A reduced-scaling density matrix-based method for the computation of the vibrational Hessian matrix at the self-consistent field level.

An analytical method to calculate the molecular vibrational Hessian matrix at the self-consistent field level is presented. By analysis of the multipole expansions of the relevant derivatives of Coulomb-type two-electron integral contractions, we show that the effect of the perturbation on the electronic structure due to the displacement of nuclei decays at least as r(-2) instead of r(-1). The perturbation is asymptotically local, and the computation of the Hessian matrix can, in principle, be performed with O(N) complexity. Our implementation exhibits linear scaling in all time-determining steps, with some rapid but quadratic-complexity steps remaining. Sample calculations illustrate linear or near-linear scaling in the construction of the complete nuclear Hessian matrix for sparse systems. For more demanding systems, scaling is still considerably sub-quadratic to quadratic, depending on the density of the underlying electronic structure.

Linear-scaling symmetry-adapted perturbation theory with scaled dispersion.

We present a linear-scaling symmetry-adapted perturbation theory (SAPT) method that is based on an atomic orbital (AO) formulation of zeroth-order SAPT (SAPT0). The non-dispersive terms are realized with linear-scaling cost using both the continuous fast multipole method (CFMM) and the linear exchange (LinK) approach for integral contractions as well as our efficient Laplace-based coupled-perturbed self-consistent field method (DL-CPSCF) for evaluating response densities. The reformulation of the dispersion term is based on our linear-scaling AO Møller-Plesset second-order perturbation theory (AO-MP2) method, that uses our recently introduced QQR-type screening [S. A. Maurer, D. S. Lambrecht, J. Kussmann, and C. Ochsenfeld, J. Chem. Phys. 138, 014101 (2013)] for preselecting numerically significant energy contributions. Similar to scaled opposite-spin MP2, we neglect the exchange-dispersion term in SAPT and introduce a scaling factor for the dispersion term, which compensates for the error and at the same time accounts for basis set incompleteness effects and intramonomer correlation. We show in extensive benchmark calculations that the new scaled-dispersion (sd-)SAPT0 approach provides reliable results for small and large interacting systems where the results with a small 6-31G** basis are roughly comparable to supermolecular MP2 calculations in a triple-zeta basis. The performance of our method is demonstrated with timings on cellulose fragments, DNA systems, and cutouts of a protein-ligand complex with up to 1100 atoms on a single computer core.

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.

Efficient linear-scaling calculation of response properties: density matrix-based Laplace-transformed coupled-perturbed self-consistent field theory.

A density matrix-based Laplace reformulation of coupled-perturbed self-consistent field (CPSCF) theory is presented. It allows a direct, instead of iterative, solution for the integral-independent part of the density matrix-based CPSCF (D-CPSCF) equations [J. Kussmann and C. Ochsenfeld, J. Chem. Phys. 127, 054103 (2007)]. In this way, the matrix-multiplication overhead compared to molecular orbital-based solutions is reduced to a minimum, while at the same time, the linear-scaling behavior of D-CPSCF theory is preserved. The present Laplace-based equation solver is expected to be of general applicability.

Convergence of Electronic Structure with the Size of the QM Region: Example of QM/MM NMR Shieldings.

The influence of the chemical environment on NMR shifts of a central molecular region is studied for several biomolecular and supramolecular systems. To investigate the long-range effects, we systematically increase the QM region until the changes of the NMR shielding tensor are negligible for the considered nuclei; that is, convergence with the selected QM size is reached. To reach size convergence, QM regions with up to about 1500 atoms and 15 000 basis functions are treated by our density matrix-based linear-scaling coupled perturbed self-consistent field methods. The results also provide insights into the locality and convergence of the electronic structure. Furthermore, we demonstrate to what extent the inclusion of the chemical environment as partial point charges within a hybrid QM/MM approach improves the convergence behavior. In addition, some benchmark data on NMR accuracies are provided using various ab initio methods.