Appendix B: The

calculation of electrostatic

energies

B

This appendix describes a method of dealing with the integrals that

arise in the calculation of the electrostatic interaction of two electrons,

as in the helium atom and other two-electron systems. Consider two

(^1) The notation indicates that the elec- electrons (^1) whose charge densities areρ 1 (r 1 )andρ 2 (r 2 ). Their energy of
trons are labelled 1 and 2, meaning that
their locations are calledr 1 andr 2.
At the same time, the electron charges
are (probably) differently distributed in
space (because the electrons’ wavefunc-
tions have different quantum numbers),
soρ 1 andρ 2 are different functions of
their arguments.
electrostatic repulsion is (cf. eqn 3.15)

J=

∫∫

ρ 1 (r 1 )

e^2

4 πε 0 r 12

ρ 2 (r 2 )d^3 r 1 d^3 r 2. (B.1)

In this expression we shall leave open the precise form of the two charge

densities, so the theorem we are to set up will apply equally to direct

(^2) We have called the integralJin antic- and to exchange integrals.^2 Also, we shall permit the charge density
ipation of working out a direct integral,
but an exchange integral is equally well
catered for if we use appropriate formu-
lae forρ 1 (r 1 )andρ 2 (r 2 ).
ρ 1 (r 1 )=ρ(r 1 ,θ 1 ,φ 1 ) to depend upon the anglesθ 1 andφ 1 ,aswellas
upon the radiusr 1 , and similarly forρ 2 (r 2 ); neither charge density is
assumed to be spherically symmetric.
In terms of the six spherical coordinates the integral takes the form

J=

∫π

0

dθ 1 sinθ 1

∫ 2 π

0

dφ 1

∫π

0

dθ 2 sinθ 2

∫ 2 π

0

dφ 2

×

∫∞

0

dr 1 r^21 ρ 1 (r 1 ,θ 1 ,φ 1 )

∫∞

0

dr 2 r^22 ρ 2 (r 2 ,θ 2 ,φ 2 )

e^2

4 πε 0 r 12

.

In rearranging this expression, we process just the two radial integrals.

Within them, we divide the range forr 2 intoapartfrom0tor 1 and a

part fromr 1 to infinity. The radial integrals become^3

(^3) The integrals are written with the in-
tegrand at the end to make clear the
range of integration for each variable.