Common Molecular Hamiltonians

A molecular Hamiltonian can mean at least four different operators: the all-particle Coulomb Hamiltonian, a clamped-nuclei electronic Hamiltonian, a nuclear-motion Hamiltonian on one or more potential-energy surfaces, or a finite effective Hamiltonian fitted to selected spectroscopic levels. Those operators are connected, but they do not act on the same Hilbert space and their parameters do not have the same status.

This page is a comparison and diagnostic reference. It gives compact forms, declares the assumptions behind them, and routes each derivation to its canonical page. Its central question is not merely “which equation looks familiar?” but:

Which degrees of freedom, frame, surface convention, angular momenta, and energy units does this Hamiltonian actually describe?

Canonical Scope

Use this entry to:

1. distinguish a laboratory-frame molecular Hamiltonian from an internal Hamiltonian;
2. translate between the two common Born–Oppenheimer surface conventions;
3. identify the assumptions behind vibrational and rigid-rotor models;
4. recognize how rotation–vibration coupling enters before a spectroscopic expansion is fitted;
5. separate electronic spin–rotation from nuclear spin–rotation;
6. identify the leading molecular hyperfine operators;
7. decide whether a weak interaction may be treated perturbatively or must be included in a coupled diagonalization;
8. compare constants only after matching units and operator normalization.

Molecular Hamiltonian owns the term-by-term Coulomb derivation, center-of-mass separation, recoil, and exact symmetries. Born–Oppenheimer in Molecules owns the electronic-channel expansion, derivative couplings, adiabatic corrections, and failure modes. The vibration, rotation, and rovibrational pages own the spectra and detailed reductions of their respective models. This reference keeps only enough of each operator to make conventions and model boundaries explicit.

Hamiltonian Declaration

Before interpreting a molecular Hamiltonian, record this ledger.

Item	Declaration required
system	molecular formula, charge, isotopologue, electronic state, and environment
particles	electrons and nuclei treated explicitly, effective cores, or selected internal levels
frame	laboratory, center-of-mass, body-fixed, rotating, or field-defined
coordinates	Cartesian, Jacobi, bond-angle, normal, or other internal coordinates
electronic convention	whether internuclear repulsion is inside the electronic eigenvalue
surface set	one adiabatic surface, several coupled surfaces, or a diabatic model
nuclear model	exact internal kinetic operator, harmonic modes, rotor, or fitted effective form
angular momenta	definitions of $\mathbf R$, $\mathbf N$, $\mathbf J$, $\mathbf I$, and $\mathbf F$
units	energy, hertz, radians per second, wavenumber, or Hartree atomic units
constants	equilibrium, vibrationally averaged, calculated, fitted, or isotope-scaled
fields	electric and magnetic amplitudes, directions, time dependence, and gauge
model space	basis states retained and states eliminated
fit domain	isotopologue, vibrational manifold, quantum-number range, and data set
approximation order	omitted couplings and highest retained distortion or perturbative order

The ledger is part of the model. A number such as $B$, $\gamma$, or $eQq$ cannot be transported safely between papers until its Hamiltonian, normalization, sign convention, vibrational state, and units have been matched.

Reduction Map

A common hierarchy is

$$\begin{aligned} H_{\mathrm{lab}} &= H_{\mathrm{cm}}+H_{\mathrm{int}}, \\ H_{\mathrm{int}} &\longrightarrow \left\{ H_e(R),\, T_N,\, \tau_{ab} \right\}, \\ &\longrightarrow H_{\mathrm{nuc}}^{(a)} \quad\text{or}\quad H_{\mathrm{coupled}}, \\ &\longrightarrow H_{\mathrm{vib}}, \ H_{\mathrm{rot}}, \ H_{\mathrm{rv}}, \\ &\longrightarrow H_{\mathrm{eff}} = H_{\mathrm{rv}} +H_{\mathrm{fine}} +H_{\mathrm{hfs}} +H_{\mathrm{field}}. \end{aligned}$$

Each arrow changes the description.

Layer	Hilbert space	Typical output	Main danger
all-particle Coulomb	electron and nuclear coordinates	exact nonrelativistic internal states	confusing laboratory and internal motion
clamped nuclei	electronic coordinates at fixed geometry	electronic energies and states	inconsistent treatment of $V_{NN}$
surface nuclear motion	nuclear coordinates on one or more surfaces	vibrational, rotational, tunneling, and reactive states	silently neglecting nonadiabatic coupling
rotor or oscillator	reduced nuclear coordinates	approximate ladders and labels	treating effective structure as exact
spectroscopic effective	selected rovibronic and spin basis	fitted line positions and intensities	extrapolation and double counting

The molecular chapter overview contains the broader physical hierarchy and its documentary figure.

