A molecular dynamics study of the far infrared spectrum of liquid waterBertrand Guillot Citation: J. Chem. Phys. 95, 1543 (1991); doi: 10.1063/1.461069 View online: http://dx.doi.org/10.1063/1.461069 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v95/i3 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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A molecular dynamics study of the far infrared spectrum of liquid water Bertrand Guillot Laboratoire de Physique Theorique des Liquides, a) Universite Pierre et Marie Curie, Tour 16 4, Place Jussieu, 75230 Paris Cedex 05-France

(Received 10 December 1990; accepted 23 April 1991)

The far infrared spectrum of liquid water at room temperature is calculated by molecular dynami~s s~mulat.ion over the spectral range 0.5-1000 cm - 1. It is shown that the experimental absorptl.on l?tenslty c.an be ~eproduced satisfactorily provided that the dipole induced dipole mechamsm IS convemently Implemented in the calculation and the classical profile corrected ~or quantum effects. The contribution due to exchange overlap dipoles between 0 and H atoms IS also investiga~ed b~t its role ~n the genesis of the far infrared (FIR) spectrum is negligible. Although the dipole mduced dipole (DID) mechanism is found to be responsible for the peculiar band shape near 200 cm - 1 by revealing the intermolecular oscillations of the hydrogen bond network, no other translational band is detected in the region 10-60 cm - 1, a result in contradistinction with data put forward recently. Moreover, it is shown that the a~sorption spectrum is the seat of various cancellation effects between permanent and induced dIpoles, effects which are described in detail.

I. INTRODUCTION

The investigation of intermolecular motions in liquid water was the purpose of a great amount of works in the last three decades. The informations obtained from neutron scat­tering experiment,1.2 infrared and Raman spectroscopies, 3

dielectric relaxation,4 and nuclear magnetic resonance (NMR)5 have permitted to draw up a catalogue of all sorts of intermolecular motions, each of them being related to a spectral feature more or less pronounced (well-defined band or shoulder). For instance, the libration motions generated by the hydrogen bond network give rise to a broad band centered about 685 cm - 1 cm in Raman9 and far infrared spectra,6 as well as in the inelastic neutron scattering spec­trum.2 In the same way, an intermediate band localized around 175 cm - 1 in the Raman spectrum7 and around 200 cm - 1 in far infrared8 and neutron spectra2 is assigned to the 0"'0 stretching mode of the O-H"'O units. A low fre­quency mode assigned to the flexion of O-H"'O units is also detected in Raman 7 and neutron data2 around 60 cm - 1, whereas its presence in the far infrared (FIR) spec­trum is more questionable.8-10 Other very weak bands are also detected in the frequency domain separating the fibra­tion band and the O' .. 0 stretching band but they seem bare­ly visible at room temperature.2.3•

7

In fact, although the aforementioned attributions are not really contestable, they are essentially based on semi­quantitative arguments 11 and on the results of a few comput­er simulation studies generally not designed for this very purpose (however, see Refs. 12-16). Each experimental technique has its own specificity and is more or less sensitive to one kind of molecular motion; a strong band appearing in one particular spectroscopy may become hardly visible by another technique. Consequently, the goal of the present study is to calculate by molecular dynamics simulation the far infrared spectrum of liquid water in some details. In this context the recent paper by Madden and Impey17 merits

a) Unite de Recherche Associee au CNRS (URA 765).

attention since it reports a molecular dynamics (MD) calcu­lation of the FIR and Rayleigh band shapes. However, al­though this study emphasizes the importance of the dipole induced dipole mechanism to make spectrally active the translational oscillations of the water molecules, several im­portant questions are not elucidated. For example, the fol­lowing points can be addressed:

(i) How sensitive is the FIR spectrum to the model po­tential used, considering the evident deficiency of the MCY potential to reproduce the dielectric constant? 15

(ii) Is it possible to reproduce in absolute value the ex­perimental FIR spectrum of water over a large domain of frequency (between 1 and 1000 crn - I)?

(iii) Quantum corrections: Can they be ignored in the case of water?

(iv) What are the respective spectral intensities coming from the permanent dipoles, the induced dipoles and the electronic overlap dipoles? What kind of motions are spec­trally active?

(v) How strong is the interference between induced di­poles and permanent dipoles, and what role does it play?

To answer these questions, we have carried out a MD simulation of liquid water with the help of the SPC model potential of Berendsen. 18 In Sec. II we present the ingredi­ents and'the method used to simulate the FIR absorption spectrum of water and, in Sec. III, a detailed review of the results is reported as well as their interpretation. In the last section, we summarize the main conclusions of this study.

II. METHOD OF CALCULATION

The MD simulations were performed with the SPC po­tential of Berendsen. 18 This water model permits to repro­duce many properties of liquid water (for a review see Ref. 19) in spite of its evident simplicity (rigid monomer, three atomic charges qH = + 0.41e and qo = - 0.82e, and one single Lennard-Jones center on the oxygen Eoolk = 78.22 K,O"oo = 3.165 A). The main features of this potential are first, to reproduce the hydrogen bond interaction by deloca-

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1544 Bertrand Guillot: Infrared spectrum of liquid water

lizing the dipole moment over three atomic charges and next, to mimic in an approximate way the polarization ef­fects in the liquid by using an enhanced dipole (2.27 D in­stead of 1.85 D in the gas phase). We have not tried to make use of the polarizable SPC potentiafo because its implemen­tation in the MD simulation is very time consuming and because the improvement expected by its use is questionable for the moment (see Ref. 20).

