Proc. Natl. Acad. Sci. USA Vol. 77, No. 4, pp. 1725-1727, April 1980 Chemistry Electron density functions for simple molecules (chemical bonding/excited states) RALPH G. PEARSON AND WILLIAM E. PALKE Department of Chemistry, University of California, Santa Barbara, California 93106 Contributed by Ralph G. Pearson, January 8, 1980

ABSTRACT Trial electron density functions have some If each component of the charge density is centered on a conceptual and computational advantages over wave functions. single point, the calculation of the potential energy is made The properties of some simple density functions for H+2 and H2 are examined. It appears that for a diatomic molecule a good much easier. However, the kinetic energy is often difficult to density function would be given by p = N(A2 + B2), in which calculate for densities containing several one-center terms. For A and B are short sums of s, p, d, etc. orbitals centered on each a wave function that is composed of one orbital, the kinetic nucleus. Some examples are also given for electron densities that energy can be written in terms of the electron density as are appropriate for excited states. T = 1/8 (VP)2dr [5] Primarily because of the work of Hohenberg, Kohn, and Sham p (1, 2), there has been great interest in studying quantum me- and in most cases this integral was evaluated numerically in chanical problems by using the electron density function rather cylindrical coordinates. The X integral was trivial in every case, than the wave function as a means of approach. Examples of and the z and r integrations were carried out via two-dimen- the use of the electron density include studies of chemical sional Gaussian bonding in molecules (3, 4), solid state properties (5), inter- quadrature. For the z integral, two 8-point molecular potentials (6), and the chemical potential or elec- Gauss-Laguerre integration regions (from z = -o to nucleus tronegativity (7, 8). A and from nucleus B to z = co) and one or more 16-point In connection with chemical bonding, it has recently been Gauss-Legendre regions between the nuclei were used. The shown latter integration range was subdivided into separate regions (9) that the so-called classical electron density between the one-center functions. The r integration was per- p = (a2+b2)/2[1] formed with a single 8-point Gauss-Laguerre integration. The gives in computer programs for these integrals were checked by cal- surprisingly good bonding the molecules H+2 and H2. culating known integrals and by variation of the number In Eq. 1, a = (a/ir)l/2e-arA and b = (a31/)l/2e-arB, a being of the effective nuclear charge. In the usual molecular orbital points. The integrals are accurate to four decimal places, so the theory, of course, the charge density per electron is written energies quoted in this work are accurate to the number of as significant figures given. Addition of the MO overlap density, 2ab, to the classical p = (2 + 2S)-I(a2+ b2 + 2ab) [2] density [1] results in such a sigpificant improvement in bonding and chemical bonding is attributed almost entirely to the that it is natural to ask what the optimal amount of overlap overlap or exchange density, 2ab. The unexpected bonding density would be. In other words, what is the best value of X in properties of Eq. 1 reside in a favorable kinetic energy contri- the trial density: bution in the region between the nuclei. This result agrees nicely p = (2 +XS)-'(a2+ b2 +Aab). [6] with Ruedenberg's views (10) about the nature of the covalent bond. The results for H+2 and H2 are shown in Table 1 where the Even though the simple density of Eq. 1 predicts substantial energy for the optimal A may be compared with that for X = bonding in H+2 or H2, it is still inferior to the predictions of 0, 2, and co. molecular orbital (MO) theory. The MO wave function For H+2, the best value of A is 8 and the resulting bonding energy of 2.75 eV is a distinct improvement over the MO value fpMO = (2 + 2S)-'/2(a + b) [3] of 2.35 eV and very near to the exact H+2 bond energy of 2.79 can be compared to the wave function that corresponds to the eV. For H2, the best value is A = 4 and the energy is again im- classical charge density of Eq. 1. proved. The binding energy of 3.59 eV in fact is rather close to the Hartree-Fock value of 3.64 eV (11) which is the best that 'PCi = (2)-'/2(a2 + b2)'/2. [4] can be obtained with a density functional approach without The latter is an acceptable orbital for H+2 or H2, but it is not a knowledge of the exchange and correlation density func- very good one. It predicts less bonding than does Eq. 3 and is tionals. far from satisfying the Hellman-Feynman theorem