J . Phys. Chem. 1991, 95, 4109-4113 therm was calculated. The global isotherm obtained was then fit to the bi-Langmuir equation (eq 9 with two terms, i = 1 and 2). A perturbed isotherm was constructed by calculating the residual corresponding to the bi-Langmuir fit, changing one of the four bi-Langmuir parameters slightly (i.e., reducing at or b2 by 5%), calculating the slightly changed bi-Langmuir isotherm, and adding the residual back to the slightly changed bi-Langmuir isotherm. Then, the energy distribution function corresponding to the new, perturbed isotherm is calculated. Figures 9 and 10 show the simulated and optimized distributions obtained for 5% perturbations of the a and 6 (respectively) bi-Langmuir parameters corresponding to the smaller peak. This small energy peak is slightly shifted in the former case, hardly changed at all in the latter. The determination of the adsorption isotherm from the experimental chromatogram of a large size sample, using the ECP method, may introduce experimental errors. A particular chromatographic characteristic point may be subject to two types of random errors: those that affect the detector response (flame current noise) and those that affect the retention time (carrier gas flow rate and/or column temperature fluctuations). To test the robustness of the method, we modified the experimental data discussed above (Figures 1 and 2), introducing higher and higher fluctuations, until a change was seen in the calculated energy distribution. First, the detector response was perturbed, holding the retention times constant. A random number generator with a rectangular distribution probability was used to superimpose a 5% random noise on each point (Figure 1111). The resulting chromatogram (symbols) is obtained by adding this noise to the data points (solid line). The noisy chromatogram is compared with the actual experimental chromatogram in Figure 111. The energy distribution calculated with the noisy chromatogram is compared to the energy distribution corresponding to the experimental data in Figure 12. The residual is extremely small, because of the smoothing effect of the integration calculation performed. In practice, the detector noise will have no effect on the determination of the adsorption energy distribution, as the signal noise for high concentration band profiles is much below 5%. Finally, the retention time of each characteristic p i n t was perturbed. Figure 131 shows the specific, corrected retention
4109
volume of diethyl ether plotted as a function of its partial pressure (solid line). A 3% random error on this volume was superimposed on each point (symbols), simulating the effect of a fluctuation of the carrier gas flow rate or of the column temperature (Figure 1311). The calculated energy distribution is compared to the energy distribution corresponding to the experimental data in Figure 14. The only effect is a very small shift, of the order of 1 X 1V2kcal/mol, in the average energy of the two peaks of the energy distribution function. Conclusion The numerical method we have developed for solving the integral equation of the problem (eq 1) offers several potential advantages over existing methods when applied to gassolid isotherm data obtained by the ECP method. These potential advantages include an improved accuracy (compared to the condensation approximation and the existing numerical methods) and a higher degree of robustness (compared to integral transforms). We have also shown that the characteristics of the energy distribution function derived from experimental results are not affected significantly by fluctuationsof the experimental conditions within a range that exceeds the normal range of noise and drift expected from modern instruments. The major objective of this paper, the introduction of a new powerful and reliable numerical method, has been achieved. Further studies, involving the systematic application of the method to characterize the surface of a number of different samples of similar materials using several probe solutes, are required before definite conclusions can be drawn regarding the practical usefulness of our approach!*
Acknowledgmenr. We acknowledge fruitful discussions with W. H. Griest (ORNL, Division of Analytical Chemistry) and M. A. Janney (ORNL, HTML). This work was supported in part by Grant DE-FG05-86ER13487 from the U.S. Department of Energy, Office of Energy Research, and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory. We acknowledge continuous support of our computational effort by the University of Tennessee Computing Center.
Entropies of Solvation of Solvated Electrons Sidney Goldent and Thomas R. Tuttle, Jr.* Department of Chemistry, Brandeis University, Waltham, Massachusetts 02254-91 10 (Received: September 13. 1990)
An expression is derived for the entropy of solvation of a solvated electron that can be evaluated directly from (1) an established correlation between the spectral energy and solvent energy of the solvated-electron system and (2) the entropy of the pure solvent. The theory predicts that entropies of solvation of solvated electrons should be positive in all pure solvents, under the usual standard conditions, in marked contrast with the negative values obtained for the hydration of conventional ions. This is supported by current experimental results: for NH', the theory yields a value of +I20 J/(mol-K), while the most recently reported experimental value is +I54 f 20 J/(mol.K); for HzO, the theory yields a value of +118 J/(mol.K), while the most recently reported experimental value is +118 f 20 J/(mol.K).
