J. Phys. Chem. 1996, 100, 17429-17433
17429
New Scale of Atomic Orbital Radii and Its Relationship with Polarizability, Electronegativity, Other Atomic Properties, and Bond Energies of Diatomic Molecules Tapan K. Ghanty and Swapan K. Ghosh* HeaVy Water DiVision, Bhabha Atomic Research Centre, Bombay 400 085, India ReceiVed: July 15, 1996X
A new scale of orbital radii is defined as the distance corresponding to the classical turning point of the electron in an orbital and is calculated for atomic systems using the self-interaction corrected version of the Kohn-Sham density functional theory with local spin-density approximation for the exchange and correlation. These orbital radii and different density and density derived quantities are shown to correlate very well with polarizability and other atomic properties of interest. A simple scheme is also proposed for the bond energy of a diatomic molecule in terms of the valence orbital radii and the electron density (at the boundary corresponding to the radii) of the constituent atoms. The calculated bond energies for simple heteronuclear diatomic molecules are shown to agree very well with the experimental values.
1. Introduction The concept of atomic size or radius is one of the most useful concepts in chemistry.1 Conventionally for neutral atoms, the so called covalent radius is obtained from the bond lengths of the corresponding homonuclear diatomic molecules, while, for the radii of the ions, some other suitable schemes are normally employed. Recently a number of interesting correlations between the radii and several other atomic properties like polarizability, electronegativity,1,2 hardness, and softness3 have been found4-10 to exist. In view of the importance of the concept of radius, it would be of immense interest to obtain an estimate of this quantity from quantum mechanical calculation for the atoms alone and without any recourse to the molecules formed. The electron density distribution of the atom can play an important role in such approaches. The distance of the outermost maxima11 of the radial distribution or the radius of a sphere enclosing a fixed fraction (say 98%) of the total number of electrons as defined12 by the integral of the radial density has often been used as an estimate of the atomic size. In recent years, more rigorous approaches based on quantum mechanical equations have been suggested.13-15 The framework that has been found highly suitable for this purpose is the density functional theory16 (DFT), where the basic variable is the electron density itself. An estimate of the atomic radii had been obtained15 earlier as the distance at which the chemical potential of the electron cloud becomes equal to the electrostatic potential. Recently we have reported17,18 other interesting schemes for the calculation of the radii of atoms as well as ions using a single DFT based framework. There can thus be many alternative routes to obtain measures of the atomic radius, and it would also be of interest to define the orbital radii corresponding to different orbitals occupied by the electrons. In fact, an angular momentum dependent orbital radii defined13 earlier within the pseudopotential framework of solids has been successfully employed19 by solid state physicists as quantum mechanical coordinates mainly for structural discrimination. The orbital radii have also been employed20 for the systematization of the heat of formation of some metal alloys. These radii which have been restricted to the valence orbitals alone have however attracted the attention of chemists21 only recently. In view of the growing importance of the concept of orbital radii for various X
Abstract published in AdVance ACS Abstracts, September 15, 1996.
