Hydrated Electiron Configuration Coordinate Model
263
~ o n f i g u ~Coordinate a ~ i ~ ~ Model for the Hydrated Electron M. r'achiya,* Y. Tabata, and K. Oshima Department of Nuclear Engineering, University of Tokyo, Tokyo, Japan (Received June 72, 7972) Publication costs assisted by the University of Tokyo
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The configuration coordinate model is developed for the hydrated electron. The configuration coordinate diagram is constructed by calculating the total energies of the hydrated electron under various orientational polarizations. The absorption spectrum of the hydrated electron is calculated by use of this diagram. The calculated absorption maximum and half-width are 2.04 and 0.52 eV, respectively. They are in seiniquantitative agreement with experiment.
I. Introduction So far various moldels have been proposed for the solvated electron. They are roughly divided into the continuum m ~ d e l : l -and ~ the structural mode1.6-8 In the continuum model the Configuration of the solvent molecules is represented by the orientational polarization f i r ) . Jortner's dielectric model1 assumes the orientational polarization given by P(r) = 0 rR (1) where t o p and cs are the optical dielectric constant and the static dielectric constant, respectively, and R is the cavity radius. The orientalional polarization given by eq 1 is not in thermal equilibrium with the average charge distribution of the solvated electron. The self-consistent field dielectric mode12.3 (SCF dielectric model) assumes the orientational polarization which is in thermal equilibrium with the average charge distribution of the solvated electron. In the SCF dielectric model the orientational polarization is given by
P ( r ) -- 0
rCR
where f is the electrostatic potential due to the average charge distribution of the solvated electron. The electrostatic potential is obtained by solving the following Poisson's equation
v2f= 4nel\L)2
The dielectric model, the SCF dielectric model, and the oriented dipole model are different from one another in the orientational polarization which is assumed. The orientational polarizations which are assumed in the three models are shown in Figure 1. As described above, the orientational polarization represents the configuration of the solvent molecules around the electron. It is possible to consider the solvated electron as a kind of molecule. If the solvated electron as a kind of molecule is compared with a diatomic molecule, the orientational polarization P corresponds to the internuclear distance R. The dielectric model and the SCF dielectric model are used to calculate the total energies of the solvated electron under the orientational polarization represented by a in Figure 1 and under the orientational polarization represented by b, respectively. This corresponds to calculating the total energies of a diatomic molecule at two different internuclear distances. The internuclear distance of a diatomic molecule is not constant. It is distributed around the internuclear distance at which the energy for the ground state is minimum. Similarly, we consider that the orientational polarization of the solvated electron is not constant but distributed around the orientational polarization under which the total energy for the ground state is minimum. This is the basic idea of the configuration coordinate model. The total energy of a diatomic molecule is calculated in the following scheme
( 3)
where $ i s the wave function for the solvated electron. In the vicinity of the cavity the orientational polarization is saturated. This effect is not taken into consideration in the dielectric model. Iguchi4 improved the dielectric model by taking into consideration the dielectric saturation effect. In his oriented dipole model the orientational polarization is given bys P(r) = 0 r1
Pz[l - ( Z2/ 4rz)]du
(20)
