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Avoided crossing

In quantum mechanics and quantum chemistry, an avoided crossing is a phenomenon in which two eigenvalues of the Hamiltonian—corresponding to energy levels of a quantum system—approach each other closely as a function of an external parameter, such as magnetic field strength or molecular geometry, but repel one another due to coupling between the states, preventing an actual degeneracy or crossing.^[1] This behavior arises from the non-crossing theorem, originally formulated by John von Neumann and Eugene Wigner, which states that eigenstates of the same symmetry in a Hermitian Hamiltonian cannot cross but instead mix, leading to a gap in the energy spectrum at the point of closest approach. The width of this energy gap is determined by the strength of the off-diagonal coupling term in the Hamiltonian, often resulting in wavefunction hybridization over a finite parameter range.^[2] Avoided crossings are ubiquitous in physical systems, manifesting in atomic and molecular spectra, solid-state band structures, and open quantum systems with resonances.^[2] In molecular quantum chemistry, they occur on potential energy surfaces where electronic states of identical symmetry interact, influencing photochemical reactions and nonadiabatic dynamics; for instance, rapid passage through an avoided crossing can lead to transitions between adiabatic states via the Landau-Zener mechanism.^[3] In condensed matter physics, they appear in phonon or electronic band dispersions, such as in nanowires or quantum billiards, where they signal state mixing and delocalization effects that impact transport properties.^[1] For non-Hermitian systems, like those with dissipation, avoided crossings may evolve into exceptional points where eigenvalues and eigenvectors coalesce, enabling unique sensing and control applications.^[2] The study of avoided crossings has evolved from theoretical predictions in the early 20th century to experimental observations in diverse platforms, including optical lattices, superconducting qubits, and acoustic analogs, highlighting their role in quantum chaos, coherence preservation, and optimization algorithms where exponentially small gaps pose computational challenges.^[4]

Fundamental Concepts

Definition and Physical Origin

In quantum mechanics, an avoided crossing is a phenomenon in which two or more energy levels that would otherwise intersect as a function of a system parameter instead approach each other closely but repel due to coupling, resulting in a finite energy gap and avoidance of degeneracy.^[5] This occurs specifically when the states belong to the same symmetry class, as dictated by the non-crossing rule for eigenvalues of Hermitian operators. The physical origin of avoided crossings lies in the interaction between nearly degenerate states through off-diagonal elements of the Hamiltonian, which introduce non-adiabatic couplings that mix the wavefunctions and split the levels.^[6] In perturbation theory, even weak couplings suffice to prevent exact degeneracy, contrasting the diabatic representation—where states are defined without considering nuclear motion and thus cross freely—with the adiabatic representation, where the full electronic Hamiltonian diagonalization enforces level repulsion. This repulsion arises because the Hamiltonian's hermiticity guarantees real eigenvalues that cannot cross for continuous parameter variations unless the coupling vanishes identically.^[5] A key illustrative example is the simple two-level system, where the Hamiltonian in the basis of the unperturbed states takes the form

\hat{H} = \begin{pmatrix} E_1 & W \\ W^* & E_2 \end{pmatrix},

with E_1 and E_2 as the diagonal (unperturbed) energies and W as the off-diagonal coupling.^[6] Diagonalization yields eigenvalues

E_\pm = \frac{E_1 + E_2}{2} \pm \sqrt{\left( \frac{E_1 - E_2}{2} \right)^2 + |W|^2},

revealing a minimum splitting of $2|W| at E_1 = E_2, where the levels avoid crossing and the eigenstates become equal superpositions.^[6] The concept was first rigorously described by John von Neumann and Eugene P. Wigner in 1929, who analyzed the conditions for discrete eigenvalues and degeneracy in quantum systems, establishing the foundational non-crossing theorem for Hermitian matrices.

