Cholesky Decomposition and the Second-Derivative Two-Electron Integrals Required for the Computation of Magnetizabilities Using Gauge-Including Atomic Orbitals

The computation of magnetizability tensors using gauge-including atomic orbitals is discussed in the context of Cholesky decomposition (CD) for the two-electron repulsion integrals with a focus on the involved doubly differentiated integrals. Three schemes for their handling are suggested: the first exploits the density fitting (DF) aspect of Cholesky decomposition, the second uses expressions obtained by differentiating the CD expression for the unperturbed two-electron integrals, while the third addresses the issue that the first two schemes are not able to represent the doubly differentiated integrals with arbitrary accuracy. This scheme uses a separate Cholesky decomposition for the cross terms in the doubly differentiated two-electron integrals. Test calculations reveal that all three schemes are able to represent the integrals with similar accuracy and yield indistinguishable results for the values of the computed magnetizability tensor elements. Thus, we recommend our first scheme which has the lowest computational cost for routine computations. The applicability of our CD schemes is further shown in large-scale Hartree-Fock calculations of the magnetizability tensor of coronene (C₂₄H₁₂) with a doubly polarized triple-ζ basis consisting of 684 basis functions.