Born–Oppenheimer in Molecules

The Born–Oppenheimer hierarchy turns the coupled quantum mechanics of electrons and nuclei into the working language of molecular structure. At each nuclear geometry, one solves an electronic eigenproblem. Its eigenvalues define potential-energy surfaces; quantum nuclear motion on one or several surfaces then produces vibration, rotation, tunneling, dissociation, and rovibronic spectra.

The approximation is not the claim that nuclei are classical or permanently fixed. “Clamped nuclei” names only the auxiliary electronic problem. Predicting a molecular spectrum also requires nuclear motion, exchange symmetry, and corrections appropriate to its precision.

Required background. The molecular Hamiltonian supplies the particles and coordinates; Born–Oppenheimer scale separation derives the electronic-channel equations. The harmonic oscillator and rigid rotor provide the leading nuclear models.

Here those models are connected to molecular structure and spectra. Potential-energy surfaces develops their geometry; nonadiabatic coupling and conical intersections treat multistate dynamics.

Starting from the Internal Hamiltonian

After overall translation has been removed, write the field-free molecular Hamiltonian schematically as

$$H_{\mathrm{int}} = T_N+H_e(R)+H_{\mathrm{recoil}}.$$

Here:

• $R$ denotes a complete set of internal nuclear coordinates;
• $T_N$ is the corresponding nuclear kinetic operator;
• $H_e(R)$ is the usual infinite-nuclear-mass electronic operator at geometry $R$;
• $H_{\mathrm{recoil}}$ collects finite-mass electronic and mixed kinetic terms introduced by the chosen internal coordinates.

For transparent formulas, it is common to write

$$T_N = -\sum_A \frac{\hbar^2}{2M_A} \nabla_A^2.$$

That Cartesian form is exact before removing laboratory translation. Afterward, nuclear relative coordinates use reduced masses and may require rotational and metric terms; electronic mass-polarization derivatives cannot simply be assigned to an operator acting only on nuclear variables. The formulas below display the leading separation, with recoil corrections treated consistently at higher order.

Surface convention

Throughout this page, $H_e(R)$ includes internuclear repulsion:

$$H_e(R) ={} -\frac12\sum_i\nabla_i^2 -\sum_{i,A}\frac{Z_A}{r_{iA}} + \sum_{i<j}\frac{1}{r_{ij}} + \sum_{A<B}\frac{Z_AZ_B}{R_{AB}}$$

in atomic units. Its eigenvalue $E_a(R)$ is therefore already a potential-energy surface. If another source excludes $V_{NN}$ from the electronic Hamiltonian, translate by

$$U_a(R) = \mathcal E_a(R)+V_{NN}(R).$$

The physics is identical when the convention is used consistently.

Fixed-Geometry Electronic States

At every admissible geometry, solve

$$H_e(R) \lvert\phi_a(R)\rangle = E_a(R) \lvert\phi_a(R)\rangle.$$

The electronic inner product holds $R$ fixed:

$$\langle\phi_a(R) \vert \phi_b(R)\rangle_e = \delta_{ab}.$$

Each state must have the correct electron antisymmetry and can be classified by spin and by the symmetries of the chosen nuclear geometry. The geometry is a parameter of this electronic calculation, not a measured classical configuration of the exact free molecule.

Geometry space has identifications

Two laboratory configurations related by an overall translation or rotation represent the same internal shape. Configurations related by exchange of identical nuclei may also represent the same physical arrangement, subject to the required permutation symmetry of the total state.

Consequently, a potential-energy surface is naturally a function on nuclear shape space, not on an arbitrary list of labeled laboratory coordinates. Cartesian coordinates remain convenient computationally, but zero modes and permutation-related copies must be handled correctly.

Electronic labels can change with geometry

Point-group irreducible representations classify states at geometries with that symmetry. Along a distortion, the point group may decrease and labels may correlate or mix. Energy ordering alone is not a reliable state label near avoided crossings. State tracking should use symmetry, overlaps, transition properties, or a carefully chosen diabatic representation.

Exact Channel Expansion and the One-Surface Step

A complete fixed-geometry electronic basis gives the exact expansion

$$\Psi(r,R) = \sum_a \chi_a(R)\phi_a(r;R).$$

The expansion itself is not the Born–Oppenheimer approximation. Nuclear derivatives act on both factors, producing diagonal and off-diagonal derivative couplings.