111. Calculation and Results A . Orientational Polarization under Which the Total Energy for the Ground State is Minimum, The orientational polarization under which the total energy i s minimum is determined by the following condition 6F/6P = 0 Since the orientational polarization is written as
P = -(l/4ae)(dV/dr)
(21)
where y is a constant. However, y for eq 18 is three or four times greater than that for eq 28. If we use eq 18 for the polarization energy, we obtain the hydration energy which is smaller tlhan the experimental value. This is probably because the polarization energy is overestimated. On the’ other hand, if we use eq 20 for the polarization energy, we obtain the hydration energy which is greater than the experimental value. This is probably because the polarization energy in underestimated. In the semicontinuum modelll.12 the polarization energy in the first solvation layer is calculated in a method similar to Iguchi’s. On the other hand, the polarization energy in the continuum is calculated by use of eq 18. Thus the polarizationt energy in the first solvation layer is underestimated and the polarization energy in the continuum is Overestimated. In the present paper we assume that the polarization energy is givlen by eq 21. The copstant y is chosen so that the calculated hydration energy is equal to the experimental value. C. TotaE Energy. ‘The total energy of the hydrated electron is given by
+
$*[- (A2/2rn)172 4n-e
JW
P(r’)dr’]$du
+
Y s P2du
(22)
Since the second term in eq 22 does riot involve # explicitly, eq 13 is equivalent to bFls/dX = 0
(23)
and eq 15 is equivalent to OF2plab.t = 0
(26)
the total energy is rewritten as
+
F = J#*[-(h2/2m)v2
V]#dv
+
[y/(4ae)P]JV’2dv (27)
Equation 25 is equivalent to 6F/6V= 0 From eq 28 the following Euler equation is derived for the potential under which the total energy for the ground state is minimum [2y/(4ae)2]( V”r2
+ 2V’r) - $qs2r2= 0
(29)
Equation 29 is rewritten as
Both eq 18 and 28 have approximately the following form
n = yJP2dv
(25)
-(y/2ae)(P’r2
+ 2Pr) - #lS2r2 = 0
The solution of eq 30 is given by
P
=
- (2re/?)(l/r2)
Lf
iLl,2r2dr
(30)
(31)
The total energy is given by
FlS= (h2X2/2m) - (5ae2XJSy)
(32)
From eq 23 and 25 we have dFis/dX = (aFls/aX) + (6F1s/bP)(GP/GX) = 0
(33)
From eq 32 and 33 we have h = ho =5a/8ya0
(34)
where a is the Bohr radius. The total energy is given by
Fls = -(h2Xo2/2m) (35) In the present paper we set y = 5.55 so that the calculated hydration energy is equal to the experimental value. The orientational polarization PO under which the total energy for the ground state is minimum is given by
PO= -(eXo2/2y) X {[l- (1 + 2Xor + 2X0213)exp( -2Xor)]/Xo2rZ] (36) The orientational polarization POis shown in Figure 2. The potential VO which is formed by the orientational polarization POis given by Vi = -(16h2X02/5m)
X
1[1 - (1 + Xor) exp(-2Xor)I/XorJ (37) The potential VOis shown in Figure 3. B. Orientational Polarization under Which the Total Energy for the Excited State is Minimum. From eq 28 the following Euler equation is derived for the potential under which the total energy for the excited state is minimum
(24)
If the orientational polarization P is given, the total energy for the ground state can be calculated by use of eq 22 and 23. The total energy for the excited state can be calculated by use of eq 22 and 24.
(11) (12)
D. A. Copeland, N. R. Kestner. and J. Jortner, J. Chem. Phys., 53, 1189 (1970). K. Fueki, D. F. Feng, L. Kevan, and R. Christoffetsen, J. Phys. Chem., 75,2297 (1971). The Journal of Physical Chemistry, Vol. 77, No. 2, 7973
M. Tachiya, Y. Tabata, and K. Oshima
266
0
7t
r, A. 10
?--
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0
12
14
Figure 3. Potentials V O , VI, and V O .formed ~ by the orientational polarizations PO,P i , and in Figure 2.
r,
a.
Figure 2. Orientationar polarizations around the electron under which the total energy For the ground state is minimum ( P O ) , under which the total energy for the excited state is minimum and given by 0.5(Po-tPi)(P0.5).
