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**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \text{geometric series:}\quad \sum_{i=0}^\infty 2^{-i}=2 }**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle W = \frac{1}{2\mu_0}\biggl( \frac{\Psi'^2 \pi^2}{b^2} \sum_{m,n} a_{mn}^2 n^2 \frac{ab}{4} + \frac{\Psi'^2 \pi^2}{a^2} \sum_{m,n} a_{mn}^2 m^2 \frac{ab}{4} + \Psi'^2ab\biggl( \mu^2 + \sum_{m,n} \frac{8a_{mn} \mu^2}{\pi^2 mn} + \frac{a_{mn}^2 \mu^2}{4} \biggr) \biggr) }**

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# Kohn–Sham equations

In physics and quantum chemistry, specifically density functional theory, the **Kohn–Sham equation** is the one electron Schrödinger equation (more clearly, Schrödinger-like equation) of a fictitious system (the "**Kohn–Sham system**") of non-interacting particles (typically electrons) that generate the same density as any given system of interacting particles.^{[1]}^{[2]} The Kohn–Sham equation is defined by a local effective (fictitious) external potential in which the non-interacting particles move, typically denoted as *v _{s}*(

**r**) or

*v*

_{eff}(

**r**), called the

**Kohn–Sham potential**. As the particles in the Kohn–Sham system are non-interacting fermions, the Kohn–Sham wavefunction is a single Slater determinant constructed from a set of orbitals that are the lowest energy solutions to

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \left(-\frac{\hbar^2}{2m}\nabla^2+v_{\rm eff}(\mathbf r)\right)\phi_{i}(\mathbf r)=\varepsilon_{i}\phi_{i}(\mathbf r)}**.

This eigenvalue equation is the typical representation of the **Kohn–Sham equations**. Here, *ε _{i}* is the orbital energy of the corresponding Kohn–Sham orbital,

*φ*, and the density for an

_{i}*N*-particle system is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \rho(\mathbf r)=\sum_i^N |\phi_{i}(\mathbf r)|^2.}**

The Kohn–Sham equations are named after Walter Kohn and Lu Jeu Sham (沈呂九), who introduced the concept at the University of California, San Diego in 1965.

## Kohn–Sham potential

In Kohn-Sham density functional theory, the total energy of a system is expressed as a functional of the charge density as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle E[\rho] = T_s[\rho] + \int d\mathbf r\ v_{\rm ext}(\mathbf r)\rho(\mathbf r) + E_{H}[\rho] + E_{\rm xc}[\rho]}**

where *T _{s}* is the

**Kohn–Sham kinetic energy**which is expressed in terms of the Kohn–Sham orbitals as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle T_s[\rho]=\sum_{i=1}^N\int d\mathbf r\ \phi_i^*(\mathbf r)\left(-\frac{\hbar^2}{2m}\nabla^2\right)\phi_i(\mathbf r)}**.

*v*_{ext} is the external potential acting on the interacting system (at minimum, for a molecular system, the electron-nuclei interaction), *E _{H}* is the Hartree (or Coulomb) energy,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle E_{H}={e^2\over2}\int d\mathbf r\int d\mathbf{r}'\ {\rho(\mathbf r)\rho(\mathbf r')\over|\mathbf r-\mathbf r'|}}**,

and *E*_{xc} is the exchange-correlation energy. The Kohn–Sham equations are found by varying the total energy expression with respect to a set of orbitals, subject to constraints on those orbitals,^{[3]} to yield the Kohn–Sham potential as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle v_{\rm eff}(\mathbf r) = v_{\rm ext}(\mathbf{r}) + e^2\int {\rho(\mathbf{r}')\over|\mathbf r-\mathbf r'|}d\mathbf{r}' + {\delta E_{\rm xc}[\rho]\over\delta\rho(\mathbf r)}}**,

where the last term

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle v_{\rm xc}(\mathbf r)\equiv{\delta E_{\rm xc}[\rho]\over\delta\rho(\mathbf r)}}**

is the exchange-correlation potential. This term, and the corresponding energy expression, are the only unknowns in the Kohn–Sham approach to density functional theory. An approximation that does not vary the orbitals is Harris functional theory.

The Kohn–Sham orbital energies *ε _{i}*, in general, have little physical meaning (see Koopmans' theorem). The sum of the orbital energies is related to the total energy as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle E = \sum_{i}^N \varepsilon_i - E_{H}[\rho] + E_{\rm xc}[\rho] - \int {\delta E_{\rm xc}[\rho]\over\delta\rho(\mathbf r)} \rho(\mathbf{r}) d\mathbf{r}}**.

Because the orbital energies are non-unique in the more general restricted open-shell case, this equation only holds true for specific choices of orbital energies (see Koopmans' theorem).

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