The simplest molecular ansatz retains one electronic channel:

$$\Psi(r,R) \approx \chi_a(R)\phi_a(r;R).$$

At leading order, the nuclear wavefunction obeys

$$\left[ T_N+E_a(R) \right] \chi_{a\nu}(R) = E_{a\nu} \chi_{a\nu}(R).$$

The index $\nu$ collectively labels vibration, rotation, tunneling, and other nuclear quantum numbers on surface $a$.

Where approximation enters

The leading one-surface equation neglects:

• coupling to other electronic surfaces;
• the geometry dependence of the electronic state inside $T_N$;
• finite-mass corrections beyond the chosen internal kinetic operator;
• relativistic, radiative, and nuclear-structure contributions unless added separately.

An accurate electronic energy at one geometry does not validate these omissions. The relevant object is the electronic eigenspace over the entire nuclear region sampled by $\chi_{a\nu}(R)$.

Potential-Energy Surfaces as Molecular Potentials

A surface $E_a(R)$ supplies the effective potential for nuclear motion in electronic state $a$.

Minima and equilibrium structures

An equilibrium geometry $R_e$ on a smooth surface satisfies

$$\left. \nabla_R E_a(R) \right|_{R_e} = 0.$$

For a nondegenerate stable minimum, the Hessian restricted to internal distortions is positive:

$$\left. \delta R^{\mathsf T} \nabla_R^2E_a \delta R \right|_{R_e} >0$$

for every nonzero vibrational displacement $\delta R$. A zero internal curvature requires higher-order analysis and may describe a soft mode rather than an instability.

The subscript $e$ means equilibrium, not electron. $R_e$ is a parameter of the effective surface. It is not generally equal to a vibrationally averaged bond length inferred from a spectrum.

Gradients and forces

For an exact, normalized, nondegenerate electronic eigenstate,

$$\frac{\partial E_a}{\partial R_\lambda} = \left\langle \phi_a \left| \frac{\partial H_e}{\partial R_\lambda} \right| \phi_a \right\rangle_e.$$

This is the Hellmann–Feynman relation. In a finite geometry-dependent basis, additional Pulay terms can appear because the basis itself changes with $R$. A numerically converged energy does not automatically imply a comparably converged gradient.

Saddles and asymptotes

A first-order saddle has one unstable internal Hessian direction and often organizes a reaction barrier. At large fragment separation, the surface must approach the correct separated-fragment channel. Surface construction, stationary points, reaction paths, and global representations deserve their own treatment; here they serve as inputs to nuclear quantum mechanics.

Small Oscillations of a Diatomic

For a diatomic molecule on an isolated electronic surface, let $R$ be the internuclear distance and $\mu_N$ the nuclear reduced mass:

H₂⁺ Ion supplies the simplest explicit example: one solves the one-electron two-center equation at each $R$, adds proton–proton repulsion, and then treats the resulting curve as the potential for nuclear vibration and rotation.

$$\mu_N = \frac{M_AM_B}{M_A+M_B}.$$

Near a stable minimum,

$$E_a(R) \approx E_a(R_e) + \frac12 k_a(R-R_e)^2,$$

where

$$k_a = \left. \frac{d^2E_a}{dR^2} \right|_{R_e}.$$

The leading harmonic frequency is

$$\omega_e = \sqrt{\frac{k_a}{\mu_N}}.$$

This relation separates electronic and isotopic ingredients:

• $k_a$ comes primarily from the electronic surface;
• $\mu_N$ comes from the isotopes;
• the zero-point width and level spacing depend on both.

The harmonic levels are

$$E_v \approx E_a(R_e) + \hbar\omega_e \left( v+\frac12 \right).$$

Anharmonicity, rotation, diagonal corrections, and surface coupling shift these levels. The exact oscillator solution is developed in Quantum Harmonic Oscillator.

Diatomic vibrations extends this reduction to anharmonic levels and infrared selection rules.

Isotope shift

If the electronic surface is held fixed while $\mu_N$ changes,

$$\omega_e\propto\mu_N^{-1/2}.$$

This leading isotope law is powerful but incomplete. High-resolution isotope shifts also probe adiabatic corrections, nonadiabatic effective masses, nuclear radii, and hyperfine interactions.

