ORIENTATION OF WATERMOLECULES IN
Dec., 1963
the p-integrals in eq. 3.7 are found to be
THE
FIELDOF
2773
=
8Z4/(4Z2
+ s2)*; 2 = 1.0 a.u.
(5.1)
Half an electron is distributed in each of the 1s orbitals of the two nuclei irt the independent atom model. The intensity functions presented in this paper have been divided by the background function
and
2
i"-l
b,(Z) j,(Z). m
=
odd
TABLE I11 THET'ALUCS m
0 1 2 3 4 3 6 7
O F THE COCFFICIEXTS
L1'awA AiYD
2m-1bt,l
z" -1bm
Pam
1 32884 1 61303
-0 71047
-0 45895 0 07351 0 04201 -0 00254 -0 00137
The integraiions over A in eq. 4.7 were evaluated by numerical integration. The interval of the integration and the maximum value of X were chosen as Ah = 0.02 and A,, = 7.00, respectively, leading to results correct to three figures. 5. Results and Conclusions The intensity of electrons scattered by Hz+ has been calculated by the use of various wave functions together with the intensity given by the independent atom model within the framework of the first Born approximation which should be reasonably good in the kilovolt energy range. The intensity expression for the independent atom model is given as (4/a2?"'s4) [3 - 4 j H 0
+ 2(1 - 2fHo) sin
I B :=
(4.13)
The values of imn,(Z) and irn-lbm(Z)are given in Table 111.
=
ALKALIIOK
where fHo
I
AN
sTAB/sTAB]
(5.0)
(4/a2T2s4)[3
- 4fHo]
(5.2)
in order to level the intensity in the region of small s and the results are shown in Fig. 1. The, electron distribution in the molecule given by the various wave functions has been shown in Fig. 2. The effect of binding is seen in the decrease of jntensity in the region of small s relative to the iiitensity given by the independent atom model. This is the same result which was obtained in the case of molecular hydrogen,lGand it is attributed to the over-all contraction of the electron distribution which results from bond formation. Although the one-center ware fuiiction ghes SL more diffuse distribution in the neighborhood of the nuclei than the exact wave function, it still gives excellent agreement with the exact results in the intensity function. This indicates that the electron scattering is not sensitive to the shape of the electron distribution at the nucleus. The simple variation function by Finkelstein and Horowitz gives the general trend of the effect of binding and the magnitude of the effect given by this function is on the average 76% of that given by the exact wave function, over the region s I5. The results obtained in this investigation also suggest that the binding energy may be an adequate criterion for judging to what extent a particular wave function will give a good description of the electron scattering. Acknowledgments.--We wish to thank the Indiana University Research Computing Center for the use of their facilities. We also wish to thank Mrs. Joanne Knight for her help in preparat,ioii of the manuscript.
THE ORIENTATIOK OF WATER MOLECULES IN THE FIELD OF *4N ALKALI 10s BY FRED VASLO'CV Chemistry Division, Oak Ridge i'ational Laboratory, Oak Ridge, Tennessee1 Receiced J u l y 11, 1963 Using the general series expansion for the potential due to an arbitrary charge distribution, and the quadrupole moment of a water molecule deduced by Buckingham, it is shown that the minimum potential energy of small positive ions is not on the dipole axis of a water molecule but a t a substantial angle to that axis. Making two different assumptions of the components of the quadrupole tensor of the water molecule, the electrostatic energies of the alkali ions in the field of a water molecule are calculated as a function of the angle of the ions a i t h the dipole axis. It is found that over large regions of the hemisphere of a water molecule, away from the hydrogen atoms, the variation in energy of an ion is smaller than the energy of breaking a hydrogen bond. Since a hydrogen bond is also somewhat flexible in length and direction, it seems reasonable that for small ions hydration is not inconsistent with normal structural groupings 1 hat may exist in liquid water. A brief discussion is given of some possible consequences of these calculations to various hydration properties.
With the exception of Verwey'sBa calculation of ionic hydration energies, it has generally been considered (1) nls pappi based ul)oI1 work Iwifoimed a t o a k Ridge xiational laboratory, nkiicli IS operated by Union Carbide Coli3oration for tile .itonnc Energy Commission. ( 2 ) (a) E. J. W.Veraey, Rec. h a u . chzm., 61,127 (1942): (b) J. 1). BernaI and R. H, FowZer, J . Chem. Phys., 1, 515 (1933).
that in aqueous solutions of electrolytes, the dipole of a water molecule adjacent to a small or highly charged ion points directly at the center of the In such a model the water molecules surrounding the ion form a tetrahedron Or octahedron the primary hydration shell of the ion.
