查看“︁局域密度近似”︁的源代码

'''局域密度近似'''（'''local-density approximation''', '''LDA'''）是[[密度泛函理论]]的其中一类交换相关[[能量]][[泛函]]中使用的近似。该近似认为交换相关能量泛函仅仅与[[电子密度]]在空间各点的取值有关（而与其[[梯度]]、[[拉普拉斯]]等无关）。尽管有多种方法都能体现局域密度近似，但在实际中最成功的是基于{{link-en|均匀电子气|UEG}}模型的泛函。下面的讨论，除非特别说明，仅限于这一类泛函。

一般地，对于非自旋极化的体系，局域密度近似的交换相关泛函可以写作：

<center><math>E_{xc}^{\mathrm{LDA}}[\rho] = \int \rho(\mathbf{r})\varepsilon_{\rm xc}(\rho)\ \mathrm{d}\mathbf{r}\ ,</math></center>

<math>\rho</math> 为电子密度，<math>\varepsilon_{\rm xc}</math> 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：

<center><math>E_{\rm xc} = E_{\rm x} + E_{\rm c}\ ,</math></center>

于是问题就变为分别寻找交换项和相关项的表达式。对于均匀电子气模型来说，交换项有着简单的解析式，而相关项只在特殊情况下有着精确的表达式。对相关作用的不同近似能够得到不同的 <math>\varepsilon_{\rm c}</math>。对于实际应用的泛函来说，相关作用能量密度项的形式总是很复杂的。

在构建泛函的过程中，局域密度近似有着重要的地位。基于局域密度近似的泛函是其它更复杂的泛函（如基于[[广义梯度近似]]（GGA）的泛函和[[杂化泛函]]）的基础。一般来说，人们要求所有的泛函都能正确处理均匀电子气模型，因此所有的泛函中都或多或少地包含局域密度近似项。

== 均匀电子气模型 ==
有多种方法构筑仅仅依赖于电子密度的交换相关能量泛函，其中最成功的模型是[[自由电子气模型]]。将 <math>N</math> 个有相互作用的电子放入体积为 <math>V</math> 的空间内，并加入正电荷背景使体系处处处于电中性。然后让 <math>N</math> 和 <math>V</math> 同时趋向无穷，同时保持电子密度 <math>\rho=N/V</math> 有限。此时的波函数可以用平面波表示。对于密度为常数的情形，交换能量密度与密度的平方根成正比。

== 交换能量密度 ==
均匀电子气模型的交换能量密度有着精确的解析解。局域密度近似把这一解析的表达式推广到了电子密度不为常数的情形。把这表达式应用于空间的每一点上，并且在对全空间积分得到下式：
<ref name="parryang">{{cite book|last=Parr|first=Robert G|coauthors=Yang, Weitao|title=Density-Functional Theory of Atoms and Molecules|publisher=Oxford University Press|location=Oxford |year=1994|isbn=978-0-19-509276-9}}</ref><ref>{{cite journal|last=Dirac|first=P. A. M.|year=1930|title=Note on exchange phenomena in the Thomas-Fermi atom|journal=Proc. Cambridge Phil. Roy. Soc.|volume=26|pages=376–385|doi=10.1017/S0305004100016108|issue=3|bibcode = 1930PCPS...26..376D }}</ref>

<center><math>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}\ .</math></center>

可以看出，这种推广只在空间处处电子密度都变化不太大的时候是有效的。<sub>請求解釋</sub>

== 相关能量密度 ==
均匀电子气模型的相关能量密度的解析表达式是未知的，但在高密度极限与低密度极限下（分别对应弱相关与强相关）的表达式是已知的。高密度极限下的表达式为：<ref name="parryang"/>

<center><math>\varepsilon_{c} = A\ln(r_{s}) + B + r_{s}(C\ln(r_{s}) + D)</math></center>

