查看“︁多组态自洽场方法”︁的源代码

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'''多组态自洽场方法'''（'''Multi-configurational self-consistent field''', '''MCSCF'''）是[[量子化学]]中的一种计算方法，主要用于在[[哈特里－福克方法]]和[[密度泛函理论]]不足以给出良好的参考态函数的时候（例如，在键断裂过程中，或者分子基态与低激发态能量近简并的情形）产生定量正确的参考态函数。它用一组[[组态态函数]]的线性组合来近似真实的电子[[波函数]]。在 MCSCF 方法中，既改变组态态函数前的线性组合系数，也改变每一个组态态函数里面的基函数前的线性组合系数，同时改变两者以使能量达到最小值，就得到变分的电子波函数。这个方法可以视作[[组态相互作用方法]]和[[哈特里－福克方法]]的组合。 

MCSCF 波函数经常用作[[多参考组态相互作用]]或多参考态微扰理论（如[[完全活性空间微扰理论]]）计算的参考态，这些方法可以处理一些很极端的情形，并且，抛开计算资源的限制不谈，这些方法能够在其它方法失效的情况下得到可靠的分子基态与激发态波函数。

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== Introduction ==

For the simplest single bond, found in the ''H''<sub>2</sub> molecule, [[molecular orbitals]] can always be written in terms of two functions χ<sub>iA</sub> and χ<sub>''iB''</sub> (which are [[atomic orbitals]] with small corrections) located at the two nuclei,

:<math>\varphi_i = N_i(\chi_{iA} \pm \chi_{iB}), \, </math>

where ''N''<sub>''i''</sub> is a normalization constant. The ground state wavefunction for ''H''<sub>2</sub> at the equilibrium geometry is dominated by the configuration (''φ''<sub>1</sub>)<sup>2</sup>, which means the molecular orbital ''φ''<sub>1</sub> is nearly doubly occupied. The [[Hartree–Fock]] model ''assumes'' it is doubly occupied, which leads to a total wavefunction of

:<math>\Phi_1 = \varphi_1(\mathbf{r}_1)\varphi_1(\mathbf{r}_2)\Theta_{2,0},</math>

where Θ<sub>2,0</sub> is the singlet (''S''&nbsp;=&nbsp;0) spin function for two electrons. The molecular orbitals in this case ''φ''<sub>1</sub> are taken as sums of 1s atomic orbitals on both atoms, namely ''N''<sub>1</sub>(1s<sub>A</sub>&nbsp;+&nbsp;1s<sub>B</sub>). Expanding the above equation into atomic orbitals yields

:<math>\Phi_1 = N_1^2 \left[ 1s_A(\mathbf{r}_1)1s_A(\mathbf{r}_2) + 1s_A(\mathbf{r}_1)1s_B(\mathbf{r}_2) + 1s_B(\mathbf{r}_1)1s_A(\mathbf{r}_2) + 1s_B(\mathbf{r}_1)1s_B(\mathbf{r}_2) \right] \Theta_{2,0}.</math>

This Hartree-Fock model gives a reasonable description of H<sub>2</sub> around the equilibrium geometry - about 0.735Å for the bond length (compared to a 0.746Å experimental value) and 84 kcal/mol for the bond energy (exp. 109 kcal/mol). This is typical of the HF model, which usually describes closed shell systems around their equilibrium geometry quite well. At large separations, however, the terms describing both electrons located at one atom remain, which corresponds to dissociation to H<sup>+</sup> + H<sup>&minus;</sup>, which has a much larger energy than H + H. Therefore, the persisting presence of ionic terms leads to an unphysical solution in this case.

