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Functional Expansion - Implicit and Explicit dependence - TDDFT

  1. Mar 4, 2014 #1
    Hello:

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

    [itex]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)] [/itex]
    w.r.t. [itex] v_{ext}(r')[/itex]

    My problem is that it depends both explicitly and implicitly and explicitly on [itex] v_{ext}[/itex]

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

    best Mikkel.
     
    Last edited: Mar 4, 2014
  2. jcsd
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