.34 The intermolecular forces were evaluated by differentiating eq 4 using the Hellmann-Feynman theorem that is exact in the SCF regime. Equilibrium simulations were carried out with 10 ps of equilibration, followed by a 400 ps trajectory from which averages were computed. In the calculations of R(3)(t), only the terms up to second order in Tij in eq 6 were included. The second-order contribution was found to be nearly negligible compared to those of the zeroth and first order; this justifies the neglect of the higher-order terms. IV. Results and Discussion The MD simulation results are summarized in Figures 1-4. We begin with Figure 1, where the distributions of the three diagonal components of 1M tensor πi (eq 6) for TAB/10 are shown.26 As noted in our previous study,22b the finite character of the distribution widths reveals the fluctuating nonlinear aspect of πi. Thus the hyperpolarizability effects are included in the TAB description, eq 2. This is in contrast with many existing polarizable models that strictly obey linear electrostatics. The ensemble averages of the diagonal components in liquid water j yy ) 1.40, and π j zz ) 1.34 Å3 in the molecular are π j xx ) 1.28, π 26 j yy and π j xx are, respectively, the largest and frame. Thus π smallest components of the liquid-phase polarizability. This is consonant with the gas-phase measurements of Murphy.21 We j ) 0.104 Å3) also note that the average anisotropy35 of πi (∆π is very close to that of Murphy’s polarizability (∆R )0.098 Å3). Thus the TAB/10 depicts nearly isotropically polarizable liquid water. The MD result for TAB/10 nuclear Kerr response, R(3)(t), is displayed in Figure 2. Also presented are two model response functions obtained with fixed electronic polarizability. For clarity, we first consider only the TAB/10 result (s); its comparison with the other models will be made below. The temporal behavior of R(3)(t) is characterized by a rapid initial rise to its maximum in ∼12 fs, followed by a fast decay with a few oscillations and an ensuing slow relaxation. In addition to the main peak, there are several distinctive structures, viz., peaks around 50 and 95 fs and a broad band centered at ∼150 fs. These features are in good qualitative agreement with recent OKE spectroscopy measurements,18-20 even though various uncertainties and resulting controversies associated with the latter make a detailed quantitative comparison difficult. To be specific, the structures around 20 and 200 fs determined from experiments are in reasonable accord with those of TAB/10 at ∼12 and ∼150 fs. Also, the existence of a peak near 50 fs for the latter is in good agreement with the experimental findings.19,20 In addition, both our MD and the Brownian oscillator fitting of OKE measurements20 yield a distinctive structure around 90-100 fs although its existence is controversial.19 Thus, despite its nearly isotropic polarizability in solution, the TAB/ 10 potential reproduces reasonably well the main characteristics of the experimental results on nuclear Kerr response. In fact, this is the first simulation study that captures the water OKE structure around 50-60 fs, to the best of our knowledge.36,37 In order to understand why the isotropic TAB/10 water fairs well, we have analyzed several models with fixed nonfluctuating solution-phase polarizability. For clarity, only the two most revealing models are considered in Figure 2 (in addition to R(3)). Their nuclear response, R h (3)(t), was evaluated by replacing i fluctuating π in eq 6 with (i) solution-phase ensemble average, π j i, of TAB/10 polarizability (‚‚‚) and (ii) gas-phase HartreeFock calculation results by Huiszoon38,39 (-‚-). To separate out the indirect dynamical effects and consider only the direct polarizability influence, we used the TAB/10 trajectory to compute R h (3)(t).40 Thus while Π of each model for a given
10996 J. Phys. Chem. B, Vol. 101, No. 51, 1997
Letters
TABLE 1: Parameters for TAB/10 Potential A. Vacuum Hamiltoniana diagonal elements ) [0, 7.6, 9.6, 10.5, 11.0, 11.5, 14.0, 16.8, 18.0, 25.0] B. Electric Dipole Operatorb,c
pˆ )
(
1.85zˆ -2.15xˆ 1.60zˆ -0.50xˆ 1.20zˆ -2.58yˆ -3.00yˆ 2.26zˆ -3.00xˆ 2.65zˆ
-2.15xˆ -1.60zˆ -0.42xˆ 2.70zˆ 4.20xˆ 0 0 0 0 0
1.60zˆ -0.42xˆ -0.70zˆ -2.52xˆ 1.80zˆ -5.00yˆ 3.92yˆ 2.66zˆ -2.40xˆ 0
-0.50xˆ 2.70zˆ -2.52xˆ 5.00zˆ 0.84xˆ 0 0 0 0 0
1.20zˆ 4.20xˆ 1.80zˆ 0.84xˆ -0.80zˆ 4.00yˆ 3.92yˆ 2.66zˆ -2.40xˆ 0
-2.58yˆ 0 -5.00yˆ 0 4.00yˆ -1.00zˆ 0 0 0 0
-3.00yˆ 0 3.92yˆ 0 3.92yˆ 0 4.00zˆ 2.35yˆ 0 0
2.26zˆ 0 2.66zˆ 0 2.66zˆ 0 2.35yˆ 3.00zˆ 0 0
-3.00xˆ 0 -2.40xˆ 0 -2.40xˆ 0 0 0 -1.60zˆ 0
2.65zˆ 0 0 0 0 0 0 0 0 -1.60zˆ
)
a In units of eV. b All unit vectors refer to the water molecular frame; zˆ is in the direction from O to the midpoint of two H’s, and xˆ is perpendicular to the molecular plane. c In units of D.
