Independent electron approximation

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). Neglecting spin, the Hamiltonian of the system takes the form:[1]

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

, where ${\displaystyle \hbar }$ is the reduced Planck's constant, e is the elementary charge, m_e is the electron rest mass, and ${\displaystyle \nabla _{i}}$ is the gradient operator for electron i. The capitalized ${\displaystyle \mathbf {R} _{I}}$ is the I-th lattice location (the equilibrium position of the I-th nuclei) and the lowercase ${\displaystyle \mathbf {r} _{i}}$ 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 (one for each electron), which greatly simplifies 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.[1] If the electron-electron interaction term is sufficiently small, however, the Coulomb interactions terms can be approximated by an effective potential term, which neglects electron-electron interactions.[1] This is known as the independent electron approximation.[1] Bloch's theorem relies on this approximation by setting the effective potential term to a periodic potential of the form ${\displaystyle V(\mathbf {r} )}$ which satisfies ${\displaystyle V(\mathbf {r} +\mathbf {R} _{j})=V(\mathbf {r} )}$, where ${\displaystyle \mathbf {R} _{j}}$ is any reciprocal lattice vector (see Bloch's theorem).[1] This approximation can be formalized using methods from the Hartree-Fock approximation or density functional theory.[1]

• Strongly correlated material

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