4951
J. Phys. Chem. 1993,97, 4951-4953
Correlation between Hardness, Polarizability, and Size of Atoms, Molecules, and Clusters Tapan K. Ghanty and Swapan K. Ghosh' Heavy Water Division, Bhabha Atomic Research Centre, Bombay 400 085, India Received: September 28, 1992
Interesting correlations between the hardness, polarizability, and size of different systems like atoms, molecules, metal clusters, and carbon clusters are demonstrated to exist. This provides support for the use of these physical properties for qualitative characterization of the hard soft acid base concept. Some of these correlations might also be useful in predicting hardness or softness or polarizability of a species from its size.
The concept of hard and soft acids and bases introduced by Pearson' has been highly successful in systematizing and rationalizing a wide variety of chemical reactions, even though the softness or hardness of a species was initially characterized only qualitatively in terms of properties like polarizability, size, electronegativity, etc. Thus, a smaller sized and less polarizable species was designated as hard while a larger and more polarizable species was considered to be soft. A quantitative formulation of the hardness and softness concept is rather recentz-5 and is due to Parr and Pearson2who defined hardness as the second derivative of the electronicenergy (E)with respect to the number of electrons (N), viz. q
= 'f2(d2E/dN2)= l / , ( d p / d N )
(1)
where p, the chemical potential6 of the electron cloud, is equal7 to the negative of the electronegativity, x = -(dE/aN). Within the finite difference approximation, the two derivatives x and q can be obtained from the ionization potential (I)and the electron affinity ( A ) using the relations x = (I+A ) / 2 and q = (I- A ) / 2 . Softness ( S ) has been defined as the reciprocal of hardness (S = l/q) and also in terms of local ~oftness.~ Thus, using the experimental values of I and A , a quantitative scale of hardness and softness has been possible. It is therefore of interest to see how these values of q or S correlates with physical properties like polarizability, electronegativity, size, etc., which were originally used to characterize qualitatively the hardness or softness of a chemical species. A few studies in this direction have already been reported in the recent past for atomic systems. Thus, Politzer9 as well as Sen et al.Io considered a linear relationship between polarizability (a)and softness, while Vela and Gazquezl have expressed the ratio (a/S)in terms of an integral, which varies from atom to atom since it involves the electron densities of the corresponding anion and cation. Nagle,12 on the other hand, has shown for atomic systems a simple linear correlation between the softness S and ( c ~ / n ) ' where / ~ , both S and a are calculated using the density functional method and n represents the number of valence electrons of the atom. Komorowski and co-workersI3have studied several aspects of electronegativityand hardness and demonstrated14 simple relations between these quantities and atomic sizes. Studies have also been reported on the correlation of polarizability with electronegativityl2 as well as ionization potentialI5 for atomic systems, where again simple approximate linear dependences have been observed. Thus, as far as the relationshipbetween polarizability,hardness, size, electronegativity, etc. are concerned, until now studies have been confined mainly to atoms. It is thus of interest to further explore the nature of the relationship among these quantities for other many-electron systems as well. While molecules are examplesof such systems of interest, there has also been a growing interest in other few-particle systems like metallic clusters and more recently carbon clusters. Harbolal6 has recently reported
a correlation of the stability of metal clusters with hardness, which can be interpreted as a manifestation of the principle of maximum hardness." Size dependencesof ionizationpotentials and electron affinities have also been reported.18 In this work, we attempt to investigate, in a unified manner, the relationships that exist between the hardness and softness, the polarizability, and the size for atoms, molecules, metal clusters, and carbon clusters. The softness, representing9essentially the charge capacity of a species, is expected13to be proportional14J9to its size. The plots of softness of a number of atoms and AB, type molecules against the corresponding atomic radii and A-B bond lengths respectively can be fitted2O with straight lines with correlation coefficients 0.91 and 0.88,respectively. In the jellium model, metal clusters are considered to be spherical with radius ( R ) given by
