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Derivation

The Born-Oppenheimer Approximation

Statement

For a molecule with light electrons (coordinates \(\mathbf r\), mass \(m_e\)) and heavy nuclei (coordinates \(\mathbf R\), masses \(M_A\)), the full stationary Schrödinger equation \(\hat H\Psi=E\Psi\) separates into (i) an electronic problem solved at each fixed nuclear geometry, \(\hat H_e(\mathbf r;\mathbf R)\,\phi_k(\mathbf r;\mathbf R)=E_k(\mathbf R)\,\phi_k(\mathbf r;\mathbf R)\), whose eigenvalue \(E_k(\mathbf R)\) is a potential-energy surface, and (ii) a nuclear problem \(\left[\hat T_n+E_k(\mathbf R)\right]\chi_k(\mathbf R)=E\,\chi_k(\mathbf R)\) that moves on that surface. The separation is controlled by the mass ratio \(m_e/M_A\ll1\); the neglected coupling between surfaces is smaller than the retained terms by a factor of order \(m_e/M_A\).

Why it matters

Nearly every quantitative statement in chemistry and condensed matter presupposes the Born–Oppenheimer (BO) separation. It is what makes a "molecular structure" — bond lengths, angles, force constants — meaningful at all: the potential-energy surface \(E_k(\mathbf R)\) is a well-defined function of the nuclei precisely because the electrons follow them adiabatically. Vibrational spectra, reaction barriers, phonons, and the entire apparatus of electronic-structure theory are computations of \(E_k(\mathbf R)\) and of nuclear motion upon it.

Equally important is knowing where it fails. Photochemistry, nonradiative decay, electron transfer, and the physics of conical intersections are all governed by the terms BO discards. The approximation is therefore both the workhorse and the boundary marker of molecular quantum mechanics.

