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A single-particle pseudo-potential that splits the effect of the electron–electron repulsive potential of Helium (He) atom into two noninteracting identical particle potentials is numerically computed. This is done by minimizing the expectation value of the difference between the approximate and exact Hamiltonians over the Hilbert space of He atom. The one-particle potential is expanded in a spatial basis set which leads to an overdetermined system of linear equation that was solved using a least square approximation. The method involves a self-consistent iterative scheme where a converged solution valid for any state of the atom can be calculated. The total ground state energy for these two noninteracting particles under the calculated potential is found to be − 2.861 68, which is the Hartree–Fock limit for the He atom.

The radial Slater-type orbitals (STO) can be simply obtained by repeated parametric differentiation of the Yukawa Potential with respect to α. A new compact two-range addition theorem (AdT) for the STO is herein derived by explicit parametric differentiation of the well-known Yukawa AdT. The resulting addition formula is combined with the single-range AdT for solid spherical harmonics to present a new AdT for three-dimensional spherical coordinate STOs. We advance the proposition that this formula is “canonical” in the same sense that the Laplace expansion of the Coulomb potential is considered canonical. We demonstrate how this procedure can be employed for all exponential-type orbitals. © 2012 Wiley Periodicals, Inc.

A numerical solution of the Kohn–Sham (KS) differential equation within the local density approximation is presented. The present method involves solving for the Hartree potential from its differential form which is the Poisson equation. Radial differential equations are derived for closed-shell atoms and are solved as initial value problems. A self-consistent procedure for solving the resulting radial KS equations based on our new algorithm for solving differential equations is also discussed. Numerical tests are done on the Helium atom and comparison with results obtained from spectral exponential and Gaussian type basis functions are shown.

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 Schrodinger 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.

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, $1/\sqrt{{x}^{2}+{\ensuremath{\beta}}^{2}}$, which becomes numerically intractable (because of extreme stiffness) as the softening parameter ($\ensuremath{\beta}$) approaches zero. We are able to maintain near machine accuracy for $\ensuremath{\beta}$ as low as $\ensuremath{\beta}={10}^{\ensuremath{-}8}$ 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 $\ensuremath{\beta}\ensuremath{\rightarrow}0$.

Flow slides in sensitive clay deposits are common phenomena in Scandinavia and Canada. These flow slides have caused catastrophes to infrastructure and human life. The post-failure movements of such flow slides usually are characterized by their retrogression distances and or by the run-out distance of the slide debris. There are empirical and numerical methods used to assess the retrogression distance of slide debris. On contrary, convincing and accurate modeling techniques for run-out of sensitive clay slide debris, which is a very complex and challenging process, is yet to be developed Keeping this in view, this work presents a preliminary study to understand the run-out process in sensitive clay slide debris. An available numerical tool called DAN3D has been used to simulate the run-out process of three large flow slides occurred in Norway. In addition, back-calculation of a laboratory scale model test has been performed. A standardized calibration and adjustments on the models based on back analysis of real cases has to be done to use such models on sensitive clay debris analysis extensively. The Study shows that a very simple plastic model in DAN3D is able to estimate the run-out distance and the process.

A new compact two-range addition theorem for Coulomb Sturmians is presented. This theorem has been derived by breaking up the exponential-type orbitals into convenient elementary functions: the Yukawa potential (e − αr ∕ r) and “evenly-loaded solid harmonics,” \(({r}^{2\nu +l}{Y }_{l}^{m}(\hat{r})\) for which translation formulas are available. The resulting two-range translation formula for the exponential orbital is presented and used to construct a new addition theorem for the Coulomb Sturmians.