Density Functional Theory

In this note we follow the definitions from the Hartree Fock post.

The Density Function

Recall that the single electron state space is defined as

\[\mathcal{H} := L^2(\mathbb{R}^3)\otimes\mathbb{C}^2\]

and that the $n$-electron state space is defined as the exterior product

\[\Lambda^n\mathcal{H}\]

The single-electron local density operator $N^1(\mathbf{r})\in\mathrm{End}(\mathcal{H})$ acts as a spatial projector (i.e a Dirac delta) and acts trivially on the spin component:

\[N^1(\mathbf{r}) = |\mathbf{r}\rangle\langle\mathbf{r}|\otimes\mathrm{Id}\]

Specifically $N^1(\mathbf{r})$ acts on a decomposable state $|\psi\rangle\otimes|s\rangle$ by:

\[N^1(\mathbf{r}) \cdot |\psi\rangle\otimes|s\rangle := \psi(\mathbf{r})|\mathbf{r}\rangle\otimes|s\rangle\]

To see why this is called the local density operator, note that the expectation value of $N^1(\mathbf{r})$ for a state $\chi = |\psi\rangle\otimes|s\rangle$ is equal to:

\[\begin{align*} \langle \chi | N^1(\mathbf{r}) | \chi \rangle &=\langle \psi|\mathbf{r}\rangle\langle\mathbf{r}|\psi\rangle \langle s | s \rangle \\ &= \psi(\mathbf{r})^*\psi(\mathbf{r}) \\ &= |\psi(\mathbf{r})|^2 \end{align*}\]

In other words, the expectation value of $N^1(\mathbf{r})$ is equal to the probability density of finding the electron at $\mathbf{r}\in\mathbb{R}^3$.

In general, we define the density function $\rho\in\mathrm{Hom}(\mathbb{R}^3,\mathbb{R})$ associated to a state $\chi\in\mathcal{H}$ in terms of the expectation of the local density operator:

\[\rho(\mathbf{r}) := \langle \chi | N^1(\mathbf{r}) | \chi \rangle\]

The $n$-electron local density operator $N^n(\mathbf{r})\in\mathrm{End}(\Lambda^n\mathcal{H})$ is defined to be the symmetric extension (or derivation) of $N^1(\mathbf{r})$ from $\mathrm{End}(\mathcal{H})$ to $\mathrm{End}(\Lambda^n\mathcal{H})$.

Similarly to the one electron case, let $|\chi_i\rangle = |\psi_i\rangle\otimes|s_i\rangle$ be orthonormal single electron states and let $\Psi = |\chi_1,\dots,\chi_n\rangle$ be their Slater determinant. Then:

\[\begin{align*} \langle \Psi | N^n(\mathbf{r}) | \Psi \rangle &= \sum_{i=1}^n \langle \chi_i | N^1(\mathbf{r}) | \chi_i \rangle \\ &= \sum_{i=1}^n |\psi_i(\mathbf{r})|^2 \end{align*}\]

This implies that the expectation value of the $N^n(\mathbf{r})$ for the Slater determinant is the sum of the probability densities of each of the spatial wavefunctions $\psi_i$.

The density function $\rho$ associated to a general state $\Psi\in\Lambda^n{\mathcal{H}}$ is defined by:

\[\rho(\mathbf{r}) := \langle \Psi | N^n(\mathbf{r}) | \Psi \rangle\]

The Reduced Density Matrix

In the previous section we defined the $n$-electron local density operator $N^n(\mathbf{r})$ as the symmetric extension of $N^1(\mathbf{r})$.

In general, if $O^1\in\mathrm{End}(V)$ is an operator and $O^n\in\mathrm{End}(V^{\otimes n})$ is its symmetric extension then, for any anti-symmetric tensor $|v\rangle\in\Lambda^n V$:

\[\begin{align*} \langle v | O^n | v \rangle &= \mathrm{Tr}(O^n |v\rangle\langle v|) \\ &= \mathrm{Tr}(O^1 \mathrm{Tr}_{n-1}(n|v\rangle\langle v|)) \end{align*}\]

Where

\[\mathrm{Tr}_{n-1}: \mathrm{End}(V^{\otimes n}) \rightarrow \mathrm{End}(V)\]

is the partial trace on the last $n-1$ factors. This motivates the introduction of the reduced density matrix $\gamma^1\in\mathrm{End}(\mathcal{H})$ associated to an $n$-electron state $|\Psi\rangle\in\Lambda^n\mathcal{H}$ defined by:

\[\gamma^1 := \mathrm{Tr}_{n-1}(n|\Psi\rangle\langle\Psi|)\]

Note that in the single-electron case, the reduced density matrix associated to $|\chi\rangle\in\mathcal{H}$ is simply the density operator $|\chi\rangle\langle\chi|$.

