Hybrid exchange–correlation potential built on the piecewise linearity conditions of both energy and electron density

The piecewise linearity condition is a fundamental concept in density functional theory (DFT) and provides powerful conditions for developing accurate approximations. The piecewise linearity condition of energy has been widely used for developing new approximations, while the piecewise linearity condition of electron density has not received much attention in these developments. In this work, we develop a method for building the exchange-correlation (XC) potential based on the piecewise linearity conditions of both energy and electron density. The XC potential is defined as a linear mixing of the exchange potentials from the exact exchange and the local density approximation. The mixing parameter is spatially dependent and is fully determined based on these two linearity conditions. The numerical tests show that the energy's linearity condition holds well as the system's electron number is varied between N and N - 1, leading to reliable predictions of the eigenvalues of the highest occupied molecular orbitals of various molecular systems. The density's linearity condition is well satisfied near N but is violated near N - 1. Finally, this method is examined with more challenging tests: calculating the Kohn-Sham correlation potentials of He, Be, and H2. The key features of these correlation potentials, such as the shell structures and the barriers at bond middle points, are semi-quantitatively captured by this new method, which demonstrates that the piecewise linearity condition of electron density is a promising condition for developing high-quality approximations in DFT.