1. Introduction For Some time,' solvated electrons in liquid ammonia were thought to have a standard entropy of ammoniation that was positive while solvated electrons in liquid water were thought to have a standard entropy of hydration that was negative, the latter sign appearing to be characteristic for the standard entropies of 'Emeritus Professor of Chemistry.
0022-3654/91/2095-4109$02.50/0
hydration of conventional ions.z This situation persisted' until quite recently: when a new determination was found to yield a positive standard entropy of hydration for solvated electrons ( I ) Lepoutre, G.; Jortner, J. J . Phys. Chrm. 1972, 76, 683. Scientific: (2) Conway, New9. York. E. Ionic 1981. Hydration in Chcmisrry and Biophysics; Elsevier (3) Schindewolf, U: Ber. Bunscn-Ges. Phys. Chrm. 1982, 86, 887. (4) Han, P.; Bartels, D. M. J . Phys. Chem. 1990, 91, 7294.
0 1991 American Chemical Society
4110 The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 comparable in value to their positive standard entropy of ammoniation. Because of the evident importance of this finding and because of the general experimental and theoretical difficulties that attend the determination of entropies of solvation for individual ionic ~pecies,~ an independent method of determining these important quantities can be useful in assessing their reliability. Just such a method is provided by a novel theory to be described here. The theory is based upon a linear relation which has been established&' between the mean frequency of the optical absorption band of a solvated electron and the mean energy of a molecule of the pure liquid solvent in which it is dissolved. It is constructed in terms of the solvent-anion-complex (SAC) model' of solvated electrons, which entity consists of a pertinent excess electron of the system and an effective number of solvent molecules that serve to localize it; the remaining solvent molecules solvate the SAC. The main result of the theory of interest here is a simple proportionality that it establishesdirectly between changes that occur in the mean energy of the SAC, as a consequence of changes in its environmental condition, and those that occur, correspondingly, in the mean energy of a molecule of the pure solvent; the proportionality constant is determinable entirely from the cited linear relation between spectral energies and solvent energies. When applied to the standard entropies of solvation of solvated electrons in liquid ammonia and in liquid water, the theory yields results that agree with the reported experimental results moderately well, estimated by the range of their uncertainty, and so provides independent support for them. In addition, the theory implies that all solvated electrons in pure solvents should have positive standard entropies of solvation. These results and a description of the theory that produces them motivate the present paper. A derivation of the basic linear relation between spectral energies and solvent energies, expressed in terms of the solventanion-complex model of solvated electrons, is given in section 11. The theory that correlates SAC energies with solvent energies is developed in section 111 and applied to calculate standard entropies of solvation of solvated electrons, some of which have yet to be determined experimentally, in section IV. The paper concludes with a discussion in section V concerning the physical basis underlying the success of the theory. 11. The Basic Linear Relation between Spectral and Solvent Energies We consider a macroscopic system consisting of a liquid of N , identical (electrically neutral) solvent molecules (S)in which is dissolved a single solvent-anion-complex (SAC) consisting of a single excess electron (e) and N l solvent molecules (I) that serve to localize it. The entire system is in equilibrium in a macroscopic volume f l and at a temperature T. The following assumptions pertain: (A) The Hamiltonian of the system is H = HSAC
+ T,+ V,
(1)
Golden and Tuttle be strongly orthogonal in the w r d i % a t F End spin of tie s_olv_ated electron,8 viz., for eigenstates *,,,(XeJI,Xs) and *.,(XeJIJ,)
