Hellmann–Feynman theorem

The Hellmann–Feynman theorem equates the derivative of an energy eigenvalue with respect to a Hamiltonian parameter to the expectation value of the Hamiltonian’s derivative, for a normalized eigenstate.

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Schrödinger equation: The equation governing how quantum states evolve or satisfy energy eigenvalue conditions. Differentiating its eigenvalue form yields the theorem’s central identity.
Potential energy surface: A function giving a molecule’s electronic energy as a function of its nuclear positions. Its gradients provide forces on nuclei through the theorem.
Hamiltonian (quantum mechanics): The operator representing a quantum system’s total energy and generating its time evolution. The theorem differentiates this operator with respect to an external parameter.
Hans Hellmann: A German theoretical physicist known for work on molecular quantum mechanics and chemical bonding. His 1937 treatment gave an early formulation of the energy-derivative result.
Eigenvalue: A scalar that an operator multiplies an associated nonzero eigenvector by. The theorem tracks how an energy eigenvalue changes as the Hamiltonian varies.
Molecular dynamics: A computational method that simulates atomic motion by integrating equations of motion. Electronic-structure simulations use theorem-based forces to propagate nuclei.
Eigenstate: A quantum state that is an eigenvector of a specified observable or operator. The theorem concerns normalized eigenstates of the parameter-dependent Hamiltonian.
Richard Feynman: An American theoretical physicist whose work ranged across quantum mechanics, electrodynamics, and computation. His 1939 paper derived the relation for forces in molecules.
Expectation value: The statistical mean of measurement outcomes for an observable in a specified quantum state. The Hamiltonian derivative’s expectation value gives the eigenvalue’s parameter derivative.
Quantum chemistry: The application of quantum mechanics to molecular structure, properties, and reactions. Molecular energy gradients are a central use of the theorem.