TABLE 2: Properties of TAB/10 Water Potential in Vacuuma µ0 0 Qxx 0 Qyy 0 Qzz 0 Rxx 0 Ryy 0 Rzz Tr r0/3 σOO OO σOH OH σHH HH
TAB/10
exptl
1.85 -2.32 2.45 -0.13 1.413 1.528 1.226 1.39 3.240 0.152 1.960 0.070 0.680 0.032
1.85b -2.5c 2.63c -0.13c 1.415d 1.528d 1.468d 1.470d, 1.429e
a
The units employed are D for dipole moment µ0, 10-26 esu cm2 for quadrupole moment Qii0 , Å3 for polarizability, Å for σRβ, and kcal mol-1 for Rβ. The polarizability and quadrupole tensors refer to the molecular frame with the center of mass at the origin. b Reference 49. c Reference 50. d Reference 21. e Reference 51.
Figure 1. Probability distributions of the solution-phase 1M polarizi i ability tensor πi for TAB/10 water: 1/3 Tr πi (s); πxx (-‚-); πyy i (-‚‚‚-); πzz (‚‚‚).
nuclear configuration is evaluated according to its own polarizability description, the time evolution of the nuclear configuration is governed by the TAB/10 potential, regardless of the models. As a consequence, TAB/10 and models i and ii differ only in their electronic response; their underlying nuclear dynamics are the same.40 Furthermore, since the ensemble average of πi for TAB/10 is identical to the model i polarizability π j i by definition, the only difference between these two is that
Figure 2. Third-order nuclear Kerr response function R(3)(t) (eq 7) for TAB/10 water (s). For comparison, response functions R h (3)(t) with fixed electronic polarizability are also shown: model i with π j i (‚‚‚); model ii with Huiszoon’s tensor (-‚-). All response functions are normalized, so that their maximum value is unity. We notice that both models with fixed polarizability fail to capture the peak around 5060 fs observed in the OKE experiments. By contrast, the TAB/10 water with dynamically fluctuating polarizability (cf. Figure 1) well describes this feature. Though not presented here, we also found that R h (3)(t) with Murphy’s tensor (ref 21) is nearly the same as that of model i.
