Evaluation of a Characteristic Atomic Radius by an Ab Initio Method

Evaluation of a Characteristic Atomic Radius by an Ab Initio Method ZHONG-ZHI YANG Department of Chemistry, Liaoning Normal University, Dalian, 116029, and Institute of Theoretical Chemistry, Jilin University, 130023, People s Republic of China ERNEST R. DAVIDSON Department of Chemistry, Indiana University, Bloomington, Indiana 47405 Received March 8, 1996; revised manuscript received May 29, 1996; accepted June 3, 1996 ABSTRACT The MELD program is employed to evaluate the Slater average potential vž. r felt by an electron at the point r within an atom. The characteristic radius R of the atom is then defined by the classical turning point equation vž R. I, where I denotes the first ionization potential of the atom. The atomic radii defined in this way have a close correlation with the van der Waals atomic radii. 1997 John Wiley & Sons, Inc. Introduction V arious kinds of atomic radii are fundamental constants in many branches of science and technology. In chemistry and physics, scientists often use the covalent, ionic, and van der Waals atomic radii to describe the atomic sizes in different circumstances. These radii are usually determined by measuring the internuclear distances in molecules or solids and then making appropriate apportionments of such distances 18. Theoretically, Slater noted that there is a close correlation between the radius at which radial electronic density reaches its maximum value and the commonly accepted covalent atomic radii and he defined such a radius as a size measure of the atom 1, which is termed the BraggSlater atomic radius in the literature 2. In a general sense, the BraggSlater atomic radii represent the radii of inner cores of electrons of atoms, which, in turn, determine how closely the two atomic species can approach each other and serve as lower bounds of the corresponding covalent radii. By using a density contour approach and accepting a suitable small value of electronic density in an atom as a criterion. Boyd explored the relative sizes of atoms 3 and gave a kind of scaled atomic radii from He through Xe by simulating Pauling univalent radii 4 of noble gas atoms. From the analysis of so-called premolecules, Spackman and Maslen proposed a type of atomic radii in molecules 5. ( ) International Journal of Quantum Chemistry, Vol. 62, 4753 1997 1997 John Wiley & Sons, Inc. CCC 0020-7608 / 97 / 010047-07

YANG AND DAVIDSON In discussing the chemical potential, Politzer et al. 6 defined a characteristic atomic radius as the radius where the electrostatic potential was equal to the chemical potential. Alternatively, in discussing hardness, Chattaraj et al. 7 defined the atomic radius as the point where the derivative of the electrostatic potential with respect to the electron number was equal to the hardness. They showed that those sorts of atomic radii are related to the covalent atomic radii. Bader et al. developed a theory of atoms in molecules in which the atomic domains in a molecule are defined by considering the charge distribution 8. In contrast to the covalent atomic radii, there are few theoretical approaches for discussing the van der Waals atomic radii. In this article, first, the MELD program 9 was used to calculate the Slater average potential vž. r felt by an electron at position r within an atom. Then, the characteristic radius R is evaluated by the classical turning point equation vž R. I, in which I stands for the first ionization potential of the atom concerned. The atomic radii defined in this way, termed boundary radii, have close correlations with the van der Waals atomic radii 2, with the effective atomic radii 10 demonstrated by various properties of atoms, such as viscosities, heat conductivities, and van der Waals equations, as well with as atomic radii 11 which are used in the simulation of the molecular sizes or volumes of large molecules. This type of characteristic atomic radii has also been evaluated by means of an empirical approach 12. Theoretical Model and Calculation Method Let us consider an atom in its ground electronic state. It has spin electrons and spin ones. The Slater average potential felt by an electron with spin at position r can be expressed as 1 Z 1 Ž r, r. 2 1 2 V Ž r. H dr, Ž 1. 