低密度极限下则为：

<center><math>\varepsilon_{c} = \frac{1}{2}\left(\frac{g_{0}}{r_{s}} + \frac{g_{1}}{r_{s}^{3/2}} + \dots\right)</math></center>

式中，维格纳－赛兹半径 <math>r_s</math> 与电子密度的关系为：

<center><math>\frac{4}{3}\pi r_{s}^{3} = \frac{1}{\rho}</math></center>

对均匀电子气模型进行的精确[[量子蒙特卡罗]]模拟得到了中等密度下的相关能量密度。<ref>{{cite journal | title = Ground State of the Electron Gas by a Stochastic Method | author = D. M. Ceperley and B. J. Alder | journal = Phys. Rev. Lett. | volume = 45 | pages = 566–569 | year = 1980 | doi = 10.1103/PhysRevLett.45.566 | bibcode=1980PhRvL..45..566C | issue = 7}}</ref> 常见的局域密度近似相关泛函是通过对这些密度值进行[[内插]]法得到的，同时需要保证在高、低密度极限下正确的行为。下面列出了一些在密度泛函计算中使用到的交换能量密度泛函的符号与其作者。

* Vosko-Wilk-Nusair (VWN) <ref name="vwn">{{cite journal | title = Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis | url = https://archive.org/details/sim_canadian-journal-of-physics_1980-08_58_8/page/1200 | author = S. H. Vosko, L. Wilk and M. Nusair | journal = Can. J. Phys. | volume = 58 | pages = 1200 |  year = 1980 | doi = 10.1139/p80-159 |bibcode = 1980CaJPh..58.1200V | issue = 8 }}</ref>

* Perdew-Zunger (PZ81) <ref>{{cite journal | title = Self-interaction correction to density-functional approximations for many-electron systems | author = J. P. Perdew and A. Zunger | journal = Phys. Rev. B | volume = 23 | pages =  5048 | year = 1981 | doi = 10.1103/PhysRevB.23.5048 |bibcode = 1981PhRvB..23.5048P | issue = 10 }}</ref>

* Cole-Perdew (CP) <ref>{{cite journal | title = Calculated electron affinities of the elements | author = L. A. Cole and J. P. Perdew | journal =  Phys. Rev. A | volume = 25 | pages =  1265  | year = 1982 | doi = 10.1103/PhysRevA.25.1265 |bibcode = 1982PhRvA..25.1265C | issue = 3 }}</ref>

* Perdew-Wang (PW92) <ref name=pw92>{{cite journal | title = Accurate and simple analytic representation of the electron-gas correlation energy | author = John P. Perdew and Yue Wang | journal = Phys. Rev. B | volume = 45 | pages = 13244–13249 | year = 1992 | doi = 10.1103/PhysRevB.45.13244 |bibcode = 1992PhRvB..4513244P | issue = 23 }}</ref>

在上面这些泛函提出之前，甚至在密度泛函理论提出之前，人们广泛使用的是对均匀电子气模型进行[[多体微扰理论|微扰]]计算得到的魏格纳相关能量泛函。<ref name=wigner>{{cite journal | title = On the Interaction of Electrons in Metals | author = E. Wigner | journal = Phys. Rev. | volume = 46 | pages = 1002–1011 | year = 1934 | url = http://link.aps.org/abstract/PR/v46/p1002 | doi = 10.1103/PhysRev.46.1002 | format = abstract |bibcode = 1934PhRv...46.1002W | issue = 11 }}</ref>
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== Spin polarization ==

The extension of density functionals to [[Spin polarization|spin-polarized]] systems is straightforward for exchange, where the exact spin-scaling is known, but for correlation further approximations must be employed. A spin polarized system in DFT employs two spin-densities, ''ρ''<sub>α</sub> and ''ρ''<sub>β</sub> with ''ρ''&nbsp;=&nbsp;''ρ''<sub>α</sub>&nbsp;+&nbsp;''ρ''<sub>β</sub>, and the form of the local-spin-density approximation (LSDA) is