Diatomic Rotation and Vibration

For an isolated electronic $\Sigma$ state with neglected spin and electronic–rotational coupling, the radial nuclear equation is

$$\Bigg[ - \frac{\hbar^2}{2\mu_N} \frac{d^2}{dR^2} + \frac{\hbar^2J(J+1)} {2\mu_NR^2} + E_a(R) \Bigg] u_{vJ}(R) = E_{vJ} u_{vJ}(R).$$

Here $u_{vJ}(R)$ is the reduced radial nuclear wavefunction. The centrifugal term couples rotation to vibration because $R$ is not fixed.

Near $R_e$, the rigid-rotor constant is

$$B_e = \frac{\hbar^2} {2\mu_NR_e^2}.$$

The leading product formula

$$E_{vJ} \approx E_v+B_eJ(J+1)$$

neglects centrifugal distortion and the variation of the rotational constant across the vibrational wavefunction. A better state-dependent constant samples

$$B_v \propto \left\langle \frac{1}{R^2} \right\rangle_v.$$

For electronic states with nonzero orbital angular momentum about the molecular axis, open shells, or strong spin couplings, the simple $J(J+1)$ term must be replaced by the appropriate coupled angular-momentum Hamiltonian. Rigid rotor solves the fixed-bond model, while Molecular Rotation Applications develops rotational tensor selection rules.

Worked example: nuclear motion on an approximate H₂⁺ surface

The minimal LCAO calculation for H₂⁺ uses two hydrogenic $1s$ orbitals with fixed exponent $1/a_0$. Its bonding curve includes proton–proton repulsion and approaches $-E_{\mathrm h}/2$ at dissociation. Minimizing that curve and differentiating it twice gives

$$\begin{aligned} R_e&\simeq2.49283\,a_0,\\ E_g(R_e)&\simeq-0.564831\,E_{\mathrm h},\\ k_g&\simeq0.0628331\,E_{\mathrm h}/a_0^2. \end{aligned}$$

These are outputs of the specified electronic trial space. Now supply the nuclear mass: taking $M_p/m_e\simeq1836.15$ from the 2022 CODATA constants gives $\mu_N=M_p/2\simeq918.075\,m_e$. In atomic units, the formulas above yield

$$\begin{aligned} \hbar\omega_e &=\sqrt{\frac{0.0628331}{918.075}}\,E_{\mathrm h} \simeq0.008273\,E_{\mathrm h},\\ B_e &=\frac{E_{\mathrm h}}{2(918.075)(2.49283)^2} \simeq8.764\times10^{-5}\,E_{\mathrm h},\\ \sqrt{\langle(R-R_e)^2\rangle_0} &=\sqrt{\frac{\hbar}{2\mu_N\omega_e}} \simeq0.257\,a_0. \end{aligned}$$

The nuclear ground state therefore samples a bond-length distribution with root-mean-square width about $10\%$ of $R_e$. The lowest spin-free rotor interval $2B_e$ is about $2.1\%$ of the harmonic vibrational spacing. These are energy scales; a spectrum also needs the transition operator and nuclear-spin restrictions. Here electronic spin and its couplings are neglected, so the rotational label plays the role usually denoted $N$ in a $^2\Sigma$ molecule.

The same model has $D_e=0.064831\,E_{\mathrm h}$. Subtracting its harmonic zero-point energy gives

$$D_0^{\mathrm{harm}} =D_e-\frac12\hbar\omega_e \simeq0.060695\,E_{\mathrm h}.$$

This is an illustrative nuclear calculation on an approximate surface, not a precision H₂⁺ dissociation energy. Even within the clamped-nuclei problem, the converged minimum is about $-0.602635\,E_{\mathrm h}$ (Fernández and Garcia, 2021, Eq. 21). The minimal-basis well-bottom error is therefore about $0.0378\,E_{\mathrm h}$, over nine times its estimated zero-point energy. Improving the electronic trial space and checking the anharmonic nuclear solution are distinct tasks; small adiabatic or radiative corrections cannot repair that basis error. The detailed diatomic vibration treatment explains how to improve the local oscillator approximation.

