Avoided crossing. The eigenvalues of a Hermitian matrix depending on N continuous real parameters cannot cross except at a manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the 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

Diatomic

Diatomic molecules are molecules composed only of two atoms, of either the same or different chemical elements. The prefix di- means two in Greek. Common diatomic molecules are hydrogen , nitrogen , oxygen , and carbon monoxide . Seven elements exist in the diatomic state in the liquid and solid...

 molecule, this means that the eigenvalues can intersect at a point only (see conical intersection

Conical intersection

In quantum chemistry, a conical intersection of two potential energy surfaces of the same spatial and spin symmetries is the set of molecular geometry points where the two potential energy surfaces are degenerate . Conical intersections are ubiquitous in both trivial and non-trivial chemical...

).

This is particularly important in quantum chemistry

Quantum chemistry

Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...

. In the Born-Oppenheimer approach, the electronic molecular Hamiltonian is diagonalized

Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...

 on a set of distinct molecular geometries (the obtained eigenvalues are the values of the adiabatic potential energy surface

Potential energy surface

A potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems...

s). The geometries for which the potential energy surfaces are avoiding to cross are the locus where the Born-Oppenheimer approximation

Born-Oppenheimer approximation

In quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer approximation, named after Max Born and J. Robert Oppenheimer. For instance the benzene molecule consists of 12 nuclei and 42...

 fails.

See also

• Avoided curve crossing section of the adiabatic theorem
  Adiabatic theorem
  The adiabatic theorem is an important concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock , can be stated as follows:...
  
   article
• Level repulsion
  Level repulsion
  A system of two coupled oscillators has two natural frequencies. As the coupling strength between the oscillators increases, the lower frequency decreases and the higher increases. This effect can be viewed as a 'repulsion' between the frequencies...