# What's the relationship between the first HK theorem and the second HK theorem?

The first Hohenberg-Kohn (HK) theorem: The external potential $$v(\vec{r})$$ is determined, within a trivial additive constant, by the ground-state electron density $$\rho(\vec{r})$$.

From basic quantum mechanics, we know that: $$v(\vec{r})\rightarrow \hat{H} \rightarrow \psi_0\rightarrow \rho$$. Accoridng to the first HK theorem, we can further know that $$\rho \rightarrow v(\vec{r})\rightarrow \hat{H} \rightarrow \psi_0,\psi_1,\cdots$$. In essence, the first HK theorem proves one-to-one mapping between the external potentials and the ground-state densities $$\rho$$ in many-electron systems.

The second HK theorem: There exists a universal functional of the density, $$F_{HK}[\rho']$$, such that for any $$N$$-representable density ($$\textit{i.e.}$$, any density that comes from some wavefunction for an $$N$$-electron system) $$\rho(\vec{r})$$, which yields a given number of electrons $$N$$, the energy functional is, $$E[\rho'] = F_{HK}[\rho']+\int \rho'(\vec{r})v(\vec{r}) d\vec{r} \geq E_g \tag{1}$$ in which $$E_g$$ is the ground-state energy and the equality holds when the density $$\rho'(\vec{r})$$ is the, possibly degenerate, ground-state density $$\rho_0'(\vec{r})$$ for the external potential $$v(\vec{r})$$.

From the two statements, I can't see any connection between the two theorems. So what's the relation between the two theorems? If $$F_{HK}(\rho')$$ is the functional of the ground-state density, I can build a connection between the two theorems. But the density in $$F_{HK}[\rho]$$ is not necessary ground-state density.

Using your notation, the definition for the universal functional is

$$F_{HK}[\rho] = \left< \psi_0[\rho] \right| \hat{T} + \hat{W} \left| \psi_0[\rho] \right>,$$

where $$\hat{T}$$ and $$\hat{W}$$ are kinetic and electron-electron interaction operators, respectively. This definition is possible because of the one-to-one mapping between densities and their corresponding ground state wavefunctions (i.e., because $$\psi_0$$ is a functional of $$\rho$$), which I believe is the connection you are seeking.

• According to the first HK theorem, $F_{HK}[\rho]$ should only be the functional of ground-state density? – Jack Jan 21 at 0:48
• Suppose $\rho'$ is any reasonable density (in a sense that can be made precise). It probably won't be the ground state density for the potential $v(r)$. However, it will be the ground state density for some other potential $v'(r)$, and it will correspond to a ground state wave function $\psi_0' = \psi_0[\rho']$. So, $F_{HK}[\rho']$ makes sense for all densities, not just the ground state density. You can find the ground state energy with $$E_g = \mathrm{min}_{\rho'} \left[ F_{HK}[\rho'] + \int v(r) \rho'(r) dr \right].$$ – wcw Jan 21 at 1:29
• So the $\rho'$ of equation (1) in my post can be considered a ground-state denisty corresponding to a external potential $v'(\vec{r})$ (guaranteed by the first HK theorem). If $v'(\vec{r})-v(\vec{r}) \neq C$, then $\rho'$ can be considered as density rather than ground-denisty of $v(\vec{r})$ (What the second HK wants to solve). Is that right? – Jack Jan 21 at 1:45
• Yes, that sounds right – wcw Jan 21 at 1:46

A formal connection is that the first theorem is used in the proof of the second one. Indeed, the second is a translation of the principle that $$E[\Psi']$$ has a minimum at the correct ground state wave function $$\Psi$$, using the one-to-one correspondence $$\rho \leftrightarrow \Psi$$ known from the first theorem.

The derivation can be found in the original paper by Kohn and Hohenberg (part I-2.). It's quite short and easy to read, so it's worth a look.

• Can $F_{HK}[\rho]$ be a functional of non-ground-state electron density? – Jack Jan 21 at 0:49