10451
J. Phys. Chem. 1994,98, 1045 1 - 10454
Hardness and Electric Dipole Polarizability. Atoms and Clusters Sanchita Hati and Dipankar Datta’ Department of Inorganic Chemistry, Indian Association for the Cultivation of Science, Calcutta 700 032, India Received: March 23, 1994; In Final Form: July 26, 1994@
A quantitative relation (q = V2(K/a)l13)between electric dipole polarizability a and hardness 9 is developed. In this relation K is a constant and r and a are expressed in au. With K = 0.792, using the available a data,
r
the above equation is found to reproduce the experimental values for a number of open shell atoms and monoatomic cations of various charges very satisfactorily. This equation applies to the clusters of nonmagic numbers of sodium atoms also. However, for closed shell species, the above equation gives hardness values much lower than the experimental data. This is probably because the equation fails to reveal the “extra” amount of stability imparted to a chemical species by a closed shell configuration. Similar lower values of 7 are produced by this equation for clusters of magic numbers of sodium atoms.
In 1963, using the idea of polarizability, Pearson classified the Lewis acids and bases in terms of “hardness”.’ A less polarizable species is hard and a more polarizable one soft, softness being the reciprocal of hardness. This hard-soft classification proved to be quite successful in rationalizing a variety of chemical observations, especially the direction of exchange reaction^.^,^ By polarizability, he meant the ease of deforming the valence electron cloud of a chemical species. A closely related experimental quantity is electric dipole polarizability, a which actually describes the response of the electron cloud of a chemical species to an external electric field much lower than what would be needed to ionize the system. The proportionality constant for the dipole moment p induced by the applied electric field F is a: p = aF.4 A connection between Pearson’s hardness and a was immediately sought by several worker^.^ However, because of the lack of a quantitative definition of hardness, no definite conclusions could be drawn then. In this context, Jorgensen even felt in 1967 that a is “a far more physical than chemical q ~ a n t i t y ” .Pearson’s ~ idea of polarizability originally rested upon the nature of the charge on and size of a chemical species-for ions of similar charge hardness decreases with the increase in its radius and for cations of the same radius hardness increases with the amount of positive charge. Fajans’ rules are helpful in this regard.6 In 1983, Pearson together with Parr gave a quantitative definition of hardness in the form of eq 1, where Eel is the electronic
(r)
1 11 = - ( S ~ E , , I S N= ~ )+scl/slv, 2
2
(1)
energy of the system having N number of electrons and p is the chemical potential of the electron cloud.’ Equation 1 is based upon density functional theory. Its finite difference approximation allows one to calculate 17 in terms of the ionization potential I and electron affinity A of a chemical species-eq 2. It should be noted that eq 1 yields eq 2 only
r = (I - A)/2
(2)
when the total energy of a system is assumed to be a quadratic function of its charge q (= Z - N , where Z is the nuclear charge). With a working formula for being available (eq 2), currently there has been a renewed interest in finding the relation between 7 and a. A brief review of the situation is available.8 Recently Ghanty and Ghosh* have empirically found that
r
@
Abstract published in Advance ACS Abstracts, September 1, 1994.
0022-3654/94/2098-1045 1 $04.50/0
softness (1/r,1) correlates linearly with all3for a number of atoms and sodium clusters. Almost at the same time, similar observations have been reported by Fuentealba and Reyes9 for certain atoms and monoatomic ions. Herein we investigate the relation between and a quantitatively. The energy E(q) required to charge a conducting sphere of radius R with q is classically given by E(q) = q2/2R. Similarly, E(q+l) = ( q 1)’/2R and E(q-1) = (q - 1)’/2R. NOW,I = E(q+l) - E(q) and A = E(q) - E(q-1). Consequently, it follows, when all the quantities are expressed in au, that
r
+
(I - A)/2 = 1/2R
(3)
This approach is not new. It was adopted by Pearson himself while discussing the hardness TI, of a species being independent of its charge.1° However, though no explicit charge dependence is indicated by eq 3, it is introduced through the radius. An inherent assumption in deriving eq 3 is that the radius R of the sphere does not change with the increase or decrease in the charge by one unit, which is valid only for a metallic sphere. Equation 3 has been obtained by Komorowski” also by applying electrodynamical equations to chemical potential assuming R to be constant. For metal clusters of finite radius R, semiclassically the same expression has been deduced.12 However, from density functional theory, for metal clusters ( I - A)/2 is found to be equal to 1/2(R d), where d, a microscopic distance, is a sort of correction to the jellium sphere. For our purpose here we shall use eq 3. For a conducting sphere, a can be equated to R3.13 But, for the atoms, because of the inhomogeneity of the electronic cloud, a more appropriate relation seems to be eq 4 , where K is the
+
a = KR3
(4)
proportionality constant. Using an approach based on the atomic oscillation theory, Dmitrieva and Plindov have derived the value of K as 0.585.14 Combining eqs 2-4, we have eq 5 , where a and 7 are
1 7 =-(~/a)’’~ 2
(5)
expressed in au. Equation 5 gives us a quantitative relation between r and a. It is really surprising to find that though relations like eqs 3 and 4 were known, the validity of a combination of them, eq 5, has never been tested earlier.
