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.