In infrared spectroscopy, the absorption coefficient per unit path length a (w) of a sample of volume Vat tempera­ture Tis given by

a(cu) = 41TW tanh (.Bw/2)

3-11n(cu)CV

X f-+ 0000 dt e - iwt (M(t)· M(O», (1)

where M is the total dipole moment of the sample and, n(w),

the refractive index. The frequency dependence of the latter one cannot be neglected since for liquid water its value changes by a factor of 7 between 0.1 and 1000 cm - 1. The total dipole moment M can be expressed as a sum of the microscopic dipoles occuring in the liquid sample, namely,

M = Lf-li + Lf-liJ + Lf-lijk + "', (2) iJ ij,k

where f-li takes into account the permanent dipole and the induced dipole at the molecule i, f-lij is the two-body dipole coming from the short range interactions (overlap exchange and dispersion) between molecules i andj, f-lijk is the irredu­cible three-body dipole coming from nonelectrostatic forces and so on. In liquid water it is conventionally accepted that only the first term of the above series has to be considered to interpret the FIR spectrum, even if no quantitative evalua­tion ofthe higher order terms is published, at our knowledge, in the literature. In order to cover up this lack we will present a rough estimation of these terms in the following. As for the major part of the absorption process, it results from the mo­tions of the individual dipoles, f-li' namely,

f-li = m i + Pi>

Pi = ai'Fi>

(3)

(4)

Fi = Ei + L Tij'pj, (5) j

where m i is the permanent dipole of the molecule i, Pi is the dipole induced on i, a i is the polarizability tensor of the wa­ter molecule i (weakly anisotropic), Ei is the electrostatic field created by all the water molecules of the liquid sample at the molecule i, and Tij = VV( l/rij) is the dipole tensor. An accurate determination of the induced dipole Pi requires us to solve the equation (5) by an iterative procedure. In order to avoid this lengthy calculation we identify the field Fi in Eq. (4) with the electric field generated only by the charge distribution due to the SPC water molecules (the atomic charges correspond to a dipole of2.27 D), and, corre­latively we assign to the permanent dipole m i its gas phase value ( 1.85 D). In this way we hope to alleviate the effect of the truncature on the equation (5). The reliability of this procedure can be estimated in noticing that Ahlstrom et al. 20 in their study of the polarizable SPC model have found,

with a fully iterated calculation, that the induced moment has an average value of 1.1 D and is virtually parallel to the permanent dipole (1m + pi = 2.9 D), a result in good agree­ment with our calculations (see next section) since we ob-

tain ~(P2) = 0.8 D and m and P almost exactly colinear (1m + pi = 2.63 D).

The evaluation of the two-body dipole, f-lij [see Eq. (2) ], due to the short range interactions in water, is based on empirical relations deduced from rare gas studies. It is known for many years that the rare gas mixtures (e.g., He­Ar, Ne-Ar, Ar-Kr, ... ) present a FIR absorption due to the dipole moment produced in a binary collision of dissimilar atoms: By reason of symmetry no net dipole appears between two identical atoms (for reviews see Ref. 21). This binary dipole is the result of the Van der Waals forces between the atoms. Recent theoretical works22

•23 have shown that at

short range (in the repulsive region) the overlap dipole is nearly exponential. By the same token, Krauss et af.24 have shown that, at the SCF level, the rare gas interaction energy curves are well described by an exponential fit. From these data it is suitable to deduce a scaling law between the overlap dipole of a rare gas pair, f-l (r) = f-lo e - ar, and its interaction energy, VCr) = Voe - br, namely,

b la = 1.36( ± 10%),

Volf-lo = 3.082 a.u.( ± 36%).

(6)

(7)

Although this scaling law is only approximate (see the de­viations in parenthesis) its accuracy is sufficient for our pur­pose. Thus in the case of water, the following mechanism takes place. When two water molecules are sufficiently close from each other (e.g., hydrogen bonded) the overlap dipole is given by the vectorial sum of the atom-atom overlap di­poles occuring between the oxygen of one molecule and the hydrogens of the other (and vice versa). Moreover the oxy­gen-hydrogen overlap dipole f-lOH is assumed to be exponen­tial and to follow the aforementioned scaling law for rare gas interactions (notice that the overlap dipole is directed from the heavier atom to the lighter one). The range aOH and the amplitude parameter f-lOH are deduced from the recent ab initio calculation ofWallqvist and Karlstrom25 on the water dimer. By using an energy partitioning scheme of the total interaction energy these authors were able to present the exchange repulsive terms under the form of atom-centered exponentials. In the case of the oxygen-hydrogen interac­tions the potential parameters are bOH = 3.992 A-I ( = 2.112 a.u.) and VOH = 5220.48 kcal!mol (= 8.321 a.u.), respectively. It is noteworthy that Wallqvist and KarI­strom add a hard core repulsive term (r - 20) to the exponen­tial term, but this term is negligible for the interatomic sepa­rations accessible during our MD simulations. Finally, from the above values of bOH and VOH ' and according to the rela­tions (6) and (7) weobtainf-loH = 6.90D (= 2.7a.u.) and aOH = 2.935A -1 ( = 1.553a.u.). These parameters yield to a value of 0.035 D for the overlap dipole of a water dimer in its minimal energy configuration (for the SPC potential roo = 2.75 A). This value already suggests that the overlap contribution might contribute very little to the infrared ab­sorption intensity (compare with 1.85 D for the permanent

