Electronic structure of vitamin B$_{\bf 12}$ within the framework of the Haldane-Anderson impurity model

We study the electronic structure of vitamin B$_{12}$ (cyanocobalamine C$_{63}$H$_{88}$CoN$_{14}$O$_{14}$P) by using the framework of the multi-orbital single-impurity Haldane-Anderson model of a transition-metal impurity in a semiconductor host. Here, our purpose is to understand the many-body effects originating from the transition-metal impurity. In this approach, the cobalt $3d$ orbitals are treated as the impurity states placed in a semiconductor host which consists of the rest of the molecule. The parameters of the resulting effective Haldane-Anderson model are obtained within the Hartree-Fock approximation for the electronic structure of the molecule. The quantum Monte Carlo technique is then used to calculate the one-electron and magnetic correlation functions of this effective Haldane-Anderson model for vitamin B$_{12}$. We find that new states form inside the semiconductor gap due to the on-site Coulomb interaction at the impurity $3d$ orbitals and that these states become the highest occupied molecular orbitals. In addition, we present results on the charge distribution and spin correlations around the Co atom. We compare the results of this approach with those obtained by the density-functional theory calculations.