# Pauli Exclusion Principle

If we consider the first ionization energy of lithium, we find that its value, at 5.39 eV, would have to result from an effective nuclear charge of 0.63. This implies that there is a higher degree of screening than can be explained, as perfect screening would result in a Zeff value of one. In fact, the ground stateelectronic configuration of lithium is 1s22s1, and the least tightly bound electron is in the 2s orbital, where Zeff = 1.26.

The electronic configuration 1s3 is prohibited by the Pauli exclusion principle, which is stated simply as:

No individual orbital may be occupied by more than two electrons, and if two electrons do occupy a single orbital, then their spins must be paired.

Many experiments, such as spectroscopic and magnetic measurements, have shown that every electron has an intrinsic rotation or angular momentum known as spin. The spin direction of an electron can have only two possible values. These are described by the spin angular momentum quantum number, ms.

 The values of the electron spin quantum number ms = +1/2 also denoted by α, or spin-up ms = -1/2 also denoted by β, or spin-down

An electron is therefore fully described by the set of four quantum numbers, n, l, ml, and ms. This leads to the alternative statement of the Pauli exclusion principle:

No two electrons in an atom may have the same values for all four quantum numbers n, l, ml, and ms.

The most important consequence of the Pauli exclusion principle for atoms is that each set of orbitals with a given n and l, known as a subshell (a shell refers to a given value of n), can hold a maximum of (4l+2) electrons, ie. two for an s subshell, 6 for a p subshell, and 10 for a d subshell.

#### The building-up principle

In the lithium atom, the third electron is in the n = 2 shell. It occupies the 2s orbital, and not a 2p orbital, giving the 1s22s1 ground state electronic configuration. Why should this be so?

The order in which the orbitals are occupied, such that the 2s orbital is occupied before the 2p orbital, is known as the building up principle, or the aufbau principle. This can be used to predict the ground-state electronic configuration of a species, which can be confirmed by spectroscopic measurements.

The building up principle gives the order of occupation of the orbitals, determined by a combination of the principle quantum number and the penetration and shielding effects, as:

1s 2s 2p 3s 3p 4s 3d 4p …

This gives the ground state electronic configurations of the neutral atoms shown in the table.

 Ground-state electron configurations of neutral atoms H 1s1 He 1s2 = [He] Li [He]2s1 Be [He]2s2 B [He]2s22p1 C [He]2s22p2 N [He]2s22p3 O [He]2s22p4 F [He]2s22p5 Ne [He]2s22p6 = [Ne] Na [Ne]3s1 Mg [Ne]3s2 Al [Ne]3s23p1 Si [Ne]3s23p2 P [Ne]3s23p3 S [Ne]3s23p4 Cl [Ne]3s23p5 Ar [Ne]3s23p6 = [Ar] K [Ar]4s1 Ca [Ar]4s2 Sc [Ar]3d14s2 Ti [Ar]3d24s2 V [Ar]3d34s2 Cr [Ar]3d54s1 Mn [Ar]3d54s2 Fe [Ar]3d64s2 Co [Ar]3d74s2 Ni [Ar]3d84s2 Cu [Ar]3d104s1 Zn [Ar]3d104s2 Ga [Ar]3d104s24p1 Ge [Ar]3d104s24p2 As [Ar]3d104s24p3 Se [Ar]3d104s24p4 Br [Ar]3d104s24p5 Kr [Ar]3d104s24p6 = [Kr]

If we consider the carbon atom, we see that there are two electrons in the 2p subshell. This subshell has three orbitals, and so the two electrons may occupy different orbitals, or be paired in one on the orbitals. The actual occupancy is each electron in a different orbital, as dictated by Hund’s rule:

When more than one orbital has the same energy, electrons occupy separate orbitals and do so with parallel spins (ie. both α, or spin up)

The electrons occupy different orbitals because the electrostatic repulsion between electrons in different orbitals is lower than that between repulsion between electrons in the same orbital, and so occupying different orbitals is a lower energy, and more stable, configuration. The reduced repulsion when the electrons are in different orbitals results from the fact that the orbitals occupy different regions of space.