Energy Level Behavior

In avoided crossings, the energy levels of two interacting quantum states that would otherwise intersect exhibit a characteristic repulsion, resulting in a minimum separation known as the energy gap. This gap occurs at the point where the uncoupled (diabatic) energies are equal and is directly proportional to the off-diagonal coupling strength V between the states, with the minimum value given by \Delta E = 2|V|.^[7] The magnitude of this gap quantifies the strength of the interaction; stronger couplings lead to wider separations, preventing actual level crossings in the adiabatic representation.^[8] The distinction between adiabatic and diabatic potential energy curves is central to understanding this behavior. Adiabatic curves, which incorporate the full Hamiltonian including couplings, follow the eigenvalues of the system and thus avoid crossing, displaying a smooth repulsion near the interaction region.^[7] In contrast, diabatic curves represent the uncoupled basis states—such as electronic configurations that ignore inter-state interactions—and cross directly at the degeneracy point, simplifying analysis of the underlying dynamics but neglecting the hybridization effects.^[7] This framework is particularly evident in molecular systems, where nuclear motion along a coordinate like internuclear distance R drives the approach of diabatic curves, leading to the observed adiabatic avoidance.^[8] At the avoidance point, the wavefunctions of the states undergo significant hybridization, where the adiabatic states become equal mixtures of the diabatic basis functions. Specifically, the coefficients reach a 50/50 character, expressed as \psi_1 = \frac{1}{\sqrt{2}} (\phi_1 - \phi_2) (up to a phase factor), reflecting maximal mixing due to the degeneracy.^[7] Away from this region, the wavefunctions gradually revert to predominantly one diabatic character each, with the mixing extent determined by the relative diabatic energy difference and coupling. This hybridization alters the electronic character across the crossing, such as transitioning from ionic to covalent configurations in diatomic molecules.^[8] Spectroscopically, avoided crossings manifest as irregular vibrational progressions in absorption or emission spectra, arising from the distorted potential surfaces that disrupt regular spacing between levels.^[8] Additionally, the state mixing enables intensity borrowing, where transitions forbidden in isolated states gain oscillator strength through coupling, often resulting in broadened lineshapes or Fano profiles due to interference between discrete and continuum pathways.^[9] These signatures are observable in systems like alkali halides or hydrogen-like ions perturbed by charge exchange, providing direct evidence of the underlying interactions.^[9]

Two-State Systems

Perturbation and Emergence

In two-state quantum systems, avoided crossings arise from the perturbative coupling between otherwise independent states, transforming potential energy level crossings into regions of repulsion. The dynamics are captured by the two-state Hamiltonian

H = \begin{pmatrix} E_1 & V \\ V & E_2 \end{pmatrix},

where E_1 and E_2 represent the unperturbed diagonal energies that depend on a system parameter (such as internuclear distance in molecules), and V denotes the off-diagonal coupling strength. In the absence of coupling (V = 0), the energy levels E_1 and E_2 may cross when they become equal at a specific parameter value, allowing degeneracy. However, even a small non-zero V prevents exact degeneracy, as the eigenvalues become

\frac{E_1 + E_2}{2} \pm \sqrt{\left( \frac{E_1 - E_2}{2} \right)^2 + V^2},

resulting in a minimum energy gap of $2|V| at the point where E_1 = E_2. This splitting ensures the levels avoid crossing and instead curve away smoothly from each other. The emergence of this avoidance can be analyzed using degenerate perturbation theory, applicable when V is small compared to the energy separation away from degeneracy but non-negligible near it. At the degeneracy point (E_1 = E_2), the first-order correction to the degenerate eigenvalues yields a splitting of exactly $2|V|, lifting the degeneracy and initiating level repulsion. As V gradually increases from zero, the unperturbed crossing evolves into a pronounced avoided crossing, with the gap broadening linearly with V in the perturbative regime; for larger V, the exact diagonalization formula shows the avoidance persisting but with a more gradual curvature. This perturbative picture highlights how weak interactions suffice to enforce non-crossing behavior, a consequence of the general level repulsion in quantum mechanics first elucidated by von Neumann and Wigner.^[10] For avoided crossings to emerge, the off-diagonal elements V must be non-zero, typically originating from symmetry-allowed interactions such as spin-orbit coupling or vibronic coupling. Spin-orbit coupling, arising from relativistic effects, introduces mixing between states of different spin multiplicities (e.g., singlet and triplet), creating V proportional to the spin-orbit matrix element and leading to avoided crossings in atomic and molecular spectra.^[11] Vibronic coupling, which intertwines electronic and nuclear vibrational motions, generates off-diagonal terms in the adiabatic potential energy surfaces of molecules, particularly in regions of near-degeneracy, thereby inducing avoidance between electronic states. Graphically, plotting the eigenvalues against a perturbation parameter—such as the detuning \delta = E_1 - E_2—illustrates the transition: for V = 0, straight lines intersect at \delta = 0; for V > 0, the curves bend apart, forming a characteristic "V-shaped" avoidance with the vertex gap of $2|V|. This visualization underscores the smooth, continuous nature of the energy levels, consistent with the hermiticity of the Hamiltonian and the repulsion observed in quantum energy spectra.

