Q-branch is absent in the spectra of heteronuclear diatomic molecules. An explicit implication of symmetry on the molecular structure can be shown in Molecular symmetry in physics and chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. 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 spectroscopi Th… Homonuclear diatomic molecules show neither pure vibrational nor pure rotational spectra. 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. For a molecule in a Σ {\displaystyle \Sigma } state, the transitions between two vibration-rotation (or rovibrational) levels ( v , ℑ ) {\displaystyle (v,\Im )} and ( v ′ , ℑ ′ ) {\displaystyle (v',\Im ')} , with vibrational quantum numbers v {\displaystyle v} and v ′ = v + 1 {\displaystyle v'=v+1} , fall into two sets according to whether Δ ℑ = + 1 {\displaystyle \Delta \Im =+1} or Δ ℑ = − 1 {\displaystyle \Delta \Im =-1} . The set corresponding to Δ ℑ = + 1 {\d If the electronic states | Φ 1 ( 0 ) ⟩ {\displaystyle \left|\Phi _{1}^{(0)}\right\rangle } and | Φ 2 ( 0 ) ⟩ {\displaystyle \left|\Phi _{2}^{(0)}\right\rangle } have the same point group symmetry, then H 12 ′ {\displaystyle H_{12}^{'}} can be, and will in general be, non-zero. Except for accidental crossing which would occur if, by coincidence, the two equations were satisfied at the same value of R {\displaystyle R} , it is in general impossible to find a single value of Δ R {\displaystyle \Delta R} (i.e., a single value of R {\displaystyle R} ) for which the two conditions are satisfied simultaneously. Thus, in a diatomic molecule, only terms of different symmetry can intersect, while the intersection of terms of like symmetry is forbidden. This is, in general, true for any case in quantum mechanics where the Hamiltonian contains some parameter and its eigenvalues are consequently functions of that parameter. This general rule is known as von Neumann - Wigner non-crossing rule. This general symmetry principle has important consequences is molecular spectra. The von Neumann-Wigner non-crossing rule must be obeyed, so that energy curves corresponding to orbitals having the same symmetry do not cross as R {\displaystyle R} varies from 0 {\displaystyle 0} to ∞ {\displaystyle \infty } . Thus, von Neumann-Wigner non-crossing rule also acts as a starting point for valence bond theory. == Observable consequences == Symmetry in diatomic molecules manifests itself directly by influencing the molecular spectra of the molecule. The effect of symmetry on different types of spectra in diatomic molecules are: === Rotational spectrum === 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 D {\displaystyle D} along the molecular axis. This is the permanent electric dipole moment. In homonuclear diatomic molecules, the permanent electric dipole moment vanishes and there is no pure rotation spectrum (but see N.B. below). {\displaystyle \hbar {{\omega }^{R}}=E(v+1,\Im +1)-E(v,\Im )=2B(\Im +1)+\hbar {{\omega }_{0}};{\text{ }}\Im =0,1,2,......} The set corresponding to Δ ℑ = − 1 {\displaystyle \Delta \Im =-1} is called the P branch. The corresponding frequencies are given by: ℏ ω P = E ( v + 1 , ℑ − 1 ) − E ( v , ℑ ) = − 2 B ℑ + ℏ ω 0 ; ℑ = 1 , 2 , 3 , . . . . . . {\displaystyle \hbar {{\omega }^{P}}=E(v+1,\Im -1)-E(v,\Im )=-2B\Im +\hbar {{\omega }_{0}};{\text{ }}\Im =1,2,3,......} Both branches make up what is called a rotational-vibrational band or a rovibrational band. These bands are in the infra-red part of the spectrum. The frequencies ω Q {\displaystyle {{\omega }^{Q}}} corresponding to the lines in this branch are given by a quadratic function of ℑ {\displaystyle \Im } if B v {\displaystyle {{B}_{v}}} and B v + 1 {\displaystyle {{B}_{v+1}}} are unequal, and reduce to the single frequency: ℏ ω Q = E ( v + 1 , ℑ ) − E ( v , ℑ ) = ℏ ω 0 {\displaystyle \hbar {{\omega }^{Q}}=E(v+1,\Im )-E(v,\Im )=\hbar {{\omega }_{0}}} if B v + 1 = B v {\displaystyle {{B}_{v+1}}={{B}_{v}}} . For a heteronuclear diatomic molecule, this selection rule has two consequences: Both the vibrational and rotational The energy change of rotation can be either subtracted from or added to the energy change of vibration, giving the P- and R- branches of the spectrum, respectively. Homonuclear diatomic molecules also show this kind of spectra. The selection rules, however, are a bit different. Conclusion: Both homo- and hetero-nuclear diatomic molecules show rovibrational spectra. A Q-branch is absent in the spectra of heteronuclear diatomic molecules. == A special example: Hydrogen molecule ion == 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, H 2 + {\displaystyle {\text{H}}_{2}^{+}} .