Polyatomic normal modes

Near a nondegenerate minimum, a Cartesian displacement vector $q$ gives $E_a\simeq E_a(R_e)+q^{\mathsf T}Kq/2$. The eigenvalues of the mass-weighted Hessian $M^{-1/2}KM^{-1/2}$ are squared harmonic frequencies. At a stationary geometry, translation and rotation supply zero modes: the remaining count is $3N_N-6$ for a nonlinear molecule and $3N_N-5$ for a linear one.

The normal-mode derivation explains the projection and symmetry labels. Large-amplitude torsion, inversion, or dissociation instead needs a potential beyond the local quadratic expansion.

Molecular Structure Is an Effective Statement

The surface minimum $R_e$ is the most common theoretical definition of equilibrium structure. Several experimentally reported structures answer different questions:

Symbol or description	Meaning
$r_e$	Minimum of an effective electronic potential surface
Ground-state average	Expectation or effective average over the $v=0$ nuclear distribution
Isotopic substitution structure	Geometry inferred from changes in rotational constants
Effective spectroscopic structure	Parameters fitted within a chosen rovibrational Hamiltonian

These need not coincide. Zero-point motion is isotope dependent, and fitted parameters can absorb higher-order corrections.

An exact field-free molecular eigenstate also has definite total angular momentum and is not pinned to one laboratory orientation. A body-fixed equilibrium geometry is an effective internal description that becomes observable through rotationally invariant quantities, aligned preparations, environmental localization, or conditional measurements.

Electronic Excitation and Vibronic Bands

Different electronic states have different surfaces and nuclear eigenfunctions:

$$\left[ T_N+E_a(R) \right] \chi_{a\nu}(R) = E_{a\nu}\chi_{a\nu}(R).$$

For a transition $a,\nu\to b,\nu'$, the Born–Oppenheimer transition amplitude contains

$$\mathcal M_{b\nu',a\nu} = \int dR\, \chi_{b\nu'}^*(R) \boldsymbol\mu_{ba}(R) \chi_{a\nu}(R),$$

where

$$\boldsymbol\mu_{ba}(R) = \langle \phi_b(R) \vert \widehat{\boldsymbol\mu} \vert \phi_a(R) \rangle_e.$$

After separating rotational and polarization factors, let $R$ here denote vibrational coordinates. If the electronic transition dipole varies slowly over their sampled region, the Condon approximation replaces it by a representative constant. The remaining vibrational line strength is governed by the Franck–Condon overlap

$$S_{\nu'\nu} = \left| \int dR\, \chi_{b\nu'}^*(R) \chi_{a\nu}(R) \right|^2.$$

The common vertical-transition picture reflects that the light–matter interaction acts on a time scale short compared with substantial nuclear displacement. It does not mean nuclear kinetic energy is zero before or after the transition.

Diagonal Adiabatic Correction

Even when one electronic channel is retained, $T_N$ differentiates the geometry-dependent electronic state. Define the diagonal connection

$$\mathbf d_{aa}^{A} = \langle \phi_a \vert \nabla_A\phi_a \rangle_e.$$

A gauge-invariant scalar correction is obtained from the part of each electronic derivative orthogonal to the retained state. Define

$$\begin{aligned} Q_a &= 1-\lvert\phi_a\rangle \langle\phi_a\rvert, \\ \lvert\nabla_A^\perp\phi_a\rangle &= Q_a \lvert\nabla_A\phi_a\rangle. \end{aligned}$$

Then

$$\Phi_a(R) = \sum_A \frac{\hbar^2}{2M_A} \left\| \nabla_A^\perp\phi_a \right\|_e^2.$$

Locally, for a nondegenerate real electronic state with

$$\mathbf d_{aa}^{A}=0,$$

this reduces to the familiar diagonal Born–Oppenheimer correction:

$$E_{\mathrm{DBOC}}(R) = \sum_A \frac{\hbar^2}{2M_A} \langle \nabla_A\phi_a \vert \nabla_A\phi_a \rangle_e.$$

The correction is geometry and isotope dependent. It modifies equilibrium structures, barriers, and vibrational levels. It is often small on a chemical-energy scale but relevant at spectroscopic accuracy.

