| dc.description |
Rigorous lower bounds for the first-gradient corrections to the kinetic and exchange-energy functionals in atoms are derived using Benson's inequality. They are formulated in terms of the expectation values and the momentum expectation value in a simple manner namelyT<sub>2</sub>[ρ] = (1/72)∫(|ρ(r)|<sup>2</sup>/ρ(r)) dr ≥ (n<sup>2</sup>/72) ⟨r<sup>n − 3</sup>⟩<sup>2</sup>⟨r<sup>2n − 4</sup>⟩<sup>−1</sup>and|K<sub>2</sub>[ρ]| = β ∫(|ρ(r)|<sup>2</sup>/ρ<sup>4/3</sup>(r)) dr ≥ 0.05105 ⟨r<sup>−1</sup>⟩<sup>2</sup>⟨p⟩<sup>−1</sup>,where n, the natural number = 1, 2, 3,..., ⟨r<sup>n − 3</sup>⟩ and ⟨r<sup>2n − 4</sup>⟩ are the expectation values, ⟨p⟩ is the momentum value, ρ(r) is the electron density with the normalization condition ∫ ρ(r) dr = N, the number of electrons. The bounds derived in this work are tested for the atoms Z = 3 to 36. A comparison is also made with the previously derived lower bound estimates for T<sub>2</sub>[ρ] and K<sub>2</sub>[ρ]. The bounds presented are sharper than the previous ones given by Pathak and Gadre (atomic units are used throughout the paper). |