Consequently, the HF model cannot be used to describe dissociation processes with open shell products. The most straightforward solution to this problem is introducing coefficients in front of the different terms in Ψ<sub>1</sub>:

:<math>\Psi_1 = C_\text{Ion}\Phi_\text{Ion} + C_\text{Cov}\Phi_\text{Cov},</math>

which forms the basis for the [[valence bond theory|valence bond]] description of [[chemical bond]]s. With the coefficients C<sub>Ion</sub> and C<sub>Cov</sub> varying, the wave function will have the correct form, with C<sub>Ion</sub>=0 for the separated limit and C<sub>Ion</sub> comparable to C<sub>Cov</sub> at equilibrium. Such a description, however, uses non-orthogonal basis functions, which complicates its mathematical structure. Instead, multiconfiguration is achieved by using orthogonal molecular orbitals. After introducing an anti-bonding orbital

:<math>\phi_2 = N_2 (1s_A - 1s_B),\,</math>

the total wave function of H<sub>2</sub> can be written as a linear combination of configurations built from bonding and anti-bonding orbitals:

:<math>\Psi_{MC} = C_1\Phi_1 + C_2\Phi_2,</math>

where Φ<sub>2</sub> is the electronic configuration (φ<sub>2</sub>)<sup>2</sup>. In this multiconfigurational description of the H<sub>2</sub> chemical bond, C<sub>1</sub>&nbsp;=&nbsp;1 and C<sub>2</sub>&nbsp;=&nbsp;0 close to equilibrium, and C<sub>1</sub> will be comparable to C<sub>2</sub> for large separations.<ref>
  {{cite book
  | last = McWeeny
  | first = Roy
  | authorlink = Roy McWeeny
  | title = [[Charles Coulson|Coulson]]'s Valence
  | publisher = Oxford University Press
  | year = 1979
  | location = Oxford
  | pages = 124–129
  | isbn = 0-19-855145-2}} 
</ref>
-->

== 完全活性空间自洽场方法 ==

一种特别重要的 MCSCF 方法是'''[[完全活性空间]]自洽场方法'''（'''CASSCF'''）。完全活性空间又称为'''全优化反应空间'''（'''full-optimized reaction space'''），相应的方法称为 '''FORS-MCSCF'''。CASSCF 与 FORS-MCSCF是同义词。在 CASSCF 方法中，展开式中包括所有给定数目的电子在给定数目的轨道上分布所得的所有[[组态态函数]]。例如，对[[一氧化氮]]分子进行 CASSCF(11,8) 计算意味着波函数展开式中包含11个价电子在8个分子轨道上自由分配所能得到的全部状态态函数。
<ref>
{{cite book
  | last = Jensen
  | first = Frank
  | title = Introduction to Computational Chemistry
  | url = https://archive.org/details/introductiontoco00jens_022
  | publisher = John Wiley and Sons
  | year = 2007
  | pages = [https://archive.org/details/introductiontoco00jens_022/page/n153 133]–158
  | location = Chichester, England
  | isbn = 0-470-01187-4}}
</ref><ref>{{cite book
  | last = Cramer  | first = Christopher J.
  | title = Essentials of Computational Chemistry
  | url = https://archive.org/details/essentialsofcomp0000cram  | publisher = John Wiley & Sons, Ltd.  | year = 2002  | location = Chichester
  | pages = [https://archive.org/details/essentialsofcomp0000cram/page/191 191]–232  | isbn = 0-471-48552-7}}</ref>

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== Restricted active space SCF ==

Since the number of CSFs quickly increases with the number of active orbitals, along with the computational cost, it may be desirable to use a smaller set of CSFs. One way to make this selection is to restrict the number of electrons in certain subspaces, done in the '''[[restricted active space SCF]] method''' ('''RASSCF'''). One could, for instance, allow only single and double excitations from some strongly-occupied subset of active orbitals, or restrict the number of electrons to at most 2 in another subset of active orbitals.
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== 参见 ==
* [[哈特里－福克方法]]

==参考文献==
{{reflist}}

==拓展阅读==
*{{cite book
  | last = Cramer
  | first = Christopher J.
  | title =`Essentials of Computational Chemistry
  | url = https://archive.org/details/essentialsofcomp0000cram
  | publisher = John Wiley and Sons
  | year = 2002
  | location = Chichester
  | isbn = 0-471-48552-7
}}


[[Category:量子化学]]
[[Category:電子結構方法]]