In quantum physics and quantum chemistry, an avoided crossing (sometimes called intended crossing,[1] non-crossing or anticrossing) is the phenomenon where two eigenvalues of a 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-3 dimensions.[2] 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 the potential energy surfaces are avoiding to cross are the locus where the Born–Oppenheimer approximation fails.
Avoided crossing also occur 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.
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.[3] The two-state Hamiltonian can be written as
H=\begin{pmatrix}E1&0\\0&E2\end{pmatrix}
The eigenvalues of which are
styleE1
styleE2
style\begin{pmatrix}1\\0\end{pmatrix}
style\begin{pmatrix}0\\1\end{pmatrix}
styleE1
E2
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
H'=H+P=\begin{pmatrix}E1&0\\0&E2\end{pmatrix}+\begin{pmatrix}0&W\\W*&0\end{pmatrix}=\begin{pmatrix}E1&W\\W*&E2\end{pmatrix}
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
styleE+
styleE-
E\pm=
| E1+E2 | |
| 2 |
\pm\sqrt{(
| E1-E2 | |
| 2 |
)2+|W|2
If a graph is plotted varying
style(E1-E2)
styleE+
styleE-
styleE1=E2
styleW
style(E1-E2)=0
styleE+=E-
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.(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 an erstwhile diagonalised Hamiltonian not only modify the energy eigenvalues but also superpose the old eigenstates into the new ones.[4] 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.
Our earlier treatment started by denoting the eigenvectors
style\begin{pmatrix}1\\0\end{pmatrix}
style\begin{pmatrix}0\\1\end{pmatrix}
style|\psi1\rangle
style|\psi2\rangle
H'
H'ij=\langle\psii|H'|\psij\rangle
i,j\in\left\{{1,2}\right\}
H'11=H'22=E
H'12=W
H'21=W*
The new eigenstates
style|\psi+\rangle
style|\psi-\rangle
H'|\psi+\rangle=E+|\psi+\rangle
H'|\psi-\rangle=E-|\psi-\rangle
|\psi+\rangle=
| 1 | |
| \sqrt{2 |
|\psi-\rangle=
| 1 | |
| \sqrt{2 |
ei\phi=W/|W|
It is evident that both of the new eigenstates are superposition of the original degenerate eigenstates and one of the eigenvalues (here
E-
style|\psi1\rangle
style|\psi2\rangle
\langle\psi1|H|\psi1\rangle=\langle\psi2|H|\psi2\rangle=E
However it turns out that the two-state Hamiltonian
H
E-<E
|\psi+\rangle
|\psi-\rangle
In molecules, the nonadiabatic couplings between two adiabatic potentials build the avoided crossing (AC) region. The rovibronic resonances in the AC region of two-coupled potentials are very special, since they are not in the bound state region of the adiabatic potentials, and they usually do not play important roles on the scatterings and are less discussed. Yu Kun Yang et al studied this problem in the New J. Phys. 22 (2020).[6] 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.
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
styleP=\begin{pmatrix}W1&W\\W&W2\end{pmatrix}
H
\begin{pmatrix}E1&0\\0&E2\end{pmatrix}+\begin{pmatrix}W1&W\\W&W2\end{pmatrix}=\begin{pmatrix}E1+W1&W\\W&E2+W2.\end{pmatrix}.
E\pm=
| 1 | |
| 2 |
(E1+E2+W1+W2)\pm
| 1 | |
| 2 |
\sqrt{(E1-E2+W1-W2)2+4W2
(E1-E2+W1-W2)=0
W=0.
P
k
{\alpha1,\alpha2,\alpha3.....\alphak}
(E1-E2+W1-W2)=F1(\alpha1,\alpha2,\alpha3.....\alphak)=0
W=F2(\alpha1,\alpha2,\alpha3.....\alphak)=0.
\alpha1
\alphak-1
\alphak
F1(\alphak-1,\alphak
| )| | |
| \alpha1,\alpha2,...,\alphak-2fixed |
=0
F2(\alphak-1,\alphak
| )| | |
| \alpha1,\alpha2,...,\alphak-2fixed |
=0.
k
k-2
\alphak
E+
E-
k-2
k
E\pm=E\pm(\alpha1,\alpha2,\alpha3.....\alphak).
k-2
k
k-2
\langle\psi1|P|\psi2\rangle ≠ \langle\psi2|P|\psi1\rangle
W=0
k-1
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 diatomicmolecule there is only one such coordinate, the bond lengthr. Thus, due to the avoided crossing theorem, in a diatomicmolecule we cannot have level crossings between electronic states of the same symmetry.[10] However, for a polyatomic molecule there is more than one geometry parameter in the electronic Hamiltonian and level crossings between electronicstates of the same symmetry are not avoided.[11]