Hellmann–Feynman theorem

Hellmann–Feynman theorem This reads like a textbook. Please see Wikipedia is not a textbook and rephrase to eliminate "we" and "us"'s tone or style may not reflect the encyclopedic tone used on Wikipedia. See Wikipedia's guide to writing better articles for suggestions. (November 2020) (Learn how and when to remove this template message) Part of a series of articles about Quantum mechanics {displaystyle ihbar {frac {partial }{partial t}}|psi (t)rangle ={hat {H}}|psi (t)rangle } Schrödinger equation IntroductionGlossaryHistory show Background show Fundamentals show Experiments show Formulations hide Equations DiracKlein–GordonPauliRydbergSchrödinger show Interpretations show Advanced topics show Scientists vte 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 {displaystyle {frac {mathrm {d} E_{lambda }}{mathrm {d} {lambda }}}={bigg langle }psi _{lambda }{bigg |}{frac {mathrm {d} {hat {H}}_{lambda }}{mathrm {d} lambda }}{bigg |}psi _{lambda }{bigg rangle },}         (1) where {displaystyle {hat {H}}_{lambda }} is a Hamiltonian operator depending upon a continuous parameter {displaystyle lambda ,} , {displaystyle |psi _{lambda }rangle } , is an eigen-state (eigenfunction) of the Hamiltonian, depending implicitly upon {displaystyle lambda } , {displaystyle E_{lambda },} is the energy (eigenvalue) of the state {displaystyle |psi _{lambda }rangle } , i.e. {displaystyle {hat {H}}_{lambda }|psi _{lambda }rangle =E_{lambda }|psi _{lambda }rangle } . Contents 1 Proof 2 Alternate proof 3 Example applications 3.1 Molecular forces 3.2 Expectation values 3.3 Van der Waals forces 4 Hellmann–Feynman theorem for time-dependent wavefunctions 4.1 Proof 5 Notes Proof 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 {displaystyle {hat {H}}_{lambda }|psi _{lambda }rangle =E_{lambda }|psi _{lambda }rangle ,} {displaystyle langle psi _{lambda }|psi _{lambda }rangle =1Rightarrow {frac {mathrm {d} }{mathrm {d} lambda }}langle psi _{lambda }|psi _{lambda }rangle =0.} 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: {displaystyle {begin{aligned}{frac {mathrm {d} E_{lambda }}{mathrm {d} lambda }}&={frac {mathrm {d} }{mathrm {d} lambda }}langle psi _{lambda }|{hat {H}}_{lambda }|psi _{lambda }rangle \&={bigg langle }{frac {mathrm {d} psi _{lambda }}{mathrm {d} lambda }}{bigg |}{hat {H}}_{lambda }{bigg |}psi _{lambda }{bigg rangle }+{bigg langle }psi _{lambda }{bigg |}{hat {H}}_{lambda }{bigg |}{frac {mathrm {d} psi _{lambda }}{mathrm {d} lambda }}{bigg rangle }+{bigg langle }psi _{lambda }{bigg |}{frac {mathrm {d} {hat {H}}_{lambda }}{mathrm {d} lambda }}{bigg |}psi _{lambda }{bigg rangle }\&=E_{lambda }{bigg langle }{frac {mathrm {d} psi _{lambda }}{mathrm {d} lambda }}{bigg |}psi _{lambda }{bigg rangle }+E_{lambda }{bigg langle }psi _{lambda }{bigg |}{frac {mathrm {d} psi _{lambda }}{mathrm {d} lambda }}{bigg rangle }+{bigg langle }psi _{lambda }{bigg |}{frac {mathrm {d} {hat {H}}_{lambda }}{mathrm {d} lambda }}{bigg |}psi _{lambda }{bigg rangle }\&=E_{lambda }{frac {mathrm {d} }{mathrm {d} lambda }}langle psi _{lambda }|psi _{lambda }rangle +{bigg langle }psi _{lambda }{bigg |}{frac {mathrm {d} {hat {H}}_{lambda }}{mathrm {d} lambda }}{bigg |}psi _{lambda }{bigg rangle }\&={bigg langle }psi _{lambda }{bigg |}{frac {mathrm {d} {hat {H}}_{lambda }}{mathrm {d} lambda }}{bigg |}psi _{lambda }{bigg rangle }.end{aligned}}} 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): {displaystyle E[psi ,lambda ]={frac {langle psi |{hat {H}}_{lambda }|psi rangle }{langle psi |psi rangle }}.