the polarizability of TAB/10 fluctuates dynamically with the solvent configuration, while that of model i remains unchanged. As mentioned above, their anisotropy is nearly the same as that of Murphy’s measurements,21 although their average polarizability defined as one-third of trace is lower than that of the latter by 0.13 Å3. By contrast, model ii is more anisotropic (∆R ) 0.297 Å3) than TAB/10 and model i, while its average polarizability is the same as that of the gas-phase experiments. With these in mind, we now compare the response functions in Figure 2. One of the most salient features there is that Kerr response of TAB/10 and model i differs markedly. After the initial peak around 15 fs, model i yields no distinctive structures in R h (3) up to ∼150 fs; this is in striking contrast with the R(3) peaks around 50 and 95 fs. Also the short-time behavior of model i is considerably slower than that of TAB/10. It should be noted that this dramatic difference in Kerr response arises solely from the polarizability fluctuations (or lack thereof). We will come back to this point below. Though not presented here, we have also studied model Kerr response using Murphy’s polarizability.21 We found that resulting R h (3)(t) is nearly the (3) same as that of model i. By contrast, R h (t) of more anisotropic model ii is much faster than that of model i. This shows that the polarizability anisotropy plays an important role in nuclear
Letters
J. Phys. Chem. B, Vol. 101, No. 51, 1997 10997 its molecular polarizability is fixed as in many existing studies. Thus they serve as a good reference for the hindered rotation and translation contributions to Kerr response, respectively.7,8,41,42 Comparison of TAB/10 and model i reveals several interesting features. The overall temporal behavior of the II-component is nearly the same for TAB/10 and model i. This indicates that RII of the former mainly arises from the translational motions and is affected little by the polarizability fluctuations. Despite this similarity, the relative amplitude of RII is considerably smaller than that of R h II. By contrast, the first peak intensity of MM is much higher than that of model i; also the former is R more oscillatory than the latter. Since the first and third peaks of RMM nearly overlap, respectively, with the first and second peaks of R h MM, we expect that the former two peaks will be mainly librational in character just like the latter two. However, the second peak of RMM, which accounts for the R(3) structure near 50 fs, does not have any counterpart in R h MM [cf. ref 41]. This means that it is directly related to the polarizability fluctuations. j i of model To see this more clearly, we expand πi around π i as
πi ) π j i + βh i‚∆ bi + ...
(10)
where β h i is the leading-order hyperpolarizability tensor of rank 3 and ∆ bi is the deviation of the local electric field from its equilibrium average, evaluated at molecule i. The second term on the right-hand side of eq 10 represents the instantaneous readjustment of molecular polarizability to the fluctuating local electric field. Neglecting cross-correlation between π j i and bi, we approximate the difference RMM - R h MM as43 βh i‚∆
Figure 3. (a) Dissected components of the normalized TAB/10 and h MM (-‚‚‚-); R h II model i response functions: RMM (s); RII (-‚-); R (‚‚‚). (b) Time correlation function F(t) (eq 11).
Kerr response; as the former increases, the latter becomes accelerated and more structured. It is interesting to observe that anisotropic model ii yields better agreement with the TAB/ 10 and experiments than model i (and also Murphy’s polarizability). Nevertheless, model ii misses the well-resolved feature at 50-60 fs observed experimentally; rather, it yields a broad structure ranging from ∼70 to ∼100 fs. In order to gain insight into the surprising role played by polarizability fluctuations, we dissect Πi into two components25
ΠM )
∑i πi;
ΠI ) Π - Π M
(8)
and examine their contributions to R(3)
R(3) ) RMM + RII + RMI; θ(t) ∂ 〈Πµ (t) Πνab(0)〉 (µ, ν ) M, I) (9) Rµν(t) ) -Cµν kBT ∂t ab where Cµν )1 if µ ) ν and 2 if µ * ν. In the literature, ΠM and ΠI are often referred to as molecular and interaction-induced contributions. However, as explicitly pointed out above near eq 6 and also evidenced by Figure 1, πi varies strongly with solvation environment in our theory due to the intermolecular Coulombic interactions. Thus care should be taken in interpreting the origin of ΠM dynamics for TAB/10 water. In Figure 3a, RMM and RII of TAB/10 are exhibited. The corresponding h II of model i are also shown.41 It should be noted R h MM and R h II represent, that in contrast to the TAB/10 case, R h MM and R respectively, the molecular and interaction-induced effects since
RMM - R h MM ≈ -
θ(t)
∂
∑∑ k T i,j c,d ∂t B
〈βh iabc(t) ∆ic(t) βh jabd(0)∆jd(0)〉 ≈-
θ(t)
∂
∑∑ 〈βh iabcβh jabd〉∂t 〈∆ic(t) ∆jd(0)〉 k T i,j c,d B
∝ θ(t) F(t) F(t) ≡ -
(11) ∂ ∂t
〈∆ bi(t)‚∆ bj(0)〉 ∑ i,j
In Figure 3b, F(t) for TAB/10 water is displayed. We first notice that its main peak location agrees with that of RMM and R h MM in Figure 3a. In view of eq 11, this will significantly increase the first RMM peak intensity of TAB/10, compared to that of model i. As a result, the relative contributions to overall nuclear response R(3)(t) from librations and translations are, respectively, enhanced and reduced, compared to a linear description with fixed polarizability. This means that the librational character of Kerr response of TAB/10 will be considerably stronger than that of model i. Another interesting feature is that the location of the second peak of F(t) is in near perfect agreement with that of RMM. This provides another evidence that the RMM structure around 50 fs arises from the nonlinear response of molecular polarizability to the fluctuating local electric field.44 Since the hindered rotational motions of water are mainly responsible for rapid changes in solvation environment,45 we expect that the major contribution to the second peak of F(t) is from libration.46 Therefore, the corresponding second RMM peak induced by the electric field fluctuations is also expected to be of librational character. To test this notion, we investigated the H/D isotope substitution effects on OKE structure. In Figure 4, Kerr response of H2O
10998 J. Phys. Chem. B, Vol. 101, No. 51, 1997
Letters Acknowledgment. The authors thank Jonggu Jeon for ab initio polarizability calculations in vacuum and in solution. This work was supported in part by NSF Grant CHE-9412035. References and Notes
Figure 4. H/D isotope substitution effects on R(3)(t) of liquid water: H2O (s); D2O (‚‚‚). For the latter, the simulation was performed for 200 ps after initial equilibration.
