D.54 Integral constraints

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

To verify that $J_{nn}=K_{nn}$, 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; $J_{n{\underline n}}=J_{{\underline n}n}$ and $K_{n{\underline n}}=K_{{\underline n}n}$.

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

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

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

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

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 $J_{n{\underline n}}\ge K_{n{\underline n}}$, note that

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

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 $J_{n{\underline n}}\ge K_{n{\underline n}}$.