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Bertrand Guillot: Infrared spectrum of liquid water 1545

dipole and 0.8 D for the induced dipole). A last remark concerns the dispersion dipole that we

have neglected in this evaluation as well as the irreducible three-body dipoles. In fact although the dispersion dipole is far to be negligible in rare gas interactions (see Ref. 26) its contribution to the two-body dipole is expected to be small with regard to the overlap dipole because the hydrogen bond involves a strong overlap between the oxygen and the hydro­gen atoms. Moreover for such distances ( roo = 2.75 A) the damping of the dispersion forces is not known with accuracy which precludes a detailed analysis. In any case, our model ofthe overlap dipole is only approximate and isjust designed to give a rough estimation of this short range contribution. As for the role played by the irreducible three-body interac­tions when the molecules are mutually overlapping, it could be estimated in following the rationale proposed for rare gas­es26.27 but no significant effect is expected from this contri­bution.

III. RESULTS AND DISCUSSION

The MD simulation (NVE ensemble) was implemented for a system composed of 256 SPC molecules in a cubic box with periodic boundary conditions, the thermodynamic state corresponding to liquid water at T = 300 K and p = 1 g/cm3

• The long range electrostatic interactions were evalu­ated with the help of an Ewald sum. The total simulation time amounts to 160 ps with a time step of 0.5 fs, and the configurations were tape recorded every ten time steps. This rather long simulation time of 160 ps is necessary to sample with accuracy the collective reorientation of the water di­poles. In particular, a good test of convergence is given by the evaluation of the dielectric constant. For a system of non polarizable polar molecules with periodic boundary condi­tions and the long range interactions treated by the Ewald lattice sum, eo is given by28

eo _ 1 = 41T (11.12

)

3 VkT' (8)

where M = k; m; and ( ) denotes an average taken over all the molecules contained in the simulation box. In the case of the SPC potential (m = 2.27 D) we obtain eo = 80, a value in excellent agreement with those given in the literature [82.5 ± 4 (Ref. 16),72 ± 7 (Ref. 19)], and with the experi­mental value [78.5 (Ref. 29)]. In the case of a polarizable polar fluid the dielectric constant is given by28

41T (11.1 2)

E -E =--- (9) o '" 3 VkT'

whereM = k; (m; +p;),andE", is the high frequency limit of the permittivity. According to the procedure described in Sec. II to calculate the induced dipole moment Pi [in that case m = 1.85 D and the polarizability tensor ax.y.z = 1.468, 1.528, 1.415 (A3

)], we obtain Eo - E", = 74.2. Since the most accurate experimental val­ues of E '" lie between 4.9 and 5.0 (Refs. 8 and 29) our calcu­lation yields to Eo = 79 a value very close to the aforemen­tioned value for the non polarizable case (the uncertainty of the computation is certainly much greater). However the contribution due to the polarization effects is far to be negli-

gible since the permanent dipoles (1.85 D) alone yield to Eo - E", = 53.4, the gap between 53.4 and 74.2 being filled by the cross correlations between permanent and induced dipoles, and to a lesser extent by the purely induced contri­bution.

Before to present the far infrared spectrum calculated in the present study, some definitions are necessary. In addi­tion to the total dipole correlation function, Cr(t) = (M(t)'M(O», which takes into account perma­nent, induced and overlap dipoles present in the simulation box, one defines the following partial correlation functions,

Cm (t) = (~mi(t)· ~ mi(o»),

Cp(t) = (~Pi(t)· ~Pi(O»),

Cov(t) = (IJ.lij(t)· IJ.lij(O») , 17'} I¥}

(10)

(11 )

(12)

where mi , Pi' and,uij are the permanent dipole (1.85 D) of the molecule i, the induced dipole on the molecule i, and the overlap dipole of the pair (iJ), respectively. In Table I are listed the values in (debye)2 of the aforementioned correla­tion functions at zero time. First of all, as expected, the over­lap contribution is very small, five orders of magnitude smaller than the permanent contribution. Next, the main contribution to Cr(O) comes from the permanent dipoles (70.6%) whereas the purely induced component Cp (0) amounts to only 2.9%. But the interference term between permanent and induced dipoles is important 26.4%, [as giv­enbyCr(O) - (Cm(O) + Cp(O»],aresultwhichcorrobo­rates the analysis made by Edwards and Madden on liquid acetonitrile. 30 An important question concerns the respec­tive role played by the collective relaxation and the single relaxation. For this very purpose we have evaluated the sin­gle and distinct correlation functions associated with C r (t), Cm (t), and Cp (t), and defined by the following generic rela­tions:

C(t) = ~ (Ai (t). Ai (0»,

Cd(t) = I (Ai(t)'Aj(O», ir'j

(13)

(14)

(15)

TABLE 1. Computed values of collective and single dipole correlation func­tions at zero time [in (debye)2 per molecule] and their associated relaxa­tion time (in ps): for definitions see text. Experimental values are given in parenthesis.

