Unique Quantum Stress Fields

We have recently developed a geometric formulation of the stress field for an interacting quantum system within the local density approximation (LDA) of density functional theory (DFT). We obtain a stress field which is invariant with respect to choice of energy density. In this paper, we explicitly demonstrate this uniqueness by deriving the stress field for different energy densities. We also explain why particular energy densities give expressions for the stress field that are more tractable than others, thereby lending themselves more easily to first-principles calculations. Understanding a material’s energetic response to strain is fundamentally coupled to understanding the physics of a broad range of phenomena, from surface reconstructions to piezoelectricity. For example, it has been demonstrated that knowledge of the spatial distribution of microscopic stress via first-principles calculations can help explain the onset of macroscopic polarization in a material [1]. Therefore it is important to understand the nature of stress at the microscopic level. Formally, the electronic contribution to the microscopic stress must be included via quantum mechanics. There have been numerous methods developed for computing the quantum stress. (See Ref. [2] and references contained within.) The stress field, σαβ(x), is a rank-two tensor field usually taken to be symmetric (torque-free). The divergence of σαβ(x) must equal the force field F (x) of the system: F α = ∇βσ. (1) (Note that the Einstein summation convention for repeated indices is used throughout the paper.) It is well-known that Eq. 1 does not uniquely define the stress field (regardless of whether a system is quantum-mechanical or classical) since one can add any tensor whose divergence is zero to σαβ(x) and still recover the same force field [3]. The volume-averaged or total stress Tαβ has been related to the energetic response of the system to uniform strain and to the integral over the stress field: T αβ = ∂E ∂eαβ (2)