Hellmann-Feynman theorem
Encyclopedia
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. Its most common application is in the calculation of forces in molecules (with the parameters being the positions of the nuclei) where it states that 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 concepts of classical electrostatics
.
The theorem has been proven independently by many authors, including Paul Güttinger (1932), Wolfgang Pauli
(1933), Hans Hellmann
(1937) and Richard Feynman
(1939).
The theorem states
where
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:
For a deep critical view of the proof see
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
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
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 force on the x-component of the given nucleus in terms of the electronic density
(ρ(r)) and the atomic coordinates and nuclear charges:
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:
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".
, the Hellmann-Feynman theorem is not valid.
However, the following identity holds:
For
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the expectation value
Expectation value (quantum mechanics)
In quantum mechanics, the expectation value is the predicted mean value of the result of an experiment. Despite the name, it is not the most probable value of a measurement...
of the derivative of the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
with respect to that same parameter. Its most common application is in the calculation of forces in molecules (with the parameters being the positions of the nuclei) where it states that once the spatial distribution of the electrons has been determined by solving the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
, all the forces in the system can be calculated using concepts of classical electrostatics
Classical electromagnetism
Classical electromagnetism is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents...
.
The theorem has been proven independently by many authors, including Paul Güttinger (1932), Wolfgang Pauli
Wolfgang Pauli
Wolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...
(1933), Hans Hellmann
Hans Hellmann
Hans Gustav Adolf Hellmann was a German theoretical physicist.Hellmann was born in Wilhelmshaven, Prussian Hanover. He began studying electrical engineering in Stuttgart, but changed to engineering physics after a semester...
(1937) and Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
(1939).
The theorem states
where
is a Hamiltonian operator depending upon a continuous parameter 
,
is a wavefunctionWavefunctionNot to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
(eigenfunctionEigenfunctionIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
) of the Hamiltonian, depending implicitly upon
,
is the energy (eigenvalue) of the wavefunction,
implies an integration over the domain of the wavefunction.
Proof
The proof of the Hellmann–Feynman theorem requires that the wavefunction be an eigenfunction of the Hamiltonian under consideration. However, the wavefunction does not need to be exact. For instance in Hartree–Fock theory the wavefunction is a relatively poor approximation to the true wavefunction, but because it is variationally optimized with respect to the Hamiltonian, the Hellmann–Feynman theorem can be applied. Notable example of where the Hellmann–Feynman is not applicable is for example finite-order Møller–Plesset perturbation theory, 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
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
these two conditions are written as
The proof then follows through an application of the derivative product rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
to the expectation value
Expectation value (quantum mechanics)
In quantum mechanics, the expectation value is the predicted mean value of the result of an experiment. Despite the name, it is not the most probable value of a measurement...
of the Hamiltonian viewed as a function of λ:
For a deep critical view of the proof see
Molecular forces
The most common application of the Hellmann–Feynman theorem is to the calculation of intramolecularIntramolecular
Intramolecular in chemistry describes a process or characteristic limited within the structure of a single molecule, a property or phenomenon limited to the extent of a single molecule.- Examples :...
forces in molecules. This allows for the calculation of equilibrium geometries
Molecular geometry
Molecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism, and biological activity.- Molecular geometry determination...
– 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
Molecular Hamiltonian
In atomic, molecular, and optical physics as well as in quantum chemistry, molecular Hamiltonian is the name given to the Hamiltonian representing the energy of the electrons and nuclei in a molecule...
is
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
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 force on the x-component of the given nucleus in terms of the electronic density
Electronic density
In quantum mechanics, and in particular quantum chemistry, the electronic density is a measure of the probability of an electron occupying an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either...
(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 atomHydrogen-like atom
A hydrogen-like ion is any atomic nucleus with one electron and thus is isoelectronic with hydrogen. Except for the hydrogen atom itself , these ions carry the positive charge e, where Z is the atomic number of the atom. Examples of hydrogen-like ions are He+, Li2+, Be3+ and B4+...
is
which depends upon the discrete azimuthal quantum number
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...
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:Van der Waals forces
In the end of Feynman's paper, he states that, "Van der Waals's forcesVan der Waals force
In physical chemistry, the van der Waals force , named after Dutch scientist Johannes Diderik van der Waals, is the sum of the attractive or repulsive forces between molecules other than those due to covalent bonds or to the electrostatic interaction of ions with one another or with neutral...
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".
Hellmann–Feynman theorem for time-dependent wavefunctions
For a general time-dependent wavefunction satisfying the time-dependent Schrödinger equationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
, 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.