Zero-temperature equation-of-motion of electron pair in the BCS theory of superconductivity

Abstract

By projecting the BCS ground state of superconducting electron condensate on the N-electron Hilbert space, a real-space equation-of-motion is obtained for the electron pair function [Formula: see text] at absolute zero temperature (T = 0):[Formula: see text]where ρN−2 denotes electron density of the (N – 2)-electron condensate given as[Formula: see text]Since the exchange-correlation potential is given as an explicit functional of electron density, this equation represents the fundamental working equation for the new density functional theory of superconductivity. The 2nd-order density matrix ΓN(1, 2|1′, 2′) projected on the N-electron Hilbert space satisfies[Formula: see text]so that asymptotically[Formula: see text]where [Formula: see text] denotes the center-of-mass coordinate of electrons e1and e2; this is considered the ODLRO (off-diagonal long-range order) at T = 0 projected on the N-electron Hilbert space. A new attractive potential analysis for the two-electron scattering problem (A. Tachibana, Bull. Chem. Soc. Jpn. 66, 3319 (1993); Int. J. Quantum Chem. 49, 625 (1994)) is straightforwardly applicable to the present equation-of-motion, and we can also plug in the vibronic interaction for the enhancement of the attractive force. Our approach is purely mathematical and basic, restricted merely at T = 0, but proves to serve as a real-space analysis of the pair function itself. Key words: equation-of-motion of electron pair, BCS theory, superconductivity, electron pair function, density functional theory.