and D2O is compared. The first three peaks of D2O are shifted by nearly a factor of x2 toward longer time with respect to the corresponding H2O peaks. This clearly exposes the librational character of the first three peaks of R(3) and thus confirms our conclusion on the dynamic origin of the 50 fs peak of H2O. We also note that a broad translational band of D2O centered around 170 fs is not clearly resolvable due to strong overlap with its more structured third librational peak near 130 fs. Finally, we briefly consider the issue associated with polarizability anisotropy in liquid water. In several previous MD studies5,8 (and also here), it was found that Huiszoon’s polarizability with enhanced anisotropy (∆R ) 0.297 Å3)38,39 yields better agreement with experiments than nearly isotropic Murphy’s tensor (∆R ) 0.098 Å3).21 This would imply that the single-molecule polarizability in liquid water would be significantly more anisotropic than that in the gas phase. However, considering the TAB/10 results (with anisotropy of 0.104 Å3) presented here, we believe that this is not the case. Rather, we think that it is the dynamically fluctuating aspect of polarizability that would make Kerr response of water rapid and oscillatory,44 compared to that obtained with gas-phase polarizability. We thus expect that the polarizability anisotropy of water would remain largely unaffected by solvation.47 In order to test this view, we have conducted several Hartree-Fock ab initio calculations in solution using the self-consistent reaction field method48 and compared with the corresponding vacuum results. We have found no significant changes in polarizability anisotropy induced by solvation. This seems to support our view even though the Hartree-Fock predictions for anisotropy are usually larger than experimental results. Thus it would be worthwhile to do high-level quantum chemistry calculations with the inclusion of electronic correlation effects for more accurate quantification of anisotropy. Also experimental measurements of H/D isotope effects on the peak locations and intensities, especially the 50 fs structure of H2O, and comparison with the trends in Figure 4 would be of considerable interest in further elucidating and clarifying the roles of polarizability fluctuations and anisotropy. In summary, we have studied the effects of polarizability fluctuations and associated hyperpolarizability on nuclear Kerr response of water. It was found that nonlinear electronic response of solvent molecules has dramatic consequences for OKE spectroscopy. It enhances the librational character of, and introduces additional short-time structures in, the Kerr response function. The details, including their effects on DRS, will be published elsewhere.23
(1) (a) Impey, R. W.; Madden, P. A.; McDonald, I. R. Mol. Phys. 1982, 46, 513. (b) Madden, P. A.; Impey, R. W. Chem. Phys. Lett. 1986, 123, 502. (2) Martı´, J.; Gua`rdia, E.; Padro´, J. A. J. Chem. Phys. 1994, 101, 10883. (3) Guillot, B.; Guissani, Y. In Collision- and Interaction-Induced Spectroscopy; Tabisz, G. C., Neuman, M. N., Eds.; Kluwer: Dordrecht, 1995. (4) Frattini, R.; Sampoli, M.; Ricci, M. A.; Ruocco, G. Chem. Phys. Lett. 1987, 141, 297. Mazzacurati, V.; Ricci, M. A.; Ruocco, G.; Sampoli, M. Chem. Phys. Lett. 1989, 159, 383. Ricci, M. A.; Ruocco, G.; Sampoli, M. Mol. Phys. 1989, 67, 19. (5) Bosma, W. B.; Fried, L. E.; Mukamel, S. J. Chem. Phys. 1993, 98, 4413. (6) Sastry, S.; Stanley, H. E. J. Chem. Phys. 1994, 100, 