Nonrelativistic Coulomb Hamiltonian

Let $N_e$ electrons have coordinates $\mathbf r_i$, and let $N_N$ nuclei have coordinates $\mathbf R_A$, charges $Z_Ae$, and masses $M_A$. In SI units, the spin-free point-particle Hamiltonian is

$$\begin{aligned} H_{\mathrm{lab}} ={}& -\sum_i \frac{\hbar^2}{2m_e} \nabla_{\mathbf r_i}^2 -\sum_A \frac{\hbar^2}{2M_A} \nabla_{\mathbf R_A}^2 \\ &+ \frac{e^2}{4\pi\epsilon_0} \left[ \sum_{i<j}\frac{1}{r_{ij}} +\sum_{A<B}\frac{Z_AZ_B}{R_{AB}} -\sum_{i,A}\frac{Z_A}{r_{iA}} \right]. \end{aligned}$$

Every coordinate is dynamical. In particular, the internuclear term is an operator on nuclear coordinates; it becomes a geometry-dependent scalar only in a clamped-nuclei electronic problem.

In Hartree atomic units,

$$\begin{aligned} H_{\mathrm{lab}} ={}& -\frac12\sum_i\nabla_{\mathbf r_i}^2 -\sum_A \frac{1}{2M_A} \nabla_{\mathbf R_A}^2 \\ &+ \sum_{i<j}\frac{1}{r_{ij}} +\sum_{A<B}\frac{Z_AZ_B}{R_{AB}} -\sum_{i,A}\frac{Z_A}{r_{iA}}, \end{aligned}$$

where every $M_A$ is expressed in electron-mass units. Atomic Units gives the unit dictionary and dimensional-restoration checks.

What this baseline omits

The displayed operator omits:

• relativistic and radiative corrections;
• finite nuclear size and nuclear internal excitation;
• external electric and magnetic fields;
• coupling to quantized radiation;
• collisions, solvents, substrates, and thermal reservoirs;
• particle creation and chemical environments not represented by the declared particles.

Electron and nuclear spins still constrain exchange symmetry even though the spin-free operator acts trivially on spin coordinates.

Remove overall translation

For an isolated molecule with no external field,

$$H_{\mathrm{lab}} = \frac{\mathbf P_{\mathrm{cm}}^2}{2M_{\mathrm{tot}}} +H_{\mathrm{int}},$$

with

$$M_{\mathrm{tot}} = N_em_e+\sum_A M_A.$$

The internal Hamiltonian contains reduced masses and, depending on the coordinate choice, mass-polarization or other kinetic cross terms. Deleting a center-of-mass coordinate from the printed laboratory equation does not by itself derive $H_{\mathrm{int}}$.

This separation is exact for a translationally invariant isolated Hamiltonian. Homogeneous magnetic fields require more careful pseudomomentum and gauge bookkeeping, especially for a charged molecule.

Born–Oppenheimer Electronic Hamiltonian

Two surface conventions

At fixed nuclear geometry $R$, define the electronic-only operator in atomic units by

$$\begin{aligned} H_{\mathrm{el}}(R) ={}& -\frac12\sum_i\nabla_i^2 -\sum_{i,A}\frac{Z_A}{r_{iA}} \\ &+ \sum_{i<j}\frac{1}{r_{ij}}. \end{aligned}$$

Solve

$$H_{\mathrm{el}}(R) \phi_a(r;R) = \mathcal E_a(R) \phi_a(r;R).$$

The corresponding potential-energy surface is

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

where

$$V_{NN}(R) = \sum_{A<B}\frac{Z_AZ_B}{R_{AB}}.$$

Another common convention includes $V_{NN}$ directly:

$$H_e(R) = H_{\mathrm{el}}(R)+V_{NN}(R), \qquad H_e(R)\phi_a = U_a(R)\phi_a.$$

Both conventions are correct. The error is to diagonalize the second operator and then add $V_{NN}$ again, or to diagonalize the first and forget it.

What is fixed and what remains quantum

During the electronic eigenproblem, $R$ is a parameter. The nuclei have not been declared classical forever; their quantum kinetic energy is restored in the nuclear equation. “Clamped nuclei” names an auxiliary problem, not a statement that a molecular eigenstate contains motionless nuclei.

At a fixed geometry, point-group labels and an electronic spin multiplicity may classify $\phi_a$. Over nuclear configuration space, labels can change, surfaces can approach one another, and a single adiabatic state may not provide a globally smooth basis.

Electronic Channels and Nuclear Motion

A complete geometry-dependent electronic basis gives

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

This expansion is exact within the electronic basis. Acting with nuclear kinetic energy generates the coupled equations

$$\sum_b \left[ \left( T_N+U_a \right)\delta_{ab} +\tau_{ab} \right] \chi_b = E\chi_a,$$

where $\tau_{ab}$ contains diagonal and off-diagonal derivative couplings. In simple Cartesian notation,

$$\begin{aligned} \tau_{ab} = -\sum_A \frac{\hbar^2}{2M_A} \Big[ 2\mathbf d_{ab,A}\mathbin{\cdot}\nabla_A +D_{ab,A} \Big], \end{aligned}$$

with

$$\mathbf d_{ab,A} = \langle\phi_a\vert\nabla_A\phi_b\rangle_e, \qquad D_{ab,A} = \langle\phi_a\vert\nabla_A^2\phi_b\rangle_e.$$

The one-surface leading Hamiltonian is

$$H_{\mathrm{nuc}}^{(a)} = T_N+U_a(R).$$

It is obtained by neglecting couplings to other electronic channels and the geometry dependence of $\phi_a$ inside $T_N$. Adiabatic diagonal corrections, geometric vector potentials, and nonadiabatic couplings can be restored systematically, but their form depends on representation and gauge.

