2566
Gary W. Schnuelle and David L. Beveridge
Faculty Research Award, and the award of a CUNY Research Associate position to G . W. Schnuelle. References and Notes (1) C. J. F. Bottcher, "Theory of Electric Polarization", Vol. I, Elsevier, Amsterdam, 1973. (2) M. Born, Z.Phys., 1, 45 (1920). (3) L. Onsager, J. Am. Chem. SOC.,58, 1486 (1936). (4) J. G. Kirkwood, J. Chem. Phys., 1, 351 (1934). (5) T. Halicioglu and 0. Slnanoglu, Ann. N.Y. Acad. Sci., 158, 308 (1969); D. L Beveridge, M. M. Kelly, and R. J. Radna, J. Am. Chem. SOC.,06, 3769 (1974) (6) B. Linder, Adv. Chem. Phys., 12, 225 (1965). (7) 0.Sinanoglu in "Molecular Associations in Biology", B. Pullman, Ed.,
(8) (9) (IO) (11) (12) (13) (14) (15) (16) (17)
Academic Press, New York, N.Y.. 1968. p 427 ff; R. J. Kassner, J. Am. Chem. SOC.,95, 2674 (1973). R . M. Noyes, J. Am. Chem. SOC.,84,513 (1962). W. Kauzmann, Adv. Protein Chem., 14, 1 (1959). H. S.Frank and W.-Y. Wen, Discuss. Faraday. SOC.,24, 133 (1957). E. Gluekauf, Trans. Faraday. Soc.. 80, 572 (1964). F. Booth, J. Chem. Phys., 19, 391 (1951). H. L. Friedman and C. V. Krishnan in "Water-A Comprehensive Treatise", F. Franks, Ed., Plenum Press, New York, N.Y., 1973. D. L. Beveridge and G. W. Schnuelle, J. Phys. Chem., 78, 2064 (1974). G. W. Schnuelle and D. L. Beveridge, J. Phys. Chem., following paper in this Issue. H. Margenau and G. M. Murphy, "The Mathematics of Physics and Chemistry", Van Nostrand, Princeton, N.J., 1956. J. Hylton, R. Christoffersen, and G. G. Hall, Chem. Phys. Lett., 24, 501 (1974).
A Statistical Thermodynamic Supermolecule-Continuum Study of Ion Hydration. I. Site Method Gary W. Schnuelle and David L. Beveridge' Department of Chemistry, Hunter College ofthe City Univefsity of New York, New York, New York 10021 (Received April 16, 1974) Publication costs assisted by the National hstitutes of Health
The methodology and results of statistical thermodynamic supermolecule-continuum calculations on hydrated ions are described. Configurational averaging is carried out using the site method. Enthalpies, free energies, and entropies are calculated for the tetrahedral and octahedral coordination of the ions Li+, Na+, K+, F-, and C1-. The calculations utilize accurate quantum-mechanical representations of all interactions in the supermolecular assembly, and include no adjustable parameters. With the exception of the entropies for anions, all values agree satisfactorily with experimental results. The possibility of using extensions of this methodology for theoretical studies of polyatomic molecules in condensed phases is discussed.
I. Introduction The statistical thermodynamic supermolecule-continuum model for the theoretical treatment of solvation energy and solvent effects on molecular structure and properties involves the calculation of a partition function for a dissolved molecule and its first solvation shell imbedded in a polarizable dielectric continuum.' The idea of treating solvent effects with a discrete representation of the solute and vicinal solvent molecules and a continuum representation for bulk solvent has been raised a number of times previously in the scientific literature, and dates back to the classic study of water and ionic solutions by Bernal and Fowler.2 Statistical aspects were first introduced with this model by Kirkwood3 in his early study of the dielectric constants of polar liquids. Contemporary use of this approach is found in theoretical studies of solvated electron systems, particularly in papers by Copeland, Kestner, and Jortner4 and by Fueki, Feng, and K e ~ a nBoth . ~ Newton6 and MOSkowitz, Boring, and Wood7 report quantum mechanical studies of hydrated electrons based on this model, but neglecting configurational averaging. Our recent interest in this approach arose in a consideration of the extension of current quantum theoretical calculations of solvent effects on biomolecular conformational stability as carried out in this laboratory using a continuum The Journal of Physical Chemistry, Vol. 79, No. 23, 1975
model8 and parallel studies elsewhere using the supermolecule modeLg When the problem is cast in the form of statistical thermodynamics, the relationship between the continuum model and supermolecule is clearly displayed and the union of these approaches into a supermolecle-continuum model is shown to accommodate, in principle, all the main factors which appear to contribute to solute-solvent interactions.' I t remains now to characterize the model in terms of (a) the methodology which renders supermolecule-continuum calculations of systems of chemical interest computationally tractable, and (b) the level of agreement to be expected between numerical supermolecule-continuum calculations and experiment. This series of two papers describes a statistical thermodynamic supermolecule continuum study of ion hydration, a prototypical solvation problem where both questions of numerical methodology and quantitative agreement between theory and experiment can be dealt with directly. Subsequent studies will deal with solvent effects on conformational stability.