R = rsN'/3
(2)
where r, is the bulk WignerSeitz radius for the electrons and N is the number of atoms in the cluster. In Figure 1, sodium metal cluster softness2' has been plotted against the radius, indicating a good linear correlation (correlation coefficient = 0.88) of softness with the cluster size. (Other metal clusters could not be studied due to nonavailability of the electron affinity values.) For carbon clusters, which are mostly nonspherical, we define the size in terms of an effective radius of an equivalent sphere with a surface area assumed to be proportional to the number of atoms. The effective radius (RN) of an N-atom carbon cluster can thus be expressed in terms of the radius of a CSo cluster (which is spherical with a radius22 of 3.531 A) as
RN = R,0(N/60)"2
(3)
In Figure 2, we have plotted the inverse of the HOMO-LUMO gapsz3against the effective radii of carbon clusters obtained from eq 3. Since the HOMO-LUMO gap is a measurez4of hardness, Figure 2 essentially indicates a very good linear correlation (correlation coefficient = 0.93)between softness and radius for carbon clusters. The other important property employed to visualize the hardsoft concept is the polarizability. For a sphere of radius R, the polarizability (a)can be approximated as a = R3. This has been verifiedz0by a very good linear plot (correlation coefficient = 0.95)of al/3 vs size for a number of atoms. For metal clusters, a similar plot, using experimental values of polarizabilities,25is shown in Figure 3, where a correlation coefficient of 0.99 is observed. It is interesting to note that such a correlation is not restricted to near spherical species like atoms and metal clusters alone, as is demonstrated in Figure 4, where a plot of all3 for molecules26 (AB, type) vs A-B bond lengths2' shows an excellent linear correlation (correlation coefficient = 0.98). From the relationship between softness and size demonstrated in Figures 1 and 2, one would expect a linear relationship between
0022-3654/93/2097-495 1%04.00/0 0 1993 American Chemical Society
4952
0.2
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993
Ghanty and Ghosh
t .
1
-,
,',-
-A
CLUSTER R4DIUS
Figure 1. Plot of softness (in (eV)-l) vs cluster radius (in au) for metal clusters.
BOND LENGTH
Figure 4. Plot of cube root of polarizability (in A) vs bond length (in A) for molecules.
I
I
35
40
~
4:
CLUSTER RADIUS
Figure 2. Plot of softness (in 8-l units) vs cluster radius (in A) for carbon clusters.
I
1
I
I
6.0
80
10.0
12.0
1.
CLUSTER RADIUS Figure 3. Plot of cube root of polarizability (in A) vs cluster radius (in au) for metal clusters. a 1 Jand 3 softness. This is demonstrated by a very good linear correlationof all3vs softnessfor metal clustersand carbon clusters in Figures 5 and 6 (with correlation coefficients 0.86 and 0.93), respectively. Similar plots for atoms and molecules also show20 very good correlation with correlation coefficients of 0.96 and 0.88, respectively. This is in contrast to poorer correlation observed in earlier studies involving plots of a vs 9 or l / q for
I
2
1
L. 5 * E
. *
1
' 5 2c CLCSTER SOFTNESS
2 5
I
?r
Figure 6. Plot of cube root of polarizability (in au) vs softness (in units) for carbon clusters.
atoms. While the values of polarizabilities for molecules and metal clusters are taken from the literature,25s26 the same values for the carbon clusters (except CSO and (270) are not yet available. Hence we have proposed a simple formula for the polarizability of an N-atom carbon cluster, viz. (4)
Correlation between Hardness, Polarizability, and Size 9.0
,
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993 4953
study provides a quantitative justification for the same idea. Some of the correlations discussed here can be used to predict one quantity from the other. For example, the very good correlation of molecular polarizability with A-B bond length, as demonstrated in Figure 4, suggests that an accurate prediction of polarizability of AB, type molecules might be possible from the corresponding A-B bond lengths. Acknowledgment. It is a pleasure to thank H. K. Sadhukhan for his kind interest and encouragement. References and Notes
COEESIVE ENERGY PER ATOM
Figure 7. Plot of cube root of polarizability (in au) vs cohesive energy (in eV) for carbon clusters.
