Molecular term symbol
In molecular physics, the molecular term symbol is a shorthand expression of the group representation and angular momenta that characterize the state of a molecule, i.e. its electronic quantum state which is an eigenstate of the electronic molecular Hamiltonian. It is the equivalent of the term symbol for the atomic case. However, the following presentation is restricted to the case of homonuclear diatomic molecules, or other symmetric molecules with an inversion centre. For heteronuclear diatomic molecules, the u/g symbol does not correspond to any exact symmetry of the electronic molecular Hamiltonian. In the case of less symmetric molecules the molecular term symbol contains the symbol of the group representation to which the molecular electronic state belongs.
It has the general form:
where
- is the total spin quantum number
- (Lambda) is the projection of the orbital angular momentum along the internuclear axis
- (Omega) is the projection of the total angular momentum along the internuclear axis
- indicates the symmetry or parity with respect to inversion () through a centre of symmetry
- is the reflection symmetry along an arbitrary plane containing the internuclear axis
Λ quantum number[edit]
For atoms, we use S, L, J and MJ to characterize a given state. In linear molecules, however, the lack of spherical symmetry destroys the relationship , so L ceases to be a good quantum number. A new set of operators have to be used instead: , where the z-axis is defined along the internuclear axis of the molecule. Since these operators commute with each other and with the Hamiltonian on the limit of negligible spin-orbit coupling, their eigenvalues may be used to describe a molecule state through the quantum numbers S, MS, ML and MJ.
The cylindrical symmetry of a linear molecule ensures that positive and negative values of a given for an electron in a molecular orbital will be degenerate in the absence of spin-orbit coupling. Different molecular orbitals are classified with a new quantum number,
Following the spectroscopic notation pattern, molecular orbitals are designated by a lower case Greek letter: for
Now, the total z-projection of L can be defined as
As states with positive and negative values of ML are degenerate, we define
Λ = |ML|,
and a capital Greek letter is used to refer to each value:
- 2S+1
Λ
and the number of electron degenerate states (under the absence of spin-orbit coupling) corresponding to this term symbol is given by:
- (2S+1)×2 if
Λ is not 0 - (2S+1) if
Λ is 0.
Ω and spin–orbit coupling[edit]
Spin–orbit coupling lifts the degeneracy of the electronic states. This is because the z-component of spin interacts with the z-component of the orbital angular momentum, generating a total electronic angular momentum along the molecule axis Jz. This is characterized by the MJ quantum number, where
- MJ = MS + ML.
Again, positive and negative values of MJ are degenerate, so the pairs (ML, MS) and (−ML, −MS) are degenerate: {(1, 1/2), (−1, −1/2)}, and {(1, −1/2), (−1, 1/2)} represent two different degenerate states. These pairs are grouped together with the quantum number
Ω =Λ + MS
is used (often
With this, a level is given by
Note that
- E = A ML MS
where A is the spin–orbit constant. For 4
When
Reflection through a plane containing the internuclear axis[edit]
There are an infinite number of planes containing the internuclear axis and hence there are an infinite number of possible reflections. For any of these planes, molecular terms with
For the
Reflection through an inversion center: u and g symmetry[edit]
Taking the molecular center of mass as origin of coordinates, consider the change of all electrons' position from (xi, yi, zi) to (−xi, −yi, −zi). If the resulting wave function is unchanged, it is said to be gerade (German for even) or have even parity; if the wave function changes sign then it is said to be ungerade (odd) or have odd parity. For a molecule with a center of inversion, all orbitals will be symmetric or antisymmetric.[2] The resulting wavefunction for the whole multielectron system will be gerade if an even number of electrons are in ungerade orbitals, and ungerade if there are an odd number of electrons in ungerade orbitals, regardless of the number of electrons in gerade orbitals.
An alternative method for determining the symmetry of an MO is to rotate the orbital about the axis joining the two nuclei and then rotate the orbital about a line perpendicular to the axis. If the sign of the lobes remains the same, the orbital is gerade, and if the sign changes, the orbital is ungerade.[3]
Wigner-Witmer correlation rules[edit]
In 1928 Eugene Wigner and E.E. Witmer proposed rules to determine the possible term symbols for diatomic molecular states formed by the combination of a pair of atomic states with given atomic term symbols.[4][5][6] For example, two like atoms in identical 3S states can form a diatomic molecule in 1
Atomic Term Symbols | Molecular Term Symbols |
---|---|
Sg + Sg or Su + Su | |
Sg + Su | |
Sg + Pg or Su + Pu | |
Sg + Pu or Su + Pg | |
Sg + Dg or Su + Du | |
Sg + Du or Su + Dg | |
Sg + Fg or Su + Fu | |
Sg + Fu or Su + Fg | |
Pg + Pg or Pu + Pu | |
Pg + Pu | |
Pg + Dg or Pu + Du | |
Pg + Du or Pu + Dg | |
Pg + Fg or Pu + Fu | |
Pg + Fu or Pu + Fg | |
Dg + Dg or Du + Du | |
Dg + Du | |
Dg + Fg or Du + Fu | |
Dg + Fu or Du + Fg |
Alternative empirical notation[edit]
Electronic states are also often identified by an empirical single-letter label. The ground state is labelled X, excited states of the same multiplicity (i.e., having the same spin quantum number) are labelled in ascending order of energy with capital letters A, B, C...; excited states having different multiplicity than the ground state are labelled with lower-case letters a, b, c... In polyatomic molecules (but not in diatomic) it is customary to add a tilde (e.g. , ) to these empirical labels to prevent possible confusion with symmetry labels based on group representations.
See also[edit]
References[edit]
- ^ p. 337, Molecular Spectra and Molecular Structure, Vol I - Spectra of Diatomic Molecules, G. Herzberg, Reprint of Second Edition w/corrections, Malabar, Florida: Krieger Publishing Company, 1989. ISBN 0-89464-268-5
- ^ Atkins, Peter; de Paula, Julio (2006). Atkins' Physical Chemistry (8th ed.). W. H. Freeman. p. 372. ISBN 0-7167-8759-8.
Fig. 11.22 The parity of an orbital is even (g) if its wavefunction is unchanged under inversion through the centre of symmetry of the molecule, but odd (u) if the wavefunction changes sign.
- ^ Lee, John David (2008). Concise Inorganic Chemistry (5th ed.). Wiley and Sons. ISBN 978-0-632-05293-6.
- ^ "Wigner-Witmer rules". Oxford Reference. Retrieved 26 August 2019.
- ^ a b Herzberg, Gerhard (1950). Molecular Spectra and Molecular Structure, Volume I.Spectra of Diatomic Molecules (2nd ed.). van Nostrand Reinhold. pp. 315–322. Reprint 2nd ed. with corrections (1989): Krieger Publishing Company. ISBN 0-89464-268-5
- ^ Wigner, Eugene (1928). "Über die Struktur der zweiatomigen Molekelspektren nach der Quantenmechanik". Zeitschrift für Physik (in German). 51 (11–12): 859–886. Bibcode:1928ZPhy...51..859W. doi:10.1007/BF01400247. S2CID 122110014.