Translational consistency

Electronic derivatives with respect to laboratory nuclear coordinates can contain apparent center-of-mass contributions. A rigorous implementation must either separate translation first or use a formulation known to reproduce the same internal correction. The simple appearance of the DBOC formula does not license mixing laboratory and body-fixed derivatives.

Mass conventions

The exact all-particle Hamiltonian contains bare nuclear masses. Leading Born–Oppenheimer nuclear motion formally starts from those masses. Practical high-accuracy rovibrational Hamiltonians may use atomic masses, geometry-dependent effective masses, or explicit nonadiabatic mass tensors to represent electronic inertia.

Changing the masses without adjusting the correction terms can double count electronic inertia. The masses must be matched to the correction terms and to the dissociation threshold convention.

Berry Connection in One Surface

Define

$$\boldsymbol{\mathcal A}_a^A(R) = i \langle \phi_a(R) \vert \nabla_A\phi_a(R) \rangle_e.$$

The gauge-covariant one-surface nuclear Hamiltonian can be written schematically as

$$H_{N,a} ={} \sum_A \frac{ \left[ -i\hbar\nabla_A -\hbar\boldsymbol{\mathcal A}_a^A \right]^2 }{2M_A} + E_a(R) + \Phi_a(R).$$

Under

$$\phi_a \longrightarrow e^{i\theta(R)}\phi_a,$$

the nuclear factor transforms as

$$\chi_a \longrightarrow e^{-i\theta(R)}\chi_a,$$

so the total molecular wavefunction is unchanged.

For a simply connected region with an isolated real electronic state, one can often choose a local gauge with $\boldsymbol{\mathcal A}=0$. Around a conical intersection, a globally single-valued real gauge may not exist. The nuclear wavefunction then acquires a geometric phase. Born–Oppenheimer Berry phase explains the resulting boundary condition and interference.

Off-Diagonal Nonadiabatic Coupling

Between different electronic channels,

$$\mathbf d_{ab}^{A}(R) = \langle \phi_a(R) \vert \nabla_A\phi_b(R) \rangle_e.$$

For nondegenerate states and an exact differentiable eigenproblem,

$$\mathbf d_{ab}^{A} = \frac{ \left\langle \phi_a \left| \nabla_AH_e \right| \phi_b \right\rangle_e }{ E_b-E_a }, \qquad a\ne b.$$

This relation exposes the central molecular diagnostic:

$$\begin{gathered} \text{small electronic gap} \\ \Downarrow \\ \text{potentially large channel coupling} \end{gathered}$$

Symmetry can force the numerator to vanish, so the gap alone is not sufficient. Conversely, a modest coupling integrated over a long trajectory or amplified by resonance can still produce substantial population transfer.

What nonadiabatic corrections do

Beyond a scalar DBOC, nonadiabatic theory can generate:

• transitions between electronic surfaces;
• geometry-dependent vibrational mass corrections;
• rotational mass corrections;
• level shifts and predissociation widths;
• intensity borrowing;
• effective vector potentials and geometric phases.

No single number called “the Born–Oppenheimer error” describes all these effects.

Approximation Hierarchy

Level	Molecular content retained
Clamped-nuclei electronic problem	$E_a(R)$ and $\phi_a(r;R)$ at specified geometries
Leading one-surface Born–Oppenheimer	Quantum nuclear motion on one bare surface
Adiabatic one-surface	Adds diagonal scalar and, when needed, geometric connection terms
Nonadiabatic perturbation theory	Adds effective masses, scalar corrections, and controlled channel effects
Coupled-surface dynamics	Retains several electronic channels and their derivative couplings
All-particle non-BO calculation	Solves electrons and nuclei together after exact translation removal

The appropriate level depends on the observable. A surface may predict an equilibrium geometry well while missing a weak transition intensity, an isotope shift, or a predissociation lifetime.

Where One Surface Works Well

The one-surface picture is usually strongest when:

• the nuclear wavefunction is localized near a smooth minimum;
• the electronic state is nondegenerate and separated by a robust gap;
• nuclear kinetic energies are small compared with relevant electronic gaps;
• the target observable is not unusually sensitive to a small coupling;
• dissociation and reaction paths do not encounter near-degeneracies.

This regime underlies much of ground-state molecular structure, thermochemistry, rotational spectroscopy, and vibrational spectroscopy.