Assumptions
Nuclei are much heavier than electrons, \(m_e/M_A\ll1\).The expansion parameter \(\kappa=(m_e/M_A)^{1/4}\) is no longer small; the nuclear kinetic operator can no longer be treated as a perturbation and the whole ordering that justifies dropping the coupling collapses. The electronic state of interest is non-degenerate and energetically well separated from all others at the geometries sampled.If two surfaces \(E_j(\mathbf R)\) and \(E_k(\mathbf R)\) touch or nearly touch, the derivative coupling \(\propto1/(E_k-E_j)\) diverges and a single-surface description is invalid.
The electronic wavefunctions \(\phi_k(\mathbf r;\mathbf R)\) can be chosen to vary smoothly (and, where possible, real and single-valued) with \(\mathbf R\).Around a conical intersection the adiabatic \(\phi_k\) acquire a geometric (Berry) phase and cannot be made single-valued; the smooth-parametrization step underlying the derivative couplings breaks and a diabatic representation is needed.
The nuclei are treated as distinguishable, effectively localized wavepackets rather than requiring full exchange antisymmetry with the electrons.For light nuclei (protons, muonium) or when nuclear indistinguishability and large-amplitude tunnelling matter, the clean product ansatz and classical-geometry intuition become quantitatively unreliable.
Derivation
1
\[ \hat H=\hat T_n+\hat T_e+V_{ne}(\mathbf r,\mathbf R)+V_{ee}(\mathbf r)+V_{nn}(\mathbf R),\qquad \hat T_n=-\sum_A\frac{\hbar^2}{2M_A}\nabla_A^2,\quad \hat T_e=-\sum_i\frac{\hbar^2}{2m_e}\nabla_i^2 \]
Write the exact non-relativistic molecular Hamiltonian. All Coulomb terms are collected by type; nothing is yet approximated. A
2
\[ \hat H_e(\mathbf r;\mathbf R)\equiv\hat T_e+V_{ne}+V_{ee}+V_{nn}=\hat H-\hat T_n \]
Define the electronic (clamped-nuclei) Hamiltonian as everything except the nuclear kinetic energy. Here \(\mathbf R\) enters only as a fixed parameter, so \(\hat H_e\) acts only on \(\mathbf r\). A
3
\[ \hat H_e(\mathbf r;\mathbf R)\,\phi_k(\mathbf r;\mathbf R)=E_k(\mathbf R)\,\phi_k(\mathbf r;\mathbf R),\qquad \langle\phi_j|\phi_k\rangle_{\mathbf r}=\delta_{jk} \]
Solve the electronic eigenproblem at each fixed geometry. The eigenvalues \(E_k(\mathbf R)\) are the potential-energy surfaces; the \(\{\phi_k\}\) form a complete orthonormal set in the electronic Hilbert space for every \(\mathbf R\). A
4
\[ \Psi(\mathbf r,\mathbf R)=\sum_k \chi_k(\mathbf R)\,\phi_k(\mathbf r;\mathbf R) \]
Expand the exact total wavefunction in the complete electronic basis with \(\mathbf R\)-dependent coefficients \(\chi_k(\mathbf R)\). This Born–Huang expansion is exact, not yet an approximation. B
5
\[ \hat T_n\big(\chi_k\phi_k\big)=\phi_k\,\hat T_n\chi_k-\sum_A\frac{\hbar^2}{2M_A}\Big[\,2(\nabla_A\chi_k)\!\cdot\!(\nabla_A\phi_k)+\chi_k\nabla_A^2\phi_k\Big] \]
Apply the product rule to \(\nabla_A^2(\chi_k\phi_k)\). Because \(\phi_k\) depends parametrically on \(\mathbf R\), \(\nabla_A\) hits it too — this is the origin of every nonadiabatic term. B
6
\[ \sum_k\Big[\hat T_n+E_k(\mathbf R)\Big]\chi_k\phi_k-\sum_A\frac{\hbar^2}{2M_A}\sum_k\Big[2(\nabla_A\chi_k)\!\cdot\!\nabla_A\phi_k+\chi_k\nabla_A^2\phi_k\Big]=E\sum_k\chi_k\phi_k \]
Substitute the ansatz into \(\hat H\Psi=E\Psi\), using \(\hat H_e\phi_k=E_k\phi_k\) for the electronic part and Step 5 for \(\hat T_n\). B
7
\[ \Big[\hat T_n+E_j(\mathbf R)\Big]\chi_j+\sum_k\hat\Lambda_{jk}\chi_k=E\,\chi_j,\qquad \hat\Lambda_{jk}=-\sum_A\frac{\hbar^2}{2M_A}\Big[2\,\mathbf F^A_{jk}\!\cdot\!\nabla_A+G^A_{jk}\Big] \]
Project onto \(\langle\phi_j|\) and integrate over \(\mathbf r\). Orthonormality collapses the diagonal, leaving exact coupled equations with the nonadiabatic coupling operators \(\hat\Lambda_{jk}\), built from the derivative couplings below. B
8
\[ \mathbf F^A_{jk}\equiv\langle\phi_j|\nabla_A\phi_k\rangle_{\mathbf r},\qquad G^A_{jk}\equiv\langle\phi_j|\nabla_A^2\phi_k\rangle_{\mathbf r} \]
Name the first- and second-order derivative couplings. \(\mathbf F^A_{jk}\) is the vector (nonadiabatic) coupling; for real, normalized \(\phi_k\) the diagonal \(\mathbf F^A_{kk}=\tfrac12\nabla_A\langle\phi_k|\phi_k\rangle=0\). C
9
\[ \mathbf F^A_{jk}=\frac{\langle\phi_j|\,(\nabla_A\hat H_e)\,|\phi_k\rangle}{E_k(\mathbf R)-E_j(\mathbf R)}\qquad(j\neq k) \]
Differentiate \(\hat H_e\phi_k=E_k\phi_k\) with respect to \(\mathbf R_A\) and project onto \(\phi_j\): the Hellmann–Feynman-type identity. The \(1/(E_k-E_j)\) makes explicit that coupling is large only near degeneracies and diverges where surfaces meet. C
10
\[ \frac{\|\hat\Lambda_{jk}\|}{\|\hat T_n\|}\sim\frac{m_e}{M_A}\cdot\frac{1}{E_k-E_j}\times(\text{electronic scale}),\qquad \hat T_n=\mathcal O\!\left(\frac{m_e}{M_A}\,E_{\text{el}}\right) \]
Estimate magnitudes. On a gently varying, well-isolated surface the coupling terms are smaller than the retained kinetic and potential energy by \(\mathcal O(m_e/M_A)\); this is the quantitative content of "electrons follow the nuclei." C
11
\[ \hat\Lambda_{jk}\to0\ (j\neq k),\qquad G^A_{kk}\to0 \quad\Longrightarrow\quad \Big[-\sum_A\frac{\hbar^2}{2M_A}\nabla_A^2+E_k(\mathbf R)\Big]\chi_k(\mathbf R)=E\,\chi_k(\mathbf R) \]
The Born–Oppenheimer step: drop all off-diagonal couplings and the diagonal correction \(G^A_{kk}\). The nuclei then move in the single effective potential \(E_k(\mathbf R)\). (Keeping only \(G^A_{kk}\) gives the Born–Huang "adiabatic" refinement.) A
Result
\[ \boxed{\;\Big[-\sum_A\frac{\hbar^2}{2M_A}\nabla_A^2+E_k(\mathbf R)\Big]\chi_k(\mathbf R)=E\,\chi_k(\mathbf R),\qquad \hat H_e\phi_k=E_k(\mathbf R)\phi_k,\qquad \Psi\approx\chi_k(\mathbf R)\,\phi_k(\mathbf r;\mathbf R)\;} \]

