Search

The Berry curvature is essential in Born-Oppenheimer molecular dynamics, describing the screening of the nuclei by the electrons in a magnetic field. Parts of the Berry curvature can be understood as the external magnetic field multiplied by an effective charge so that the resulting Berry force behaves like a Lorentz force during the simulations. Here, we investigate whether these effective charges can provide insight into the electronic structure of a given molecule or, in other words, whether we can perform a population analysis based on the Berry curvature. To develop our approach, we first rewrite the Berry curvature in terms of charges that partially capture the effective charges and their dependencies on the nuclear velocities. With these Berry charges and charge fluctuations, we then construct our population analysis yielding atomic charges and overlap populations. Calculations at the Hartree-Fock level reveal that the atomic charges are similar to those obtained from atomic polar tensors. However, since we additionally obtain an estimate for the fluctuations of the charges and a partitioning of the atomic charges into contributions from all atoms, we conclude that the Berry population analysis is a useful alternative tool to analyze the electronic structures of molecules.

Recent work has established Moreau-Yosida regularization as a mathematical tool to achieve rigorous functional differentiability in density-functional theory. In this article, we extend this tool to paramagnetic current-density-functional theory, the most common density-functional framework for magnetic field effects. The extension includes a well-defined Kohn-Sham iteration scheme with a partial convergence result. To this end, we rely on a formulation of Moreau-Yosida regularization for reflexive and strictly convex function spaces. The optimal L p -characterization of the paramagnetic current density L 1 ∩ L 3/2 is derived from the N-representability conditions. A crucial prerequisite for the convex formulation of paramagnetic current-density-functional theory, termed compatibility between function spaces for the particle density and the current density, is pointed out and analyzed. Several results about compatible function spaces are given, including their recursive construction. The regularized, exact functionals are calculated numerically for a Kohn-Sham iteration on a quantum ring, illustrating their performance for different regularization parameters.

We study the effects of magnetic fields in the context of magnetic field density-functional theory (BDFT), where the energy is a functional of the electron density ρ and the magnetic field B. We show that this approach is a worthwhile alternative to current-density functional theory (CDFT) and may provide a viable route to the study of many magnetic phenomena using density-functional theory (DFT). The relationship between BDFT and CDFT is developed and clarified within the framework of the four-way correspondence of saddle functions and their convex and concave parents in convex analysis. By decomposing the energy into its Kohn-Sham components, we demonstrate that the magnetizability is mainly determined by those energy components that are related to the density. For existing density functional approximations, this implies that, for the magnetizability, improvements of the density will be more beneficial than introducing a magnetic-field dependence in the correlation functional. However, once a good charge density is achieved, we show that high accuracy is likely only obtainable by including magnetic-field dependence. We demonstrate that adiabatic-connection (AC) curves at different field strengths resemble one another closely provided each curve is calculated at the equilibrium geometry of that field strength. In contrast, if all AC curves are calculated at the equilibrium geometry of the field-free system, then the curves change strongly with increasing field strength due to the increasing importance of static correlation. This holds also for density functional approximations, for which we demonstrate that the main error encountered in the presence of a field is already present at zero field strength, indicating that density-functional approximations may be applied to systems in strong fields, without the need to treat additional static correlation.

We compare the performance of three approximate methods for speeding up evaluation of the exchange contribution in Hartree-Fock and hybrid Kohn-Sham calculations: the chain-of-spheres algorithm (COSX; Neese , F. Chem. Phys. 2008 , 356 , 98 - 109 ), the pair-atomic resolution-of-identity method (PARI-K; Merlot , P. J. Comput. Chem. 2013 , 34 , 1486 - 1496 ), and the auxiliary density matrix method (ADMM; Guidon , M. J. Chem. Theory Comput. 2010 , 6 , 2348 - 2364 ). Both the efficiency relative to that of a conventional linear-scaling algorithm and the accuracy of total, atomization, and orbital energies are compared for a subset containing 25 of the 200 molecules in the Rx200 set using double-, triple-, and quadruple-ζ basis sets. The accuracy of relative energies is further compared for small alkane conformers (ACONF test set) and Diels-Alder reactions (DARC test set). Overall, we find that the COSX method provides good accuracy for orbital energies as well as total and relative energies, and the method delivers a satisfactory speedup. The PARI-K and in particular ADMM algorithms require further development and optimization to fully exploit their indisputable potential.

Plots of electronic energy vs electron number, determined using approximate density functional theory (DFT) and Hartree-Fock theory, are typically piecewise convex and piecewise concave, respectively. The curves also commonly exhibit a minimum and maximum, respectively, in the neutral → anion segment, which lead to positive DFT anion HOMO energies and positive Hartree-Fock neutral LUMO energies. These minima/maxima are a consequence of using basis sets that are local to the system, preventing fractional electron loss. Ground-state curves are presented that illustrate the idealized behavior that would occur if the basis set were to be modified to enable fractional electron loss without changing the description in the vicinity of the system. The key feature is that the energy cannot increase when the electron number increases, so the slope cannot be anywhere positive, meaning frontier orbital energies cannot be positive. For the convex (DFT) case, the idealized curve is flat beyond a critical electron number such that any additional fraction of an electron added to the system is unbound. The anion HOMO energy is zero. For the concave (Hartree-Fock) case, the idealized curve is flat up to some critical electron number, beyond which it curves down to the anion energy. A minimum fraction of an electron is required before any binding occurs, but beyond that, the full fraction abruptly binds. The neutral LUMO energy is zero. Approximate DFT and Hartree-Fock results are presented for the F → F(-) segment, and results approaching the idealized behavior are recovered for highly diffuse basis sets. It is noted that if a DFT calculation using a highly diffuse basis set yields a negative LUMO energy then a fraction of an electron must bind and the electron affinity must be positive, irrespective of whether an electron binds experimentally. This is illustrated by calculations on Ne → Ne(-).

