Rules for Molecular State Symbols

1. Orbital Angular Momentum. The quantum number  represents the projection of angular momentum L on the molecular axis. It is a sum of the  quantum numbers for each individual electron.   i i Note that this is a scalar addition rather than the vector addition required for atomic states.

The states have the following symbols: Symbol:  for   0  for   1  for   2  for   3 etc…

2 For example, the ground state for H 2 with a configuration of  g 1s must be  because there is no net angular momentum.

2. Spin Angular Momentum. Molecular states use the same classification scheme for spin angular momentum as the atomic states. Spin degeneracy is also determined by the same method as for atomic states:

spin degeneracy  g s  2S 1 and is written as an upper left superscript of the term symbol. In our previous example, the two 1 electrons in the ground state of H 2 have S = 0. Therefore, the state symbol is  .

3. Inversion Symmetry. The wave functions are products of orbitals, which must be either even (gerade) or odd (ungerade) with respect to inversion. The product of two gerade orbitals is also gerade. The product of two ungerade orbitals is gerade; this applies to any case with an even number of ungerade orbitals.

Gerade (g) or ungerade (u) is given as a right subscript of the term symbol.

1 For H 2 , the symbol is  g .

4. The Vertical Mirror Plane (ˆ v ). A vertical mirror plane is a plane of symmetry containing the bond axis.  states are labeled either   or   according to their symmetry with respect to a reflection in a vertical mirror plane.     ˆ v     and ˆ v     The  molecular orbitals are always symmetric for this operation: ˆ v   . Any molecular wave function that is a product of  molecular orbitals will be   . For example, the 1  complete symbol for ground state of H 2 is  g .  and  states are not classified by the vertical mirror plane. These types always occur as degenerate pairs, which can be made into linear combinations that are either symmetrical or unsymmetrical with respect to ˆ v .

Note: Molecular orbitals have symbols corresponding to the quantum number  as follows: Symbol:  for   0 Symbol:  for   1 Symbol:  for   2 etc…

Further discussions of these topics can be found in Mortimer pages 651 and 663.

Term symbols can be used to determine the allowed electronic transitions for diatomics. The selection rules for electronic transitions in diatomic molecules are as follows:   0,1 S  0 g  u    and    only;    not allowed