where ? , stands for the coordinates and spins of the pertinent particles. (C) The absorption spectrum ascribable to the solvated electron is that of the SAC. Because its electric dipole moment operator can be expressed as GSAC
the subscripts denoting the pertinent particle dependence, it follows from (3) that
which enables us to utilize the model-free thermalized spectral moment theory of solvated-electron spectra that we have described el~ewhere,~ without any modification needed. (D) The absorption spectrum consists predominantly of a single band of bound-continuum transitions with a nonzero threshold for photoionization of the SAC9Joand exhibits essential spectral shape stability.'lJ2 At the threshold frequency for photoionization, we have (in au) u?h
HSAC
= Te +
vel
+
ves
+
+
+
(HSAC)f
- (HSAC)i
u%= ( V , l + V , s ) f + ( T l + V , l + ~ s ) f - ( T e + V e I + V ~ ) i (7'1+ VI + Vb)i (6)
since the mean kinetic energy of the excess electron in the threshold state vanishes. From the first moment of the absorption spectrum? the mean absorption frequency is uav
= (4/3)(7'e)i
the subscripts denoting the entities to which the terms refer; T represents the kinetic energy of all the constituent particles of the species indicated and V represents the totality of pairwise Coulombic terms comprising the pertinent interparticle interactions. The interaction between the SAC and the solvating solvent is assigned entirely to the former so as to ascribe all solvation properties of the system to the SAC. (B) The eigenstates of H are confined to the volume ll and are thus quadratically integrable. They are, in addition, assumed to (5) BocLris, J. O M . , Conway, B. E., Eds. Modern Aspccrs o/ Electrochemistry; Butterworthr Scientific Publications: London, 1954; Chapter 2. (6) Golden, S.; Tuttle, T. R. Jr. J . Chcm. Sa.,Faraday Trans. 2 1981, 77, 1421. (7) Golden, S.; Tuttle, T. R. Jr. J . Chcm. Soc., Faraday Trans. 2 1982, 78, 1581.
(7)
and from the virial theorem for the entire system9 it follows that for the threshold state (Vel + V,,)r = -(2Tl
+ VI, + 2T, + V,, + Vs)r
-6f (8)
while for the initial state
+ Vel + ves)i = (T,)i+ (27'1 + V,, + V,,
-(Te
+ 2T,+ V,)i
(3/ 4)~,,
+ 6i
(9)
the 6's being the virial-theorem contribution of the entire solvent in the indicated states. After some rearrangement, we obtain V,s)i
- (TI +
VI + V,s)d = ((3/4)uav
(2)
(5)
where u;,,is the photoionization threshold frequency of the optical absorption band of the SAC, (...) signifies the equilibrium average of the indicated quantities, in accord with the selection rules that prevail, and i and f refer to the initial and final equilibrium-averaged states in the transition. With the aid of (2), we can get
{(TI+ VII+
where
= 3, + GI
- w?h) + (6i - 8,) (10)
(8) We have assumed in previous work that the solvated electron was *distinct" from all others in the system. The strongly orthogonal assumption here is a far weaker one and allows the Pauli exclusion principle to be fulfilled. It is also weaker than the adiabatic approximation that leads to the FranckCondon principle for electronic spectra. (9) Golden, S.;Tuttle, T. R. Jr. J . Chem. Soc. Faraday Tram. 2 1979, 75, 474. A contribution from the container confining the solution has been omitted as being negligible. See ref 6. (IO) The possibility that there may be some bound-bound transitions of very low intensity near the photoionization threshold will not seriously affect the conclusions reached here as a result of neglecting them. (1 1) Tuttle, T. R. Jr.; Golden, S., J . Chem. Soc., Faraday Trans. 2 1981, 77, 873. Golden, S.; Tuttle, T. R. Jr., J. Chem. Soc., Faraday Trans. 2 1981, 77, 889. Tuttle. T. R. Jr.; Golden, S.;Lwenje, S.; Stupak, C. J . fhys. Chem. 1984,88, 3811. (12) There is some indication that spectral shape stability does not prevail in the vicinity of the absorption threshold of ND,. See, for example: Tuttle. T. R. Jr.; Golden, S.;Rosenfeld, G. Radial. Phys. Chem. 1988.32, 525. This will not seriously affect the conclusions reached here.
