Question

How to derive the classical Hartree potential for a slab system?

Answer 1

I am now working on a slab system, but encountered some problems on the classical Hartree potential. This slab system is infinity along x-y plane, and has finite size along zz axis z∈[0,L]z∈[0,L]. I found a paper [PHYSICAL REVIEW B 80, 235101 (2009)], in which the Hartree potential reads:

VH=−2πe2∫∞−∞dz′ |z−z′|n(z′)VH=−2πe2∫−∞∞dz′ |z−z′|n(z′)

In this equation, n(z)n(z) is the position-dependent induced electron density in the system (or, can be interpreted as the net charge density ne(z)−niron(z)ne(z)−niron(z). In other words, usually for n(z′)n(z′), it satisfies ∫dz′n(z′)=0∫dz′n(z′)=0, this is the condition that I saw in most papers for such Hartree potential). Does anyone have any idea of the derivation of this equation? Or, some references are also very helpful.

Besides, in the above equation it doesn't account for the influence of permittivity. My system has the dielectric constant for surrounding medium ϵ0ϵ0 and also for slab ϵslabϵslab. In this case, what's the equation for the classical Hartree potential

Answer 2

Explanation:

The Fock operator is a one-electron operator and solving a Hartree-Fock equation gives the energy and Hartree-Fock orbital for one electron.