}         (2) The eigenvalues are the values that the Schrödinger functional takes at the stationary points: {displaystyle E_{lambda }=E[psi _{lambda },lambda ],}         (3) where {displaystyle psi _{lambda }} satisfies the variational condition: {displaystyle left.{frac {delta E[psi ,lambda ]}{delta psi (x)}}right|_{psi =psi _{lambda }}=0.}         (4) By differentiating Eq. (3) using the chain rule, one obtains: {displaystyle {frac {dE_{lambda }}{dlambda }}={frac {partial E[psi _{lambda },lambda ]}{partial lambda }}+int {frac {delta E[psi ,lambda ]}{delta psi (x)}}{frac {dpsi _{lambda }(x)}{dlambda }}dx.}         (5) 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 ≤ i ≤ N 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 {displaystyle {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 }|}}.} 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 {displaystyle F_{X_{gamma }}=-{frac {partial E}{partial X_{gamma }}}=-{bigg langle }psi {bigg |}{frac {partial {hat {H}}}{partial X_{gamma }}}{bigg |}psi {bigg rangle }.} Only two components of the Hamiltonian contribute to the required derivative – the electron-nucleus and nucleus-nucleus terms. Differentiating the Hamiltonian yields[6] {displaystyle {begin{aligned}{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),\&=-Z_{gamma }sum _{i=1}^{N}{frac {x_{i}-X_{gamma }}{|mathbf {r} _{i}-mathbf {R} _{gamma }|^{3}}}+Z_{gamma }sum _{alpha neq gamma }^{M}Z_{alpha }{frac {X_{alpha }-X_{gamma }}{|mathbf {R} _{alpha }-mathbf {R} _{gamma }|^{3}}}.end{aligned}}} 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: {displaystyle 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).} 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 {displaystyle {hat {H}}_{l}=-{frac {hbar ^{2}}{2mu 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}},} 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: {displaystyle {frac {partial {hat {H}}_{l}}{partial l}}={frac {hbar ^{2}}{2mu r^{2}}}(2l+1).} The Hellmann–Feynman theorem then allows for the determination of the expectation value of {displaystyle {frac {1}{r^{2}}}} for hydrogen-like atoms:[7] {displaystyle {begin{aligned}{bigg langle }psi _{nl}{bigg |}{frac {1}{r^{2}}}{bigg |}psi _{nl}{bigg rangle }&={frac {2mu }{hbar ^{2}}}{frac {1}{2l+1}}{bigg langle }psi _{nl}{bigg |}{frac {partial {hat {H}}_{l}}{partial l}}{bigg |}psi _{nl}{bigg rangle }\&={frac {2mu }{hbar ^{2}}}{frac {1}{2l+1}}{frac {partial E_{n}}{partial l}}\&={frac {2mu }{hbar ^{2}}}{frac {1}{2l+1}}{frac {partial E_{n}}{partial n}}{frac {partial n}{partial l}}\&={frac {2mu }{hbar ^{2}}}{frac {1}{2l+1}}{frac {Z^{2}mu e^{4}}{hbar ^{2}n^{3}}}\&={frac {Z^{2}mu ^{2}e^{4}}{hbar ^{4}n^{3}(l+1/2)}}.end{aligned}}} In order to compute the energy derivative, the way {displaystyle n} depends on {displaystyle l} 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 {displaystyle n-l+1} , so {displaystyle partial n/partial l=1} .

