I 844
B. R. Lentz, A. T. Hagler, and H. A. Scheraga
(11) L. D. Kispert, K. Chang, and C. M. Bogan, J. Phys. Chem., 77, 629 (1973). L. D.Kispert and M.7 . Rogers, J. Chem. Phys., 58, 2065 (1973). b. D. Kispert and K. Chang, J. Magn. Resonance, 10, 162 (1973). M. Iwasaki, K. Toriyarna, and K. Nunome, J. Chem. Phys., in press. N. M. Atherton, Cheim. SOC., Spec. Pub/.,No. 1, 32 (1973): No. 2, in press. (16) J. S. Hyde, R . C. Snoed, Jr., and G. H. Rist, J. Chem. Phys., 51, 1404 (1969). (12) (13) (14) (15)
(17) R. C.McCalleyand A. L. Kwiram, Phys. Rev. Lett, 24, 1279(1970). (18) R. C. McCalley. Ph.D. Thesis, Harvard University, 1971. (19) H. M. McConnell, C. Heller. T. Cole, and R. W. Fessenden, J. Amer. Chem. SOC., 82, 766 (1960); A, Horsfield. J. R. Morton, and D. H. Wiffen, Mol. Phys., 4, 329 (1961). (20) R. C. McCalley and A. L. Kwiram, J. Amer Chem. Soc., 92, 1441 (1970). (21) H. Yoshida, D.F. Feng. and L. Kevan, J. Chem. Phys., 58,4924 (1973). (22) M. Nechtschein and J. S. Hyds, Phys. Rev. Lett. 24, 672 (1972).
Vibrationtal Frequencies of Water Clusters1 arry R. Lentz,2a Arnold T. Hagler, and Harold A. Scheraga*2b Department of Chemistry, Cornel1University, Ithaca, New York 14850 and Weizmann Institute of Science, Rehcvot, Israel (Received March 5, 1974)
Calculations are presented which lead to the intermolecular normal mode frequency distributions for a series of water clusters, starting with the Ben-Naim-Stilllinger water-molecule interaction potential function. In addition, the ability of this potential to account qualitatively for the intermolecular frequency spectrum of ice 1is investigated. The librational modes of a water molecule in the center of a pentamer having an icelike configuration are found to be too stiff. This is attributed to the positions of the point charges representing the lone-pair orbitals. Finally, the results of the calculations are used to suggest possible interpretations of the intermolecular normal mode frequency spectrum of ice I.
1. Introduction In a recent theoretical treatment3 of the structure of liquid water, the normd mode vibrational frequencies were assumed to be independent of cluster size. This treatment has now been improved4 by taking into account, among other things, the dependence of the normal mode vibrational frequencies of‘a cluster on its size and shape (or connectivity), making use of the potential function of BenNaim and Stillinger5 (BNS potential). The BNS potential is used here to calculate the normal mode vibrational frequencies of A tetracoordinated water molecule (whose normal mode motions should be roughly similar to those of a water molecule in ice I). A comparison of the calculated frequencies with the experimental ones of ice I provides a test .of the intermolecular potential function used to describe water rnolecuie interactions.
