Symmetry breaking in water molecule interactions Frank H. Stillinger and Howard L. Lemberg Bell Laboratories, Murray Hill. New Jersey 07974 (Received 10 October 1974) We discuss the phenomenon of symmetry breaking observed in minimum-energy structures for water dimers. Based on two explicit representations of the intermolecular pair potential, the ST2 and PKC forms, lowest energy dimers are evaluated as a function of oxygen-oxygen separation, and it is concluded that a critical distance R, exists which separates mechanically stable dimers of differing symmetries. For small distances, the familiar linear hydrogen bond is manifested, while for large distances dipole--dipole forces promote alignment of the molecular axes. At R c' eight equivalent small-distance minima of low symmetry and four large-distance high-symmetry configurations undergo confluence. Thorough investigation of constrained minima illustrates how this transition occurs and invites comparison with the theory of critical phenomena. The second-order phase transition of the Ising ferromagnet provides a useful analogy to the structural symmetry breaking of the water dimer. Finally, we consider the relation of the present case to the larger issue of interactions in condensed phases, and to the interaction of more complex molecular pairs; one thereby discovers analogs for (a) Ising models in external fields, (b) upper and lower consolute points in liquid mixtures, and (c) first-order phase transitions. I. INTRODUCTION curate and extensive Hartree-Fock calculations for the Substantial research effort has recently been expended ground-state energy hypersurface for two (rigid) water in attempting to understand the interactions between molecules. Although this "PKC" potential differs from water molecules. This activity has comprised both di­ "ST2" in several quantitative details, calculations pre­ rect quantum-mechanical studies, 1-8 as well as semi­ sented in the next two sections show that both lead to empirical approaches to determine closed form approxi­ the same type of singular but continuous symmetry mations to the potential. 9-12 At present, a consensus breaking at finite intermolecular distance. This agree­ seems to have been reached concerning at least the ment probably indicates that the exact water-molecule qualitative character of water molecule interactions: pair potential shares identical behavior. At small distances, linear hydrogen bonds tend to form Section IV stresses the analogy between spontaneous between neighbors, while at large distances the inter­ symmetry breaking in the present context, and that action between permanent molecular dipole moments encountered in the statistical mechanics of phase transi­ dominates. tions. At the very least, this analogy provides an ele­ This paper focuses attention on one special aspect of mentary mechanical model in terms of which critical the water-molecule pair potential which seems to have phenomena and phase transitions may be partially under­ received little attention thus far. Eight equivalent lin­ stood. ear hydrogen bond structures exist for a molecular pair The basic mechanism for occurrence of spontaneously at small separation, with the optimal geometry exhibit­ broken symmetry is already present with hypothetical ing only a single symmetry plane. On the other hand, point molecules possessing permanent dipole and axial dominant dipole-dipOle forces at large separation would quadrupole moments. Details are provided in the Ap­ cause alignment of molecular symmetry axes, so that pendix. However, this level of simplicity is not ap­ only four equivalent optimal pair structures arise, each propriate for description of water molecules. with two symmetry planes. The manner in which the former transforms into the latter as intermolecular II. ST2 INTERACTION distance increases is not obvious. We illustrate that The ST2 interaction is based upon a rigid four-point­ transformation by citing concrete numerical results charge model for each water molecule. Figure 1 (a) for two published water-molecule potential functions. shows the relevant geometrical structure. The point For both, we conclude that upon bringing two molecules charges are located along four directions emanating together from infinite distance, a critical distance is from the oxygen nucleus, each pair of which occurs reached at which a spontaneous change in symmetry at the ideal tetrahedral angle 109.47°. Two of the point occurs for the minimum-energy structure. charges are positive (+q) and can be identified as pro­ The first pair potential examined (Sec. II) is the "ST2" tons; they are placed 1 A from the oxygen nucleus. The potential, which was devised for use in the simulation two negative charges (- q) are present for charge neu­ of liquid water by molecular dynamics. l1,IS-15 This po­ trality; they reside 0.8 A from the oxygen nucleus. tential is to be interpreted as an "effective pair poten­ The ST2 potential consists of two principal parts; tial,,18 whose form incorporates the principal structural effects of potential nonadditivity. V(2) (ST2) = V LJ (R I2>+ S (R I2 ) Vel (1, 2). (2.1) The second pair potential used in the present study The first, VLJ, is a Lennard-Jones 12-6 potential act­ (Sec. m) was suggested by Popkie, Kistenmacher, and ing between the oxygen nuclei, whose separation has Clementi. 