S0022-3654(96)02092-8 CCC: $12.00
applications as indicated in recent works,21,22 it would be of interest to obtain these quantities through more rigorous ab initio all-electron self-consistent procedures rather than the pseudopotential schemes. Also of interest would be to study their usefulness in predicting various atomic and molecular properties. In this work, we propose a new scheme to obtain a set of orbital radii corresponding to each spin-orbital within the framework of self-interaction23 corrected (SIC) Kohn-Sham density functional theory.16 The calculated values of the orbital radii are then correlated with a number of other ground state atomic properties. We also employ these radii to propose a new model for predicting the binding energies of simple diatomic molecules. 2. New Scheme of Orbital Radii and Its Calculation In the spin-polarized SIC version of DFT used here, one solves the single-particle Kohn-Sham equation given by (in atomic units) iσ SIC [- 1/2∇2 + Veff (r,{Fσ})]ψiσ(r) ) iσ ψiσ(r)
(1)
where Fσ(r) ) ∑Fiσ(r) ) ∑niσ|ψiσ(r)|2 with the occupation numbers niσ satisfying ∑iniσ ) Nσ, the number of electrons with spin state σ (R or β), and the spin and orbital dependent KohnSham effective potential is given by iσ (r) ) V(r) + ∫ Veff
dr′ [F(r′) - Fiσ(r′)] LSD (r; [FR, Fβ]) + µXC |r - r′| LSD (r;[Fiσ]) (2) µXC
Here V(r) is the external potential due to the nuclei, the integral on the right hand side denotes the self-interaction corrected classical electrostatic potential due to the electron distribution, LSD LSD /δFσ) denotes the exchange-correlation (XC) ( ) δEXC and µXC potential within the local spin-density approximation. We define the atomic orbital radii, Ri for the ith orbital φi ≡ φnlσ with n, l, and σ as the principal, azimuthal, and spin quantum numbers, respectively, as the distance corresponding to the classical turning point of the electron in this orbital, which here corresponds to the condition of the effective potential being equal to the orbital energy for the orbital concerned. The orbital radius Ri for the ith orbital is thus determined from the equation © 1996 American Chemical Society
17430 J. Phys. Chem., Vol. 100, No. 43, 1996
Ghanty and Ghosh
TABLE 1: Calculated Values of Valence Orbital Radii, Electron Density (G), Gradient of Density (∇G), and Laplacian of Density (∇2G) at the HOMO Radiia atom
(n)b
He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba
2 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 2 2 2 2 2 2 2 2 2 2 3 4 5 6 7 8 1 2 2 2 2 2 2 2 2 2 2 2 3 4 5 6 7 8 1 2
χc 3.01 4.9 4.29 6.27 7.3 7.54 10.41 10.78 2.85 3.75 3.23 4.77 5.62 6.22 8.3 7.89 2.42 2.2 3.34 3.45 3.6 3.72 3.72 4.06 4.3 4.4 4.48 4.45 3.2 4.6 5.3 5.89 7.59 6.99 2.34 2.0 3.19 3.64 4.0 3.9 4.5 4.3 4.45 4.44 4.33 3.1 4.3 4.85 5.49 6.76 6.07 2.18 2.40
RHOMO
RS
F
-(∇F/F)
(∇2F/F)
(qR)d
1.1638 5.0934 3.3767 2.7830 2.1224 1.7201 1.5214 1.2838 1.1168 5.2905 4.0347 4.3263 3.4593 2.9050 2.5610 2.2481 2.0104 6.3475 5.1134 4.8180 4.6098 4.4407 4.3046 4.1832 3.9556 3.7750 3.6236 3.4926 3.3765 4.2074 3.5486 3.1109 2.8154 2.5572 2.3520 6.6867 5.5417 5.1551 4.9144 4.7389 4.6012 4.4880 4.2594 4.0831 3.9387 3.8160 3.7095 4.5587 3.9723 3.5639 3.2696 3.0198 2.8155 7.3719 6.2191
