Trouble understanding equation from density functional theory

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](https://doi.org/10.1590/S0103-50532009000700002).


