Energy density and stress fields in quantum systems

TL;DR

This work resolves longstanding ambiguities in local energy density and stress fields for nonrelativistic quantum systems by splitting kinetic energy into a density-dependent part and a uniquely defined exchange–correlation part, and by treating interactions either as potentials or as field-based quantities with a Hartree plus xc decomposition. It shows that the ground-state variational principle uniquely fixes the energy density as $e(f r)=-\tfrac{1}{2}s(f r)\nabla^2 s(f r)+\tau_{xc}(f r)+V_{ext}(f r)+\tfrac{1}{2}V_{Hart}(f r)+\epsilon^V_{xc}(f r)$, naturally leading to density functional theory (DFT) and the Kohn–Sham formalism, where the effective potential $V_{eff}(f r)$ contains the functional derivatives of the xc contributions. In contrast, the stress field is tied to forces and admits a distinct, uniquely defined kinetic-stress form $oldsymbol{oldsymbol{\sigma}}^t(f r)=\boldsymbol{oldsymbol{\sigma}}^{t,n}(f r)+\boldsymbol{oldsymbol{\sigma}}^{t,xc}(f r)$, with the kinetic part given by the appropriate average of kinetic-energy operators and the Coulomb interactions expressed via electric fields through Maxwell stress. The clear separation between energy density and stress, along with the insistence on physically motivated forms (e.g., setting a minimal $B=0$ and using Maxwell stress for Coulomb interactions), reconciles the historic ambiguities and provides well-defined formulations for energy density and stress fields applicable to DFT analyses of bonding and material properties.

Abstract

There has been an enduring interest and controversy about whether or not one can define physically meaningful energy density and stress fields, $e(\bf{r})$ and $σ_{αβ}(\bf{r})$, since the two forms of the kinetic energy, $\frac{1}{2}|\nabla Ψ|^2$ and $-\frac{1}{2}Ψ\nabla^2 Ψ$, lead to different densities, and analogous issues arise for interactions. This paper considers the ground state of a system of many interacting particles in an external potential, and presents a resolution in steps. 1) For the kinetic energy all effects of exchange and correlation are shown to be unique functions of position $\bf{r}$; all issues of non-uniqueness involve only the density $n(\bf{r})$ and are equivalent to an effective single-particle problem with wavefunction $s(\bf{r}) = \sqrt{n(\bf{r})/N}$. 2) Interactions can be considered as potentials acting on particles or interaction fields, e.g., the Maxwell form in terms of electric fields. In each case, there is a mean field part that is a function of the density and a part due to correlation that is uniquely defined. 3) The final results follow from the nature of energy and stress. Because the energy determines the ground state itself through the variational principle, the kinetic energy must involve $-\frac{1}{2}s\nabla^2 s$ and interactions in terms of potentials. This leads to density functional theory interpreted as energy density $e(\bf{r})$ equilibrated to minimize fluctuations with the same chemical potential at all points $\bf{r}$. However, stress is related to forces, and the only acceptable expressions for the stress field involve the combination $\frac{1}{2}[s\nabla^2 s - |\nabla s|^2]$, and Coulomb interactions in terms of electric fields. Together these results lead to well-defined formulations of energy density and stress fields that are physically motivated and based on a clear set of arguments.

Figures (1)

• Figure 1: Example of a finite square well, shown in the top line along with a characteristic wave function $\psi(x)$ that is sinusoidal inside the well and decays exponentially outside the well. The third and fourth lines show the two forms of the kinetic energy density. The bottom line shows the stress for the case $A=1/2$ which is the average of the two forms for the kinetic energy density. Any other combination leads to an unphysical result for the stress field.