Symmetry of diatomic molecules

Molecular symmetry is a fundamental concept in the application of quantum mechanics in physics and chemistry, for example, it can be used to predict or explain many of a molecule's properties, such as its dipole moment and its allowed spectroscopic transitions (based on selection rules), without doing the exact rigorous calculations (which, in some cases, may not even be possible).

Among all the molecular symmetries, diatomic molecules show some distinct features and are relatively easier to analyze.

It can predict degeneracies of eigenstates and gives insights about the matrix elements of the Hamiltonian without calculating them.

A group is a mathematical structure (usually denoted in the form (G,*)) consisting of a set G and a binary operation

Common to all symmetry operations is that the geometrical center-point of the molecule does not change its position; hence the name point group.

One can determine the elements of the point group for a particular molecule by considering the geometrical symmetry of its molecular model.

The symmetry classification of the rotational levels, the eigenstates of the full (rovibronic nuclear spin) Hamiltonian, requires the use of the appropriate permutation-inversion group as introduced by Longuet-Higgins.

In addition to axial reflection symmetry, homonuclear diatomic molecules are symmetric with respect to inversion or reflection through any axis in the plane passing through the point of symmetry and perpendicular to the inter-nuclear axis.

The internuclear axis chooses a specific direction in space and the potential is no longer spherically symmetric.

The electronic Hamiltonian for a diatomic molecule is also invariant under reflections in all planes containing the internuclear line.

So the Complete Set of Commuting Operators (CSCO) for a general heteronuclear diatomic molecule is

Molecular term symbol is a shorthand expression of the group representation and angular momenta that characterize the state of a molecule.

By analogy with the spectroscopic notation S, P, D, F, ... used for atoms, it is customary to associate code letters with the values of

) are thus doubly degenerate, each value of the energy corresponding to two states which differ by the direction of the projection of the orbital angular momentum along the molecular axis.

This twofold degeneracy is actually only approximate, and it is possible to show that the interaction between the electronic and rotational motions leads to a splitting of the terms with

The subscripts g or u are therefore added to the term symbol, so that for homonuclear diatomic molecules electronic states can have the symmetries

The complete Hamiltonian of a homonuclear diatomic molecule also commutes with the operation of permuting (or exchanging) the coordinates of the two (identical) nuclei and rotational levels gain the additional label s or a depending on whether the total wavefunction is unchanged (symmetric) or changed in sign (antisymmetric) by the permutation operation.

The nuclear hyperfine Hamiltonian can mix the rotational levels of g and u vibronic states (called ortho-para mixing) and give rise to ortho-para transitions[4][5] If S denotes the resultant of the individual electron spins,

are the eigenvalues of S and as in the case of atoms, each electronic term of the molecule is also characterised by the value of S. If spin-orbit coupling is neglected, there is a degeneracy of order

In fact, in the applications of valence bond method in case of diatomic molecules, three main correspondence between the atomic and the molecular orbitals are taken care of: Thus, von Neumann-Wigner non-crossing rule also acts as a starting point for valence bond theory.

The effect of symmetry on different types of spectra in diatomic molecules are: In the electric dipole approximation the transition amplitude for emission or absorption of radiation can be shown to be proportional to the vibronic matrix element of the component of the electric dipole operator

In homonuclear diatomic molecules, the permanent electric dipole moment vanishes and there is no pure rotation spectrum (but see N.B.

Heteronuclear diatomic molecules possess a permanent electric dipole moment and exhibit spectra corresponding to rotational transitions, without change in the vibronic state.

Symmetry considerations require that the electric dipole moment of a diatomic molecule is directed along the internuclear line, and this leads to the additional selection rule

So, the transition matrix is non-zero only if the molecular dipole moment varies with displacement, for otherwise the derivatives of

For small displacements, the electric dipole moment of a molecule can be expected to vary linearly with the extension of the bond.

This would be the case for a heteronuclear molecule in which the partial charges on the two atoms were independent of the internuclear distance.

Now, the matrix elements can be expressed in position basis in terms of the harmonic oscillator wavefunctions: Hermite polynomials.

However, as the absorption of a photon requires the molecule to take up one unit of angular momentum, vibrational transitions are accompanied by a change in rotational state, which is subject to the same selection rules as for the pure rotational spectrum.

An explicit implication of symmetry on the molecular structure can be shown in case of the simplest bi-nuclear system: a hydrogen molecule ion or a di-hydrogen cation,