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),[1] Wolfgang Pauli (1933),[2] Hans Hellmann (1937)[3] and Richard Feynman (1939).
Note that there is a breakdown of the Hellmann-Feynman theorem close to quantum critical points in the thermodynamic limit.
[5] 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.
[6] 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 The proof then follows through an application of the derivative product rule to the expectation value of the Hamiltonian viewed as a function of
This is also why it holds, e.g., in density functional theory, which is not wave-function based and for which the standard derivation does not apply.
(3) using the chain rule, the following equation is obtained: Due to the variational condition, Eq.
In one sentence, the Hellmann–Feynman theorem states that the derivative of the stationary values of a function(al) with respect to a parameter on which it may depend, can be computed from the explicit dependence only, disregarding the implicit one.
[citation needed] Because the Schrödinger functional can only depend explicitly on an external parameter through the Hamiltonian, Eq.
The most common application of the Hellmann–Feynman theorem is the calculation of intramolecular forces in molecules.
-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 Only two components of the Hamiltonian contribute to the required derivative – the electron-nucleus and nucleus-nucleus terms.
Differentiating the Hamiltonian yields[7] Insertion of this in to the Hellmann–Feynman theorem returns the
and the atomic coordinates and nuclear charges: A comprehensive survey of similar applications of the Hellmann-Feynman theorem in quantum chemistry is given in B. M. Deb (ed.)
An alternative approach for applying the Hellmann–Feynman theorem is to promote a fixed or discrete parameter which appears in a Hamiltonian to be a continuous variable solely for the mathematical purpose of taking a derivative.
As an example, the radial Schrödinger equation for a hydrogen-like atom is which depends upon the discrete azimuthal quantum number
to be a continuous parameter allows for the derivative of the Hamiltonian to be taken: The Hellmann–Feynman theorem then allows for the determination of the expectation value of
In the end of Feynman's paper, he states that, "Van der Waals' forces can also be interpreted as arising from charge distributions with higher concentration between the nuclei.
, large compared to the radii of the atoms, leads to the result that the charge distribution of each is distorted from central symmetry, a dipole moment of order
The negative charge distribution of each atom has its center of gravity moved slightly toward the other.
However, the following identity holds:[9][10] For The proof only relies on the Schrödinger equation and the assumption that partial derivatives with respect to λ and t can be interchanged.