Intra-atomic exchange energy

The elections tend to go into different orbitals with their spins parallel, since this results in a lower intra-atomic repulsion energy between them and in a more favorable exchange energy. 

A simplified closed-shell expression for the Breit interaction can be derived as well [201]. For open-shell atoms, an average of configurations yields a similar total energy expression except for the intra-shell exchange energy [57, p. 164]. 

Here, denotes the energy of the orbital a (a = s, p, d) at atomic site i and hiao are the corresponding occupation-number operators. The are the hopping integrals describing electronic transitions between atoms i and j. The third term in Eq. (1) approximates the electron-electron interactions [30, 31]. The prime in the sum indicates that the terms with a = P and a = a are to be excluded. is the effective on-site interaction between electrons with spin a and a in the orbitals a and P of atom i. In Eq. (1) only the Coulomb and exchange intra-atomic integrals Uafi = otP /r 2 otP) and J fi = otP lr 2 Pot) are... 

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