0 1994 American Chemical Society
10452 J. Phys. Chem., Vol. 98, No. 41, 1994
Hati and Datta
TABLE 1: Application of Eq 5 to Atom#
0.65
7
I;lcal
atom a K=1 K = 0.585 H 4.521 0.302 (1.28) 0.252 (1.07) He 1.383 0.449 (0.72) 0.375 (0.60) Li 163.981 0.091 (1.03) 0.076 (0.86) Be 37.790 0.149 (0.90) 0.125 (0.76) B 20.447 0.183 (1.24) 0.153 (1.04) C 11.877 0.216 (1.19) 0.181 (0.98) N 7.423 0.256 (0.96) 0.214 (0.80) 0 5.412 0.285 (1.28) 0.238 (1.07) F 3.779 0.321 (1.24) 0.268 (1.04 Ne 2.672 0.360 (0.63) 0.300 (0.53) Na 159.257 0.092 (i.09j 0.077 (0.92j 71.531 0.120 (0.84) 0.100 (0.70) 56.280 0.130 (1.27) 0.109 (1.07) Si 36.305 0.151 (1.22) 0.126 (1.02) P 24.496 0.172 (0.96) 0.144 (0.80) S 19.570 0.185 (1.22) 0.155 (1.02) C1 14.711 0.204 (1.19) 0.170 (0.99) Ar 11.074 0.224 (0.57) 0.187 (0.48) K 292.872 0.075 (1.06) 0.063 (0.89) Ca 161.282 0.092 (0.63) 0.077 (0.52) Sc 120.118 0.101 (0.86) 0.084 (0.71) Ti 98.524 0.108 (0.87) 0.090 (0.73) V 83.678 0.114 (1.00) 0.095 (0.83) Cr 78.279 0.117 (1.04) 0.098 (0.87) Mn 63.433 0.125 (0.91) 0.104 (0.76) Fe 56.685 0.130 (0.93) 0.109 (0.78) Co 50.611 0.135 (1.02) 0.113 (0.86) Ni 45.888 0.140 (1.18) 0.117 (0.98) Cu 45.247 0.140 (1.18) 0.117 (0.98) Zn 42.851 0.143 (0.79) 0.119 (0.66) Ga 54.795 0.132 (1.23) 0.110 (1.03) Ge 40.962 0.145 (1.16) 0.121 (0.97) As 29.085 0.163 (0.99) 0.136 (0.82) Se 25.441 0.170 (1.20) 0.142 (1.00) Br 20.582 0.182 (1.17) 0.152 (0.98) Kr 16.762 0.195 (0.56) 0.163 (0.47) Rb 319.190 0.073 (1.07) 0.061 (0.90) Sr 186.250 0.087 (0.64) 0.073 (0.54) Y 153.184 0.093 (0.79) 0.078 (0.84) Zr 120.793 0.101 (0.86) 0.084 (0.71) Nb 105.947 0.106 (0.96) 0.089 (0.81) Mo 86.377 0.113 (0.99) 0.094 (0.82) Ru 64.783 0.124 (1.13) 0.104 (0.94) Rh 58.034 0.129 (1.11) 0.108 (0.93) Pd 32.391 0.157 (1.10) 0.131 (0.92) Ag 53.176 0.133 (1.16) 0.111 (0.96) Cd 48.587 0.137 (0.80) 0.114 (0.67) In 68.832 0.122 (1.18) 0.102 (0.99) Sn 51.961 0.134(1.20) 0.112 (1.00) Sb 44.538 0.141 (1.01) 0.118 (0.84) Te 37.115 0.150 (1.16) 0.125 (0.97) I 33.134 0.156 (1.15) 0.130 (0.96) Xe 27.290 0.166 (0.55) 0.139 (0.46) Cs 402.193 0.068 (1.08) 0.057 (0.90) Ba 267.903 0.078 (0.73) 0.065 (0.61) La 209.869 0.084 (0.88) 0.070 (0.74) Hf 109.321 0.105 (0.95) 0.088 (0.80) Ta 88.401 0.112 (0.81) 0.094 (0.68) W 74.905 0.119 (0.90) 0.099 (0.75) Re 65.457 0.124 (0.87) 0.104 (0.73) Os 57.360 0.130 (0.93) 0.109 (0.78) Ir 51.286 0.135 (0.96) 0.113 (0.81) Pt 43.863 0.142 (1.10) 0.119 (0.92) Au 39.139 0.147 (1.16) 0.123 (0.97) Hg 38.465 0.148 (0.72) 0.124 (0.61) Tl 51.286 0.135 (1.26) 0.113 (1.06) Pb 45.888 0.140 (1.08) 0.117 (0.90) Bi 49.937 0.136 (0.99) 0.114 (0.83)
