# A universal density criterion for correlating the radii and other properties of atoms and ions

TL;DR: In this paper, an improved Thomas-Fermi-Dirac (TFD) method was used to define a characteristic radius r D for a free atom or ion, where the electron density acquires the universal value of 0.008714.

Abstract: On the basis of an improved Thomas-Fermi-Dirac (TFD)-type method, a characteristic radius r D is defined for a free atom or ion, where the electron density acquires the universal value of 0.008714. This value is obtained from the ratio of the Dirac exchange constant to the TF kinetic energy constant. Among vertical groups in the Periodic Table, r D calculated from near-Hartree-Fock densities shows striking linear relationships with the empirical covalent radius, the van der Waals radius, the Wigner-Seitz radius, the ionic radius, the first ionization potential, the electronegativity, the softness, the dipole polarizability and the London dispersion coefficient. The potential at r D due to the net charge inside a sphere of radius r d also varies linearly as Mulliken's electronegativity, among vertical groups. On the average, a sphere of radius r d contains more than 95% of the total electronic charge. The horizontal variations in r d among non-transition, transition, and inner transition atoms are sensitive to the changes in electronic configuration, identifying half-filled and completely filled Subshells and also reflecting the lanthanide contraction.

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TL;DR: This chapter discusses the development of DFT as a tool for Calculating Atomic andMolecular Properties and its applications, as well as some of the fundamental and Computational aspects.

Abstract: I. Introduction: Conceptual vs Fundamental andComputational Aspects of DFT1793II. Fundamental and Computational Aspects of DFT 1795A. The Basics of DFT: The Hohenberg−KohnTheorems1795B. DFT as a Tool for Calculating Atomic andMolecular Properties: The Kohn−ShamEquations1796C. Electronic Chemical Potential andElectronegativity: Bridging Computational andConceptual DFT1797III. DFT-Based Concepts and Principles 1798A. General Scheme: Nalewajski’s ChargeSensitivity Analysis1798B. Concepts and Their Calculation 18001. Electronegativity and the ElectronicChemical Potential18002. Global Hardness and Softness 18023. The Electronic Fukui Function, LocalSoftness, and Softness Kernel18074. Local Hardness and Hardness Kernel 18135. The Molecular Shape FunctionsSimilarity 18146. The Nuclear Fukui Function and ItsDerivatives18167. Spin-Polarized Generalizations 18198. Solvent Effects 18209. Time Evolution of Reactivity Indices 1821C. Principles 18221. Sanderson’s Electronegativity EqualizationPrinciple18222. Pearson’s Hard and Soft Acids andBases Principle18253. The Maximum Hardness Principle 1829IV. Applications 1833A. Atoms and Functional Groups 1833B. Molecular Properties 18381. Dipole Moment, Hardness, Softness, andRelated Properties18382. Conformation 18403. Aromaticity 1840C. Reactivity 18421. Introduction 18422. Comparison of Intramolecular ReactivitySequences18443. Comparison of Intermolecular ReactivitySequences18494. Excited States 1857D. Clusters and Catalysis 1858V. Conclusions 1860VI. Glossary of Most Important Symbols andAcronyms1860VII. Acknowledgments 1861VIII. Note Added in Proof 1862IX. References 1865

3,890 citations

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TL;DR: The method chosen is a set of two-parameter correlations of Bondi's radii with repulsive-wall distances calculated by relativistic coupled-cluster electronic structure calculations, which results in new atomic radii for 16 main-group elements in the periodic table.

Abstract: Atomic radii are not precisely defined but are nevertheless widely used parameters in modeling and understanding molecular structure and interactions. The van der Waals radii determined by Bondi from molecular crystals and data for gases are the most widely used values, but Bondi recommended radius values for only 28 of the 44 main-group elements in the periodic table. In the present Article, we present atomic radii for the other 16; these new radii were determined in a way designed to be compatible with Bondi's scale. The method chosen is a set of two-parameter correlations of Bondi's radii with repulsive-wall distances calculated by relativistic coupled-cluster electronic structure calculations. The newly determined radii (in A) are Be, 1.53; B, 1.92; Al, 1.84; Ca, 2.31; Ge, 2.11; Rb, 3.03; Sr, 2.49; Sb, 2.06; Cs, 3.43; Ba, 2.68; Bi, 2.07; Po, 1.97; At, 2.02; Rn, 2.20; Fr, 3.48; and Ra, 2.83.

1,215 citations

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TL;DR: The atomic radii as defined in this way correlate well with van der Waals radii derived from crystal structures, and are derived using relativistic all-electron density functional theory calculations, close to the basis set limit.

Abstract: Atomic and cationic radii have been calculated for the first 96 elements, together with selected anionic radii. The metric adopted is the average distance from the nucleus where the electron density falls to 0.001 electrons per bohr3, following earlier work by Boyd. Our radii are derived using relativistic all-electron density functional theory calculations, close to the basis set limit. They offer a systematic quantitative measure of the sizes of non-interacting atoms, commonly invoked in the rationalization of chemical bonding, structure, and different properties. Remarkably, the atomic radii as defined in this way correlate well with van der Waals radii derived from crystal structures. A rationalization for trends and exceptions in those correlations is provided.