1.1638 5.0934 3.3767 2.4801 1.9755 1.6472 1.4433 1.2549 1.1127 5.2905 4.0347 3.2272 2.7307 2.3843 2.1274 1.9228 1.7567 6.3475 5.1134 4.8180 4.6098 4.4407 4.3046 4.1832 3.9556 3.7750 3.6236 3.4926 3.3765 2.9750 2.6835 2.4554 2.2549 2.1066 1.9785 6.6867 5.5417 5.1551 4.9144 4.7389 4.6012 4.4880 4.2594 4.0831 3.9387 3.8160 3.7095 3.3547 3.0851 2.8673 2.6605 2.5153 2.3867 7.3719 6.2191
0.058 285 19 0.000 430 97 0.002 982 81 0.006 791 08 0.021 756 64 0.052 102 03 0.085 892 26 0.172 016 96 0.301 780 97 0.000 392 28 0.001 834 15 0.001 376 10 0.004 487 92 0.010 553 38 0.019 017 69 0.034 654 56 0.057 331 58 0.000 239 72 0.000 957 67 0.001 163 13 0.001 310 87 0.001 434 67 0.001 546 09 0.001 650 69 0.002 043 60 0.002 418 51 0.002 787 50 0.003 156 16 0.003 529 44 0.001 385 14 0.004 004 18 0.008 455 91 0.014 526 04 0.023 998 20 0.036 736 17 0.000 210 00 0.000 772 66 0.001 057 42 0.001 231 24 0.001 347 00 0.001 428 98 0.001 490 63 0.001 838 82 0.002 147 97 0.002 430 78 0.002 694 27 0.002 942 64 0.001 201 31 0.003 053 39 0.005 956 38 0.009 866 12 0.015 424 51 0.022 631 84 0.000 161 54 0.000 564 37
3.2178 0.9479 1.3626 1.7190 2.0720 2.4371 2.7790 3.1378 3.5051 0.9204 1.1860 1.3266 1.5290 1.7437 1.9369 2.1377 2.3429 0.8001 0.9895 1.0822 1.1476 1.1985 1.2406 1.2771 1.3334 1.3816 1.4238 1.4629 1.4995 1.3713 1.5159 1.6750 1.8120 1.9572 2.1046 0.7713 0.9339 1.0512 1.1394 1.2034 1.2501 1.4670 1.2844 1.3378 1.3802 1.4148 1.4432 1.3047 1.4067 1.5275 1.6311 1.7421 1.8548 0.7172 0.8568
5.1851 0.4494 0.9287 1.6581 2.2727 3.0468 4.1365 5.0708 6.1903 0.4246 0.7055 1.1582 1.4275 1.7688 2.1648 2.5536 2.9937 0.3253 0.4977 0.6417 0.7535 0.8443 0.9213 0.9888 1.0830 1.1708 1.2484 1.3257 1.4007 1.3000 1.4671 1.6995 1.9472 2.2085 2.4975 0.3080 0.4496 0.6240 0.7813 0.9092 1.0092 1.0851 1.1842 1.2705 1.3451 1.4092 1.4637 1.1797 1.2758 1.4341 1.5970 1.7770 1.9705 0.2702 0.3833
1.46 (72.8) 2.80 (93.2) 3.55 (88.8) 4.43 (88.5) 5.07 (84.5) 5.72 (81.7) 6.51 (81.4) 7.06 (78.4) 7.64 (76.4) 10.80 (98.1) 11.56 (96.3) 12.65 (97.3) 13.36 (95.4) 14.06 (93.8) 14.80 (92.5) 15.45 (90.9) 16.12 (89.6) 18.80 (98.9) 19.58 (97.9) 20.58 (98.0) 21.58 (98.1) 22.59 (98.2) 23.60 (98.3) 24.60 (98.4) 25.57 (98.4) 26.55 (98.3) 27.53 (98.3) 28.52 (98.3) 29.50 (98.3) 30.66 (98.9) 31.38 (98.1) 32.11 (97.3) 32.83 (96.5) 33.52 (95.8) 34.22 (95.1) 36.80 (99.5) 37.58 (98.9) 38.55 (98.8) 39.56 (98.9) 40.57 (98.9) 41.58 (99.0) 42.59 (99.0) 43.56 (99.0) 44.54 (99.0) 45.52 (99.0) 46.50 (98.9) 47.49 (98.9) 48.65 (99.3) 49.38 (98.8) 50.12 (98.3) 50.84 (97.8) 51.56 (97.3) 52.28 (96.8) 54.80 (99.6) 55.58 (99.3)
a
All quantities are in atomic units except electronegativity (χ) which is in electronvolts. b Number of s and p valence electrons. c Mulliken electronegativity values are from refs 9 and 32. d Values in parentheses denote the percentage of electronic charge enclosed in a sphere of radius RH.