zg
K = 0.792 0.279 (1.18) 0.415 (0.67) 0.084 (0.95) 0.138 (0.84) 0.169 (1.15) 0.200 ( 1.09) 0.237 (0.89) 0.264 (1.18) 0.297 (i.isj 0.333 (0.59) 0.085 (i.oij 0.111 (0.78) 0.120 (1.18) 0.140 (1.13) 0.159 (0.89) 0.171 (1.12) 0.189 (1.10) 0.207 (0.53) 0.069 (0.97) 0.085 (0.58) 0.093 (0.79) 0.100 (0.81) 0.105 (0.92) 0.108 (0.96) 0.116 (0.85) 0.120 (0.86) 0.125 (0.95) 0.129 (1.08) 0.129 (1.08) 0.132 (0.73) 0.122 (1.14) 0.134 (1.07) 0.151 (0.91) 0.157 (1.11) 0.168 (1.08) 0.180 (0.52) 0.067 (0.98) 0.080 (0.59) 0.086 (0.92) 0.093 (0.79) 0.098 (0.89) 0.104 (0.91) 0.115 (1.04) 0.119 (1.03) 0.145 (1.01) 0.123 (1.07) 0.127 (0.74) 0.113 (1.10) 0.124 (1.11) 0.130(0.93) 0.139 (1.08) 0.144 (1.06) 0.153 (0.51) 0.063 (1.00) 0.072 (0.67) 0.078 (0.82) 0.097 (0.88) 0.104 (0.75) 0.110 (0.83) 0.115 (0.81) 0.120 (0.86) 0.125 (0.89) 0.131 (1.01) 0.136 (1.07) 0.137 (0.67) 0.125 (1.17) 0.129 (0.99) 0.126 (0.92)
rl,,,
0.236 0.62ob 0.088 0.165 0.147 0.184 0.266 0.223 0.258 0.568* 0.084 0.143 0.102 0.124 0.179 0.152 0.172 0.3936 0.071 0.147 0.118 0.124 0.114 0.112 0.137 0.140 0.132 0.119 0.119 0.181 0.107 0.125 0.165 0.142 0.155 0.347b 0.068 0.136 0.093 0.118 0.110 0.114 0.110 0.116 0.143 0.115 0.171 0.103 0.112 0.140 0.129 0.136 0.302* 0.063 0.107 0.095 0.110 0.139 0.132 0.142 0.140 0.140 0.129 0.127 0.204 0.107 0.130 0.137
For the meanings of the symbols, see text. All quantities are given in au. Sources of data: a,ref 15a; qexp,ref 10 unless otherwise specified. Within the parentheses are given the qcaJvexpvalues. The average value of vcdvexp:K = 1, 1.00 f 0.20; K = 0.585, 0.84 f 0.16; K = 0.792, 0.93 f 0.17. Excluding the closed shell atoms, the average value of vcal/qexp:K = 1, 1.07 f 0.13; K = 0.585, 0.90 f 0.11; K = 0.792,0.99 f 0.12. Value obtained theoretically by Robles and Bart~lotti.'~
0.05
0.05
0.45
0.25
0.65
experimental hardness Figure 1. Correspondencebetween experimental hardness and hardness calculated (in au) by eq 5 with K = 0.792 for some 63 atoms. The closed shell atoms are marked by dark circles. For data, see Table 1. The line drawn has a slope of unity.