Collective Single

Cr(O) 21.97 6.94 Cm(O) 15.52 3.42 Cpe O) 0.64 0.68 Coo(O) 0.0003 n.c.

1'1' 7.7 (8.2[29]) 2.8(2.1 [41])

1'm 7.8 2.8 1'p 7.8 2.8

1'00 7.6 n.c.

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1546 Bertrand Guillot: Infrared spectrum of liquid water

where Ai symbolizes the dipole moment under investigation (total, permanent, induced).

At zero time, the magnitude of C T (0) and C m (0) comes in major part (68% and 78%, respectively) from their cor­responding distinct components C~(O) and C~ (0), respec­tively (see Table I). This result indicates that in liquid water where the molecules are mostly tetrahedrally oriented, the correlations between the dipoles are responsible for the high dielectric constant Eo = 80. On the contrary the result of the decomposition of the purely induced component Cp (0) is more puzzling since the distinct part C ~ (0) contributes only for 5.7% and is negative. This means that the induced di­poles are the seat of strong cancellation effects between each others, a phenomenon which also occurs in non polar liquids (see Ref. 31 and for a general discussion see Ref. 32). In fact we will see in the following that these cancellation effects play an important role in the genesis of the FIR absorption spectrum of liquid water.

The absorption spectrum was evaluated between 0.5 and 1000 cm - 1 by taking the Fourier transform of CT(t) [see Eq. (1)]. The result is presented in Fig. 1 and is com­pared with the experimental values of the quantity a(cu) Xn(cu) (in Np/cm) given in the literature. 8,9,29,33-36 The experimental spectrum is characterized by a well de­fined band around 200 cm - 1 resting against the libration band which peaks about 625 cm - 1. The calculated spectrum exhibits only a shoulder in the 200 cm - 1 region, whereas the line shape is dominated by a broad band culminating at 600

'T 2500

:I U I: 2000

11/ L 11/ 1500

Z

1000

500

• •

... •

100 200 300 400 500 600 700 800 900 1000

CM-1

FIG. I. Results for the absorption spectrum of liquid water at 25 'C: the reported quantity is the absorption coefficient a(cu) in Npcm-', multi­plied by the refractive index n (cu). The black dots are the experimental val­ues taken from Ref. 33 between 6 and 450 cm -, and from Refs. 34-36 between 450 and 1000 cm - '; the maximum of the absorption band peaks about 625 cm -, and amounts to 4250 Np cm - '. The heavy curve is the classical profile as produced by our MD simulation, and the thin curve is the quantum corrected profile (see text). The small intermediate peaks visible along the calculated profiles are due to the statistical noise.

200

150

100 L L LLJ..._.L----LL-l LLUc-LL-LLLLc-LL-'--.LL.W.-L.L.LL.LW.-L.L.L.LJLu.-L.L.Lw-'-'-.l

o 10 15 20 2S 30 3S 40 4S 50

CM-1

FIG. 2. As Fig. I but for the frequency range 0-50 cm - '. Notice that only the classical simulated profile is shown since the quantum corrected profile is mostly indistinguishable from the former one in this frequency range. The squares represent the experimental values taken from Ref. 29, the black dots correspond to Ref. 33, the stars correspond to Ref. 9 and the circles corre­spond to Ref. 10.

cm - 1 (the small peaks visible along the spectrum are just due to the statistical inaccuracy). However, the overall ab­sorption intensity is badly reproduced: e.g., at 600 cm - 1 the calculated intensity is only 41 % of the observed intensity. But this disagreement tends to disappear at low frequency (below 50 cm - 1) where the calculation reproduces quite well the absorption intensity, see Fig. 2. In particular, the sharp rise of the absorption intensity in the first wave numbers is well reproduced and corresponds to a Debye re­laxation time of 7.7 ps a value very close to the experimental measurement of8.2 pS.29 In fact, above 0.5 ps the correlation function CT(t) is quasi-exponential as C m (t), and its time evolution is governed by the collective reorientation of the water molecules. The same finding holds for Cp (t) and Cov(t), see Table I.

