J . Phys. Chem. 1990, 94, 5669-5671
-
u* anion states would lead one to expect large (1-2 eV) splittings u* bands.' It is hoped that the present between the various n study, which provides further evidence of large splittings between the C-CI u* anion states of CH2C12, CHC13, and CC14, will promote further experimental and theoretical work to obtain more definitive assignments of the inner-shell and valence excitation spectra of the chloromethanes.
5669
Acknowledgment. This research has been carried out with the support of the National Science Foundation. The calculations were performed on the Chemistry Department's FPSSOOEA computer, funded by the NSF. We thank Drs. P. Burrow and G. Gallup for valuable comments on the manuscript. We also thank N. Nystrom for providing the QCPLOT program used for plotting the orbitals.
"Flexible" Water Molecules in External Electrostatic Potentials Uri Dinur Department of Chemistry, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel (Received: May 14, 1990)
Effective pairwise potentials for "flexible" water molecules are discussed. It is pointed out that the dependence of the atomic charges on the valence bonds and angle (charge flux) leads to a strong coupling of the inter- and intramolecular coordinates. This coupling significantly modifies the electrostatic forces on the nuclei of the water molecule. It is further pointed out that the contribution of atomic dipoles to the electrostatic forces is comparable to the contribution of the molecular polarizability. Ab initio 6-31G** Hartree-Fock parameters for charge flux and atomic dipoles that can be used in water potentials are suggested.
Charge Flux and Intermolecular Electrostatic Forces In recent years there appeared several molecular dynamic simulations of liquid water'-' in which standard intermolecular potentials were augmented with an intramolecular term, thus treating the water molecule as a flexible entity. The interaction between two water molecules is written in these studies as
where Uintra,i denotes the internal energy of molecule i and is a function of the internal valence coordinates, i.e., the bond lengths and valence angle, and where Vinmis a rigid water-water potential that is a function of the intermolecular coordinates. Toukan and Rahmanl and Anderson et al.' have used the SPC model potential4 for the intermolecular coordinates and a Morse potential for the internal O H stretches plus a harmonic force field for the valence H O H angle. Lie and Clementi2 have used a fourth-order expansion of the intramolecular potential and the MCYS potential for the intermolecular interactions. Both potentials are pairwise additive with respect to the intermolecular interactions and thus do not include polarization and three-body effects. In fact, the SPC model was specifically parametrized with the intention to reflect the properties of the water molecule in the mean field that exists in the liquid. Although efforts are being invested in improving current otentials with respect to the representation of it appears that in the application of (1) to molecular dynamics there hides another approximation, possibly more severe than others, and that is the neglect of the coupling between Uinter and Uintra. ( I ) Toukan, K.; Rahman, A. Phys. Reo. B 1985, 31, 2643. (2) Lie. G. C.; Clementi, E. Phys. Reo. A 1986, 33. 2679. (3) Anderson, J.; Ullo, J. J.; Yip, S. J . Chem. Phys. 1987, 87, 1726. (4) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Hermans, J. In Infermolecular Forces; Pullman, B., Ed.; Reidel: Dordrecht, Holland, 1981; p 331. Clementi, E.; Yoshimine, M. J . Chem. Phys. 1976,64, (5) Matsuoka, 0.; 1351. (6) Sprik, M.; Klein, M. L. J . Chem. fhys. 1988, 89, 7556. (7) Rullmann, J. A. C.; van Duijnen, P. Th. Mol. Phys. 1988, 63, 451. (8) Berendsen, H. J. C.;Grigera, J. R.; Straatsma, T. P. J. fhys. Chem. 1987, 91,6269. (9) Watanabc, K.; Klein, M. L. Chem. Phys. 1989, 131, 157. (10) Kuwajima, S.; Warshel, A. J . fhys. Chem. 1990, 94, 460.
