Reply to "Comment on 'Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit' ".

This is a response to the comment we received on our recent paper "Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit." In that paper, we introduced a computational algorithm that is appropriate for solving stiff initial value problems, and which we applied to the one-dimensional time-independent Schrödinger equation with a soft Coulomb potential. We solved for the eigenpairs using a shooting method and hence turned it into an initial value problem. In particular, we examined the behavior of the eigenpairs as the softening parameter approached zero (hard Coulomb limit). The commenters question the existence of the ground state of the hard Coulomb potential, which we inferred by extrapolation of the softening parameter to zero. A key distinction between the commenters' approach and ours is that they consider only the half-line while we considered the entire x axis. Based on mathematical considerations, the commenters consider only a vanishing solution function at the origin, and they question our conclusion that the ground state of the hard Coulomb potential exists. The ground state we inferred resembles a δ(x), and hence it cannot even be addressed based on their argument. For the excited states, there is agreement with the fact that the particle is always excluded from the origin. Our discussion with regard to the symmetry of the excited states is an extrapolation of the soft Coulomb case and is further explained herein.

Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit.

An efficient way of evolving a solution to an ordinary differential equation is presented. A finite element method is used where we expand in a convenient local basis set of functions that enforce both function and first derivative continuity across the boundaries of each element. We also implement an adaptive step-size choice for each element that is based on a Taylor series expansion. This algorithm is used to solve for the eigenpairs corresponding to the one-dimensional soft Coulomb potential, 1/sqrt[x(2)+ß(2)], which becomes numerically intractable (because of extreme stiffness) as the softening parameter (ß) approaches zero. We are able to maintain near machine accuracy for ß as low as ß = 10(-8) using 16-digit precision calculations. Our numerical results provide insight into the controversial one-dimensional hydrogen atom, which is a limiting case of the soft Coulomb problem as ß â†’ 0.

Structure and properties of the aluminum borates Al(BO2)n and Al(BO2)n(-), (n = 1-4).

The geometrical and electronic structures of Al(BO(2))(n) and Al(BO(2))(n)(-) (n = 1-4) clusters are computed at different levels of theory including density functional theory (DFT), hybrid DFT, double-hybrid DFT, and second-order perturbation theory. All aluminum borates are found to be quite stable toward the BO(2) and BO(2)(-) loss in the neutral and anion series, respectively. Al(BO(2))(4) belongs to the class of hyperhalogens composed of smaller superhalogens, and should possess a large adiabatic electron affinity (EA(ad)) larger than that of its superhalogen building block BO(2). Indeed, the aluminum tetraborate possesses the EA(ad) of 5.6 eV, which, however, is smaller than the EA(ad) of 7.8 eV of the AlF(4) supehalogen despite BO(2) is more electronegative than F. The EA(ad) decrease in Al(BO(2))(4) is due to the higher thermodynamic stability of Al(BO(2))(4) compared to that of AlF(4). Because of its high EA and thermodynamic stability, Al(BO(2))(4) should be capable of forming salts with electropositive counter ions. We optimized KAl(BO(2))(4) as corresponding to a unit cell of a hypothetical KAl(BO(2))(4) salt and found that specific energy and energy density of such a salt are competitive with those of trinitrotoluol (TNT).

Interactions of a Mn atom with halogen atoms and stability of its half-filled 3d-shell.

Using density functional theory with hybrid exchange-correlation potential, we have calculated the geometrical and electronic structure, relative stability, and electron affinities of MnX(n) compounds (n = 1-6) formed by a Mn atom and halogen atoms X = F, Cl, and Br. Our objective is to examine the extent to which the Mn-X interactions are similar and to elucidate if/how the half-filled 3d-shell of a Mn atom participates in chemical bonding as the number of halogen atoms increases. While the highest oxidation number of the Mn atom in fluorides is considered to be +4, the maximum number of halogen atoms that can be chemically attached in the MnX(n)(-) anions is 6 for X = F, 5 for X = Cl, and 4 for X = Br. The MnCl(n) and MnBr(n) neutrals are superhalogens for n ≥ 3, while the superhalogen behavior of MnF(n) begins with n = 4. These results are explained to be due to the way different halogen atoms interact with the 3d electrons of Mn atom.

A systematic study of neutral and charged 3d-metal trioxides and tetraoxides.

Using density functional theory with generalized gradient approximation, we have performed a systematic study of the structure and properties of neutral and charged trioxides (MO(3)) and tetraoxides (MO(4)) of the 3d-metal atoms. The results of our calculations revealed a number of interesting features when moving along the 3d-metal series. (1) Geometrical configurations of the lowest total energy states of neutral and charged trioxides and tetraoxides are composed of oxo and∕or peroxo groups, except for CuO(3)(-) and ZnO(3)(-) which possess a superoxo group, CuO(4)(+) and ZnO(4)(+) which possess two superoxo groups, and CuO(3)(+), ZnO(3)(+), and ZnO(4)(-) which possess an ozonide group. While peroxo groups are found in the early and late transition metals, all oxygen atoms bind chemically to the metal atom in the middle of the series. (2) Attachment or detachment of an electron to∕from an oxide often leads to a change in the geometry. In some cases, two dissociatively attached oxygen atoms combine and form a peroxo group or a peroxo group transforms into a superoxo group and vice versa. (3) The adiabatic electron affinity of as many as two trioxides (VO(3) and CoO(3)) and four tetraoxides (TiO(4), CrO(4), MnO(4), and FeO(4)) are larger than the electron affinity of halogen atoms. All these oxides are hence superhalogens although only VO(3) and MnO(4) satisfy the general superhalogen formula.