### D.53 Integral constraints

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

To verify that , just check their definitions.

The fact that

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

The fact that

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 and are symmetric matrices; and .

That is positive is a bit trickier; write it as

with . The part within parentheses is just the potential of a distribution of charges with density . Sure, may be complex but that merely means that the potential is too. The electric field is minus the gradient of the potential, , and according to Maxwell’s equation, the divergence of the electric field is the charge density divided by : . So and the integral is

and integration by parts shows it is positive. Or zero, if is zero wherever is not, and vice versa.

To show that , note that

is nonnegative, for the same reasons as but with replacing . If you multiply out the inner product, you get that is nonnegative, so .