# 局域密度近似

${\displaystyle E_{xc}^{\mathrm {LDA} }[\rho ]=\int \rho (\mathbf {r} )\varepsilon _{\rm {xc}}(\rho )\ \mathrm {d} \mathbf {r} \ ,}$

${\displaystyle \rho }$ 为电子密度，${\displaystyle \varepsilon _{\rm {xc}}}$ 为交换相关能量密度，它仅仅是电子密度的函数。交换相关能可以分解为交换项与相关项：

${\displaystyle E_{\rm {xc}}=E_{\rm {x}}+E_{\rm {c}}\ ,}$

## 交换能量密度

${\displaystyle E_{x}^{\mathrm {LDA} }[\rho ]=-{\frac {3}{4}}\left({\frac {3}{\pi }}\right)^{1/3}\int \rho (\mathbf {r} )^{4/3}\ \mathrm {d} \mathbf {r} \ .}$

## 相关能量密度

${\displaystyle \varepsilon _{c}=A\ln(r_{s})+B+r_{s}(C\ln(r_{s})+D)}$

${\displaystyle \varepsilon _{c}={\frac {1}{2}}\left({\frac {g_{0}}{r_{s}}}+{\frac {g_{1}}{r_{s}^{3/2}}}+\dots \right)}$

${\displaystyle {\frac {4}{3}}\pi r_{s}^{3}={\frac {1}{\rho }}}$

• Vosko-Wilk-Nusair (VWN) [4]
• Perdew-Zunger (PZ81) [5]
• Cole-Perdew (CP) [6]
• Perdew-Wang (PW92) [7]

## 交换相关势

${\displaystyle v_{xc}^{\mathrm {LDA} }(\mathbf {r} )={\frac {\delta E^{\mathrm {LDA} }}{\delta \rho (\mathbf {r} )}}=\varepsilon _{xc}(\rho (\mathbf {r} ))+\rho (\mathbf {r} ){\frac {\partial \varepsilon _{xc}(\rho (\mathbf {r} ))}{\partial \rho (\mathbf {r} )}}}$

## 参考文献

1. Parr, Robert G; Yang, Weitao. Density-Functional Theory of Atoms and Molecules. Oxford: Oxford University Press. 1994. ISBN 978-0-19-509276-9.
2. ^ Dirac, P. A. M. Note on exchange phenomena in the Thomas-Fermi atom. Proc. Cambridge Phil. Roy. Soc. 1930, 26 (3): 376–385. Bibcode:1930PCPS...26..376D. doi:10.1017/S0305004100016108.
3. ^ D. M. Ceperley and B. J. Alder. Ground State of the Electron Gas by a Stochastic Method. Phys. Rev. Lett. 1980, 45 (7): 566–569. Bibcode:1980PhRvL..45..566C. doi:10.1103/PhysRevLett.45.566.
4. ^ S. H. Vosko, L. Wilk and M. Nusair. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can. J. Phys. 1980, 58 (8): 1200. Bibcode:1980CaJPh..58.1200V. doi:10.1139/p80-159.
5. ^ J. P. Perdew and A. Zunger. Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B. 1981, 23 (10): 5048. Bibcode:1981PhRvB..23.5048P. doi:10.1103/PhysRevB.23.5048.
6. ^ L. A. Cole and J. P. Perdew. Calculated electron affinities of the elements. Phys. Rev. A. 1982, 25 (3): 1265. Bibcode:1982PhRvA..25.1265C. doi:10.1103/PhysRevA.25.1265.
7. ^ John P. Perdew and Yue Wang. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B. 1992, 45 (23): 13244–13249. Bibcode:1992PhRvB..4513244P. doi:10.1103/PhysRevB.45.13244.
8. ^ E. Wigner. On the Interaction of Electrons in Metals (abstract). Phys. Rev. 1934, 46 (11): 1002–1011. Bibcode:1934PhRv...46.1002W. doi:10.1103/PhysRev.46.1002.
9. ^ Fiolhais, Carlos; Nogueira, Fernando; Marques Miguel. A Primer in Density Functional Theory. Springer. 2003: 60. ISBN 978-3-540-03083-6.
10. ^ Perdew, J. P.; Zunger, Alex. Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B. 1981, 23 (10): 5048–5079. Bibcode:1981PhRvB..23.5048P. doi:10.1103/PhysRevB.23.5048.