
In the description of the energies of transition of the hydrogen
atom, the n values for the different
energies are known as the principal quantum
number for that energy level.
Each atomic orbital is described by a set of quantum numbers: the
principal quantum number, and three others, the orbital
angular momentum quantum number, l, the magnetic
quantum number, m, and the spin angular
momentum quantum number, s.
These have the ranges of values:
l: 0, 1, 2,..., (n1)
m: l, l+1,.., 0, ..., l1, l. (May have upto (2l+1) values)
s: 1/2 and +1/2
Atomic orbitals are named according to the values
of their principal and orbital angular momentum quantum numbers.
Atomic Orbitals with n up to 4 
n 
l 
name 
Number of different m orbitals 
1 
0 
1s 
1 
2 
0 
2s 
1 
1 
2p 
3 (m = 1, 0, 1) 
3 
0 
3s 
1 
1 
3p 
3 (m = 1, 0, 1) 
2 
3d 
5 (m = 2, 1, 0, 1, 2) 
4 
0 
4s 
1 
1 
4p 
3 (m = 1, 0, 1) 
2 
4d 
5 (m = 2, 1, 0, 1, 2) 
3 
4f 
7 (m = 3, 2, 1, 0, 1,
2, 3) 
In a many electron atom, each of these atomic orbitals
can hold two electrons, and the spin quantum number is different
for these two electrons.
Energy Levels
The energy of the transitions in the hydrogen ion
is given by:
We can interpret this in terms of a transition
between two energy levels, and hence the transition energy is
the difference between the energies of the two levels. Therefore,
each of the terms in the above expression corresponds to the
energies of the levels given by the values of n_{1}
and n_{2} (In the expression below, n and m are used
instead of n_{1} and n_{2}).
Transition energies and energy levels in hydrogen 


The energy of a given atomic orbital is therefore
proportional to the inverse square of the principal quantum number.
When we consider hydrogenic atoms with nuclear charges
greater than one, we must allow for the increased attraction between
the nucleus and the electron, and the resultant change in the
energy. The energies of the allowed states now depend on the nuclear
charge, Z, according to:
The sizes of the orbitals also decrease with increasing
Z. A useful approximate guide to the size of an orbital is: size
~ n^{2}/Z (in Bohr radii).
In hydrogen, the energy depends only upon the principal
quantum number, n, but this is not true in atoms with more than
one electron.
This is due to the fact that orbitals with different
values of the orbital angular momentum quantum number, l, have
different shapes,
and so there is a different degree of interaction between electrons
and the nucleus for the different l numbers, and so in the presence
of other electrons the orbitals with different l numbers have
different energies. For species with more than one electron, for
example, the s (l = 0) level is lower in energy than the p (l
= 1) level for a given value of n.
However, it is always true that in free atoms orbitals
with the same values of n and l have the same energies: they are
called degenerate. For example, all
three p orbitals with a given n value are degenerate, as are all
five d orbitals, and all seven f orbitals, etc.
When a field is applied to an ion, this rule is
also not obeyed, as in the splitting in energy of the d orbitals
in a ligand field. This splitting is described by Ligand
and Crystal Field Theory.

