D.53 Integral constraints

This note verifies the mentioned constraints on the Coulomb and exchange integrals.

To verify that $J_{nn}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $K_{nn}$, just check their definitions.

The fact that

\begin{eqnarray*}
J_{n{\underline n}} & = &
\langle
\pe n/{\skew0\vec r}_i...
... d}^3{\skew0\vec r}_i {\,\rm d}^3{\skew0\vec r}_{\underline i}.
\end{eqnarray*}

is real and positive is self-evident, since it is an integral of a real and positive function.

The fact that

\begin{eqnarray*}
K_{n{\underline n}}
& = &
\langle
\pe n/{\skew0\vec r}...
...m d}^3{\skew0\vec r}_i {\,\rm d}^3{\skew0\vec r}_{\underline i}
\end{eqnarray*}

is real can be seen by taking complex conjugate, and then noting that the names of the integration variables do not make a difference, so you can swap them.

The same name swap shows that $J_{n{\underline n}}$ and $K_{n{\underline n}}$ are symmetric matrices; $J_{n{\underline n}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $J_{{{\underline n}}n}$ and $K_{n{\underline n}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $K_{{{\underline n}}n}$.

That $K_{n{\underline n}}$ is positive is a bit trickier; write it as

\begin{displaymath}
\int_{{\rm all}\;{\skew0\vec r}_i}
-e f^*({\skew0\vec r}...
...vec r}_{\underline i}
\right)
{\,\rm d}^3{\skew0\vec r}_i
\end{displaymath}

with $f$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\pe{\underline n}////\strut^*\pe{n}////$. The part within parentheses is just the potential $V({\skew0\vec r}_i)$ of a distribution of charges with density $\vphantom0\raisebox{1.5pt}{$-$}$$ef$. Sure, $f$ may be complex but that merely means that the potential is too. The electric field is minus the gradient of the potential, $\skew3\vec{\cal E}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom0\raisebox{1.5pt}{$-$}$$\nabla{V}$, and according to Maxwell’s equation, the divergence of the electric field is the charge density divided by $\epsilon_0$: $\div\skew3\vec{\cal E}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom0\raisebox{1.5pt}{$-$}$$\nabla^2V$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom0\raisebox{1.5pt}{$-$}$$ef$$\raisebox{.5pt}{$/$}$$\epsilon_0$. So $\vphantom0\raisebox{1.5pt}{$-$}$$ef^*$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-\epsilon_0\nabla^2{V}^*$ and the integral is

\begin{displaymath}
- \epsilon_0 \int_{{\rm all}\;{\skew0\vec r}_i} V \nabla^2 V^* {\,\rm d}^3{\skew0\vec r}_i
\end{displaymath}

and integration by parts shows it is positive. Or zero, if $\pe{\underline n}////$ is zero wherever $\pe{n}////$ is not, and vice versa.

To show that $J_{n{\underline n}}$ $\raisebox{-.5pt}{$\geqslant$}$ $K_{n{\underline n}}$, note that

\begin{displaymath}
\langle
\pe n/{\skew0\vec r}_i///\pe{\underline n}/{\ske...
...0\vec r}_i///\pe n/{\skew0\vec r}_{\underline i}///
\rangle
\end{displaymath}

is nonnegative, for the same reasons as $J_{n{\underline n}}$ but with $\pe{n}////\pe{\underline n}////-\pe{\underline n}////\pe{n}////$ replacing $\pe{n}////\pe{\underline n}////$. If you multiply out the inner product, you get that $2J_{n{\underline n}}-2K_{n{\underline n}}$ is nonnegative, so $J_{n{\underline n}}$ $\raisebox{-.5pt}{$\geqslant$}$ $K_{n{\underline n}}$.