# 科恩－沈吕九方程

${\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+v_{\rm {eff}}(\mathbf {r} )\right)\phi _{i}(\mathbf {r} )=\varepsilon _{i}\phi _{i}(\mathbf {r} )}$

${\displaystyle \rho (\mathbf {r} )=\sum _{i}^{N}|\phi _{i}(\mathbf {r} )|^{2}.}$

## 科恩－沈势

${\displaystyle E[\rho ]=T_{s}[\rho ]+\int d\mathbf {r} \ v_{\rm {ext}}(\mathbf {r} )\rho (\mathbf {r} )+V_{H}[\rho ]+E_{\rm {xc}}[\rho ]}$

${\displaystyle T_{s}[\rho ]=\sum _{i=1}^{N}\int d\mathbf {r} \ \phi _{i}^{*}(\mathbf {r} )\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\right)\phi _{i}(\mathbf {r} ),}$

${\displaystyle v_{\rm {ext}}}$ 是作用在真实系统上的外势（至少包括原子核与电子之间的相互作用势），${\displaystyle V_{H}}$ 是哈特里（库仑）能：

${\displaystyle V_{H}={e^{2} \over 2}\int d\mathbf {r} \int d\mathbf {r} '\ {\rho (\mathbf {r} )\rho (\mathbf {r} ') \over |\mathbf {r} -\mathbf {r} '|}.}$

${\displaystyle E_{\rm {xc}}}$ 是交换相关能量。对虚拟体系总能量表达式右端除动能项之外的部分取电子密度的泛函微商[3]，就得到科恩－沈势的表达式：

${\displaystyle v_{\rm {eff}}(\mathbf {r} )=v_{\rm {ext}}(\mathbf {r} )+e^{2}\int {\rho (\mathbf {r} ') \over |\mathbf {r} -\mathbf {r} '|}d\mathbf {r} '+{\delta E_{\rm {xc}}[\rho ] \over \delta \rho (\mathbf {r} )}.}$

${\displaystyle v_{\rm {xc}}(\mathbf {r} )\equiv {\delta E_{\rm {xc}}[\rho ] \over \delta \rho (\mathbf {r} )}}$

${\displaystyle E=\sum _{i}^{N}\varepsilon _{i}-V_{H}[\rho ]+E_{\rm {xc}}[\rho ]-\int {\delta E_{\rm {xc}}[\rho ] \over \delta \rho (\mathbf {r} )}\rho (\mathbf {r} )d\mathbf {r} }$

## 参考文献

1. ^ Kohn, Walter; Sham, Lu Jeu. Self-Consistent Equations Including Exchange and Correlation Effects. Physical Review. 1965, 140 (4A): A1133–A1138. Bibcode:1965PhRv..140.1133K. doi:10.1103/PhysRev.140.A1133.
2. ^ Parr, Robert G.; Yang, Weitao. Density-Functional Theory of Atoms and Molecules. Oxford University Press. 1994. ISBN 978-0-19-509276-9.
3. ^ http://muchomas.lassp.cornell.edu/P480/Notes/dft/node11.html