Floating Wigner crystal with no boundary charge fluctuations

Abstract

We modify the "floating crystal" trial state for the classical Homogeneous Electron Gas (also known as Jellium), in order to suppress the boundary charge fluctuations that are known to lead to a macroscopic increase of the energy. The argument is to melt a thin layer of the crystal close to the boundary and consequently replace it by an incompressible fluid. With the aid of this trial state we show that three different definitions of the ground state energy of Jellium coincide. In the first point of view the electrons are placed in a neutralizing uniform background. In the second definition there is no background but the electrons are submitted to the constraint that their density is constant, as is appropriate in Density Functional Theory. Finally, in the third system each electron interacts with a periodic image of itself, that is, periodic boundary conditions are imposed on the interaction potential.

Large coupling-strength expansion of the Møller–Plesset adiabatic connection: From paradigmatic cases to variational expressions for the leading terms

“…where W DFT ∞ [ρ] is the λ → ∞ limit of the DFT density-fixed adiabatic connection, 32,33 which for the UEG corresponds to the bcc Madelung energy. 57,63 We thus see that for the case of a uniform density, we have the equality…”

“…However, very recently, the equivalence between the two models has been fully established, including for the strong-coupling (low-density) regime. 56,57 The jellium Hamiltonian readŝ…”

“…where we have N electrons immersed in the background of positive charge density ρ inside the volume V, and we are interested in the thermodynamic limit N, V → ∞ with ρ = N/V kept fixed, which can be done in different equivalent ways. 57 The relevant length scale in H jel is rs, defined for D = 3 as rs = 3 √ 3/(4πρ): if we use scaled coordinates si = ri/rs, we see that the kinetic energy scales as 1/r 2 s , while all the potential energy terms scale as 1/rs. The low-density regime rs → ∞ is thus equivalent to the large-λ case and the electrons are believed to localize in lattice points to form a bcc Wigner crystal.…”

“…The low-density regime rs → ∞ is thus equivalent to the large-λ case and the electrons are believed to localize in lattice points to form a bcc Wigner crystal. 35,53 In the case of the UEG, the uniform density is recovered by making a linear superposition of all orientations and elementary translations of the lattice, 57,58 which is a special case of the strictly correlated-electron (SCE) limit 32 of DFT.…”

Effect of Periodic Arrays of Defects on Lattice Energy Minimizers

“…Notice that this sum is absolutely convergent as a simple consequence of the definition of F d . We could also define E f without such decay assumption by renormalizing the sum using, for instance, a uniform background of opposite charges (see, e.g., [35]) or an analytic continuation in case of parametrized potential such as r −s (see [18]). The fact that the origin is excluded from the above sum is motivated by two reasons: 0 is a fixed point of L when L varies in L d and f is not necessarily defined for r = 0 (e.g., when f is an inverse power laws or a Lennard-Jones-type potentials).…”