Nonadiabatic Coupling is the canonical home for derivative couplings, adiabatic and diabatic representations, avoided crossings, and coupled nuclear dynamics.

One-surface validity

A useful local diagnostic for eliminating channel $b$ from a retained channel $a$ is

$$\epsilon_{ab} \sim \frac{ \left\lVert \tau_{ab} \right\rVert }{ \left| U_b-U_a \right| }.$$

Small $\epsilon_{ab}$ over the nuclear region actually sampled supports a one-surface treatment. A large electronic gap at the equilibrium geometry alone is not enough: dissociation paths, avoided crossings, conical intersections, and highly excited nuclear states can sample very different regions.

Nuclear Vibrational Hamiltonians

General internal-coordinate form

For internal coordinates $q^\mu$ with mass metric $G_{\mu\nu}(q)$, a coordinate-invariant kinetic operator has the schematic Laplace–Beltrami form

$$T_{\mathrm{nuc}} = -\frac{\hbar^2}{2} G^{-1/2} \partial_\mu \left[ G^{1/2} G^{\mu\nu} \partial_\nu \right],$$

where

$$G=\det G_{\mu\nu}.$$

The precise internal measure, removal of translation and rotation, body-fixed frame, and any ordering or pseudopotential terms must be derived for the selected coordinates. Replacing this operator by a Cartesian Laplacian in curvilinear coordinates is generally wrong.

Diatomic radial Hamiltonian

For a spin-free diatomic on one electronic surface, the reduced radial equation at rotational quantum number $N$ is

$$\begin{aligned} \Bigg[ &- \frac{\hbar^2}{2\mu} \frac{d^2}{dR^2} +U_a(R) \\ &+ \frac{\hbar^2N(N+1)} {2\mu R^2} \Bigg] u_{vN}(R) = E_{vN}u_{vN}(R), \end{aligned}$$

where

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

Setting $N=0$ gives the pure radial vibrational problem. Freezing $R=R_e$ gives the leading rigid rotor. The full equation already contains rotation–vibration coupling because the centrifugal term depends on the vibrational coordinate.

Vibrations of Diatomics owns anharmonic potentials, isotope scaling, Morse levels, and infrared transition moments.

Harmonic normal modes

Near a stable equilibrium, mass-weighted normal coordinates $Q_k$ reduce the quadratic vibrational Hamiltonian to

$$H_{\mathrm{vib}}^{(2)} = \frac12 \sum_{k=1}^{f} \left( P_k^2+\omega_k^2Q_k^2 \right),$$

with energies

$$E_{\{n_k\}} = \sum_{k=1}^{f} \hbar\omega_k \left( n_k+\frac12 \right).$$

For $N_N$ nuclei,

$$f = \begin{cases} 3N_N-5, & \text{linear molecule},\\ 3N_N-6, & \text{nonlinear molecule}. \end{cases}$$

The normal-mode basis is a local quadratic approximation. Anharmonicity, large-amplitude motion, resonances, tunneling, and vibronic coupling require a larger nuclear model. Normal Modes of Polyatomics owns the mass-weighted Hessian, symmetry classification, and activity criteria.

Rigid Rotor Hamiltonians

General top

At a fixed reference geometry, let $\mathbf I$ be the inertia tensor about the center of mass. Neglecting shape motion gives

$$H_{\mathrm{rot}} = \frac12 \mathbf J_{\mathrm{rot}}^{\mathsf T} \mathbf I^{-1} \mathbf J_{\mathrm{rot}}.$$

In principal axes,

$$H_{\mathrm{rot}} = \frac{J_a^2}{2I_a} +\frac{J_b^2}{2I_b} +\frac{J_c^2}{2I_c}.$$

If $A$, $B$, and $C$ are frequency constants,

$$\frac{H_{\mathrm{rot}}}{h} = A\frac{J_a^2}{\hbar^2} +B\frac{J_b^2}{\hbar^2} +C\frac{J_c^2}{\hbar^2},$$

where

$$A=\frac{h}{8\pi^2I_a}, \qquad B=\frac{h}{8\pi^2I_b}, \qquad C=\frac{h}{8\pi^2I_c}.$$

Rotational constants in energy or wavenumber units differ by factors of $h$ or $hc$.

