DFPT second order energy variational form

[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.

 

[eljose79]

Thank you for bringing up this topic on the perturbation expansion of density functional Kohn Sham energy expression. I understand that you are having trouble understanding the last three terms involving the XC energy and density in equation (92) of Phys. Rev. A 52, 1096.

Firstly, let me clarify that the expansion you are referring to is known as the perturbation expansion of the XC energy, which is a common technique used in density functional theory (DFT). This expansion is used to approximate the XC energy, which is notoriously difficult to calculate exactly.

The expression for v_{HXC}^{(i)} that you have listed is derived from the general expression for the XC energy given in Eq (50) of Phys. Rev. A 52, 1096. In this expression, the XC energy is written as a functional of the density, E_{HXC}[n(r)]. The variational form of the second order energy (Eq (92)) is obtained by expanding the XC energy in a Taylor series around the reference density n^(0), and keeping only terms up to second order.

The first term in this expansion is the interaction energy potential v_{HXC}^{(1)}, which is given by the integral in the first term of your second equation. This term represents the direct effect of the perturbation on the XC energy.

The second term in the expansion is the first derivative of the XC energy with respect to the perturbation parameter, evaluated at the reference density. This term represents the indirect effect of the perturbation on the XC energy.

The third term, which is the one you are having trouble with, involves the second derivative of the XC energy with respect to the density, evaluated at the reference density. This term represents the second order indirect effect of the perturbation on the XC energy.

I hope this explanation helps you understand the rationale behind the expansion and how the terms in the variational expression are obtained. If you have any further questions, please do not hesitate to ask.