查看“︁费曼-海尔曼定理”︁的源代码

{{量子力学}}
[[量子力学]]中，'''费曼–海尔曼定理'''描述的是一个体系的能量对某个参量的导数与[[哈密顿量]]算符对同一参量的导数的期望值之间的关系。根据这一定理，通过求解[[薛定谔方程]]得到电子密度的空间分布后，体系中的所有力都能通过经典[[静电学]]求出。

该理论分别被不同的物理学家独立地证明过，包括[[Paul Güttinger]]（1932）<ref>{{cite journal|last=Güttinger|first=P.|date=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>、[[沃尔夫冈·泡利|泡利]]（1933）<ref>{{cite book|last=Pauli|first=W.|title=Handbuch der Physik|chapter=Principles of Wave Mechanics|volume=24|publisher=Springer|location=Berlin|date=1933|page=162}}</ref>、[[Hans Hellmann|海尔曼]] （1937）<ref>{{cite book|last=Hellmann|first=H|title=Einführung in die Quantenchemie|publisher=Franz Deuticke|location=Leipzig|ol=21481721M|date=1937|page=285}}</ref>以及[[理查德·費曼|费曼]]（1939）。<ref>{{cite journal|last=Feynman|first=R. P.|date=1939|title=Forces in Molecules|url=https://archive.org/details/sim_physical-review_1939-08-15_56_4/page/n50|journal=Phys. Rev.|volume=56|issue=4|pages=340|doi=10.1103/PhysRev.56.340 |bibcode = 1939PhRv...56..340F }}</ref>

该定理的表达式如下：

<center><math>\frac{{\rm d} E}{{\rm d} {\lambda}}=\int{\psi^{*}(\lambda)\frac{{\rm d}{\hat{H}_{\lambda}}}{{\rm d}{\lambda}}\psi(\lambda)\ {\rm d}\tau},</math></center>

式中

*<math>\hat{H}_{\lambda}</math> 表示依赖于连续变化的参变量<math>\lambda</math>的哈密顿量；
*<math>\psi(\lambda)\,</math> 是该哈密顿量的本征函数，通过哈密顿量间接依赖于<math>\lambda</math>；
*<math>E\,</math> 为能量，即哈密顿量的本征值；
*<math>{\rm d}\tau</math>为积分微元。上述积分在全空间进行。
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==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.<ref>{{cite book|last=Jensen|first=Frank|title=Introduction to Computational Chemistry|url=https://archive.org/details/introductiontoco00jens_022|publisher=John Wiley & Sons|location=West Sussex|date=2007|isbn=0-470-01186-6|page=[https://archive.org/details/introductiontoco00jens_022/page/n342 322]}}</ref>

The proof also employs an identity of normalized wavefunctions&nbsp;– 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

:<math>\hat{H}_{\lambda}|\psi(\lambda)\rangle = E_{\lambda}|\psi(\lambda)\rangle,</math>
:<math>\langle\psi(\lambda)|\psi(\lambda)\rangle = 1 \Rightarrow \frac{\partial}{\partial\lambda}\langle\psi(\lambda)|\psi(\lambda)\rangle =0.</math>

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:

:<math>
\begin{align}
\frac{d E_{\lambda}}{d\lambda} &= \frac{d}{d\lambda}\langle\psi(\lambda)|\hat{H}_{\lambda}|\psi(\lambda)\rangle \\
&=\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 \\
&=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 \\
&=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 \\
&=\bigg\langle\psi(\lambda)\bigg|\frac{d\hat{H}_{\lambda}}{d\lambda}\bigg|\psi(\lambda)\bigg\rangle.
\end{align}
</math>

For a deep critical view of the proof see <ref>{{cite journal|last=Carfì|first=David|date=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>

==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 [[molecular geometry|equilibrium geometries]]&nbsp;– 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

:<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>

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

:<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>

Only two components of the Hamiltonian contribute to the required derivative&nbsp;– 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|date=2006|page=620|isbn=0-444-52227-1}}</ref>

:<math>
\begin{align}
\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{align}
</math>

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:

:<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>

===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 [[Hydrogen-like atom|radial Schrödinger equation for a hydrogen atom]] is

:<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>

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:

:<math>\frac{\partial \hat{H}_{l}}{\partial l} = \frac{\hbar^{2}}{2\mu r^{2}}(2l+1).</math>

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|date=2002|isbn=0-521-65124-7|page=186}}</ref>

:<math>
\begin{align}
\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}|\psi_{nl}\bigg\rangle \\
&=\frac{2\mu}{\hbar^{2}}\frac{1}{2l+1}\frac{\partial E_{n}}{\partial l} \\
&=\frac{2\mu}{\hbar^{2}}\frac{1}{2l+1}\frac{\partial E_{n}}{\partial n}\frac{\partial n}{\partial l} \\
&=\frac{2\mu}{\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{align}
</math>

===Van der Waals forces===

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 
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/''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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== 随时间变化的波函数的费曼–海尔曼定理 ==
因为一个一般的随时间变化的波函数满足[[含时薛定谔方程]]，所以费曼–海尔曼定理'''不再'''适用。

但是，该波函数满足以下关系：

:<math>
\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
</math>
其中&psi;满足：
:<math>
i\hbar\frac{\partial\Psi_\lambda(t)}{\partial t}=H_\lambda\Psi_\lambda(t)
</math>

=== 证明 ===

以下证明只依赖于薛定谔方程，以及对于&lambda;和t求偏导时，可以互换顺序的假设。

:<math>
\begin{align}
\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 \\
&= i\hbar \frac{\partial}{\partial\lambda}\bigg\langle\Psi_\lambda(t)\bigg|\frac{\partial\Psi_\lambda(t)}{\partial t}\bigg\rangle
 - i\hbar\bigg\langle\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg|\frac{\partial\Psi_\lambda(t)}{\partial t}\bigg\rangle
+  i\hbar\bigg\langle\frac{\partial\Psi_\lambda(t)}{\partial t}\bigg|\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg\rangle \\
&= i\hbar \bigg\langle\Psi_\lambda(t)\bigg| \frac{\partial^2\Psi_\lambda(t)}{\partial\lambda \partial t}\bigg\rangle
+  i\hbar\bigg\langle\frac{\partial\Psi_\lambda(t)}{\partial t}\bigg|\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg\rangle \\
&= i \hbar \frac{\partial}{\partial t}\bigg\langle\Psi_\lambda(t)\bigg|\frac{\partial \Psi_\lambda(t)}{\partial \lambda}\bigg\rangle
\end{align}
</math>

== 参考 ==
{{reflist}}

[[Category:量子力学]]
[[Category:物理定理]]
[[Category:理查德·费曼]]
<!--[[Category:Intermolecular forces]]
[[Category:Richard Feynman]]-->