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In the review by Oliviera et. al., I do not understand how the authors get equation 14: \begin{equation} E[\rho]=\sum_i^Mn_i\langle\Psi_i|-\frac{1}{2}\nabla^2+v_{ext}(\vec{r})+\frac{1}{2}\int\frac{\rho(\vec{r'})}{|\vec{r}-\vec{r'}|}|\Psi_i\rangle+E_{xc}+\frac{1}{2}\sum_\beta^N\sum_{\beta\neq\alpha}^N\frac{Z_\alpha Z_\beta}{|\vec{R_\alpha}-\vec{R_\beta}|}\tag{14.} \end{equation} To me the energy should be in this form:

$$E = \langle\psi|T + V_\mathrm{ext} + V_\mathrm{ee}|\psi\rangle$$

How did they manage to put some of term out of the bracket? I also have trouble retrieving (16) as they cite, are they using a Taylor expansion?

References:

  1. Oliveira, A. F.; Seifert, G.; Heine, T.; Duarte, H. A. Density-functional based tight-binding: an approximate DFT method. J. Braz. Chem. Soc. 2009, 20 (7), 1193–1205. DOI: 10.1590/S0103-50532009000700002.
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    $\begingroup$ Could you please duplicate equations (14) and (16) in your question, as well as add some context and define all the variables in case the full-text article becomes unavailable? $\endgroup$ – andselisk♦ Apr 14 '20 at 15:30
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    $\begingroup$ I think they pull E_xc[\rho] out of bracket because it is a functional. V_ee is composed of coulombic and exchange interaction right, the coloumbic term can be calculated represented easily with the 1-RDM (?) operator, whereas the exchange is more complicated so they pulled it out for simplicity. $\endgroup$ – Cody Aldaz Apr 14 '20 at 17:42

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