Rotor classes

Model	Moment relation	Spectral consequence
linear rotor	$I_a=0$, $I_b=I_c$	one scalar constant and a $J(J+1)$ ladder
spherical top	$I_a=I_b=I_c$	orientation has extra degeneracy
prolate symmetric top	$I_a<I_b=I_c$	exact body-axis projection $K$
oblate symmetric top	$I_a=I_b<I_c$	exact symmetry-axis projection $K$
asymmetric top	$I_a<I_b<I_c$	no exact $K$ and matrix diagonalization for each $J$

For a closed-shell linear rotor,

$$\frac{H_{\mathrm{rot}}}{h} = B\frac{\mathbf J^2}{\hbar^2}, \qquad \frac{E_J}{h} = BJ(J+1).$$

For open-shell molecules, $J$ usually includes electronic spin while $N$ does not, so the spin-free rotational term is often written $B\mathbf N^2/\hbar^2$. Rotations of Molecules develops the conventions, rotor classes, centrifugal distortion, and spectroscopic inference in depth.

Vibration–Rotation Hamiltonians

The coupling is kinematic

In a semirigid polyatomic molecule, the inverse inertia tensor changes with internal coordinates. In a molecule-fixed frame, a useful schematic form is

$$\begin{aligned} H_{\mathrm{rv}} ={}& H_{\mathrm{vib}} +\frac12 \sum_{\alpha,\beta} \mu_{\alpha\beta}(\mathbf q) \\ &\times \left( J_\alpha-\pi_\alpha \right) \left( J_\beta-\pi_\beta \right) +V_{\mathrm{ord}}(\mathbf q). \end{aligned}$$

Here $\mu_{\alpha\beta}$ is a coordinate-dependent inverse inertia tensor, $\boldsymbol\pi$ is vibrational angular momentum, and $V_{\mathrm{ord}}$ denotes convention-dependent ordering or pseudopotential terms. An Eckart frame suppresses avoidable local coupling near a reference geometry; it does not remove physical Coriolis coupling.

Diatomic effective form

For a simple closed-shell diatomic, a common wavenumber expansion is

$$\frac{E_{vJ}}{hc} = \sum_{k,l} Y_{kl} \left( v+\frac12 \right)^k \left[ J(J+1) \right]^l.$$

The leading truncated form is

$$\frac{E_{vJ}}{hc} \approx G(v) +\widetilde B_vX -\widetilde D_vX^2, \qquad X=J(J+1),$$

with

$$\widetilde B_v = \widetilde B_e -\widetilde\alpha_e \left( v+\frac12 \right) +\cdots.$$

Mixed Dunham coefficients and the $v$ dependence of $\widetilde B_v$ are rotation–vibration coupling, not evidence that the underlying kinetic energy has literally separated.

Effective Hamiltonians are representation dependent

For semirigid asymmetric tops, Watson $A$ and $S$ reductions remove redundant centrifugal-distortion parameters by different contact transformations. They predict the same spectrum to the retained order when used consistently, but their fitted constants need not agree term by term. Axis representations and sign conventions must also match.

Rovibrational Coupling owns Dunham organization, Coriolis effects, $P/Q/R$ branches, combination differences, branch heads, and fit diagnostics.

Angular-Momentum Dictionary

No molecular spin Hamiltonian is portable until its angular momenta are defined. A common diatomic convention is:

Symbol	Meaning
$\mathbf R$	end-over-end nuclear rotation
$\mathbf L$	total electronic orbital angular momentum
$\mathbf S$	total electronic spin
$\mathbf N$	total angular momentum excluding electronic spin and nuclear spin; often $\mathbf N=\mathbf J-\mathbf S$
$\mathbf J$	total angular momentum excluding nuclear spin
$\mathbf I_A$	spin of nucleus $A$
$\mathbf I$	coupled resultant of the nuclear spins
$\mathbf F$	total angular momentum including nuclear spin, usually $\mathbf F=\mathbf J+\mathbf I$

For a closed-shell $^1\Sigma$ state, $S=0$ and the simple rotational labels $N$ and $J$ coincide. In open-shell states, they do not. Some authors use $\mathbf N$ for bare rotation and others use it for all angular momentum except spin; the operator definitions, rather than the letter alone, are authoritative.

Fine and Spin–Rotation Terms

Electronic spin couplings

Within a selected electronic manifold, common effective fine-structure operators include

$$\begin{aligned} H_{\mathrm{fine}} ={}& A_{\mathrm{so}} \frac{L_zS_z}{\hbar^2} + \frac{ \mathbf S \mathbin{\boldsymbol\cdot} \mathbf D \mathbin{\boldsymbol\cdot} \mathbf S }{\hbar^2} \\ &+ \frac{ \mathbf N \mathbin{\boldsymbol\cdot} \boldsymbol\gamma \mathbin{\boldsymbol\cdot} \mathbf S }{\hbar^2} +\cdots. \end{aligned}$$

The terms represent, respectively:

• effective spin–orbit coupling in an axial basis;
• electron spin–spin or zero-field splitting;
• electronic spin–rotation coupling.