where 9 a P is given by Equation 39 i;3rewritten as -(y/2ne)(P'r2
+ 2Pr) - ( 9 z p 2 / 3 ) r 2= 0
(40)
The solution of eq 40 is given by
P
:= -
Sr
( 2 ~ e / ~ ) ( l / r (pZP2/3)r2dr ~) 0
(41)
(42)
From eq 24 and 25 we have dF2p/dp = (a bh,/ap)
+ (6F2p/6P)(6P/6p) = 0
(43)
93~/256yao
(44)
From eq 42 and 43 we have p
=: po
The orientational polarization PIunder which the total energy for the excited state is minimum is given by PJ.= - ( e p , ~ 2 / 6 ~ 1 { ( 3 / , &-r ~[2p02r2 ) + 4por + 6 +- (6/por) i (3/p$r2)] exp( - 2 w ) l (45) The orientational polarization PI is shown in Figure 2. The potential 1 ;' which is formed by the orientational polarization PI is given by
VI = -(256h"2/279n2)((6/por) [2p02rr2iL IjMor -t9 + ( 6 / m r ) ]exp( -2por)l
(46)
The potential V-i is shc~wnin Figure 3. C. Configurntion Coordinate Diagram. As is well known, the absorption spectrum of the hydrated electron is very broad. 'That is, the intensity is strong in a wide range of the excitation energy hv. The excitation energy of The Journal ol P h y s k a l Chemistry, VoL 77,No. 2, 1973
h 4 P ) = F~P(P) -J'IP(~
(47)
If FlS(P)is far away from F l S ( P 0 ) ,the thermal population of F I S ( P ) is small and the intensity of hv(P) is weak. If the intensity of hv(P) is not weak, Fl,(P) must be close to Fl,(PO) and P must be close to PO.The total energy for the excited stat,e changes most rapidly when the orientational polarization changes from PO toward PI. Thus the excitation energy also changes most rapidly when the orientational polarization changes from POtoward PI. In the present paper we assume that the variation of the orientational polarization, given by
P = (1- x)Po + xP1
The total energy is given by
,FzP = ( h 2 f i 2 / 2 n ).- (93ne2p/256y)
the hydrated electron is given by
(48)
where x is a variable, contributes most to the width of the absorption spectrum. We calculated the total energies of the hydrated electron under various orientational polarizations represented by eq 48, by varying x. For example, the total energies of the hydrated electron for x = 0.5 are calculated in the following way. The orientational polarization P0.5 which corresponds to x = 0.5 is given by
po.5 = 0.5(Po + A) (49) The orientational polarization P0.5 is shown in Figure 2 . The total energy for the ground state under P0.5 is calculated by use of eq 22 and 23. The total energy for the excited state under Po.5 is calculated by use of eq 22 and 24. The total energies of the hydrated electron for other x's are calculated in the same way. We show the configuration coordination diagram of the hydrated electron in Figure 4. In Figure 4 the abscissa x represents the orientational polarization. For example, x = 0 represents the orientational polarization PO in Figure 2. x = 0.5 represents the orientational polarization Po.5 and x = 1 represents the orientational polarization PI. D . Absorption Spectrum. The intensity distribution function I(hu) of the absorption spectrum is given by where hu is given by
Hydrated Electron Configuration Coordinate Model 1.5
267
1
------
E --I
L
i
1.4
I
I
,
I
I
.l
I
1.8 2.0 2.2 2.4 2.6 2.8 Excitation energy, eV. Calculated absorption spectrum of the hydrated elec-
1.6
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Figure 5. tron (300°K).
