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Configuration state function

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Quantum_chemistry" title="Quantum chemistry">quantum chemistry</a>, a configuration state function (CSF), is a symmetry-adapted linear combination of <a href="Slater_determinant" title="Slater determinant">Slater determinants</a>. A CSF must not be confused with a <a href="Electron_configuration" title="Electron configuration">configuration</a>. In general, one configuration gives rise to several CSFs; all have the same total quantum numbers for spin and spatial parts but differ in their intermediate couplings.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

A configuration state function (CSF), is a symmetry-adapted linear combination of <a href="Slater_determinant" title="Slater determinant">Slater determinants</a>. It is constructed to have the same quantum numbers as the wavefunction, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }">

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<annotation encoding="application/x-tex">{\displaystyle \Psi }</annotation>

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</math><img src="./f5471531a3fe80741a839bc98d49fae862a6439a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }" loading="lazy">, of the system being studied. In the method of <a href="Configuration_interaction" title="Configuration interaction">configuration interaction</a>, the wavefunction<a href="#cite_note-1">[1]</a> can be expressed as a linear combination of CSFs, that is in the form

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<annotation encoding="application/x-tex">{\displaystyle \Psi =\sum _{k}c_{k}\psi _{k}}</annotation>

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</math><img src="./c8786d652e7eabef8da28ea711fbb941ddde1bcb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.346ex; height:5.509ex;" alt="{\displaystyle \Psi =\sum _{k}c_{k}\psi _{k}}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{k}}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{k}}</annotation>

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</math><img src="./21677536dadbcbaa5026d3253578f2ea29cffc39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.602ex; height:2.509ex;" alt="{\displaystyle \psi _{k}}" loading="lazy"> denotes the set of CSFs. The coefficients, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{k}}</annotation>

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</math><img src="./3d2f8052630e67b00d04e3487e1d68ed7070470b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.096ex; height:2.009ex;" alt="{\displaystyle c_{k}}" loading="lazy">, are found by using the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }">

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<annotation encoding="application/x-tex">{\displaystyle \Psi }</annotation>

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</math><img src="./f5471531a3fe80741a839bc98d49fae862a6439a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }" loading="lazy"> to compute a Hamiltonian matrix. When this is diagonalized, the eigenvectors are chosen as the expansion coefficients. CSFs rather than just Slater determinants can also be used as a basis in <a href="Multi-configurational_self-consistent_field" title="Multi-configurational self-consistent field">multi-configurational self-consistent field</a> computations.

In atomic structure, a CSF is an eigenstate of

<ul><li>the square of the <a href="Angular_momentum#Angular_momentum_in_quantum_mechanics" title="Angular momentum">angular momentum</a> operator, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {L}}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {L}}^{2}}</annotation>

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</math><img src="./5ae5c94b79ed25964f21af007cd459fb7689d475.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:3.343ex;" alt="{\displaystyle {\hat {L}}^{2}}" loading="lazy"></li>

<li>the z-projection of angular momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {L}}_{z}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {L}}_{z}}</annotation>

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</math><img src="./f890c759326e0b9e75b24931dcf2a53862ab309c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.585ex; height:3.176ex;" alt="{\displaystyle {\hat {L}}_{z}}" loading="lazy"></li>

<li>the square of the <a href="Spin_(physics)" title="Spin (physics)">spin operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {S}}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {S}}^{2}}</annotation>

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</math><img src="./f525fc62601674c7d99f4926642ca9eac29703a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:3.343ex;" alt="{\displaystyle {\hat {S}}^{2}}" loading="lazy"></li>

<li>the z-projection of the spin operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {S}}_{z}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {S}}_{z}}</annotation>

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</math><img src="./7ac67cb8315ba14707385264d2f64e1e65a965f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:3.176ex;" alt="{\displaystyle {\hat {S}}_{z}}" loading="lazy"></li></ul>

In linear molecules, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {L}}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {L}}^{2}}</annotation>

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</math><img src="./5ae5c94b79ed25964f21af007cd459fb7689d475.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:3.343ex;" alt="{\displaystyle {\hat {L}}^{2}}" loading="lazy"> does not commute with the Hamiltonian for the system and therefore CSFs are not eigenstates of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {L}}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {L}}^{2}}</annotation>