Under this displayed normalization, $A_{\mathrm{so}}$ and the components of $\mathbf D$ and $\boldsymbol\gamma$ carry energy units. If the constants are quoted in hertz, divide the entire Hamiltonian by $h$.

For an idealized $^3\Sigma$ diatomic with constants in frequency units, a frequently used leading model is

$$\begin{aligned} \frac{H_{\mathrm{eff}}}{h} ={}& B\frac{\mathbf N^2}{\hbar^2} +\frac{2\lambda}{3} \frac{ 3S_z^2-\mathbf S^2 }{\hbar^2} \\ &+ \gamma \frac{ \mathbf N\mathbin{\boldsymbol\cdot}\mathbf S }{\hbar^2}. \end{aligned}$$

This compact form is state-specific. Centrifugal corrections, vibrational dependence, spin–orbit mixing with other electronic states, and hyperfine terms may be required at higher resolution.

Electronic versus nuclear spin–rotation

The electronic term is

$$H_{\mathrm{esr}} = \frac{ \mathbf N \mathbin{\boldsymbol\cdot} \boldsymbol\gamma \mathbin{\boldsymbol\cdot} \mathbf S }{\hbar^2}.$$

The nuclear spin–rotation term is instead

$$H_{\mathrm{nsr}} = \sum_A \frac{ \mathbf I_A \mathbin{\boldsymbol\cdot} \mathbf C_A \mathbin{\boldsymbol\cdot} \mathbf N }{\hbar^2}.$$

The first can exist only when $S\ne0$. The second can split a closed-shell $^1\Sigma$ rotational level whenever a nucleus has $I_A\ne0$. Both are often called “spin–rotation,” so the spin operator must always be printed.

Molecular Hyperfine Hamiltonian

At leading electromagnetic multipole order, a schematic molecular hyperfine Hamiltonian is

$$\begin{aligned} H_{\mathrm{hfs}} ={}& \sum_A b_{F,A} \frac{ \mathbf I_A\mathbin{\boldsymbol\cdot}\mathbf S }{\hbar^2} \\ &+ \sum_A \frac{ \mathbf I_A \mathbin{\boldsymbol\cdot} \mathbf T_A \mathbin{\boldsymbol\cdot} \mathbf S }{\hbar^2} \\ &+ \sum_A T^{(2)}(Q_A) \mathbin{\boldsymbol\cdot} T^{(2)}(\nabla\mathbf E_A) \\ &+ \sum_A \frac{ \mathbf I_A \mathbin{\boldsymbol\cdot} \mathbf C_A \mathbin{\boldsymbol\cdot} \mathbf N }{\hbar^2} \\ &+ \sum_{A<B} \frac{ \mathbf I_A \mathbin{\boldsymbol\cdot} \mathbf D_{AB} \mathbin{\boldsymbol\cdot} \mathbf I_B }{\hbar^2} +\cdots. \end{aligned}$$

The displayed normalization is schematic: authors distribute factors of $h$, $\hbar$, nuclear moments, and tensor normalizations differently.

Interaction	Physical source	Required angular momentum
Fermi contact	electron spin density at a nucleus	$S\ne0$, $I_A\ne0$
electron–nuclear dipolar	anisotropic magnetic dipole coupling	$S\ne0$, $I_A\ne0$
nuclear electric quadrupole	nuclear quadrupole with molecular electric-field gradient	$I_A\ge1$
nuclear spin–rotation	nuclear magnetic moment with field generated by molecular rotation	$I_A\ne0$
nuclear spin–spin	magnetic coupling between nuclei	at least two nonzero nuclear spins

Nuclear spin–orbit and higher nuclear multipoles may also enter specialized models. The nuclear electric-quadrupole term is a rank-2 tensor interaction; the familiar scalar constant $eQq$ is obtained only after choosing molecular axes and a tensor convention.

State dependence and vibrational averaging

Hyperfine constants are generally functions of geometry. A constant reported for vibrational level $v$ represents a matrix element such as

$$b_{F,v} = \langle \chi_v \vert b_F(R) \vert \chi_v \rangle,$$

possibly after further electronic-state projection and effective transformations. It is not automatically the value at $R_e$.

The NIST $^1\Sigma$ diatomic compilation uses separate constants for nuclear quadrupole, nuclear spin–rotation, and nuclear spin–spin effects. The NIST $^3\Sigma$ compilation also displays the electron spin–spin and electron spin–rotation terms. These are useful evaluated conventions, not universal replacements for declaring the Hamiltonian.

External-Field Additions

For a neutral molecule in slowly varying classical fields, common leading terms are

$$\begin{aligned} H_{\mathrm{field}} ={}& -\boldsymbol\mu \mathbin{\boldsymbol\cdot} \mathbf E -\frac12 \mathbf E \mathbin{\boldsymbol\cdot} \boldsymbol\alpha \mathbin{\boldsymbol\cdot} \mathbf E \\ &- \boldsymbol\mu_{\mathrm{mag}} \mathbin{\boldsymbol\cdot} \mathbf B +\cdots. \end{aligned}$$

The molecular dipole and polarizability are body-fixed tensors, while the applied fields are usually specified in the laboratory frame. Direction cosines or spherical tensors connect the two. A static electric field mixes opposite-parity rotor states; a magnetic field can decouple angular momenta that were useful at zero field.