The presence of H atoms suggests that for frequencies cu ;;: kT liz the use of quantum corrections is necessary even at room temperature (kT liz = 208 cm - 1) since a major part of the spectrum (the lib ration band) is located above 200 cm - 1. The correction of classical line shapes for quantum effects can be done in following one of the various proce­dures given in the literature and which all are semi empirical in nature. Recently, Borysow et al.37 have tested different expressions to correct a classical lineshape in using as a benchmark, an ab initio calculation of the quantum line shape. They have found that the Egelstaffprocedure38 simu­lates the quantum profile with the best accuracy (the devi­ation is less than 1 % at 300 cm - 1). This procedure consists to replace the time t by the complex time (t 2 - ilzt I kn 112 in the expression of the classical correlation function to cor­rect. In the present case the quantum corrections are so im-

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Bertrand Guillot: Infrared spectrum of liquid water 1547

portant that they eliminate a major part of the aforemen­tioned discrepancy. Thus the quantum corrected profile is approximately two times more intense at 600 cm - I than the classical profile, see Fig. 1, and the calculated maximum intensity represents now as much as 91 % of the observed magnitude (a"b. = 4250 Np/cm - I at 630 cm - I). It may be notice that the Egelstaff procedure is, in any case, an ap­proximate procedure and its validity at very high frequency (aJ> 600 cm - I) is certainly questionable. In particular a rough estimation based upon the quantum harmonic oscilla­tor shows that the Egelstaff correction underestimates the true quantum contribution by a fair amount above 500 cm - t, which could explain the remaining discrepancy. As expected the quantum corrections diminish drastically with the frequency since the quantum profile is only 11 % more intense than the classical profile at 200 cm - 1 and becomes virtually identical to it below 100 cm - I. Correlatively, the quantum corrections desymmetrize the profile and induce a small shift of the maximum of the band located now around 650 cm - 1 instead of 600 cm - 1 previously. Thus the overall agreement between calculated and experimental spectra is now more satisfying over a large domain offrequency (0.5-1000 cm - 1) even if the MD simulation does not reproduce with accuracy the well identified band at 200 cm - I and pro­duces only a shoulder in this region.

In order to analyze in more details the genesis of the FIR band we have calculated the absorption intensities associat­ed with Cm (t), Cp (t), Cov(t) and the interference term CrU) - CmU) - Cp(t), respectively: The results are pre­sented in Figs. 3 and 4 (only the classical profiles are shown). First of all, the cross correlations between perma-

QL-~L-~L-~L-~~~~~~~~~~J-~~

o \1,0 200 300 qOO 500 600 700 800 900 1000

CM-1

FIG. 3. Simulation results for partial absorption profiles. The thin curve corresponds to the absorption spectrum associated with the permanent di­pole correlation function, the heavy curve corresponds to the absorption spectrum due to the DID mechanism alone, and the dashed curve is the overlap contribution. For the sake of clarity, the DID spectrum was multi­plied by a factor of 16 and the overlap spectrum by a factor of 35000.

nent dipoles and induced dipoles play an important role for all frequencies: They give a negative contribution to the ab­sorption spectrum above 250 cm - 1 and give a positive con­tribution on this side, see Fig. 4. Moreover, when the absorp­tion spectrum due to the permanent dipoles shows essentially a high frequency band centered around 600 cm - 1, the dipole induced dipole (DID) contribution and the overlap contribution exhibit two bands, one at 200 cm - 1

and the other in the 600 cm - 1 region. The high frequency band corresponds to the libration motions of the water mole­cules in the hydrogen bond network. In fact, the frequency distribution of these libration modes is fairly broad as shown in Fig. 5, where the power spectra associated with the reor­ientational motions around the three principal axes of a mol­ecule are presented. The power spectra of the X and Z com­ponents exhibit broad and unsymmetrical profiles between 400 and 900 cm - 1 whereas the Y component is mostly sym­metric and sharp, and peaks around 430 cm - 1. In this con­text, it is noteworthy that the power spectrum associated with the collective correlation function Cm (t) (see Fig. 3) is sensibly different from the one deduced from the single com­ponent C;" (t) (the X component in Fig. 5) since in the for­mer case the libration frequencies are redistributed in such manner that the maximum of the band is located around 600 cm - 1. On the other hand, the band at 200 cm - 1 which char­acterizes the DID spectrum and the overlap spectrum can be associated with the O-H" ·0 stretching mode. As a matter offact, in the case of the overlap spectrum, the overlap dipole between 0 and H atoms is so rapidly decreasing (the effec­tive range is less than 1 A) that only the nearest neighbors are involved in the interaction. Furthermore, although the stretching and the bending motions of the O-H"O bridges

-100

-200

'i -::5UU

I "-400

I: III-SOl)

IL 111-6ilO Z

-700

-800

-900

-1000 L-L-L-L-'--~"--~L-~"---'--~-'--~-,-:-L--,-:~-,--:-:, o lUO 200 300 400 500 600 700 800 900 1000

CM-l

FIG. 4. As Fig. 3 but the thin curve is the absorption profile due to the interference term between induced and permanent dipoles (see text) and the heavy curve is the purely induced absorption profile (DID spectrum of Fig. 3) shown for comparison.