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</math><img src="./5ae5c94b79ed25964f21af007cd459fb7689d475.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:3.343ex;" alt="{\displaystyle {\hat {L}}^{2}}" loading="lazy">. However, the z-projection of angular momentum is still a <a href="Good_quantum_number" title="Good quantum number">good quantum number</a> and CSFs are constructed to be eigenstates of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {L}}_{z},{\hat {S}}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {L}}_{z},{\hat {S}}^{2}}</annotation>

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</math><img src="./8145922b2d6fa47862cdc657746ec42f2bafa0f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.245ex; height:3.676ex;" alt="{\displaystyle {\hat {L}}_{z},{\hat {S}}^{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {S}}_{z}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {S}}_{z}}</annotation>

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</math><img src="./7ac67cb8315ba14707385264d2f64e1e65a965f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:3.176ex;" alt="{\displaystyle {\hat {S}}_{z}}" loading="lazy">. In non-linear (which implies polyatomic) molecules, neither <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {L}}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {L}}^{2}}</annotation>

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</math><img src="./5ae5c94b79ed25964f21af007cd459fb7689d475.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:3.343ex;" alt="{\displaystyle {\hat {L}}^{2}}" loading="lazy"> nor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {L}}_{z}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {L}}_{z}}</annotation>

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</math><img src="./f890c759326e0b9e75b24931dcf2a53862ab309c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.585ex; height:3.176ex;" alt="{\displaystyle {\hat {L}}_{z}}" loading="lazy"> commutes with the Hamiltonian. The CSFs are constructed to have the spatial transformation properties of one of the irreducible representations of the point group to which the nuclear framework belongs. This is because the Hamiltonian operator transforms in the same way.<a href="#cite_note-2">[2]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {S}}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {S}}^{2}}</annotation>

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</math><img src="./f525fc62601674c7d99f4926642ca9eac29703a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:3.343ex;" alt="{\displaystyle {\hat {S}}^{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {S}}_{z}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {S}}_{z}}</annotation>

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</math><img src="./7ac67cb8315ba14707385264d2f64e1e65a965f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:3.176ex;" alt="{\displaystyle {\hat {S}}_{z}}" loading="lazy"> are still valid quantum numbers and CSFs are built to be eigenfunctions of these operators.

<div class="mw-heading mw-heading2"><h2 id="From_configurations_to_configuration_state_functions">From configurations to configuration state functions</h2></div>

CSFs are derived from configurations. A configuration is just an assignment of electrons to orbitals. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1s^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle 1s^{2}}</annotation>

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</math><img src="./e9b12f0314f16fc34749146e27b4581392564c5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.307ex; height:2.676ex;" alt="{\displaystyle 1s^{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\pi ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle 1\pi ^{2}}</annotation>

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</math><img src="./b1ed650984cf99a52d0bb194006486a967d8c4e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.551ex; height:2.676ex;" alt="{\displaystyle 1\pi ^{2}}" loading="lazy"> are example of two configurations, one from atomic structure and one from molecular structure.

From any given configuration we can, in general, create several CSFs. CSFs are therefore sometimes also called N-particle symmetry adapted basis functions. For a configuration the number of electrons is fixed; let's call this <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">. When we are creating CSFs from a configuration we have to work with the spin-orbitals associated with the configuration.

For example, given the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1s}">

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<annotation encoding="application/x-tex">{\displaystyle 1s}</annotation>

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</math><img src="./59a6d7a59b12f82f6b269b57b2d9eca77c99d4cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.253ex; height:2.176ex;" alt="{\displaystyle 1s}" loading="lazy"> orbital in an atom we know that there are two spin-orbitals associated with this,

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<annotation encoding="application/x-tex">{\displaystyle 1s\alpha \;\;\;1s\beta }</annotation>

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</math><img src="./946cf2f6c8b48a53b32adb7ee3cb37cfc89fcedb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.261ex; height:2.509ex;" alt="{\displaystyle 1s\alpha \;\;\;1s\beta }" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle \alpha ,\;\;\;\beta }</annotation>

