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///...
...m 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}_i//...
...rm 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}_i...
...w0\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}/{\skew0...
...ew0\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$}$ $K_{n{\underline n}}$.