1 2 r Ž r. r r 1 1 1 2 where Z is the nuclear charge of the atom; N, the total electron number of the atom; Ž r. 1, the electron density of spin electrons at position r 1; and Ž r, r. 2 1 2, the probability of finding one electron with spin at point r1 and another one with arbitrary spin at point r 2. For an electron with spin, there is a similar expression. This potential is not quite the usual local potential of KohnSham theory. It could be a reasonable approximation if the functional variation of 12 2 with respect to could be replaced by Ž r, r. Ž r. 2 1 2 1. This would be strictly true if 2 were a quadratic function of, but it is generally recognized that the part of 2 leading to the exchange energy is not quadratic. In spin-restricted HartreeFock self-consistent molecular orbital calculations, 2 2 r 1, r 2 r1 r 2 1 r 1; r 2, 2 Ž. Ž. Ž. Ž. Ž. where Ž r ; r. 1 a b is the one-body reduced density matrix for spin. Hence, Eq. Ž. 1 can be written as in which Z Ž. V r, Ž 3. 1 es ex r 1 Ž r 2. H dr Ž 4. es 2 r r 1 2 1 ÝÝn n Ž r. Ž r. ex i j i 1 j 1 Ž r 1. i j Ž r. Ž r. i 2 j 2 H dr, Ž 5. 2 r r 1 2 where i and j are the orbitals, ni and n j are the occupation numbers of spin electrons and spin electrons in the orbitals i and j, respec- tively, and Ý Ž r. Ž n n. Ž r. Ž r. Ž 6. 2 i i i 2 i 2 i Ž. r n Ž r. Ž r.. Ž 7. 1 Ý i i 1 i 1 i Notice that this equation is the same as Eqs. Ž 17. Ž 22. in 13, in which Slater defined this average exchange potential. The spin- and equivalence-restricted Hartree Fock method in the MELD program 9 was used with near-hartreefock quality Gaussian basis sets for a series of calculations for atoms from He through K. A separate program was made to evaluate,, and then vž r. es ex 1. For instance, for the oxygen atom, the Partridge Ž 20 s,12p. Gaussian basis set was used 14. To evaluate the potential along the z axis, the single-determinant wave function with M S and nž p. nž p. nž p. s z y x 48 VOL. 62, NO. 1

CHARACTERISTIC ATOMIC RADIUS was used. Once the potential vž z. for an atom was known, we then evaluated the characteristic atomic radius R where vž R. I and I is the first experimental ionization potential of the atom concerned. Justification Ž. As Slater noted 13, the exchange operator 5 gives a close approximation to the local exchange K except near the nodes in. Hence, the Fock operator may be approximated by F t V h. Ž 8. If the orbitals are chosen as eigenfunctions of this F, then Fii is close to the HartreeFock orbital energy. Slater also noted that x ie x can be approx- imated very well by the Dirac exchange / es 13 3 3. Ž 9. D ž 4 Orbital energies obtained by this method are not quite as negative as the corresponding HartreeFock energies. With this approximation, the exchange energy in Eq. Ž. 9 for E can be rewritten as ex so, as is well known, the i do not approximate the HartreeFock orbital energies and Koopmans theorem does not apply to the i. This minimum E is only a little lower than the Slater E with 1 and is still higher than the HartreeFock energy. In the X method, D is replaced by D everywhere with chosen empirically to reproduce the HartreeFock energy Ž with 23.. While is slightly different for each atom, a good average value has been found to be 16 1.05. On the other hand, a much larger value, 1.95 with 23, is required to reproduce the average of the orbital energies, F ii, for krypton. In Figure 1, we show the exchange potential K for the 3 p orbital of krypton along with the Slater average exchange Ž. 5 and the Dirac exchange Ž. 9 evaluated with the HartreeFock density. It is clear