In physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include atomic physics

Atomic physics

Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

 and neutron scattering

Neutron scattering

Neutron scattering,the scattering of free neutrons by matter,is a physical processand an experimental technique using this processfor the investigation of materials.Neutron scattering as a physical process is of primordial importance...

.

Atomic physics

The pseudopotential is an attempt to replace the complicated effects of the motion of the core

Core electron

Core electrons are the electrons in an atom that are not valence electrons and therefore do not participate in bonding. An example: the carbon atom has a total of 6 electrons, 4 of them being valence electrons. So the remaining 2 electrons must be core electrons.They are so tightly bound to the...

 (i.e. non-valence

Valence electron

In chemistry, valence electrons are the electrons of an atom that can participate in the formation of chemical bonds with other atoms. Valence electrons are the "own" electrons, present in the free neutral atom, that combine with valence electrons of other atoms to form chemical bonds. In a single...

) electrons of an atom

Atom

The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

 and its nucleus with an effective potential

Potential

*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...

, or pseudopotential, so that 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....

 contains a modified effective potential term instead of the Coulombic

Coulomb's law

Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...

 potential term for core electrons normally found in the Schrödinger equation. The pseudopotential approximation was first introduced by 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...

 in the 1934. By construction of this pseudopotential, the valence wavefunction

Wavefunction

Not 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...

 generated is also guaranteed to be orthogonal to all the core states.

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 nodeless pseudo-wavefunctions. In this approach 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.
Norm-conserving 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 cutoff radius .
Pseudopotentials with larger cutoff 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:

1. Reduction of basis set size
2. Reduction of number of electrons
3. Inclusion of relativistic and other effects

Approximations:

1. One-electron picture.
2. 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.

Norm-conserving pseudopotentials enforce the condition that, outside of a cutoff radius, the norm of each pseudo-wavefunction be identical to its corresponding all-electron wavefunction. Ultrasoft pseudopotentials relax the norm-conserving constraint to reduce the basis-set size further.

Fermi pseudopotential

Enrico Fermi

Enrico Fermi

Enrico Fermi was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics...

 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

Planck constant

The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...

 divided by , is the mass

Mass

Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

 , is the Dirac delta function

Dirac delta function

The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

, is the neutron scattering length, and the center of mass

Center of mass

In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

 of the nucleus

Atomic nucleus

The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

. The Fourier transform of this -function leads to the constant neutron form factor.