The Journal of Physical Chemistry, Vol. 95, No. lo, 1991 4111
Entropies of Solvation of Solvated Electrons
TABLE I: StrnQrd Entropies of Sdr~th,A8", ( s d r ~ t l (J/(msl.K)) ~) solvent NHia
Nan'
A$," ( s ) ~ 84.9 69.9 126.8 160.7 150.2 190.7
HZO
CH3OH CIHSOH CH3NH3 THF
0.18 0.95 0.69 0.65 0.83 0.38
do,(i3)b -k In 15.4 21.0 21.0 21.0 21.0 21.0
xO&
+ k In 2
theoryd
36.5 39.0 32.4 29.3 31.1 26.6
ABo,(solvation) expt 154 f 203~21 I 18 4 204 na na
120 118 143 165 197 1 I4
na na
#Standardconditions at 233 K see ref 16. *Primary source of data refs 17 and 18. CCalculatedusing solvent densities obtained from refs 16, 18, 19, and 20. dCalculated from eq 30; na, not available. Since each 6 would vanish were there no excess electron, their difference can be expected to be small in value and not to change much with changing equilibrium conditions of the system. When the &difference is negligible, an examination of experimental solvated-electron spectra indicates that the right-hand side of (10) is essentially positive6and corresponds to a decrease in the energy of the localizing solvent molecules during the optical transition. For our purposes, we consider only those changes in the condition of the system that maintain shape stability of the solvated-electron spectrum. Then presuming that the energetic and spectral properties of the system can be extrapolated to the absolute zero of temperature and that (Si - 6J is essentially constant, we can obtain from (10) Ne&
* -(1/4)A~w
(11)
the A-differences being understood to be taken with respect to the values at the absolute zero of temperature. In obtaining (1 l), we have taken as an energy unit the equilibrium-averaged energy (measured relative to an appropriate reference value) of a typical solvent molecule in solution, E,, and have introduced an effective number of typical solvent molecules that are energetically equivalent to the localizing ones, viz.,
VI + V 8 ) i - (TI + 61 + Vs)d/E, (12) The utility of the defined ne^ is clearly contingent on our being Nefi I {(TI+
able to determine its value. Since energies in condensed phases do not change appreciably with changes in volume that are not too great, we will also take it that (1 1) holds for all densities (NJQ) and temperatures T, and hence pressures P, that maintain shape stability of the solvated electron's absorption spectrum. Equation I 1 is the basic linear relation we need." It has been tested: with the result that solvent-dependentessentially constant positive values less than unity were obtained for Ndl for the solvents examined.
(HSAC)i
-
TI)^
- (27'9 + vm)i
(13)
The last term is the equilibrium-averaged virial-theorem contribution of the solvating solvent. Its value is assumed not to change much, as previously assumed for the entire solvent. Then, substituting the values of the quantities in (13) at absolute zero, we get (3/4)&v
* -AESAC- A( T1)i
(14)
We have introduced &AC
E (HSAC)i
(15)
as the equilibrium-averaged energy of the SAC, since the latter's absorption spectrum reflects the presence of a single equilibrium-averaged initial state that is bound. Upon combining with (1 l), we obtain
GAC * 3Ne&s - A( T1)i
A(Tl)i
(1/2)Ne&g
(17)
A&AC
(5/2)NeffAEs
(18)
and get This is the correlation that we seek between SAC energies and solvent energies. It provides a basic correlation of the spectral and thermodynamic properties of solvated electrons since, by (1 l), it follows that
SAC * -(5/8)Auav
(19) Depending on the data that are available, either (18) or (19) may be used to determine changes in SAC energies. For most situations, however, (18) would appear to be of greater utility. Then, all thermodynamic properties that are the result of linear operations on the energy functions will maintain the proportionality of (18). Thus, we have the heat-capacity changes
with an analogous expression for the heat capacities at constant pressure, since the relevant energy changes are essentially the same as the enthalpy changes in condensed phases. Also, we have the entropy changes
which is of primary interest here, and will be dealt with in the following section. In addition, we have Helmholtz function changes
111. Correlation of SAC Energies and Solvent Energies From (2). (7), and (9), we can obtain
(3/4)% = -
From what we know about the energetic properties of molecules, the kinetic energy of their electrons will not change significantly if the temperatures involved are not too large. Changes in the kinetic energy of their nuclei will (by the virial theorem) be approximately one-half the changes in their total energy. As a result, we take
(16)
( I 3) Other characteristic frequencies of the absorption band, e.g., the frequency of maximum absorption. the threshold a h r p t i o n frequency, etc., may be used as long as spectral shape stability prevails.
