We propose that spatial density matrices, which are singularly important in the study of quantum entanglement, encode the electronic fluctuations and correlations responsible for covalent bonding. From these density matrices, we develop tools that al
low us to analyse how many body wave functions can be broken up into real space pieces. We apply these tools to the first row dimers, and in particular, we address the conflicting evidence in the literature about the presence of an inverted fourth bond and anti-ferromagnetic correlations in the $text{C}_2$ molecule. Our results show that many body effects enhance anti-ferromagnetic fluctuations but are not related to the formation of an inverted fourth bond. We identify two inverted bonds in the $text{C}_2$ molecule and establish their correspondence to the bonds in the $text{Be}_2$ molecule. Additionally, we provide a new interpretation of the Mayer index, introduce partial bonds to fix deficiencies in molecular orbital theory, and prove the Hartree-Fock wave function for C$_{2}$ is not a triple bond. Our results suggest that entanglement-based methods can lead to a more realistic treatment of molecular and extended systems than possible before.
Several algorithms have been proposed to calculate the spatial entanglement spectrum from high order Renyi entropies. In this work we present an alternative approach for computing the entanglement spectrum with quantum Monte Carlo for both continuum
and lattice Hamiltonians. This method provides direct access to the matrix elements of the spatially reduced density matrix and we determine an estimator that can be used in variational Monte Carlo as well as other Monte Carlo methods. The algorithm is based on using a generalization of the Swap operator, which can be extended to calculate a general class of density matrices that can include combinations of spin, space, particle and even momentum coordinates. We demonstrate the method by applying it to the Hydrogen and Nitrogen molecules and describe for the first time how the spatial entanglement spectrum encodes a covalent bond that includes all the many body correlations.
