Analytic and Variational Xα in the Slater−Roothaan Method

Sb@Ni12@Sb20-/+ and Sb@Pd12@Sb20n Cluster Anions, Where n = +1, -1, -3, -4: Multi-Oxidation-State Clusters of Interpenetrating Platonic Solids

K₅Sb₄ and K₃Sb₇ Zintl ion precursors react with Pd(PPh₃)₄ in ethylenediamine/toluene/PBu₄⁺ solutions to give crystals of Sb@Pd₁₂@Sb₂₀^n-/PBu₄⁺ salts, where n = 3, 4. The clusters are structurally identical in the two charge states, with nearly perfect I_h point symmetry, and can be viewed as an Sb@Pd₁₂ icosahedron centered inside of an Sb₂₀ dodecahedron. The metric parameters suggest very weak Sb-Sb and Pd-Pd interactions with strong radial Sb-Pd bonds between the Sb₂₀ and Pd₁₂ shells. All-electron DFT analysis shows the 3- ion to be diamagnetic with I_h symmetry and a 1.33 eV HOMO-LUMO gap, whereas the 4- ion undergoes a Jahn-Teller distortion to an S = 1/2 D_3d structure with a small 0.1 eV gap. The distortion is predicted to be small and is not discernible by crystallography. Laser desorption-ionization time-of-flight mass spectrometry (LDI-TOF MS) studies of the crystalline samples show intense parent Sb@Pd₁₂@Sb₂₀^- ions (negative ion mode) and Sb@Pd₁₂@Sb₂₀⁺ (positive ion mode) along with series of Sb@Pd_12-y@Sb_20-x^-/+ ions. Ni(cyclooctadiene)₂ reacts with K₃Sb₇ in en/tol/Bu₄PBr solvent mixtures to give black precipitates of Sb@Ni₁₂@Sb₂₀^n- salts that give similar Sb@Ni₁₂@Sb₂₀^-/+ parent ions and Sb@Ni_12-y@Sb_20-x^-/+ degradation series in the respective LDI-TOF MS studies. The solid-state and gas-phase studies of the icosahedral Sb@M₁₂@Sb₂₀^n-/n+ ions show that the clusters can exist in the -4, -3, -1, +1 (M = Pd) and +1, -1 (M = Ni) oxidation states. These multiple-charge-state clusters are reminiscent of redox-active fullerenes (e.g., C₆₀ⁿ, where n = +1, 0, -1, -2, -3, -4, -5, -6).

Optical excitation energies, Stokes shift, and spin-splitting of C24H72Si14

As an initial step toward the synthesis and characterization of sila-diamondoids, such as sila-adamantane (Si(10)H(16),T(d)), the synthesis of a fourfold silylated sila-adamantane molecule (C(24)H(72)Si(14),T(d)) has been reported in literature [Fischer et al., Science 310, 825 (2005)]. We present the electronic structure, ionization energies, quasiparticle gap, and the excitation energies for the Si(14)(CH(3))(24) and the exact silicon analog of adamantane Si(10)H(16) obtained at the all-electron level using the delta-self-consistent-field and transitional state methods within two different density functional models: (i) Perdew-Burke-Ernzerhof generalized gradient approximation and (ii) fully analytic density functional (ADFT) implementation with atom dependent potential. The ADFT is designed so that molecules separate into atoms having exact atomic energies. The calculations within the two models agree well, to within 0.25 eV for optical excitations. The effect of structural relaxation in the presence of electron-hole-pair excitations is examined to obtain its contribution to the luminescence Stokes shift. The spin-influence on exciton energies is also determined. Our calculations indicate overall decrease in the absorption, emission, quasiparticle, and highest occupied molecular orbital-lowest unoccupied molecular orbital gaps, ionization energies, Stokes shift, and exciton binding energy when passivating hydrogens in the Si(10)H(16) are replaced with electron donating groups such as methyl (Me) and trimehylsilyl (-Si(Me)(3)).

Dipole moments from atomic-number-dependent potentials in analytic density-functional theory

Molecular dipole moments of analytic density-functional theory are investigated. The effect of element-dependent exchange potentials on these moments are examined by comparison with conventional quantum-chemical methods and experiment for the subset of the extended G2 set of molecules that have nonzero dipole moment. Fitting the Kohn-Sham [Phys. Rev. 140, A1133 (1965)] potential itself makes a mean absolute error of less than 0.1 D. Variation of alpha (Slater's [Phys. Rev. 81, 385 (1951)] exchange parameter) values has far less effect on dipole moments than on energies. It is argued that in variable alpha methods one should choose the smaller of the two rather than the geometric mean of the two alpha values for the heteroatomic part of the linear-combination-atomic-orbital density. Calculations on the dipole moment of NH(2)(CH)(24)NO(2) are consistent with earlier calculations and show that varying the differences between alpha values for atoms with different atomic numbers has only short-ranged electrostatic effects.

The limitations of Slater’s element-dependent exchange functional from analytic density-functional theory

Our recent formulation of the analytic and variational Slater-Roothaan (SR) method, which uses Gaussian basis sets to variationally express the molecular orbitals, electron density, and the one-body effective potential of density-functional theory, is reviewed. Variational fitting can be extended to the resolution of identity method, where variationality then refers to the error in each two-electron integral and not to the total energy. However, a Taylor-series analysis shows that all analytic ab initio energies calculated with variational fits to two-electron integrals are stationary. It is proposed that the appropriate fitting functions be charge neutral and that all ab initio energies be evaluated using two-center fits of the two-electron integrals. The SR method has its root in Slater's Xalpha method and permits an arbitrary scaling of the Slater-Gàspàr-Kohn-Sham exchange-correlation potential around each atom in the system. The scaling factors are Slater's exchange parameters alpha. Of several ways of choosing these parameters, two most obvious are the Hartree-Fock (HF) alpha(HF) values and the exact atomic alpha(EA) values. The former are obtained by equating the self-consistent Xalpha energy and the HF energies, while the latter set reproduces exact atomic energies. In this work, we examine the performance of the SR method for predicting atomization energies, bond distances, and ionization potentials using the two sets of alpha parameters. The atomization energies are calculated for the extended G2 set of 148 molecules for different basis-set combinations. The mean error (ME) and mean absolute error (MAE) in atomization energies are about 25 and 33 kcal/mol, respectively, for the exact atomic alpha(EA) values. The HF values of exchange parameters alpha(HF) give somewhat better performance for the atomization energies with ME and MAE being about 15 and 26 kcal/mol, respectively. While both sets give performance better than the local-density approximation or the HF theory, the errors in atomization energy are larger than the target chemical accuracy. To further improve the performance of the SR method for atomization energies, a new set of alpha values is determined by minimizing the MAE in atomization energies of 148 molecules. This new set gives atomization energies half as large (MAE approximately 14.5 kcal/mol) and that are slightly better than those obtained by one of the most widely used generalized-gradient approximations. Further improvements in atomization energies require going beyond Slater's functional form for exchange employed in this work to allow exchange-correlation interactions between electrons of different spins. The MAE in ionization potentials of 49 atoms and molecules is about 0.5 eV and that in bond distances of 27 molecules is about 0.02 A. The overall good performance of the computationally efficient SR method using any reasonable set of alpha values makes it a promising method for study of large systems.