## Fang-Howard variational approach The main idea of this variational approach is to assume the wave function has the form of $\psi_0(z) = \sqrt{\frac{b^3}{2}} \, z e^{-\frac{bz}{2}}$ and take $b$ as the variational term. The minimum energy is given by, $b= \left( \frac{33 \pi}{2} \, n_s \, a_B^{*2} \right)^{\frac{1}{3}} \frac{1}{a_B^*}$ and ground state energy is, $E_0(n_s) = \frac{5}{4} E_{Ry}^{*} \left( \frac{33 \pi}{2} \, n_s \, a_B^{*2} \right)^{\frac{2}{3}}$ The capacitance is, $\frac{1}{e^2} \frac{dE_0(n_s)}{dn_s} = \frac{55}{96} \frac{1}{\varepsilon \varepsilon_0} \langle z \rangle = \frac{1}{C_{so}}$ The $\langle z \rangle$ is the center of mass of the wave function (WF), or the stand-off distance. So the combined form of the total capacitance becomes what we presented before, $\frac{1}{C_{\text{tot}}} = \frac{1}{C_{\text{geo}}} + \frac{1}{C_q} + \frac{1}{C_{so}}$ Fang-Howard approach is a very cleaver approach without self-consistency calculation or [[Begin of DFT, Hohenberg-Kohn theorems|DFT]]. >[!Info] >See he book [[Semiconductor Nanostructures]] chapter 9.4 for more information.