Problems of Self-Interaction in Standard Quantum Mechanics

Self-interaction in Density Functional Theory

In Newton's theory of gravity gravitational potential $\phi (x)$ and mass distribution $\rho (x)$ with $x$ a spatial coordinate are coupled through the differential equation: 

• $-\Delta\phi (x) = 4\pi\rho (x),$                                                       (1)

or equivalently through the integral equation: 

• $\phi (x)=\int\frac{\rho (y)}{\vert x-y\vert }dy.$                         (2) 

We here assume a setting of a continuum with no smallest scale in space, in which the integration variable $y$ in (2) is never equal to $x$, and we can argue that a mass at an isolated point $x$ does not contribute to the gravitational potential $\phi (x)$ present at $x$ and so that there is no self-interaction on a microscopic scale. 

In the setting of $N$ distinct mass points with location $x_i$ and mass $m_i$ for $i=1,...,N$, (2) takes the form 

• $\phi (x_i) =\sum_{j\neq i}\frac{m_j}{\vert x_i-x_j\vert}$,         (3)

which does not involve self-interaction because $j\neq i$. Choosing $j=i$ would cause break down of (3) into infinity by dividing by 0.  

But on a macroscopic scale there is self-interaction in the sense that Earth's gravitational potential is present everywhere on and inside the Earth. Similarly the Universe interacts with itself by gravitational forces. But that is because the Earth and the Universe are made up of elementary particles (protons and electrons) which must be free of self-interaction as in (3) to avoid break-down into infinity.  

Standard quantum mechanics stdQ for an atom/molecule with $N$ electrons is governed by an electronic wave function $\Psi (x_1,...,x_N)$ depending on $N$ 3d spatial variables $x_1, x_2,...,x_N$ satisfying a Schrödinger equation with an electron-electron repulsion term of the following form connecting to  (2) and (3):

•  $\sum_i\sum_{j<i}\frac{1}{\vert x_i-x_j\vert}\Psi (x_1,...,x_N)$.       (4)

This expression does not carry any infinity since $j\neq i$, but is it free of self-interaction? This is not clear since the coordinates $x_i$ do not represent actual presence of individual electrons which be tracked by index, but only possible presence in a $3N$ dimensional configurations space without electron individuality where a specific coordinate is not tied to a specific electron, thus without Exclusive Occupancy.  The wave function $\Psi (x_1,...,x_N)$ of stdQM is anti-symmetric (assuming for simplicity that all electrons have the same spin) thus asking $\Psi (x_1,...,x_N)$ to vanish if $x_j=x_i$ for some $i$ and $j$, but that is not enough to prevent self-interaction.  

In fact, in density functional theory derived from the standard Schrödinger equation self-interaction is present and has to be handled by some trick. More precisely, the wave function is built from products of overlapping electron orbitals generating a contribution to Coulomb repulsion energy from exchange correlation expressing self-interaction.   

RealQM offers a different version of quantum mechanics keeping electron individuality as non-overlapping charge clouds with Exclusive Occupancy, which does not suffer from self-interaction.