Reading. The electrons solve their own Schrödinger equation at every frozen nuclear geometry, producing an energy \(E_k(\mathbf R)\) that acts back on the nuclei as a potential-energy surface. The nuclei then move quantum-mechanically on that single surface, blind to the other electronic states. Molecular structure, force constants, and reaction paths are all features of \(E_k(\mathbf R)\); the approximation is controlled by \(m_e/M_A\ll1\).

Units check. \(E_k(\mathbf R)\) is an energy (J). The kinetic term has \(\left[\frac{\hbar^2}{M}\nabla^2\right]=\frac{(\mathrm{J\,s})^2}{\mathrm{kg}}\,\mathrm{m^{-2}}=\frac{\mathrm{J^2\,s^2}}{\mathrm{kg\,m^2}}\); since \(\mathrm{J}=\mathrm{kg\,m^2\,s^{-2}}\), this reduces to \(\mathrm{kg\,m^2\,s^{-2}}=\mathrm{J}\). Both terms are energies acting on the dimensionless \(\chi_k\), matching \(E\chi_k\).

Limiting cases
  • \(M_A\to\infty\) (clamped nuclei): \(\hat T_n\to0\), the nuclear equation degenerates to \(E=E_k(\mathbf R)\) at fixed \(\mathbf R\) — the pure electronic-structure limit, exact in this limit.
  • Small vibrations about a minimum \(\mathbf R_0\): expand \(E_k(\mathbf R)\approx E_k(\mathbf R_0)+\tfrac12\sum(\mathbf R-\mathbf R_0)^\top\mathbf H(\mathbf R-\mathbf R_0)\); the nuclear equation becomes coupled harmonic oscillators (normal modes) with \(\hbar\omega\sim(m_e/M)^{1/2}E_{\text{el}}\).
  • Rigid rotation: the lowest excitations of \(\chi_k\) on a flat direction of \(E_k\) give rotational levels with spacing \(\sim(m_e/M)\,E_{\text{el}}\), the smallest of the three scales.
  • Well-separated surfaces: when \(\min_{\mathbf R}|E_k-E_j|\gg\|\hat\Lambda_{jk}\|\), the single-surface result is essentially exact and BO corrections are perturbatively tiny.
Breaks when
  • Conical intersections and true degeneracies. Where \(E_k(\mathbf R)=E_j(\mathbf R)\), the derivative coupling \(\mathbf F^A_{jk}\propto1/(E_k-E_j)\) diverges. A single surface is meaningless; the nuclear wavepacket splits between surfaces and the adiabatic \(\phi_k\) pick up a geometric phase.
  • Avoided crossings and fast nuclei. Near a narrow avoided crossing, or when nuclei move quickly (high collision energy, light H atoms), the Massey parameter \(\xi=\Delta E\,a/(\hbar v)\lesssim1\) and the system hops surfaces (Landau–Zener, nonadiabatic transitions).
  • Light nuclei / strong vibronic coupling. For protons, muonium, and Jahn–Teller-active systems the parameter \(m_e/M_A\) is at its largest and vibronic coupling mixes electronic states; the clean separation degrades even away from crossings.
  • Metals and dense manifolds. A gapless or near-continuous electronic spectrum (metals, small-gap semiconductors) offers no energy denominator to suppress coupling, so electron–phonon nonadiabaticity is intrinsic.
Failure modes
  • Treating \(\phi_k(\mathbf r;\mathbf R)\) as \(\mathbf R\)-independent. Then \(\nabla_A\phi_k=0\) and all coupling vanishes by fiat — but the parametric \(\mathbf R\)-dependence is the physics; this is the crude "diabatic-by-neglect" error.
  • Adding \(V_{nn}(\mathbf R)\) to the nuclear equation a second time. \(V_{nn}\) is already inside \(\hat H_e\), hence already in \(E_k(\mathbf R)\). Double-counting it shifts every surface.
  • Using the wrong mass. Placing \(m_e\) (or a total mass) in the nuclear kinetic operator instead of the nuclear/reduced mass \(M_A\); or using atomic instead of nuclear masses when a mass-scaling convention demands the latter.
  • Assuming BO gives the exact energy. \(E_k(\mathbf R_0)\) is the electronic minimum, not the molecular ground-state energy — the nuclear zero-point energy \(\tfrac12\sum\hbar\omega_i\) must be added.
  • Ignoring the diagonal Born–Oppenheimer correction near light atoms. Dropping \(G^A_{kk}\) is fine for heavy nuclei but introduces visible error for H/D isotope effects and high-accuracy spectroscopy.
  • Applying a single surface through a curve crossing. Forcing one \(E_k(\mathbf R)\) across a conical intersection where two states must be treated together.
Discussion

