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| In [[quantum mechanics]], the '''Hellmann–Feynman theorem''' relates the derivative of the total energy with respect to a parameter, to the [[Expectation value (quantum mechanics)|expectation value]] of the derivative of the [[Hamiltonian (quantum mechanics)|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 electromagnetism|classical electrostatics]].
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| The theorem has been proven independently by many authors, including [[Paul Güttinger]] (1932),<ref>{{cite journal|last=Güttinger|first=P.|year=1932|journal=Z. Phys.|volume=73|pages=169 |bibcode = 1932ZPhy...73..169G |doi = 10.1007/BF01351211|title=Das Verhalten von Atomen im magnetischen Drehfeld|issue=3–4 }}</ref> [[Wolfgang Pauli]] (1933),<ref>{{cite book|last=Pauli|first=W.|title=Handbuch der Physik|chapter=Principles of Wave Mechanics|volume=24|publisher=Springer|location=Berlin|year=1933|page=162}}</ref> [[Hans Hellmann]] (1937)<ref>{{cite book|last=Hellmann|first=H|title=Einführung in die Quantenchemie|publisher=Franz Deuticke|location=Leipzig|ol=OL21481721M|year=1937|page=285}}</ref> and [[Richard Feynman]] (1939).<ref>{{cite journal|last=Feynman|first=R. P.|year=1939|title=Forces in Molecules|journal=Phys. Rev.|volume=56|issue=4|pages=340|doi=10.1103/PhysRev.56.340 |bibcode = 1939PhRv...56..340F }}</ref> | |
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| The theorem states
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| {{NumBlk|:|<math>\frac{d E}{d {\lambda}}=\int{\psi^{*}(\lambda)\frac{d{\hat{H}_{\lambda}}}{d{\lambda}}\psi(\lambda)\ d\tau},</math>|{{EquationRef|1}}}}
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| where
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| *<math>\hat{H}_{\lambda}</math> is a Hamiltonian operator depending upon a continuous parameter <math>\lambda\,</math>,
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| *<math>\psi(\lambda)\,</math> is an eigen-[[wavefunction]] ([[eigenfunction]]) of the Hamiltonian, depending implicitly upon <math>\lambda\,</math>,
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| *<math>E\,</math> is the energy (eigenvalue) of the wavefunction,
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| *<math>d\tau\,</math> implies an integration over the domain of the wavefunction.
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| ==Proof==
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| 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.<ref>{{cite book|last=Jensen|first=Frank|title=Introduction to Computational Chemistry|publisher=John Wiley & Sons|location=West Sussex|year=2007|isbn=0-470-01186-6|page=322}}</ref>
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| 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
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| :<math>\hat{H}_{\lambda}|\psi(\lambda)\rangle = E_{\lambda}|\psi(\lambda)\rangle,</math>
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| :<math>\langle\psi(\lambda)|\psi(\lambda)\rangle = 1 \Rightarrow \frac{\partial}{\partial\lambda}\langle\psi(\lambda)|\psi(\lambda)\rangle =0.</math>
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| The proof then follows through an application of the derivative [[product rule]] to the [[Expectation value (quantum mechanics)|expectation value]] of the Hamiltonian viewed as a function of λ:
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| :<math>
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| \begin{align}
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| \frac{d E_{\lambda}}{d\lambda} &= \frac{d}{d\lambda}\langle\psi(\lambda)|\hat{H}_{\lambda}|\psi(\lambda)\rangle \\
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| &=\bigg\langle\frac{d\psi(\lambda)}{d\lambda}\bigg|\hat{H}_{\lambda}\bigg|\psi(\lambda)\bigg\rangle + \bigg\langle\psi(\lambda)\bigg|\hat{H}_{\lambda}\bigg|\frac{d\psi(\lambda)}{d\lambda}\bigg\rangle + \bigg\langle\psi(\lambda)\bigg|\frac{d\hat{H}_{\lambda}}{d\lambda}\bigg|\psi(\lambda)\bigg\rangle \\
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| &=E_{\lambda}\bigg\langle\frac{d\psi(\lambda)}{d\lambda}\bigg|\psi(\lambda)\bigg\rangle + E_{\lambda}\bigg\langle\psi(\lambda)\bigg|\frac{d\psi(\lambda)}{d\lambda}\bigg\rangle + \bigg\langle\psi(\lambda)\bigg|\frac{d\hat{H}_{\lambda}}{d\lambda}\bigg|\psi(\lambda)\bigg\rangle \\
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| &=E_{\lambda}\frac{d}{d\lambda}\bigg\langle\psi(\lambda)\bigg|\psi(\lambda)\bigg\rangle + \bigg\langle\psi(\lambda)\bigg|\frac{d\hat{H}_{\lambda}}{d\lambda}\bigg|\psi(\lambda)\bigg\rangle \\
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| &=\bigg\langle\psi(\lambda)\bigg|\frac{d\hat{H}_{\lambda}}{d\lambda}\bigg|\psi(\lambda)\bigg\rangle.
