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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?

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    $\begingroup$ I think the point is that there is no physical meaning to a self-interaction, indeed the self-interaction must physically be zero, and hence firstly any functional that contains such an interaction can not be the exact one and secondly the exact functional must respect the condition of zero self-interaction as required by the physics. But I'm sure somebody can give a more formal answer than this. $\endgroup$
    – Ian Bush
    Jan 2 at 22:31
  • $\begingroup$ @IanBush thanks for your comment, I am going to extend this question regarding the physical meaning of self-interaction into another question, if you have a full answer then you may provide it in the new question. $\endgroup$ Jan 4 at 2:51

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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.

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  • $\begingroup$ @JaafarMehrez the question about whether or not there is a physical meaning for the self-interaction, is probably better suited as a separate post, so that we can get opinions not only from this user, but by others too. Please see this. $\endgroup$ Jan 4 at 2:25
  • $\begingroup$ @NikeDattani, sure thing I have updated my question, and will ask another one seperately. $\endgroup$ Jan 4 at 2:48
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    $\begingroup$ @elutionary, It is a very thoughtful answer, thank you for the Bonus as well. $\endgroup$ Jan 4 at 2:49
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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

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  • $\begingroup$ Thank you for raising this perspective, correct me if I am wrong, the self-interaction is avoided with this functional because It recognizes that different wavefunctions can describe the same electron distribution and accounts for the energetically favorable configurations within that set by including the minimization step? As if the functional is capturing the collective behavior of electrons represented by the density, rather than focusing on individual wavefunctions? $\endgroup$ Jan 4 at 11:03
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    $\begingroup$ I think your intuition is correct. The underlying principle is there are many different wavefunctions that can produce the same fixed electron density. The form of the functional given in the first equation is basically saying for a fixed density, search through all possible wavefunctions and measure T + Vee in the wavefunction picture (NOT USING approximations, i.e using the proper Hamiltonian). In a hand waving manner, you are mapping a DFT problem to a wavefunction problem and using the proper operators so the Coulomb part and exchange part for self interaction will cancel out. $\endgroup$
    – lex2763
    Jan 5 at 10:50

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