Quantum Numbers and Energy Levels

n 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,…, (n-1)

m: -l, -l+1,.., 0, …, l-1, 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 n1 and n2 (In the expression below, n and m are used instead of n1 and n2).

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 ~ n2/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.