Quantum chemistry in this context refers to numerical simulations of how atoms and their electrons behave. The paper says that they used density functional theory as implemented in VASP (Vienna Ab initio Simulation Package), a common way to approximate electron-atom and electron-electron interactions:

https://en.wikipedia.org/wiki/Density_functional_theory

https://www.vasp.at/info/about/

Another common way to simulate behavior of materials dissolved in water is "classical" molecular dynamics, using only Newtonian physics and a set of lumped empirical parameterizations to model molecules as a collection of "balls and springs." This is much faster than ab initio molecular dynamics but less usable for exotic materials like promethium complexes, where it is unlikely that anyone has ever generated/validated a good set of parameters.

Solving the Schrodinger equation for multiple electrons in a molecule. This becomes very computationally intensive as you add more electrons.

For example, there’s not enough energy in the universe to simulate a human with the full Schrodinger equation with our existing technology for a meaningful amount of time.