Self-Consistent Single-Particle Equations and Ground-State Energies

Self-Consistent Single-Particle Equations and Ground-State Energies 

Since r is now written as an orbital functional one cannot directly minimize (46) with respect to n. Instead, one commonly employs a scheme suggested by Kohn and Sham for performing the minimization indirectly. This scheme starts by writing the minimization as 

As a consequence of (34), 8Vjhn = w(r), the external potential the electrons move in. This potential is called external because it is external to the electron system and is not generated self-consistently from the electron-electron interaction, as i h and Uxc- It comprises the lattice potential and any additional truly external field applied to the system as a whole. The term 8Es.l8n simply yields the Hartree potential, introduced in (21). For the term bE-s j n, which can only be calculated explicitly once an approximation for xo has been chosen, one commonly writes Uxc- 

Consider now a system of noninteracting particles moving in the potential Us(r). For this system the minimization condition is simply 

Equations (61 )-(63) are the celebrated Kohn-Sham equations. They replace the problem of minimizing E[ri by that of solving a single-body Schrodinger equation. (Recall that the minimization of E[n originally replaced the problem of solving the many-body Schrodinger equation ) 

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