2774
FRED J T
The evidence for this type of structure is not experimental since experiments have thus far only been able to say that the aforesaid theory is not inconsistent with the findings. The foundation of this hydration concept is theoretical and rests on energetic considerations of the interaction of the dipole of the water molecule with the spherically symmetrical charge of the ion. At the interatomic distances of ion and oxygen occurring in ciystal hydrates and with the experimental dipole moment of a water molecule, the ion-dipole interaction energy according to this concept is large enough to disrupt any inherent structure of liquid water and limit the ion to a ielatively small angular region around the dipole axis of each adjacent mater molecule. While this is a plausible theory, from the point of view of electrostatic theory, there are serious defects. Firstly, the water molecule is a rather complex system of electric charges combining the diffuse charge of the orbital electrons with the point charges of the nuclei. At the short distances involved, neither an ideal point dipole model nor a representation of the individual orbital charges by point charges can give a valid description of the electrostatic field surrounding the water molecule. Within the limits of a classical electrostatic treatment and where polarization and static effects can be treated separately, a rigorous description of the electrostatic field requires a series expansion either in spherical harmonics or the closely related expansion in terms of m ~ l t i p o l e s . ~The latter treatment has recently been applied to calculate hydration energies of ions by Buckingham4 mho in doing so estimated the quadrupole moment of the water molecule. Accepting the quadrupole moment estimated by Buckingham and the rigorous potential energy equation, it is shown later in this paper that for small positive ions, the position of minimum potential energy does not lie on the dipole axis of the water molecule but at a substantial angle to that axis. A second defect of the classical hydration picture is actually a consequence of the inexact electrostatic treatment used. For ion-water molecule distances occurring in the crystal hydrates, and for small ions as Li+ and Ka+, hydration energies calculated on the basis of the water dipole moment alone are substantially too large. For example, Bernal and Fowler2bin their calculation of ionic hydration energies used an overly large energy of making a hole in the water, for compensation, which was later strongly criticized by Verwey, who attempted calculating hydration energies only for larger ions. I n another type of calculation, Latimer, Pitzer, and Slansky6 had to add an arbitrary length to the ionic radii in order to obtain reasonable hydration energies. Providing that the various charge moments can be estimated and that the basic assumptions are valid, then the series expansion gives reasonable energy values without further arbitrary assumptions. As already stated, a consequence of this treatment is that the most stable energetic position of a positive ion is not on the dipole axis. Based on this treatment, it is firstly the purpose of this paper to point out the aforementioned (3) J. A. Stratton, “Eleotiomagnetic Theory,” XcGuaw-Hill Book Co., lnc., New I’ork, N. P., 1941, p. 172. (4) A. D. Buckingliam, Dzscussions Paradag SOL.,24, 151 (1957). ( 5 ) W. M . Latimer, K. S.Pitzer, and C . M. Slansky, J . Chem. Phys., 7 , 108 (1939).
~
s
~
~
~
Vol. 67
~
fact, secondly to calculate for several plausible values of the quadrupole tensor components (consistent with Buckingham’s value of the quadrupole moment) potential energy profiles for the alkali ions in the static electric field of a water molecule, and thirdly to briefly discuss some possible consequences of the energy behavior on the hydration structure and other properties of the ions. The model used here is essentially that of the ion and ad3acent water molecule under vacuum (or spherical cavity in bulk water), and the calculation is only of anisotropic contributions to the energy rather than of the total hydration energy as Buckingham4 has done. It is implicitly assumed that the dipole and quadrupole terms are the only large anisotropic contributions to the energy and following Buckingham, that both the polarization interaction (due to the polarizability of the ad,jaceiit water molecule) and the interaction of the ion with bulk water outside the cavity are isotropic. According to classical electrostatic theory, the potential a t a point outside an arbitrary charge distribution is given by a series expansion in terms of the net charge, alrd the dipole, quadrupole, and higher moments of the di~tribution.~For a water molecule, a system of coordinates can be taken with the Z-axis along the dipole (twofold axis), the X-axis is a t right angles to the plane of the molecule, and the Y-axis in the plane with the 0 nucleus as the origin. I n this coordinate system the quadrupole tensor is diagonal with components B X X , Byy, and BZZ. For a continuous charge distribution, a typical component is given by BXX = ‘/JpXZdV where p is the charge density and dV the volume element with the integration over the entire charge volume of the molecule. Since the direction angles +x, + y j and 4~ of the field-point vector with respect to the coordinate axes are not independent, the tensor is further reduced to two components BA = Bzz - OXX and OB = Ozz
- eyy.
