0198506961.pdf

304 Appendix B: The calculation of electrostatic energies

by electron 2 owing to the presence of electron 1. Whenρ(r 1 )=ρ(r 2 ),

as in the calculation of the direct integral for the 1s^2 configuration (Sec-

tion 3.3.1), these two parts of eqn B.3 are equal (and only one integral

(^5) A similar simplification can be used in needs to be calculated). 5
the calculation of exchange integrals, as
in Exercise 3.6.
ThefactthatthepotentialsV 12 andV 21 arepartialpotentials deserves
comment. Had we defined aV 21 (r 1 ) to represent the whole of the elec-
trostatic potential fromρ 2 (r 2 ) acting on electron 1, then the interaction
energyJwould have been given by the single term
∫π
0
dθ 1 sinθ 1
∫ 2 π
0
dφ 1

∫∞

0

dr 1 r^21 ρ 1 (r 1 )V 21 (r 1 )

=

∫

ρ 1 (r 1 )V 21 (r 1 )d^3 r 1 ,

(B.4)

or by a similar expression with the labels 1 and 2 interchanged through-

(^6) The physical idea that each electron out. (^6) Such a presentation is, of course, physically correct, but it is much
should ‘feel’ a potential due to the other
is so intuitive that there can be a strong
temptation to write the interaction en-
ergy down twice—once in the form B.4
and a second time with 1 and 2 inter-
changed. If this is done, the interaction
energy is double-counted. The advan-
tage of the form B.3 is precisely that it
conforms to the intuitive idea without
double-counting.
less convenient mathematically than the version presented earlier; it
precludes the separation of the Schr ̈odinger equation into two equations,
each describing the behaviour of one electron.
The simplified form forJgiven in eqn B.3 is used in the evaluation of
direct and exchange integrals for helium in Chapter 3, and is generally
applicable to electrostatic energies.