that the potential V evaluated with HartreeFock orbitals approximates the HartreeFock exchange, and, hence, by Koopmans theorem, eigenvalues using this potential H Ž E 12. d. Ž 10. x D D As noted by Kohn and Sham 15 and others, determination of the orbitals by the equation Ž. F 11 with the approximation Ž. 9 to ex does not mini- mize the energy using approximation Ž 10. to E x. The equation of minimization using the approximation Ž 11. to E leads to x Ž. Ž. G F 1, 12 D with 23. The equation Ž. G 13 i i i leads to a new density with a slightly lower total energy. This equation is the exchange-only local spin density approximation 15. From Ž 12., Ž.² : Ž. F 1, 14 ii i i D i FIGURE 1. Approximate exchange potentials of krypton. K is evaluated for the 3 p orbital. VXavg is the Slater average ( 5) evaluated with the HartreeFock density. Ž. is the Dirac potential ( 10) D HF evaluated with the HartreeFock potential. As noted by Slater, these two are nearly equal. Ž. D HFS is the Dirac potential ( 10) evaluated with a self-consistent density. 2 / 3 Ž. is 2 / 3 of ( 10) D LDA evaluated with a self-consistent density. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 49

YANG AND DAVIDSON should approximate the ionization energies. Thus, the equation used above to define an atomic radius as a classical turning point with I being the experimental ionization potential should be approximately valid. Also shown in Figure 1 is the Dirac exchange evaluated with the density determined self-consistency using 1 Ž HFS. and 23 Ž LDA.. From this figure, it is clear that use of the HFS or LDA exchange in place of the Slater average exchange would not be expected to work in the proposed definition of the atomic radius since they differ greatly from the Slater exchange for R greater than 3 A where the classical turning point occurs. Results and Discussion As an example, carbon is chosen to display the numerical results that we have obtained. In Table I, six quantities are listed for the values of r Ž measured along the z axis. given in column 1. TABLE I A list of a series of the electron properties at various points within a carbon atom. z vr es ex den pot HOMO pot 0.1000 60.0000 13.1693 5.0099 38.9295 51.8407 49.8913 0.2000 30.0000 10.9582 3.9008 12.2409 22.9426 21.0819 0.3000 20.0000 9.1938 2.9674 4.0038 13.7736 12.2748 0.4000 15.0000 7.9282 2.1948 1.4391 9.2666 8.2648 0.5000 12.0000 7.0262 1.5884 0.6453 6.5622 6.0485 0.6000 10.0000 6.3269 1.2460 0.4000 4.8832 4.6666 0.7000 8.5714 5.8522 1.1127 0.3170 3.8319 3.7318 0.8000 7.5000 5.4403 1.0650 0.2774 3.1247 3.0627 0.9000 6.6667 5.0940 1.0402 0.2469 2.6128 2.5639 1.0000 6.0000 4.7935 1.0177 0.2174 2.2243 2.1809 1.1000 5.4545 4.5263 0.9923 0.1885 1.9205 1.8801 1.2000 5.0000 4.2851 0.9632 0.1611 1.6782 1.6396 1.3000 4.6154 4.0647 0.9311 0.1361 1.4818 1.4445 1.4000 4.2857 3.8621 0.8970 0.1139 1.3205 1.2843 1.5000 4.0000 3.6749 0.8616 0.0947 1.1866 1.1512 1.6000 3.7500 3.5015 0.8258 0.0782 1.0743 1.0396 1.7000 3.5294 3.3404 0.7901 0.0644 0.9791 0.9452 1.8000 3.3333 3.1906 0.7551 0.0528 0.8979 0.8647 1.9000 3.1579 3.0512 0.7212 0.0433 0.8279 0.7955 2.0000 3.0000 2.9213 0.6884 0.0354 0.7671 0.7356 2.1000 2.8571 2.8002 0.6570 0.0289 0.7140 0.6834 2.2000 2.7273 2.6871 0.6272 0.0236 0.6673 0.6376 2.3000 2.6087 2.5815 0.5988 0.0192 0.6260 0.5972 2.4000 2.5000 2.4828 0.5720 0.0157 0.5892 0.5614 2.5000 2.4000 2.3904 0.5467 0.0128 0.5563 0.5294 2.6000 2.3077 2.3039 0.5228 0.0104 0.5266 0.5008 2.7000 2.2222 2.2228 0.5004 0.0085 0.4998 0.4750 2.8000 2.1429 2.1467 0.4793 0.0069 0.4755 0.4516 2.9000 2.0690 2.0751 0.4595 0.0056 0.4533 0.4305 3.0000 2.0000 2.0078 0.4409 0.0046 0.4331 0.4111 3.1000 1.9355 1.9445 0.4234 0.0037 0.4145 0.3935 3.2000 1.8750 1.8847 0.4071 0.0031 0.3973 0.3773 3.3000 1.8182 1.8284 0.3917 0.0025 0.3815 0.3623 3.4000 1.7647 1.7751 0.3772 0.0020 0.3668 0.3485 3.5000 1.7143 1.7247 0.3636 0.0017 0.3532 0.3357 3.6000 1.6667 1.6770 0.3508 0.0014 0.3405 0.3239 3.7000 1.6216 1.6317 0.3388 0.0011 0.3287 0.3128 3.8000 1.5789 1.5888 0.3274 0.0009 0.3176 0.3025 3.9000 1.5385 1.5480 0.3167 0.0007 0.3072 0.2928 50 VOL. 62, NO. 1

