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

Area of Science:

• Computational Physics
• Quantum Chemistry
• Materials Science

Background:

• The gradient expansion approximation (GEA) in density functional theory (DFT) relies on analytic expansions in the reduced density gradient (s).
• This analytic structure is derived from the long-wavelength limit (q→0) and may not fully capture the behavior of real materials.
• Existing generalized gradient approximation (GGA) functionals often contain non-analytic features.

Purpose of the Study:

• To develop a more rigorous and systematic extension of the GEA.
• To incorporate non-analytic behavior observed in the exchange-hole structure of inhomogeneous jellium.
• To enable the construction of improved exchange-correlation functionals.

Main Methods:

• Relaxing the analyticity constraint of the GEA.
• Developing a generalized gradient expansion (GGE) based on Puiseux series (integer and fractional powers).
• Analyzing the exchange-hole structure of inhomogeneous jellium, considering finite-wavevector effects and Kohn anomalies.

Main Results:

• The GGE naturally incorporates non-analytic contributions, including a leading s^(3/2) behavior.
• Finite-wavevector singularities at the Fermi surface are identified as sources of non-analyticity.
• The GGE framework is shown to be consistent with existing GGA functionals.

Conclusions:

• The GGE provides a minimal and rigorous extension of the GEA by including essential non-analytic terms.
• This framework facilitates the systematic development of new and improved GGA exchange-correlation functionals.
• The GGE approach offers a path towards more accurate predictions in DFT calculations for condensed matter systems.