Precision hydrogen as a layered success

Molecular hydrogen illustrates both the success and the limits of the hierarchy. A highly accurate Born–Oppenheimer surface provides the leading energy. Adiabatic, nonadiabatic, relativistic, radiative, and finite-size corrections are then added with a consistent Hamiltonian and error estimate. Agreement with precision spectra tests the entire layered calculation, not the bare surface alone.

Water and polyatomic spectroscopy

High-accuracy water spectra likewise require more than an electronic minimum: a global surface, variational nuclear motion, an accurate dipole-moment surface, and small adiabatic, nonadiabatic, relativistic, and radiative corrections. The Born–Oppenheimer organization remains indispensable even when empirical refinement is used to reach the final spectroscopic accuracy.

Where One Surface Fails

Conical intersections

In the spin-free real-Hamiltonian setting, two electronic surfaces of the same spin can form a conical-intersection seam. Derivative couplings become singular in an adiabatic basis, and a wavepacket reaching the seam can undergo strong electronic transfer. The dynamics depends on its path and symmetry; a geometric phase can also alter interference.

Avoided crossings

States of the same symmetry generally repel along a one-dimensional coordinate. If the nuclear wavepacket traverses a narrow avoided crossing, population transfer can be appreciable even for heavy nuclei.

Dissociation and rearrangement

At large separation, electronic gaps can close or several fragment arrangements can compete. A single state label followed from the equilibrium geometry may not correlate smoothly with the physically relevant asymptotic channel.

Light-particle transfer

Hydrogen transfer, proton-coupled electron transfer, and systems containing muons or positrons can weaken a naive heavy–light hierarchy. Tunneling may also sample geometries far from the minimum where gaps are smaller.

Excited-state photochemistry

Photoexcited wavepackets often reach intersections on femtosecond time scales. Internal conversion and intersystem crossing then require several electronic states and, for the latter, spin–orbit coupling beyond the spin-free Hamiltonian.

Near-threshold and resonance observables

A tiny energy correction can strongly shift a weakly bound level, scattering length, or predissociation width. “Small compared with an electronic energy” is not the right accuracy test for a near-threshold observable.

Nuclear Permutation Symmetry

Identical nuclei remain identical after electronic separation. Suppose $P_{AB}$ exchanges two identical nuclei. The electronic factor can transform nontrivially:

$$P_{AB}\phi_a(r;R) = \lambda_a \phi_a(r;R),$$

with $\lambda_a=\pm1$ in a one-dimensional symmetry sector. The nuclear spatial and spin factors must compensate so that the total state has the bosonic or fermionic symmetry required by the isotope.

Combining these exchange symmetries produces ortho and para species and restricts allowed rotational levels. Treating nuclei as permanently labeled classical centers erases those restrictions and can give the wrong statistical weights or selection rules.

Which electronic data a spectrum needs

Energies determine the nuclear potential, gradients locate equilibrium, and Hessians give harmonic frequencies. Intensities additionally need dipole or higher-multipole surfaces; lifetimes and nonadiabatic dynamics may require multiple surfaces and coupling vectors. An accurate electronic energy at one geometry therefore does not determine a complete molecular spectrum.

Exercises

Exercise 1: Change the nuclei, retain the surface

In the H₂⁺ model above, approximate deuterons by $M_D=2M_p$ and keep the electronic surface fixed. Predict the vibrational spacing, rotational constant, ground-state width, and increase in $D_0^{\mathrm{harm}}$ for D₂⁺. Does a measured vibrational isotope ratio different from $1/\sqrt2$ alone demonstrate failure of a one-surface Born–Oppenheimer description?