0022-3654/90/2094-5669%02.50/0
In order to demonstrate the problem, we first note that the quantity required in molecular dynamics is not the energy but the energy derivatives, namely, the forces on the atoms. From (1) we have for the force on atom i in molecule 1: For example, consider the force on the oxygen atom in water due to an external point charge. This external point charge can be an atom of another water molecule or simply a proton. At sufficiently large distances Qnvsis fairly independent of the external charge and results from the momentary deviation of the molecule from its equilibrium configuration. Because this term is irrelevant to the discussion here, we can further simplify the analysis by considering the case of a water molecule in its equilibrium position so that ViWintra = 0. The force on the oxygen atom is thus purely electrostatic and within a simple point charge model is expected to be
(3)
In (3) p denotes the external point charge and j denotes the atoms in the water molecule. The first term on the right-hand side is the usual Coulombic force while the second term comes from the internal charge j7ux which results from the dependence of the atomic charges q on the internal bond lengths and angle. This is the dependence that is neglected in the above-mentioned potential~'-~ but in fact can be quite large. Figure 1 shows 6-31G** ab initio Hartree-Fock calculations of Fo when an external proton approaches a water molecule. Two different cases are shown. In one the proton approaches the oxygen atom on top, along the line perpendicular to the molecular plane, while in the second case the proton is in-plane and approaches the oxygen along the bisector of the H O H angle. In both cases the water molecule is in its C,, equilibrium configuration so that the force arises solely from the action of the external proton. As is clear from the figure, the forces in the two cases are significantly different. This sharply contradicts the model of rigid point charges which predicts of course that the 0 1990 American Chemical Society
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Letters
The Journal of Physical Chemistry, Vol. 94, No. 15. 1990
LL
40
?
2
2
30
2
-
0
0
Fop
H
/
H 0
\
Fo,r
H
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200
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1
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8
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7
8
9
calculations").
0
e
0
4
Figure 2. Electrostatic force on the oxygen in water due to two external point charges. qp = 0.25 e and qp.-- 1 .O e. The distance O.-P' is always twice the distance 0-P' so that the total electrostaticfield at the oxygen is identically zero. The gradient of the field is however nonzero. The abscissa shows the distance from P. The squares are the a b initio points and the dots are the contributions of the FR atomic dipoles according to (7). mo,* = 0.3715 D;& = 0.1456 e (from 6-31G** Hartree-Fock
0
-
3
'
8
'
I
'
'
-
two forces have to be equal. Also, the ratio of the two forces 1.8, irrespective of the maintain a fairly constant value of distance (Figure I , top). This shows that the large anisotropy is not due to polarization since the latter would decay at large distances. To see how this behavior relates to (3), note that in the C,, symmetry the atomic charges on the hydrogens equal -90/2 each. The flux term may thus be rewritten as
where d is the distance from 0 to H along the bisector of the HOH angle. Comparing with the standard form of the Coulomb force (see the first term on the right hand side of ( 3 ) ) , it appears that the oxygen atom (as well as the hydrogen) carries an effective 'anisotropic" charge as already noted for the coupling second derivatives in dimers." This effective charge is composed of the "true" charge qo and an anisotropic flux term ( V 0 9 o ) d which is zero by symmetry in the direction perpendicular to the molecular plane, but quite large in plane. The ratio between the in-plane and the perpendicular forces is just the ratio of these two "charges"
As is clear from Figure 1, the error associated with the neglect of the charge flux is -80% for the in-plane force at all distances. Although this result was obtained from a 6 - 3 1 G * * ab initio calculation, it is in qualitative agreement with higher level cal(11) Dinur, U.; Hagler, A. T. J . Am. Chem. SOC.1989, 1 1 1 , 5149