11. Procedure In treating liquid water,^ selected clusters containing from two to nine water molecules were considered, and their intermolecular normal mode vibrational frequencies were computed with the BNS potential and reported in the earlier4paper. For the purpose of investigating the properties of the RhiS potential, we consider here only a cluster of five water molecules, containing a central tetracoordinated water molecule (the “star” pentamer), The intermolecular potential energy of a cluster is calcu, ~ the essential features of lated with the BNS p ~ t e n t i a land this function are summarized here. The BNS potential consists of a Lennard-Jones 6-12 potential between the oxygens of the two interacting water molecules, together with The Journal of Physical Chemistry, Vol. 78, No. 18, 1974
an electrostatic interaction term. The latter is based on the Bjerrum point-charge model6 which positions point charges (two positive and two negative) tetrahedrally around the oxygen nucleus at a distance of 1 A from the center of the oxygen nucleus. Thus
where roo is the distance between the centers of the oxygens of the two interacting water molecules, VLJ is the Lennard-Jones potential term, Vel is the electrostatic interaction term (being a function of the distance between all the point charges in the two interacting water molecules), and S is a “switching” function which “switches” off the Vel term at small roo to prevent the overlapping of the point charges. The total potential energy of a cluster is minimized with respect to the coordinates of all atoms (and point charges) in the cluster, starting from configurations that yield the maximum number of hydrogen bonds for a given size cluster4 (or from an ice-like geometry in the case of the star pentamer). The minimum-energy coordinates of each cluster (obtained from our starting configurations) are presented in the Appendix. The second derivatives of the potential energy at the minimum-energy configuration are then used to solve the equations of motion of all of the atoms in the cluster, in the harmonic a p p r ~ x i m o t i o n . ~In- ~this manner, the normal mode frequencies for the coupled motions of all the water molecules in a given cluster are calculated. For most of the cluster species considered, the computed frequencies are reported el~ewhere,~ while we report here only the normal mode frequencies of the ice-like star pen-
Vibrational Frequencies of Water Clusters tamer. For the star pentamer, we also assigned very large masses ( BO8 amu) to the atoms of the four peripheral water molecules to zlncouple the motions of the central, tetrahedrally coordinated water molecule from the motions of the peripheral water molecules. This allows us to compare the computed normal mode frequencies of this tetrahedrally coordinated molecule with those observed for ice I, and to observe the effects of coupling of the motions of the central water molecule with those of the peripherals. This comparison cannot be an ex act one since only the effects of nearest neighbors are included. Instead, in using this criterion to investigate the properties of a potential function, we can ask only whether the computed frequencies lie within the range of the experimentally observed ones.
111. Results The minimum-energy configurakion of the star pentamer, resulting from the BNS potential, is shown in Figure 1, and the intermolecular normal mode frequencies of this cluster (along with those for the motions of the central water molecule unsoupled from the motions of its neighbors) are presented in Table I. The nature of the motions corresponding to these normal modes can be described roughly as Folliows: (a) UL,librational motions of the central water molecule in the field of i t s neighbors; VL’ and VL”, coordinated librational motions of both the central and the peripheral water molecules (UL” involves more coupling than does ~JL’J; (b) VT,hindered translations of essentially only the central water molecule in the field of its neighbors; UT’,translational motion of the peripheral molecules couto some translational motion of the central molecule, resulting mainly in the stretching of hydrogen bonds; (c) utor, mainly torsional rotations of the peripheral molecules about their hydrogen bonds; (d) Ub, coordinated wagging of the peripheral1 molecules coupled to motions of the central water molecuie, resulting in hydrogen-bond bending; ub/, coordinated wagging of mainly the peripheral molecules. The importance of coupling of the motions of neighboring molecules is illustrated clearly by these assignments. For example, coupling has produced a spread in librational frequencies 01’ -300 cm-l between VL’ and VL”; as noted above YL” imotiorts are more highly coupled than VL’ motions. Also, coupling raises the frequencies ( V L ) of the librational motions of the central water molecule, as can be seen by comparing the values of V L for the complete pentamer with those fo- the loentamer with heavy-mass peripheral atoms. The calculated normal mode frequencies of Table I may be compared with recent experimental values for ice I, given in Table IP. ‘The frequencies of the hindered-translational