Avoided crossing

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An avoided energy level crossing in a two level system subjected to an external magnetic field. Note the energies of the diabatic states, $\scriptstyle{|1\rangle}$ and $\scriptstyle{|2\rangle}$ , and the eigenvalues of the Hamiltonian giving the energies of the eigenstates, $\scriptstyle{|\phi_1\rangle}$ and $\scriptstyle{|\phi_2\rangle}$ (the adiabatic states).

Avoided crossing (sometimes misleadingly called intended crossing^[1]). The eigenvalues of a Hermitian matrix depending on N continuous real parameters cannot cross except at a manifold of N-2 dimensions. In the case of a diatomic molecule (one parameter, which describes the bond length), this means that the eigenvalues do not cross. In the case of a triatomic molecule, this means that the eigenvalues can intersect at a point only (see conical intersection).

This is particularly important in quantum chemistry. In the Born-Oppenheimer approach, 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.