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] {displaystyle {bigg langle }Psi _{lambda }(t){bigg |}{frac {partial H_{lambda }}{partial lambda }}{bigg |}Psi _{lambda }(t){bigg rangle }=ihbar {frac {partial }{partial t}}{bigg langle }Psi _{lambda }(t){bigg |}{frac {partial Psi _{lambda }(t)}{partial lambda }}{bigg rangle }} For {displaystyle ihbar {frac {partial Psi _{lambda }(t)}{partial t}}=H_{lambda }Psi _{lambda }(t)} Proof The proof only relies on the Schrödinger equation and the assumption that partial derivatives with respect to λ and t can be interchanged.

{displaystyle {begin{aligned}{bigg langle }Psi _{lambda }(t){bigg |}{frac {partial H_{lambda }}{partial lambda }}{bigg |}Psi _{lambda }(t){bigg rangle }&={frac {partial }{partial lambda }}langle Psi _{lambda }(t)|H_{lambda }|Psi _{lambda }(t)rangle -{bigg langle }{frac {partial Psi _{lambda }(t)}{partial lambda }}{bigg |}H_{lambda }{bigg |}Psi _{lambda }(t){bigg rangle }-{bigg langle }Psi _{lambda }(t){bigg |}H_{lambda }{bigg |}{frac {partial Psi _{lambda }(t)}{partial lambda }}{bigg rangle }\&=ihbar {frac {partial }{partial lambda }}{bigg langle }Psi _{lambda }(t){bigg |}{frac {partial Psi _{lambda }(t)}{partial t}}{bigg rangle }-ihbar {bigg langle }{frac {partial Psi _{lambda }(t)}{partial lambda }}{bigg |}{frac {partial Psi _{lambda }(t)}{partial t}}{bigg rangle }+ihbar {bigg langle }{frac {partial Psi _{lambda }(t)}{partial t}}{bigg |}{frac {partial Psi _{lambda }(t)}{partial lambda }}{bigg rangle }\&=ihbar {bigg langle }Psi _{lambda }(t){bigg |}{frac {partial ^{2}Psi _{lambda }(t)}{partial lambda partial t}}{bigg rangle }+ihbar {bigg langle }{frac {partial Psi _{lambda }(t)}{partial t}}{bigg |}{frac {partial Psi _{lambda }(t)}{partial lambda }}{bigg rangle }\&=ihbar {frac {partial }{partial t}}{bigg langle }Psi _{lambda }(t){bigg |}{frac {partial Psi _{lambda }(t)}{partial lambda }}{bigg rangle }end{aligned}}} Notes ^ 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. S2CID 124962011. ^ Pauli, W. (1933). "Principles of Wave Mechanics". Handbuch der Physik. Vol. 24. Berlin: Springer. p. 162. ^ Hellmann, H (1937). Einführung in die Quantenchemie. Leipzig: Franz Deuticke. p. 285. OL 21481721M. ^ Feynman, R. P. (1939). "Forces in Molecules". Physical Review. 56 (4): 340–343. Bibcode:1939PhRv...56..340F. doi:10.1103/PhysRev.56.340. ^ Jensen, Frank (2007). Introduction to Computational Chemistry. West Sussex: John Wiley & Sons. p. 322. ISBN 978-0-470-01186-7. ^ Piela, Lucjan (2006). Ideas of Quantum Chemistry. Amsterdam: Elsevier Science. p. 620. ISBN 978-0-444-52227-6. ^ 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. ^ Epstein, Saul (1966). "Time-Dependent Hellmann-Feynman Theorems for Variational Wavefunctions". The Journal of Chemical Physics. 45 (1): 384. doi:10.1063/1.1727339. ^ Hayes, Edward F.; Parr, Robert G. (1965). "Time-Dependent Hellmann-FeynmanTheorems". The Journal of Chemical Physics. 43 (5): 1831. doi:10.1063/1.1697020. show vte Richard Feynman Categories: Intermolecular forcesTheorems in quantum mechanicsRichard Feynman