where 6 is an effective distance outside the radius of a carbon cluster (cf. jellium boundary in case of metal clusters) up to which the electron density extends in the presence of an applied electric field. The value of 6, determined from the polarizability28 (0’60) of c 6 0 along with eqs 3 and 4, leads to the expression for polarizability ( a ~given ) by
The value of 0‘70 (384.2 au) obtained from eq 5 agrees very well with the (ab initio) calculated value (383 au) reported in l i t e r a t ~ r e . ~For ~ other carbon clusters, one can get a rough estimate of the polarizabilities from eq 5 . In Figure 6,the values of the polarizabilities used correspond to results from eq 5. The size dependenceof the calculated polarizability of carbon clusters is given by eq 5 , and hence the corresponding plot is omitted. Figures 1-6 thus establish the desired relationships between the hardness and softness, the polarizability, and the size of molecules and clusters. The polarizability is a measure of the response of the outer shell electrons toward an external perturbation. Chemical binding also can be viewed30 as resulting from reorganization and redistribution of the outermost electron density of atoms due to mutual perturbation, when different atoms are brought closer. Thus it would be of interest to see if there exists some correlation between polarizability and the binding energy in multiatom systems. We have therefore plotted in Figure 7 the cube root of the polarizabilities (all3) against cohesive energies for carbon clusters,3’ where the linear fit is excellent (correlation coefficient = 0.98). A similar study20 for metal clusters shows a correlation coefficient of 0.89. These observations of correlations provide support for our recent inclusion of the polarizability parameter in developing32 a simple picture of chemical binding. It has been shown by Parr et al.33 that a-resonance energy, which is a measure of aromaticity of organic molecules,correlates very well with the absolute hardness of molecules. So it is of interest to see whether such correlation exists in carbon clusters which are candidates for three-dimensional aromaticity. We have observedzothat the HOMO-LUMO gaps23 (which are a measure of the absolute hardness of a species) multiplied by Nz correlate very well (correlation coefficient = 0.93) with a-resonance energies23 of N-atom carbon clusters. It is interesting to note that the aromaticity in carbon clusters is higher34 than that in normal aromatic compounds. Also, the hardness of the carbon clusters for N 3 60 decreases much faster as the cluster size increases. This work thus demonstrates interesting correlations among the hardness, polarizability, and size of different many-electron systemslike atoms, molecules,metal clusters, and carbon clusters. Although such a correlation was inherent in the qualitative characterization of the hard soft acid base concept, the present
