K. D. Sen

Characteristic features of the electrostatic potentials of singly negative monoatomic ions

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.

Characteristic features of Shannon information entropy of confined atoms.

The Shannon information entropy of 1-normalized electron density in position and momentum space Sr and Sp, and the sum ST, respectively, are reported for the ground-state H, He+, Li2+, H-, He, Li+, Li, and B atoms confined inside an impenetrable spherical boundary defined by radius R. We find new characteristic features in ST denoted by well-defined minimum and maximum as a function of confinement. The results are analyzed in the background of the irreducible lower bound stipulated by the entropy uncertainty principle [I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)]. The spherical confinement model leads to the ST values which satisfy the lower bound up to the limits of extreme confinements with the interesting new result displaying regions over which a set of upper and lower bounds to the information entropy sum can be locally prescribed. Similar calculations on the H atom in 2s excited states are presented and their novel characteristics are discussed.

On the importance of the "density per particle" (shape function) in the density functional theory

The central role of the shape function sigma(r) from the density functional theory (DFT), the ratio of the electron density rho(r) and the number of electrons N of the system (density per particle), is investigated. Moreover, its relationship with DFT based reactivity indices is established. In the first part, it is shown that an estimate for the chemical hardness can be obtained from the long range behavior of the shape function and its derivative with respect to the number of electrons at a fixed external potential. Next, the energy of the system is minimized with the constraint that the shape function should integrate to unity; the associated Lagrange multiplier is shown to be related to the electronic chemical potential micro of the system. Finally, the importance of the shape function for both molecular structure, reactivity, and similarity is outlined.

Information entropies for eigendensities of homogeneous potentials

For homogeneous potentials, the sum S(T), of position and momentum Shannon information entropies Sr and Sp is shown to be independent of the coupling strength scaling. The other commonly used uncertainty like products also follow similar behavior. The ramifications of this scaling property in the cases of hydrogenlike, harmonic oscillator, Morse, and Poeschl-Teller potentials are discussed with the example of S(T).

Confined helium atom low-lying S states analyzed through correlated Hylleraas wave functions and the Kohn-Sham model

Calculation including the electron correlation effects is reported for the ground 1 1S and lowest triplet 1 3S state energies of the confined helium atom placed at the center of an impenetrable spherical box. While the adopted wave-functional treatment involves optimization of three nonlinear parameters and 10, 20, and 40 linear coefficients contained in wave functions expressed in a generalized Hylleraas basis set that explicitly incorporates the interelectronic distance r12, via a Slater-type exponent and through polynomial terms entering the expansion, the Kohn-Sham model employed here uses the Perdew and Wang exchange-correlation functional in its spin-polarized version within the local-density approximation (LDA) with and without the self-interaction correction. All these calculations predict a systematic increase in the singlet-triplet energy splitting toward the high confinement regime, i.e., when the box radius is reduced. By using the variational results as benchmark, it is found that the LDA underestimates the singlet-triplet energy splitting, whereas the self-interaction correction overestimates such a quantity.

Mean excitation energy, static polarizability, and hyperpolarizability of the spherically confined hydrogen atom

Calculations of mean excitation energy, Im, static polarizability, α, and hyperpolarizability, γ, using the variation perturbation procedure are reported for the spherically confined hydrogen atom. The electric response properties α and γ have been found to strongly depend upon the radius of confinement. The hyperpolarizabilty changes sign and becomes negative under strong confinement.

Ground- and excited-state cusp conditions for the electron density

Higher-order cusp relations are derived for the wave function and the electron density of the ground and the excited states of atoms, ions or molecules. The total energy is expressed in terms of the electronic potential and density terms defined at the nucleus. It is proved that the linear term of the spherical part in the expansion of the Kohn–Sham potential, the classical Coulomb and the exchange correlation potentials around a nucleus are all equal to zero. A relationship involving the values of the density and its second and third derivatives at the nucleus is derived.

Relativistic effects on information measures for hydrogen-like atoms

Position and momentum information measures are evaluated for the ground state of the relativistic hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Renyi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large r, but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of p. Both features yield divergent Renyi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac-Fock vs. Hartree-Fock complexity measures for many-electron neutral atoms.

Shell-confined hydrogen atom.

Calculations of electronic energy and static dipole polarizability are reported for the hydrogen atom in the ns states (n = 1-6) confined between two impenetrable concentric spheres of inner and outer radii placed at the locations of the radial nodes corresponding to the free hydrogen ns (n = 2-7) orbitals. Interesting new conditions of degeneracy arising due to the shell confinement are discussed. Shell-confined states of unusually high polarizability are predicted for hydrogen atom as the inner sphere radius is shifted towards the outer nodal points of the free atom corresponding to the higher principal quantum numbers.

An accurate local exchange potential for atomic one- and two-electron excited states

An orbital-generated local exchange potential, obtained as the work done in moving an electron in the electric field of its Fermi hole, is found to give Hartree-Fock quality results for the one- and two-electron excited states of Na and Be, respectively. The calculated total energies are found to provide upper bounds to the Hartree-Fock energies.