5361. (7) Saito, S.; Ohmine, I. J. Chem. Phys. 1995, 102, 3566. (8) Saito, S.; Ohmine, I. J. Chem. Phys. 1997, 106, 4889. (9) Rusk, A. N.; Williams, D.; Querry, M. R. J. Opt. Soc. Am. 1971, 61, 895. (10) Afsar, M. N.; Hasted, J. B. J. Opt. Soc. Am. 1977, 67, 902. (11) Kaatze, U.; V. Uhlendorf, V. Z. Phys. Chem., Neue Folge 1981, 126, 151. (12) Hasted, J. B.; Husain, S. K.; Frescura, F. A. M.; Birch, J. R. Chem. Phys. Lett. 1985, 118, 622. (13) Krishnamurthy, S.; Bansil, R.; Wiafe-Akenten, J. J. Chem. Phys. 1983, 79, 5863. (14) Walrafen, G. E.; Fisher, M. R.; Hokmabadi, M. S.; Yang, W.-H. J. Chem. Phys. 1986, 85, 6970. Walrafen, G. E.; Hokmabadi, M. S.; Yang, W.-H. J. Chem. Phys. 1988, 88, 4555. Walrafen, G. E. J. Phys. Chem. 1990, 94, 2237. (15) See, e.g.: Walrafen, G. E. In Water: A ComprehensiVe Treatise; Franks, F., Ed.; Plenum: New York, 1972; Vol. 1. (16) Shen, Y. R. The Principles of Nonlinear Optics; Wiley: New York, 1984. (17) Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford University Press: New York, 1995. (18) Chang, Y. J.; Castner, E. W., Jr. J. Chem. Phys. 1993, 99, 7289. (19) Castner, E. W., Jr.; Chang, Y. J.; Chu, Y. C.; Walrafen, G. E. J. Chem. Phys. 1995, 102, 653. (20) Palese, S.; Schilling, L.; Miller, R. J. D.; Staver, P. R.; Lotshaw, W. T. J. Phys. Chem. 1994, 98, 6308. Palese, S.; Buontempo, J. T.; Schilling, L.; Lotshaw, W. T.; Tanimura, Y.; Mukamel, S.; Miller, R. J. D. J. Phys. Chem. 1994, 98, 12466. (21) Murphy, W. F. J. Chem. Phys. 1977, 67, 5877. (22) (a) Bursulaya, B. D.; Kim, H. J. Generalized Molecular Mechanics Including Quantum Electronic Structure Variation of Polar Solvents. I. Theoretical Formulation via a Truncated Adiabatic Basis Set Description. J. Chem. Phys., in press. (b) Bursulaya, B. D.; Jeon, J.; Zichi, D. A.; Kim, H. J. Generalized Molecular Mechanics Including Quantum Electronic Structure Variation of Polar Solvents. II. A Molecular Dynamics Simulation Study of Water. J. Chem. Phys., in press. (23) Bursulaya, B. D.; Kim, H. J. To be submitted. (24) While the point-dipole approximation is invoked in the theoretical formulation, the extended charge distributions via the interaction site model description are used in the actual simulations. (25) For a recent review, see, e.g.: Madden, P. A. In Spectroscopy and Relaxation of Molecular Liquids; Steele, D., Yarwood, J., Eds.; Elsevier: Amsterdam, 1991. See also: Ladanyi, B. M. Ibid. (26) According to our previous MD simulations, the dipole-induced contributions to 1M water polarizability is nearly insignificant.22b Thus it is safe to identify πi as molecular polarizability in liquid water. (27) Berendsen, H. J. C.; Gigera, J. R.; Straatsma, T. P. J. J. Phys. Chem. 1987, 91, 6269. (28) Our previous MD study shows that the TAB/10 water results for structure, dynamics, dielectric properties, and static electronic spectroscopy agree very well with experiments.22b For example, its translational diffusion coefficient and static dielectric constant are found to be, respectively, Dtr ≈ 2.4 Fick and 0 ) 79; the corresponding experimental results are Dtr ≈ 2.3 Fick and 0 ) 78. Also its average internal energy per molecule, -10.07 kcal mol-1, compares very well with the experimental estimate of -9.92 kcal mol-1. (29) Nose´, S. J. Chem. Phys. 1984, 81, 511. (30) Adams, D. J. Chem. Phys. Lett. 1979, 62, 329. (31) Ryckaert, J. P.; Ciccotti, G.; Berendsen, H. J. C. J. Comput. Phys. 1977, 23, 327. (32) Verlet, L. Phys. ReV. 1967, 159, 98. (33) Heyes, D. M. J. Chem. Phys. 1982, 74, 1924.
