Origin of gradient corrections to the Thomas — Fermi approximation: the Kirzhnits method

The article presents the Kirzhnits method for calculating quantum corrections to the Thomas — Fermi approximation in many-fermion systems.The method is based on an operator approach employing the density matrix and does not rely on perturbation theory, which ensures its applicability in the strongly nonlinear regime. A key aspect is the treatment of the non-commutativity between the momentum operator and the potential by reducing the density matrix to normal form. Explicit expressions are derived for the fourth-order correction to the electron density and to the total energy. Based on these results, a fourth-order gradient correction for the kinetic energy functional is obtained. The method is universal for both local and self-consistent potentials, including exchange-correlation effects, and is promising for density functional theory calculations and applications in nanoelectronics, particularly for modeling the energy of nanostructures.

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