Page 77 - Electrical Properties of Materials

60 The hydrogen atom and the periodic table

coordinate of the electron. If we have two electrons, we need two differential
operators. Thus,
2
∇ ψ
is replaced by
2
2
∇ ψ + ∇ ψ,
1 2
where the indices 1 and 2 refer to electrons 1 and 2, respectively. We may take
∗
∗ It is a separate story how positively the protons as if they are infinitely heavy again and put them at the origin of
charged protons and neutrons can peace-
the coordinate system. Thus, the potential energy of electron 1 at a distance r 1
fully coexist in the nucleus. 2
from the protons is –2e /4π 0 r 1 , and similarly for electron 2. There is, how-
ever, one more term in the expression for potential energy: the potential energy
due to the two electrons. If the distance between them is r 12 , then this potential
2
energy is e /4π 0 r 12 . It is of positive sign because the two electrons repel each
other. We can now write down Schrödinger’s equation for two protons and two
Note that the wave function ψ now electrons:
depends on six variables, namely
on the three spatial coordinates of 2 2 2 1 2e 2 2e 2 e 2
– (∇ ψ + ∇ ψ)+ – – + ψ = Eψ. (4.30)
each electron. 2m 1 2 4π 0 r 1 r 2 r 12
Can this differential equation be solved? The answer, unfortunately, is no.
No analytical solutions have been found. So we are up against mathematical
difficulties even with helium. Imagine then the trouble we should have with
tin. A tin atom has 50 protons and 50 electrons; the corresponding differential
equation has 150 independent variables and 1275 terms in the expression for
potential energy. This is annoying. We have the correct equation, but we cannot
solve it because our mathematical apparatus is inadequate. What shall we do?
Well, if we can’t get exact solutions, we can try to find approximate solutions.
This is fortunately possible. Several techniques have been developed for solv-
ing the problem of individual atoms by successive approximations. The math-
ematical techniques are not particularly interesting, and so I shall mention only
the simplest physical model that leads to the simplest mathematical solution.
In this model we assume that there are Z positively charged protons in the
nucleus, and the Z electrons floating around the nucleus are unaware of each
other. If the electrons are independent of each other, then the solution for each
of them is the same as for the hydrogen atom provided that the charge at the
2
2
centre is taken as Ze. This means putting Ze instead of e into eqn (4.2) and
4
2 4
Z e instead of e into eqn (4.18). Thus, we can rewrite all the formulae used
for the hydrogen atom, and in particular the formula for energy, which now
stands as
Z 2
E n = –13.6 . (4.31)
n 2
That is, the energy of the electrons decreases with increasing Z. In other
words, the energy is below zero by a large amount; that is, more energy is
needed to liberate an electron. This is fairly easy to understand; a large positive
charge in the nucleus will bind the electron more strongly.
The model of entirely independent electrons is rather crude, but it can go
a long way towards a qualitative explanation of the chemical properties of the