212 citations

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TL;DR: In this paper, the authors used the double hump curve of the experimental radii of 3 d-block transition metal ions to calculate the diamagnetic susceptibility, polarizability and chemical hardness of the ions and compared with available experimental data.

Abstract: The theoretical method of determination of absolute atomic size, discussed in Int. J. Mol. Sci. 2002, 3, 87-113, is exploited to calculate absolute radii of the ions whose experimental radii are published by Shanon. The computed radii are found to reproduce the expected periodic variation of size in periods and in groups and nicely reproduce the d-block and f-block contractions in the respective series. It is pointed out that experimental radii of d and f block transition metal ions make erroneous and misleading representation of the size behaviour of the respective series. A detailed comparative study of the crystal radii vis-a-vis the theoretical radii is reported. A rationale of the double hump curve of the experimental radii of 3 d-block transition metal ions is put forward in terms of the crystal field theory and Jahn-Teller distortion. The theoretical radii are exploited to calculate the diamagnetic susceptibility, polarizability and chemical hardness of the ions and compared with available experimental data. The fact of good agreement between the experimental and computed global hardness of ions and correct demonstration of d-block and f-block contraction by the computed radii are used as benchmark to test the validity of the values of the computed theoretical radii of the ions as their representative sizes. It is concluded that the theoretically computed radii of ions are visualizable size representation of ions and can be used as their absolute radii at the respective oxidation states.

208 citations

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TL;DR: In this article, a set of effective ab initio van der Waals radii for free and covalently bonded atoms and ions of H-Ar (Z=1-18) were determined using a helium atom probe.

Abstract: We employ natural steric analysis (introduced in a previous paper) to evaluate a set of effective ab initio van der Waals radii for free and covalently bonded atoms and ions of H–Ar (Z=1–18) determined using a helium atom probe. We critically examine the degree of anisotropy, dependence on charge state, and other intrinsic limitations of a simple atomic van der Waals hard sphere representation of the accurate steric surface. We also evaluate the ab initio steric force (gradient of steric energy at van der Waals contact) as a measure of “hardness” of the atomic van der Waals spheres. Comparison with empirical van der Waals radii shows reasonable agreement (within the acknowledged uncertainties of the latter values in the most important cases), but suggest a wider range of variability and anisotropy than could be adequately represented by any fixed constant radius. Simple expressions for incorporating the dependence on natural atomic charge or correcting for other types of intermolecular contact are given, extending the accuracy and usefulness of the atomic van der Waals sphere concept.

166 citations

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^{1}TL;DR: In this paper, expansion wave functions of double-zeta-and triple-Zeta-valence dimensions are computed from the formalism of Roothaan and Bagus.

319 citations

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TL;DR: In this paper, it was shown that a monoatomic negative ion has a minimum in its electrostatic potential V(r) occurring at the radial distance that encompasses a quantity of electronic charge exactly equal to the nuclear charge.

Abstract: We show that a monoatomic negative ion has a minimum in its electrostatic potential V(r) occurring at the radial distance rm that encompasses a quantity of electronic charge exactly equal to the nuclear charge. Thus, V(rm) is due entirely to the excess electronic charge on the ion. We suggest that rm can be identified with the radius of the ion, while V(rm) indicates the strength of its interactions with positive ions.

111 citations

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TL;DR: In this paper, a kinetic energy density functional based on a locally linearized approximation for the potential has been investigated through calculation using accurate Hartree-Fock densities for several atoms.

Abstract: A kinetic energy density functional based on a locally linearized approximation for the potential has been investigated through calculation using accurate Hartree–Fock densities for several atoms. The integrated and the local values of this kinetic energy density as well as the local behavior of its functional derivative are compared with corresponding quantities for Hartree–Fock and other existing kinetic energy functionals including a newly suggested one. With an N‐dependent correction factor, this functional shows better agreement.

41 citations

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TL;DR: The radii and electrostatic potential of singly negative ions of the 3d, 4d and 5d transition series have bee calculated in this article, which gives a quantitative estimate of the size of these ions.

Abstract: The radii and electrostatic potential of singly−negative ions of the 3d,4d and 5d transition series have bee calculated. This gives a quantitative estimate of the size of these ions.(AIP)

33 citations

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TL;DR: This paper reports the first quadratic, nondifferential, and self-consistent solution to a Thomas-Fermi-Dirac (TFD)-type equation for many-electron atoms, which is easier to perform than other TFD-type calculations, while the radial density and energy obtained compare very well with similar calculations.

Abstract: This paper reports the first quadratic, nondifferential, and self-consistent solution to a Thomas-Fermi-Dirac (TFD)-type equation for many-electron atoms. The essential feature in the Euler-Lagrange equation is the inclusion of a "first-gradient" correction to the TF kinetic energy, which leads to chemical binding in molecules and solids. The calculations are easier to perform than other TFD-type calculations, while the radial density and energy obtained compare very well with similar calculations.

26 citations