An Empirical Intermolecular Potential Energy Function for H 2 0
then the Walden product for the electrolyte A m will be a highly complex function of composition. It seems unlikely that the variation of W with composition could be utilized for the measurements of the various K values. Rather what is needed is the independent determination of K from spectral measurements combined with conductance measurements of W in both pure solvents if possible and then in mixtures. Naturally any method for estimating the diffusion coefficients (polarography, chronopotentiometry, etc.) could also be used by suitable modification of the above equations.
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Acknowledgment. The author wishes to thank the Research Council of Rutgers University for a grant which helped support this work. References and Notes (1) P.Walden, Z. Phys. Chem., 55, 207, 246 (1906). (2) R . Zwanzig, J. Chem. Phys., 52, 3625 (1970). (3) V. Kacena and L. Matousek, Collect. Czech. Chem. Commun., 18, 294 (1953), (4) Z. Zabransky, Collect. Czech. Chem. Commun., 24, 3075 (1959) (5) N. Tanaka and A . Yomada, Z. Anal. Chem., 224,117 (1967). ( 6 ) W. Lynessand P. Hemmes,J. Inorg. Nucl. Chem., 35, 1392 (1973).
An Empirical Intermolecular Potential Energy Function for Water' Lester L. Shipman* and Harold A. Scheraga" Department of C h e m N r y , Cornel/ University, lthaca, New York 74850 (Received November 7, 7973)
An empirical intermolecular potential energy function for water has been derived using data from pertinent available experimental and theoretical studies. The data utilized include the lattice energy of ice, X-ray structure of ice, compressibility of ice, intermolecular vibrational frequencies of ice, gas-phase dipole moment, gas-phase quadrupole moments, gas-phase infrared and microwave structure of the water monomer, and the localized molecular orbital structure of the water monomer. Expressions have been derived which relate various macroscopic thermodynamic properties to the intermolecular potential energy function. The contributions of the zero-point vibrational energy of ice to several of the above proper0 distance curves for various water dimers have been ties have been examined. Potential energy us. 0 calculated and compared with large basis set ab initio quantum mechanical results. The need for an adequate treatment of the quantum mechanical nature of the intermolecular motions in the water dimer in the calculation of the second virial coefficient is discussed.
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I. Introduction In studies of the interactions of water molecules with each other and with other (solute) molecules, it is helpful to have an approximate empirical intermolecular potential for water. Among the many empirical intermolecular potentials that have been developed, the one upon which attention has been centered most recently is that of BenNaim and Stillinger3 (BNS), which has been applied to liquid water by Rahman and Stillinger,4,5 Weres and Rice,6 and Lentz, et a1.7 These studies with the BNS potential have furthered our understanding of the structure and dynamics of liquid water. Although the BNS potential has been applied with considerable success to studies of liquid water, several defects in the potential have been identified,6-8 including the overestimation of the magnitude of the librational frequencies of tetracoordinated water molecules and the related overly strong tendency to form perfect tetrahedral coordination through hydrogen bonding. Also, the BNS potential was derived on a classical mechanical basis, whereas the intermolecular motions among water molecules are definitely of a quantum mechanical nature,6 as will be shown in section VIII. In the present paper, we present a derivation of a physically reasonable potential for water, based on a wide variety of experimental and theoretical data, with a view toward being able to reproduce and account for the most
pronounced features of the structure and dynamical behavior of water in isolated dimers, trimers, etc., as well as in the liquid and solid states. The data utilized include the equilibrium 0 . 0 distance of ice,s the lattice energy of ice, compressibility of ice, intermolecular vibrational frequencies of ice, gas-phase dipole moment, gas-phase quadrupole moments, gas-phase infrared and microwave structure of the water monomer, and the localized molecular orbital structure of the water monomer. In order to relate macroscopic quantities such as the lattice energy of ice to the intermolecular potential, it has been necessary to start with relations from equilibrium thermodynamics and derive formulas which relate certain macroscopic data to the intermolecular potential. Contributions from the zero-point intermolecular vibrational energies have been considered explicitly in this derivation. The potential is used to study the potential energy of a water dimer, the second virial coefficient of water vapor and, in another paper,1° the intermolecular vibrational modes of ice I.
