费曼－海尔曼定理

该理论分别被不同的物理学家独立地证明过，包括Paul Güttinger（1932）、泡利（1933）、海尔曼 （1937）以及费曼（1939）。

该定理的表达式如下： （公式略）

式中 H_(λ) 表示依赖于连续变化的参变量 λ 的哈密顿量； ψ (λ) 是该哈密顿量的本征函数，通过哈密顿量间接依赖于 λ； E 为能量，即哈密顿量的本征值； dτ 为积分微元。上述积分在全空间进行。

随时间变化的波函数的费曼–海尔曼定理

因为一个一般的随时间变化的波函数满足含时薛定谔方程，所以费曼–海尔曼定理不再适用。

但是，该波函数满足以下关系： （公式略）

其中ψ满足： （公式略）

证明

以下证明只依赖于薛定谔方程，以及对于λ和t求偏导时，可以互换顺序的假设。 （公式略）

In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.

The theorem has been proven independently by many authors, including Paul Güttinger (1932), Wolfgang Pauli (1933), Hans Hellmann (1937) and Richard Feynman (1939).

The theorem states （公式略）

where H_(λ) is a Hermitian operator depending upon a continuous parameter λ,

| |ψ _(λ)⟩, is an eigenstate (eigenfunction) of the Hamiltonian, depending implicitly upon λ,

E_(λ) is the energy (eigenvalue) of the state |ψ _(λ)⟩, i.e. H_(λ)|ψ _(λ)⟩ =E_(λ)|ψ _(λ)⟩.

Note that for systems with degenerate states, a refined version of the Hellmann–Feynman theorem is needed.

Proof

This proof of the Hellmann–Feynman theorem requires that the wave function be an eigenfunction of the Hamiltonian under consideration; however, it is also possible to prove more generally that the theorem holds for non-eigenfunction wave functions which are stationary (partial derivative is zero) for all relevant variables (such as orbital rotations). The Hartree–Fock wavefunction is an important example of an approximate eigenfunction that still satisfies the Hellmann–Feynman theorem. Notable example of where the Hellmann–Feynman is not applicable is for example finite-order Møller–Plesset perturbation theory, which is not variational.

The proof also employs an identity of normalized wavefunctions – that derivatives of the overlap of a wave function with itself must be zero. Using Dirac's bra–ket notation these two conditions are written as H_(λ)|ψ _(λ)⟩ =E_(λ)|ψ _(λ)⟩,

⟨ ⟨ ψ _(λ)|ψ _(λ)⟩ =1⇒ d/(d λ)⟨ ψ _(λ)|ψ _(λ)⟩ =0.

Alternate proof

The Hellmann–Feynman theorem is actually a direct, and to some extent trivial, consequence of the variational principle (the Rayleigh–Ritz variational principle) from which the Schrödinger equation may be derived. This is why the Hellmann–Feynman theorem holds for wave-functions (such as the Hartree–Fock wave-function) that, though not eigenfunctions of the Hamiltonian, do derive from a variational principle. This is also why it holds, e.g., in density functional theory, e.g. in the adiabatic connection fluctuation dissipation theorem, which is not wave-function based and for which the standard derivation does not apply.

According to the Rayleigh–Ritz variational principle, the eigenfunctions of the Schrödinger equation are stationary points of the functional (which is nicknamed Schrödinger functional for brevity): （公式略） The eigenvalues are the values that the Schrödinger functional takes at the stationary points: E_(λ)=E[ψ _(λ),λ ], where ψ _(λ) satisfies the variational condition:.(δ E[ψ,λ ])/(δ ψ (x))|_ψ =ψ _(λ)=0.

Example applications

Molecular forces

The most common application of the Hellmann–Feynman theorem is the calculation of intramolecular forces in molecules. This allows for the calculation of equilibrium geometries – the nuclear coordinates where the forces acting upon the nuclei, due to the electrons and other nuclei, vanish. The parameter λ corresponds to the coordinates of the nuclei. For a molecule with 1≤ i≤ N electrons with coordinates （公式略）, and 1≤ α ≤ M nuclei, each located at a specified point （公式略） and with nuclear charge Z_(α), the clamped nucleus Hamiltonian is （公式略）

The x -component of the force acting on a given nucleus is equal to the negative of the derivative of the total energy with respect to that coordinate. Employing the Hellmann–Feynman theorem this is equal to （公式略）

Only two components of the Hamiltonian contribute to the required derivative – the electron-nucleus and nucleus-nucleus terms. Differentiating the Hamiltonian yields （公式略）

Hellmann–Feynman theorem for time-dependent wavefunctions

For a general time-dependent wavefunction satisfying the time-dependent Schrödinger equation, the Hellmann–Feynman theorem is not valid.

However, the following identity holds: （公式略）

For （公式略）

Proof

The proof only relies on the Schrödinger equation and the assumption that partial derivatives with respect to λ and t can be interchanged. （公式略）