For charged molecules, center-of-mass and internal electric or magnetic couplings require an explicit gauge and coordinate derivation. A neutral internal dipole is origin independent, while an ionic dipole is not unless the center-of-mass convention is fixed.

Symmetries and Good Labels

Model	Exact or robust labels	Labels that are conditional
isolated spin-free Coulomb molecule	total spatial angular momentum, parity, permutation symmetry	equilibrium point group and body-fixed geometry
fixed-geometry electronic problem	geometry point-group irrep, electron exchange symmetry	energy ordering as a state label
one smooth surface	total nuclear angular momentum, parity, permutation–inversion symmetry	separate vibrational and rotational quanta
harmonic normal modes	mode occupation within the quadratic model	local-mode character outside the harmonic region
linear or symmetric rigid rotor	$J$ and an allowed projection label	vibrational state independence of constants
asymmetric rotor	$J$, $M$, parity, exact molecular symmetry	$K_a$ and $K_c$ as exact projections
field-free spin-resolved model	$F$ and parity when all retained terms are rotational scalars	intermediate couplings such as $N$, $J$, or individual $I_A$
molecule in a dc field	projection on the field axis under axial symmetry	zero-field parity and coupled angular momenta

Exact labels belong to the declared Hamiltonian, not to the molecule’s name. When an interaction is added, states with the same exact symmetry may mix and their approximate labels may exchange character.

Molecular Symmetry owns point groups, irreducible representations, permutation–inversion issues, and spectroscopic selection rules.

Perturb or Diagonalize

Consider two unperturbed levels coupled by $V$:

$$H = \begin{pmatrix} E_1 & V\\ V^* & E_2 \end{pmatrix}.$$

The mixing angle obeys

$$\tan 2\theta = \frac{2|V|}{E_1-E_2}.$$

The perturbative regime requires

$$2|V| \ll |E_1-E_2|.$$

Near a degeneracy, even a numerically small fine, hyperfine, Coriolis, or Stark matrix element can produce order-one state mixing. The coupled states must then be retained in the same block and diagonalized. A good workflow is:

1. block the basis by exact symmetry;
2. estimate coupling norms and nearest allowed energy gaps;
3. include all quasi-degenerate states in the retained model space;
4. transform both Hamiltonian and observables consistently if states are eliminated;
5. test convergence as the model space grows.

Fitted effective constants can absorb eliminated-state effects. Adding an explicit coupling to a Hamiltonian whose constants already encoded that coupling is double counting.

Unit Conventions

The same operator may be reported in four common forms.

Printed object	Eigenvalue unit	Evolution factor
$H$	joule, electronvolt, or Hartree	$\exp(-iEt/\hbar)$
$H/h$	hertz	$\exp(-i2\pi\nu t)$
$H/\hbar$	radians per second	$\exp(-i\omega t)$
$H/(hc)$	inverse length, usually $\mathrm{cm}^{-1}$	convert through $E=hc\widetilde\nu$

Thus

$$E = h\nu = \hbar\omega = hc\widetilde\nu, \qquad \omega=2\pi\nu.$$

For a moment of inertia $I$,

$$B_E = \frac{\hbar^2}{2I}, \qquad B_\nu = \frac{h}{8\pi^2I}, \qquad \widetilde B = \frac{h}{8\pi^2cI}.$$

Never insert a hertz-valued $B_\nu$ into an energy Hamiltonian without multiplying by $h$. Constants and Conversions gives the corresponding $\mathrm{Hz}$, $\mathrm{cm}^{-1}$, electronvolt, kelvin, and angular-frequency conversions.

Model Assembly Workflow

1. Name the observable. A dissociation energy, microwave interval, hyperfine component, Stark shift, and reaction rate need different model spaces.
2. Declare the particles and isotope. Nuclear masses and spins are part of the Hamiltonian.
3. Remove or retain center-of-mass motion deliberately.
4. Choose the electronic manifold. State whether $V_{NN}$ is included in the electronic eigenvalue.
5. Choose one surface or a coupled-surface representation.
6. Choose the nuclear coordinates and kinetic operator.
7. Define every angular momentum before adding spin terms.
8. Add corrections by expected scale. Include relativistic, fine, hyperfine, field, and environmental terms only as the uncertainty target requires.
9. Match units and parameter conventions.
10. Block by exact symmetry and diagonalize each block.
11. Converge the basis and perturbative order.
12. Validate against limiting cases, withheld transitions, isotopologues, and independent observables.

For a fitted Hamiltonian, archive the line list, assignments, weights, excluded data, parameter covariance, software version, and residuals. A small root-mean-square residual does not prove that the model extrapolates.