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1548 Bertrand Guillot: Infrared spectrum of liquid water

550

sao

450

400

350

::j ci 300

250

200

ISO

100

So

100 200 300 400 500 600 700 800 900 1000

CM-1

FIG. 5. Power spectra in arbitrary units associated with the reorientational motions around the three principal axes of a water molecule. The thin curve (X component) corresponds to an unitary vector lying along the permanent dipole moment, the heavy curve ( Y component) corresponds to an unitary vector in the plane of the molecule and perpendicular to the dipole vector and the dashed curve (Z component) corresponds to an unitary vector out of plane.

can modulate the electronic overlap mechanism to an appre­ciable extent, the spherical symmetry of the dipole might strengthen the contribution coming from the stretching mode to the prejudice of the bending mode. This is clearly illustrated in Fig. 3 where the O-H"'O oscillations gener­ates a sharp peak around 200 cm - 1 whereas the libration mode gives only a broad shoulder on the high frequency side. Nevertheless the contribution of the overlap mechanism to the total absorption spectrum is extremely weak and we do not think that the approximate character of our model inter­action can invalidate our conclusions. For the DID spec­trum (see Fig. 3) the modulation of the induced dipoles by the stretching mode gives rise to the same sharp peak at 200 cm - I, but the libration band is now much more intense than in the overlap spectrum. Interestingly enough, no low fre­quency band is observed on the flank of the 200 cm - 1 peak, the statistical inaccuracy of our MD run being taken into account. This contrasts with the computer simulation re­sults of Madden and Impeyl7 where a weak shoulder ap­pears near 60 cm - I. This could be due to the proximity of the intense 200 cm - 1 band which could blurred an eventual weak band. Nevertheless it must be noticed that this low frequency band is barely visible in the experimental spec­trum the authors detecting it either at 30 cm - 1 (Ref. 10) or at 50 cm - 1 (Ref. 8) and' virtually disappearing above 303 K (see the scattering of the experimental values in Fig. 2).

One of the key results of the present calculation is the occurence of a strong dynamical cancellation effect in the DID spectrum. To illustrate our purpose, we present in Fig. 6 the power spectrum associated with the single correlation

220

200

180

1 160

I U 140

II: Ul120

L Ulloa Z

I~'~· I •

I • I •

I • I' ' , ,~,

...,L .... ''--- ".... \

/':-; ~\ ,',: \\-, " " , 80 I ',,' "

I, \\ ,--',' '.

~ ~ , I "-I ,

40 I ,

I '.

m ' ~ o 100 200 300 400 500 1600 700 800 900 1000

CM-

FIG. 6. Comparison between the collective DID spectrum (dashed curve) and the self-contribution (thin curve).

function C;(t) = l:, (p,(t)·p,(O», where p, is the dipole induced on molecule i. The line shape exhibits three peaks at 225, 450, and 850 cm - I, respectively. According to the pre­vious discussion the two latter peaks are reminiscent of the libration motions (Fig. 5), and the former one is the O' . ·0 stretching band slightly shifted by the proximity of the 450 cm - 1 libration mode. In comparing this self-absorption spectrum with the collective absorption profile (see Fig. 6) one remarks that the libration contour of the self part seems to be due to a splitting of the high frequency band of the collective spectrum. To understand this phenomenon it is worthwhile to have recourse to some algebra.

It is possible to decompose the DID correlation func­tions, Cp (t) and C~ (t) into 2-,3-, and 4-body terms31

•32 by

noticing that the dipole induced on molecule i can be written, p, = l:J 1Tij' where 1Tij is the dipole induced on iby the mole­cule j (in fact the index j takes into account the Coulomb fields coming from the atomic charges of the molecule j), namely,

Cp (t) = C2a (t) + C2b (t) + C3a (t) + C3b (t)

+2C3c (t)+C4 (t), (16)

C;(t) = C2a (t) + C3a (t), (17)

where

C2a (t) = L (1Tij(t)'1Tij(O»,

''''J C2b (t) = L (1Tij (t) '1TJ, (0»,

''''J C3a (t) = L (1TU(t)'1T,dO»,

ih",k

C3b (t) = L (1Tij(t)' 1TkJ(O», ''''J",k

(18)

(19)

(20)

(21)

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Bertrand Guillot: Infrared spectrum of liquid water 1549

C3c (t) = L (1Tij (thTki (0», ih~k

C4 (t) = L (1Tij(t)· 1Tk/(O». i~Nk~/

(22)

(23)

In order to simplify the notations we will replace in the fol­lowing the delocalized permanent dipole moment (coming from the atomic charges) of each molecule i by an equivalent point dipolem i (in our case Iml = 2.27 D) and we will con­sider the water polarizability a was isotropic. These assump­tions are immaterial on the present discussion. Hence, the dipole moment 1Tij induced on the molecule iby the perma­nent dipole mj = mUj can be written

~ = (maw) Tf/u1 (a,/3=x,y,z),

where

(24)

Tf/=(3uijuf-Oa{3)IRt. (25)

In these equations uj is an unitary vector lying along the permanent dipole moment mj and Rij = Rijuij is the vector connecting the center of mass of molecules i and j. For the sake of clarity, consider just C3a (t) [Eq. (20)], then with the expressions (24) and (25) we have

