![Sketch Pseudopotentials](https://upload.wikimedia.org/wikipedia/commons/f/fb/Sketch_Pseudopotentials.png)
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include
atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
and
neutron scattering
Neutron scattering, the irregular dispersal of free neutrons by matter, can refer to either the naturally occurring physical process itself or to the man-made experimental techniques that use the natural process for investigating materials. Th ...
. The pseudopotential approximation was first introduced by
Hans Hellmann in 1934.
Atomic physics
The pseudopotential is an attempt to replace the complicated effects of the motion of the
core
Core or cores may refer to:
Science and technology
* Core (anatomy), everything except the appendages
* Core (manufacturing), used in casting and molding
* Core (optical fiber), the signal-carrying portion of an optical fiber
* Core, the centra ...
(i.e. non-
valence)
electrons
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
of an
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
and its nucleus with an effective
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
, or pseudopotential, so that 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 ...
contains a modified effective potential term instead of the
Coulombic
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
potential term for core electrons normally found in the Schrödinger equation.
The pseudopotential is an effective potential constructed to replace the atomic all-electron potential (full-potential) such that core states are eliminated ''and'' the valence electrons are described by pseudo-wavefunctions with significantly fewer nodes. This allows the pseudo-wavefunctions to be described with far fewer
Fourier modes, thus making
plane-wave basis sets practical to use. In this approach usually only the chemically active valence electrons are dealt with explicitly, while the core electrons are 'frozen', being considered together with the nuclei as rigid non-polarizable ion cores. It is possible to self-consistently update the pseudopotential with the chemical environment that it is embedded in, having the effect of relaxing the frozen core approximation, although this is rarely done. In codes using local basis functions, like Gaussian, often effective core potentials are used that only freeze the core electrons.
First-principles pseudopotentials are derived from an atomic reference state, requiring that the pseudo- and all-electron valence eigenstates have the same energies and amplitude (and thus density) outside a chosen core cut-off radius
.
Pseudopotentials with larger cut-off radius are said to be ''softer'', that is more rapidly convergent, but at the same time less ''transferable'', that is less accurate to reproduce realistic features in different environments.
Motivation:
# Reduction of basis set size
# Reduction of number of electrons
# Inclusion of relativistic and other effects
Approximations:
# One-electron picture.
# The small-core approximation assumes that there is no significant overlap between core and valence wave-function. Nonlinear core corrections
or "semicore" electron inclusion
deal with situations where overlap is non-negligible.
Early applications of pseudopotentials to atoms and solids based on attempts to fit atomic spectra achieved only limited success. Solid-state pseudopotentials achieved their present popularity largely because of the successful fits by Walter Harrison to the nearly free electron Fermi surface of aluminum (1958) and by
James C. Phillips to the covalent energy gaps of silicon and germanium (1958). Phillips and coworkers (notably
Marvin L. Cohen and coworkers) later extended this work to many other semiconductors, in what they called "semiempirical pseudopotentials".
Norm-conserving pseudopotential
''Norm-conserving'' and ''ultrasoft'' are the two most common forms of pseudopotential used in modern
plane-wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, t ...
electronic structure codes. They allow a basis-set with a significantly lower cut-off (the frequency of the highest Fourier mode) to be used to describe the electron wavefunctions and so allow proper numerical convergence with reasonable computing resources. An alternative would be to augment the basis set around nuclei with atomic-like functions, as is done in
LAPW. Norm-conserving pseudopotential was first proposed by Hamann, Schlüter, and Chiang (HSC) in 1979. The original HSC norm-conserving pseudopotential takes the following form:
:
where
projects a one-particle wavefunction, such as one Kohn-Sham orbital, to the angular momentum labeled by
.
is the pseudopotential that acts on the projected component. Different angular momentum states then feel different potentials, thus the HSC norm-conserving pseudopotential is non-local, in contrast to local pseudopotential which acts on all one-particle wave-functions in the same way.
Norm-conserving pseudopotentials are constructed to enforce two conditions.
1. Inside the cut-off radius
, the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
of each pseudo-wavefunction be identical to its corresponding all-electron wavefunction:
:
,
:where
and
are the all-electron and pseudo reference states for the pseudopotential on atom
.
2. All-electron and pseudo wavefunctions are identical outside cut-off radius
.
Ultrasoft pseudopotentials
Ultrasoft pseudopotentials relax the norm-conserving constraint to reduce the necessary basis-set size further at the expense of introducing a generalized eigenvalue problem.
With a non-zero difference in norms we can now define:
:
,
and so a normalised eigenstate of the pseudo Hamiltonian now obeys the generalized equation
:
,
where the operator
is defined as
:
,
where
are projectors that form a
dual basis
In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with th ...
with the pseudo reference states inside the cut-off radius, and are zero outside:
:
.
A related technique is the
projector augmented wave (PAW) method.
Fermi pseudopotential
Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and ...
introduced a pseudopotential,
, to describe the scattering of a free neutron by a nucleus. The scattering is assumed to be
''s''-wave scattering, and therefore spherically symmetric. Therefore, the potential is given as a function of radius,
:
:
,
where
is the
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
divided by
,
is the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
,
is the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
,
is the bound coherent neutron
scattering length The scattering length in quantum mechanics describes low-energy scattering. For potentials that decay faster than 1/r^3 as r\to \infty, it is defined as the following low-energy limit (mathematics), limit:
:
\lim_ k\cot\delta(k) =- \frac\;,
wher ...
, and
the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of the
nucleus
Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to:
*Atomic nucleus, the very dense central region of an atom
*Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA
Nucle ...
. The Fourier transform of this
-function leads to the constant
neutron form factor.
Phillips pseudopotential
James Charles Phillips
James Charles Phillips (born March 9, 1933) is an American physicist and a member of the National Academy of Science (1978). Phillips invented the exact theory of the ionicity of chemical bonding in semiconductors, as well as new theories of compa ...
developed a simplified pseudopotential while at
Bell Labs
Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984),
then AT&T Bell Laboratories (1984–1996)
and Bell Labs Innovations (1996–2007),
is an American industrial research and scientific development company owned by mult ...
useful for describing silicon and germanium.
See also
*
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 ...
*
Projector augmented wave method
The projector augmented wave method (PAW) is a technique used in ab initio electronic structure calculations. It is a generalization of the pseudopotential and linear augmented-plane-wave methods, and allows for density functional theory calculat ...
*
Marvin L. Cohen
*
Alex Zunger
References
Pseudopotential libraries
Pseudopotential Library: A community website for pseudopotentials/effective core potentials developed for high accuracy correlated many-body methods such as quantum Monte Carlo and quantum chemistry
: This webpage maintained by th
provides a searchable database of pseudopotentials for density functional codes as well as links to pseudopotential generators, converters, and other online databases.
Vanderbilt Ultra-Soft Pseudopotential Site: Website of
David Vanderbilt with links to codes that implement ultrasoft pseudopotentials and libraries of generated pseudopotentials.
GBRV pseudopotential site: This site hosts the GBRV pseudopotential library
PseudoDojo: This site collates tested pseudo potentials sorted by type, accuracy, and efficiency, shows information on convergence of various tested properties and provides download options.
SSSP: Standard Solid State Pseudopotentials
Further reading
*
*
*
*
*
*
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Computational physics
Electronic structure methods
Quantum mechanical potentials