We have a porphyrin-like molecule with a transition metal atom in the center, like the one shown in the figure.
Consider first the porphyrin without the metal. In such a case, several orbitals around the Fermi level are purely $\pi$-states, so that the porphyrin can be very accurately described by constructing the effective Hamiltonian in the subspace of $p_z$ orbitals:
$$\hat{H}_\pi= \sum_i t_{i,j} \hat{c}^\dagger_i\hat{c}_j \tag{1}\label{1}$$
where the indices $i,j$ run over sites of the lattice defining the molecule (I omit the explicit spin indices here for simplicity of notation, you can imagine they are absorbed in the indices $i,j$).
If one does this, the spectrum that results from diagonalizing $H_\pi$ coincides around Fermi level with the spectrum of the full Hamiltonian, and so do the corresponding eigenstates.
I would like to do something similar when the porphyrin has the metal atom added. Here the idea is to describe the molecule by an effective Hamiltonian that includes only the $p_z$ orbitals of the organic ring and the $d$ orbitals of the metal. Something like:
$$\hat{H}_{\pi-d}= \sum_i t_{i,j} \hat{c}^\dagger_i\hat{c}_j + \sum_{i,n} V_{i,n} \hat{c}^\dagger_i\hat{d}_n + \sum_{n} \varepsilon_{n} \hat{d}^\dagger_n\hat{d}_n \tag{2}\label{2}$$
If we do this the resulting spectrum is a disaster, it does not match in any region the spectrum of the overall hamiltonian. I have tried this from DFT Hamiltonians with minimal basis of atomic orbitals (using the Löwdin orthonormal basis) and from the Tight-binding Slater-Koster Hamiltonians. In summary, just the sub_$\pi-d$ hamiltonian cannot describe the molecule, but I am thinking if such a Hamiltonian could be made to match the full one just by adding a on-site term in the $d$ shell that would reproduce the crystal-field splitting of the $d$ orbitals in the original hamiltonian; that is, if for some $e_1,e_2,e_3,e_4,e_5$ it happens that
$$\hat{H}_{\pi-d} + \sum_{n} e_{n} \hat{d}^\dagger_n\hat{d}_n \tag{3}\label{3}$$
indeed reproduces the spectrum and eigenvectors around Fermi level.
If this was possible, it would be very interesting to develop simple modelistic many-body approaches to such system, where one would pick just a few $\pi$ orbitals to describe the organic ring (like in a CAS calculation) and add something like a Kanamori Hamiltonian to the $d-$shell. In such simple models it would be easy to keep track of the observables in the metal atom, like total spin or charge, unlike in ab initio CAS where the active space orbitals might have a lot of hybridization between the $\pi$ and $d$ orbitals.
How to find such coefficients $e_1,e_2,e_3,e_4,e_5$ in the most effective way? For now I was trying pure brute force for a simple system, but probably there is some way using the symmetries of the system and a calculation of the crystal-field splitting or local Density of States to fit these parameters.
