Efficient implementation of the analytic second derivatives of Hartree–Fock and hybrid DFT energies: a detailed analysis of different approximations

In this paper, various implementations of the analytic Hartree–Fock and hybrid density functional energy second derivatives are studied. An approximation-free four-centre implementation is presented, and its accuracy is rigorously analysed in terms of self-consistent field (SCF), coupled-perturbed SCF (CP-SCF) convergence and prescreening criteria. The CP-SCF residual norm convergence threshold turns out to be the most important of these. Final choices of convergence thresholds are made such that an accuracy of the vibrational frequencies of better than 5 cm^?1 compared to the numerical noise-free results is obtained, even for the highly sensitive low frequencies (<100–200 cm^?1). The effects of the choice of numerical grid for density functional exchange–correlation integrations are studied and various weight derivative schemes are analysed in detail. In the second step of the work, approximations are introduced in order to speed up the computation without compromising its accuracy. To this end, the accuracy and efficiency of the resolution of identity approximation for the Coulomb terms and the semi-numerical chain of spheres approximation to the exchange terms are carefully analysed. It is shown that the largest performance improvements are realised if either Hartree–Fock exchange is absent (pure density functionals) and otherwise, if the exchange terms in the CP-SCF step of the calculation are approximated by the COSX method in conjunction with a small integration grid. Default values for all the involved truncation parameters are suggested. For vancomycine (176 atoms and 3593 basis functions), the RIJCOSX Hessian calculation with the B3LYP functional and the def2-TZVP basis set takes ～3 days using 16 Intel® Xeon® 2.60GHz processors with the COSX algorithm having a net parallelisation scaling of 11.9 which is at least ～20 times faster than the calculation without the RIJCOSX approximation.     

Testing Lorentz invariance and CPT conservation with NuMI neutrinos in the MINOS near detector

A search for a sidereal modulation in the MINOS near detector neutrino data was performed. If present, this signature could be a consequence of Lorentz and CPT violation as predicted by the effective field theory called the standard-model extension. No evidence for a sidereal signal in the data set was found, implying that there is no significant change in neutrino propagation that depends on the direction of the neutrino beam in a sun-centered inertial frame. Upper limits on the magnitudes of the Lorentz and CPT violating terms in the standard-model extension lie between 10(-4) and 10(-2) of the maximum expected, assuming a suppression of these signatures by a factor of 10(-17).     

On the Riemannian geometry of Seiberg–Witten moduli spaces

We construct a natural L²$L^2$-metric on the perturbed Seiberg–Witten moduli spaces _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$ of a compact 4-manifold M$M$, and we study the resulting Riemannian geometry of _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$. We derive a formula which expresses the sectional curvature of _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$ in terms of the Green operators of the deformation complex of the Seiberg–Witten equations. In case M$M$ is simply connected, we construct a Riemannian metric on the Seiberg–Witten principal U(1)$U\left(1\right)$ bundle P→_Mμ+$\mathfrak{P}\to {\mathfrak{M}}_{{\mu }^{+}}$ such that the bundle projection becomes a Riemannian submersion. On a Kähler surface M$M$, the L²$L^2$-metric on _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$ coincides with the natural Kähler metric on moduli spaces of vortices.     

Point defects and orientation-dependent transport of matter and charge in iron-containing olivines

The variation of the oxygen content, x_O, of synthetic fayalite (Fe₂SiO₄) single crystals was investigated thermogravimetrically at 1130 °C as a function of the oxygen activity, a_O2 (= P_O2/P_O2° ≈ f_O2/f_O2° with P_O2° ≈ f_O2° = 1 bar ≈ 1 atm). It was found that x_O varies less in fayalite single crystals than in polycrystalline Fe₂SiO₄ studied earlier. The majority defects are most likely cation vacancies, (V_Me²⁺)″, ferric ions on M-sites, (Fe³⁺_Me²⁺), and ferric ions on Si-sites, (Fe³⁺_Si⁴⁺)′. Furthermore, the diffusion of iron in synthetic olivine single crystals ((Fe_xMg_1 − x)₂SiO₄) was studied at 1130 °C as a function of orientation, oxygen activity, and cationic composition. The observed oxygen activity dependencies suggest that cations move via different types of cation vacancies, most likely isolated vacancies, (V_Fe²⁺)″, and possibly neutral associates, {2(Fe³⁺_Me²⁺) ⋅ (V_Me²⁺)^′ ? ′}^x, the latter being minority defects. In addition, the electrical conductivity, σ, of fayalite single crystals was investigated as a function of orientation and oxygen activity within the stability field of fayalite at 1130 °C. The observed oxygen activity dependencies are compatible with (V_Me²⁺)^′ ? ′, (Fe³⁺_Me²⁺), and (Fe³⁺_Si⁴⁺)′ being the majority point defects at high a_O2 and with h^⋅ and e′ as the majority defects at low a_O2. The electrical conduction in fayalite is governed by contributions of electrons and holes. This extended point defect model for fayalite is also compatible with data for the variation of the oxygen content and for the iron tracer diffusion.     

Probing electron-electron interaction in quantum Hall systems with scanning tunneling spectroscopy

Using low-temperature scanning tunneling spectroscopy applied to the Cs-induced two-dimensional electron system (2DES) on p-type InSb(110), we probe electron-electron interaction effects in the quantum Hall regime. The 2DES is decoupled from bulk states and exhibits spreading resistance within the insulating quantum Hall phases. In quantitative agreement with calculations we find an exchange enhancement of the spin splitting. Moreover, we observe that both the spatially averaged as well as the local density of states feature a characteristic Coulomb gap at the Fermi level. These results show that electron-electron interaction can be probed down to a resolution below all relevant length scales.