The Hellmann-Feynman Theorem: Forces from a Fixed Wavefunction

What the theorem actually claims

The Hellmann-Feynman theorem concerns a quantum system whose Hamiltonian H(λ) depends on some continuous parameter λ. That parameter can be almost anything physical: a nuclear coordinate, an applied electric or magnetic field, a coupling constant, a particle mass, or even the dimensionality of space.

• H(λ) — the parameter-dependent energy operator.
• |ψ(λ)⟩ — a normalized eigenstate, so ⟨ψ|ψ⟩ = 1.
• E(λ) — its eigenvalue, satisfying H(λ)|ψ⟩ = E(λ)|ψ⟩.

The theorem states simply that dE/dλ = ⟨ψ | ∂H/∂λ | ψ⟩. In words: the rate of change of the energy with the parameter equals the expectation value of the rate of change of the Hamiltonian, evaluated in the unperturbed eigenstate. You do not need to know how ψ shifts — the fixed wavefunction is enough. When λ is a nuclear position R, ∂H/∂R is just the gradient of the Coulomb potential, and −dE/dR is the physical force pushing the atoms toward their equilibrium geometry.

The derivation: why the wavefunction terms cancel

The magic is a one-line cancellation. Start from the eigenvalue equation and differentiate the energy expectation value E = ⟨ψ|H|ψ⟩ with respect to λ using the product rule:

dE/dλ = ⟨∂ψ/∂λ | H | ψ⟩ + ⟨ψ | ∂H/∂λ | ψ⟩ + ⟨ψ | H | ∂ψ/∂λ⟩.

Now use that |ψ⟩ is an eigenstate: H acting on |ψ⟩ (or ⟨ψ| on the left, since H is Hermitian) gives the number E. The first and third terms become E(⟨∂ψ/∂λ|ψ⟩ + ⟨ψ|∂ψ/∂λ⟩). But that parenthesis is exactly the derivative of the normalization:

• ⟨∂ψ/∂λ|ψ⟩ + ⟨ψ|∂ψ/∂λ⟩ = ∂/∂λ ⟨ψ|ψ⟩ = ∂/∂λ (1) = 0.

So both wavefunction-derivative terms vanish, leaving only dE/dλ = ⟨ψ|∂H/∂λ|ψ⟩. The result is exact — no perturbation expansion, no small parameter. The single essential requirement is that |ψ⟩ be a genuine eigenstate (or, more generally, a stationary point of the energy functional), which is why the theorem also holds for variational wavefunctions optimized to stationarity.

Feynman's electrostatic theorem and a worked number

In his 1939 paper Forces in Molecules, Feynman specialized λ to a nuclear coordinate and drew a striking conclusion: once the electron density n(r) is known, the force on a nucleus is purely classical electrostatics. There are no mysterious quantum or exchange forces acting on the nuclei themselves; the quantum mechanics is entirely contained in the shape of n(r).

The force on nucleus A of charge Z_A is:

• F_A = Z_A e² [ Σ_B (Z_B (R_A − R_B)/|R_A − R_B|³) − ∫ n(r) (R_A − r)/|R_A − r|³ d³r ],

the Coulomb repulsion from the other nuclei minus the attraction from the electron cloud. At equilibrium these cancel and F_A = 0. As a scale check: two protons in H₂ sit about 0.74 Å apart; the bare nuclear repulsion there is roughly e²/(4πε₀ d) ≈ 19 eV, exactly balanced by the electron density piled up in the bond. Displace an atom by 0.1 Å in a typical covalent solid and the restoring force is of order 1 eV/Å ≈ 1.6 nN — the regime probed by force-field and DFT calculations.

How it is used: forces in modern computation

The theorem is the computational engine behind almost all atomistic simulation. To relax a molecule or crystal to its lowest-energy geometry, or to run ab initio molecular dynamics, you need the force on every atom at every step — and finite-difference (nudge each atom, recompute the energy) would cost 3N energy evaluations for N atoms. Hellmann-Feynman gives all 3N force components from a single converged wavefunction as an expectation value, an enormous saving.