APsAc E AESAC- TA
SAC
=
(5/2)Ne&',
(22)
It is to be emphasized that ne^ in the preceding equations is determinable from (1 1) directly or via Neff = -(I / 4 ) t a A % / m * / G (23) where * denotes the constraint under which (aAo,,/aT). is determined, e.g., n, P, or appropriate combinations of them. As a result, the proportionality in (18) and (20)-(23) is fixed and nonadjustable. Values of Neffhave been determined from (1 1) or via (23) for several solvents; the values are listed in Table I. IV. Entropies of Solvation for Solvated Electrons Presuming the applicability of the third law of thermodynamics, the intrinsic molecular entropy of a SAC is, from (21)
+
( 5 / 2 ) N e d 8 k In 2 (24) where k In 2 is the entropy contribution due to spin degeneracy. This entropy does not take the pertinent entropies of mixing into account. To do so, and to obtain molecular entropies that correspond to conventional standard entropies, we express the latter for the SAC as $SAC
4112 The Journal of Physical Chemistry, Vol. 95, No. 10, 1991
where xoSAcis a standard mole fraction of the SAC, usually taken to be equivalent to a concentration of (1 mol/L) in the solution. For the solvent, we express BO&)
= 3,
(26)
corresponding to xo, = 1 . Then, we obtain
go&
=
( 5 / 2 ) N e 9 0 s ( ~ )- k In XOSAC
+ k In 2
(27)
The process of solvating an excess electron will be represented by 43)
+ 0" + N,rr)s(S) -,SAC(s) + N,S(s)
(28)
where temperature and pressure are held fvred during the solvation. The associated standard entropy of solvation is then A8",(solvation) * [gosAc(s) + Ns$Os(s)][a0r(g) + ( ~ +s Nerr)g0s(s)I (29) We note that the replacement of N, by Nerrin (28) is dictated by the energetic behavior of the SAC expressed in (1 6). Accordingly, we obtain from (25)-(27) A8Oe(solvatian) = (3/2)Ner,30s(~) - k In XOSAC + k In 2 g 0 m (30) Table I gives values of P$O,(solvation) calculated from (30) for several solvents. Experimental values have been reported only for ammonia and water and are evidently accounted for moderately well by the theoretical values. Explicitly, the most recent experimental standard entropy of ammoniation (at 233 K) is reported to be3 Agor(ammoniation) = 154 f 20 J/(mol.K)
(31)
while the theoretical value from (30) is Ago,(ammoniation) * 120 J/(mol.K) (32) which is slightly outside of the estimated range of experimental uncertainty. Likewise, the most recent experimental standard entropy of hydration is reported to be4 Ago,(hydration) = 118 f 20 J/(mol.K) (33) while the theoretical value from (30) is A$O,(hydration) = 118 J/(mol.K) (34) which is quite well within the estimated range of experimental uncertainty. For standard conditions not too different from those indicated in Table I, ( k In 2 - k In x0SAC - $O,(g)J is demonstrably positiue. Since so is A,#O,(s), we can see that (30) implies that solvated electrons should exhibit standard entropies of solvation that are positioe for all pure solvents. This remains to be verified experimentally.
V. Discussion At first sight, it may seem surprising that the theory we have described can produce calculated values of the standard entropies of ammoniation and of hydration for solvated electrons which, in view of their approximateness, agree as well as they do with the corresponding experimental values that have been reported. After all, apart from the standard entropy of gaseous electrons, 8O,(g), and the effective numbers of localizing solvent molecules, N#, the calculated values require only a knowledge of the standard enpopies of the pure solvents. Indeed, from the form derived for A8°,(solvation), it might even appear that the standard entropy of solvation for solvated electrons is primarily a property of their solvents and that the electron's properties play only a minor role. On the reflection, however, it will be realized that the essential solvated-electron property upon which the theory is based is its optical absorption spectrum. It then may still seem surprising that the spectral property of a solvated electron suffices to determine any of its thermodynamic properties. Yet, because of
Golden and Tuttle special characteristics that are possessed by solvated electrons, it can be shown that the results we have obtained are entirely reasonable even if originally unexpected. To do so, we imagine some sort of a species-solvent complex in solution which yields an absorption spectrum that changes with changing conditions of the solution. These changes reflect changes occurring in both the initial and final states involved in the spectral transitions. Suppose that a spectral frequency is chosen for which the final state involves the species component in such a condition that it is hardly affected by changing conditions of the solution. Then, the changes which can occur in that frequency are to be identified with the changes