The deepest point is what "molecular geometry" even means. Quantum mechanically the nuclei are not at fixed positions; the concept of a bond length or a bond angle only survives because, to leading order in \(m_e/M_A\), the electrons instantaneously relax to the ground state of whatever configuration the slow nuclei present. The potential-energy surface \(E_k(\mathbf R)\) is the emergent object that turns a many-body electron problem into a landscape over which classical or quantum nuclei move. Chemistry is, to first approximation, the topography of these surfaces.

Born and Oppenheimer's original 1927 argument was a systematic expansion in \(\kappa=(m_e/M)^{1/4}\), and the quartic root is not arbitrary: electronic energies scale as \(\kappa^0\), vibrational energies as \(\kappa^2\), and rotational energies as \(\kappa^4\). The three great length/energy scales of molecular spectroscopy — electronic transitions in the visible/UV, vibrations in the infrared, rotations in the microwave — are a direct consequence of successive powers of a single small parameter. Seeing the spectrum organized this way is one of the triumphs of the approximation.

The terms BO discards define a research field. The off-diagonal \(\hat\Lambda_{jk}\) are what drive internal conversion, intersystem crossing (with spin–orbit added), and radiationless decay in photochemistry and vision. At a conical intersection the double-cone topology forces the electronic wavefunction to change sign on encircling the seam — the molecular Berry phase — which must be compensated by a sign change in \(\chi_k\) for single-valuedness of \(\Psi\). Diabatic representations trade the divergent derivative couplings for smooth off-diagonal potential couplings, and the choice between adiabatic and diabatic pictures is a practical, geometry-dependent one rather than a matter of principle. Surface-hopping and multiconfigurational time-dependent methods exist precisely to propagate amplitude between the surfaces BO decouples.

Common misconceptions. BO is not the statement that electrons are "faster" in a naive velocity sense — it is a statement about the energy separation of electronic states relative to nuclear kinetic energy, encoded in the denominator \(E_k-E_j\). A system can have slow-moving electrons and still be adiabatic if its gap is large; a light, fast nucleus can break BO even with a modest gap. Nor does "adiabatic" here mean the thermodynamic sense — it means the nuclei do not induce electronic transitions.

Worked examples
1
Estimate the electronic, vibrational, and rotational energy scales of H\(_2\) from the mass ratio, taking the electronic scale as \(E_{\text{el}}\approx10\ \text{eV}\).
Use the Born–Oppenheimer scaling \(E_{\text{vib}}\sim(m_e/M)^{1/2}E_{\text{el}}\), \(E_{\text{rot}}\sim(m_e/M)\,E_{\text{el}}\) with \(M\) the proton mass. A
2
\[ \frac{m_e}{M_p}=\frac{9.109\times10^{-31}}{1.673\times10^{-27}}=5.45\times10^{-4}=\frac{1}{1836} \]
Form the small parameter. A
3
\[ E_{\text{vib}}\sim(5.45\times10^{-4})^{1/2}\times10\ \text{eV}=0.0233\times10\ \text{eV}=0.23\ \text{eV} \]
Vibrational scale from the \(\kappa^2\) power. (Measured H\(_2\) fundamental \(\approx0.55\) eV — right order of magnitude.) B
4
\[ E_{\text{rot}}\sim(5.45\times10^{-4})\times10\ \text{eV}=5.4\times10^{-3}\ \text{eV}=5.4\ \text{meV} \]
Rotational scale from the \(\kappa^4\) power. (Measured \(2B\approx15\) meV — again the right order.) B
\[ E_{\text{el}}:E_{\text{vib}}:E_{\text{rot}}\ \approx\ 10\ \text{eV}:0.23\ \text{eV}:5.4\ \text{meV}\ \approx\ 1:0.023:5.4\times10^{-4} \]

