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P. P. S. Saluja and H. A. S c h e r a g a
(24) S.W. Benson and E. Gerjuoy,J. Chem. Phys., 17,914(1949). (25) W, E. Strickfaden and L. desobrino, Can. J. Phys., 48, 2507 (1970). (26) H. C. Longuet-Higgins and E. Widom, M o l , Phys., 8,549 (1964). (27) E. J. Alder and C. E. Hecht, J. Chem. Phys.. 50,2032 (1969). (28) H. Reiss, Advan. Chem. Phys.. 9, 1 (1965). (29) R. M. Gibbons, Mol. Phys., 17, 81 (1969). (30) G. R. Dowling and H. T. Davis, J. Stat. Phys., 4,1 (1972). (31)A. lsiharaand T. Hayashida, J . Phys. SOC.Jap.: 6,46 (1951). (32)J. 0.Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids," Wiley, New York, N. Y. 1954,p 184. (33) M. Rigby, J. Chem. Phys., 53, 1021 (1970). (34) M . Rigby, J. Phys. Chem., 76,2014 (1972). (35) J. F. Connolly and G. A. Kandalic, Phys. Fluids, 3,463 (1960). (36) J. G. Eberhart, W. Kremsner, and E. Hathaway, Argonne National Laboratory Report No 4NL-7923,15 (1972). (37) H. Eyring and M. S Jhon, "Significant Liquid Structures," Wiley New York, N, Y., 1969, (38) H. Eyring and R. P. Marchi, J. Chem. Educ.. 40,562 (1963). (39) M. S.Jhon, J, Grosh. T. Ree, and H. Eyring, J. Chem. Phys., 44, 1465 (1966). (40) M. Biander, private communication, 1973. (41) R . E. Apfel, Nature (London), Phys. Sci., 238(82),63 (1972). (42) L. J. Briggs, J . Appl. Phys., 21,721 (1950). (43) L.J. Briggs, J. Appl. Phys., 24, 488 (1953). (44) J. L. Katz, d. Chem. Phys., 52, 4733 (1970). (45) G. M. Pound, J. Phys. Chem. Ref. Data, 1, 119 (1972).
(46) R. M. Gibbons, J. Phys. Chem., 76,1479 (1972). (47)J. A. Barker and D. Henderson, J. Chem. Phys.; 47,4714 (1967). (48)J. Hansen, Phys. Rev. A, 2, 221 (1970). (49) L. Veriet, Phys. Rev., 159,98 (1967). (50) i. R. McDonald and K. Singer, J. Chem. Phys., 50,2308 (1969). (51) J. P. Hansen and L. Veriet, Phys. Rev., 184,151 (1969). (52) G. A. Kozlov and A. A. Ravdel', Koiloid-Z., 33,847 (1971). (53) I. Karutzand I. N. Stranski, Z. Anorg. Ailg. Chem., 292,330 (1957). (54) E. Raedder, Science, 155,1413 (1967). (55) M. Blackrnan, S.J. Peppiatt, and J. R. Sambles, Nature (London), Phys. Sci., 239,61 (1972). (56) J. R. Asay and B. M . Butcher, Bull. Amer. Phys. SOC.. Ser. / / , 17, 1105,Abstract ID5 (1972). (57) E. J. Fuller, T. Ree, and H. Eyring, Proc. Nat. Acad. Sci. U.S., 45, 1594 (1959). (58) K. Liang, H. Eyring, and R. P. Marchi, Proc. Nat. Acad. Sci. U.S., 52, 1107 (1964). (59) D. R. McLaughlin and H. Eyring, Proc. Nat. Acad. Sci. U.S., 55, 1031 (1966). (60)J. Grosh, M. S. Jhon, T. Ree, and H. Eyring, Proc. Nat. Acad. Sci. U. S.,58,2196 (1967). (61) A. Fuiinski, Acta Phys. Polon., A37, 177 (1970). (62) A. Fulinski,Acta Phys. Polon., A37, 185 (1970). (63) A. Fulinski,Acfa Phys. Polon., A39, 181 (1971). (64) A. Fulinski,Acta Phys. Poion., A40,221 (1971). (65) A. Fulinski, Phys. Left., 31A,176 (1970). (66) A. Fulinski and M. Jurkiewicz, Phys. Lett., 32A,126 (1970).
ion-Water Interactions in the Gas Phase' P. P. S. Saluja and H. A. Scheraga" Departmenf o f Chemistry, Cornell University. Ithaca, New York 74850 (ReceivedJune 4, 7973)
A quasi-experimental potential function for ion-water interactions in the gas phase is derived from massspectrometric and crystallographic data. Since ab initio computations on large ion-water complexes are not yet available. except for only a few systems, an attempt was made to use a semiempirical molecular orbital method to compute the energies of such complexes to obtain a theoretical basis for the potential function. Because the agreement between the values calculated by the molecular orbital method and the observed ones is poor, we rely on an empirical procedure to obtain a potential function for the interaction between an ion and the surrounding water molecules.