TABLE I : Properties of the Hydrated Electron
Present
-0.5
0 X.
0.5
1-0
Absorption maximum (300°K). eV Absorption half-width (300°K), eV Hydration energy, eV
Figure 4. Configuration coordinate diagram of the hydrated electron. The abscissa x represents the orientational polarization P x . For example, x = 0, x = 0.5, and x = 1 represent Po,Po.5,and P, in Figure 2, respectively. (I
The Calculated absorption spectrum of the hydrated electron is shown in Figure 5. Table I summarizes the calculated absorption maximum, absorption half-width, and hydration energy with the experimental values. The calculated half-width is by a factor of 2 or 3 greater than Fueki, et al., value, but still smaller than the experimental value. In the present calculation only the variation of the orientational polarization given by eq 48 is considered. If other types of variation of the orientational polarization are taken into consideration, agreement between theory and experiment will be better. Recently, Copeland, et uL.,l1 have developed the semicontinuum model for the solvated electron. Fueki, et u L . , ~ ~ have applied the semicontinuum model to the hydrated electron. In the semicontinuum model the total energies of the solvated electron are calculated as a function of the cavity radius H. The configuration coordinate diagram of the solvated electron is constructed by plotting the total energies against the cavity radius. In the semicontinuum model12 the solvated electron is stabilized by expanding the cavity after it is formed. In general the excitation energy of the solvated electron decreases with increasiing cavity radius. Thus the semicontinuum model suggests that the transient absorption spectrum of the solvated electron shifts toward the red with time. However, in reality the transient absorption spectrum of the solvated electron shifts toward the blue with time. 13~4 In the codiguration coordinate model the solvated electron is stabilized by changing the orientational polarization, for example, from PI to PO after it is formed (see Figure 4). The change of the orientational polarization from .PIto RDimplies the increase of the orientational polarization around the electron (see Figure 2). The increase
calculation
Experiment
2.04 0.52 ('I .7)
1.72a 0.92a
1.7b
* W . C. Gottschail and E. J. Hart, J. Phys. Chem.. 71, 2102 (1967): J. H. Baxendale, Radiat. Res. Suppi.. 4, 139 (1964).
of the orientational polarization implies the reorientation o f the solvent molecules. Thus in the configuration coordinate model the solvated electron is stabilized by reorienting the solvent molecules around the electron. When the orientational polarization changes from PI to PO,the excitation energy of the solvated electron increases (see Figure 4). Thus the configuration coordinate model suggests that the transient absorption spectrum of the solvated electron shifts toward the blue with time. The semicontinuum model12 uses the orientational polarization represented by eq 2. As described in section I, the orientational polarization around the electron is thermally distributed. The orientational polarization represented by eq 2 is the thermal average of this distribution and depends on temperature. By use of the orientational polarization which is already thermally averaged, Fueki, et al., calculated the total energies of the solvated electron as a function of the cavity radius, and tried to explain the width of the absorption spectrum by taking into consideration the thermal distribution of the cavity racfius. That is, in their calculation, after the distribution of the orientational polarization is thermally averaged, the thermal distribution of the cavity radius is taken into consideration. However, if the thermal distribution of the cavity radius is taken into consideration, the thermal distribution of the orientational polarization should also be taken into consideration. In the configuration coordinate model, since the cavity is not assumed, only the thermal distribution of the orientational polarization is taken into consideration. The configuration of the solvent molecules in which the total energy for the excited state is minimum is different from the configuration in which the total energy for the (13) J T Richardsand J K Thomas, J Cherr Phys , 53, 218 (1970) (14) L Kevan, J Chem Phys. 56,838 (1972) The Journal of Physical Chemistry, Voi. 7 7 , No. 2. 1973
Kenneth S. Pitzer
26
ground state is minimum. Thus the optical excitation of the solvated electron from the ground state to the excited state i s followed by the rearrangement of the solvent molecules. In the semicontinuum model12 the optical excitation is followed by the expansion of the cavity. On the other hand, in bhe configuration coordinate model, the optical excitation is followed by the change of the orientational polarization from Po to PI (see Figure 4). The change of the orientational polarization from Po to PI implies the decrease of the orientational polarization around the electron (see Figure 2). The decrease of the
orientational polarization implies the deorientation of the solvent molecules. Thus in the configuration coordinate model the optical excitation is followed by the deorientation of the solvent molecules. In the present calculation the effect of the electronic polarization is neglected. This effect will be taken into consideration in our later paper.
Acknowledgments. The authors wish to thank Professor T. Watanabe and Dr. M. Natori for helpful discussions.