because Whereas Eq. 6 is an excellent function, it is desirable to avoid there is not enough charge density in the region between the the two-center overlap density, and so its role was simulated by nuclei. The advantages of the classical density are its conceptual several one-center functions. Two additional Is orbitals c and and computational simplicity because it is a simple sum of d, centered at distance A from nuclei A and B, respectively, one-center functions. The purpose of this work was to see what were added to the classical density to give improvements can be made to Eq. 1 while retaining as simple a form as possible. The objective was not to obtain exact results p = (2 + 2A)-[a2 + b2 + A(c2 + d2)]. [7] but to determine the magnitude of the improvement obtained The results are included in Table 1. For H+2 the best value of with the simplest functions. Calculations are restricted to H+2 the variable parameter A is 0.59 and the best A = 0.6 a.u. For and H2, and computations were made for the experimental internuclear distances except as noted. Abbreviation: MO, molecular orbital. 1725 Downloaded by guest on October 2, 2021 1726 Chemistry: Pearson and Palke Proc. Natl. Acad. Sci. USA 77 (1980) Table 1. Calculated bonding energies for H+2 and H2 at the equilibrium internuclear distances Bonding Density function Parameters energy, eV (2 + XS)-1(a2 + b2 + Xab) X = 0 a = 1.145 1.51 H+2 a = 1.130 2.64 H2 X = 2 a = 1.228 2.35 H+2 a = 1.193 3.47 H2 X= 4 a = 1.20 3.59 H2 X = 8 a = 1.30 2.75 H+2 A= 0 a = 1.35 2.20* H+2 a = 1.22 2.95t H2 (2 + 2X)-1[a2 + b2 + X(C2 + d2)] X = 0.59 a = 1.20 2.40 H+2 A = 0.6 X = 0.53 a = 1.13 3.42 H2 A = 0.4 (2 + X)-'(a2 + b2 + XC2)t A = 0.68 a = 1.20 2.23 H+2 X= 0.60 a = 1.13 3.32 H2 X = 0.68 a = 1.305 2.36 H+2 a' = 1.02 (2 + X)-'(a2 + b2 + X33/2a-371r-1/2e-r2) A = 1.0 a = 1.20 2.61 H2 0= 1.00 X= 0.5 a = 1.10 3.42 H2 = 1.30 (2 + 2X2/a2)-1[a2(1 + XzA)2 + b2(1 + XZB)2] X = 0.29 a = 1.175 2.42 H+2 X = 0.20 a = 1.143 3.42 H2 Exact 2.79 H+2 4.75 H2 * At R = 1.78 a.u. t AtR= 1.38a.u. Ils orbital at bond center. H2, X = 0.53 and A = 0.4 a.u. The energies of function 7 are not centered on one nucleus. Each term would be the square of a particularly good. short sum of s, p, d, etc. orbitals for that atom. The two additional orbitals in function 7 may also be col- Slater orbitals form a complete set, so the exact H+2 or H2 lapsed into a single orbital at the midpoint of the bond. For H+2, wave function can be expanded in terms of Slater orbitals this gives a bonding energy of only 2.23 eV and for H2, one of centered at any point. Such an expansion has been carried out 3.32 eV. Varying the orbital exponent of the added orbital at the bond midpoint (14) and on one nucleus (15). If the sum improved the energy to 2.36 eV for H+2. It appears that floating is restricted to a small number of terms, an expansion on both is orbitals are not a good substitute for the overlap density. This nuclei should give the best results. apparently results from increased kinetic energy at the cusps The question naturally arises as to the form of the anti- of the added orbitals. bonding orbital corresponding to the bonding orbital (5). An In order to avoid this cusp, a Gaussian function was added apparent candidate bond to the center of the giving uP = LN(ja2 -b 21)1/2 [10] p = (2 + X)-'(a2 + b2 + Xf3/217r-/2a-3e-r2) [8] is not an acceptable wave function because the average kinetic with r being the distance from the bond midpoint. As is shown energy becomes infinite. in Table 1, this improved the energy considerably. The density The square root of Eq. 9, however, gives both a bonding and of Eq. 8 does indeed have a simple form, but integrals over a an anti-bonding wave function mixture of hydrogenic and Gaussian orbitals are awkward to ( = + XZA)2 + b2(1 + XZB)2]1/2. [11] evaluate. +N'/2[a2(l In the spirit of the Rosen and Dickenson functions (12, 13) The bonding orbital is given by positive values of the mixing for H2 and H+2, the electron densities a2 and b2 were polarized coefficient, X. The anti-bonding orbital is found by setting X in the direction of the other nucleus to give = -2/R, R being the internuclear separation. This function has a node at ZA + ZB = R/2. If we take the positive sign for Eq. p = (2 + 2X2/a2)-'[a2(1 + XZA)2 + b2(1 + XZB)2] [9] 11 on one side of the nodal plane and the negative sign on the in which ZA and ZB are the z coordinates measured inward from other side, the function is automatically orthogonal to the each nucleus. bonding function, where the sign is kept constant. It is also or- As shown in Table 1, this gives a best value of the binding thogonal to function 3, or