Beyond the gradient expansion approximation: A generalized gradient expansion for exchange

Abstract

The gradient expansion approximation (GEA) exchange in density functional theory is derived from the long-wavelength response of jellium and yields an analytic expansion in even powers of the reduced density gradient s. This structure reflects the perturbative q→0 limit rather than an exact constraint. Relaxing analyticity in s leads naturally to a generalized gradient expansion (GGE) based on a Puiseux series containing both integer and fractional powers. The exchange-hole structure of inhomogeneous jellium indicates that finite-wavevector singularities associated with the Kohn-anomaly at the Fermi surface generate non-analytic contributions, including a leading s3/2 behavior. Non-analytic, non-polynomial, and mixed structures are already intrinsic to widely used generalized gradient approximation (GGA) exchange functionals. With natural extensions to correlation, the GGE provides a minimal and rigorous extension of the GEA, enabling systematic construction of new GGA exchange–correlation functionals.