Overview of the Electron Spin

In this article, the concept of the spin is discussed, and in particular the electron spin due to its importance to the understanding of the theory of chemical bond and spectroscopic information of atoms and molecules.  Spin is a fundamental property of all particles on the subatomic level.  However, this article concentrates on the electron spin.  The spin is a relativistic quantum mechanical phenomenon that does not have a classical analogue.  This is in contrast to many other physical concepts which have classical as well as quantum mechanical analogues such as the energy and momentum which are associated with a particle. 

The spin of the electron is obtained mathematically by solving the relativistic quantum mechanical equation known as the Dirac equation. the relativistic quantum mechanical equation that fits the spin of non-integer values such as electrons which are fermions is the Dirac equation.  On the other hand, the quantum mechanical equation for integer values of the spin is called the Klein-Gordon equation. 

The spin of the electron is obtained mathematically by solving the Dirac equation for an electron.  The non-relativistic quantum mechanical equation known as the Schroedinger equation does not give any information that is related to the spin of the electron.  There are, in quantum mechanics, linear operators that represent the spin of the electron.  This is in an analogy to the angular momentum operators which are physical properties that exist in classical as well as quantum mechanical systems.

There are correspondingly operators that depict the spin angular momentum.   The operator that depicts the square of the total angular momentum is designated L*2.  The analogous operator for the spin is the square of total spin angular momentum and is designated S*2.  L*2 and S*2 commute with the Hamiltonian.  Therefore, they each have a common set of eigen functions common with the appropriate Hamiltonian.  The eigen values of L*2 and S*2 are also very similar in the mathematical formula with the difference in the variable that represent each operator.

The spin operator Sz in the direction of the z axis and the square of the total spin angular momentum operator can both be represented using 2*2 matrices.  These matrices are called Pauli matrices and are convenient representation of the spin operators in terms of matrices.  The spin of the electron has a non-integer spin value that can be either 1/2 or -1/2.  The electron is therefore a Fermion that obeys the Fermi-Dirac statistical mechanics.

This is in contrast to the integer value of the spin of bozons which can all fill the ground state energy level.  Electrons in particular follow the Pauli principle of exclusion.  This principle states that only two electrons with different spins can occupy the same orbital in an atom at a time. 

The transition between electronic levels in atoms and molecules must obey the rule of spin conservation, namely that an electron in a singlet state can be promoted only to another singlet state.  Also an electron in a triplet state can be promoted only to a triplet state.  The spin of the electron is important in spectroscopic studies such as in transition metals chemistry.  It is responsible for example for giving the magnetic properties of organometallic complexes.  This is in addition to its being partly responsible for the color of the complex.

The spin as well as the angular movement of the electron in the atom gives rise to magnetic moments that can interact with magnetic fields. This in turn gives rise to additional energies of the atoms and molecules, an example of which is the Zeeman effect.  The Zeeman effect is the result of the interaction of the magnetic moment of the electron due to its angular movement with the external magnetic field.  Also important is the phenomenon of spin-orbit coupling which involves the spin as well as the angular magnetic moment of the electron.