C3a (t) = (ma w )2 L (Tij(t)uj(t)Tik(O)Uk(O» i~#k

(26)

where T,j contains the relative translational coordinates of the pair (iJ) and uj expresses the rotation of the molecule j, and similarly for Tik and Uk' To go a step further we make use of the decoupling approximation which writes

(T'j (t)uj (I) Tik (O)u k (0»

",,(T,j(t)Tik(O»(Uj(t)Uk(O». (27)

For liquid water which presents strong orientational correla­tions between the nearest neighbors it seems hard to justify such a decoupling between translational motions and rota­tional motions. Nevertheless the fact that the translational band (200 cm - I) and the librational band (600 cm - I) are rather well separated in the FIR range suggests that for this frequency domain the Eq. (27) is a useful approximation. In any case, our purpose being just to shed some light on the complicated cancellations occuring in Cp (t), the use of the decoupling approximation is relevant. Consequently, in in­troducing the approximate relation (27) into Eqs. (18)­(23) and in using the property of invariance by permutation of identical particles as well as the even parity of Tij (T'j = ~i)' we obtain the following expressions:

C2a (t) = L (T,j(t)Tij(O»(Ui(t)Ui(O», (27) i~j

C2b (t) = L (Tij(t)Tij(O»(ui(t)uj(O», (28) i#.}

C3a (t) = I (Tij(t)TidO»(ui(t)uj(O», (29) i~j~k

C3b (t) = I (Tij(t) TidO»(u, (t)u,(O», (30) i~#k

Ck (t) = C 3a (t),

C4 (t) = L (Tij(t)Tk/(O»(ui(t)uj(O», (31) i##k#l

where the constant factor (maw)2 was omitted as well as other unimportant factors. In introducing the expressions (27)-( 31) into the definitions (16) and (17) we obtain

Cp(t) = N(N - 1)(TI2 (t)T12 (0» [(u 1 (t)u 1 (0» + (u\

and

X (t)uz (0) >] + N(N - I)(N - 2) «'12 (t)T\3 (0»

X [(u\ (t)u\ (0» + 3(u\ (t)u 2 (0»] + N(N - 1)

X (N - 2) (N - 3) (T12 (t) T34 (0» (u l (t)u z (0» (32)

C~ (t) = N(N - 1) (T12 (t) T12 (0» (u l (t)u I (0»

+N(N-l)(N-2)(T12 (t)T\3(0»

X(U I (t)U2 (0», (33)

where N is the number of molecules in a liquid sample of volume V.

The equations (32) and (33) have important conse­quences. First of all, Cp (t) and C~ (t) are expressed in terms of 2-, 3-, and 4-body translational correlation functions. These functions are related to those found in studies on colli­sion induced spectra by rare gas liquids.zl It is now well understood that these translational correlation functions are the seat of a strong cancellation between positive 2- and 4-body contributions and a negative 3-body contribution. This cancellation effect proceeds from the macroscopic property of isotropy in a fluid (inversion symmetry). Thus, the ob­served spectral intensity associated with translational vari­ables is nothing but the result of the local fluctuations mea­sured in the neighborhood of a given molecule: longer is the range of the interaction smaller is the deviation from iso­tropy. In the present case one expects that (T12 (t) T12 (0» is partly canceled by the negative component (Tlz (t) TI3 (0» (the 4-body term being generally very small, it is neglected in our discussion). A very crude estimation based upon a lat­tice gas model32 gives the following relation:

N(Tlz (t)TI3 (O»"" -plPo(T12 (t)T12 (0», (34)

where p = N IV is the number density and Po a reference density for the lattice. Although this estimation is very ap­proximate,39 the very essence of the phenomenon is well accounted for.

These nobody functions appear in Cp (t) and Cs (t) through a product involving the self and distinct orienta­tional correlation functions (u I (t) U 1 (0» and (u l (t)u z (0», respectively. If we suppose, for the sake of argument, that the cross correlation function (u I (t) Uz (0» is negligible with respect to the self term (u 1 (t) u 1 (0» then one might observe the same libration band in the power spec­tra of Cp (t), C~ (t) and (u I (t) u l (0»; just a glance to Figs 3,5, and 6 shows that is far to be the case. Actually, in liquid water the cross angular correlations are particularly impor­tant as indicated by the high value of the kirkwood Gk factor measured in our simulation basic cell,

N

Gk = N -I L (u i 'u) ",,4.5. iJ= I

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1550 Bertrand Guillot: Infrared spectrum of liquid water

Nevertheless, the angular correlations probed by the DID mechanism do not extend over all the liquid sample since they are weighted by the range of the dipolar field. Conse­quently the angular correlation function (U I (t) U2 (0» fig­uring in Eqs. (32) and (33) takes only into account the correlations between molecules separated by a distance smaller than the range of the DID mechanism. For this rea­son the value at zero time of (u l (t)u2 (0» is not given by Gk but can be estimated with the help of a local G[ factor such as