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</math><img src="./9f7b4cb085e60ac3719ed495aad77dfec02cc431.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.789ex; height:2.509ex;" alt="{\displaystyle \alpha ,\;\;\;\beta }" loading="lazy"></dd></dl>

are the one electron spin-eigenfunctions for spin-up and spin-down respectively. Similarly, for the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\pi }">

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<annotation encoding="application/x-tex">{\displaystyle 1\pi }</annotation>

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</math><img src="./ab11cdddeb3c618d3df1a8490363d9a5acfa5427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 1\pi }" loading="lazy"> orbital in a linear molecule (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\infty v}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{\infty v}}</annotation>

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</math><img src="./f5470f23a62a7a5991a5bc5d4a45c7dd09f7c31f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.335ex; height:2.509ex;" alt="{\displaystyle C_{\infty v}}" loading="lazy"> point group) we have four spin orbitals:

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<annotation encoding="application/x-tex">{\displaystyle 1\pi (+)\alpha ,\;1\pi (+)\beta ,\;1\pi (-)\alpha ,\;1\pi (-)\beta }</annotation>

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</math><img src="./f9493202367242a5ed4a889e965bc45982dfec5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.124ex; height:2.843ex;" alt="{\displaystyle 1\pi (+)\alpha ,\;1\pi (+)\beta ,\;1\pi (-)\alpha ,\;1\pi (-)\beta }" loading="lazy">.</dd></dl>

This is because the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> designation corresponds to z-projection of angular momentum of both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}">

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<annotation encoding="application/x-tex">{\displaystyle +1}</annotation>

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</math><img src="./d04cf05c67d41d9f39dabf6a90722ce860a76958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">

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<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

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</math><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy">.

We can think of the set of spin orbitals as a set of boxes each of size one; let's call this <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> boxes. We distribute the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> electrons among the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> boxes in all possible ways. Each assignment corresponds to one Slater determinant, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation>

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</math><img src="./9f07b53d3212e08ca316a536c8aac0bbefa79ee1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}" loading="lazy">. There can be great number of these, particularly when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N<<M}">

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<annotation encoding="application/x-tex">{\displaystyle N<<M}</annotation>

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</math><img src="./df5c953ec6bef6d7837c6adeed24db1fb5e99639.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.412ex; height:2.176ex;" alt="{\displaystyle N<<M}" loading="lazy">. Another way to look at this is to say we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> entities and we wish to select <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> of them, known as a <a href="Combination" title="Combination">combination</a>. We need to find all possible combinations. Order of the selection is not significant because we are working with determinants and can interchange rows as required.

If we then specify the overall coupling that we wish to achieve for the configuration, we can now select only those Slater determinants that have the required quantum numbers. In order to achieve the required total spin angular momentum (and in the case of atoms the total orbital angular momentum as well), each Slater determinant has to be premultiplied by a coupling coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation>

</semantics>

</math><img src="./01acb7953ba52c2aa44264b5d0f8fd223aa178a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}" loading="lazy">, derived ultimately from <a href="Clebsch%E2%80%93Gordan_coefficients" title="Clebsch–Gordan coefficients">Clebsch–Gordan coefficients</a>. Thus the CSF is a linear combination

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i}c_{i}\;D_{i}}</annotation>

</semantics>

</math><img src="./9ac5508b58f5bd4e77dbd0ffa0f0bf9d6f84e877.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:8.918ex; height:5.509ex;" alt="{\displaystyle \sum _{i}c_{i}\;D_{i}}" loading="lazy">.</dd></dl>

The Lowdin projection operator formalism<a href="#cite_note-3">[3]</a> may be used to find the coefficients. For any given set of determinants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation>

</semantics>

</math><img src="./9f07b53d3212e08ca316a536c8aac0bbefa79ee1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}" loading="lazy"> it may be possible to find several different sets of coefficients.<a href="#cite_note-4">[4]</a> Each set corresponds to one CSF. In fact this simply reflects the different internal couplings of total spin and spatial angular momentum.

