When and Why Do Semilocal Density Functionals Work?


The simplest and most computationally efficient density functionals for the exchange-correlation energy are semilocal. The local spin density approximation and the generalized gradient approximation (GGA) are semilocal in the electron density. The meta-GGA is fully nonlocal in the electron density, but still computationally semilocal in any Kohn-Sham calculation, since it can be expressed as a single integral over 3D space of a function of ingredients that are available at each point r of space: the electron density, its gradient, and the positive orbital kinetic energy density. Since the exact exchange-correlation energy can be regarded as the electrostatic interaction between the electron density at r and the density at r’ of the exchange-correlation hole around an electron at r, semilocal approximations are expected to work when the exact exchange-correlation hole is well localized around its electron. This is the case in the electron gas of uniform or slowly-varying density, and also in an atom or other single-center system, where semi-local approximation can be accurate for exchange alone and for correlation alone. Since the exact exchange-correlation hole is typically deeper and more short-ranged than the separate exchange hole and correlation hole, semilocal approximations can still work by error cancellation between exchange and correlation in many molecules and solids near equilibrium geometries. But semilocal approximations necessarily fail when the exact exchange-correlation hole is delocalized over two or more centers, as when electrons are shared over stretched bonds (e.g., in stretched H2+). In that case, fully nonlocal functions (hybrids, self-interaction-corrected functionals, etc.) are needed, at considerably higher computational cost. We review the construction and performance of the SCAN (strongly constrained and appropriately normed) meta-GGA [1], which is designed to be accurate when the exact exchange-correlation hole is indeed well-localized around its electron, and is constructed without fitting to any bonded system.


[1] J. Sun, A. Ruzsinszky, and J.P. Perdew, Phys. Rev. Lett. 115, 036402 (2015).

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