# Functional derivatives in density functional theory

I am studying density functional theory and I am currently dealing with manipulating the intrinsic free energy, $$\mathcal{F}$$, which is defined as $$\mathcal{F} = F - \int dr \rho ^{(1)}(r)\phi (r)$$ which can be expressed in it's differential form as $$\delta \mathcal{F} = -S\delta T+ \int dr \delta \rho ^{(1)}(r) \psi (r)$$ In the above equations, $$F$$ is the Helmholtz free energy, $$\phi (r)$$ is some external field, $$\psi$$ is the intrinsic chemical potential defined as $$\psi (r) = \mu - \phi (r)$$, where $$\mu$$ is the chemical potential, and $$\rho ^{(1)}$$ is the single-particle density given in terms of the local activity as

$$\rho ^{(1)} (r_1)=\Xi ^{-1} \sum _{N=1}^{\infty} \frac{1}{(N-1)!} \int dr^{(N-1)} \exp [-\beta U(r^N)] \prod _{i=1}^N \left\{ z^{*}(r_i) \right\}$$ which can be generalized to $$\rho ^{(n)} (r_1)=\Xi ^{-1} \sum _{N=n}^{\infty} \frac{1}{(N-n)!} \int dr^{(N-n)} \exp [-\beta U(r^N)] \prod _{i=1}^N \left\{ z^{*}(r_i) \right\}$$ where $$z^{*}(r) = z\exp [-\beta \phi (r)]$$, where $$z$$ is the fugacity.

Defining the grand potential $$\Omega = \mathcal{F} - N\mu + \int dr \rho ^{(1)}(r) \phi (r)$$ we obtain $$\delta \Omega = -S\delta T - \int dr \rho ^{(1)}(r) \delta \psi (r)$$.

This is all fine so far. Now, in the notes I have been provided, they calculate the second derivative of $$\Omega$$, we take the $$\psi$$ derivative of $$\rho ^{(1)}$$, $$\frac{\delta ^2 \Omega}{\delta \psi (r_1) \delta \psi (r_2)} = -\frac{\delta \rho ^{(1)}(r_1)}{\delta \psi (r_2)}$$.

The first derivative makes sense, but now, they claim that \begin{align*} -\frac{\delta \rho ^{(1)}(r_1)}{\delta \psi (r_2)} &= \beta [\rho^{(1)}(r_1)\rho ^{(1)}(r_2) - \rho ^{(1)}(r_1) \delta (r_1-r_2) -\rho ^{(2)}(r_1,r_2)] \\ &= -\beta[\rho ^{(1)}(r_1) \rho ^{(1)}(r_2) h^{(2)}(r_1,r_2)+\rho ^{(1)}(r_1) \delta (r_1-r_2)] \end{align*}

Where $$h(r_1,r_2)$$ is the total correlation function.

I have no idea how they performed these derivatives and used the correlations. How did they simplify $$d\rho ^{(1)}(r_1)/d\psi (r_2)$$? I do not understand how the chain rule leads to the terms on the RHS. Nor how they managed to get $$\rho ^{(2)}$$ in the mix.

Any help with these functional derivatives would be appreciated!

• Which notes? Are they online? Which pages? Sep 18, 2021 at 5:48
• Sep 18, 2021 at 8:50
• @Qmechanic, these are from my professor's notes for the class. They are online, but not available publicly... Sep 18, 2021 at 12:48