<div class="mw-heading mw-heading2"><h2 id="A_genealogical_algorithm_for_CSF_construction">A genealogical algorithm for CSF construction</h2></div>

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At the most fundamental level, a configuration state function can be constructed from a set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> orbitals and a number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> of electrons using the following genealogical algorithm:

<ol><li>distribute the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> electrons over the set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> orbitals giving a configuration</li>

<li>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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{z}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{z}}</annotation>

</semantics>

</math><img src="./bf685f2b7d1c4266b32c1e16fdc6a41a610ab21e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.427ex; height:2.509ex;" alt="{\displaystyle S_{z}}" loading="lazy"> undefined.</li>

<li>check that the spatial coupling of all orbitals matches that required for the system wavefunction. For a molecule exhibiting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\infty v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{\infty v}}</annotation>

</semantics>

</math><img src="./f5470f23a62a7a5991a5bc5d4a45c7dd09f7c31f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.335ex; height:2.509ex;" alt="{\displaystyle C_{\infty v}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\infty h}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{\infty h}}</annotation>

</semantics>

</math><img src="./215df5462bb3f5f3a9e29e4f6fc4bacfea8c6262.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.746ex; height:2.509ex;" alt="{\displaystyle D_{\infty h}}" loading="lazy"> this is achieved by a simple linear summation of the coupled <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> value for each orbital; for molecules whose nuclear framework transforms according to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{2h}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{2h}}</annotation>

</semantics>

</math><img src="./6507e6f6cf8eb341c49d5bbae3898348e5bddd44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.925ex; height:2.509ex;" alt="{\displaystyle D_{2h}}" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> orbitals.</li>

<li>couple the total spins of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> orbitals from left to right; this means we have to choose a fixed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{z}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{z}}</annotation>

</semantics>

</math><img src="./bf685f2b7d1c4266b32c1e16fdc6a41a610ab21e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.427ex; height:2.509ex;" alt="{\displaystyle S_{z}}" loading="lazy"> for each orbital.</li>

<li>test the final total spin and its z-projection against the values required for the system wavefunction</li></ol>

The above steps will need to be repeated many times to elucidate the total set of CSFs that can be derived from the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> electrons and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> orbitals.

<div class="mw-heading mw-heading2"><h2 id="Single_orbital_configurations_and_wavefunctions">Single orbital configurations and wavefunctions</h2></div>

Basic quantum mechanics defines the possible single orbital wavefunctions. In a software implementation, these can be provided either as a table or through a set of logic statements. Alternatively group theory may be used to compute them.<a href="#cite_note-5">[5]</a>

Electrons in a single orbital are called equivalent electrons.<a href="#cite_note-6">[6]</a> They obey the same coupling rules as other electrons but the <a href="Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a> makes certain couplings impossible. The <a href="Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a> requires that no two electrons in a system can have all their quantum numbers equal. For equivalent electrons, by definition the principal quantum number is identical. In atoms the angular momentum is also identical. So, for equivalent electrons the z components of spin and spatial parts, taken together, must differ.

The following table shows the possible couplings for a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> orbital with one or two electrons.

<th width="150pt">Orbital Configuration

</th>

<th width="150pt">Term symbol

</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{z}}</annotation>

</semantics>

</math><img src="./bf685f2b7d1c4266b32c1e16fdc6a41a610ab21e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.427ex; height:2.509ex;" alt="{\displaystyle S_{z}}" loading="lazy"> projection

</th></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{1}}</annotation>

</semantics>

</math><img src="./08028f7316b8139f221c65258ef382798257679b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{1}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{2}\Sigma ^{+}}</annotation>

</semantics>

</math><img src="./925a8fd9961c1695a4c4eef382ce04a0415a7a73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.243ex; height:2.676ex;" alt="{\displaystyle ^{2}\Sigma ^{+}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;\;{\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./08a49dc5c6601dd071b608719710f07b020af070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.289ex; height:5.176ex;" alt="{\displaystyle \;\;{\frac {1}{2}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{1}}</annotation>

</semantics>

</math><img src="./08028f7316b8139f221c65258ef382798257679b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{1}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{2}\Sigma ^{-}}</annotation>

</semantics>

</math><img src="./246eb0550c063c083b4d4c587f07b0dcb2c51733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.243ex; height:2.676ex;" alt="{\displaystyle ^{2}\Sigma ^{-}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./0dfd9bc2c74bd2b31dc85201e2857845703b2613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.807ex; height:5.176ex;" alt="{\displaystyle -{\frac {1}{2}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>

</semantics>

</math><img src="./53a5c55e536acf250c1d3e0f754be5692b843ef5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{1}\Sigma ^{+}}</annotation>

</semantics>

</math><img src="./3d10cc0be483eddba3fe19b4aeb254a93b49c6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.243ex; height:2.676ex;" alt="{\displaystyle ^{1}\Sigma ^{+}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

</td></tr></tbody></table>

The situation for orbitals in Abelian point groups mirrors the above table. The next table shows the fifteen possible couplings for a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> orbital.

