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avoided crossing
In quantum physics and quantum chemistry, an avoided crossing (sometimes called intended crossing,〔for a less mathematical explanation see 〕 ''non-crossing'' or anticrossing) is defined as the case when the eigenvalues of an Hermitian matrix representing an observable for a system and depending on ''N'' continuous real parameters cannot cross (that is, two or more eigenvalues cannot become equal in value) except at a manifold of ''N''-2 dimensions when the states are symmetric.〔Landau,Lifshitz(1981),Quantum Mechanics, p.305〕 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 only at a 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 in two-state systems==
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