查看“︁组态态函数”︁的源代码

'''组态态函数'''（'''configuration state function'''，'''CSF'''）在[[量子化学]]中用于指称一组[[斯莱特行列式]]的具有确定对称性的线性组合（也可以是单个具有确定对称性的斯莱特行列式）。组态态函数不应与[[电子组态]]的概念相混淆。

==定义==
CSF 在[[量子化学]]中用于指称一组[[斯莱特行列式]]的具有确定对称性的线性组合。在构建时需要使得得到的 CSF 具有与体系波函数<math> \Psi</math>相同的量子数。在[[组态相互作用方法]]中，体系波函数可以表示为 CSF 的线性组合。<ref>
{{cite book
|last=Engel |first=T.
|year=2006
|title=Quantum Chemistry and Spectroscopy
|url=https://archive.org/details/physicalchemistr0000enge |publisher=[[Pearson PLC]]
|isbn= 0-8053-3842-X
}}</ref>如下式所示：

<center><math> \Psi = \sum_k c_k \psi_k </math></center>

式中 <math>\psi_k</math> 表示 CSF。式中的线性组合系数 <math>c_k</math> 可以通过下面的方法求得：用上面展开式给出的 <math>\Psi</math> 来计算哈密顿量的矩阵，将矩阵对角化后，相应的本征矢就给出了所有的展开系数。CSF 还用于代替单个斯莱特行列式来进行[[多组态自洽场方法]]的计算。

对于原子结构，CSF 同时是下列算符的本征函数：
* 轨道[[角动量]]平方算符，<math> \hat{L}^2 </math>；
* 轨道角动量 <math>z</math> 分量算符，<math> \hat {L}_z </math>；
* [[自旋]]角动量平方算符，<math> \hat{S}^2 </math>；
* 自旋角动量 <math>z</math> 分量算符，<math> \hat {S}_z </math>。

在线型分子中，<math> \hat{L}^2 </math>与哈密顿不对易，因此 CSF 不再是 <math> \hat{L}^2 </math> 的本征函数。但是，轨道角动量 <math>z</math> 分量量子数仍然是好量子数，CSF 仍然是 <math> \hat {L}_z </math>， <math> \hat{S}^2 </math> 和 <math> \hat {S}_z </math> 的本征函数。在多原子非线型分子中，CSF 必须与分子点群的不可约表示具有相同的对称性性质。这是因为哈密顿量与相应的点操作算符对易。<ref>
{{cite book
 |last=Pilar |first = F. L.
 |year=1990
 |title= Elementary Quantum Chemistry
 |edition=2nd
 |publisher=[[Dover Publications]]
 |isbn= 0-486-41464-7
}}</ref> 在这种情况下，<math> \hat{S}^2 </math> and  <math> \hat {S}_z </math> 对应的量子数仍然是好量子数。CSF 仍是它们的本征函数。

==从组态到组态态函数==
CSF 是从组态里面得到的。组态是电子在轨道上分布的一种方式。<math>1s^2</math> 和 <math>1\pi^2</math> 都是组态的例子，前者是原子组态，后者是分子组态。

对于每个给定的组态，我们可以构造数个 CSF。CSF 有时也叫做 N 粒子对称匹配基函数（N-particle symmetry adapted basis functions）。与给定的组态相关联的电子数目是一定的（用 <math>N</math> 表示）。从组态构造 CSF 的时候，需要考虑与该组态相关的[[自旋轨道]]。

例如，与 <math>1s</math> 轨道相关的自旋轨道有两个：

<center><math>1s\alpha \;\;\; 1s\beta</math></center>

式中 <math>\alpha, \;\;\; \beta</math> 分别表示单电子的自旋向上和向下的自旋本征函数。类似地，对线型分子（<math>C_{\infty\rm v}</math> 点群）的 <math>1\pi</math> 轨道，有四个对应的自旋轨道：

<center><math>1\pi(+)\alpha, \; 1\pi(+)\beta, \; 1\pi(-)\alpha, \; 1\pi(-)\beta</math></center>

这是因为 <math>\pi</math> 轨道对应的轨道角动量 <math>z</math> 分量量子数有两个：<math>+1</math> 与 <math>-1</math>。.