From the available electric dipole polarizability data,lSaJ6J7 we have calculated the 7 values (l;lcd)by eq 5 for some 63 atoms (Table l), where the experimental hardnesses (uexp) are known,10~18~19 with K = 1 and 0.585. As revealed by the ratio of qcd and qexp, the calculated values are close to the experimental ones (see Table 1). Only the closed shell atoms (the alkaline earth metals, Zn, Cd, Hg, and the noble gases) are in general found to deviate significantly. In these cases the calculated 17 values are uniformly much lower than the experimental data. The situation improves if we assume K = 0.792, which is the average of 1 and 0.585. Then, excepting the closed shell atoms, the experimental data are mostly reproduced within &12% (Table 1; see Figure 1 also). Very recently Politzer and co-workers have studied empirically the relation between a and ab initio SCF volume at the 6-31G* level for some 25 molecules of different shapes and sizes.13 From their study, the volume V corresponding to the 0.01 au density contour can be considered as roughly proportional to a (eq 6, correlation coefficient = 0.962; a and V both are
+
a = 0.010 0.1674V
(6)
expressed in A3). Neglecting the constant 0.010 in eq 6 and replacing V by an equivalent sphere of effective radius R, the value of K (eq 4) comes out as 0.701, which is very near to the average value of K assumed above. Equation 5 is found to apply in Na clusters (Na,) also, as the 7 values calculated from the experimental a data20 match satisfactorily (Table 2 and Figure 2) with the 7 values obtained theoretically by Beck21except when n is a magic numberz2(e.g. 18, 20, 34,40, etc.). For clusters of magic numbers of sodium atoms, our qcd values are somewhat lower than those of Beck, as found in the case of the closed shell atoms. To assess the effect of charge on 7,we have calculated the 7 values of some monoatomic ions also by eq 5. For some 13 open shell monoatomic cations of various charges, using the theoretical a values of Fuentealba and re ye^,^ eq 5 reproduces the experimental 7 valueslSbvery well with K = 0.792 (Table 3 and Figure 3). Thus eq 5 shows that the effect of charge on the 7 value of a species is absorbed in its a value. However, for closed shell monoatomic cations like Li+, Be2+, B3+, etc.,
J. Phys. Chem., Vol. 98, No. 41, 1994 10453
Hardness and Electric Dipole Polarizability
TABLE 2: Application of Eq 5 to Sodium Clusters Nanu
TABLE 3: Application of Eq 5 to Some Open SheU Monoatomic IoW
VCd
a
n
4 5 6 7
545.929 725.418 823.678 810.459 10 1296.356 12 1494.470 15 1751.036 17 1822.291 18‘ 1874.778 19 2024.325 20‘ 2076.718 21 2090.255 34‘ 3422.127 4OC 4013.289
K= 1 0.061 (1.07) 0.056 (0.92) 0.053 (0.96) 0.054 (1.00) 0.046 (1.02) 0.044 (1.00) 0.041 (0.95) 0.041 (0.98) 0.040 (0.87) 0.039 (0.95) 0.039 (0.80) 0.039 (1.05) 0.033 (0.80) 0.031 (0.91)
K = 0.585 0.051 (0.89) 0.047 (0.77) 0.044 (0.80) 0.045 (0.83) 0.038 (0.84) 0.037 (0.84) 0.034 (0.79) 0.034 (0.81) 0.033 (0.72) 0.033 (0.80) 0.033 (0.67) 0.033 (0.89) 0.028 (0.68) 0.026 (0.76)
K = 0.792 0.056 (0.98) 0.052 (0.85) 0.049 (0.89) 0.050 (0.93) 0.042 (0.93) 0.041 (0.93) 0.038 (0.88) 0.038 (0.90) 0.037 (0.80) 0.036 (0.88) 0.036 (0.73) 0.036 (0.97) 0.030 (0.73) 0.029 (0.85)
vb 0.057 0.061 0.055 0.054 0.045
Be+ BZ+ C3+ N4+
0.044
05+
0.043 0.042 0.046 0.041 0.049 0.037 0.041 0.034
F6+
“The meanings of the symbols are the same as in the text. All quantities are given in au. The a values are taken from ref 20. Within the parentheses are given the vcd/vexpvalues. The average value of vcdlvexp: K 1, 0.95 f 0.08; K = 0.585, 0.79 f 0.07; K = 0.792, 0.87 f 0.08. Excluding the magic numbers (see footnote c): K = 1, 0.99 f 0.05; K = 0.585,0.83 f 0.04; K = 0.792,0.91 f 0.04. Values obtained theoretically by Beck.*l Magic number; see text.