By the above result, it immediately follows that if $O^1\in\mathrm{End}(\mathcal{H})$ is a single-electron operator, $O^n\in\mathrm{End}(\Lambda^n\mathcal{H})$ is its symmetric extension, $|\Psi\rangle\in\Lambda^n\mathcal{H}$ is an $n$-electron state and $\gamma^1$ is its reduced density matrix then:

\[\langle \Psi | O^n | \Psi \rangle = \mathrm{Tr}(O^1\gamma^1)\]

As a special case, the density function associated to a state $|\Psi\rangle$ contains the diagonal elements of $\gamma^1$ in the basis $|\mathbf{r}\rangle$:

\[\begin{align*} \rho(\mathbf{r}) &= \langle \Psi | N^n(\mathbf{r}) | \Psi \rangle \\ &= \mathrm{Tr}(N^1(\mathbf{r})\gamma^1) \\ &= \mathrm{Tr}(|\mathbf{r}\rangle\langle\mathbf{r}|\gamma^1) \\ &= \langle \mathbf{r} | \gamma^1 | \mathbf{r} \rangle \end{align*}\]

Expectation Values

A key observation of density functional theory is that the expectation values of some of the operators that appear in the electronic Hamiltonian can be expressed in terms of density functions.

Electron Nuclear Attraction

First consider the electron-nuclear attraction operator. In the single-electron case, the operator $V_{\mathrm{en}}^1\in\mathrm{End}(\mathcal{H})$ is defined by

\[V_{\mathrm{en}}^1 |\psi\rangle\otimes|s\rangle = |v_{\mathrm{ne}} \psi \rangle|s\rangle\]

where $v_{\mathrm{ne}}(\mathbf{r})\in L^2(\mathbb{R}^3)$ is the electron-nuclear attraction potential:

\[v(\mathbf{r}) = -\sum_{i=1}^m \frac{Z_i}{||\mathbf{R}_i - \mathbf{r}||}\]

Note that by definition:

\[V_{\mathrm{en}}^1 |\mathbf{r}\rangle = v_{\mathrm{ne}}(\mathbf{r})|\mathbf{r}\rangle\]

In other words, $V_{\mathrm{en}}^1$ is diagonal in the position basis $|\mathbf{r}\rangle$ with eigenvalues $v_{\mathrm{ne}}(\mathbf{r})$.

The $n$-electron electron-nuclear attraction operator $V_{\mathrm{en}}^n\in\mathrm{End}(\Lambda^n\mathcal{H})$ is defined to be the symmetric extension of $V^1$.

Now let $|\Psi\rangle\in\Lambda^n\mathcal{H}$ be an $n$-electron state and let $\gamma^1$ be its reduced density matrix. By the previous section, the expectation value of $V_{\mathrm{en}}^n$ with respect to $|\Psi\rangle$ is given by

\[\langle \Psi | V_{\mathrm{en}}^n | \Psi \rangle = \mathrm{Tr}(V_{\mathrm{en}}^1 \gamma^1)\]

Since $V_{\mathrm{en}}^1$ is diagonal in the positional basis $|\mathbf{r}\rangle$, the trace is equal to the inner product of the eigenvalues of $V_{\mathrm{en}}^1$ and the diagonal elements of $\gamma^1$ in that basis. But in the previous section we saw that by the diagonal elements of $\gamma^1$ are given by the density function:

\[\langle\mathbf{r}|\gamma^1|\mathbf{r}\rangle = \rho(\mathbf{r})\]

Together this implies that:

\[\langle \Psi | V_{\mathrm{en}}^n | \Psi \rangle = \langle v_{\mathrm{ne}}(\mathbf{r}) | \rho(\mathbf{r}) \rangle_{L^2(\mathbb{R}^3)}\]

In summary, the expectation value of $V_{\mathrm{en}}^n$ in the state $|\Psi\rangle$ can be computed in terms of the potential $v_{\mathrm{ne}}$ and the density function associated to $|\Psi\rangle$.

The Kohn-Sham Equations

Let $\Psi Consider an electronic state with density

\[\rho: \mathbb{R}^3 \rightarrow\]