CHARACTERISTIC ATOMIC RADIUS FIGURE 2. The potentials V for Li, Be, B, and C with dots showing the place where V = I. The quantities in the rest of the columns are, respectively, properties as follows: vr, the nuclear potential Zr; es, the electrostatic repulsive potential defined in Eq. Ž. 4 ; ex, the exchange potential defined in Eq. Ž. 5 ; den, the one-electron density defined in Eq. Ž. 7 ; pot, the potential expressed in Eq. Ž. 2 ; and HOAO pot, the local potential for an electron that is in the HOAO orbital Žhighest occupied atomic orbital. with ex evaluated as K using the HartreeFock exchange operator. All these quantities are given in atomic units in Table I: bohr for distance and Hartrees for energy. The experimental value of the first ionization potential of the carbon atom is 0.41383 Hartree. According to the assumption mentioned above, its minus corresponds to the point where R 3.10 ao 164 pm, which we have defined here as a characteristic measurement of the carbon atom, termed its boundary radius. For a detailed comparison and illustration, the potentials for Li, Be, B, and C atoms are plotted in Figure 2, and potentials for Na, Mg, Si, P, and S atoms, in Figure 3. All atoms have been considered up to Z 18, but the plots of the other atoms are omitted for clarity. In these two figures, the black points on the curves denote the locations where the negatives of the potentials are equal to the corresponding first ionization potentials and these FIGURE 3. The potentials V for Na, Mg, P, Si, and S with dots showing the place where V = I. values of r define the characteristic atomic radii of the corresponding atoms. The shapes of these curves are all quite similar. Obviously, when r becomes large, all the potential curves become FIGURE 4. The functions V +1/r showing the contribution to V from incomplete screening. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 51

YANG AND DAVIDSON similar. This implies that an electron within an atom will feel an effective nuclear charge which contains more and more screening effects from other electrons as r increases, the limit of which is full screening by Z 1 electrons as the electron goes further and further away from the nucleus. To show this fact more clearly, we just add 1r to the potentials and plot the results in Figure 4. The curve for the Na atom in this picture rises rapidly at the beginning and the curves for other atoms rise a bit slower. This can be easily understood since for an electron in the outermost orbital for the Na atom there is little penetration as it moves far away from the nucleus, whereas for the electrons in other atoms, there does exist the penetration of other valence electrons to some extent even when the test electron is fairly far away from the nuclei. The dependence on the atomic number appears somewhat irregular since the exchange term depends on the number and angular momentum of the singly occupied orbitals and these do not change in a simple way with the atomic number. The characteristic atomic radii, termed the atomic boundary radii, evaluated in this approach for He through K atoms are collected in Table II, with the comparison to the van der Waals radii, the effective atomic radii, and the atomic radii TABLE II Comparison between the different atomic radii ( pm ). Boundary radii van der Waals Effective Ab initio Empirical Rp Rb Rvis Rvdw Rhc Atomic radii Reference [ ] [ ] [ ] [ ] [ ] [ ] [ ] Element This work 12 4 2 10 10 10 11 He 62 100 140 95 133 115 Ne 94 114 154 Ar 134 155 188 144 147 143 Kr 147 175 202 185 157 Xe 202 205 216 201 171 Rn 228 H 106 106 106 120 100 Li 267 267 Na 279 280 K 333 332 Rb 345 Cs 370 Be 164 170 B 188 208 165 C 164 166 153 170 170 N 136 139 146 155 165 O 127 159 142 152 160 F 107 132 140 147 Mg 200 207 Al 249 289 Si 226 230 193 210 P 175 192 186 180 S 171 209 180 180 190 Cl 146 178 175 175 As 206 194 185 Se 222 190 190 Br 195 187 185 Te 240 208 206 I 220 204 198 52 VOL. 62, NO. 1