Solution

Doubling both nuclear masses doubles the reduced mass while leaving $R_e$, $k_g$, and $D_e$ unchanged in this model. Hence $\omega_e$ scales by $2^{-1/2}$, $B_e$ by $2^{-1}$, and the root-mean-square width by $2^{-1/4}$:

$$\begin{aligned} \hbar\omega_e(\mathrm D_2^+)&\simeq0.005850\,E_{\mathrm h},\\ B_e(\mathrm D_2^+)&\simeq4.382\times10^{-5}\,E_{\mathrm h},\\ \sqrt{\langle(R-R_e)^2\rangle_0}(\mathrm D_2^+)&\simeq0.216\,a_0. \end{aligned}$$

The smaller zero-point energy increases the harmonic dissociation energy by

$$D_0^{\mathrm{harm}}(\mathrm D_2^+)-D_0^{\mathrm{harm}}(\mathrm H_2^+) =\frac{\hbar\omega_e(\mathrm H_2^+)}{2} \left(1-\frac1{\sqrt2}\right) \simeq0.001212\,E_{\mathrm h}.$$

No change in electronic well depth was needed. A different observed isotope ratio is not by itself evidence against one surface: the actual deuteron mass is not exactly twice the proton mass, and anharmonic nuclear levels on the same surface already deviate from the harmonic scaling. Those effects must be included before attributing a residual to adiabatic or nonadiabatic corrections.

Exercise 2: Rotation–vibration coupling

Explain why replacing $R$ by $R_e$ in the centrifugal term suppresses rotation–vibration coupling.

Solution

The exact one-surface centrifugal contribution is

$$\frac{\hbar^2J(J+1)} {2\mu_NR^2}.$$

Because $R$ is an operator, its expectation value depends on the vibrational state. Replacing it by the constant $R_e$ gives

$$B_eJ(J+1),$$

which is independent of vibration. The rigid-rotor approximation therefore removes the variation of rotational energy across the vibrational wavefunction and misses centrifugal distortion.

Exercise 3: Gauge-invariant diagonal correction

Show that

$$\left\langle \nabla\phi \left| 1-\lvert\phi\rangle\langle\phi\rvert \right| \nabla\phi \right\rangle$$

is invariant under $\phi\to e^{i\theta(R)}\phi$.

Solution

The derivative transforms as

$$\nabla\phi' = e^{i\theta} \left( \nabla\phi +i(\nabla\theta)\phi \right).$$

The projector

$$Q=1-\lvert\phi\rangle\langle\phi\rvert$$

annihilates $\phi$. Hence

$$Q\nabla\phi' = e^{i\theta}Q\nabla\phi.$$

Taking the norm removes the phase, so the projected derivative norm and $\Phi_a(R)$ are gauge invariant.

Exercise 4: Vertical excitation

Starting from the Born–Oppenheimer transition amplitude, state the additional assumption needed to reduce intensity to a Franck–Condon factor.

Solution

The amplitude is

$$\mathcal M = \int dR\, \chi_{b\nu'}^*(R) \boldsymbol\mu_{ba}(R) \chi_{a\nu}(R).$$

The Condon approximation assumes

$$\boldsymbol\mu_{ba}(R) \approx \boldsymbol\mu_{ba}(R_0)$$

over the nuclear region carrying appreciable amplitude. The constant electronic factor then leaves the overlap

$$\int dR\, \chi_{b\nu'}^*(R) \chi_{a\nu}(R),$$

whose squared magnitude is the Franck–Condon factor. Rapid geometry dependence of the transition moment produces non-Condon corrections.

Exercise 5: Diagnose an avoided crossing

A wavepacket approaches a region where $E_b-E_a$ decreases by two orders of magnitude while $\langle\phi_a|\nabla_RH_e|\phi_b\rangle$ remains comparable. What happens to the derivative-coupling estimate?

Solution

For nondegenerate states,

$$d_{ab} \sim \frac{ \langle\phi_a| \nabla_RH_e |\phi_b\rangle }{ E_b-E_a }.$$

If the numerator is unchanged and the gap decreases by a factor of $100$, the estimated coupling grows by a factor of $100$. A one-surface treatment becomes less credible. One should inspect the wavepacket velocity and symmetry, then use coupled surfaces or a suitable diabatic representation if transfer is appreciable.

Exercise 6: Identical nuclei

Two identical spin-$1/2$ nuclei are exchanged. If the electronic-vibrational-rotational spatial factor is symmetric, what symmetry must their nuclear-spin factor have?

Solution

Spin-$1/2$ nuclei are fermions, so the complete molecular state must be antisymmetric under their exchange. If the electronic-vibrational-rotational spatial factor contributes $+1$, the nuclear-spin factor must contribute $-1$. It must therefore lie in the antisymmetric nuclear-spin sector. For two spin-$1/2$ particles, that sector is the singlet.

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