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Thermodytriarnics of Electrolytes. I . Theoretical Basis and General Equation Kenneth S. Pitter inorganic Materials Research Division of the Lawrence Berkeley Laboratory and Department of Chemistry, University of Caiifornia, Berkeiey, California 94720 (Received May 72, 7972) Publication cosfs assisted by The U.S. Atomic Energy Commission
A system of equations for the thermodynamic properties of electrolytes is developed OR the basis of theoretical insights from improved analysis of the Debye-Huckel model as well as recently published numerical calculations for more realistic models. The most important result is the recognition of an ionic strength dependence of the effect of short-range forces in binary interactions. By modifying the usual second virial coefficients to include this feature, one obtains a system of equations which are only slightly more complex than those of Guggenheim but yield agreement within experimental error to concentrations of several molal instead of 0.1 M . If one compares instead with the recently proposed equations of Scatchard, Rush, and Johnson, the present equations are very much simpler for mixed electrolytes (and somewhat simpler for single electrolytes) yet appear to yield comparable agreement with experimental results for both single electrolytes and mixtures.
The thermodynamic properties of aqueous electrolytes have been extensively investigated both experimentally and theoretically. The monographs of Harned and Owen1 and of Robinson and Stokes2 provide excellent summaries. While the detailed nature of these solutions is so complex that an ab inibio quantum-statistical theory is not feasible, the data appear to relate to few enough independent parameters 'to make relatively exact semiempirical representation possible. It is the present objective to develop equations which reproduce the measured properties substantially within experimental accuracy, which are compact and convenient in that only a very few parameters need be tabulated for each substance and the mathematical calculations are simple, which have appropriate form for mixed electrolytes as well as for solutions of a single solute, and whose parameters have physical meaning as far as possible. In 1960 Brewer and the writer3 selected as the best available system one proposed and applied to dilute solutions by Guggenkeid with modifications suggested by Scatchard5 for concentrated solutions. While this system was useful in providing a simple and compact summary of experimental data, it did not fully satisfy the other desired qualities. Recent theoretical advances of Friedman and collaborators"? and of Card and Valleaus provide imThe Journal of Physical Chemistry, Voi. 77, No. 2, 1973
portant insights and support for the greatly improved semiempirical treatment proposed below. Indeed it is interesting that a key idea can be obtained by introducing the Debye-Huckel model and distribution function into modern equations relating such functions to the osmotic pressure. But first we review the Guggenheim equations. (1) H. S. Harned and 8. 8.Owen, "The Physical Chemistry of Electrolytic Solutions," Reinhold, New York, N. Y., 1950. (2) R. A. Robinson and R. H, Stokes, "Electrolytic Solutions," Butterworths, London, 1959. (3) K. S. Pitzer and L. Brewer, revised edition of "Thermodynamics" by G. N. Lewis and M. Randall, McGraw-Hili, New York, N. Y., 1961. (4) (a) E. A. ,Guggenheim, Phil. Mag., [7] 18, 588 (1035); (b) E. A , . Guggenheim and J. C. Turgeon, Trans. Faraday Soc., 51, 747 (1955); (c) E. A. Guggenheim, "Applications of Statistical Mechanics, Clarendon Press, Oxford, 1966, pp 166-170; here Guggenhelm surprisingly abandons the limitation to 0.1 M for validity of his equations for the difference in ~pfrom a reference salt. A5 is obvious from Figure 1 or the figures and tables of ref 4, this is a crude approximation. (5) G. Scatchard. Chem. Rev., 19, 909 (1939); also in W. J. Hamer, Ed., "The Structure of Electrolytic Solutions." Wilev. New York. N. Y., 1959, p 9. (6) J. C. Rasaiah and H. L. Friedman, J. Chem. Phys., 48, 7242 (1968); 50, 3965 (1969). (7) P. S . Ramanathan and H. L. Friedman, J Chem. Phys.. 54, 1086 (1971). (8) (a) D: N. Card and J. P. Valleau, J. Chem. Plrys., 52, 6232 (1970); (b) J. C. Rasaiah, D. N. Card, and J. P. Valleau, J. Chem. Phys., 56, 248 (1972).