culations as well as with the analysis of experimental IR intensities of which imply a substantially anisotropic atomic polar tensors VAp. (It may be verified that the right-hand side of ( 6 ) is equal to the ratio of the following elements of this tensor: (alr,/dxo)/(a~,/azo)). A similar situation is observed for many other polar molecules, and it thus appears that the charge flux term is a first-order contribution to the electrostatic force in general and cannot be neglected in molecular dynamic simulations of flexible molecules. As long as polarization and intermolecular charge transfer are ignored, the forces that arise from charge flux do not influence the translation and rotation of the molecule as a whole. They only distort the molecules about the center of mass. Differences from rigid-body simulations are to be found in the deviation of the molecules from their equilibrium structure with concomitant changes in the pair correlation functions. Results reported later on in this Letter indicate that the structural changes are probably small and that the main effect of the charge flux will be noticed in the dynamics of the liquid,
Atomic Dipoles Another aspect of electrostatic interactions that may be considered in future improvements of molecular potentials is the existence of atomic dipoles. There are various ways of defining these quantities,I4-l6and in what follows we use the formalism presented in ref 17 in which the atomic multipoles are related to the electrostatic f o r c e ~ (hence l ~ ~ ~they ~ are referred to as forcerelated (FR) m ~ l t i p o l e s ' ~ ) . Figure 2 displays the force Fo for a water molecule and two point charges arranged such that the total electrostatic field at the oxygen, Eo, is zero, while the field gradient is not zero. Because is zero, there is no contribution to Fo from the atomic charge of the oxygen (qoEo = 0). Likewise, the polarization is negligible. The main source for the force in this case is the interaction between the gradient of the field and the atomic dipole of the water oxygen and its flux. Utilizing the definition and properties of the FR atomic dipoles," one can derive the following expression for the force in this case:
In (7) j & is the flux amo,,/ax0, and d has been defined in ( 5 ) above. As is seen in Figure 2, the force F0,,is small but not entirely negligible, and is accurately reproduced by ( 7 ) . Yamaoka, Y.; Machida, K.J . Mol. Spectrosc. 1983, 100, 234. Zilles, B. A,; Person, W.B. J . Chem. Phys. 1983, 79, 6 5 . Buckingham, A. D.; Fowler, P. W. Can. J . Chem. 1985, 63, 2018. Spackman, M. A. J . Chem. Phys. 1986,85, 6587. Carpenter, J . E.; Yets, W. T., III; Carpenter, I. L.; Hehre, W. J. J. Phys. Chem. 1990, 9 4 , 4 4 3 . (17) Dinur, U.; Hagler, A. T. J . Chem. Phys. 1989, 91, 2949. (18) Dinur, U.; Hagler, A. T. J . Chem. Phys. 1989, 91, 2959. ( 1 9) Dinur, U. Chem. Phys. Lett. 1990. 166, 21 1 . (12) (13) (14) (15) (16)
The Journal of Physical Chemistry, Vol. 94, No. 15, 1990 5671
Letters I
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TABLE I: 6-31C** Electrostatic Parameters of Water Molecule' $= -1.0
05
~
*
~
.
.
.
0
.
H
-0.787 0.3935
0.25
-0.25
0.286 -0.143
-0.068 -0.068
0.0075
Punits: qo, a, and @ in e; jA,in e/A and jA, in e/Az. The 631G** Hartree-Fock equilibrium bond length and angle are 0.9431 A and 105.95'. The charges are given by qo = qoo + jo,(A + A'); 9 H = qHo jHJ, where j,,, (A = 0, H) is the flux aqA/arand where A
+
denotes a bond stretch. Higher order flux terms and angle-dependent fluxes were found to be sufficiently small to be neglected. (However, the largest quadratic term-jAJf-is included in the table.) The atomic dipoles are given by ino = a(rOH + IOH'); i n H = - a r O H + @rHH', where rOHis the vector from 0 to H, rHHP is the vector from H to H', and a and @ are functions of the internal coordinates. A crude but sufficient approximation for present purposes is to assume constant a and 0.
ab initio -20
$= 1.0
I -5 2.0
D
3.0
5.0
4.0
6.0
7.0
8.0
R, A
Figure 3. Multipolar and polarizability contributions to the electrostatic
with a point charge model to account for both the dipole and the quadrupole moments by adjusting the location of one of the charges, as done in the Bernal-Fowler,Zi MCY,5 and TIP4Pz2 potentials. Indeed, the FR atomic dipole of the water oxygen, as obtained from 6-31G** ab initio calculations,18has its negative pole pointing toward the hydrogens, which is consistent with the shift of the negative point charge in the above-mentioned fourcenter potentials from the oxygen toward the hydrogens. Including atomic dipoles in an SPC-like potential can thus modify the quadrupolar interactions while keeping the three-center form.
force on the oxygen due to an in-plane external point charge (orientation
6-31G**Electrostatic Parameters for Flexible Water
as in Figure I ) . The solid line interpolates between the forces obtained from the ab initio calculation^.^^ Open circles: contributions to the force from atomic charge and charge flux. qo = -0.7875 e; jo, = 0.5817 e/A. Open squares: contributions from atomic dipoles and dipolar flux (parameters as in Figure 2). Open diamonds: polarizability contribution. axx= 0.7267 A3; da,,/dx = 1.501 A2. (All values are from ab initio 6-31G** Hartree-Fock c a l ~ u l a t i o n s ~ ~ Filled - ~ ~ ) . circles: sum of all contributions. Bottom: the magnitude of the external charge is 1 .Oe (as in Figure I ) . Top: the external point charge is -1.0 e.