We present the self-consistent implementation of current-dependent (hybrid) meta-generalized gradient approximation (mGGA) density functionals using London atomic orbitals. A previously proposed generalized kinetic energy density is utilized to implement mGGAs in the framework of Kohn-Sham current density functional theory (KS-CDFT). A unique feature of the nonperturbative implementation of these functionals is the ability to seamlessly explore a wide range of magnetic fields up to 1 au (∼235 kT) in strength. CDFT functionals based on the TPSS and B98 forms are investigated, and their performance is assessed by comparison with accurate coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) data. In the weak field regime, magnetic properties such as magnetizabilities and nuclear magnetic resonance shielding constants show modest but systematic improvements over generalized gradient approximations (GGA). However, in the strong field regime, the mGGA-based forms lead to a significantly improved description of the recently proposed perpendicular paramagnetic bonding mechanism, comparing well with CCSD(T) data. In contrast to functionals based on the vorticity, these forms are found to be numerically stable, and their accuracy at high field suggests that the extension of mGGAs to CDFT via the generalized kinetic energy density should provide a useful starting point for further development of CDFT approximations.

Kohn-Sham density-functional calculations are used in many branches of science to obtain information about the electronic structure of molecular systems and materials. Conventional algorithms for minimization of the Kohn-Sham energy have certain deficiencies, however, that may cause divergence or, worse, convergence to unphysical saddle points. We here present a three-level hierarchical minimization strategy which is both more efficient and robust than the conventional algorithms and which does not suffer from the flaws of these algorithms. Using the three-level minimization strategy, the molecular density is built up in a hierarchical fashion in accordance with chemical insight: First, the molecular density is composed by a superposition of atomic densities; next, bonds are formed by performing a simple valence-shell optimization; finally, the molecular description is refined by an optimization in the full molecular basis. Importantly, the density matrix generated at each of the two lower levels in this hierarchy is transferred to the next without loss of information. Examples demonstrate the efficacy and robustness of the proposed scheme.

Density-functional theory (DFT) and coupled-cluster singles-and-doubles (CCSD) theory are applied to compute the nuclear magnetic resonance (NMR) shielding and indirect nuclear spin-spin coupling constants of o-benzyne, whose biradical nature makes it difficult to study both experimentally and theoretically. Because of near-equilibrium triplet instabilities that follow from its biradical character, the calculated DFT NMR properties of o-benzyne are unusually sensitive to details of the exchange-correlation functional. However, this sensitivity is greatly reduced if these properties are calculated at the equilibrium of the chosen functional. A strong correlation is demonstrated between the quality of the calculated indirect spin-spin coupling constants and the quality of the calculated lowest triplet excitation energy in o-benzyne. Orbital-unrelaxed coupled-cluster theory should be less affected by such instabilities, and the CCSD NMR properties were only calculated at the experimental equilibrium geometry. For the shielding constants, the results in best agreement with experimental results are obtained with CCSD theory and with the Keal-Tozer KT1 and KT2 functionals. For the triply bonded carbon atoms, these models yield an isotropic shielding of 1.3, -3.3, and -1.2 ppm, respectively, compared with the experimentally observed shielding of 3.7 ppm for incarcerated o-benzyne. For the indirect spin-spin coupling constants, the CCSD model and the Perdew-Burke-Ernzerhof functional both yield reliable results; for the most interesting spin-spin coupling constant, (1)J (C⋮C), we obtain 210 and 209 Hz with these two models, respectively, somewhat above the recently reported experimental value of 177.9 ± 0.7 Hz for o-benzyne inside a molecular container, suggesting large incarceration effects.

The calculation of rotational g tensors using density functional theory (DFT) with hybrid exchange-correlation functionals is considered. A total of 143 rotational g tensor elements in 58 molecules (67 isotopic combinations) are calculated using three standard hybrid functionals. Tensor elements determined using an uncoupled approach with orbitals and eigenvalues calculated from the multiplicative optimized effective potential (OEP) constitute a significant improvement over those determined in the conventional coupled manner with a nonmultiplicative exchange-correlation operator. Relative to experimental results, mean absolute errors are reduced by a factor of 2; mean errors and standard deviations are reduced by more than a factor of 3. The results are also an improvement over those determined using a generalized gradient-approximation functional optimized for magnetic response properties. The influence of orbital exchange is investigated for a representative subset of molecules, yielding an optimal amount near 0.3. Rotational g tensors are also determined from coupled-cluster electron densities using a combined DFT/wave-function approach. Being substantially more expensive, they do not offer a notable improvement on the pure DFT values from OEP-based hybrid calculations.

The rotational g factor for a large number of organic compounds has been investigated with density-functional theory. Rapid convergence toward the basis-set limit is ensured by the use of London atomic orbitals. A statistical analysis of the results has been carried out in comparison with accurate experimental data. It is shown that gradient-corrected and hybrid functionals reproduce experimental results most closely, with the Keal-Tozer KT2 functional being the most accurate.