Chemistry Reference
In-Depth Information
Table B.1
Details of quantum numbers associated with energy levels of an isolated atom
Name
Symbol
Values allowed
Physical property
Principal
n
n
=
1, 2, 3,
...
Determines radial extent and energy
Orbital
l
l
=
0, 1,
...
,
(
n
−
1
)
Angular momentum and orbit shape
Magnetic
m
−
l
,
−
l
+
1,
...
,
l
−
1,
l
Projection of orbital angular momentum
along quantisation axis
1
2
1
2
Spin
s
z
and
−
Projection of electron spin
along quantisation axis
+
Table B.2
Spectroscopic labels associated with different orbital
quantum numbers (atomic subshells) in an isolated atom
Orbital quantum number
l
0
1
2
3
4
5
Spectroscopic label
s
p
d
f
g
h
The degeneracy we have found for states with the same principal
quantum number
n
, but different orbital quantum numbers
l
,isan
'accidental' consequence of the hydrogen atompotential
V
(
r
)
, which varies
as 1
r
. This 'accidental'
l
-degeneracy is removed when most other central
potentials are used in (B.11), including the potential of any multi-electron
atom. Each shell of states with particular principal quantum number
n
therefore breaks up into a set of subshells in the multi-electron atom, with
a different orbital quantumnumber
l
associatedwith each subshell. Histor-
ically, the states in different subshells were identified using spectroscopic
techniques, and the different subshells were given the spectroscopic labels
shown in Table B.2. States with
n
/
=
1and
l
=
0 are referred to as 1s states,
while states with
n
=
2and
l
=
0 and 1 are referred to as 2s and 2p states,
respectively.
We recall from eq. (B.11) that the effect of increasing angular momentum
(increasing
l
) is described by an increasingly strong centrifugal potential
barrier (proportional to
l
r
2
) which pushes the electron away from
the nucleus. As a consequence, the 2s wavefunction (with
l
(
l
+
1
)/
0) will have
larger amplitude close to the nucleus than does the 2p wavefunction (with
l
=
. The 2s states therefore experience on average a stronger attrac-
tive potential, and so will be at a lower energy than the 2p states. The
energies of the different electron states in a multi-electron atom are clearly
affected by the presence of other electrons. The 1s orbital will always have
lowest energy and, because it largely experiences the full nuclear attrac-
tion (proportional to
Ze
for atomic number
Z
), its binding energy will be
close to
Z
4
times the binding energy of the 1s hydrogen state. The 1s states
will then partially screen the higher lying levels, modifying their energies
accordingly.
=
1
)
