Accurate Approximations of Density Functional Theory for Large Systems with Applications to Defects in Crystalline Solids

Abstract

This chapter presents controlled approximations of Kohn–Sham density functional theory (DFT) that enable very large scale simulations. The work is motivated by the study of defects in crystalline solids, though the ideas can be used in other applications. The key idea is to formulate DFT as a minimization problem over the density operator, and to cast spatial and spectral discretization as systematically convergent approximations. This enables efficient and adaptive algorithms that solve the equations of DFT with no additional modeling, and up to desired accuracy, for very large systems, with linear and sublinear scaling. Various approaches based on such approximations are presented, and their numerical performance is demonstrated through selected examples. These examples also provide important insights into the mechanics and physics of defects in crystalline solids.

Appendix: Crystalline solids and the Cauchy–Born Rule

A Bravais lattice is a lattice with a single atom in its unit cell:

$$\displaystyle \begin{aligned} {\mathcal L}_B (a_i,o) = \{r \in {\mathbb R}^3: r = \sum_{i=1}^3 \nu^i a_i, \ \nu_i \text{ integers} \} \end{aligned}$$

where a set of linearly independent vectors or lattice vectors$\{a_i\}_{i=1}^3$ describes the unit cell, or translational symmetry, and o signifies the presence of an atom at the origin. A crystal (also called lattice with a basis) is a periodic arrangement of atoms (points) in ${\mathbb R}^3$ with a finite number M of atoms in the unit cell. It may be regarded as a union of P congruent Bravais lattices which are displaced from each other:

$$\displaystyle \begin{aligned} {\mathcal L} (a_i,p_\alpha) = \cup_{\alpha=1}^M {\mathcal L}_B (a_i,p_\alpha) \end{aligned}$$

where $\{a_i\}_{i=1}^3$ are the lattice vectors and the shift vectors p_α, α = 1, …, M describe the relative positions of the atoms within the unit cell. It is conventional to take p₁ = o, but this is not necessary. The underlying Bravais lattice is often referred to as the skeletal lattice.

A crystalline solid is a restriction of a lattice to a domain $\Omega \subset {\mathbb R}^3$. Let $\{r_a\}_{a=1}^A$ denote the positions of the atoms in a crystalline solid ${\mathcal L} (a_i^0,p_\alpha ^0) \cap \Omega$ in the reference domain $\Omega \subset {\mathbb R}^3$. As the solid deforms, the current position of the atoms are given by $\{y_a\}_{a=1}^A \subset {\mathbb R}^3$. Let $y: \Omega \to {\mathbb R}^3$ denote a smooth deformation that maps the positions of the underlying skeletal lattice, i.e., $y_a = y (r_a) \ \forall \ r_a \in {\mathcal L}_B(a_i^0, p_1^0)$. We call y the macroscopic deformation. Now, if the scale of the lattice is small compared to the size of the domain, and if the deformation y varies slowly on the scale of the lattice, i.e., it may be approximated by an affine map of a scale large compared to that of a_i, then at any r₀ ∈ Ω, the current positions of the atoms in the neighborhood of y(r₀) are arranged in a lattice ${\mathcal L} (a_i, q_\alpha )$ where

$$\displaystyle \begin{aligned} a_i = \nabla y (r_0) a_i^0. \end{aligned}$$

In other words, for moderate macroscopic deformations, the deformation gradient convects the lattice vectors. This is known as the Cauchy–Born rule. Note that the macroscopic deformation only constrains the skeletal Bravais lattice and the atoms are free to “shuffle” within the unit cell.