Is there a way to find one electron density matrices for the eigen states of a given CI matrix generated from a Hartree-Fock reference so that one can visualize the resultant states?
I am trying to find a way to project the density matrix in CSF (or say slater determinant) basis into a basis that can be easily passed to cubeprop() to generate cube files to visulalise the electron density for a particular electronic state obtained from a CI calculation.
Note: In this case, I have explicitly constructed the CI matrix using the 1e and 2e integrals from MinstHelper using molecular orbitals and their corresponding energies from an HF calculation.
Are you asking how the 1RDMs of a given wavefunction are defined, in what format cubeprop accepts densities, or for a high-level overview of how to go from an arbitrary wavefunction to cube files? Or maybe something else?
Actually a mix of what format of densities or other input cubeprop accepts to generate .cube files for an arbitrary CI wave function and the electronic states represented either in terms of configuration state functions (CSFs) or directly in-terms of slater determinants as it is the case when using detci for getting the first first few roots for the CI Hamiltonian.
I wanted to know how I can represent the density matrix of an eigenstate of the CI Hamiltonian(which is either in CSF or slater determinant basis) in terms of orbitals of the reference slater determinant used to generate the configuration interaction wavefunction.
It sounds like you’re not asking a question about how to use Psi4 but a question about the theory of reduced density matrices.
The conceptually simplest way to get the 1RDM is to express the wavefunction in terms of determinants of orthonormal orbitals, write the 1RDM in terms of second quantization, use second quantization algebra to write the formulas for the RDMs in terms of a sum of tensor contractions of CI coefficients, and then perform the tensor contractions. I haven’t personally done any work with highly optimized computation of 1RDMs, but I know people who have. I could reach out to them, if speed is your concern.
Of course, Psi4 insists on SO basis densities (this is what Roothaan-Hall HF gives you), not the MO basis densities that are more natural for CI. But that conversion is a simple matter of contracting both indices of the MO basis RDM against your orbital coefficient matrix.
I’m sure that doesn’t fully answer your question, so what part(s) do you need more detail on? I don’t know your background and am purposefully giving a high-level overview.
Okay, I am actually confused, on how I could get to Da and Db from a many-body wave function expanded in terms of CSFs.
I get the part on how to evaluate the matrix elements for 1RDMs using slater-condon rules(I am not sure, if that’s what you meant, when you were referring to tensor contractions), considering it to be one electron operator. But, then I still have a matrix represented in the CSF basis.
Your matrix elements are gamma_pq = <Psi | a^p a_q | Psi>, which is both clearly an MO basis matrix and clearly not in Slater-Condon Rule form. You can get it in Slater-Condon Rule form with a “trick.” Starting from <Psi | a^p a_q | Psi>, insert the CI expansion for the bra and the ket. For each bra state, define that as the reference for purposes of the Slater-Condon rules, which will change the excitation level for the ket. You then have a (linear combination of) expectation values of the form <Reference Determinant | a^p a_q | Excited Determinant>, which you can evaluate with Slater-Condon rules.
gamma^i_j looks like c^i_a c^j_a + c^ik_ab c^jk_ab + …
gamma^a_b looks like c^i_a c^i_b + c^ij_ad c^ij_bd + …
…if you insist on the i < j < k convention for all indices other than those appearing in the RDM element. Don’t forget the permutational parity factor.
Incidentally, specialists in deriving methods have a formalism to solve these kinds of problems that is much faster than Slater-Condon rules for cases like this. That’s the second quantization algebra that I mentioned previously.