Hellmann-Feynman theorem

The Hellmann-Feynman theorems are used to calculate these single derivatives.

The first order derivative is given by the first Hellmann-Feynman theorem directly.

The most common application of the Hellmann-Feynman theorem is to the calculation of intramolecular forces in molecules.

With the differential rules given by the Hellmann-Feynman theorems, the perturbative correction to the energies and states can be calculated systematically.

Whenever a state derivative is encountered, resolve it by inserting the complete set of basis, then the Hellmann-Feynman theorem is applicable.

The Hellmann-Feynman theorem then allows for the determination of the expectation value of for hydrogen-like atoms:

The Hartree-Fock wavefunction is an important example of an approximate eigenfunction that still satisfies the Hellmann-Feynman theorem.

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

Insertion of this in to the Hellmann-Feynman theorem returns the force on the x-component of the given nucleus in terms of the electronic density (ρ(r)) and the atomic coordinates and nuclear charges:

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.

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.

In science, his name is primarily associated with the Hellmann-Feynman theorem, as well as with one of the first-ever textbooks on quantum chemistry ('Kvantovaya Khimiya', 1937; translated into German as 'Einfuehrung in die Quantenchemie', Vienna, 1937).

This proof of the Hellmann-Feynman theorem requires that the wavefunction be an eigenfunction of the Hamiltonian under consideration; however, one can also prove more generally that the theorem holds for non-eigenfunction wavefunctions which are stationary (partial derivative is zero) for all relevant variables (such as orbital rotations).