Quantum solution for the one-dimensional Coulomb problem

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)+β(2)], which becomes numerically intractable (because of extreme stiffness) as the softening parameter (β) approaches zero. We are able to maintain near machine accuracy for β as low as β = 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 β → 0.

show abstract

“…The implications of our results compared with previous analytic approaches can be traced starting with the recent work described in Ref. [16].…”

Section: Discussionmentioning

confidence: 80%

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

Gebremedhin

Weatherford

2014

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…To construct the LDA, E LDA XC is constructed from Eqs. (17) and (25). In practice, we always use spin-DFT, and the local spin density approximation (LSDA).…”

Section: A Atomic Energiesmentioning

confidence: 99%

One-dimensional mimicking of electronic structure: The case for exponentials

et al. 2015

View full text Add to dashboard Cite

An exponential interaction is constructed so that one-dimensional atoms and chains of atoms mimic the general behavior of their three-dimensional counterparts. Relative to the more commonly used soft-Coulomb interaction, the exponential greatly diminishes the computational time needed for calculating highly accurate quantities with the density matrix renormalization group. This is due to the use of a small matrix product operator and to exponentially vanishing tails. Furthermore, its more rapid decay closely mimics the screened Coulomb interaction in three dimensions. Choosing parameters to best match earlier calculations, we report results for the one dimensional hydrogen atom, uniform gas, and small atoms and molecules both exactly and in the local density approximation.

show abstract

“…The one-dimensional hydrogen molecular ion consists of one electron restricted to move between two stationary protons along the x-axis. The two protons act as impenetrable walls [4,5]. The Schrödinger equation for this molecular ion is…”

Section: Introductionmentioning

confidence: 99%