Quantum Resonance Effects

In two-state quantum systems, quantum resonance arises when the energy levels approach near-degeneracy at an avoided crossing, enabling coherent oscillations of the population between the states, analogous to Rabi oscillations observed in driven two-level systems under resonant conditions.^[12] This oscillatory behavior emerges from the coupling term in the Hamiltonian, which mixes the states and induces periodic transfers when the detuning is minimal, leading to full population inversion at specific times determined by the gap size.^[12] The Landau-Zener model provides a foundational description of the dynamic evolution through an avoided crossing in a time-dependent sweep, quantifying the probability of non-adiabatic transitions between adiabatic states. In this model, the diabatic energies approach linearly with time, and the off-diagonal coupling V lifts the degeneracy, resulting in the transition probability P = \exp\left(-\frac{2\pi V^2}{\hbar |\alpha|}\right), where \alpha represents the rate of relative energy change near the crossing point.^[13] This formula highlights how slow sweeps (|\alpha| small) favor adiabatic following with P \approx 0, while rapid sweeps increase non-adiabatic jumping with P \to 1.^[13] When multiple closely spaced levels interact near an avoided crossing, quantum resonances produce complex splitting patterns in the energy spectrum, often manifesting as asymmetric line shapes resembling Fano profiles due to interference between discrete and continuum-like states.^[14] These profiles arise from the phase-sensitive coupling across the gap, altering absorption or emission spectra with characteristic dips and peaks that reflect the underlying resonance structure.^[14] Time-dependent dynamics across the avoided crossing enable coherent population transfer, where the system evolves unitarily to shift probability from one state to the other while accumulating a geometric phase dependent on the path encircling the gap in parameter space.^[15] This transfer is most efficient in adiabatic passages that traverse the crossing slowly enough to avoid diabatic jumps, preserving coherence and enabling applications in quantum control.^[15]

Theoretical Foundations

General Avoided Crossing Theorem

The general avoided crossing theorem, also known as the von Neumann–Wigner non-crossing rule and first formulated by John von Neumann and Eugene Wigner in 1929,^[16] asserts that for a Hermitian matrix depending analytically on a single real parameter, two eigenvalues cannot cross unless the off-diagonal coupling between the corresponding eigenstates vanishes at the would-be crossing point.^[17] This vanishing coupling typically occurs when the states belong to different symmetry sectors, leading to a degeneracy where the Hamiltonian is block-diagonal, with a multiple eigenvalue and a higher-dimensional eigenspace. In generic systems without such symmetries, achieving zero coupling demands fine-tuning of the parameter, rendering exact crossings improbable; instead, the eigenvalues repel each other, resulting in an avoided crossing characterized by hyperbolic veering of the eigenvalue curves.^[17] The proof outline, originating from the work of von Neumann and Wigner, employs perturbation theory to demonstrate this behavior: near a near-degeneracy, non-zero coupling leads to a first-order splitting that varies linearly with respect to deviations in the parameter, ensuring no true intersection occurs for indecomposable matrix flows. This linear splitting arises from the off-diagonal elements in the perturbed Hamiltonian basis, which generically prevent degeneracy unless explicitly zero.^[18] The theorem holds in full generality for any finite-dimensional Hilbert space equipped with a parameter-dependent Hermitian operator H(\lambda), where \lambda varies continuously, encompassing quantum mechanical systems from atomic spectra to molecular potentials.^[17] Exceptions arise in systems with additional symmetries, such as parity or angular momentum conservation, which decompose the Hilbert space into orthogonal subspaces; in these cases, eigenvalues from distinct subspaces can cross exactly, though such structured symmetries are atypical in fully generic, symmetry-unconstrained Hamiltonians.^[17]

