Beating the egg-box effect in plane-wave DFT simulations

Abstract The ‘egg-box effect’ is a known challenge in density functional theory (DFT) calculations which arises from the discretisation of continuous quantities, e.g. the electron density. This effect is observed when the system is moved relative to the underlying computational grid and causes an unphysical change in the system’s total energy, violating translational invariance. The pattern of energy change with translation is reminiscent of an egg-box. This effect can cause unphysical results such as geometry relaxations finding incorrect crystal symmetries or imaginary phonon modes in vibrational calculations. The egg-box effect can be mitigated by using finer grids for all continuous quantities, but this greatly increases computational cost. For plane-wave DFT the effect’s origin is the evaluation of the exchange and correlation (XC) energy ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi>XC</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> ). We present a novel technique for estimating the violation of translational invariance in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi>XC</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , using a Fourier interpolation scheme, providing an estimate of the uncertainty. Our results also show that the numerical behaviour of the XC approximations is strongly linked to the magnitude of the violation of the translation invariance. The more numerically ill behaved XC functionals, such as the more advanced meta-generalised-gradient approximations functionals, exhibit changes in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi>XC</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> (due to violation of translation invariance) that are orders of magnitude worse than less advanced functionals. Performing this analysis at an early stage of a workflow can inform the user about the expected accuracy of subsequent calculations. Further, our results demonstrate that by selectively computing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi>XC</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and its corresponding potential ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>XC</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> ) on a finer grid, the egg-box effect can be significantly reduced. Coupled with our uncertainty quantification method, egg-box related inaccuracies can be avoided more conveniently and efficiently than just increasing grid resolution until inaccuracies appear suppressed. This work offers a promising pathway towards mitigating the egg-box effect in a diverse range of materials modelling applications.