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

Abstract

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.

Real-time ab initio simulation of inelastic electron scattering using the exact, density functional, and alternative approaches

“…Ref. [41] has constructed an exponential interaction, which takes into account cusp of the potential at the origin. Although compared with the soft-Coulomb form, the exponential interaction greatly reduces computational cost for calculating accurate quantities with the density matrix renormalization group, they exhibit a similar quality in terms of energy components for various systems.…”

Accurate correlation energies in one-dimensional systems from small system-adapted basis functions

“…We define the 'exact' solution as solving this Hamiltonian on a very fine grid, which is close to the continuum limit. 21,69 For both the grid and for basis functions, we find the exact manyparticle ground state of these 1D reference systems using DMRG. The one-electron integrals are…”

Functional derivative of the zero-point-energy functional from the strong-interaction limit of density-functional theory

“…if ω(x) diverges to +∞ then, for every fixed λ, there is a point x after which the second term in (43) becomes dominant and the minimum of v λ (x) is at x = ±∞ (since v SCE (x) ∼ −(N − 1)/|x| for large x for the chosen interaction). To make sense of (43), one has to be careful in taking the correct order of limits: what we mean here is that for each fixed x the expansion of v λ as a function of λ follows (43). On the other hand, 1D models often assume an effective electron-electron interaction depending on the physics they aim to describe, often leading to short range interactions.…”

“…From the preceding discussion, it is clear that a short-range interaction should lead to a better behaviour of ω(x) and hence, the behaviour of v ZPE . We have tried two different short range interactions, namely a modified Yukawa potential v Yuk ee (x) = e −α|x| 1 + |x| (44a) and a purely exponential one, popular in DMRG calculations [43], v exp ee (x) = Ae −κ|x| ,…”