modes of the tetrahedrally coordinated water molecule not only falls within the range of the experimentally observed hindered-translational band, but also correspond roughly to the 200--220-~m-~ peak observed in the ice spectrum. It has been suggested‘l that the band (or bands) in the vicinity of 300 cm-l might be due to long-range forces; if this is so, such an effect would of course not be included in our simple nod el. In contrast to the good agreement between the calculated and observed values in the hindered translational region, the highest calculated librational frequency of thti tetrehedrally coordinated water molecule (1346 cm-l) is seen to be out of the range of the experimental values ( 4 3 Q to p-1100 em-l for most of the results reported in Table 11) The apparent reason for this high librational freqiiency is that the charges on the orbitals have
1845
Y
Figure 1. Minimum-energy geometry of the star pentamer obtained with the ENS potential. The oxygen separations and dimer interaction energies for dimers formed by the central and each peripheral water molecule are 2.672 A and -5.625 kcal/moi of dimer, respectively, for 1 4 1 and 1-IV, and 2.732 A and -6.495 kcal/mol of dimer, respectively, for 1-~111 and I-V. Plus and minus indicate point charges used in this potential to account for the hydrogen atoms and the oxy-
gen orbitals, re~pectively.~
been placed so far out (1 A) in the orbital direction. For a linear hydrogen bond with an O--O distance of 2.1 A, this would place the positive charge of the hydrogen only 0.7 A away from the negative lone-pair charge of its neighboring water molecule. This situation appears to be responsible for the high calculated frequencies. In order to determine whether this can account for the discrepancy, we have calculated the normal mode frequencies of our model cluster (of course, with reminimization of the configuration) using a modified version of the BNS potential in which the lonepair point charges are moved closer to the center of the oxygen ( i e . , at 0.9 A instead of 1.0 A from the center of the oxygen, in the tetrahedral direction), with the hydrogen left at 1 A. These results, shown in Table I (for normalmass peripherals), correspond more closely to the experimental values in that the YL frequencies fall within the experimental band. This would seem to confirm that the orbital charges were placed too far from the oxygen in the BNS potential. It is interesting to note that the frequencies of the highly coupled librational modes (uL”)fall below the expected range of librational frequencies in ice (400-1100 em-l) but are approximately in the range of the 300-cm-l hindered translational peak. These results (obtained with the modified BNS potential) are open to at least two possible interpretations. On one hand, the 300-cm-l experimental peak might be due to highly coupled librational motions, rather than to a hindered translation involving longrange f0rces.l‘ Alternatively, the imperfections of our model might be expected to lower the frequencies of highly coupled modes involving motion of the peripheral waters (such as UL”)relative to their frequencies in ice.22
IV. Discussion The BNS potential leads to a local minimum-energy structure with essentially the tetrahedral configuration predicted for water molecule interactions by recent extensive basis set ah initio LCAO-MO-SCF calculations.23~2* However, even though the tetrahedrally arranged point charges of the BNS potential confer the proper directionality on the structure, our results indicate that the location of The Journal of Physical Chemistry, Val. 78. No. 18. 1974
B. R. Lentz, A. T. Hagler, an
1846
TABLE I: Normal Mode Frequenciesa Calculated with the BNS Potential
_-
Complete pentarner Weavy-ma& peripherals Modified lt71NSc potentia X
VL
VL'
YL"
UT
UT'
vtor
Vb
Vb'
1409 1045 968 1346 904 760 1083 796 753
809 706 647,610
510 430 407,390
157 132 125
96 79
5l 50
39 29
607 551 521,486
394 343 334,326
214 207 205 231 228 218 191 186
139 119 112
8
38
71
38
183
47
31 19
34
Units of frequency are em-'. Assignments (VL, VT, etc.) are discussed in the text. Frequencies for motions of central water molecule uncoupled from motion of the peripherals by assigning a mass of 108 amu to the peripheral atoms. C Frequencies calculated using BNS potential with oxygen point-charge distance taken as 0.9A (and normal masses for the peripheral atoms). a
TABLE II: Summary of Experimental Data on Intermolecular Motions in Ice I Description of results Method
Ref
Temp, OK
Hindered translations (UT and vb)&
Librations
(PL)~
_ . I -
Band reported centered at 229, 190, 164, and Main band peak reported a t 840 cm-l. Shoulders a t 900, 770, 600, 555 cm-1. 65 cm-l; possibly also a t 305 and 260 Width ~ 4 0 to 0 1050 cm-1; highest cm-1. The upper limit of the absorption librational frequency reported, 1050 is at -328 cm-1 Cm-1 Poorly defined. Band center appears Main peak appears to be a t -230 cm-l; Not to be a t -750 cm-1. Width appears t o with another peak at -320 cm-1 specified be 500--1000em-1 Peak centered at 650 cm-'. Gaussian Majority of translational modes have 06 half-width, u, is 200 crn-1 frequencies of