我们可以把这些自旋轨道（设其总个数为 <math>M</math>）视作各自可以容纳一个电子的箱子。考虑将 <math>N</math> 个电子分配到 <math>M</math> 个箱子中的所有方式。每一种方式对应一个斯莱特行列式 <math>D_i</math>。这样的斯莱特行列式的数目由[[组合数]]给出。由于电子不可分辨，电子与箱子的相对顺序是无关紧要的。

下一步是构造 CSF，为了得到 <math> \hat{S}^2 </math> 的本征函数（对于原子结构，同时还要求是 <math> \hat{L}^2 </math> 的本征函数），需要对这些斯莱特行列式进行线性组合，线性组合的系数 <math>c_i</math> 可以从[[克莱布施－戈登系数]]得出。于是每一个 CSF 都具有下列形式：

<center><math>\sum_i c_i \; D_i</math></center>

Löwdin投影算符法<ref>
{{Cite journal
 |last= Crossley |first=R. J. S.
 |year=1977
 |title=On Löwdin's projection operators for angular momentum. I
 |url= https://archive.org/details/sim_international-journal-of-quantum-chemistry_1977-06_11_6/page/917 |journal=[[International Journal of Quantum Chemistry]]
 |volume= 11 | issue= 6 |pages =917–929
 |doi=10.1002/qua.560110605
}}</ref>也可以用来求解线性组合的系数。对于给定的任意一组行列式 <math>D_i</math> 能够找到几组不同的线性组合系数。<ref>
{{cite book
 |last=Nesbet |first=R. K.
 |year=2003
 |chapter= Section 4.4
 |editor1-last=Huo |editor1-first=W. M .
 |editor2-last=Gianturco |editor2-first=F. A.
 |title=Variational principles and methods in theoretical physics and chemistry
 |url=https://archive.org/details/variationalprinc00nesb_191 |page=[https://archive.org/details/variationalprinc00nesb_191/page/n63 49]
 |publisher=[[Cambridge University Press]]
 |isbn =0-521-80391-8
}}</ref> 每一组对应一个 CSF。这实际上体现了总自旋角动量与总轨道角动量之间的内在耦合。

<!--
==A genealogical algorithm for CSF construction==
At the most fundamental level, a configuration state function can be constructed  
* from a set of <math>M</math> orbitals 
and 
* a number <math>N</math> of electrons 
using the following genealogical algorithm:
# distribute the <math>N</math> over the set of <math>M</math> orbitals giving a configuration
# for each orbital the possible quantum number couplings (and therefore wavefunctions for the individual orbitals) are known from basic quantum mechanics; for each orbital choose one of the permitted couplings but leave the z-component of the total spin, <math>S_z</math> undefined.
# check that the spatial coupling of all orbitals matches that required for the system wavefunction. For a molecule exhibiting <math>C_{\infty v}</math> or <math>D_{\infty h}</math> this is achieved by a simple linear summation of the coupled
<math>\lambda</math> value for each orbital; for molecules whose nuclear framework transforms according to <math>D_{2h}</math> symmetry, or one of its sub-groups, the group product table has to be used to find the product of the irreducible representation of all <math>N</math> orbitals. 
# couple the total spins of the <math>N</math> orbitals from left to right; this means we have to choose a fixed <math>S_z</math> for each orbital. 
# test the final total spin and its z-projection against the values required for the system wavefunction
The above steps will need to be repeated many times to elucidate the total set of CSFs that can be derived from
the <math>N</math> electrons and <math>M</math> orbitals.

==Single Orbital configurations and wavefunctions==
Basic quantum mechanics defines the possible single orbital wavefunctions. In any software implementation,
these will be provided either as a table or thourgh a set of logic statements. 

The following table shows the possible couplings for a <math>\sigma</math> orbital with one or two 
electrons.  

{| class="wikitable"
|-
! width="150pt" | Orbital Configuration 
! width="150pt" | Term symbol  
! width="150pt" | <math>S_z</math> projection
|- align="center"
| <math>\sigma^1</math>
| <math>^{2}\Sigma^{+}</math> 
| <math> \;\;\frac{1}{2}</math>
|- align="center"
| <math>\sigma^1</math> 
| <math>^{2}\Sigma^{+}</math> 
| <math>-\frac{1}{2}</math>
|- align="center"
| <math>\sigma^2</math> 
| <math>^{1}\Sigma^{+}</math> 
| <math>0</math>
|}

The next table shows the fifteen possible couplings for a <math>\pi</math> orbital. 
The <math>\delta, \phi, \gamma, \ldots</math> orbitals also each generate fiftenn possible
couplings, all of which can be easily inferred from this table.

