# DFPT second order energy variational form

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1. Nov 11, 2015

### Sheng

I am referring to perturbation expansion of density functional Kohn Sham energy expression in
Phys. Rev. A 52, 1096.

In equation (92) the variational form of the second order energy is listed, but I cannot seem to work out the last 3 terms involving XC energy and density. Particularly, the interaction energy potential is given as:
$$v_{HXC}^{(i)} = \frac{1}{i!} \frac{d^i}{d\lambda^i} \left[ \frac{\delta E_{HXC} \left[ \sum_{j=0}^i \lambda^j n^{(j)} \right] }{\delta(n(r))} \right] \Bigg|_{\lambda=0}$$

In Phys. Rev. B 55, 10337 by the same author, it is listed that
$$v_{HXC}^{(1)} = \int \frac{\delta^2 E_{HXC}}{\delta(n(r))\delta(n(r'))} \bigg|_{n^(0)} n^{(1)}(r') \,dr' + \frac{d}{d\lambda} \frac{\delta E_{HXC}}{\delta(n(r))} \bigg|_{n^(0)}$$
which I don't know the rationale behind the expansion.

From the general expression Eq (50) in Phys. Rev. A 52, 1096, I am lost at finding the terms which contribute to the last 3 terms in the variational expression.

Thank you for any help.

2. Nov 16, 2015