Avoided crossing on arvau

Wikipedia / Avoided crossing

In quantum physics and quantum chemistry, an avoided crossing (AC, sometimes called intended crossing, non-crossing or anticrossing) is the phenomenon where two eigenvalues of a Hermitian matrix representing a quantum observable continuous real parameters cannot become equal in value ("cross") except on a manifold. 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 crossings of the potential energy surfaces are avoided are the locus where the Born–Oppenheimer approximation fails.

Avoided crossing also occurs 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.

In two-state systems

Emergence

Study of a two-level system is of vital importance in quantum mechanics because it embodies simplification of many physically realizable systems. The effect of perturbation on a two-state system Hamiltonian is manifested through avoided crossings in the plot of individual energy versus energy difference curve of the eigenstates. The two-state Hamiltonian can be written as

and the eigenvectors there will be a twofold degeneracy in the Hamiltonian. In that case any superposition of the degenerate eigenstates is evidently another eigenstate of the Hamiltonian. Hence the system prepared in any state will remain in that forever.

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

are

and the levels cross. Thus with the effect of the perturbation these level crossings are avoided.

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 corresponds to increasing stability (see: Energy minimization). 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 a previously diagonalised Hamiltonian not only modify the energy eigenvalues but also superpose the old eigenstates into the new ones. These effects are more prominent if the original Hamiltonian had degeneracy. This superposition of eigenstates to attain more stability is precisely the phenomena of chemical bond resonance.

of a two-state system. Using bra–ket are actually the terms

. From simple calculations it can be shown that

) is less than the original unperturbed eigenenergy. So the corresponding stable system will naturally mix up the former unperturbed eigenstates to minimize its energy. In the example of benzene.

. such that it requires more energy for the system to achieve the higher energy configuration.

Resonances in avoided crossing

In molecules, the nonadiabatic couplings between two adiabatic potentials build the AC region. Because they are not in the bound state region of the adiabatic potentials, the rovibronic resonances in the AC region of two-coupled potentials are very special and usually do not play important roles on the scatterings. Exemplified in particle scattering, resonances in the AC region are comprehensively investigated. The effects of resonances in the AC region on the scattering cross sections strongly depend on the nonadiabatic couplings of the system, it can be very significant as sharp peaks, or inconspicuous buried in the background. More importantly, it shows a simple quantity proposed by Zhu and Nakamura to classify the coupling strength of nonadiabatic interactions, can be well applied to quantitatively estimate the importance of resonances in the AC region.

General avoided crossing theorem

affecting a two-dimensional subspace, we may write the effective Hamiltonian matrix in that subspace as

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 are given by

The terms under the square root are squared real numbers. So for these two levels to cross we simultaneously require

we may in general vary these numbers to satisfy these two equations.

then both of the equations above have one single free parameter such that both of the equations are satisfied. However, if we allow another parameter to be free, both of these two equations will now be controlled by the same two parameters

freely varying coordinates (while the remaining two coordinates are fixed from the condition equations). Geometrically the eigenvalue equations describe a surface dimensional space.

Since their intersection is parametrized continuous real parameters controlling the perturbed Hamiltonian, the levels (or surfaces) can only cross at a manifold..

In polyatomic molecules

In an N-atomic polyatomic molecule there are 3N-6 vibrational coordinates (3N-5 for a linear molecule) that enter into the electronic Hamiltonian as parameters. For a diatomic molecule there is only one such coordinate, the bond length r. Thus, due to the avoided crossing theorem, in a diatomic molecule we cannot have level crossings between electronic states of the same symmetry. However, for a polyatomic molecule there is more than one geometry parameter in the electronic Hamiltonian and level crossings between electronic states of the same symmetry are not avoided.