# Functional Expansion - Implicit and Explicit dependence - TDDFT

1. Mar 4, 2014

### dikmikkel

Hello:

I am trying to find a functional derivative of the following functional:

$v_{s}[n[v_{ext}(r)], v_{ext}(r)] = \int \dfrac{n(r')}{|r-r'|}\,\text{d}r' + v_{ext}(r) + v_{xc}[n(r)]$
w.r.t. $v_{ext}(r')$

My problem is that it depends both explicitly and implicitly and explicitly on $v_{ext}$

My idea was to write
$\delta v_{s}[v_{ext}(r)] = \int \dfrac{\delta v_{s}(r)}{\delta v_{ext}(r')}\delta v_{ext}(r') + \dfrac{\delta v_{s}(r)}{\delta n(r')}\delta n(r')\,\text{d}r'$
And then varying also n:
$\delta n(r') = \int \dfrac{\delta n(r')}{\delta v_{ext}(r_2)}\delta v_{ext}(r_2)\,\text{d}r_2$
Sorry for my very non-mathematical way of explaining my problem.

best Mikkel.

Last edited: Mar 4, 2014