that occur in the energy of the species component in the initial state and in the energy difference of the complexed-solvent component that is involved in the transition. This is the situation that prevails at the photoionization threshold of a solvated electron, as a consequence of assumption D; it is expressed by ( 5 ) and (6). If, then, the energy of the species component in the initial state can be determined in some way, changes in the energy difference of the complexed-solvent components can be expressed in terms of the changes which occur in the threshold frequency. This is the situation that prevails for a solvated electron as a consequence of assumptions B, C, and D and the use of the virial theorem for the system described in assumption A; it is expressed by (1 1) and (12) in terms of the mean frequency of the absorption band.13 Finally, by combining the changes that can occur in the initial-state energy of the species component and in the associated energy difference of the solvent component, in such a way that eliminates the explicit dependence on spectral frequencies, we can obtain a relation between the changes in the energy of the species complex and those of the pure solvent. For solvated electrons, the result is expressed in (18). Thus, for solvated electrons, the final result we have obtained is indeed in accord with reasonable expectations. But, we may ask, can the same sort of relationship also hold for any speciessolvent complex? This does not appear to be likely for several reasons: (1) The absorption spectrum of the species-solvent complex must be ascribable to a single electron which undergoes bound-continuum transitions from a single equilibrium-averaged initial state. (2) The spectrum must exhibit shape stability with changing solvent conditions. (3) There must be a linear correlation between characteristic frequencies of the spectrum and the equilibrium-averaged energies of molecules of the pure solvent. All three requirements appear to be fulfilled by solvated electrons:+" but it does not seem likely that the first two requirements can be met by conventional anions which are stable in the gas phase. However, as evidenced by the correlation that obtains14 between iodide spectra and solvated-electron spectra, meeting the third requirement may be possible. If so, we may be able to construct a theory for some conventional anions along the lines considered here for solvated electrons, but one with somewhat more complexity than the present one. If such a theory were to be constructed that would yield the form of (30),but with the gas-phase standard entropy of the excess electron replaced by the gas-phase standard entropy of the relevant conventional anion, the standard entropy of solvation of the anion likely would then be negatioe. This would occur because the gas-phase standard entropies of conventional anions turn out to be very large compared to that of gaseous electrons. This is, perhaps, one of the main features that produces positive standard entropies of solvation for solvated electrons while the values for conventional anions are negative. Finally, we consider the role played by the SAC model in the theory we have described. For this purpose, we note that the solvation process represented by (28) reduces (with an appropriate change of notation) to
4%) + ")
-
4 s ) + N,S(S)
(35) by arbitrarily setting New equal to zero. Similarly, (29) then reduces to (14) Fox, M. F.;Hayon, E.;Chem. Phys. Leu. 1974, 25, 51 1; J. Chcm. Soc., Faraday Trans. I 1976, 72, 1990.
J. Phys. Chem. 1991, 95, 41 13-41 19 A8°f(solvation) = 8O,(s)
- Bof(g)
(36)
as for conventional ions,5 with
go,(,) = af(s) - k In xo,
(37)
analogous to (25). Guided by the large positive standard entropies of solvation which have been determined for the ammoniation and the hydration of solvated electrons, we can infer that large p i t i v e values for the intrinsic entropies of the solvated electrons, FPt(s), are required in these solvents. Because of the spatial confinement that a solvated electron experiences, we can further infer that such cannot be the case.I5 Any large positive contribution to the (15) The intrinsic entropy of a solvated electron is related to its motion relative to its mean location. It may be estimated from its spectral-momentanstrained maxentropic density matrix. Values about 4 J/(mol.K) are obtaincd. See: Golden, S.;Tuttle, T. R.Jr. J . Chem. Soc., Faraday Trans. 2 1988,84, 1913. (16) Haar, L.;Gallagher, J. S.J. Phys. Chem. Ref Data 1978,7, 635. (17) Wagman, D. D., et al. J. Phys. Chem. Ref Data 1982,II, Suppl. No. 2. (18) Karapet’yants, M.Kh.; Karapet’yants, M. L. Thermodynamic Consronrs of Inorganic and Organic Compounds; Humphrey Science Publishers: Ann Arbor, MI, 1970. (19) Wilhoit, R. C.; Zwolinski, B. J. J . Phys. Chem. Ref Dara 1973, 2, Suppl. No. 1.
