Truncated CI Wavefunction

In a calculation of for example a Li-Atom in a DZ basis we have 4 MO.


For the alpha string (containing two electrons), that means we have for a FCI calculation the determinants: 
|12>, |13>, |14>, |23>, |24>, |34> (electrons in MO 1 and 2, 1 and 3 and so on).


If I wanted to do a truncated CI calculation (CIS for example) only singly excited wavefunctions would contribute. 


My (maybe extremely dumb) question now is: Are the determinants |23> and |24> singly excited or doubly excited? Is it the excitation 
$$a^\dagger\_3a\_1|12> = |32> = -|23>$$
 or is it 
$$a^\dagger\_2a\_1 a^\dagger\_3a\_2|12> = |23>$$
 Or is it both? And the coupling (1,-1) so to say determines whether this in used as a single or doubly excited determinant?


EDIT: From the "Molecular Electronic Structure Theory" book by Helgaker, Jorgensen and Olsen, from the chapter 11.8.5 (Direct CI algorithms for RAS Calculations) I read "*In CISD theory, there are three such classes (string classes): class 0, with no electrons in virtual space; class 1, with one electron in virtual space; and class 2, with two electrons in virtual space*".


This for me reads as if |23> would count as a class 1 string and not a class 2.




In configuration interaction (and in coupled cluster) we typically think of excited determinant being caused by single particle excitations of the form $\hat{t}^a\_i = \hat{a}^\dagger\_a\hat{a}\_i$ where $i$ labels a virtual orbital and $a$ labels an occupied orbital, causing an excitation of the Hartree Fock reference determinant $\Psi\_0$.


The resulting CI expansion would go:
$$
\Phi\_\text{CI} = \Psi\_0 + \sum\_{i\in occ\\ a \in vir} \hat{t}^a\_i \Psi\_0 + \sum\_{i,j\in occ\\ a,b \in vir} \hat{t}^{ab}\_{ij} \Psi\_0 + \sum\_{i,j,k\in occ\\ a,b,c \in vir} \hat{t}^{abc}\_{ijk} \Psi\_0 + \: ...
$$


Hence, in the calculation on Li mentioned above, we would not consider the excitation $\hat{t}^2\_1 = \hat{a}^\dagger\_2\hat{a}\_1$ in the first place as it would annihilate the Hartree Fock determinant $|12>$ containing a particle in orbital 2.


By this logic the determinant $|23>$ is a singly excited determinant caused by $\hat{t}^3\_1= \hat{a}^\dagger\_3\hat{a}\_1$, or class 1 as described by Helgaker.


### Note:


As determinants $|23>$ and $|32>$ only differ by a multiplicative constant (-1) only one needs to be considered and the resulting coefficient/cluster amplitude will take care of the relative sign.


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