11. Functional Form of the Potential The empirical potential energy function derived in this study is pairwise additive; Le., the potential energy of an aggregate of water molecules is the sum of the potential energy of all pairs of water molecules in the aggregate. The Journai of Physicai Chemistry. Vol. 78, No. 9. 1974
Lester L. Shipman and Harold A. Scheraga
91 0
Since three-body and higher order interactions have not been considered explicitly in the derivation of the potential here, the potential is best classified as an “effective” pair potential. The potential consists of two basic components, an electrostatic and an exp-6 component, as shown in eq 1,where Uis the total potential energy.
u = u,, +
UexpG
(1)
The exponential form for the nonbonded repulsion was chosen, rather than the R-12 form, because of the need to fit the zeroth, first, and second derivatives to experimental quantities, and the exponential form was found to be more satisfactory for this than the R-12 form. The electrostatic component has the form h’ ues
ues(i, jj
=
+le
U
(2)
Figure 1. Positions of the seven point charges within the water Tolecule. All distances have been rounded to the nearestoO.OO1 A and all angles have been rounded to the nearest 0.01 , The hydrogens and bonding pairs are in the xz plane and the lone pairs in the x y plane.
LN + UR
(21)
The roles of UNNand UR in determining the equilibrium value of R are quite different, in that UNNleads to a repulsive force and U, to an attractive force. From eq 16 and 21, we have
This is not a strict equality because of the omission of the zero-point vibrational energy contributions. The forces, F" and FR, corresponding to the energy components, U" and U,, are the negatives of the first derivatives in the first and second terms, respectively, in eq 22. Rearranging, and rewriting eq 22 in terms of the forces, we obtain FKS -FR (23) Le., the nearest-neighbor force is repulsive to about the same extent as the force from the rest of the crystal is attractive. This leads to a "compressive effect" wherein nearest neighbors are closer together at equilibrium in ice than they are a t equilibrium as isolated For example, the equilibrium value of R in ice (both experimentally and from the SS potential) is = 2.75 A and the equilibrium value of R in the TLD (computed with the SS poThe Journai of Physicai Chemistry. Vol. 78. No. 9. 1974
A.
A, a compressive effect in ice of = 0.11
X. Second Virial Coefficient The second virial coefficient, B ( T ) ,is a measure of the intermolecular pair potential, and is a quantity that can be determined experimentally. The results of section VI11 show that the intermolecular vibrations of the water dimer are quantum mechanical, not classical, in nature, and it follows that a quantum mechanical formulation should be used to calculate B ( T ) . Unfortunately, a rigorous calculation using a quantum formulation is not computationally feasible at this time since a knowledge of the intermolecular vibrational states from the ground state to the dissociation limit is needed for the quantum treatment. In view of this, B(T) has been calculated here for various potentials, using the classical formulation. The classical expression48 for water is
l]r2 sin 19 sin fl d r d8 d 4 d a dfl dy (24) where NO is Avogadro's number, U is the potential energy of interaction between water molecule no. 1 and water molecule no. 2 , the spherical coordinates r, 8, and cp define the position of the center of mass of water molecule no. 2 (the center of mass of water molecule no. 1 being at the origin), a, /3, and y are the Euler angles which specify the rotation of water molecule no. 2 with respect to water molecule no. 1, k is Boltzmann's constant, and T is the absolute temperature. The Euler angle rotations are carried out in a very specific manner; water molecule no. 2 is rotated by a about its c axis, then by (3 about its b axis, and finally by y about its c axis, where b and c are principal axes. The calculation of B ( T ) may be simplified by breaking the integral dawn into four intervals in r as shown in Table 111. U is so positive in interval no. 1 that e - ~ ( r , B : @ , ~ , f l , ~