[
iσ Veff (Ri,{Fσ}) +
]
l(l + 1) Ri2
SIC ) iσ
(3)
where the orbital-dependent effective potentials and the orbital energies are determined through self-consistent iterative procedures. 3. Results and Discussion Through the numerical solution of the Kohn-Sham equation (1) with local spin density approximation16,23 for the XC potential, we have calculated the radii of all of the orbitals of
a number of atoms (up to Z ) 56). Since the valence orbitals play a more important role in chemistry, we have reported in Table 1 the calculated values for the highest occupied orbital (HOMO) radii (denoted by RH) and also the s-orbital radii corresponding to the same principal quantum number. Obviously, for atoms with the s-orbital as the HOMO, both these radii are the same. We have also calculated other different atomic properties at these orbital radii; viz., electron density (F), gradient of density (∇F), Laplacian of density (∇2F), the charge encompassed within these orbital radii (qR), and the values of these quantities at r ) RH are also reported in Table 1.
New Scale of Atomic Orbital Radii
Figure 1. Plot of the cube root of polarizability against the calculated values of HOMO radius for atoms.
Figure 2. Plot of the cube root of polarizability against the calculated values of HOMO radius for ionic systems.
Now we consider the HOMO radius RH as the measure of the atomic size (radius) which is correlated here with different atomic properties. In view of the recent interest4-9 on the correlation of polarizability with other atomic properties, we have plotted in Figure 1 the cube root of experimental values of polarizability6 against the HOMO radii for the atoms mentioned above (except hydrogen and noble gas atoms). The linear least squares fit is found to be very good with a correlation coefficient of 0.987. It is interesting to note that the slope of this least squares fit is 0.987, which is very close to unity, and the intercept is 0.058, which is also close to zero, indicating a quantitative correspondence between these two quantities. It may be noted that a similar least squares fit of R1/3 vs the Zunger orbital radius13 (average of s- and p-orbital radii) leads to a slope of 2.656 and an intercept equal to 1.255 with a correlation coefficient 0.934. We have also calculated the HOMO radii for a number of ions of main group atoms (see ref 9), and, in Figure 2, we have plotted the cube root of polarizability against these values of the ionic radii. The slope and the intercept of the best fitted line are +0.922 and -0.077, respectively, and the correlation coefficient is 0.988. For a direct proportionality relationship of the type R1/3 ) KR, the value of K for a best fit for the atomic systems considered here using the present results of R is found to be 1.000. In this context, it may be noted that this relationship with K ) 0.836 had been obtained earlier by Dimitrieva and Plindov24 through some approximations in a theoretical derivation, and more recently a similar relation has been proposed by Hati and Datta8 in their calculation of hardness (η) from polarizability using the relation η ) (1/2)K/R1/3 for a
J. Phys. Chem., Vol. 100, No. 43, 1996 17431
Figure 3. Plot of the HOMO radius against the radius corresponding to the outermost maxima of the HOMO radial density.
number of atoms, open shell monoatomic ions, and sodium clusters. With K ) 0.925, the above relation has been found8 to reproduce the experimental hardness values for a number of open shell species quite satisfactorily. It is also of interest to consider other scales of atomic size. For this purpose, in the spirit of the work of Waber and Cromer,11 we have calculated RM which corresponds to the position of the outermost maxima of the radial density of the outermost orbital, using the Kohn-Sham DFT scheme. These values plotted against the HOMO radii in Figure 3 show an excellent linear correlation with a correlation coefficient of 0.993. It is also interesting that the present radii values are of magnitudes comparable to radii obtained by Politzer et al.12 based on the criterion that 98% of electronic charge is enclosed inside the corresponding sphere. This is reflected in the present values of qR ()∫4πr2F(r) dr) reported in Table 1, which indicate that more than 95% electronic charge is enclosed within the sphere of radius equal to RH except very light atoms. In view of the importance of the HOMO emphasized in these studies as well as in chemical binding and reactivity,25 we have studied