In quantum physics and quantum chemistry, an avoided crossing (sometimes called intended crossing, non-crossing or anticrossing) is the phenomenon where two eigenvalues of an Hermitian matrix representing a quantum observable and depending on N continuous real parameters cannot become equal in value ("cross") except on a manifold of N-2 dimensions. 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).
Contents
- Emergence of avoided crossing
- Avoided crossing and quantum resonance
- The general avoided crossing theorem
- Avoided crossing in polyatomic molecules
- References
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.
Emergence of avoided crossing
Study of a two-level system is of vital importance in quantum mechanics because it embodies simplification of many of physically realizable systems. The effect of perturbation on a two-state system Hamiltonian is manifested through avoided crossings in the plot of individual energy vs energy difference curve of the eigenstates. The two-state Hamiltonian can be written as
The eigenvalues of which are
However, when subjected to an external perturbation, the matrix elements of the Hamiltonian change. For the sake of simplicity we consider a perturbation with only off diagonal elements. Since the overall Hamiltonian must be Hermitian we may simply write the new Hamiltonian
Where P is the perturbation with zero diagonal terms. The fact that P is Hermitian fixes its off-diagonal components. The modified eigenstates can be found by diagonalising the modified Hamiltonian. It turns out that the new eigenvalues are,
If a graph is plotted varying
Avoided crossing and quantum resonance
The immediate impact of avoided level crossing in a degenerate two state system is the emergence of a lowered energy eigenstate. The effective lowering of energy always correspond to increasing stability. Bond resonance in organic molecules exemplifies the occurrence of such avoided crossings. To describe these cases we may note that the non-diagonal elements in an erstwhile diagonalised Hamiltonian not only modify the energy eigenvalues but also mix the old eigenstates into the new ones. These effects are more prominent if the original Hamiltonian had degeneracy. This mixing of eigenstates to attain more stability is precisely the phenomena of chemical bond resonance.
Our earlier treatment started by denoting the eigenvectors
where
The new eigenstates
It is evident that both of the new eigenstates are mixture of the original degenerate eigenstates and one of the eigenvalues (here
However it turns out that the two-state Hamiltonian
The general avoided crossing theorem
The above illustration of avoided crossing however is a very specific case. From a generalised view point the phenomenon of avoided crossing is actually controlled by the parameters behind the perturbation. For the most general perturbation
Here the elements of the state vectors were chosen to be real so that all the matrix elements become real. Now the eigenvalues of the system for this subspace is given by
The terms under the square root are squared real numbers. So for these two levels to cross we must simultaneously require
Now if the perturbation
If we choose the values of
And generally there will be two such values of them for which the equations will simultaneously satisfy. So with
Since their intersection is parametrized by
Avoided crossing in polyatomic molecules
In polyatomic molecules, there are various parameters which determine the Hamiltonian of the system. The mutual distances between the atoms are one set of parameters. If both of the atoms of a diatomic molecule is same, the symmetry suggests that different configurations keeping their mutual distance fixed will result into same electronic states. So it is the relative distance