Introduction For an understanding of the properties of dilute aqueous solutions of electrolytes it is essential to know the mutual effects of the ion and the surrounding water molecules on each other. Thus, it is logical to consider first the strong interactions of the ion with the nearest solvent molecules and then examine the influence of the ion on the structure and energies of the water molecules a t greater distances. Therefore, in this paper, we compute the interaction energies between an ion M and those n water molecules next to it in the gas phase. The modification of such M(H20), complexes by the surrounding water molecules in the liquid phase is a separate problem, which is presently under investigation in our laboratory. Since ab inztio computation^^-^ on large hydrated complexes have been carried out thus far5s6 for only a few monohydrate complexes, a comparison of a b initio results with experiment7-11 is not yet possible. Because of this, we used a semiempirical molecular orbital m e t h ~ d ~ (CNDO/S) to compute the energy of M(H20), complexes of varying n for several cations and anions. While a few calculations have been reported in the literature for severThe Journal of Physical Chemistry, Voi. 77, No. 23, 1973
al of the complexes treated here, the present paper extends these to other complexes not considered heretofore. Our calculations are summarized e1~ewhere.l~ The poor agreement between the calculated14 and observed vaiues7-l1 is discussed in light of the inadequacies of primarily the C N D 0 / 2 method, and an empirical procedure is presented to obtain potential functions for the interaction between an ion and the surrounding water molecules. Potential Function from Experimental Data Ion-water interaction energies have recently7-11 become available from mass-spectrometric experiments on the energetics of ion-water equilibria in the gas phase. A combination of these results14 with equilibrium ion-water distances, taken from crystal ionic radii,15 is used here to deduce a potential function ( e . g . , A,/rP - B,/i-q)> the coefficients of which may be determined provided that the form ( i e . . p and q ) is obtainable from theoretical consid~erations. J ~ In the following section, we use an approximate molecular orbital method to determine the values of p and q, and then deduce empirical values of A , and R , for various hydrated ions.
Ion-Water
2737
!nteractions in the Gas Phase
Molecular Orbital Theory The method used here is the CNDO/2 procedure of Pople and coworkers.12 The detailed hydration models and geometrical parameters for the ion-water complexes investigated here are given in the supplementary microfilm edition of this j o ~ r n a 1 . l ~ For any complex M(H20),, the energy is calculated as a function of E^ (the same value of r being kept for each water molecule for a particular value of n ) , and the minimum-energy values (1.0)~and ( E o ) , determined. The computed values of (Eo), are augmented by the dispersion energy16J7 (to take account of correlation effects); the dispersion energy is calculated for all pairs (i.e., ion-HzO and HzO-HzO) in each complex, using London's formulal8 and experimental values of polarizabilitie~l~ and ionization potentials.20 In any event, the dispersion energy is very small compared to being approximately 0.4 kcal/mol peI pair interaction. In the remainder of this paper, (Eo), includes the dispersion energy. Then 9En-1,, for the equilibrium M(H-O),,-i
$.
