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Suppose, one knows the ground state wave function $|\Phi\rangle$ in terms of a slater determinant(SD) basis say, from a Configuration interaction calculation.

$$|\Phi\rangle= c_{\text{HF}}\left|\Psi_{\text{HF}}\right\rangle+\sum_{r a} c_{a}^{r}\left|\Psi_{a}^{r}\right\rangle+\sum_{a<b, r<s} c_{a b}^{r s}\left|\Psi_{a b}^{r s}\right\rangle+\sum_{r<s<t, a<b<c} c_{a b c}^{r s t}\left|\Psi_{a b c}^{r s t}\right\rangle+\ldots $$

How can one transform to the molecular orbital(MO) basis? (it is implied that one knows the MOs used to build the SDs in the first place.)

Do any of the ab initio packages already have the functionality where I can pass the eigen vectors of the CI Hamiltonian and the MO coefficients to get the density matrix or the wavefunction in terms of the MO basis?

I am trying to calculate the density matrices from the CI wavefunction in terms of MO basis, so that I can visualize the states from my CI calculation.

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  • $\begingroup$ I'll hopefully have a chance to answer soon, but you might want to look into natural orbitals $\endgroup$
    – Tyberius ♦
    Commented Mar 24, 2022 at 13:53
  • $\begingroup$ @Tyberius, yes I am familiar with NBOs and have some experience using the software suite nbo7.chem.wisc.edu $\endgroup$
    – user784
    Commented Mar 24, 2022 at 13:57
  • $\begingroup$ Note there are multiple things referred to as natural orbitals including natural atomic orbitals (NAOs), bond orbitals(NBOs), and transition orbitals (NTOs, not listed in the link). I'm referring to just plain natural orbitals which are the eigenfunctions of the one particle reduced density matrix. $\endgroup$
    – Tyberius ♦
    Commented Mar 24, 2022 at 14:03

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This is impossible (except in the trivial case where there is only one electron), because MOs span the space of single electron wavefunctions, while SDs span the space of multielectron wavefunctions. As MOs and SDs span different spaces, you cannot transform one to the other. You can transform a single electron wavefunction between MO, AO, orthogonal AO, plane wave, and Wannier function bases, etc., and you can transform a multielectron wavefunction between SD and CSF bases, but you cannot do any basis transformations between these two categories of bases because such transformations are not even defined.

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  • $\begingroup$ Then how can one visualize the MOs from Post HF methods? Aren't SDs made up of MOs themselves? I am just trying to find a way to visualize the eigen states you get from diagonalising the CI Hamiltonian. $\endgroup$
    – user784
    Commented Mar 24, 2022 at 13:35
  • $\begingroup$ @EverydayFoolish One can only talk about natural orbitals of post HF methods, which are obtained by diagonalizing the density matrix. But this does not count as a basis set transform, one of the reasons being that the natural orbitals contain much less information than the original SDs, while basis transforms always preserve all information as long as the transformation matrix is not singular. SDs are linear combinations of products of MOs, not linear combination of MOs themselves (unless there is only one electron), and only linear combinations of MOs can be transformed to the MO basis. $\endgroup$
    – wzkchem5
    Commented Mar 24, 2022 at 14:09
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    $\begingroup$ @wzkchem5 I think mentioning natural orbitals in your answer, along with the caveats from your comment, would make for a good addition. I think the OP is more interested in having a visualization for the state, even if it an approximate picture. $\endgroup$
    – Tyberius ♦
    Commented Mar 24, 2022 at 14:13
  • $\begingroup$ @wzkchem I don't disagree. But, from the CSF or SD basis, to constructing 1-RDMs that can be visualised as orbitals. I am missing something on how its done. $\endgroup$
    – user784
    Commented Mar 24, 2022 at 14:14
  • $\begingroup$ @EverydayFoolish I think this depends heavily on the program with which you obtained the multielectron wavefunction. The transformation of the multielectron wavefunction to 1-RDM is typically done in the same program that generated the wavefunction, one reason being that it is costly to write the wavefunction to disk to be read by another program. So while you may always write the wavefunction to disk, transform it to the desired format and calculate the 1-RDM by an external program, this is almost never the optimal way to do this. $\endgroup$
    – wzkchem5
    Commented Mar 25, 2022 at 8:51

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