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.

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

For an example of the Independent electron approximation's usefulness in quantum mechanics, consider an N-atom crystal with one free electron per atom (each with atomic number Z), which has the Hamiltonian:

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

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

The first term in parentheses is called the kinetic energy operator while the last two are simply the Coulomb interaction terms for electron-nucleus and electron-electron interactions, respectively. If the electron-electron term were negligible, the Hamiltonian could be decomposed into a set of N decoupled Hamiltonians, greatly simplifying analysis. The electron-electron interaction term, however, prevents this decomposition by ensuring that the Hamiltonian for each electron will include terms for the position of every other electron in the system. If the electron-electron interaction term is sufficiently small, however, the Coulomb interactions terms can be approximated by a new effective potential term, which neglects electron-electron interactions. This is known as the independent electron approximation. Bloch's theorem relies on this approximation by setting the effective potential term to a periodic potential of the form $V(r+R_{j})=V(r)$, where $R_{j}$ is any reciprocal lattice vector (see Bloch wave).

See also

• Strongly correlated electrons

References

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