56 Helium
unlm=Rnl(r)Ylm(θ, φ) only the radial part depends onZ.Wewrite
the exchange integral as (cf. eqn 3.28)K1snl=
e^2
4 π 0∫∫
K(r 1 ,r 2 )R1s(r 1 )Rnl(r 1 )R1s(r 2 )Rnl(r 2 )r 12 dr 1 r^22 dr 2.
(3.32)
The functionK(r 1 ,r 2 ) containing the angular integrals is (cf. eqn 3.29)K(r 1 ,r 2 )=∫∫∫∫
1
r 12
Ylm∗(θ 1 ,φ 1 )1
4 π
Ylm(θ 2 ,φ 2 )×sinθ 1 dθ 1 dφ 1 sinθ 2 dθ 2 dφ 2.(3.33)
For the 1snp configuration only the second term of the expansion in
eqn 3.30, withk= 1, survives in the integration because of the orthog-
onality of the spherical harmonic functions (see Exercise 3.7), to giveK(r 1 ,r 2 )={
r 1 / 3 r^22 forr 1 <r 2 ,
r 2 / 3 r^21 forr 2 <r 1.(3.34)
Carrying out the integration over the radial wavefunctions in eqn 3.32 for
the 1s2p configuration gives the splitting between^3 Pand^1 Pas2K1s2p
0 .21 eV (close to the measured value of 0.25 eV).
The assumption that the excited electron lies outside the 1s wave-
function does not work so well for 1sns configurations sinceψns(0) has
a finite value and the above method of calculatingJ andK is less(^25) At smallrthe wavefunction of an accurate. (^25) The 1s2s configuration of helium has a singlet–triplet sepa-
ns-electron deviates significantly from
uZns=1; for this reason 1s2p was chosen
as an example above.
ration ofE
( 1
S
)
−E
( 3
S
)
=2K1s2s 0 .80 eV and the direct integral is
also larger than that for 1s2p—these trends are evident in Fig. 3.4 (see26 also Exercise 3.7).^26
The overlap of the 1s andnlwave-
functions becomes smaller as nand
lincrease. In Heisenberg’s treatment
where screening is taken into account,
the direct integral gives the deviation
from the hydrogenic levels (which could
be characterised by a quantum de-
fect as in the alkalis, see Chapter 4).
For electrons withl =0theterm
^2 l(l+1)/ 2 mr^2 in the Schr ̈odinger
equation causes the electron’s wave-
function to lie almost entirely outside
the region whereuZ1s=2=R1s(r)/√ 4 π
has a significant value.
In some respects, helium is a more typical atom than hydrogen. The
Schr ̈odinger and Dirac equations can be solved exactly for the one-
electron system, but not for helium or other atoms with more electrons.
Thus in a careful study of helium we encounter the approximations
needed to treat multi-electron atoms, and this is very important for
understanding atomic structure in general. Helium also gives a good
example of the influence of identical particles on the occupation of the
states in quantum systems. The energy levels of the helium atom (and
the existence of exchange integrals) do not depend on the fact that the
two electrons are identical, as demonstrated in Exercises 3.3 and 3.4;
however, this is a common point of confusion. The books recommended
for further reading give clear and accurate descriptions of helium that
reward careful study.Further reading
The recommended books are divided into two categories corresponding
to the two main themes in this chapter: (a) a description of how to
calculate the electrostatic energy in an atom with more than one elec-
tron, which introduces principles that can be used in atoms with more