Nc

G[ = 1 + L (u l 'uj ), r lj <RDID' (35) j,,< I

where RDID is the range of the DID mechanism. For exam­ple, a very approximate evaluation can be made by consider­ing only the correlations between a central molecule and its neighbors of the first shell; our simulation results give G[ =.2.5 and Nc = 4.2 for the number of neighbours. The latter values lead to (u l u2 ) =0.36, a value one order of mag­nitude greater than in the case of a quadrupolar liquid as N 2 which presents very weak angular correlations.40 Hence, bearing in mind the key role devoted to the orientational correlations in this problem, we can rewrite the Eqs. (32) and (33) under the following form:

Cp (t) = N (T12 (t) TI2 (0» {( 1 - :a)<U I (t)u I (0»

+ (1 - ~ )(UI (t)u 2 (O»} + 4-body term,

(36)

C~ (t) = N (T12 (t) Tl2 (0»

X {(UI (t)u I (0» - :a (ul (t)u2 (O»}, (37)

where we have introduced the relationship (34). The equa­tion (37) permits to understand why the libration band asso­ciated to C ~ (t) (Fig. 6) presents two maxima at 430 and 850 cm -I, respectively. As a matter of fact this peculiar line shape is just the result of the substraction of two libration bands (the convolution with the translational spectrum only affects the bandwidth) one associated with (u l (t)u I (0», (see in Fig. 5 the x component) and the other one associated with (U I (t)U2(0». But (UI(t)UI(O» and (U I (t)U 2 (0»

being proportional to the self-correlation function C;" (t) and the distinct correlation function C:;. (t), respectively [see definitions (10), (13), (14), and (15)], one obtains with the help of Eg. (35)

C~ (t) ex: (Tl2 (t) TI2 (0»

X{(1 +~.E...)C;"(t) -.E...~Cm(t)}. GkNc Po Po GkNc

(38)

Consequently, the libration band ofC~ (t) is nothing but the difference band between the line shape of the single dipole correlation function C;" (t) and the line shape generated by the collective dipole correlation function Cm (t), the net re­sult being the appearing of an hollow in the profile around 600 cm - I (see Fig. 6). The same rationale applies to the case ofCp(t), namely,

X {[(I _.E...) -~ (1 - 3.E...)] Po GkNc Po

XC;"(t) + (1-~)~Cm(t)}, (39) Po GkNc

where the balance between collective and single reorienta­tion favors the latter contribution; see Fig. 6 and the x com­ponent in Fig. 5 and compare the respective libration bands. In summary, the DID spectrum is the seat ofintricated can­cellation effects between the 2-, 3-, and 4-body translational contributions from one hand and the angular correlations from the other hand. The angular correlations reflect the tetrahedral symmetry of the hydrogen bond network where­as the translational correlations are mostly sensitive to the symmetry of inversion which is a common property to all molecular liquids.

IV. CONCLUSIONS

Now it is possible to answer the questions that we ad­dressed in the Introduction. First of all, we have shown that the intermolecular motions generated by the SPC model per­mit to reproduce satisfactorily the overall line shape of the FIR absorption spectrum of liquid water. In particular, the introduction of the DID mechanism is crucial to obtain at once the correct magnitude of the absorption intensity over a large domain of frequency (0.5 -+ 1000 cm - I) and the ap­pearing of a shoulder in the 200 cm - I region which is the signature of the O-H"'O stretching mode. If the DID mechanism is neglected (then only the effective dipoles of 2.27 0 are considered) the translational band disappears and the high frequency libration band exhibits an absolute intensity which is approximately two times greater than the experimental value. Moreover, this work has put forward the role devoted to quantum corrections for frequencies greater than 200 cm - I: Their effect is to desymmetrize the absorption profile and to enhance the intensity of the libra­tion band by roughly a factor of 2. We have also shown that the modulation of the overlap dipoles, occuring between 0 and H atoms, by the oscillations of the hydrogen bond network is not sufficiently important to affect significantly the spectral intensity. Even if one can argue on the simplicity of the exponential dipole used in the MD simulation, we believe that our conclusion is robust.

Nevertheless, the MD simulation has not revealed the weak bands reported in the experimental literature around 10 and 30-60 cm - I. In spite of the controversial character of these bands,8,lo.29 it should be worthwhile to simulated the FIR spectrum at lower temperature (this work is under way). Furthermore, the stretching band at 200 cm - I which is prominent on the experimental spectrum is only qualita­tively reproduced by the MD simulation. The reason for that could lie in the magnitude of the interference term between permanent and induced dipoles. This interference term is governed, as the DID spectrum, by very intricated cancella­tion effects between n-body translational contributions and the angular correlations produced by the tetrahedral sym­metry of the first coordination shell. The balance is so deli-

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Bertrand Guillot: Infrared spectrum of liquid water 1551

cate that to restore the back polarization in the calculation of the induced dipole could properly change the spectrum in the 200 cm - I region (see in Fig. 4 the rapid drop off of the interference term after 1 SO cm - I). We believe that a com­puter simulation taking into account a full treatment of the polarization mechanism in the interaction potential and in the evaluation of the induced dipole should answer this ques­tion.

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