Avoided crossing

In quantum physics and quantum chemistry, an avoided crossing (AC, sometimes called intended crossing,^[1] non-crossing or anticrossing) is the phenomenon where two eigenvalues of a Hermitian matrix representing a quantum observable and depending on ${\displaystyle k}$ continuous real parameters cannot become equal in value ("cross") except on a manifold of dimension ${\displaystyle k-2}$.^[2] The phenomenon is also known as the von Neumann–Wigner theorem. In the case of a diatomic molecule (with one parameter, namely the bond length), this means that the eigenvalues cannot cross at all. In the case of a triatomic molecule, this means that the eigenvalues can coincide only at a single point (see conical intersection).

This is particularly important in quantum chemistry. In the Born–Oppenheimer approximation, the electronic molecular Hamiltonian is diagonalized on a set of distinct molecular geometries (the obtained eigenvalues are the values of the adiabatic potential energy surfaces). The geometries for which the potential energy surfaces are avoiding to cross are the locus where the Born–Oppenheimer approximation fails.

Avoided crossing also occur in the resonance frequencies of undamped mechanical systems, where the stiffness and mass matrices are real symmetric. There the resonance frequencies are the square root of the generalized eigenvalues.