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| \end{align}
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| </math>
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| For a deep critical view of the proof see <ref>{{cite journal|last=Carfì|first=David|year=2010|title=The pointwise Hellmann–Feynman theorem|journal=AAPP Physical, Mathematical, and Natural Sciences|volume=88|issue=1|at=no. C1A1001004|doi=10.1478/C1A1001004|issn=1825-1242 }}</ref>
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| ==Alternate proof==
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| The Hellmann-Feynman theorem is actually a direct, and to some extent trivial, consequence of the variational principle (the [[Rayleigh-Ritz_method|Raleigh-Ritz variational principle]]) from which the Schrödinger equation can be made to derive. 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|density-functional theory]], which is not wave-function based and for which the standard derivation does not apply.
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| According to the Raleigh-Ritz variational principle, the eigenfunctions of the Schrödinger equation are stationary points of the functional (which we nickname ''Schrödinger functional'' for brevity):
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| {{NumBlk|:|<math>E[\psi,\lambda]=\frac{\langle\psi|\hat{H}_{\lambda}|\psi\rangle}{\langle\psi|\psi\rangle}.</math>|{{EquationRef|2}}}}
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| The eigenvalues are the values that the Schrödinger functional takes at the stationary points:
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| {{NumBlk|:|<math>E_{\lambda}=E[\psi_{\lambda},\lambda],</math>|{{EquationRef|3}}}}
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| where <math>\psi_{\lambda} </math> satisfies the variational condition:
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| {{NumBlk|:|<math>\left.\frac{\delta E[\psi,\lambda]}{\delta\psi(x)}\right|_{\psi=\psi_{\lambda}}=0.</math>|{{EquationRef|4}}}}
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| Let us differentiate Eq. (3) using the [[Chain_rule|chain rule]]:
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| {{NumBlk|:|<math> \frac{dE_{\lambda}}{d\lambda}=\frac{\partial E[\psi_{\lambda},\lambda]}{\partial\lambda}+\int\frac{\delta E[\psi,\lambda]}{\delta\psi(x)}\frac{d\psi_{\lambda}(x)}{d\lambda}dx. </math>|{{EquationRef|5}}}}
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| 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''. On account of the fact that the Schrödinger functional can only depend explicitly on an external parameter through the Hamiltonian, Eq. (1) trivially follows. As simple as that.