The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ,\phi ,\gamma ,\ldots }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta ,\phi ,\gamma ,\ldots }</annotation>

</semantics>

</math><img src="./488880bcc940080f3eca50f7dfc6e4e460c712ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.521ex; height:2.843ex;" alt="{\displaystyle \delta ,\phi ,\gamma ,\ldots }" loading="lazy"> orbitals also each generate fifteen possible couplings, all of which can be easily inferred from this table.

<th width="150pt">Orbital Configuration

</th>

<th width="150pt">Term symbol

</th>

<th width="150pt">Lambda coupling

</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{z}}</annotation>

</semantics>

</math><img src="./bf685f2b7d1c4266b32c1e16fdc6a41a610ab21e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.427ex; height:2.509ex;" alt="{\displaystyle S_{z}}" loading="lazy"> projection

</th></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{1}}</annotation>

</semantics>

</math><img src="./fad59792d2bd1e07735b69d35d81c7e8ad6e626d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{1}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{2}\Pi }</annotation>

</semantics>

</math><img src="./ec66fbecb9d50e3ee4d5364db139c74608d54e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle ^{2}\Pi }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +1}</annotation>

</semantics>

</math><img src="./d04cf05c67d41d9f39dabf6a90722ce860a76958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./a11cfb2fdb143693b1daf78fcb5c11a023cb1c55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{1}}</annotation>

</semantics>

</math><img src="./fad59792d2bd1e07735b69d35d81c7e8ad6e626d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{1}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{2}\Pi }</annotation>

</semantics>

</math><img src="./ec66fbecb9d50e3ee4d5364db139c74608d54e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle ^{2}\Pi }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +1}</annotation>

</semantics>

</math><img src="./d04cf05c67d41d9f39dabf6a90722ce860a76958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./0dfd9bc2c74bd2b31dc85201e2857845703b2613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.807ex; height:5.176ex;" alt="{\displaystyle -{\frac {1}{2}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{1}}</annotation>

</semantics>

</math><img src="./fad59792d2bd1e07735b69d35d81c7e8ad6e626d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{1}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{2}\Pi }</annotation>

</semantics>

</math><img src="./ec66fbecb9d50e3ee4d5364db139c74608d54e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle ^{2}\Pi }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./a11cfb2fdb143693b1daf78fcb5c11a023cb1c55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{1}}</annotation>

</semantics>

</math><img src="./fad59792d2bd1e07735b69d35d81c7e8ad6e626d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{1}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{2}\Pi }</annotation>

</semantics>

</math><img src="./ec66fbecb9d50e3ee4d5364db139c74608d54e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle ^{2}\Pi }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./0dfd9bc2c74bd2b31dc85201e2857845703b2613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.807ex; height:5.176ex;" alt="{\displaystyle -{\frac {1}{2}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{2}}</annotation>

</semantics>

</math><img src="./0f2197d9f1fe0e87e860af1b6db71b269e20683e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{2}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{3}\Sigma ^{-}}</annotation>

</semantics>

</math><img src="./e72c84f09cda36a33a08797fcbf5028bdc37cfe9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.243ex; height:2.676ex;" alt="{\displaystyle ^{3}\Sigma ^{-}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +1}</annotation>

</semantics>

</math><img src="./d04cf05c67d41d9f39dabf6a90722ce860a76958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{2}}</annotation>

</semantics>

</math><img src="./0f2197d9f1fe0e87e860af1b6db71b269e20683e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{2}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{3}\Sigma ^{-}}</annotation>

</semantics>

</math><img src="./e72c84f09cda36a33a08797fcbf5028bdc37cfe9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.243ex; height:2.676ex;" alt="{\displaystyle ^{3}\Sigma ^{-}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{2}}</annotation>

