Hellmann–Feynman theorem

Last updated


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). [4]

The theorem states








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). 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. [5]

The proof also employs an identity of normalized wavefunctions – that derivatives of the overlap of a wavefunction 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 λ:

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, 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[ by whom? ]Schrödinger functional for brevity):






The eigenvalues are the values that the Schrödinger functional takes at the stationary points:






where satisfies the variational condition:






By differentiating Eq. (3) using the chain rule, one obtains:






Due to the variational condition, Eq. (4), the second term in Eq. (5) vanishes. 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 ] On account of the fact that the Schrödinger functional can only depend explicitly on an external parameter through the Hamiltonian, Eq. (1) trivially follows.

Example applications

Molecular forces

The most common application of the Hellmann–Feynman theorem is to 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 ≤ iN electrons with coordinates {ri}, and 1 ≤ α ≤ M nuclei, each located at a specified point {Rα={Xα,Yα,Zα}} and with nuclear charge Zα, the clamped nucleus Hamiltonian is

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

Only two components of the Hamiltonian contribute to the required derivative – the electron-nucleus and nucleus-nucleus terms. Differentiating the Hamiltonian yields [6]

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

Expectation values

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. Possible parameters are physical constants or discrete quantum numbers. As an example, the radial Schrödinger equation for a hydrogen-like atom is

which depends upon the discrete azimuthal quantum number l. Promoting l 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 for hydrogen-like atoms: [7]

In order to compute the energy derivative, the way depends on has to be known. These quantum numbers are usually independent, but here the solutions must be varied so as to keep the number of nodes in the wavefunction fixed. The number of nodes is , so .

Van der Waals forces

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. The Schrödinger perturbation theory for two interacting atoms at a separation R, 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 1/R7 being induced in each atom. The negative charge distribution of each atom has its center of gravity moved slightly toward the other. It is not the interaction of these dipoles which leads to van der Waals's force, but rather the attraction of each nucleus for the distorted charge distribution of its own electrons that gives the attractive 1/R7 force."[ excessive quote ]

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: [8] [9]



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


  1. Güttinger, P. (1932). "Das Verhalten von Atomen im magnetischen Drehfeld". Zeitschrift für Physik. 73 (3–4): 169–184. Bibcode:1932ZPhy...73..169G. doi:10.1007/BF01351211.
  2. Pauli, W. (1933). "Principles of Wave Mechanics". Handbuch der Physik. 24. Berlin: Springer. p. 162.
  3. Hellmann, H (1937). Einführung in die Quantenchemie. Leipzig: Franz Deuticke. p. 285. OL   21481721M.
  4. Feynman, R. P. (1939). "Forces in Molecules". Physical Review. 56 (4): 340–343. Bibcode:1939PhRv...56..340F. doi:10.1103/PhysRev.56.340.
  5. Jensen, Frank (2007). Introduction to Computational Chemistry. West Sussex: John Wiley & Sons. p. 322. ISBN   978-0-470-01186-7.
  6. Piela, Lucjan (2006). Ideas of Quantum Chemistry. Amsterdam: Elsevier Science. p. 620. ISBN   978-0-444-52227-6.
  7. Fitts, Donald D. (2002). Principles of Quantum Mechanics : as Applied to Chemistry and Chemical Physics. Cambridge: Cambridge University Press. p. 186. ISBN   978-0-521-65124-0.
  8. Epstein, Saul (1966). "Time-Dependent Hellmann-Feynman Theorems for Variational Wavefunctions". The Journal of Chemical Physics. 45 (1): 384. doi:10.1063/1.1727339.
  9. Hayes, Edward F.; Parr, Robert G. (1965). "Time-Dependent Hellmann-FeynmanTheorems". The Journal of Chemical Physics. 43 (5): 1831. doi:10.1063/1.1697020.

Related Research Articles

Dirac equation Relativistic quantum mechanical wave equation

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-½ massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.

Schrödinger equation Linear partial differential equation whose solution describes the quantum-mechanical system.

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. The approach is named after Max Born and J. Robert Oppenheimer who proposed it in 1927, in the early period of quantum mechanics.

In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product is a linear map A that is its own adjoint: for all vectors v and w. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.

In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an additional "perturbing" Hamiltonian representing a weak disturbance to the system. If the disturbance is not too large, the various physical quantities associated with the perturbed system can be expressed as "corrections" to those of the simple system. These corrections, being small compared to the size of the quantities themselves, can be calculated using approximate methods such as asymptotic series. The complicated system can therefore be studied based on knowledge of the simpler one. In effect, it is describing a complicated unsolved system using a simple, solvable system.

Blochs theorem Fundamental theorem in condensed matter physics

In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. Mathematically, they are written:

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

The adiabatic theorem is a concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928), was stated as follows:

LSZ reduction formula

In quantum field theory, the LSZ reduction formula is a method to calculate S-matrix elements from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann.

Two-state quantum system Quantum system that can be measured as one of two values; sought for "quantum bits" in quantum computing

In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.

The Pulay stress or Pulay forces is an error that occurs in the stress tensor obtained from self-consistent field calculations due to the incompleteness of the basis set.

The time-evolving block decimation (TEBD) algorithm is a numerical scheme used to simulate one-dimensional quantum many-body systems, characterized by at most nearest-neighbour interactions. It is dubbed Time-evolving Block Decimation because it dynamically identifies the relevant low-dimensional Hilbert subspaces of an exponentially larger original Hilbert space. The algorithm, based on the Matrix Product States formalism, is highly efficient when the amount of entanglement in the system is limited, a requirement fulfilled by a large class of quantum many-body systems in one dimension.

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals. The basis for this method is the variational principle.


A phonovoltaic (pV) cell converts vibrational (phonons) energy into a direct current much like the photovoltaic effect in a photovoltaic (PV) cell converts light (photon) into power. That is, it uses a p-n junction to separate the electrons and holes generated as valence electrons absorb optical phonons more energetic than the band gap, and then collects them in the metallic contacts for use in a circuit. The pV cell is an application of heat transfer physics and competes with other thermal energy harvesting devices like the thermoelectric generator.

Tau functions are an important ingredient in the modern theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form. The term Tau function, or -function, was first used systematically by Mikio Sato and his students in the specific context of the Kadomtsev–Petviashvili equation, and related integrable hierarchies. It is a central ingredient in the theory of solitons. Tau functions also appear as matrix model partition functions in the spectral theory of Random Matrices, and may also serve as generating functions, in the sense of combinatorics and enumerative geometry, especially in relation to moduli spaces of Riemann surfaces, and enumeration of branched coverings, or so-called Hurwitz numbers.