Independent electron approximation

The independent electron approximation is a simplification used in complex systems, consisting of many electrons, that approximates the electron-electron interaction in crystals as null. It is used in both the free electron model and the nearly-free electron model and is important in condensed matter physics, where it is used in Bloch's theorem.[1]

While this simplification holds for many systems, electron-electron interactions may be very important for certain properties in materials. For example, the theory covering much of superconductivity is BCS theory, in which the attraction of pairs of electrons to each other, termed "Cooper pairs", is the mechanism behind superconductivity. (See Strongly correlated electrons) One major effect of electron-electron interactions is that electrons distribute around the ions so that they screen the ions in the lattice from other electrons.

Quantum Treatment

In Hamiltonian mechanics, a system is modeled by taking into account the kinetic energy and potential energy of each particle in the system. The Hamiltonian is then written as a sum over each of these particles. In the quantum formulation, for an N-atom crystal with one free electron per atom, this is given by: [1]

${\displaystyle {\mathcal {H}}=\sum _{i=1}^{N}\left({\frac {-\hbar ^{2}\nabla _{i}^{2}}{2m_{e}}}-\sum _{I=0}^{N}{\frac {e^{2}Z_{I}}{\left|{\overrightarrow {r_{i}}}-{\overrightarrow {R_{I}}}\right|}}+{\frac {1}{2}}\sum _{i\neq j}^{N}{\frac {e^{2}}{\left|{\overrightarrow {r_{i}}}-{\overrightarrow {r_{j}}}\right|}}\right)}$

Where ${\displaystyle \hbar }$ is the reduced Planck's constant, ${\displaystyle \nabla _{i}}$ is the gradient operator for electron i, m_e is the mass of the electron, and ${\displaystyle Z_{I}}$ is the atomic number of atom I. The capitalized ${\displaystyle {\overrightarrow {R_{I}}}}$ is the I'th lattice location (the equilibrium position of the I'th nuclei) and the lowercase ${\displaystyle {\overrightarrow {r_{i}}}}$ is the i'th electron position.

Note, the first term is simply the kinetic energy of the i'th electron, given by the square of the quantum mechanical momentum operator divided by 2 times the mass of the electron. The last two terms are simply the Coulomb interaction terms for electron-nuclei and electron-electron interactions, respectively.

The third term in this Hamiltonian represents the electron-electron interaction of the system. If that term were negligible, the Hamiltonian could be decomposed into a set of N decoupled Hamiltonians which can each be solved independently, simplifying the problem dramatically. The electron-electron interaction term, however, prevents this decomposition by ensuring that the Hamiltonian for each electron will include terms for the positions of every other electron in the system. If the system under consideration has only a small electron-electron interaction term, however, the second and third terms can be combined into a new effective potential, which neglects electron-electron interactions.[1] This is known as the independent electron approximation.[1]

See also

• Strongly correlated electrons

References

• Omar, M. Ali (1994). Elementary Solid State Physics, 4th ed. Addison Wesley. ISBN 978-0-201-60733-8.

1. Girvin, Steven M.; Yang, Kun (2019). Modern Condensed Matter Physics (1 ed.). Cambridge University Press. pp. 105–117. ISBN 978-1-107-13739-4.