:<math>E_{xc}^{\mathrm{LSDA}}[\rho_{\alpha},\rho_{\beta}] = \int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r})\epsilon_{xc}(\rho_{\alpha},\rho_{\beta})\ .</math>

For the exchange energy, the exact result (not just for local density approximations) is known in terms of the spin-unpolarized functional<ref>{{cite journal|last=Oliver|first=G. L.|coauthors=Perdew, J. P. |year=1979|title=Spin-density gradient expansion for the kinetic energy|journal=Phys. Rev. A|volume=20|pages=397–403|doi=10.1103/PhysRevA.20.397|bibcode = 1979PhRvA..20..397O|issue=2 }}</ref>:

:<math>E_{x}[\rho_{\alpha},\rho_{\beta}] = \frac{1}{2}\bigg( E_{x}[2\rho_{\alpha}] + E_{x}[2\rho_{\beta}] \bigg)\ .</math>

The spin-dependence of the correlation energy density is approached by introducing the relative spin-polarization:

:<math>\zeta(\mathbf{r}) = \frac{\rho_{\alpha}(\mathbf{r})-\rho_{\beta}(\mathbf{r})}{\rho_{\alpha}(\mathbf{r})+\rho_{\beta}(\mathbf{r})}\ .</math>

<math>\zeta = 0\,</math> corresponds to the paramagnetic spin-unpolarized situation with equal
<math>\alpha\,</math> and <math>\beta\,</math> spin densities whereas <math>\zeta = \pm 1</math> corresponds to the ferromagnetic situation where one spin density vanishes. The spin correlation energy density for a given values of the total density and relative polarization, ''ε''<sub>c</sub>(''ρ'',''ς''), is constructed so to interpolate the extreme values. Several forms have been developed in conjunction with LDA correlation functionals.<ref name="vwn"/><ref>{{cite journal|last=von Barth|first=U.|coauthors=Hedin, L.|year=1972|title=A local exchange-correlation potential for the spin polarized case|journal=J. Phys. C: Solid State Phys.|volume=5|pages=1629–1642|doi=10.1088/0022-3719/5/13/012|bibcode = 1972JPhC....5.1629V|issue=13 }}</ref>
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== 交换相关势 ==

与局域密度近似相对应的交换相关势由下式给出：<ref name="parryang"/>

<center><math>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})}</math></center>

在有限体系中，局域密度近似交换相关势在无穷远处以指数形式衰减，这种渐近行为是错误的。真实的交换相关势以慢得多的与距离成反比的速度衰减。这种不正常的渐近行为会影响[[束缚态]]的轨道数，并且无法用来描述[[里德堡态]]。这导致在计算中高估[[HOMO]]能量，使得基于[[库普曼斯定理]]进行的电离能计算结果不正确。进一步地，局域密度近似在描述富电子体系如[[离子#阴离子|负离子]]的时候表现不佳，常常因为无法将额外的电子纳入到束缚态中而给出体系不能稳定存在的错误结论。<ref>{{cite book|last=Fiolhais|first=Carlos|coauthors=Nogueira, Fernando; Marques Miguel|title=A Primer in Density Functional Theory|url=https://archive.org/details/primerdensityfun00fiol_216|publisher=Springer|year=2003|isbn=978-3-540-03083-6|page=[https://archive.org/details/primerdensityfun00fiol_216/page/n72 60]}}</ref><ref>{{cite journal|last=Perdew|first=J.  P. |coauthors=Zunger, Alex |year=1981|title=Self-interaction correction to density-functional approximations for many-electron systems|journal=Phys. Rev. B|volume=23|issue=10|pages=5048–5079|doi=10.1103/PhysRevB.23.5048 |bibcode = 1981PhRvB..23.5048P }}</ref>

== 参考文献 ==
{{reflist|2}}

[[Category:密度泛函理论]]