4113
intrinsic entropy of the species, it would seem, can come only from solvent molecules with which it is associated. This, it will be recognized, is naturally provided by the SAC model. Although the present theory has given to good account of itself, further tests of it are warranted in order to determine the range of its validity. In particular, tests of (1 l), the relation basic to the construction of the theory, and of (30),of more direct interest, should provide this. Of particular interest is the positivity that it predicts for the standard entropy of solvation of solvated electrons in all pure solvents. We hope that such tests will not be long in coming.
Acknowledgment. We are indebted to Dr. Charles Jonah and Dr. David Bartels of the Argonne National Laboratory: the former for bringing to our attention the work of Han and Bartels on the entropy of the hydrated electron prior to its publication and the latter for useful discussions on the subject. Registry No. NH,, 7664-41-7; H20,7732-18-5. (20) TRC Thermodynamic Tables. Texas A & M University System: College Station, TX 77843-3111. Non-Hydrocarbons. Suppl. No.63,June 30, 1990. (21) An earlier value of 133 J/(mol.K) at 238 K, of unspecified uncertainty, was reported in ref 1. The value calculated from (30) for this temperature is 121 J/(mol.K).
Performance of the Diffusion Equation Method in Searches for Optimum Structures of Clusters of LennarbJones Atoms Jaroslaw Kostrowicki: Lucjan Piela,* Binny J. Cherayi1,S and Harold A. Scheraga* Baker Laboratory of Chemistry, Cornel1 University, Ithaca, New York 14853-1 301 (Received: September 14, 1990)
Our previously described diffusion equation method for global optimization was applied here to find minimum-energystructures of several clusters of identical atoms. In this method, the potential energy surface of the system, which contains a large number of minima, is deformed (the deformation parameter being analogous to ‘time”) in such a way that the number of minima is decreased dramatically. The deformation is carried out by finding a solution F of the diffusion equation for some time I , with the original potential function f as the boundary condition. In the present calculations, the atoms are assumed t o interact by means of a Lennard-Jones potential, with parameters pertaining to argon; however, the results can easily be rescaled to any other Lennard-Jones parameters. The Lennard-Jonespotential was fitted by a linear combination of Gaussians, so chosen to reproduce the potential accurately at physically relevant distances in clusters. The choice of Gaussians as fitting functions allowed F to be determined analytically. Further, by enforcing certain constraints on the Gaussian approximation off, the following desirable behavior of F was guaranteed: at sufficiently large (known) time, the interaction potential is purely attractive so that the minimum of F can easily be recognized as the only existing one; it corresponds to a collapsed configuration (i.e., one with all atoms occupying the same position in space). Subsequent local minimizations for gradually decreasing times (the so-called time-reversing procedure) led rapidly to the global minimum of the original function. The resulting cluster configuration did not depend on the number of reversing time steps when this number was sufficiently large (- 100). The method was applied to several systems of up to m = 55 atoms and successfully found the global minimum, even in the largest of these (the 55-atom system, which involves minimization in the space of dimension 3m - 6 = 159 for a function having about local minima).
1. Introduction We recently introduced a diffusion equation method’ for global optimization of a function with many minima and subsequently applied it to several standard test functions.2 In this paper, the method is used to locate minimum-energy structures of clusters ‘On leave from the Department of Chemistry, Technical University of Gdansk, Majakowskiego 11/12, 80-952Gdansk, Poland, 1989-1992. *Present address: Quantum Chemistry Laboratory, Department of Chemistry, University of Warsaw, Pasteura 1, 02-093Warsaw, Poland. 1Resent address: Department of Inorganic and Physical chemistry, Indian Institute of Science, Bangalore-560012,Karnataka, India. *To whom correspondence should be addressed.
containing various numbers of identical atoms that interact by means of a Lennard-Jones potential. No symmetry was assumed for the clusters. 2. Diffusion Equation Method The diffusion equation method transforms a function to be minimized according to this partial differential equation, thereby (1) Piela, L.; Kostrowicki, J.; Scheraga, H.A. J . Phys. Chem. 1989,93, 3339. (2) Kostrowicki, J.; Piela. L. J . Oprimizlrrion Theory and Applications, in press.
0022-365419112095-4113f02.5010 0 1991 American Chemical Society
