Chemistry Reference
In-Depth Information
Appendix 9
The Atomic Orbitals of Hydrogen
The discussion of chemical bonding in the main text depends on the description of molec-
ular orbitals as linear combinations of atomic orbitals. In this appendix we show how
solutions of the Schrödinger equation for H-like atoms give us the atomic orbitals that are
used as the building blocks in this approach. We will also take the opportunity to cover
some basic ideas in quantum mechanics.
By 'H-like' we mean that a solitary electron moves in the field of a positively charged
nucleus. This avoids the complication of considering electron-electron interactions. For a
qualitative insight into chemical bonding, these can be reintroduced later.
To calculate wavefunctions for any stationary state of an H-like atom we would like
solutions to the Schrödinger equation:
−
2
2
m
e
∇
2
χ
p
+
V
χ
p
=
E
p
χ
p
(A9.1)
χ
p
the wavefunc-
tion for an electron experiencing the potential
V
with total energy
E
p
;
p
is just an index to
allow us to tell the many possible solutions apart from one another. As we go through out-
lining the solution of Equation (A9.1) we will replace
p
with the quantum numbers for the
atomic orbitals. In this appendix we will assume that the nuclear mass is so much greater
than the electron mass that it can be treated as a stationary point around which the electron
moves. The effects of removing this assumption and introducing the reduced mass of the
system are very minor.
Equation (A9.1) assumes nothing about the units of the quantities used. If we switch
to atomic units (au) then the manipulation and solution of this formula becomes clearer
because we remove the clutter of the physical constants. In au
π
where
is the Planck constant divided by 2
,
m
e
the electron mass and
1,
m
e
= 1 and the elec-
tron charge
e
= 1, i.e. these quantities are used to define the units of angular momentum,
mass and charge respectively. These units are actually derived from the solution of the H
=
