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.[3] The two-state Hamiltonian can be written as

${\displaystyle H={\begin{pmatrix}E_{1}&0\\0&E_{2}\end{pmatrix}}\,\!}$

The eigenvalues of which are ${\displaystyle \textstyle E_{1}}$ and ${\displaystyle \textstyle E_{2}}$ and the eigenvectors, ${\displaystyle \textstyle {\begin{pmatrix}1\\0\end{pmatrix}}}$ and ${\displaystyle \textstyle {\begin{pmatrix}0\\1\end{pmatrix}}}$. These two eigenvectors designate the two states of the system. If the system is prepared in either of the states it would remain in that state. If ${\displaystyle \textstyle E_{1}}$ happens to be equal to ${\displaystyle E_{2}}$ 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

${\displaystyle H'=H+P={\begin{pmatrix}E_{1}&0\\0&E_{2}\end{pmatrix}}+{\begin{pmatrix}0&W\\W^{*}&0\end{pmatrix}}={\begin{pmatrix}E_{1}&W\\W^{*}&E_{2}\end{pmatrix}}\,\!}$

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 are,

${\displaystyle E_{+}={\frac {1}{2}}(E_{1}+E_{2})+{\frac {1}{2}}{\sqrt {(E_{1}-E_{2})^{2}+4|W|^{2}}}}$

${\displaystyle E_{-}={\frac {1}{2}}(E_{1}+E_{2})-{\frac {1}{2}}{\sqrt {(E_{1}-E_{2})^{2}+4|W|^{2}}}}$

If a graph is plotted varying ${\displaystyle \textstyle (E_{1}-E_{2})}$ along the horizontal axis and ${\displaystyle \textstyle E_{+}}$ or ${\displaystyle \textstyle E_{-}}$ along the vertical, we find two branches of a hyperbola (as shown in the figure). The curve asymptotically approaches the original unperturbed energy levels. Analyzing the curves it becomes evident that even if the original states were degenerate (i.e. ${\displaystyle \textstyle E_{1}=E_{2}}$ ) the new energy states are no longer equal. However, if ${\displaystyle \textstyle W}$ is set to zero we may find at ${\displaystyle \textstyle (E_{1}-E_{2})=0}$, ${\displaystyle \textstyle E_{+}=E_{-}}$ and the levels cross. Thus with the effect of the perturbation these level crossings are avoided.

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 ${\displaystyle \textstyle {\begin{pmatrix}1\\0\end{pmatrix}}}$ and ${\displaystyle \textstyle {\begin{pmatrix}0\\1\end{pmatrix}}}$ as the matrix representation of eigenstates ${\displaystyle \textstyle |\psi _{1}\rangle }$ and ${\displaystyle \textstyle |\psi _{2}\rangle }$ of a two-state system. Using bra–ket notation the matrix elements of ${\displaystyle H'}$ are actually the terms

${\displaystyle H'_{ij}=\langle \psi _{i}|H'|\psi _{j}\rangle }$ with ${\displaystyle i,j\in \left\{{1,2}\right\}}$

where ${\displaystyle H'_{11}=H'_{22}=E}$ due to the degeneracy of the unperturbed Hamiltonian and the off-diagonal perturbations are ${\displaystyle H'_{12}=W}$ and ${\displaystyle H'_{21}=W^{*}}$.

The new eigenstates ${\displaystyle \textstyle |\psi _{+}\rangle }$ and ${\displaystyle \textstyle |\psi _{-}\rangle }$ can be found by solving the eigenvalue equations ${\displaystyle H'|\psi _{+}\rangle =E_{+}|\psi _{+}\rangle }$ and ${\displaystyle H'|\psi _{-}\rangle =E_{-}|\psi _{-}\rangle }$. From simple calculations it can be shown that

${\displaystyle |\psi _{+}\rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}e^{i\phi }\\1\end{pmatrix}}={\frac {1}{\sqrt {2}}}(e^{i\phi }|\psi _{1}\rangle +|\psi _{2}\rangle )}$ and

${\displaystyle |\psi _{-}\rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}-e^{i\phi }\\1\end{pmatrix}}={\frac {1}{\sqrt {2}}}(-e^{i\phi }|\psi _{1}\rangle +|\psi _{2}\rangle )}$ where ${\displaystyle e^{i\phi }=W/|W|}$

It is evident that both of the new eigenstates are superposition of the original degenerate eigenstates and one of the eigenvalues (here ${\displaystyle E_{-}}$) 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 the experimental evidences of probable bond structures give rise to two different eigenstates, ${\displaystyle \textstyle |\psi _{1}\rangle }$ and ${\displaystyle \textstyle |\psi _{2}\rangle }$. The symmetry of these two structures mandates that ${\displaystyle \langle \psi _{1}|H|\psi _{1}\rangle =\langle \psi _{2}|H|\psi _{2}\rangle =E}$.

However it turns out that the two-state Hamiltonian ${\displaystyle H}$ of benzene is not diagonal. The off-diagonal elements result into lowering of energy and the benzene molecule stabilizes in a structure which is a superposition of these symmetric ones with energy ${\displaystyle E_{-}<E}$.[5] For any general two-state system avoided level crossing repels the eigenstates ${\displaystyle |\psi _{+}\rangle }$ and ${\displaystyle |\psi _{-}\rangle }$ such that it requires more energy for the system to achieve the higher energy configuration.

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.