A procedure was developed to automatically generate auxiliary basis sets (ABSs) for use with the resolution of the identity (RI) approximation, starting from a given orbital basis set (OBS). The goal is to provide an accurate and universal solution for cases where no optimized ABSs are available. In this context, "universal" is understood as the ability of the ABS to be used for Coulomb, exchange, and correlation energy fitting. The generation scheme (denoted AutoAux) works by spanning the product space of the OBS using an even-tempered expansion for each atom in the system. The performance of AutoAux in conjunction with different OBSs [def2-SVP, def2-TZVP, def2-QZVPP, and cc-pwCVnZ (n = D, T, Q, 5)] has been evaluated for elements from H to Rn and compared to existing predefined ABSs. Due to the requirements of simplicity and universality, the generated ABSs are larger than the optimized ones but lead to similar errors in MP2 total energies (on the order of 10 to 10 E/atom).

Analytic calculation of nuclear magnetic resonance chemical shielding tensors, based on gauge-including atomic orbitals, is implemented for double-hybrid density functional theory (DHDFT), using the resolution of the identity (RI) approximation for its second order Møller-Plesset perturbation theory (MP2) correlation contributions. A benchmark set of 15 small molecules, containing H,C, N,O, F, andP nuclei, is used to assess the accuracy of the results in comparison to coupled cluster and empirical equilibrium reference data, as well as to calculations with MP2, Hartree-Fock, and commonly used pure and hybrid density functionals. Investigated are also errors due to basis set incompleteness, the frozen core approximation, different auxiliary basis sets for the RI approximation, and grids used for the chain-of-spheres exchange integral evaluation. The DSD-PBEP86 double-hybrid functional shows the smallest deviations from the reference data with mean absolute relative error in chemical shifts of 1.9%. This is significantly better than MP2 (4.1%), spin-component-scaled MP2 (3.9%), or the best conventional density functional tested, M06L (5.4%). A protocol (basis sets, grid sizes, etc.) for the efficient and accurate calculation of chemical shifts at the DHDFT level is proposed and shown to be routinely applicable to systems of 100-400 electrons, requiring computation times 1-2 orders of magnitude longer than for equivalent calculations with conventional (pure or hybrid) density functionals.

The chain-of-spheres method (COS) for approximating two-electron integrals is applied to Hartree-Fock and density functional theory calculations of nuclear magnetic resonance chemical shielding tensors, based on gauge-including atomic orbitals. The accuracy of the approximation is compared to that of the resolution of the identity (RI) approach, using a benchmark test set of 15 small molecules. Reasonable auxiliary basis sets and grid sizes are selected on the basis of a careful investigation of how approximating each of the two-electron terms in the self-consistent field (SCF) and coupled perturbed SCF equations affects the calculated shielding constants. It is found that the errors are linearly additive but can have either sign. The mean absolute relative error due to applying the RI/COS approximations with the chosen settings to all two-electron terms is on the order of 0.01% and therefore negligible compared to the errors due to basis set incompleteness (∼1%) and the method used (10-50%). Several larger organic systems are used to assess the efficiency of the RI approximation for both Coulomb- and exchange-type integrals (RIJK) as well as a combination of RI for Coulomb and COS for exchange contributions (RIJCOSX). The RIJK approximation is more efficient for small molecules, while for systems of over 100 electrons and 1000 basis functions, the RIJCOSX approximation is superior.

We present a derivation and efficient implementation of the formally complete analytic second derivatives for the domain-based local pair natural orbital second order Møller–Plesset perturbation theory (MP2) method, applicable to electric or magnetic field-response properties but not yet to harmonic frequencies. We also discuss the occurrence and avoidance of numerical instability issues related to singular linear equation systems and near linear dependences in the projected atomic orbital domains. A series of benchmark calculations on medium-sized systems is performed to assess the effect of the local approximation on calculated nuclear magnetic resonance shieldings and the static dipole polarizabilities. Relative deviations from the resolution of the identity-based MP2 (RI-MP2) reference for both properties are below 0.5% with the default truncation thresholds. For large systems, our implementation achieves quadratic effective scaling, is more efficient than RI-MP2 starting at 280 correlated electrons, and is never more than 5–20 times slower than the equivalent Hartree–Fock property calculation. The largest calculation performed here was on the vancomycin molecule with 176 atoms, 542 correlated electrons, and 4700 basis functions and took 3.3 days on 12 central processing unit cores.

In this work, we explore the accuracy of post-Hartree–Fock (HF) methods and double-hybrid density functional theory (DFT) for the computation of solid-state NMR chemical shifts. We apply an embedded cluster approach and investigate the convergence with cluster size and embedding for a series of inorganic solids with long-range electrostatic interactions. In a systematic study, we discuss the cluster design, the embedding procedure, and basis set convergence using gauge-including atomic orbital (GIAO) NMR calculations at the DFT and MP2 levels of theory. We demonstrate that the accuracy obtained for the prediction of NMR chemical shifts, which can be achieved for molecular systems, can be carried over to solid systems. An appropriate embedded cluster approach allows one to apply methods beyond standard DFT even for systems for which long-range electrostatic effects are important. We find that an embedded cluster should include at least one sphere of explicit neighbors around the nuclei of interest, given that a sufficiently large point charge and boundary effective potential embedding is applied. Using the pcSseg-3 basis set and GIAOs for the computation of nuclear shielding constants, accuracies of 1.6 ppm for 7Li, 1.5 ppm for 23Na, and 5.1 ppm for 39K as well as 9.3 ppm for 19F, 6.5 ppm for 35Cl, 7.4 ppm for 79Br, and 7.5 ppm for 25Mg as well as 3.8 ppm for 67Zn can be achieved with MP2. Comparing various DFT functionals with HF and MP2, we report the superior quality of results for methods that include post-HF correlation like MP2 and double-hybrid DFT.