First-order differential equations for single-particle quantum mechanical systems.

Coupled first-order differential forms of a single-particle Schrödinger equation are presented. These equations are convenient to solve efficiently using the widely available ordinary differential equation solvers. This is particularly true because the solutions to the differential equation are two sets of complementary functions that share simple and consistent mathematical relationships at the boundary and across the domain for a given potential.

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.

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.

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).

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.

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.