Appendix B: The calculation of electrostatic energies 303∫∞
0dr 1 r^21 ρ 1 (r 1 )∫r 10dr 2 r^22 ρ 2 (r 2 )
e^2
4 πε 0 r 12+∫∞
0dr 1 r^21 ρ 1 (r 1 )∫∞
r 1dr 2 r^22 ρ 2 (r 2 )e^2
4 πε 0 r 12=∫∞
0dr 1 r^21 ρ 1 (r 1 )∫r 10dr 2 r^22 ρ 2 (r 2 )e^2
4 πε 0 r 12+∫∞
0dr 2 r^22 ρ 2 (r 2 )∫r 20dr 1 r^21 ρ 1 (r 1 )e^2
4 πε 0 r 12=∫∞
0dr 1 r^21 ρ 1 (r 1 )V 21 (r 1 )+∫∞
0dr 2 r^22 ρ 2 (r 2 )V 12 (r 2 ).
(B.2)Figure B.1 illustrates the rearrangement of the ranges of integration.
HereV 21 (r 1 )isthepartial electrostatic potential atr 1 caused by the
charge densityρ(r 2 ), partial because it is caused by only that part of
ρ(r 2 ) that lies at radii less thanr 1 ; and a similar definition applies to
V 12 (r 2 ). The entire electrostatic energyJis now obtained by integrating
expression B.2 over the anglesθ 1 ,φ 1 ,θ 2 andφ 2 .Weobtain
J=
∫
ρ 1 (r 1 )V 21 (r 1 )d^3 r 1 +∫
ρ 2 (r 2 )V 12 (r 2 )d^3 r 2. (B.3)Expression B.3 can be applied to any charge densities, however com-
plicated their dependence on the anglesθandφ. But the significance of
our result is most easily explained if we take the special case where the
potentialsV 21 (r 1 )andV 12 (r 2 ) are spherically symmetric—independent
of angles—either in fact or as the result of imposing an approximation.
In such a case we can think ofV 21 (r 1 ) as the (partial) potential result-
ing from a radial electric field^4 that is felt by electron 1 owing to the
(^4) In the trade, the potential itself is usu-
ally referred to as a radial, orcentral
field. We should point out that the
functionsV 12 andV 21 are here not nec-
essarily the same functions of their ar-
guments; indeed, they will generally be
different unlessρ 1 (r)=ρ 2 (r). Later
in this book we shall discussthecen-
tral field, which is a singleV(r), acting
alike on all electrons. The potentials
introduced here are different from this
presence of electron 2; and likewiseV 12 (r 2 ) is the (partial) potential felt common-to-all central field.
(a) (b)
Fig. B.1Integration over the region
r 2 >r 1 >0 can be carried out in two
ways: (a) integration with respect to
r 2 fromr 2 =r 1 to∞, followed by in-
tegration fromr 1 =0to∞;or(b)in-
tegration with respect tor 1 from 0 to
r 1 =r 2 , followed by integration from
r 2 =0to∞. The latter is convenient
for calculating the electrostatic interac-
tion in eqn B.2 and it leads to two con-
tributions that are related by an inter-
change of the particle labelsr 1 ↔r 2.
For a symmetrical configuration such
as 1s^2 in helium these two contribu-
tions are equal and this feature reduces
the amount of calculation. By defini-
tion, exchange integrals are unchanged
by swapping the labelsr 1 ↔r 2 and so
the same comments apply to their eval-
uation.