11. Background We have previously described the supermolecule-continuum model in the context of solvent effects on conforma-
Supermolecule-Continuum Study of Ion Hydration tional stabi1ity.l The general expression for the partition function representative of the system is
where 2; is the partition function for the ith conformation of solute, Eieffis the energy of the supermolecular assembly of solute (particle 1) and vicinal solvent molecules (particles 2 - M ) imbedded in a polarizable dielectric. The integrations in eq 1 are effected over solvent configurational coordinates qi. The relatively hard modes of the solute are considered to be integrated out, leaving the partition function dependent on the set of conformational coordinates TI (dihedral angles) of solute. The thermodynamic properties of the system follow from the partition function in the usual manner. Maintaining a statistical thermodynamic supermoleculecontinuum calculation within computationally tractable limits requires special consideration of the evaluation of the configurational energies Eieffand the evaluation of the integrals over configurational coordinates of solvent molecules (configurational averaging). The most accurate computational method for configurational energies is molecular quantum mechanics. The direct evaluation of individual energy grid points for a molecular assembly of,chemical interest using quantum mechanics directly is computationally feasible but relatively time consuming. Within this framework there are a variety of alternative methods for evaluating the configurational energy, including the various approximate and ab initio molecular orbital schemes for the supermolecular part and the various approaches to Born-Onsager-Kirkwood polarization energy for the continuum part. Some general ideas on approximation methods for configurational averaging were developed in an earlier paper, where a progression of computational methods was proposed for keeping this aspect of the calculation within tractable limits. Implicit in eq 1 is the shell method, where the range of configurational integration is restricted to the first solvation shell. The cell method involves restricting the configurational freedom of solvent molecules to mutually exclusive regions of the configurational space of the first solvation shell. The site method, elaborated herein, involves assuming particular sites in the first solvation shell for solvent molecules, disposing the solvent molecules with their centers of gravity coincident with these sites, and carrying out a configurational averaging over orientational coordinates. In the site method, eq 1reduces to
The errors accumulated in passing from the shell method to the cell and site method for configurational averaging are known as communal errors. The characterization of the statistical thermodynamic supermolecule-continuum model requires a study of the capabilities and limitations of the alternative methods for energy evaluation methods and configurational averaging for a small system for which experimental data are simultaneously available for comparison. The chemical process most suitable for prototype studies of the statistical thermodynamic supermolecule-continuum model is the hydration of select monoatomic anions and cations. A convenient approach to accurate energy evaluation is available from
2567
the large basis set ab initio molecular orbital study of ionwater and water-water interactions by Popkie, Kistenmacher, and Clementi.loJ1 Here a data base of interaction energies a t various points in configuration space was developed, and analytical functions were generated with disposable parameters chosen to reproduce the quantum mechanically calculated interaction energies within a prescribed numerical tolerance. These functions can be used directly for rapid evaluation of the supermolecular contribution to Eieff.The ion hydration system is small enough that configurational averaging can be fully carried out by the shell method, cell method, and site method, and the communal errors quantified. Finally, the extensive scientific literature available on the ion hydrstion process allows us to study our methodology in the context of reasonably well-understood experimental data. The present state of theory and experiment on ion hydration has been extensively reviewed in a recent article by Friedman and Krishnan,12 and a number of theoretical studies of the problem based on a continuum model, a totally discrete model, and hybrids of the two have been reported; the statistical mechanical supermolecule-continuum model under consideration herein falls in the hybrid category. The studies based on the continuum model applied the analysis of Born13 on the reversible work necessary to charge a conducting sphere in a polarizable dielectric. Debye and Pauling14 carried out early work along this line. There are essentially two disposable parameters