different density related functions at the HOMO radii. One such function which involves the density at the boundary and is found to correlate well (correlation coefficient ) 0.963) with the ionization potential is the quantity [F(RH)/n]1/3, where n is the number of valence electrons. Another quantity of interest is the logarithmic derivative of density (|∇F|/F), which amounts to a measure of the exponent of an exponentially decaying electron density (e.g., in the long range). The calculated values of this density exponent (|∇F|/F) at r ) RH are also found to correlate well (correlation coefficient ) 0.965) with the ionization potentials of atoms. In view of an approximate inverse relation5 between the ionization potential and the atomic size, we have plotted in Figure 4 the cube root of polarizability against the quantity (|∇F|/F)-1 at r ) RH, which shows a linear correlation with a correlation coefficient of 0.987. The quantity (|∇F|/F) at r ) RH has also been plotted against the Allen electronegativity values26 in Figure 5, showing an excellent linear correlation with a correlation coefficient of 0.985. Very recently, Hati and Datta9 have demonstrated a linear correspondence between the Allen electronegativity and the inverse of the cube root of the polarizability. Thus, the density exponent (|∇F|/F) can be a good measure of the electronegativity of an atom. The Mulliken electronegativity values are also found to correlate well with the density at r ) RM with a correlation coefficient of 0.970. The other quantity reported in Table 1 is (|∇2F|/F) at r ) RH, related to the curvature
17432 J. Phys. Chem., Vol. 100, No. 43, 1996
Ghanty and Ghosh TABLE 2: Experimental and Calculated Values of Bond Energy (kcal/mol) of Simple Diatomic Molecules
Figure 4. Plot of the cube root of polarizability against the inverse of density exponent (|∇F|/F) at the HOMO radii.
molecule (AB)
exp a (DAB )
cal b (DAB )
cal c (DAB )
LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI CsF CsCl CsBr CsI
138.0 112.2 100.2 82.6 124.2 98.6 87.9 72.8 119.0 103.6 90.9 77.8 118.2 102.3 91.1 76.3 124.2 107.2 93.1 80.7
126.2 107.3 93.9 79.1 123.3 103.1 90.4 76.2 124.6 103.3 91.2 77.7 124.8 103.3 91.3 77.8 126.5 104.9 93.0 79.6
129.8 109.3 100.6 81.4 125.3 103.6 95.5 77.1 123.4 100.6 93.1 75.2 123.1 100.1 92.8 75.0 123.8 100.7 93.4 75.6
a Experimental bond energies are from ref 31. b Bond energies calculated using eq 5. c Bond energies calculated using eq 6.
Figure 5. Plot of the density exponent (|∇F|/F) at the HOMO radii against the Allen electronegativity.
of the density function which shows the general trend of increasing in a period and decreasing in a group in most of the cases. Thus, the HOMO radius can be considered to be a good measure of the atomic size, and the density and the density exponent at this radius have direct correlations with various atomic properties of significance. We now explore the possibility of using these quantities for the prediction of molecular properties. For simplicity, we consider only the chemical binding in simple heteronuclear diatomic molecules, for which extensive work has been done recently27 by using the concepts of electronegativity and hardness. The first such prescription for the prediction of binding energy DAB for a diatomic molecule AB is due to Pauling,1 which has been modified by many others, and one of the simplest recent prescription due to Reddy et al.28 is given by COV DAB ) DAB + c|χA - χB|
(4)
where the last term denotes the ionic contribution represented in terms of the difference in electronegativity (χ) and the COV covalent contribution DAB is normally obtained as the geometric mean of the corresponding homonuclear bond energies DAA and DBB. We, however, modify eq 4 by adding an extra term in the spirit of the work of Miedema et al.29 and propose the bond energy DAB to be given by COV + c1|χA - χB| + c2|FA1/3 - FB1/3| DAB ) DAB
(5)
where FA (or FB) denotes the value of the electron density of atom A (or atom B) at the HOMO radius and c1 and c2 are two