Atomic charge flux parameters for gas-phase water have been previously derived by Yamaoka and Machida7 based upon analysis of 1R intensities. The related experimental atomic polar tensor and 4-31G ab initio calculations of these tensors for the water monomer and dimer have been published by Zilles and Person8 Here we report charge flux parameters and atomic dipoles as obtained from 6-31G** ab initio calculations. This particular basis set yields for the water molecule (in its 6-31G** equilibrium geometry) a dipole moment and molecular quadrupole moments that are very similar to those of the TIP4P potential. In addition, the equilibrium structure of water as calculated with the 6-31G** basis set is also much closer to that of TIP4P than to the SPC structure. It appears therefore that the 6-12 potential of TIP4P plus electrostatic parameters taken from 6-3 1G** ab initio calculations and stretching and bending force constants taken from refs 1 and 3 will result in a three-center potential that has similar properties to the original TIP4P and can be applied to flexible water molecules. The increase in computational demand is somewhat offset by the elimination of the fourth site. The 6-31G** electrostatic parameters are listed in Table I. Taken together with the above-mentioned parameters, they yield a nearly linear water dimer in which the O-.O distance is 2.69 A and the angle 0 of the O-.O line with the bisector of H O H is 65.8' (TIP4P values are 2.75 A and 46'). These structural parameters are found to be sensitive to the atomic dipoles. Excluding the latter from the potential yields an 0-0 distance of 2.78 A and an angle B of 21', which is fairly similar to the results of the SPC model potential. On the other hand, the exclusion or inclusion of the flux term influences only the bond lengths and only by 0.005 A. The above outlined potential improves current potentials in the electrostatic regime. Further modifications regarding the charge flux along the hydrogen bondI3 as well as the dependence of the 6-1 2 parameters on the internal coordinate are still required for a proper description of the close contact regime.
In general the electrostatic field is not zero and the contribution of atomic dipoles to the electrostatic forces in water is much smaller than that of the atomic charges. This is both because the latter contribution behaves like E while the contribution of the atomic dipoles behaves like V E and because the atomic charges in water are larger in magnitude than the atomic dipoles. However, the dipolar contribution may be similar in magnitude or even larger than the effect of polarizability since the latter behaves like E2:
In ( 8 ) E, is (arbitrarily) evaluated at the center of mass. For example, in the case of an external point charge at a distance R the charge, dipole and polarizability contributions to the force respectively behave like RZ, R3,and Figure 3 (bottom) shows the ab initio force FoJ from Figure 1 along with the separate contributions from the charges (9,atomic dipoles (7), and the molecular polarizability (8). Clearly, the contribution from the charge is significantly larger than the other two. The latter terms are similar in magnitude but opposite in sign and thus cancel each other. This situation changes when the external point charge is -1 e instead of 1 e (Figure 3, top). All contributions to the force due to the permanent atomic multipoles change sign, while the contribution of the polarizability remains unchanged. Consequently, the polarizability and the atomic dipole contributions do not cancel each other in this case. As a result, the charge and charge flux alone are no Ion er sufficient to describe the force at distances smaller than 5 . In addition to the above considerations we note that the FR atomic dipoles are necessary for the reproduction of the molecular quadrupole moments17which have been observed to influence the dielectric constant of ~ a t e r . ~ *As ~ Ois well-known, it is possible
e.
1
(20) Carney, S. L.; Patey, G. N. Mol. Phys. 1982, 47, 1129. (21) Bernal. J. D.; Fowler, R. H. J . Chem. Phys. 1933, I , 515. (22) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. J . Chem. Phys. 1983, 79, 926. (23) Amos, R. D.; Rice, J. E. CADPAC: The Cambridge Analytic Derivative Package, issue 4.0, Cambridge, 1987.