Mathematical Formulation

In the simplest case of a two-state quantum system, an avoided crossing arises from the parameter-dependent Hamiltonian matrix in the diabatic basis,

H(\lambda) = \begin{pmatrix} E_1(\lambda) & V \\ V & E_2(\lambda) \end{pmatrix},

where \lambda is a continuous parameter (such as internuclear distance or magnetic field strength), E_1(\lambda) and E_2(\lambda) are the unperturbed (diabatic) energy levels that would cross in the absence of coupling, and V is the constant off-diagonal coupling term assuming Hermitian symmetry with V real.^[19] The eigenvalues of this matrix, representing the adiabatic energy levels, are obtained by solving the characteristic equation and given by

E_{\pm}(\lambda) = \frac{E_1(\lambda) + E_2(\lambda)}{2} \pm \sqrt{\left( \frac{E_1(\lambda) - E_2(\lambda)}{2} \right)^2 + V^2}.

This expression shows that the levels repel each other, with the minimal energy gap \Delta = E_+ - E_- = 2|V| occurring at the parameter value \lambda_0 where E_1(\lambda_0) = E_2(\lambda_0), preventing an actual crossing as required by the general avoided crossing theorem for Hermitian operators under generic parameter variations.^[19]^[20] For systems involving more than two near-degenerate states, the mathematical treatment employs degenerate perturbation theory to handle the near-degeneracy at the avoided crossing point. In this approach, the Hilbert space is partitioned into the degenerate (or quasi-degenerate) subspace spanned by the unperturbed states near \lambda_0 and the orthogonal complement. The perturbation Hamiltonian restricted to this subspace is diagonalized to obtain the leading-order corrections to the energies and the proper linear combinations of basis states that form the good zeroth-order adiabatic states, lifting the degeneracy and producing the characteristic level repulsion.^[21] The eigenvalues of this reduced matrix yield the first-order energy shifts, while higher-order corrections from coupling to non-degenerate states are computed using non-degenerate perturbation theory on the transformed basis. This method is particularly effective when the coupling V is small compared to the separation from other states, ensuring the subspace captures the essential dynamics of the crossing.^[21] The parameter dependence of the avoided crossing can be analyzed via Taylor expansion around the diabatic crossing point \lambda_0. Assuming linear behavior in the diabatic energies near \lambda_0, E_1(\lambda) - E_2(\lambda) \approx \frac{d(E_1 - E_2)}{d\lambda} (\lambda - \lambda_0), the adiabatic eigenvalues simplify to

E_{\pm}(\lambda) \approx \frac{E_1(\lambda) + E_2(\lambda)}{2} \pm \frac{1}{2} \sqrt{ \left( \frac{d(E_1 - E_2)}{d\lambda} (\lambda - \lambda_0) \right)^2 + 4 V^2 },

revealing the minimal gap \Delta \approx 2|V| at \lambda = \lambda_0. Higher-order terms in the expansion (e.g., quadratic deviations) introduce small shifts in the position of the minimal gap, but the dominant effect is the Lorentzian-like broadening of the repulsion region. The width of this region in parameter space, where significant mixing occurs (|E_1 - E_2| \lesssim 2|V|), is approximately \Delta\lambda \approx 2|V| / |d(E_1 - E_2)/d\lambda|, quantifying the scale over which the adiabatic approximation may break down.^[5]^[20] For complex multi-state systems beyond analytic tractability, numerical diagonalization of the full Hamiltonian matrix as a function of \lambda is employed to compute the eigenvalues and visualize avoided crossings. Methods such as full matrix diagonalization for small systems or iterative techniques like the Lanczos algorithm for larger sparse matrices efficiently resolve the spectrum, with the focus remaining on analytic insights from the two-state and perturbation frameworks for interpretation.^[5]