in a Born charging expression: radius of the ion and dielectric constant of the continuum. Latimer, Pitzer, and Slansky15 found that a good fit to experimental hydration energies could be obtained with experimental ionic radii adjusted by a constant depending only on the type, anion or cation. NoyesI6 treated the dielectric constant as a disposable parameter and gave an account of the experimental data involving a much reduced value of the dielectric constant in regions vicinal to the ion, a consequence of dielectric saturation. Recently, Laidler and Muirhead-G~uldl~ have reported additional calculations based on an optimized continuum model. The early discrete and hybrid models take explicit account solvent molecules, with interaction energies computed empirically or semiempirically based on classical electrostatics. The Bernal-Fowler paper cited above2 appears to be the first of this genre. An important paper in this line of development is that of Eley and Evans,lBwho were the first to give a statistical thermodynamic account of ion hydration. Buckinghamlg pointed out the importance of considering solvent molecules explicitly by accounting for the differential solvation of K+ and F- on the basis of ion-quadrupole interactions, assuming a bifurcated anion-water gemetry. Laidler and Muirhead-GouldZ0aas well as Goldman and Bates20b have also described extensive applications of this approach. All of these studies incorporate a continuum model for long-range interactions, and in the context of the present work are regarded as supermoleculecontinuum calculations. Only the studies of solvation entropy by Eley and Evans and by Goldman and Bates include effects of configurational averaging. The advent of molecular quantum mechanics in recent years has made possible calculations using the discrete model with a more reliable evaluation of interaction energies. Several studies have been reported using a totally discrete model, including the approximate molecular orbital studies of Burton and DalyZ1 and Lischka, Plesser, and The Journal of Physical Chemistry, Vol. 79, No. 23, 1975
2568
I
Gary W. Schnuelle and David L. Beveridge
Schuster.22 Ab initio and approximate molecular orbital calculations were compared with classical energy calculations for ion-water interactions by Breitschwerdt and Kist e n m a ~ h e r The . ~ ~ most extensive totally discrete study to date is that of Kistenmacher, Popkie, and ClementiZ4 where the accurate ion-water and water-water analytical functions were used on extensive study of the configurations and solvation numbers of monatomic ions based on energy optimization. The study described herein serves then a dual purpose: characterization of the statistical thermodynamic supermolecule-continuum model as described above and extending theoretical description of ion hydration along the line of Eley and Evans with more accurate interaction energies and an approach to configurational averaging which can be readily generalized to other problems. The calculations presented in this paper are based on the site method, and an article to follow will report results using the cell method and shell method for configurational averaging.
111. Theory and Methodology In this section we present details of the configurational averaging, energy evaluation, and the assumptions inherent in statistical thermodynamic supermolecule-continuum calculations on hydrated ions. In this problem the solute is devoid of conformational coordinates, so the variable T may be dropped from eq 1 and 2. When the site method is used for configurational averaging, one must assume a number A4 - 1 of vicinal solvent molecules and also the disposition of their centers of gravity in the first solvation shell. The configurational averaging over solvent molecules described by eq 1 reduces to the orientational averaging as specified in eq 2. For numerical calculation the multiple integration is replaced by summation e-Eeff(wZ,...,wM)/kT d q . . . dwM -+
(4)
and the internal energy U is computed simultaneously with Z as 1 N Eieffe-ELeff/kT u=-
NZ i = i
(5)
The entropy consistent with eq 4 and 5 is then computed as
S = (U - A)/T
(6)
For the evaluation of the energies for the individual configurations of the system, we assume that Eieff(wz. . . W M ) may be partitioned into a contribution from the dissolved ion and its first solvation shell EiSmand a contribution resulting from the interaction of this supermolecular assembly with the polarizable dielectric, &Po', the supermolecule, and the polarization energy, respectively
The Journal of Physical Chemistry, Val. 79, No. 23, 1975
Eism(W2.
.. W M )
= Esrni-"(Wz.
. .WM)
-k Eamw-w(W~. .. WM)
(8)
where the first term on the right of eq 8 represents ionwater interactions and the second term represents waterwater interactions. Assuming pairwise additivity, Esmi-w and E,,"-" can be written as M
ESmi-"(w2... W M ) =
Vj'-"(wj)
j=2
and
k