Reading. A single small parameter \(m_e/M_p\) generates the three-tier hierarchy of molecular spectra, with vibrations in the infrared and rotations in the far-infrared/microwave. Units check. Every quantity is an energy in eV; the ratios are dimensionless powers of the dimensionless \(m_e/M_p\).

1
Show that nuclear motion is small compared with the electronic energy that sets the well: compute the harmonic zero-point energy of H\(_2\) and compare with the dissociation depth \(D_e=4.75\ \text{eV}\). Force constant \(k=575\ \text{N m}^{-1}\), reduced mass \(\mu=m_p/2\).
The curvature of \(E_k(\mathbf R)\) at its minimum gives \(\omega=\sqrt{k/\mu}\); ZPE \(=\tfrac12\hbar\omega\). A
2
\[ \mu=\frac{m_p}{2}=\frac{1.673\times10^{-27}}{2}=8.36\times10^{-28}\ \text{kg} \]
Reduced mass of two identical protons. A
3
\[ \omega=\sqrt{\frac{k}{\mu}}=\sqrt{\frac{575}{8.36\times10^{-28}}}=\sqrt{6.88\times10^{29}}=8.29\times10^{14}\ \text{rad s}^{-1} \]
Angular frequency of the normal mode on the surface. B
4
\[ \tfrac12\hbar\omega=\tfrac12(1.055\times10^{-34})(8.29\times10^{14})=4.37\times10^{-20}\ \text{J}=0.27\ \text{eV} \]
Zero-point energy; convert with \(1\ \text{eV}=1.602\times10^{-19}\ \text{J}\). B
\[ \frac{\text{ZPE}}{D_e}=\frac{0.27\ \text{eV}}{4.75\ \text{eV}}=0.057\approx6\% \]

Reading. Nuclear zero-point motion is a few percent of the electronic well depth, confirming that the surface \(E_k(\mathbf R)\) (an electronic quantity) dominates and nuclear dynamics is a small correction upon it — exactly the ordering BO exploits. The BO energy of the molecule is \(-D_e+\text{ZPE}\), so the true dissociation energy is \(D_0=D_e-\text{ZPE}=4.48\ \text{eV}\). Units check. \(\sqrt{(\text{N m}^{-1})/\text{kg}}=\sqrt{\text{s}^{-2}}=\text{s}^{-1}\); \(\hbar\omega\) has \(\text{J s}\cdot\text{s}^{-1}=\text{J}\), an energy.