Molecular Applications

Diatomic Molecules

In diatomic molecules, avoided crossings arise primarily from vibronic coupling, where electronic states interact through nuclear motion along the internuclear distance, preventing direct crossings between adiabatic potential energy curves of the same symmetry. This coupling mixes electronic and vibrational degrees of freedom, altering the shapes of the potentials and influencing molecular dissociation pathways and spectral features.^[22] For instance, in alkali halide diatomics such as NaI, Σ-Π vibronic interactions occur between the ionic ^1Σ^+ state and the covalent ^1Π state, resulting in an avoided crossing around 6-7 Å that facilitates predissociation by allowing wave packets to transition from bound to dissociative regions.^[23]^[24] In the H_2^+ ion, multiple avoided crossings appear in the correlation diagram between adiabatic molecular orbitals, connecting Rydberg-like states to dissociative limits and affecting the overall potential landscape.^[25] Spectroscopically, avoided crossings in diatomics manifest as perturbed rotational constants and anomalous isotope effects in absorption or emission spectra. For example, in the C_2 molecule, interactions near 30,000 cm^{-1} cause irregular vibrational spacings and deviations in rotational parameters (B_v), attributed to near-avoided crossings between ^1Σ_g^+ and ^1Δ_u states.^[26] Similarly, in O_2, predissociation via avoided crossings in the F ^3Π_u (v=1) state produces isotope-dependent linewidth asymmetries and shifts, with heavier isotopologues showing broader lines due to enhanced non-adiabatic transitions.^[27] These observables arise because the crossing region's position and coupling strength vary with reduced mass, altering vibrational wavefunction overlap and transition intensities. The general mathematical formulation of avoided crossings, involving a diabatic basis with off-diagonal coupling terms, quantifies the minimum energy gap as twice the coupling matrix element at the nominal crossing point.^[22]

Polyatomic Molecules

In polyatomic molecules, the multidimensional nature of the nuclear coordinate space allows for conical intersections, which are degeneracy points where two adiabatic potential energy surfaces touch and form a double cone. These intersections occur in coordinates beyond the single-bond distance typical of diatomics, and along most paths approaching the intersection, the surfaces exhibit avoided crossings due to nonadiabatic coupling. This geometric feature enables ultrafast radiationless transitions and is central to understanding excited-state dynamics in complex systems. The Jahn-Teller theorem asserts that nonlinear molecules with electronically degenerate ground states undergo spontaneous geometric distortions to lower their symmetry and energy, thereby stabilizing the system. In such cases, the degeneracy is lifted through vibronic coupling, leading to avoided crossings along specific distortion coordinates; for near-degenerate states, pseudo-Jahn-Teller effects similarly produce avoidance in high-symmetry configurations. These distortions are particularly pronounced in transition metal complexes and organic polyatomics, where they dictate the branching of potential energy surfaces.^[28] Conical intersections are exemplified in the photodissociation of formaldehyde (H₂CO), where an extended seam of S₁/S₀ intersections facilitates internal conversion and dissociation into H₂ + CO via nonadiabatic pathways on the excited surface. In ozone (O₃) isomerization, intersections between low-lying singlet states near the transition state enable photoinduced cis-trans conversion, influencing atmospheric photochemistry. Encircling these intersections in nuclear space generates a Berry phase of π, observable as a sign inversion in the wavefunction that governs quantum interference effects in scattering and spectroscopy.^[29]^[30]^[31] Non-adiabatic dynamics at conical intersections promote efficient population transfer, or "funneling," from photoexcited states to the ground state, with transition probabilities approaching unity over femtosecond timescales. This mechanism underpins key photochemical processes, such as ultrafast energy dissipation in retinal proteins or selective bond breaking in polyatomic reactions, highlighting the intersections' role in controlling product yields and quantum efficiencies.^[32]

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