On the use of effective core potentials in the calculation of magnetic properties, such as magnetizabilites and magnetic shieldings

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State-of-the art effective core potentials (ECPs) that replace electrons of inner atomic cores involve non-local potentials. If such an effective core potential is added to the Hamiltonian of a system in a magnetic field, the resulting Hamiltonian is not gauge invariant. This means, magnetic properties such as magnetisabilities and magnetic shieldings (or magnetic susceptibilities and nuclear magnetic resonance chemical shifts) calculated with different gauge origins are different even for exact solutions of the Schrödinger equation. It is possible to restore gauge invariance of the Hamiltonian by adding magnetic field dependent terms arising from the effective core potential. Numerical calculations on atomic and diatomic model systems (potassium mono-cation and potassium dimer) clearly demonstrate that the standard effective core potential Hamiltonian violates gauge invariance, and this affects the calculation of magnetisabilities more strongly than the calculation of magnetic shieldings. The modified magnetic field dependent effective core potential Hamiltonian is gauge invariant, and therefore it is the correct starting point for distributed gauge origin methods. The formalism for gauge including atomic orbitals (GIAO) and individual gauge for localized orbitals methods is worked out. ECP GIAO results for the potassium dimer are presented. The new method performs much better than a previous ECP GIAO implementation that did not account for the non-locality of the potential. For magnetic shieldings, deviations are clearly seen, but they amount to few ppm only. For magnetisabilities, our new ECP GIAO implementation is a major improvement, as demonstrated by the comparison of all-electron and ECP results.