CHARACTERISTIC ATOMIC RADIUS evaluated by means of various methods. It is clearly shown that this definition of the boundary atomic radius has a close correlation to the empirical atomic radii listed in the table. Naturally, the atomic boundary radii should have a certain correlation with the van der Waals radii. In the solid state, the nonmetallic elements exist as aggregates of monoatomic Žnoble gas elements. or polyatomic molecules Že.g., Cl 2,O 2,N, 2 S, P. 8 4. The intermolecular forces in molecular solids are London dispersion forces or van der Waals forces. Half of the nearest distance of two atoms which belong to the same element and are in different molecules is defined as the van der Waals radius of this atom. In Figure 5, the diagram of van der Waals radii 2 vs. the boundary radii is plotted, where the upper straight line is the correlation line for rare gas atoms and the lower one is for the atoms of the rest of the nonmetallic elements. The reason for separate correlation lines is that atoms that exist in polyatomic molecules have bonding situations which vary from one atom to another and affect the van der Waals radii. Nevertheless, the correlation between the van der Waals radii and the boundary radii is quite good as demonstrated in the figure. FIGURE 5. A comparison of the boundary radii with the empirical van der Waals radii. To conclude, the characteristic atomic radius defined and evaluated by setting the potential felt by an electron at this radius point within an atom equal to the minus of its first ionization potential gives a measure of the atomic size which has close correlation with other atomic radii used in the literature; particularly the van der Waals atomic radius. ACKNOWLEDGMENTS This research was supported in part by Grant CHE-9007393 from the U.S. National Science Foundation and was carried out at Indiana University. Z.-Z. Y. would like to thank his home institution and the National Science Foundation of China for partial support of his visit to Indiana University. References 1. J. C. Slater, J. Chem. Phys. 41, 3199 Ž 1964.. 2. Ž. a T. Moeller, Inorganic Chemistry Ž A Modern Introduction. Ž Wiley, New York, 1982., p. 70. Ž b. A. Bondi, J. Phys. Chem. 68, 441 Ž 1964.. Ž c. C. W. Robert, Ed., CRC Handbook of Chemistry and Physics Ž CRC Press, Boca Raton, FL, 1989., p. D190. 3. R. J. Boyd, J. Phys. B: Atom. Mol. Phys. 10, 2283 Ž 1977.. 4. L. Pauling, The Nature of the Chemical Bond, 3rd ed. ŽCornell University Press, Ithaca, NY, 1960.. 5. M. A. Spackman and E. N. Maslen, J. Phys. Chem. 90, 2020 Ž 1986.. 6. P. Politzer, R. G. Parr, and D. R. Murphy, J. Chem. Phys. 79, 3859 Ž 1983.. 7. P. K. Chattaraj, A. Cedillo, and R. G. Parr, J. Chem. Phys. 103, 10621 Ž 1995.. 8. Ž. a R. F. W. Bader and T. T. Nguyen-Dang, Adv. Quantum Chem. 14, 63 Ž 1981.. Ž b. R. F. W. Bader and P. M. Beddall, J. Chem. Phys. 56, 3320 Ž 1972.. 9. The MELD series of electronic structure code was developed by L. E. McMurchie, S. T. Elbert, S. R. Langhoff, and E. R. Davidson and was extensively modified by D. Feller and D. C. Rawlings. 10. C. W. Robert, Ed., CRC Handbook of Chemistry and Physics Ž CRC Press, Boca Raton, FL, 1983., p. F162. 11. J. A. Grant and B. T. Pickup, J. Phys. Chem. 99, 3503 Ž 1995.. 12. Ž. a Z. Z. Yang and S. Y. Niu, Chin. Sci. Bull. 36, 964 Ž 1991.. Ž b. S. Y. Niu and Z. Z. Yang, Acta Chim. Sin. 52, 551 Ž 1994.. 13. J. C. Slater, Quantum Theory of Atomic Structure, Vol. 2 Ž McGraw-Hill, New York, 1960., pp. 1115. See also, J. C. Slater, Phys. Rev. 81, 385 Ž 1951.. 14. H. J. Partridge, J. Chem. Phys. 87, 6643 Ž 1987.. 15. W. Kohn and L. J. Sham, Phys. Rev. 140, A133 Ž 1965.. 16. K. Schwarz, Phys. Rev. B 5, 2466 Ž 1972.. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 53