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Abstract The Morse potential is an important problem to examine due to its applications in describing vibrations and bond breaking in molecules. It also shares some properties with the simpler harmonic oscillator, at the same time displaying differences, allowing for an interesting contrast to its well-studied counterpart. The solution of the Morse potential is not usually taught in a quantum mechanics class, since using differential equations makes it very tedious. Here, we illustrate how to solve the Morse potential using the Schrödinger factorization method. This operator method is a powerful tool to find the energy eigenvalues, eigenstates, and wavefunctions without using differential equations in position space, allowing us to solve more problems without requiring a discussion of hypergeometric or confluent hypergeometric functions. 

For molecules and solids, we developed efficient MPI-parallel algorithms for evaluating the second-order exchange (SOX) term with bare, statically screened, and dynamically screened interactions. We employ the resulting term in a fully self-consistent manner together with self-consistent GW (scGW), resulting in the following vertex-corrected scGW schemes: scGWSOX, scGWSOSEX, scGW2SOSEX, and scG3W2 theories. We show that for the vertex evaluation, the reduction of scaling by tensor hypercontraction has two limiting execution regimes. We used the resulting code to perform the largest (by the number of orbitals) fully self-consistent calculations with the SOX term. We demonstrate that our procedure allows for a reliable evaluation of even small energy differences. Utilizing a broken-symmetry approach, we explore the influence of the SOX term on the effective magnetic exchange couplings. We show that the treatment of SOX has a significant impact on the obtained values of the effective exchange constants, which we explain through a self-energy dependence on an effective dielectric constant. We confirm this explanation by analyzing natural orbitals and local changes in charge transfer, quantifying superexchange. Our analysis explains the structure of weak electron correlation responsible for the modulation of superexchange in both molecules and solids. Finally, for solids, we evaluate Néel temperatures utilizing the high-temperature expansion and compare the results obtained with experimental measurements. In addition, we prove a lack of Φ-derivability of the considered theories.