{| class="wikitable"
|-
! width="150pt" | Orbital Configuration 
! width="150pt" | Term symbol   
! width="150pt" | Lambda coupling                 
! width="150pt" | <math>S_z</math> projection
|- align="center"
| <math>\pi^1</math>
| <math>^{2}\Pi</math>
| <math>+1</math> 
| <math> \frac{1}{2}</math>
|- align="center"
| <math>\pi^1</math> 
| <math>^{2}\Pi</math>
| <math>+1</math> 
| <math>-\frac{1}{2}</math>
|- align="center"
| <math>\pi^1</math> 
| <math>^{2}\Pi</math>
| <math>-1</math> 
| <math>\frac{1}{2}</math>
|- align="center"
| <math>\pi^1</math> 
| <math>^{2}\Pi</math>
| <math>-1</math> 
| <math>-\frac{1}{2}</math>
|- align="center"
| <math>\pi^2</math>
| <math>^{3}\Sigma^{-}</math>
| <math>0</math> 
| <math>+1</math>
|- align="center"
| <math>\pi^2</math>
| <math>^{3}\Sigma^{-}</math>
| <math>0</math> 
| <math>0</math>
|- align="center"
| <math>\pi^2</math>
| <math>^{3}\Sigma^{-}</math>
| <math>0</math> 
| <math>-1</math>
|- align="center"
| <math>\pi^2</math>
| <math>^{1}\Delta</math>
| <math>+2</math> 
| <math>0</math>
|- align="center"
| <math>\pi^2</math>
| <math>^{1}\Delta</math>
| <math>-2</math> 
| <math>0</math>
|- align="center"
| <math>\pi^2</math>
| <math>^{1}\Sigma^{+}</math>
| <math>0</math> 
| <math>0</math>
|- align="center"
| <math>\pi^3</math>
| <math>^{2}\Pi</math>
| <math>+1</math> 
| <math> \frac{1}{2}</math>
|- align="center"
| <math>\pi^3</math> 
| <math>^{2}\Pi</math>
| <math>+1</math> 
| <math>-\frac{1}{2}</math>
|- align="center"
| <math>\pi^3</math> 
| <math>^{2}\Pi</math>
| <math>-1</math> 
| <math>\frac{1}{2}</math>
|- align="center"
| <math>\pi^3</math> 
| <math>^{2}\Pi</math>
| <math>-1</math> 
| <math>-\frac{1}{2}</math>
|- align="center"
| <math>\pi^4</math> 
| <math>^{1}\Sigma^{+}</math> 
| <math>0</math>
| <math>0</math>
|}

Similar tables can be constructed for atomic systems, which transform according to the point group of the sphere,
that is for s, p, d, f <math>\ldots</math> orbitals. The number of term sysmbols and therefore possible couplings 
is significantly larger in the atomic case.

==Computer Software for CSF generation==
Computer programs are readily available to generate CSFs for atoms<ref>
{{Cite journal
 |last=Sturesson |first=L.
 |last2=Fischer |first2=C. F.
 |year=1993
 |title=LSGEN - a program to generate configuration-state lists of LS-coupled basis functions
 |journal=[[Computer Physics Communications]]
 |volume=74 |issue=3 |pages=432–440
 |bibcode = 1993CoPhC..74..432S
 |doi=10.1016/0010-4655(93)90024-7
}}</ref> for molecules <ref>
{{cite book
 |last1=McLean |first1=A. D.
 |coauthors=''et al.''
 |year=1991
 |chapter=ALCHEMY II, A Research Tool for Molecular Electronic Structure and Interactions
 |editor-last=Clementi |editor-first=E.
 |title=Modern Techniques in Computational Chemistry (MOTECC-91)
 |publisher=[[ESCOM Science Publishers]]
 |isbn=90-72199-10-3
}}</ref> and for electron and positron scattering by molecules .<ref>
{{cite journal
 |last1=Morgan |first=L. A.
 |last2=Tennyson |first2=J.
 |last3=Gillan |first3=C. J.
 |year=1998
 |title=The UK molecular R-matrix codes
 |journal=[[Computer Physics Communications]]
 |volume=114 |issue= 1–3 |pages =120–128
 |bibcode = 1998CoPhC.114..120M
 |doi=10.1016/S0010-4655(98)00056-3
}}</ref> A popular computational method for CSF construction is the [[Graphical Unitary Group Approach]].
-->
==参考文献==
{{reflist}}

{{DEFAULTSORT:Configuration State Function}}
[[Category:量子化学]]