Common Mistakes

• Calling the laboratory Coulomb operator an internal Hamiltonian after merely suppressing the center-of-mass coordinate.
• Treating $V_{NN}$ as both part of the electronic eigenvalue and an additional surface term.
• Saying that Born–Oppenheimer theory makes nuclei classical or motionless.
• Assuming a single surface is valid because the equilibrium electronic gap is large.
• Using a Cartesian Laplacian in curvilinear internal coordinates without the metric and measure.
• Calling the diatomic radial equation a sum of independent vibration and rotation even though its centrifugal term depends on $R$.
• Comparing $B$, $D$, $\gamma$, or $eQq$ without matching $H$, $H/h$, or $H/(hc)$ conventions.
• Using $J$ and $N$ interchangeably in an open-shell molecule.
• Confusing electronic spin–rotation $\mathbf N\mathbin{\boldsymbol\cdot}\mathbf S$ with nuclear spin–rotation $\mathbf I\mathbin{\boldsymbol\cdot}\mathbf N$.
• Calling every small spin splitting “hyperfine”; electron spin–orbit, electron spin–spin, and electronic spin–rotation are fine structure.
• Treating equilibrium constants as identical to vibrationally averaged constants.
• Comparing Watson reduction parameters term by term across different reductions.
• Adding explicit interactions to fitted constants that already absorb them.
• Extrapolating an effective Hamiltonian beyond its fitted vibrational, rotational, isotopic, or field domain.
• Treating point-group labels of one geometry as exact labels of a freely rotating, vibrating molecule.

Worked Diagnostics

Closed-shell polar diatomic

For a $^1\Sigma$ molecule with one quadrupolar nucleus:

• $S=0$, so electronic spin–orbit, electron spin–spin, electronic spin–rotation, Fermi-contact, and electron–nuclear dipolar terms vanish in the isolated-state model;
• rotation, nuclear electric quadrupole, and nuclear spin–rotation remain;
• a static electric field can mix opposite-parity rotational states through $-\boldsymbol\mu\mathbin{\boldsymbol\cdot}\mathbf E$;
• at zero field, coupling $\mathbf F=\mathbf J+\mathbf I$ is useful when the retained hyperfine Hamiltonian is rotationally invariant.

“Closed shell” therefore does not mean “no hyperfine structure.”

Isotopic substitution

At leading Born–Oppenheimer order, replacing one isotope changes nuclear masses but not the electronic surface $U_a(R)$. Consequently,

$$\omega_e\propto\mu^{-1/2}, \qquad B_e\propto\mu^{-1}$$

for a diatomic with approximately unchanged equilibrium distance. Deviations at precision level can reveal adiabatic, nonadiabatic, finite-size, and hyperfine effects. An isotope shift is not evidence that the electronic bond potential changed unless those corrections have been separated.

Exercises

Exercise 1: Audit the surface convention

A program reports the eigenvalue of

$$H_e = T_e+V_{eN}+V_{ee}+V_{NN}$$

as $-75.0\,E_h$ at one geometry. A script then adds $V_{NN}=9.0\,E_h$ to obtain the potential. Identify the error and the reported value that should enter the leading nuclear Hamiltonian.

Solution

Because $V_{NN}$ is already inside $H_e$, its eigenvalue is already the surface value in this convention. Adding $9.0\,E_h$ again double counts internuclear repulsion. The leading nuclear Hamiltonian should use

$$U(R)=-75.0\,E_h.$$

If the electronic operator had excluded $V_{NN}$, then adding it once would have been required.

Exercise 2: Expose rotation–vibration coupling

Expand the diatomic centrifugal factor $1/R^2$ through second order in $q=R-R_e$. Identify the terms that couple vibration to rotation.

Solution

Using $(1+x)^{-2}=1-2x+3x^2+\cdots$,

$$\frac{1}{R^2} = \frac{1}{R_e^2} \left[ 1 -2\frac{q}{R_e} +3\frac{q^2}{R_e^2} +\cdots \right].$$

Therefore,

$$\frac{\hbar^2J(J+1)}{2\mu R^2} = B_eJ(J+1) +\left[ -2B_e\frac{q}{R_e} +3B_e\frac{q^2}{R_e^2} +\cdots \right] J(J+1).$$

The constant term is the equilibrium rigid rotor. The terms proportional to $qJ(J+1)$, $q^2J(J+1)$, and higher powers are explicit rotation–vibration couplings.

Exercise 3: Rotational units

A paper gives a linear-molecule rotational constant $B_\nu=10.0\,\mathrm{GHz}$. What are the $J=0\to1$ transition frequency and the energy spacing?

Solution

For

$$\frac{E_J}{h} = B_\nu J(J+1),$$

the transition is

$$\nu_{1\leftarrow0} = B_\nu[1(2)-0] = 20.0\,\mathrm{GHz}.$$

The energy spacing is

$$\Delta E = h(20.0\,\mathrm{GHz}).$$

It is not $10.0\,\mathrm{GHz}$, and the hertz value is not itself an energy.