8 It represents an analytical fit to their ac- been denoted by R 12 • The second part contains the total 1340 The Journal of Chemical Physics, Vol. 62, No.4, 15 February 1975 Copyright © 1975 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.112.66.66 On: Sat, 18 Jan 2014 03:22:55 F. H. Stillinger and H. L. Lemberg: Symmetry breaking in water molecule interactions 1341 80°r-------------------------------------~ H'''v<q0 qH 60' 109.47 0 2 0 ,0 IA o 52.5 40' o " 0.8AI \ 0.2307A , .-q 20' -q 0 0 (0) (b) -20' FIG.!. Geometric arrangement of point charges used to de­ -40' fine water-molecule pair interactions; (a) ST2 effective pair potential, (b) analytical fit to Hartree-Fock calculations con­ -60' structed by Popkie, Kistenmacher, and Clementi. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 R1z(AJ electrostatic interaction v.I, sixteen terms in all, for point charges on the two different molecules. This elec­ FIG. 3. Values of angles 9 and qJ (see Fig. 2) which provide trostatic contribution is modulated by a cubic spline constrained energy minima for the ST2 potential. function of the oxygen-oxygen distance R 1Z: flection symmetry in the plane containing the proton S(R ) =0, 1Z donor molecule and the symmetry axis of the acceptor _ (R 1Z - RL)2 (3Ru- RL - 2R12 ) molecule. The dimer is prevented from twisting out - (R u -RL )3 of this reflection-symmetric arrangement in part by proton-proton repulsions. = 1 (Ru ::S Rt2)' (2.2) Including the Lennard-Jones parameters E: and (J, the Direct calculationll shows that the dimer energy mini­ ST2 potential utilizes the following quantities: mum exhibits the following geometric parameters (see Fig. 2): q=O. 2357 e, E: = 7.5750 X lO-z kcal/mole, R12 = 2. 852 A, (J=3.1000 A, (2.3) 9=51.8°, (2.4) RL = 2. 0160 A, rp = 53. 6° . Ru =3.1287 A. The ST2 energy at this minimum is - 6.839 kcal/mole. In order to specify the relative configuration for two Equivalent minima may be generated by any combina­ rigid water molecules, six coordinates are required. tion of the following operations: These can be chosen to be the distance R 12, and five angles describing the relative orientation of the mole­ (a) rotate the acceptor molecule 180° about its sym­ cules. The ST2 interaction provides an approximate metry axis; value for the intermolecular interaction for any given (b) rotate the donor molecule 180 0 about its sym­ set of the six relative coordinates. metry axis; Figure 2 shows schematically the structure of the (c) exchange roles of donor and acceptor molecules. minimum-energy dimer implied by the ST2 function. In all, eight distinct structures arise from these inde­ The single hydrogen bond is obvious (proton Hz donated pendent operations. to acceptor oxygen Ca). This dimer possesses a re- The molecular arrangement shown in Fig. 2 is also qualitatively correct for constrained energy minima, in which R 12 has a preassigned value not necessarily equal to that shown in Eq. (2.4) for a global minimum. Of course, the angles 9 and qJ would then exhibit dis­ placed values. Figure 3 shows computed values of 9 (R 12 ) and rp (R 1g ), 0 1 O2 8 H2 the angles which provide constrained minima, versus H4 R 12• As R 1a increases from the distance 2.852 A cor­ responding to the global minimum, these minimizing . H3 angles decline monotonically and continuously to zero. FIG. 2. Minimum-energy dimer structure. The plane con­ As a result of these declines, the molecular symmetry taining the proton donor molecule (Hl01H2) serves as a reflec­ axes become more and more nearly aligned with the tion symmetry plane. oxygen-oxygen axis. Suddenly, at the critical distance J. Chem. Phys., Vol. 62, No.4, 15 February 1975 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.112.66.66 On: Sat, 18 Jan 2014 03:22:55 1342 F. H. Stillinger and H. L. Lemberg: Symmetry breaking in water molecule interactions Re (ST2) =4.964 A, (2.5) 6 6 the alignment becomes complete. When R 1Z exceeds 4 this value, no further change in the angles () and rp is 4 required; the molecular symmetry axes remain rigor­ 2 ~ "0 ously aligned.