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ExchangeThe best suggestion is to read the original Phys Rev 82 (1951) 538 paper of Slater, briefly summarized below. Atoms, ions and molecules are described perturbatively in the Hartree-Fock scheme, starting from single electron wave functions, {$u_i(x)$}. Each electron moves in a potential made up by [...] all electrons, with a correction because it does not act on itself. The correction represents a charge density of total magnitude one electron, concentrated about the electron in question and is given by {$ (1) \quad\quad -e\sum_{k=1}^n \frac {u_i^*(x_1)u_k^*(x_2)u_i(x_2)u_k(x_1)}{|{u_i(x_1)}|^2} $} The contribution to the total Coulomb energy of this density is its product with the electron probability density {$|{u_i(x_1)}|^2$} and the Coulomb factor {$1/r_{12}$}, integrated over coordinates {$x_1, x_2$}: it is the so-called exchange energy. For most atoms the density of Eq. (1), hence the exchange energy, do not depend much on which electron state i we discuss. They strongly depend on the type of state when there are different numbers of electron in the spin-up and spin-down states, i.e. for magnetic atoms. Qualitatively, the exchange energy reduces the total potential energy by an amount proportional to the local charge. Therefore if there are more spin-up electrons their Hartree-Fock potential will be lower. |