H@
r=t
M(HD)n
TABLE I : Coefficients A, and B, in the Empirical Potential An/r'' - B,/r6 Obtained from Experimental Dataa
System
Li+(H*O)n
Na"(HPO),,
F-(HzO)n
for n = 1, 2 , 3, . . . . The values of ENzO and EO)^ ( i . e , for n = 0) are those of Table I of the supplementary material.14 While there is no vibrational excitation of an H20 molecule at room temperature, we have no information about the intermolecular vibrational excitation in the complex; in the absence of this knowledge, we shall assume that there is none, and hence that the computed values of A E n . I,, pertain t o 0°K. Since the experimental values? of LHn-1,, are independent of temperature in a range which includes room temperature, we assume that the observed values also pertain to 0°K. Thus, since the I ( p V ) term is small compared to AH,-I,~, we may compare the computed values of AE,-1,, directly with the experimental values of AHn-1.n. Results The results for AE,-l,, and ro obtained here are given in Tables 111-V as supplementary material in the microfilm edition of this j0urna1.l~Table I11 contains results for complexes of Li' and Na- containing one to six water molecules; Table IV contains the same information for complexes of F- (with one to four water molecules), C1(with one to four water molecules), and C N - (with one water molecule); Table V contains data for complexes of Be2+, Mg2+, and A13+ (all with one to four water molecules). The computed values of AErL-l,,,become less negative with increasing n. This indicates that the binding of water molecules in the hydrated complex becomes weaker as n increases. Although the results for stabilization energies, AE,-l,,, are in qualitative agreement with experiment, the magnitudes are somewhat overestimated by the CND0/2 method. For example, the computed values of AE,-,,, for Li-(H20), with n varying from 1 to 6 are -45.7, -42.3, -40.7. -37.0, -27.8. and -27.6 kcal/mol, respectively; and the corresponding experimental values9 are -34.0, -26.8, -20.7, -16.4, -13.9 and -12.1 kcal/ mol. Thus, the computed values are on the average 17, 15, 16, and 9 kcal/mol more negative than the experimental values for the hydrated complexes Li+(HzO),, Na+(H[zO),, F-(H20),, and Cl-(H20),, respectively. The
1 2 3 4 5 6 1 2 3 4 5 6 I'
2 3 4 5
(1)
is computed as
n
CI-(H20)n
1
CN-(H*O)n
2 3 4 1
An X 10-6, kcalA'2
0.139 0.245 0.330 0.397 0.454 0.503 0.681 1.243 1.691 2.082 2.431 2.735 4.553 7.796 10.473 13.111 15.690 15.679 30.880 44.884 58.170 61.161
Bn X 10--3. kcalAG
4.352 7.654 10.304 12.403 14.182 15.731 8.084 14,754 20.076 24.725 28.868 32.472 20.599 35.274 47.386 59.321 70.990 28.699 56.452 82.052 106.340 58.104
An/n X 10-6, kcal A''
0.139 0.122 0.110 0.099 0.091 0.084 0.681 0.621 0.564 0.521 0.486 0.456 4.553 3.898 3.491 3.278 3.138 15.679 15.440 14.961 14.543 61.161
Bn/n,X
io-,
kcalAG
4.352 3.827 3.435 3.101 2.836 2.622 8.084 7.377 6.692 6.181 5.774 5.41 2 20.599 17.637 15.795 14.830 14.198 28.669 28.226 27.351 26.585 58.104
nThe values of 1 0 are taken from crystal ionic radii,15 and the experimental data on the equilibrium energy are taken from ref 7-1 1.
most likely explanation for the overestimation of the ionwater interaction energy by the CNDO/2 method is given in the Discussion section. Discussion The computed values of AE,z.-l.n and ro agree with those from the C N D 0 / 2 calculations of some worker^,^^-^^ but not with the results obtained for a few systems by ab initio treatment^.^-^ ,24-27 The computed vaiues of A E n - l , , are more negative than the experimental ones in all cases. This overestimation of the stabilization energies, AEn-1,,: of the hydrated complexes by the CNDO/2 procedure arises from the approximations in the method, the values assigned to the parameters,21 and also to some extent on the size of the basis set used. First, no attempt was made to optimize the program parameters at this stage. For example, in a study of heats of formation of hydrocarbons and their cations, Wibergz8 achieved better agreement with experiment after reparameterization. This was not attempted here because the parameterization (particularly in orbital exponents) for ions is probably different from that for neutral species, and no criteria are available for assessing the validity of the orbital exponents for ions. Second, the basis sets used for the CNDO/2 method are small (valence basis set), i.e., the IS and 3d orbitals were omitted from the second row atomic basis sets (Li, Be, 0, C, N). Lastly, the detailed ab initio results for nonohydrated complexes of L i t , N a f , and K + obtained by Kistenmacher, e t a1.,3 indicate that the neglect of three- and four-center integrals in the CNDO/2 approximation13 gives rise to an overestimation of two-center terms. A breakdown3 of the total energy of the hydrated complex into one-, two-, three-, and four-center contributions shows that threecenter terms are mainly responsible for the repulsion in The Journal of Physical Chemistry, Voi. 77, No. 23, 11073