0.05
VCd
ion
Mg+ AIZ+ Si3+
p+ S5+
C16+ Ar7+
a 24.70 7.92 3.50 1.81 1.05 0.65 35.10 14.40 9.03 4.50 2.95 2.00 1.50
K=l 0.172 (1.05) 0.251 (1.07) 0.329 (1.05) 0.410 (1.09) 0.492 (1.10) 0.577 (1.12) 0.153 (1.12) 0.205 (1.16) 0.240 (1.12) 0.303 (1.21) 0.349 (1.22) 0.397 (1.23) 0.437 (1.22)
K = 0.585 0.144 (0.88) 0.210 (0.89) 0.275 (0.88) 0.343 (0.91) 0.411 (0.92) 0.482 (0.94) 0.128 (0.94) 0.171 (0.97) 0.201 (0.93) 0.253 (1.01) 0.292 (1.02) 0.332 (1.03) 0.365 (1.02)
K = 0.792 0.159 (0.97) 0.232 (0.99) 0.304 (0.97) 0.379 (1.01) 0.455 (1.02) 0.534 (1.04) 0.141 (1.04) 0.190 (1.07) 0.222 (1.03) 0.280 (1.11) 0.323 (1.13) 0.367 (1.14) 0.404 (1.13)
vexp 0.163 0.235 0.312 0.376 0.446 0.515 0.136 0.177 0.215 0.251 0.286 0.323 0.357
a For the meanings of the symbols, see text. All quantities are given in au. Sources of data: a,ref 9; vexp,ref 15b. Within the parentheses are given the vc&exp values. The average value of 7;lcd/vexp:K = 1, 1.13 f 0.07; K = 0.585, 0.95 f 0.05; K = 0.792, 1.05 f 0.06.
2 1
4
0.40
aJ
cl
p 0.04 i
0.30
4 4
0.03
1
/. 0.10 0.10
0.02
0.03
0.04
0.05
0.06
hardness values of Beck Figure 2. Correspondence between hardness values obtained by Beck and our hardness values calculated (in au) by eq 5 with K = 0.792 for some clusters of sodium. The closed shell clusters are marked by dark circles. For data, see Table 2. The slope of the line drawn is unity.
eq 5 (using the a values of Pauling4) gives 7 values which are much lower than the experimental ones. Similar lower values of 17 are obtained by eq 5 for a number of closed shell molecules also. A general reason for the failure of eq 5 deals with the validity of eq 3. It can be stated that in a case where eq 5 does not reproduce qexpproperly there is a marked dependence of R on q in the range q f 1. For molecules which cannot be considered spherical, eq 3 is probably not applicable. However, a notable feature of our present study is that, only for the closed shell systems, eq 5 predicts hardness values significantly lower than the actual ones. This observation can be reasoned qualitatively. It is now well understood that a closed shell configuration always gives rise to an extra stability (thermodynamic) of a system.23 From mass spectra, it has been observed that the clusters of magic numbers of lithium atoms are extra table.^^.^^ Similar is the case with the clusters of sodium, another alkali meta1.21,23,24There are indirect evidences that the stability
0.30 0.40 0.50 0.60 experimental hardness Figure 3. correspondencebetween experimental hardness and hardness calculated (in au) by eq 5 with K = 0.792 for some 13 open shell monoatomic cations. For data, see Table 3. The line is drawn with a slope of unity to emphasize the fit. 0.20
(thermodynamic) of a chemical system increases with the We feel that in a increase in the value of hardness.22,23*25-27 way eq 5 fails to reveal this “extra” stability in the closed shell systems and sodium clusters of magic numbers of atoms. Earlier from density functional theory, for an atom Vela and GazquezZ8have derived eq 7, where (9)is the expectation