• Density functional theory (DFT) codes (VASP, Quantum ESPRESSO, Gaussian) compute analytic forces this way for systems from small molecules to 10⁴–10⁵ atoms.
• Car-Parrinello and Born-Oppenheimer MD integrate Newton's equations using these forces at ~1 fs timesteps.
• Phonons, elastic constants, and reaction paths all derive from the same force expression.

A crucial practical caveat: with atom-centered basis sets that move with the nuclei, extra Pulay forces appear because ∂ψ/∂λ ≠ 0 for an incomplete basis. Plane-wave codes avoid them (the basis doesn't move with the atoms), which is one reason plane-wave DFT is popular for solids.

Related theorems and where it breaks down

The Hellmann-Feynman theorem sits in a family of exact quantum relations, and knowing its neighbors sharpens what it does and doesn't say.

• The virial theorem relates average kinetic and potential energy (2⟨T⟩ = −⟨V⟩ for Coulomb systems) and can itself be derived by applying Hellmann-Feynman with λ a uniform scaling of coordinates.
• First-order perturbation theory gives ΔE = ⟨ψ⁰|H'|ψ⁰⟩ — the same expression, but as an approximation for a fixed perturbation, whereas Hellmann-Feynman is exact for a continuously varied parameter.
• The generalized/off-diagonal form connects ⟨ψ_m|∂H/∂λ|ψ_n⟩ to matrix elements, underpinning response theory.

The theorem fails when its assumptions fail: if |ψ⟩ is not a true eigenstate (an unconverged or non-variational trial function), if the energy is not stationary, at level crossings and degeneracies where dE/dλ is not single-valued, or when the domain of H itself depends on λ (moving boundaries). In these cases the neglected ⟨∂ψ/∂λ|H−E|ψ⟩ term no longer vanishes and must be added back.

History, significance, and open questions

The result carries two names but was discovered at least four times. Paul Güttinger proved it in 1932, Wolfgang Pauli in 1933, and Hans Hellmann presented it in his 1937 textbook Einführung in die Quantenchemie — the first comprehensive quantum-chemistry book. Richard Feynman rederived it as an MIT undergraduate; on John Slater's suggestion it became his 1939 senior thesis and the paper Forces in Molecules, published in Physical Review when Feynman was 21.

Its significance is conceptual as much as computational: it shows that chemical bonding, at the level of forces, is electrostatic — the quantum weirdness lives in the charge density, not in some extra force law. That reframing made the covalent bond intuitive and gave simulation a rigorous, cheap force.

• Extensions exist to finite temperature (free-energy derivatives), to time-dependent and open systems, and to relativistic Hamiltonians.
• Open practical questions center on Pulay-force cancellation in machine-learning potentials and on force accuracy in strongly correlated systems where approximate wavefunctions violate the stationarity condition.

The Hellmann-Feynman force versus the correction terms that appear with approximate or basis-set-dependent wavefunctions.

Term	Physical origin	Present when ψ is exact?	Typical size
Hellmann-Feynman force	⟨ψ|∂H/∂λ|ψ⟩ — bare Coulomb attraction of electrons and nuclei	Yes — this is the whole force	0 to a few eV/Å per atom
Pulay (wavefunction) force	∂ψ/∂λ terms that survive because basis functions move with the nuclei	No (vanishes for exact ψ)	Comparable to HF force; up to ~1 eV/Å
Incomplete-basis error	⟨∂ψ/∂λ|H−E|ψ⟩ ≠ 0 for non-eigenstates	No	0.1–1 eV/Å, shrinks with basis size
Electrostatic theorem check	Force reduced to pure classical electrostatics on nuclei	Yes (Feynman 1939)	Matches HF force exactly