COV empirical constants. For evaluating DAB , one can either use the geometric mean or other representations30 which have been derived from a consideration of charge accumulation in the bond region. While the second term in eq 5 represents the ionic contribution and favors binding, the last term corresponds to a repulsive contribution arising from the mismatch of the electron density at the boundary of the two atoms A and B. The necessity of such a term has been highlighted through successful application of an empirical scheme proposed by Miedema et al.29 for predicting the heat of formation of binary alloys in terms of the work function and the electron density at the boundary of the Wigner-Seitz cell of the constituent metal atoms. While the electron density at the atomic boundary appears explicitly in eq 5, it would also be of interest to see if the atomic size itself can be used directly in the prediction of bond energy. For this purpose we replace χ in eq 5 by the quantity (n1/3/R), where n is the number of valence electrons and R is the radius of the valence shell s-orbital of the concerned atom. The replacement is motivated by the correlation between analogous quantities as demonstrated recently by Nagle6 and by Hati and Datta.9 Thus, the modified bond energy equation can be written as
COV DAB ) DAB + c3|nA1/3/RA - nB1/3/RB| + c4|FA1/3 - FB1/3| (6)
where c3 and c4 are two empirical constants. We have calculated the bond energies of a number of heteronuclear diatomic molecules using eqs 5 and 6, and the calculated values are compared with experimental bond energies31 in Table 2. For the quantities expressed in units as mentioned in Tables 1 and 2, the constants c1 and c2 in eq 5 or c3 and c4 in eq 6, as determined from two separate least squares fits of the experimental bond energies with the calculated ones, are +13.608 and -10.119 or +184.933 and -304.200, respectively. The negative sign of c2 and c4 indicates that the third term in eq 5 or eq 6 is a repulsive term. The average percentage errors in the bond energy for the molecules considered here are only 2.8 and 3.3, respectively. It may be noted that the average errors increase if the last term in eq 5 or eq 6 is dropped. It indicates that the repulsive contribution due to the density mismatch at the atomic boundaries plays an important role in the prediction of chemical binding through these types of
New Scale of Atomic Orbital Radii approaches. Since the energy cost due to density reorganization is expected to be less for bonding atom pairs with comparable values for the density quantity at the atomic boundary, a consideration of this parameter can lead to a conjecture of the form “like prefers like” somewhat in the spirit of the widely established statements of facts such as “hard likes hard” and “soft likes soft” in hard and soft acid and base principle.3 4. Concluding Remarks DFT has not only introduced computational simplicity and economy in quantum mechanical calculations but also provided foundation to many widely used chemical concepts16 and introduced generalized electronegativity equalization procedures for chemical binding. In the present work, through a simple scheme based on the concept of a classical turning point within the framework of DFT, a good measure of the atomic size which correlates well with a number of atomic properties has been shown to be possible. The present scheme which uses an ab initio all-electron self-consistent potential through density functional exchange-correlation contribution is parameter free and works very well for all the atoms, including even the transition metal ones. One of the most significant results in this work is that the polarizability values can be predicted quite accurately, which can therefore be very useful for those atomic species for which reliable values are not available. Although many earlier estimates for radii have been proposed, the present results show a better correlation with other atomic properties as discussed. We have also proposed new schemes for the prediction of binding energies of simple diatomic molecules by using these radii or the electron density values at these distances. The predicted values are in quite good agreement with the experimental results. Although we have reported here only the valence orbital radii, we have calculated the radii corresponding to other orbitals as well. It would be of interest to see if they can be correlated with orbital electronegativity and other related concepts. The concepts and the results reported here are of importance not only for atoms and ions, but also for molecules, clusters, and solids. Extensions of the present schemes to these systems are straightforward and would be of considerable interest. Acknowledgment. It is a pleasure to thank T. G. Varadarajan and H. K. Sadhukhan for their kind interest and encouragement. References and Notes (1) Pauling, L. The Nature of the Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, NY, 1960.
