Self-interaction in density functional approximations for many-electron systems

Question

In their famous paper Self-interaction correction to density-functional approximations for many-electron systems Perdew and Zunger said

The exact density functional for the ground-state energy is strictly self-interaction-free (i.e., orbitals demonstrably do not self-interact), but many approximations to it, including the local-spin-density (LSD) approximation for exchange and correlation, are not.

How to prove that the exact density functional is self-interaction-free if the exact form of the exchange-correlation functional is not known?

Answer 1

No self-interaction in the exact functional

The electronic structure Hamiltonian in real space is

$$\tag{1} H = T + V_{ee} + V_{\text{ext}} = -\frac12 \sum_i \nabla_i^2 + \sum_{i<j} \frac{1}{\lvert r - r' \rvert} + V_{\text{ext}}(r), $$ where $i, j$ index the electrons in your favorite system.

The relevant term is the electron-electron interaction $V_{ee}$, and especially the index in its sum: $i < j$ means that electron $i$ does not interact with itself.

Because the (unknown) Hohenberg–Kohn functional is exact, it must replicate the behavior of the Schrödinger equation. No self-interaction is part of that.

Self-interaction in Hartree–Fock theory and Kohn–Sham DFT

Neither Hartree–Fock (HF) theory nor Kohn–Sham DFT calculate $V_{ee}$ directly. HF approximates the electron-electron interaction between one-particle orbitals $\lvert \psi_i \rangle$ and $\lvert \psi_j \rangle$ as

$$ \begin{align} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ \langle \psi_i \vert V_{ee} \vert \psi_j \rangle &\small{\approx J[\psi_i, \psi_j] - K[\psi_i, \psi_j] }\tag{2}\\ &\small{= \iint dr\, dr'\, \frac{\psi_i^*(r) \psi_i(r) \psi_j^*(r') \psi_j^*(r')}{\lvert r - r' \rvert} - \iint dr\, dr'\, \frac{\psi_i^*(r) \psi_j(r) \psi_j^*(r') \psi_i(r')}{\lvert r - r' \rvert}.}\tag{3} \end{align} $$ $J$ is called the Hartree energy (it's the classical electrostatic repulsion) and $K$ is called the exchange energy; they differ by a swapping of indices. A notable property of HF is that $J[\psi_i, \psi_i] - K[\psi_i, \psi_i] = 0$, as you can verify by inspection. For this reason, HF theory is said to be self-interaction–free.

Kohn–Sham DFT's treatment of $V_{ee}$ is not as simple; it's something like $$\tag{4} \langle \psi_i \vert V_{ee} \vert \psi_j \rangle \approx J[\psi_i, \psi_j] + E_{xc}[\rho], $$ except that the exchange-correlation energy $E_{xc}$ also includes some information about the many-body kinetic energy. But you can check whether a one-electron system, like the hydrogen atom, actually satisfies $J[\psi_1, \psi_1] - E_{xc}[\rho] = 0$ for a given density functional approximation's $E_{xc}$: usually the answer is no. This is usually called self-interaction error.

Bonus: Many-electron self-interaction error

There are actually some density functional approximations, such as those described here, that are free of (one-electron) self-interaction error, just like HF is. However, they still suffer from most of the DFT problems traditionally associated with self-interaction error: too-small band gaps and reaction energy barriers, overly delocalized charges, and so on. This more general issue with DFT is known as delocalization error (it's sometimes been called many-electron self-interaction error too!) and mitigating it is an active area of research.

Answer 2

The way I like to think about this is the Levy-Lieb functional. The universal functional takes the following form:

$F[\rho] = \underset{\substack{\Psi \mapsto \rho(\vec{r}) \\ \rho \in \mathcal{D}}}{\mathrm{min}} \bigg[ \langle{\Psi}| T + V_{ee} |{\Psi}\rangle ]\bigg],$

where $\Psi \mapsto \rho$ constrains the search to any wavefunction that yields the fixed density $\rho$. Here the set $\mathcal{D}$ denotes all the $N$-representable densities coming from a particular wavefunction. Note the set of all $N$-representable densities coming from different wavefunctions is a larger set than $v$-representable densities.

The variational theorem decomposes the energy minimization over $\Psi$ in two steps. The outer step requires defining a set of $\rho$ that define all $N$-electron densities $\mathcal{N}$. For each $\rho \in \mathcal{N}$, the inner loop performs a minimization over all wavefunctions that generate that density. The ground state energy is given by the minimum energy from this search. Overall:

\begin{equation} \label{eq:levy_energy} \begin{aligned} E_{0} &= \underset{\substack{ \rho \in \mathcal{N}}}{\mathrm{min}} \bigg[ \langle{\Psi}| T + V_{ee} + V_{Ne} |{\Psi}\rangle \bigg] \\ &= \underset{\rho \in \mathcal{N}}{\mathrm{min}} \Bigg[ \underset{\substack{\Psi \mapsto \rho(\vec{r}) \\ \rho \in \mathcal{D}\subset \mathcal{N}}}{\mathrm{min}} \bigg( \langle{\Psi}| T + V_{ee} + V_{Ne} |{\Psi}\rangle \bigg) \Bigg] \\ &= \underset{\rho \in \mathcal{N}}{\mathrm{min}} \Bigg[ \underset{\substack{\Psi \mapsto \rho(\vec{r}) \\ \rho \in \mathcal{D} \subset \mathcal{N}}}{\mathrm{min}} \bigg( \langle{\Psi}| T + V_{ee} |{\Psi}\rangle \bigg) + \int v_{ext}(\vec{r}) \: \rho(\vec{r}) \: d\vec{r} \Bigg] \\ &= \underset{\rho \in \mathcal{N}}{\mathrm{min}} \Bigg[ F[\rho]+ \int v_{ne}(\vec{r}) \: \rho(\vec{r}) \: d\vec{r} \Bigg]. \end{aligned} \end{equation} The minimum of this search is obtained for ground-state density $\rho_{0}(\vec{r})$ corresponding to the potential $v_{ne}(\vec{r})$.

In practice this search is not viable... but is still formally exact and solves the issue of self-interaction