(1) Pearson, R. G. J . Am. Chem. SOC.1963,85, 3533. For a recent review, see: Pearson, R. G. Coord. Chem. Rev.1990,100,403and references therein. See also: Datta, D. Inorg. Chem. 1992, 31, 2797. (2) Parr, R. G.; Pearson, R. G. J . Am. Chem. Soc. 1983,105,7512. (3) Berkowitz, M.; Ghosh, S.K.; Parr, R. G. J . Am. Chem. SOC.1985, 107,6811. Ghosh, S.K. Chem. Phys. Lett. 1990, 172,77. (4) Yang, W.;Parr, R. G. Proc. Natl. Acad. Sci. U S A . 1985,82,6723. (5) Harbola, M. K.; Chattaraj, P. K.; Parr, R. G. Isr. J . Chem. 1991, 31, 395. (6) March, N. H., Deb, B. M., Eds.; Single Particle Density in Physics and Chemistry; Academic Press: New York, 1987. Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford Univ. Press: New York, 1989. (7) Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J. Chem. Phys. 1978,68, 3801. (8) Jorgensen, C. K. Srruct. Bonding (Berlin) 1967,3, 106. (9) Politzer, P.J. Chem. Phys. 1987,86, 1072. (10) Sen, K. D.; Bohm, M. C.; Schmidt, P. C. Struct. Bonding (Berlin) 1987,66,99. (11) Vela, A.; Gazquez, J. L. J. Am. Chem. SOC.1990, 112, 1490. (12) Nagle, J. K. J . Am. Chem. SOC.1990,112,4741. (13) Komorowski, L.; Lipinski, J. Chem. Phys. 1991, 157,45. (14) Komorowski, L. Chem. Phys. 1987,114,55; Chem. Phys. Lett. 1987, 134,536;2.Naturforsch. 1987,42a,167. (15) Fricke, B. J . Chem. Phys. 1986,84,862.See also: Dmitrieva, I. K.; Plindov, G. I. Phys. Scr. 1983,27,402. (16) Harbola, M. K. Proc. Natl. Acad. Sci. U.S.A. 1992,89, 1036. (17) Parr, R. G.; Chattaraj, P. K . J . Am. Chem. Soc. 1991, 113, 1854. (18) Harbola, M. K. J . Chem. Phys. 1992,97,2578. (19) For an earlier paper discussing this aspect for atoms, see: Ray, N. K.; Samuels, L.; Parr, R. G. J. Chem. Phys. 1979,70, 3680. (20) Ghanty, T.K.; Ghosh, S.K. Unpublished results. (21) Softness of Na metal clusters (r, = 4.0)calculated using I and A values from the following: Beck, D. E . Solid State Commun. 1984,49,381. (22)Dunlap, B. I.; Brenner, D. W.; Mintmire, J. W.; Mowrey, R. C.; White, C. T. J. Phys. Chem. 1991,95,8737. (23) Values of r energy and HOMO-LUMO gaps are taken from the following: Liu, X.;Schmalz, T. G.; Klein, D. J. Chem. Phys. Lert. 1992,188, 550. In Figures 2 and 6, the quantity plotted is the average of the inverse HOMO-LUMO gaps for all isomers with a particular N. (24) Pearson, R. G. Proc. Narl. Acad. Sci. U.S.A. 1986,83,8440. (25) Polarizabilities of Na metal clusters are taken from the following: Selby, K.; Vollmer, M.; Masui, J.; Kresin, V.; De Heer, W. A.; Knight, W. D. Phys. Rev. 1989,840, 5417. (26) For values of polarizability of molecules, see: Miller, T. M. In CRC Handbook of Chemistry and Physics, 70th ed.; CRC Press: Boca Raton, FL, 1989;pp E70-E79. (27) Bond lengths of molecules are taken from the following: Huheey, J. E. Inorganic Chemistry, 3rd ed.; Harper & Row: New York, 1983. (28) Fowler, P. W.; Lazzeretti, P.; Zanasi, R. Chem. Phys. Lett. 1990, 165,79. (29) Baker, J.; Fowler, P. W.; Lazzeretti, P.; Malagali, M.; Zanasi, R. Chem. Phys. Lett. 1991,184,182. (30) Ghanty,T. K.;Ghosh,S. K. J.Phys. Chem. 1991,95,6512.Ghanty, T.K.; Ghosh, S. K. Inorg. Chem. 1992,31,1951. Ghanty, T.K.;Ghosh, S. K. J. Chem. SOC.,Chem. Commun. 1992, 1502. (31) Values of carbon clusters cohesive energy are taken from the following: Zhang, B. L.; Wang, C. Z.; Ho, K. M. Chem. Phys. Lett. 1992, 193,225. (32) Ghanty, T. K.; Ghosh, S. K. J . Am. Chem. SOC.,submitted for publication. See also: Pitzer, K. S . J . Chem. Phys. 1955,23, 1735. (33) Zhou, Z.; Parr, R. G.;Garst, J. F. TetrahedronLett. 1988,29,4843. Zhou, Z . ; Parr, R. G. J. Am. Chem. SOC.1989,111, 7371. (34)Amic, D.; Nenand, T. J . Chem. SOC.,Perkin Trans. 2 1990,1595.