The potential, v, to the quadrupole term of the series is then given by eq. 1 where r is the distance from the origin to the field point, and p~ is the dipole moment.
8 B ( 3 cos2 4Y
- 111 (1)
This equatioii is identical mith Buckingham’s eq. G for 0 net charge. Mathematically the choice of origin, as stated, and following Buckingham4 is arbitrary, requiring only that the various moments be evaluated from the same origin, values of moments higher than the dipole being dependent on the origin. It is not obvious from Buckingham’s4method of estimation of the quadrupole moment that his choice of origin is consistent with the aforesaid requirements or could be used as a basis of calculation here. That Buckingham’s choice is reasonable, however, is shown by the apparent consistency of his calculated energy values for both positive (47 = 0) and negative ( 4 ~= 9) ions of very different sizes, which require that the origin chosen has been a consistent one. From eq. 1 the angles of the energy minima can be evaluated by differentiation with respect to angle and setting the derivatives equal to 0. For a fixed value
Dec., 1963
ORIENTATIONOF WATERMOLECULES ISTEE FIELD OF AX ALKALIIori
2775
of cpz, the energy minimum is in the plane of the larger of 0.k (X-axis) or 8~ (Y-axis) and the Z-axis, or for Ba = BB, independent of the angle around the Z-axis. Assuming that Ba is the larger component, then in the / X-Y plane, if r p / 3 0 ~< 1 an energy minimum exists si J 2 - 1.2 and at the minimum, cos $21 = r p / 3 0 ~ . Also for r p / 3 0 ~< 1 energy maximurns exist a t +Z = 0 and T. $-kO I n order to actually evaluate the variation of energy 0 with angle, it is necessary to make assumptions of t,he E >-0.0 relative values of 0~ and 813 consistent with this sum being 3.9 X e.8.u. as given by B ~ c k i n g h a m . ~ P -0.6 Since actual values are unknown, what will be attempted here i s to select values which “bracket” the range -0.4 of plausible vahes of 0~ and 8~ with a correct value presumably lying somewhere in this range. -0.2 A first possible choice is that of Bi\. = On = 1.95 X e.s.u. and which attributes cylindrical symmetry 0 10 20 30 40 50 60 70 00 90 $00 to the electric field of the water molecule. Since the +I . energy minimum lies in the plane corresponding to the Fig. 1.-Potential energy of Li”, Ka+, and Cs+ ions in tlie larger of BA and 8~ (with the Z-axis), this choice give the electrostatic field of a water molecule in the X-Z plane: ( I ) Li+, smallest possible larger component and gives, therefore, BA = 2 6 ~ ;(2) L i t , BA = BB; (3) Na+, 6a = 2 6 ~ ;(4) N a + @A = the s~na,llestpossible deviation of the energy minimum OB; ( 5 ) COS 62; (6) CS“, 6.4 = 2633; ( 7 ) cS+, 6A = OB, direction from the dipole axis. This value is taken as TABLD I the lower limit, of 0g. POTENTIAL E K C R GOF Y ~IONS IN FIELD OF WATERMOLECCLE Since water is not cylindrically symmetrical, two -Cylindrical symmetry-. C-OA = 2 8 symmetry-~ further choices were tried giving increasing degrees of dz i Z positive potential along the Y-axis. One choice in (mm.), bin.), Ion Z-axis deg. E,,, X-axis cieg. E,,, X-axis = 3.9 X e.s.u. and 0~ = 0 was found to which Li+ -1 05 51 62 -1 69 -1 22 -1 29 -0 55 give unreasonably large negative potentials along the S a + -0 89 42 -0 96 36 56 -1 17 -0 72 X-axis and was therefore discarded. A third choice of K + - .72 30 - 74 - .23 50 -0.85 - .46 BA = 2.6 x 1 0 - 2 6 e.s.u. and 0~ = 1.3 X e.s.u. or Cs+ - 61 12 - 61 - .16 47 - .66 - 32 BA = 2O3 gives results which are more reasonable alIn ergs/molecule >< lo’*. though still perhaps high, and has therefore been used as a tentative upper bound of BA. energy of the ion is low are much more extended than in The energies3 are cadculated by substitution in eq. 1 the point dipole model. Aside from the curve for Li+ of tlie values of the quadrupole components, the dipole ion in OA = 243 symmetry, which shows a fairly sharp D. and the Pauling radii6 and large energy minimum, the curves are rather flat moment of 1.85 X of the ions rather than the Goldschmidt radii used by and low energies extend over large arcs of the X-Z Buckingham. These radii are 0.60, 0.95, 1.33, and plane. 1.69 A. for Lid, Na+, I