Exercise 4: Leading isotope scaling

Two isotopologues have reduced masses $\mu'$ and $\mu''=4\mu'$. Assuming the same Born–Oppenheimer surface and equilibrium geometry, find $\omega_e''/\omega_e'$ and $B_e''/B_e'$.

Solution

The harmonic frequency scales as $\mu^{-1/2}$, while the rotational constant scales as $\mu^{-1}$:

$$\frac{\omega_e''}{\omega_e'} = \left( \frac{\mu'}{\mu''} \right)^{1/2} = \frac12,$$

and

$$\frac{B_e''}{B_e'} = \frac{\mu'}{\mu''} = \frac14.$$

These are leading mass laws, not exact high-precision isotope identities.

Exercise 5: Which spin–rotation term survives?

Consider a closed-shell $^1\Sigma$ diatomic with one nucleus of spin $I=1$. Which of

$$\gamma\, \mathbf N\mathbin{\boldsymbol\cdot}\mathbf S$$

and

$$C_I\, \mathbf I\mathbin{\boldsymbol\cdot}\mathbf N$$

can split a rotational level?

Solution

For a $^1\Sigma$ state, $S=0$. The electronic spin–rotation term therefore vanishes. Because $I=1$, the nuclear spin–rotation term can remain and split states with different coupling of $\mathbf I$ and $\mathbf N$. The same nucleus may also possess an electric quadrupole moment, so quadrupole hyperfine structure can be larger than the nuclear spin–rotation splitting.

Exercise 6: Hyperfine angular momentum

For a closed-shell rotational level with $J=2$ and one nucleus with $I=1$, list the allowed $F$ values. Which label is exact for a field-free rotationally invariant hyperfine Hamiltonian?

Solution

Angular-momentum addition gives

$$F = |J-I|, |J-I|+1, \ldots, J+I,$$

so

$$F=1,2,3.$$

For a field-free scalar Hamiltonian that includes the hyperfine coupling, $F$ is exact. $J$ and $I$ can remain useful coupling labels, but interactions may mix basis states sharing the same exact $F$ and parity.

Exercise 7: Perturb or diagonalize

Two same-symmetry rovibronic levels are separated by $\Delta/h=4\,\mathrm{MHz}$ and coupled by $V/h=3\,\mathrm{MHz}$. Is nondegenerate perturbation theory reliable?

Solution

The mixing criterion is

$$\frac{2|V|}{|\Delta|} = \frac{6}{4} = 1.5.$$

This is not small. The mixing angle satisfies

$$\tan 2\theta=1.5,$$

so both levels belong in the retained model space and the $2\times2$ block should be diagonalized. Calling the coupling “only a few megahertz” is irrelevant without comparing it with the allowed-state gap.

Exercise 8: Detect double counting

A spectroscopic fit uses effective constants $B_v$, $D_v$, and $\gamma_v$ obtained from measured levels. A later model keeps those fitted constants and also adds an explicit electronic state whose elimination was the dominant source of $\gamma_v$. What must be checked?

Solution

The fitted $\gamma_v$ may already contain the second-order effect of the eliminated electronic state. Restoring that state explicitly while retaining the full fitted constant can count the same coupling twice. One must derive or refit the effective parameters in the enlarged model space, subtract the contribution represented explicitly, and transform observables consistently. Agreement of one fitted interval does not establish that the partition is sound.

Key Takeaways

• The all-particle, electronic, nuclear-motion, and spectroscopic effective Hamiltonians act on different spaces.
• Internuclear repulsion may be inside or outside the electronic eigenvalue; the convention must be declared.
• The Born–Oppenheimer approximation begins with truncation or neglect of electronic-channel couplings, not with the fixed-geometry eigenproblem itself.
• Rigid rotors and harmonic vibrations are controlled local reductions of a joint nuclear Hamiltonian.
• Rotation–vibration coupling is already present through coordinate-dependent inertia and centrifugal terms.
• Electronic spin–rotation and nuclear spin–rotation use different spin operators and belong to different structural layers.
• Hyperfine constants are state- and convention-dependent effective matrix elements.
• Small interactions require coupled diagonalization near degeneracy.
• Constants can be compared only after matching Hamiltonian, units, normalization, isotope, and fit domain.

Cross-Links

• Term Symbol Reference translates the angular-momentum and symmetry labels attached to these Hamiltonians.
• Molecular Hamiltonian for the exact Coulomb starting point, center-of-mass separation, and recoil.
• Born–Oppenheimer in Molecules for channel expansions, adiabatic corrections, and validity.
• Potential Energy Surfaces for geometry space, stationary points, crossings, and validation.
• Nonadiabatic Coupling for derivative couplings and coupled-surface dynamics.
• Vibrations of Diatomics and Normal Modes of Polyatomics for nuclear vibration.
• Rotations of Molecules and Rovibrational Coupling for rotor and coupled term values.
• Selection Rule Tables for transition diagnostics after the Hamiltonian and operator are fixed.
• Common Atomic Hamiltonians for the corresponding atomic operator ledger.

References

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