(7) value of 9 for the corresponding anion and ($)+ is that for the corresponding cation. However, the a values calculated from eq 7 using the experimental ( I - A ) data and Hartree-Fock values of (?)+ are found to be half of the actual ones. Moreover, the observed dependence8q9of a on the third power of softness cannot be discerned from eq 7. Here we have shown that at least for open shell systems there exists a rather simple relation between a and 17 like eq 5. In such cases the experimental 17 values are reproduced best when K = 0.792. Thus, in reality, an atom, as far as the relation between a and R is concerned, is found to behave as in between
10454 J. Phys. Chem., Vol. 98, No. 41, 1994 a hard sphere and a Dmitrieva-Plindov atom. In analogy with the closed shell atoms and monoatomic ions, the failure of eq 5 in the clusters of magic numbers of sodium atoms in a way supports the existence of a shell structure for the metal clusters which is a subject matter of much current interest.29
References and Notes (1) Pearson, R. G. J . Am. Chem. SOC. 1963, 85, 3533. (2) Pearson, R. G. Coord. Chem. Rev. 1990, 100, 403 and references therein. (3) Dam, D.; Singh, S. N. J . Chem. Soc., Dalton Trans. 1991, 1541. (4) Pauling, L. Proc. R. SOC.A 1927, 114, 181. (5) Jorgensen, C. K. Stmcr. Bonding (Berlin) 1967, 3, 106 and references therein. (6) Huheey, J. E. Inorganic Chemistry: Principles of Structure and Reactivity, 3rd ed.; Harper and Row: New York, 1983; pp 129-131. (7) Parr, R. G.; Pearson, R. G. J . Am. Chem. SOC.1983, 105, 7512. (8) Ghanty, T. K.; Ghosh, S. K. J . Phys. Chem. 1993, 97, 4951. (9) Fuentealba, P.; Reyes, 0. THEOCHEM 1993, 282, 65. (IO) Pearson, R. G. Inorg. Chem. 1988, 27, 734. (11) Komorowski, L. Chem. Phys. 1987, 114, 55. (12) Wood, D. M. Phys. Rev. Lett. 1981, 46, 749. (13) Brinck, T.; Murray, J. S.; Politzer, P. J . Chem. Phys. 1993, 98, 4305. (14) Dmitrieva, I. K.; Plindov, G. I. Phys. Scr. 1983, 27, 402.
Hati and Datta (15) CRC Handbook of Chemistry and Physics, 71st ed.; CRC Press: Boca Raton, FL, 1990-1991; (a) pp 10-193-10-209; (b) pp 10-210-10211. (16) It should be mentioned that the average error in most of the atomic a values used here is &50%.'5a Recently these have been corrected to f20-30% by Fricke." Since the correction is of an empirical nature, here we have not used the a values of Fricke. (17) Fricke, B. J. Chem. Phys. 1986, 84, 862. (18) In the case of the noble gases where values are not available, we have compared OUT qCd with the r] values obtamed theoretically by Robles and Bart~lotti.'~ (19) Robles, J.; Bartolotti, L. J. J . Am. Chem. SOC.1984, 106, 3723. (20) Selby, K.;Vollmer, M.; Masui, J.; Kresiu, V.; Heer, W. A. de; Knight, W. D. Phys. Rev. B 1989, 40, 5417. (21) Beck, D. E. Solid State Commun. 1984, 49, 381. (22) Harbola, M. K. Proc. Natl. Acad. Sci. U.S.A.1992, 89, 1036 and references therein. (23) Parr, R. G.; Zhou, Z. Acc. Chem. Res. 1993, 26, 256. (24) Knight, W. D.; Clemenger, K.; Heer, W. A. de; Saunders, W. A.; Chou, M. Y.; Cohen, M. L. Phys. Rev. Lett. 1984, 52, 2141. (25) Zhou, Z.; Parr, R. G. J. Am. Chem. SOC.1989, 111, 7371. (26) Pearson, R. G. Acc. Chem. Res. 1993, 26, 250. (27) Datta, D. J. Phys. Chem. 1992, 96, 2409; Inorg. Chem. 1992, 31, 2796 and references therein. (28) Vela, A.; Gazquez, J. L. J. Am. Chem. SOC. 1990, 112, 1490. (29) Baguenard, B.; Pellarin, M.; Lerme, J.; Vialle, J. L.; Broyer, M. J . Chem. Phys. 1994, 100, 754 and references therein.
