Analytic second variational derivative of the exchange-correlation functional

A general analytic expression for the second variational derivative of gradient-corrected exchange-correlation energy functionals is derived, and the terms for the widely used Becke/Perdew, Becke/Lee-Yang-Parr, and Perdew-Burke-Ernzerhof exchange-correlation functionals are given. These analytic derivatives can be used for all applications employing linear-response theory or time-dependent density-functional theory. Calculations are performed in a plane-wave scheme and shown to be numerically more stable, more accurate, and computationally less costly than the most widely used finite-difference scheme. © 2004 The American Physical Society.