Contraction relations for Grassmann products of reduced density matrices and implications for density matrix reconstruction

We consider, for systems of indistinguishable fermions, approximate reconstruction of the three- and four-particle reduced density matrices (RDMs) from the one- and two-particle RDMs, $\mathit{\gamma }$ and $\mathit{\Gamma }.$ Our ansatz for reconstructing the four-particle RDM is the linear combination $a\left(\mathit{\Gamma }\wedge \mathit{\Gamma }\right)+b\left(\mathit{\gamma }\wedge \mathit{\gamma }\wedge \mathit{\Gamma }\right)+c\left(\mathit{\gamma }\wedge \mathit{\gamma }\wedge \mathit{\gamma }\wedge \mathit{\gamma }\right),$ where “∧” denotes the antisymmetrized (Grassmann) product. This is a generalization of reconstruction functionals employed recently to perform direct RDM calculations without wave functions via the contracted Schrödinger equation. Here we consider relationships between the parameters a, b, and c that are required in order for the reconstruction functionals to respect the hierarchy of contraction relations between RDMs. To this end we establish several general theorems concerning contractions of antisymmetrized tensor products of $\mathit{\gamma },$ $\mathit{\Gamma },$ and various products thereof. The accuracy of proposed reconstruction functionals is evaluated using accurate density matrices for the ground state of Be.