In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the
expectation value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of the derivative of the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified 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
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 ...
, all the forces in the system can be calculated using
classical electrostatics.
The theorem has been proven independently by many authors, including
Paul Güttinger
Paul may refer to:
*Paul (given name), a given name (includes a list of people with that name)
*Paul (surname), a list of people
People
Christianity
* Paul the Apostle (AD c.5–c.64/65), also known as Saul of Tarsus or Saint Paul, early Chri ...
(1932),
Wolfgang Pauli
Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics fo ...
(1933),
Hans Hellmann
Hans Gustav Adolf Hellmann (14 October 1903 – 29 May 1938) was a German theoretical physicist.
Biography
Hellmann was born in Wilhelmshaven, Prussian Hanover. He began studying electrical engineering in University of Stuttgart, Stuttgart, b ...
(1937) and
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...
(1939).
The theorem states
where
*
is a Hamiltonian operator depending upon a continuous parameter
,
*
, is an eigen-
state
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of State
* ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States
* ''Our S ...
(
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
) of the Hamiltonian, depending implicitly upon
,
*
is the energy (eigenvalue) of the state
, i.e.
.
Proof
This proof of the Hellmann–Feynman theorem requires that the
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
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
Møller–Plesset perturbation theory (MP) is one of several quantum chemistry post–Hartree–Fock ab initio methods in the field of computational chemistry. It improves on the Hartree–Fock method by adding electron correlation effects by m ...
, which is not variational.
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
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathema ...
these two conditions are written as
:
:
The proof then follows through an application of the derivative
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
to the
expectation value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of the Hamiltonian viewed as a function of
:
:
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
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
, 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 ''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
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, 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''. 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 force
An intramolecular force (or primary forces) is any force that binds together the atoms making up a molecule or compound, not to be confused with intermolecular forces, which are the forces present between molecules. The subtle difference in the na ...
s 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
corresponds to the coordinates of the nuclei. For a molecule with
electrons with coordinates
, and
nuclei, each located at a specified point
and with nuclear charge
, the
clamped nucleus Hamiltonian is
:
The
-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
:
Insertion of this in to the Hellmann–Feynman theorem returns the
-component of the force on the given nucleus in terms of the
electronic density
In quantum chemistry, electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial va ...
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
The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe t ...
. Promoting
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:
:
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
, 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
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
force."
Hellmann–Feynman theorem for time-dependent wavefunctions
For a general time-dependent wavefunction satisfying the time-dependent
Schrödinger equation
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 ...
, the Hellmann–Feynman theorem is not valid.
However, the following identity holds:
:
For
:
Proof
The proof only relies on the Schrödinger equation and the assumption that partial derivatives with respect to λ and t can be interchanged.
:
Notes
{{DEFAULTSORT:Hellmann-Feynman theorem
Intermolecular forces
Theorems in quantum mechanics
Richard Feynman