any other function of ag sym- energy of 2.52 eV for H+2, with X = 0.29. For H2, the binding metry. energy is 3.42 eV, with X = 0.20. Just as for the Rosen and Furthermore, Eq. 11 with X = 2/R is clearly a reasonable Dickenson functions, polarization is not quite as useful for H2 approximation to the au molecular orbital of H+2 or H2. When as for H+2. Nevertheless, the function typified by Eq. 9 seems R = 0, it becomes the 2pz orbital of helium. When R = oo, it to be promising for further development. For a diatomic mol- reverts to the is orbital of a hydrogen atom. ecule the electron density would be given by two terms, each However, calculation of the energy for this au orbital for H+2 Downloaded by guest on October 2, 2021 Chemistry: Pearson and Palke Proc. Natl. Acad. Sci. USA 77 (1980) 1727

shows that only for R <1.4 a.u. does it compare favorably with optimization of approximate energies and electron densities of the exact values given by Bates et al. (16). At intermediate molecules by use of the variation condition that holds for or- values of R, too much p orbital character is enforced and the thogonal Hartree-product wave functions (18, 19). energy is high. The use of a Hartree-product function eliminates the ex- In order to get an orbital with the correct au character, it is change integrals, which would be very difficult to evaluate for necessary to have a node at ZA = ZB = R/2. This can be achieved wave functions such as those described here. It would then be in a number of ways. The function necessary to estimate exchange (and correlation) energies by other methods, such as the Xa pioneered by Slater (20, 21). (p = Nl/2[e-2arA(1 + Xea'ZA)2 + e-2arB(1 + Xeat'ZB)21/2 [12]

is such a function, if A = -e-a'R/2. Calculations with function 1. Hohenberg, P. & Kohn, W. (1964) Phys. Rev. B 136,864-871. 12 show that it gives good values of the energy, compared to the 2. Kohn, W. & Sham, L. J. (1965) Phys. Rev. A 140, 1133-1138. 3. Harris, J. & Jones, R. 0. (1979) J. Chem. Phys. 70, 830-841. exact, R. It exact at R = for intermediate values of is also ao. 4. Jones, R. 0. (1979) J. Chem. Phys. 71, 1300-1308. The strategy outlined above can be extended to higher or- 5. Gunnarson, 0. & Lundquist, B. I. (1976) Phys. Rev. B 13, bitals of a symmetric molecule like H+2. An approximation to 4274-4298. the 7ru orbital would be given by 6. Kim, Y. & Gordon, R. G. (1974) J. Chem. Phys. 61, 1-16. 7. Parr, R. G., Donnelly, R. A., Levy, M. & Palke, W. E. (1978) J. = AN'/2[a2X2(1 + XA)2 + b2X2(1 + XB)2] [13] Chem. Phys. 68,3801-3802. with the sign changed at x = 0 and with X positive. To obtain 8. Donnelly, R. A. & Parr, R. G. (1978) J. Chem. Phys. 69,4431- the lrg orbital, X would again be set equal to -2/R. Eq. 12 4439. corresponds to an electron density at nucleus A given by the 9. Pearson, R. G. (1979) Theoret. Chim. Acta 52, 253-257. 10. Ruedenberg, K. (1962) Rev. Mod. Phys. 34, 326-376. square of a Px - hybrid on A, and similarly for nucleus B. dz 11. Kolos, W. & Roothaan, C. C. J. (1960) Rev. Mod. Phys. 32, For goes to 0 as R goes to we a the 7ru orbital, X 0, and obtain 219-232. 2Px orbital of helium. The 7rg orbital would go to the 3d. or- 12. Rosen, N. (1931) Phys. Rev. 38,2099-2114. bital of helium in the united atom limit. 13. Dickenson, B. N. (1933) J. Chem. Phys. 1, 317-319. In the general case, the density function must go to 0 at least 14. Joy, H. W. & Handler, G. S. (1965) J. Chem. Phys. 42, 3047- as fast as the square of the distance from the necessary nodal 3051. surfaces. An acceptable wave function can be written as the 15. Keefer, J. A., Su Fu, J. K. & Belford, R. L. (1969) J. Chem. Phys. square root of such a density with alternating signs for adjacent 50, 160-173. regions. Wilson (17) has given a useful analysis of the nodal 16. Bates, D. R., Ledsham, K. & Stewart, A. L. (1953) Philos. Trans. surfaces of molecular orbitals for the common point groups. For R. Soc. London 246, 215-235. 17. E. B. molecules of no symmetry or for excited states of a particular Wilson, (1975) J. Chem. Phys. 63, 4870-4879. 18. Levy, M., Nee, T. S. & Parr, R. G. (1975) J. Chem. Phys. 63, symmetry, the problem appears to be considerably more dif- 316-318. ficult. However, if electron densities corresponding to a series 19. Christiansen, P. A. & Palke, W. E. (1977) J. Chem. Phys. 67, of orthogonal wave functions can be constructed, then the ki- 57-63. netic energies could be calculated for more than two electrons 20. Slater, J. C. (1951) Phys. Rev. 81, 385-390. via Eq. 5 for each density component. This would facilitate the 21. Slater, J. C. (1972) Adv. Quantum. Chem. 6, 1-92. Downloaded by guest on October 2, 2021