Letters
J. Phys. Chem. B, Vol. 101, No. 51, 1997 10999
(34) Thus the solvent charge distributions in the image cells are identical to those in the central cell, and the former fluctuate in exactly the same way as the latter during the simulation. (35) The anisotropy ∆R of a polarizability tensor r is defined as
∆R )
1 [(Rxx - Ryy)2 + (Ryy - Rzz)2 + (Rzz - Rxx)2]1/2 x2
where Rii (i ) x, y, z) are the three principal components of r. (36) We also note that the polarizability effects on intermolecular forces (and thus nuclear dynamics) are correctly reflected in our simulation through exact differentiation of eq 4, whereas they are totally neglected in previous MD studies.5,8 As pointed out in ref 5, this could affect the spectroscopy results in a nonnegligible manner. This indirect polarizability influence on Kerr response by way of nuclear dynamics will be studied in ref 23. (37) Due to the truncated character of its electronic description, the vacuum polarizability of TAB/10 water, especially the zz-component, is smaller than the experimental values (Table 2).22 To examine potential consequences of this, we have performed test simulations employing a 0 0 different parameter set with enhanced Rzz (Rzz ) 1.295 Å3). We found that 0 the enhancement of Rzz (without any changes in solution-phase anisotropy) does not alter the main features of Kerr response. (38) Huiszoon, C. Mol. Phys. 1986, 58, 865. (39) For perspective, we point out that the polarized basis sets (Sadlej, A. J. Theor. Chim. Acta 1991, 79, 123) usually yield smaller anisotropy than Huiszoon’s result. Also the inclusion of electronic correlation significantly reduces anisotropy, compared to the corresponding SCF results (see, e.g.: Sekino, H.; Bartlett, R. J. Chem. Phys. 1993, 98, 3022). (40) We used 100 ps of the 400 ps TAB/10 trajectory to calculate Kerr response of models i and ii. Thus their statistics are different from those of TAB/10. (41) Though not presented here, we note that the dynamical behaviors of R h µν(t) for model ii are nearly the same as those of the corresponding
components for model i. The main difference between the two is the relative h (3)(t). Thus the contribution of each R h µν component to overall response R respective locations of the R h MM peaks, for example, are the same for both models, while their relative intensities for model ii are much higher than those for model i. (42) Since RII can relax via both rotation and translation (Ladanyi, B. M.; Liang, Y. Q. J. Chem. Phys. 1995, 103, 6325), a more precise assignment of the RII dynamics would require a more accurate decomposition, e.g., a projection method.25 However, in this initial study focusing on qualitative aspects of polarizability fluctuations, we use a simple decomposition in eq 9. (43) Despite its crude nature, this approximation helps to expose essential consequences of hyperpolarizability for RMM. (44) We note that the electronic polarizability modulations through ultrafast intramolecular vibrational modes (e.g., bond stretch) not considered here could also contribute. (45) Maroncelli, M.; Fleming, G. R. J. Chem. Phys. 1988, 89, 5044. (46) The first t-derivative (not shown here) of single-molecule orientational time correlation functionsi.e., d/dt 〈P1[nˆ (t)‚nˆ (0)]〉 where P1 is the first-order Legendre polynomial and nˆ is the molecular orientationsyields structures around 12 and 60 fs. This also indicates that both the first and second peaks of F(t) are due to librational motions. (47) Considering the fact that the 50 fs peak of TAB/10 response is less distinctive than experiments,19,20 slight enhancement of anisotropy induced by solvation would be expected. (48) For a very recent review, see: Tomasi, J.; Persico, M. Chem. ReV. 1994, 94, 2027. See also: Cramer, C. J.; Truhlar, D. G. In SolVent Effects and Chemical ReactiVity; Tapia, O., Bertran, J., Eds.; Kluwer: Dordrect, 1995. (49) Clough, S. A.; Beers, A.; Klein, G. P.; Rothman, L. S. J. Chem. Phys. 1973, 59, 2254. (50) Verhoeven, J.; Dynamus, A. J. Chem. Phys. 1970, 52, 3222. (51) Zeiss, G. D.; Meath, W. J. Mol. Phys. 1977, 33, 1155.