J. Phys. Chem., Vol. 100, No. 43, 1996 17433 (2) ElectronegatiVity: Structure and Bonding; Sen, K. D., Jorgensen, C. K., Eds.; Springer-Verlag: Berlin, 1987; Vol. 66. (3) Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512. See also: Sen, K. D., Ed. Chemical Hardness: Structure and Bonding; Springer-Verlag: Berlin, 1993; Vol. 80. (4) Ghanty, T. K.; Ghosh, S. K. J. Phys. Chem. 1993, 97, 4951. (5) Rosseinsky, D. R. J. Am. Chem. Soc. 1994, 116, 1063. (6) Nagle, J. K. J. Am. Chem. Soc. 1990, 112, 4741. (7) Fricke, B. J. Chem. Phys. 1986, 84, 862. (8) Hati, S.; Datta, D. J. Phys. Chem. 1994, 98, 10451. (9) Hati, S.; Datta, D. J. Phys. Chem. 1995, 99, 10742. (10) Roy, R. K.; Chandra, A. K.; Pal, S. J. Phys. Chem. 1994, 98, 10447. Roy, R. K.; Chandra, A. K.; Pal, S. J. Mol. Struct. (THEOCHEM) 1995, 331, 261. (11) Waber, J. T.; Cromer, J. T. J. Chem. Phys. 1963, 42, 4116. (12) Politzer, P.; Murray, J. S.; Grice, M. E.; Brinck, T.; Ranganathan, S. J. Chem. Phys. 1991, 95, 6699. Politzer, P.; Murray, J. S.; Grice, M. E. In Chemical Hardness: Structure and Bonding; Sen, K. D., Ed.; Springer-Verlag: Berlin, 1993; Vol. 80. (13) Zunger, A. Phys. ReV. 1980, B22, 5839. (14) Garcia, A.; Cohen, M. L. Phys. ReV. 1993, B47, 4221. Zhang, S. B.; Cohen, M. L.; Phillips, J. C. Phys. ReV. 1987, B36, 5861. (15) Politzer, P.; Parr, R. G.; Murphy, D. R. J. Chem. Phys. 1983, 79, 3859. Politzer, P.; Parr, R. G.; Murphy, D. R. Phys. ReV. 1985, B31, 6809. (16) Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press: New York, 1989. (17) Ghanty, T. K.; Ghosh, S. K. J. Am. Chem. Soc. 1994, 116, 8801. (18) Ghanty, T. K.; Ghosh, S. K. J. Phys. Chem. 1994, 98, 9197. (19) O’Keeffe, M., Navrotsky, A., Eds. Structure and Bonding In Crystals; Academic Press: New York, 1981; Vol. 1. See, particularly, the articles by J. C. Phillips, A. N. Bloch and G. C. Schatteman, and A. Zunger. (20) Chelikowsky, J. R.; Phillips, J. C. Phys. ReV. 1978, B17, 2453. (21) Ganguly, P. J. Am. Chem. Soc. 1993, 115, 9287. Ganguly, P. J. Am. Chem. Soc. 1995, 117, 1772. Ganguly, P. J. Am. Chem. Soc. 1995, 117, 2655. (22) O’Keeffe, M.; Brese, N. E. J. Am. Chem. Soc. 1991, 113, 3226. (23) Perdew, J. P.; Zunger, A. Phys. ReV. 1981, B23, 5048. (24) Dmitrieva, I. K.; Plindov, G. I. Phys. Scr. 1983, 27, 402. (25) Fukui, K. Science 1982, 218, 747. Parr, R. G.; Yang, W. J. Am. Chem. Soc. 1984, 106, 4049. (26) Allen, L. C. J. Am. Chem. Soc. 1989, 111, 9003. Allen, L. C. Int. J. Quantum Chem. 1994, 49, 253. (27) See, for example: Ghanty, T. K.; Ghosh, S. K. J. Phys. Chem. 1994, 98, 1840. Ghanty, T. K.; Ghosh, S. K. J. Am. Chem. Soc. 1994, 116, 3943. (28) Reddy, R. R.; Rao, T. V. R.; Biswanath, R. J. Am. Chem. Soc. 1989, 111, 2914. (29) Miedema, A. R.; de Chatel, P. F.; de Boer, F. R. Physica 1980, 100, 1. For a recent review, see: Pettifor, D. G. Solid State Phys. 1987, 40, 43. (30) Ghanty, T. K.; Ghosh, S. K. Inorg. Chem. 1992, 31, 1951. (31) C.R.C. Handbook of Chemistry and Physics, 74th ed.; CRC Press: Boca Raton, FL, 1993-4. (32) Pearson, R. G. Inorg. Chem. 1988, 27, 734.
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