Problems
  1. Mass-ratio hierarchy for a heavier nucleus. Compute \(\kappa=(m_e/M)^{1/4}\) for a \(^{12}\)C nucleus (\(M=12\,u\)) and give the vibrational and rotational scaling factors relative to a \(10\ \text{eV}\) electronic scale.
    Solution \(M=12\times1.6605\times10^{-27}=1.993\times10^{-26}\ \text{kg}\); \(m_e/M=9.109\times10^{-31}/1.993\times10^{-26}=4.57\times10^{-5}\). \(\kappa=(4.57\times10^{-5})^{1/4}=0.0822\). Vibrational \(\sim\kappa^2=6.76\times10^{-3}\Rightarrow0.068\ \text{eV}\); rotational \(\sim\kappa^4=m_e/M=4.57\times10^{-5}\Rightarrow4.6\times10^{-4}\ \text{eV}=0.46\ \text{meV}\). Heavier nuclei give a smaller \(\kappa\) and a better BO separation.
  2. Zero-point energy of CO. The C–O stretch has \(\tilde\nu=2170\ \text{cm}^{-1}\). Find the zero-point energy in eV and its ratio to \(D_e=11.2\ \text{eV}\).
    Solution \(\hbar\omega=hc\tilde\nu\): \(E=hc\tilde\nu=(6.626\times10^{-34})(3.00\times10^{10}\ \text{cm s}^{-1})(2170\ \text{cm}^{-1})=4.31\times10^{-20}\ \text{J}=0.269\ \text{eV}\) per quantum. ZPE \(=\tfrac12(0.269)=0.135\ \text{eV}\). Ratio \(=0.135/11.2=1.2\times10^{-2}\approx1.2\%\). Even smaller than H\(_2\) because the nuclei are heavier and the well deeper.
  3. Derivative coupling from a gap. Two surfaces are separated by \(E_k-E_j=0.50\ \text{eV}\) and the coupling matrix element is \(\langle\phi_j|\partial_R\hat H_e|\phi_k\rangle=1.2\ \text{eV\,\AA}^{-1}\). Estimate \(|F_{jk}|=|\langle\phi_j|\partial_R\phi_k\rangle|\), and comment on what happens as the gap shrinks to \(0.02\ \text{eV}\).
    Solution \(|F_{jk}|=|\langle\phi_j|\partial_R\hat H_e|\phi_k\rangle|/|E_k-E_j|=1.2/0.50=2.4\ \text{\AA}^{-1}\). At a \(0.02\ \text{eV}\) gap, \(|F_{jk}|=1.2/0.02=60\ \text{\AA}^{-1}\), a 25-fold increase; the coupling operator \(\hat\Lambda_{jk}\propto F_{jk}\) becomes large and single-surface BO is no longer justified — the surfaces must be treated together.
  4. Diagonal derivative coupling vanishes. Prove that for a real, normalized electronic state \(\phi_k(\mathbf r;\mathbf R)\), the first-order diagonal coupling \(\mathbf F^A_{kk}=\langle\phi_k|\nabla_A\phi_k\rangle=0\), and state the more general result for complex \(\phi_k\).
    Solution Normalization gives \(\langle\phi_k|\phi_k\rangle=1\) for all \(\mathbf R\). Differentiate: \(\nabla_A\langle\phi_k|\phi_k\rangle=\langle\nabla_A\phi_k|\phi_k\rangle+\langle\phi_k|\nabla_A\phi_k\rangle=0\). For real \(\phi_k\) the two terms are equal, so \(2\langle\phi_k|\nabla_A\phi_k\rangle=0\Rightarrow\mathbf F^A_{kk}=0\). For complex \(\phi_k\), the identity gives \(\mathbf F^A_{kk}+(\mathbf F^A_{kk})^*=0\), i.e. \(\mathbf F^A_{kk}\) is purely imaginary; it is a gauge (phase) quantity whose loop integral is the Berry phase and can be removed locally but not globally around a conical intersection.
  5. Massey parameter and surface hopping. A nucleus of mass \(M=2.0\times10^{-26}\ \text{kg}\) crosses an avoided crossing with gap \(\Delta E=0.10\ \text{eV}\) over a coupling length \(a=0.30\ \text{\AA}\) at kinetic energy \(1.0\ \text{eV}\). Compute the nuclear speed \(v\) and the Massey parameter \(\xi=\Delta E\,a/(\hbar v)\); is the passage adiabatic (BO-valid) or not?
    Solution \(v=\sqrt{2E_k/M}=\sqrt{2(1.0\times1.602\times10^{-19})/(2.0\times10^{-26})}=\sqrt{1.602\times10^{7}}=4.00\times10^{3}\ \text{m s}^{-1}\). With \(\Delta E=0.10\ \text{eV}=1.602\times10^{-20}\ \text{J}\), \(a=3.0\times10^{-11}\ \text{m}\): \(\xi=\dfrac{(1.602\times10^{-20})(3.0\times10^{-11})}{(1.055\times10^{-34})(4.00\times10^{3})}=\dfrac{4.81\times10^{-31}}{4.22\times10^{-31}}=1.1\). Since \(\xi\sim1\), the crossing is borderline: neither cleanly adiabatic (\(\xi\gg1\), BO holds, nucleus stays on one surface) nor cleanly diabatic (\(\xi\ll1\)); a significant Landau–Zener hopping probability \(P\sim e^{-2\pi\xi}\approx e^{-6.9}\approx1\times10^{-3}\) — small but nonzero, so BO is marginal here.