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| ==Example applications==
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| ===Molecular forces===
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| The most common application of the Hellmann–Feynman theorem is to the calculation of [[intramolecular]] forces in molecules. This allows for the calculation of [[molecular geometry|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 {'''r'''<sub>''i''</sub>}, and 1 ≤ α ≤ ''M'' nuclei, each located at a specified point {'''R'''<sub>α</sub>={''X''<sub>α</sub>,''Y''<sub>α</sub>,''Z''<sub>α</sub>)} and with nuclear charge ''Z''<sub>α</sub>, the [[molecular Hamiltonian|clamped nucleus Hamiltonian]] is | |
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| :<math>\hat{H}=\hat{T} + \hat{U} - \sum_{i=1}^{N}\sum_{\alpha=1}^{M}\frac{Z_{\alpha}}{|\mathbf{r}_{i}-\mathbf{R}_{\alpha}|} + \sum_{\alpha}^{M}\sum_{\beta>\alpha}^{M}\frac{Z_{\alpha}Z_{\beta}}{|\mathbf{R}_{\alpha}-\mathbf{R}_{\beta}|}.</math>
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| The force acting on the x-component of 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
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| :<math>F_{X_{\gamma}} = -\frac{\partial E}{\partial X_{\gamma}} = -\bigg\langle\psi\bigg|\frac{\partial\hat{H}}{\partial X_{\gamma}}\bigg|\psi\bigg\rangle.</math>
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| Only two components of the Hamiltonian contribute to the required derivative – the electron-nucleus and nucleus-nucleus terms. Differentiating the Hamiltonian yields<ref name="piela">{{cite book|last=Piela|first= Lucjan|title=Ideas of Quantum Chemistry|publisher=Elsevier Science|location=Amsterdam|year=2006|page=620|isbn=0-444-52227-1}}</ref>
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| :<math>
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| \begin{align}
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| \frac{\partial\hat{H}}{\partial X_{\gamma}} &= \frac{\partial}{\partial X_{\gamma}} \left(- \sum_{i=1}^{N}\sum_{\alpha=1}^{M}\frac{Z_{\alpha}}{|\mathbf{r}_{i}-\mathbf{R}_{\alpha}|} + \sum_{\alpha}^{M}\sum_{\beta>\alpha}^{M}\frac{Z_{\alpha}Z_{\beta}}{|\mathbf{R}_{\alpha}-\mathbf{R}_{\beta}|}\right), \\
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| &=Z_{\gamma}\sum_{i=1}^{N}\frac{x_{i}-X_{\gamma}}{|\mathbf{r}_{i}-\mathbf{R}_{\gamma}|^{3}}
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| -Z_{\gamma}\sum_{\alpha\neq\gamma}^{M}Z_{\alpha}\frac{X_{\alpha}-X_{\gamma}}{|\mathbf{R}_{\alpha}-\mathbf{R}_{\gamma}|^{3}}.
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| \end{align}
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| </math>
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| 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:
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| :<math>F_{X_{\gamma}} = -Z_{\gamma}\left(\int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r})\frac{x-X_{\gamma}}{|\mathbf{r}-\mathbf{R}_{\gamma}|^{3}} - \sum_{\alpha\neq\gamma}^{M}Z_{\alpha}\frac{X_{\alpha}-X_{\gamma}}{|\mathbf{R}_{\alpha}-\mathbf{R}_{\gamma}|^{3}}\right).</math>
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| ===Expectation values===
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| 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 [[Hydrogen-like atom|radial Schrödinger equation for a hydrogen-like atom]] is
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| :<math>\hat{H}_{l}=-\frac{\hbar^{2}}{2\mu r^2}\left(\frac{\mathrm{d}}{\mathrm{d}r}\left(r^{2}\frac{\mathrm{d}}{\mathrm{d}r}\right)-l(l+1)\right) -\frac{Ze^{2}}{r},</math>
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| 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:
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| :<math>\frac{\partial \hat{H}_{l}}{\partial l} = \frac{\hbar^{2}}{2\mu r^{2}}(2l+1).</math>
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| The Hellmann–Feynman theorem then allows for the determination of the expectation value of <math>\frac{1}{r^{2}}</math> for hydrogen-like atoms:<ref>{{cite book|last=Fitts|first=Donald D.|title=Principles of Quantum Mechanics : as Applied to Chemistry and Chemical Physics|publisher=Cambridge University Press|location=Cambridge|year=2002|isbn=0-521-65124-7|page=186}}</ref>
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| :<math>
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| \begin{align}
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| \bigg\langle\psi_{nl}\bigg|\frac{1}{r^{2}}\bigg|\psi_{nl}\bigg\rangle &= \frac{2\mu}{\hbar^{2}}\frac{1}{2l+1}\bigg\langle\psi_{nl}\bigg|\frac{\partial \hat{H}_{l}}{\partial l}\bigg|\psi_{nl}\bigg\rangle \\
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| &=\frac{2\mu}{\hbar^{2}}\frac{1}{2l+1}\frac{\partial E_{n}}{\partial l} \\
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| &=\frac{2\mu}{\hbar^{2}}\frac{1}{2l+1}\frac{\partial E_{n}}{\partial n}\frac{\partial n}{\partial l} \\
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| &=\frac{2\mu}{\hbar^{2}}\frac{1}{2l+1}\frac{Z^{2}\mu e^{4}}{\hbar^{2}n^{3}} \\
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| &=\frac{Z^{2}\mu^{2}e^{4}}{\hbar^{4}n^{3}(l+1/2)}.
