M17 The Aufbau principle and orbital energy levels

Illustration M17 shows the orbital energy levels as a function of r (the distance from the nucleus to the electron) or increasing principal quantum number, n, the integer associated with the energy levels e.g. 1s, 2p or 5f. The principle quantum number, n, is an integer, which determines the angular momentum, J, of electrons in permitted orbits, according to the expression:

[the integer n in the expression describes the wavelengths of the series of spectral lines emitted by hydrogen see comments regarding illustrations M09 and M10)].

The energy of an electron in an atom can be characterised, by four quantum numbers:

• the principal quantum number, n, an integer;
  
• the subsidiary quantum number, l, having an integral value from 0,1,2,3,….(n-1)
  
• the magnetic quantum number, m, having an integral number varying from -l, -(l-1), -(l-2),…0…(l-2), (l-1), l;
  
• the spin quantum number, s, having values of +1/2 and -1/2. The electron can be regarded as spinning on its axis, like a top, in a clockwise and anticlockwise direction.

This method of filling the orbitals’ energy levels was first proposed by Wolfgang Pauli (1900-1958). He stated in his “Exclusion Theory” that no two electron can have the same four quantum numbers. In other words no two electrons in any one atom behave in an identical manner. If the helium atom is considered in its lowest energy state, the two electrons are assigned the identical quantum numbers n=1, l=0 and m=0 which means that according to the Pauli exclusion principle the fourth quantum number, the spin number, must be different for the two electrons i.e. one has a spin of +1/2 and the other a spin of -1/2. A pairing of electrons with opposite spin is represented by the notation:

According to Hund’s rule electrons occupy each level singly before electron pairing takes place.

According to the Aufbau principle the available electrons are filled in pairs in the available orbitals in the order of increasing orbital energy level. For orbitals with the same principle quantum number, n, e.g. 2s and 2p orbitals or 3s, 3p and 3d orbitals the orbital energy level increases in the order: s< p< d< f.

All orbitals of a particular shell or subshell (group of orbitals with the same principle and subsidiary quantum number) e.g.; the 2p-orbitals, have the same energy.

The Aufbau principle is not always followed at the higher energy levels: e.g. Ac: 7s² 6d¹ instead of
7s² 5f¹: the various energy levels overlap each other.

References:

C.E. Mortimer, ‘Chemistry’, Wadsworth Publishing
Company, Belmont, California.