</semantics>

</math><img src="./0f2197d9f1fe0e87e860af1b6db71b269e20683e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{2}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{3}\Sigma ^{-}}</annotation>

</semantics>

</math><img src="./e72c84f09cda36a33a08797fcbf5028bdc37cfe9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.243ex; height:2.676ex;" alt="{\displaystyle ^{3}\Sigma ^{-}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{2}}</annotation>

</semantics>

</math><img src="./0f2197d9f1fe0e87e860af1b6db71b269e20683e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{2}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{1}\Delta }</annotation>

</semantics>

</math><img src="./85aabae5b0d545a2dcbf68b9d03f1f8696895eb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.99ex; height:2.676ex;" alt="{\displaystyle ^{1}\Delta }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +2}</annotation>

</semantics>

</math><img src="./425979b37896d1911b58956146264cc1fb29aec0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +2}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{2}}</annotation>

</semantics>

</math><img src="./0f2197d9f1fe0e87e860af1b6db71b269e20683e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{2}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{1}\Delta }</annotation>

</semantics>

</math><img src="./85aabae5b0d545a2dcbf68b9d03f1f8696895eb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.99ex; height:2.676ex;" alt="{\displaystyle ^{1}\Delta }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -2}</annotation>

</semantics>

</math><img src="./46e5b5b462e546b1d3d7e5f9a23efece405b2e78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -2}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{2}}</annotation>

</semantics>

</math><img src="./0f2197d9f1fe0e87e860af1b6db71b269e20683e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{2}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{1}\Sigma ^{+}}</annotation>

</semantics>

</math><img src="./3d10cc0be483eddba3fe19b4aeb254a93b49c6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.243ex; height:2.676ex;" alt="{\displaystyle ^{1}\Sigma ^{+}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{3}}</annotation>

</semantics>

</math><img src="./3f6b28d4d8e575ad755a4728ecfe26776a30ae04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{3}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{2}\Pi }</annotation>

</semantics>

</math><img src="./ec66fbecb9d50e3ee4d5364db139c74608d54e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle ^{2}\Pi }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +1}</annotation>

</semantics>

</math><img src="./d04cf05c67d41d9f39dabf6a90722ce860a76958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./a11cfb2fdb143693b1daf78fcb5c11a023cb1c55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{3}}</annotation>

</semantics>

</math><img src="./3f6b28d4d8e575ad755a4728ecfe26776a30ae04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{3}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{2}\Pi }</annotation>

</semantics>

</math><img src="./ec66fbecb9d50e3ee4d5364db139c74608d54e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle ^{2}\Pi }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +1}</annotation>

</semantics>

</math><img src="./d04cf05c67d41d9f39dabf6a90722ce860a76958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./0dfd9bc2c74bd2b31dc85201e2857845703b2613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.807ex; height:5.176ex;" alt="{\displaystyle -{\frac {1}{2}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{3}}</annotation>

</semantics>

</math><img src="./3f6b28d4d8e575ad755a4728ecfe26776a30ae04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{3}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{2}\Pi }</annotation>

</semantics>

</math><img src="./ec66fbecb9d50e3ee4d5364db139c74608d54e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle ^{2}\Pi }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./a11cfb2fdb143693b1daf78fcb5c11a023cb1c55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{3}}</annotation>

</semantics>

</math><img src="./3f6b28d4d8e575ad755a4728ecfe26776a30ae04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{3}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{2}\Pi }</annotation>

</semantics>

</math><img src="./ec66fbecb9d50e3ee4d5364db139c74608d54e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle ^{2}\Pi }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./0dfd9bc2c74bd2b31dc85201e2857845703b2613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.807ex; height:5.176ex;" alt="{\displaystyle -{\frac {1}{2}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{4}}</annotation>

</semantics>

</math><img src="./075a6ffb2efbc34e0711ebf8fe13549844766c5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.676ex;" alt="{\displaystyle \pi ^{4}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{1}\Sigma ^{+}}</annotation>

</semantics>

</math><img src="./3d10cc0be483eddba3fe19b4aeb254a93b49c6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.243ex; height:2.676ex;" alt="{\displaystyle ^{1}\Sigma ^{+}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

</td></tr></tbody></table>

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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots }">

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<annotation encoding="application/x-tex">{\displaystyle \ldots }</annotation>

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</math><img src="./3b8619532e44ee1ccae3ab03405a6885260d09ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \ldots }" loading="lazy"> orbitals. The number of term symbols and therefore possible couplings

is significantly larger in the atomic case.

