Nuclei-selected NMR shielding calculations: a sublinear-scaling quantum-chemical method.

An ab initio method for the direct calculation of NMR shieldings for selected nuclei at the Hartree-Fock and density-functional theory level is presented. Our method shows a computational effort scaling only sublinearly with molecular size, as it is motivated by the physical consideration that the chemical shielding is dominated by its local environment. The key feature of our method is to avoid the conventionally performed calculation of all NMR shieldings but instead to solve directly for specific nuclear shieldings. This has important implications not only for the study of large molecules, but also for the simulation of solvent effects and molecular dynamics, since often just a few shieldings are of interest. Our theory relies on two major aspects both necessary to provide a sublinear scaling behavior: First, an alternative expression for the shielding tensor is derived, which involves the response density matrix with respect to the nuclear magnetic moment instead of the response to the external magnetic field. Second, as unphysical long-range contributions occur within the description of distributed gauge origin methods that do not influence the final expectation value, we present a screening procedure to truncate the B-field dependent basis set, which is crucial in order to ensure an early onset of the sublinear scaling. The screening is in line with the r(-2) distance decay of Biot-Savarts law for induced magnetic fields. Our present truncation relies on the introduced concept of "individual gauge shielding contributions" applied to a reformulated shielding tensor, the latter consisting of gauge-invariant terms. The presented method is generally applicable and shows typical speed-ups of about one order of magnitude; moreover, due to the reduced scaling behavior of O(1) as compared to O(N), the wins become larger with increasing system size. We illustrate the validity of our method for several test systems, including ring-current dominated systems and biomolecules with more than 1000 atoms.