# Is this notation for single and double excited Slater determinants correct?

Is the following notation

$$\Phi^\mathrm{HF}_0 \equiv | \phi_1, \phi_2, \cdots, \phi_i, \phi_j, \cdots, \phi_N \rangle$$

$$\Phi_{ia} \equiv | \phi_1, \phi_2, \cdots, \phi_a, \phi_j, \cdots, \phi_N \rangle$$

$$\Phi_{ia,jb} \equiv | \phi_1, \phi_2, \cdots, \phi_a, \phi_b, \cdots, \phi_N \rangle$$

$$\cdots$$

where $$\Phi_{ia}$$ refers to single excited Slater determinant (SD) and $$\Phi_{ia,jb}$$ refers to a double excited SD correct?

I want to use this notation to define the operators

$$\hat a^a \hat a_i \Phi^\mathrm{HF}_0 = \Phi_{ia}$$

$$\hat a^a \hat a^b \hat a_j \hat a_i \Phi^\mathrm{HF}_0 = \Phi_{ia,jb}$$

which I then want to use to define $$T_1$$ (part of the cluster operator) as:

$$\hat T_1 = \sum_i \sum_a t_a^i \hat{a}^a \hat{a}_i$$

Is there something wrong with how I define $$\Phi^\mathrm{HF}_0$$, $$\Phi_{ia}$$, and $$\Phi_{ia,jb}$$?

In Configuration Interaction and Couple Cluster theory, the widely used convention for singles, doubles, triples Wave functions(Slater determinants, coefficients etc) is to denote occupied orbitals in subscripts and virtual orbitals in superscript.

For example, $$|\Phi_{i}^{a}\rangle$$ represents a singles Slater determinant where the electron is excited from occupied orbital $$i$$ is excited to virtual orbital $$a$$. Also, note that its a widely used convention to use indicies $$i,j,k,l....$$ and $$a,b,c,d...$$ for occupied and virtual molecular orbitals respectively.

The ordering of the indices may not matter, i.e. $$|\Phi_{ijk}^{abc}\rangle$$ is the same as $$|\Phi_{kji}^{abc}\rangle$$ as by definition we consider anti-symmetrical wave functions. This can also be understood by playing around with the combinations of second quantized notation using commutation relations of your fermionic creation-annihilation operators.

$$\hat{a_{a}}^{\dagger} \hat{a^{}_{i}}\hat{a_{b}}^{\dagger}\hat{a^{}_{j}}\hat{a_{c}}^{\dagger}\hat{a^{}_{k}} | \Phi_{0}\rangle = |\Phi_{ijk}^{abc}\rangle$$

You can find more information on such issues in chapter two of Szabo and Ostlund.

Szabo, Attila, and Neil S. Ostlund. Modern quantum chemistry: introduction to advanced electronic structure theory. Courier Corporation, 2012.

Note: The convention for indices is different from the ones found in Szabo, which seems to follow a convention that's considered outdated now.

• No, it's the exact opposite: i,j,k,l,... denote occupied indices and a,b,c,d,... denote virtual orbitals... i,j,k,l etc are typically lower indices and a,b,c,d etc upper indices. Oct 28 '21 at 18:07
• @SusiLehtola My bad going by what Szabo has, it has to be r,s,.... for upper indices and a, b , c.... for lower indices. Besides, my point was that lower indices are for occupied orbitals and upper indices for virtual orbitals. And that is only for wavefunctions, and not for creation/annihilation operators. Both of these things are missing in OPs notations. Oct 30 '21 at 19:33
• p, q, r, s conventionally denote general indices which can be either occupied or virtual. Nov 1 '21 at 17:50
• Sure, would you like to edit the answer, reflect these points? Nov 2 '21 at 5:07
• Szabo is overall a good reference, but it does use an odd/outdated convention for the orbital labels. It seems to use a,b,c,... for occupied, r,s,t,... for virtual, and i,j,k... for general orbitals. I think your answer is fine, but it may be worth including a disclaimer that the index conventions are specific to that reference and the conventions Susi listed are the norm now.
– Tyberius
Nov 7 '21 at 1:50