<div class="mw-heading mw-heading2"><h2 id="Computer_software_for_CSF_generation">Computer software for CSF generation</h2></div>

Computer programs are readily available to generate CSFs for atoms<a href="#cite_note-7">[7]</a> for molecules<a href="#cite_note-8">[8]</a> and for electron and positron scattering by molecules.<a href="#cite_note-9">[9]</a> A popular computational method for CSF construction is the <a href="Graphical_Unitary_Group_Approach" class="mw-redirect" title="Graphical Unitary Group Approach">Graphical Unitary Group Approach</a>.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFEngel2006" class="citation book cs1">Engel, T. (2006). Quantum Chemistry and Spectroscopy. <a href="Pearson_PLC" class="mw-redirect" title="Pearson PLC">Pearson PLC</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8053-3842-X</bdi>.</cite>

</li>

<cite id="CITEREFPilar1990" class="citation book cs1">Pilar, F. L. (1990). Elementary Quantum Chemistry (2nd ed.). <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-41464-7</bdi>.</cite>

</li>

<cite id="CITEREFCrossley1977" class="citation journal cs1">Crossley, R. J. S. (1977). "On Löwdin's projection operators for angular momentum. I". <a href="International_Journal_of_Quantum_Chemistry" title="International Journal of Quantum Chemistry">International Journal of Quantum Chemistry</a>. 11 (6): 917–929. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fqua.560110605">10.1002/qua.560110605</a>.</cite>

</li>

<cite id="CITEREFNesbet2003" class="citation book cs1">Nesbet, R. K. (2003). "Section 4.4". In Huo, W. M .; Gianturco, F. A. (eds.). Variational principles and methods in theoretical physics and chemistry. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 49. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-80391-8</bdi>.</cite>

</li>

<cite id="CITEREFKarayianis1965" class="citation journal cs1">Karayianis, N. (1965). "Atomic Terms for Equivalent Electrons". J. Math. Phys. 6 (8): 1204–1209. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1965JMP.....6.1204K">1965JMP.....6.1204K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1704761">10.1063/1.1704761</a>.</cite>

</li>

<cite id="CITEREFWise1976" class="citation journal cs1">Wise, J.H. (1976). "Spectroscopic terms for equivalent electrons". J. Chem. Educ. 53 (8): 496. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1976JChEd..53..496W">1976JChEd..53..496W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fed053p496.2">10.1021/ed053p496.2</a>.</cite>

</li>

<cite id="CITEREFSturessonFischer1993" class="citation journal cs1">Sturesson, L.; Fischer, C. F. (1993). "LSGEN - a program to generate configuration-state lists of LS-coupled basis functions". <a href="Computer_Physics_Communications" title="Computer Physics Communications">Computer Physics Communications</a>. 74 (3): 432–440. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993CoPhC..74..432S">1993CoPhC..74..432S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0010-4655%2893%2990024-7">10.1016/0010-4655(93)90024-7</a>.</cite>

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<cite id="CITEREFMcLean1991" class="citation book cs1">McLean, A. D.; et al. (1991). "ALCHEMY II, A Research Tool for Molecular Electronic Structure and Interactions". In Clementi, E. (ed.). Modern Techniques in Computational Chemistry (MOTECC-91). ESCOM Science Publishers. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>90-72199-10-3</bdi>.</cite>

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<cite id="CITEREFMorganTennysonGillan1998" class="citation journal cs1">Morgan, L. A.; Tennyson, J.; Gillan, C. J. (1998). "The UK molecular R-matrix codes". <a href="Computer_Physics_Communications" title="Computer Physics Communications">Computer Physics Communications</a>. 114 (1–3): 120–128. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998CoPhC.114..120M">1998CoPhC.114..120M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0010-4655%2898%2900056-3">10.1016/S0010-4655(98)00056-3</a>.</cite>

</li>

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