2738
Li+(HZO) and Na+(HzO) for ion-water distances in the range of 2-3 ti. Thus, the use of the CKDO/P method appears to lead to spurious attraction a t such short ionwater distances in the hydrated complexes studied here. In order to account for these higher terms explicitly, it would be necessary to modify the CND0/2 method and use a larger basis set.17 But, if one is going to use a larger basis set (requiring more computer time), one might as well use a more exact a b initio method. However, a b initio calculations on large complexes are prohibitively expensive,j and have thus far been carried out extensively2, 3 only for the species LiT(HzO), NaT(HZO), and KT(HzO); thus, a comparison of a b initio results with experiment will not be possible until the computations on higher ionwater complexes become available. As can be seen from Tables 111-V of the supplementary material,l* the computed values of ro are larger for cations and smaller for anions than the experimental values of ro. This discrepancy is attributed to the same shortcoming of the CNDO/Z procedure, which led to discrepancies between the calculated and observed energies. Since the trend in AE+l,n with n appears to be similar in both the theoretical and experimental results, we may resort to an empirical method to obtain the dependence of AE,L-I,non r. We assume that the functional form for the dependence of the CNDO/2 values of AE,-l.,L on r is correct, even though the abso!ute values are not, and represent this function as
Equation 3 represents the total interaction energy of the hydrated complex WI(HZO)~ when n water molecules are brought into the first hydration shell of, and equidistant from, the central ion. Using the Cr\rD0/2 results, A , and B , can be expressed in terms of p, q, ( r o ) n , and Z n ( A E o ) n - l , n .Then optimization of p and q for all of the CXD0/2 data gives the best values as 12 and 6, respectively. Having determined p and q from the CNDOI2 results, we can compute empirical values of A E n - I , n , 1 e , of An and 13, by requiring that the empirical curve agree with the experimental values of AE,-1,, of Kebarle, et and of ro from crystal ionic radii.15 The resulting values of A, and B , are shown in Table I. It can be seen that ( A n / n - A l / 1 ) and (B,/n - Bl/l) become more negative as n increases, ~ e . the , stabilization energies are nonadditive, probably because of the combined effects of H20*.-HzO interactions in the first hydration layer and the increased shielding of the ionic chargez9 as HzQ molecules are added stepwise to the first hydration layer of the ion. These empirical functions will be of use in considering the effect of additional hydration layers when the M(HzO), complexes are placed in liquid water.
The Journal of Physical Chemistry, Vol. 77, No. 23, 1973
P. P. S. Saluja and ti. A.
Scheraga
Achnouledgment. We are indebted to Dr. Frank A. Momany for helpful discussion. Supplementary Material Available: A listing of' the experimental data on interaction energies, the hydration models, geometrical parameters, and results of the computations will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 x 148 mm, 20X reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from the Journals Department, American Chemical Society, 1155 16th St., N.W., Washington, D. C. 20036. Remit check or money order for $5.00 for photocopy or $2.00 for microfiche, referring to code number J P C -73-2736.
References and Notes (1) This work was supported by research grants from the National lnstitute of General Medical Sciences of the National Institutes of Health, U. S. Public Heaith Service (GM-14312). and from the National Science Foundation (GB-28469x2). (2) E. Clementi and H. Popkie, J. Chern. Phys . 57,1077 (1972). (3) H. Kistenmacher, H. Popkie, and E. Clementi, J. Cilem. Phys.. 58,
1689 ( 1 973). (4) W. P. Kraemer and G . H. F. Diercksen, Theor. Chim. Acta, 27,265 (1972), (5) Ab initio computations on large complexes are prohibitively expensive. However, Clementi6 has recently pointed out that these computations for higher ion-water complexes, such as MiH20)6, can now be carried out more rapidly and thus less expensively with a revised version of his computer program. Such ab initio results have not yet been reported in the literature, and therefore we have resorted to an approximate molecular orbital method. (6) E. Clementi, Proc. Nat. Acad. Sci. U.S., 69, 2942 (1972). (7) P. Kebarle in "Ions and Ion Pairs in Organic Reactions," Vol. I . M. Szwarc, Ed., Wiley-lnterscience, New York, N. Y.. 1972, Chaper 2. ( 8 ) I. Dzidic and P. Kebarle, J. Phys. Chem., 74. 1466 (1970). (9) M. Arshadi. R. Yamdagni. and P. Kebarle. J. Phys. Chem., 74, 1475 11 9701 \ , - . - , .
(10) J. D. Payzant, R. Yarndagnt. and P. Kebarle, Can. J . Chem., 49, 3308 11971). (1 1) S. K. Searies and P. l
