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Density Functional Theory (DFT)

A method based on the electron density rather than the wavefunction to solve the Schrödinger equation, offering computational efficiency.
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The statement of the theorem

The ground state energy EE of a system is a universal functional of the electron density ρ(r)\rho(\mathbf{r}): \nE[ρ]=T[ρ]+Vext[ρ]+Vee[ρ]E[\rho] = T[\rho] + V_{ext}[\rho] + V_{ee}[\rho]\nwhere T[ρ]T[\rho] is the kinetic energy functional, Vext[ρ]V_{ext}[\rho] is the external potential energy functional, and Vee[ρ]V_{ee}[\rho] is the electron-electron interaction functional. The Kohn-Sham equations are used to solve this functional:\n\left(-\frac{\hbar^2}{2m}\nabla^2 + V_{eff}(\mathbf{r})) \phi_i(\mathbf{r}) = \epsilon_i \phi_i(\mathbf{r})\nwith Veff(r)=Vext(r)+12VHartree(r)+VXC(r)V_{eff}(\mathbf{r}) = V_{ext}(\mathbf{r}) + \frac{1}{2}V_{Hartree}(\mathbf{r}) + V_{XC}(\mathbf{r}).