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| \end{align}
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| </math>
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| ===Van der Waals forces===
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| In the end of Feynman's paper, he states that, "[[Van der Waals force|Van der Waals's forces]] can also be interpreted as arising from charge distributions
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| 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
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| symmetry, a dipole moment of order 1/''R''<sup>7</sup> 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/''R''<sup>7</sup> force."
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| ==Hellmann–Feynman theorem for time-dependent wavefunctions==
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| For a general time-dependent wavefunction satisfying the time-dependent [[Schrödinger equation]], the Hellmann–Feynman theorem is '''not''' valid.
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| However, the following identity holds:
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| :<math>
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| \bigg\langle\Psi_\lambda(t)\bigg|\frac{\partial H_\lambda}{\partial\lambda}\bigg|\Psi_\lambda(t)\bigg\rangle = i \hbar \frac{\partial}{\partial t}\bigg\langle\Psi_\lambda(t)\bigg|\frac{\partial \Psi_\lambda(t)}{\partial \lambda}\bigg\rangle
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| </math>
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| For
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| :<math>
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| i\hbar\frac{\partial\Psi_\lambda(t)}{\partial t}=H_\lambda\Psi_\lambda(t)
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| </math>
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| ===Proof===
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| The proof only relies on the Schrödinger equation and the assumption that partial derivatives with respect to λ and t can be interchanged.
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| :<math>
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| \begin{align}
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| \bigg\langle\Psi_\lambda(t)\bigg|\frac{\partial H_\lambda}{\partial\lambda}\bigg|\Psi_\lambda(t)\bigg\rangle &=
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| \frac{\partial}{\partial\lambda}\langle\Psi_\lambda(t)|H_\lambda|\Psi_\lambda(t)\rangle
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| - \bigg\langle\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg|H_\lambda\bigg|\Psi_\lambda(t)\bigg\rangle
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| - \bigg\langle\Psi_\lambda(t)\bigg|H_\lambda\bigg|\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg\rangle \\
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| &= i\hbar \frac{\partial}{\partial\lambda}\bigg\langle\Psi_\lambda(t)\bigg|\frac{\partial\Psi_\lambda(t)}{\partial t}\bigg\rangle
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| - i\hbar\bigg\langle\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg|\frac{\partial\Psi_\lambda(t)}{\partial t}\bigg\rangle
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| + i\hbar\bigg\langle\frac{\partial\Psi_\lambda(t)}{\partial t}\bigg|\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg\rangle \\
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| &= i\hbar \bigg\langle\Psi_\lambda(t)\bigg| \frac{\partial^2\Psi_\lambda(t)}{\partial\lambda \partial t}\bigg\rangle
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| + i\hbar\bigg\langle\frac{\partial\Psi_\lambda(t)}{\partial t}\bigg|\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg\rangle \\
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| &= i \hbar \frac{\partial}{\partial t}\bigg\langle\Psi_\lambda(t)\bigg|\frac{\partial \Psi_\lambda(t)}{\partial \lambda}\bigg\rangle
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| \end{align}
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| </math>
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| ==Notes==
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| {{reflist}}
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| {{DEFAULTSORT:Hellmann-Feynman theorem}}
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| [[Category:Quantum mechanics]]
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| [[Category:Intermolecular forces]]
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| [[Category:Theorems in quantum physics]]
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| [[Category:Richard Feynman]]
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