Optimizing Molecular Geometries in Strong Magnetic Fields

An efficient implementation of geometrical derivatives at the Hartree-Fock (HF) and current-density-functional theory (CDFT) levels is presented for the study of molecular structure in strong magnetic fields. The required integral derivatives are constructed using a hybrid McMurchie-Davidson and Rys quadrature approach, which combines the amenability of the former to the evaluation of derivative integrals with the efficiency of the latter for basis sets with high angular momentum. In addition to its application to evaluating derivatives of four-centre integrals, this approach is also applied to gradients using the resolution-of-the-identity approximation, enabling efficient optimization of molecular structure for many-electron systems under a strong magnetic field. The CDFT contributions have been implemented for a wide range of density-functionals up to and including the meta-GGA level with current-density dependent contributions and (range-separated) hybrids for the first time. Illustrative applications are presented to the OH and benzene molecules, revealing the rich and complex chemistry induced by the presence of an external magnetic field. Challenges 1 for geometry optimization in strong fields are highlighted, along with the requirement for careful analysis of the resulting electronic structure at each stationary point. The importance of correlation effects is examined by comparison of results at the HF and CDFT levels. The present implementation of molecular gradients at the CDFT level provides a cost-effective approach to the study of molecular structure under strong magnetic fields, opening up many new possibilities for the study of chemistry in this regime

Modeling Ultrafast Electron Dynamics in Strong Magnetic Fields Using Real-Time Time-Dependent Electronic Structure Methods

An implementation of real-time time-dependent Hartree-Fock (RT-TDHF) and current-density-functional theory (RT-TDCDFT) for molecules in strong uniform magnetic fields is presented. In contrast to earlier implementations, the present work enables the use of the RT-TDCDFT formalism, which explicitly includes field dependent terms in the exchange-correlation functional. A range of current dependent exchange-correlation functionals based on the TPSS functional are considered, including a range-separated variant, which is particularly suitable for application to excited state calculations. The performance of a wide range of propagator algorithms for real-time methods is investigated in this context. A recently proposed molecular orbital pair decomposition analysis allows for assignment of electronic transitions, providing detailed information about which molecular orbitals are involved in each excitation. 1 The application of these methods is demonstrated for the electronic absorption spectra of N 2 and H 2 O both in the absence and in the presence of a magnetic field. The dependence of electronic spectra on the magnetic field strength and its orientation relative to the molecule is studied. The complex evolution of the absorption spectra with magnetic field is rationalised using the molecular orbital pair decomposition analysis, which provides crucial insight in strong fields where the spectra are radically different from their zero-field counterparts

Topological Analysis of Magnetically Induced Current Densities in Strong Magnetic Fields Using Stagnation Graphs

Stagnation graphs provide a useful tool to analyze the main topological features of the often complicated vector field associated with magnetically induced currents. Previously, these graphs have been constructed using response quantities appropriate for modest applied magnetic fields. We present an implementation capable of producing these graphs in arbitrarily strong magnetic fields, using current-density-functional theory. This enables us to study how the topology of the current vector field changes with the strength and orientation of the applied magnetic field. Applications to CH4, C2H2 and C2H4 are presented. In each case, we consider molecular geometries optimized in the presence of the magnetic field. The stagnation graphs reveal subtle changes to this vector field where the symmetry of the molecule remains constant. However, when the electronic state and symmetry of the corresponding equilibrium geometry changes with increasing field strength, the changes to the stagnation graph are extensive. We expect that the approach presented here will be helpful in interpreting changes in molecular structure and bonding in the strong-field regime

Exchange-correlation functionals via local interpolation along the adiabatic connection

The construction of density-functional approximations is explored by modeling the adiabatic connection locally, using energy densities defined in terms of the electrostatic potential of the exchange−correlation hole. These local models are more amenable to the construction of size-consistent approximations than their global counterparts. In this work we use accurate input local ingredients to assess the accuracy of a range of local interpolation models against accurate exchange−correlation energy densities. The importance of the strictly correlated electrons (SCE) functional describing the strong coupling limit is emphasized, enabling the corresponding interpolated functionals to treat strong correlation effects. In addition to exploring the performance of such models numerically for the helium and beryllium isoelectronic series and the dissociation of the hydrogen molecule, an approximate analytic model is presented for the initial slope of the local adiabatic connection. Comparisons are made with approaches based on global models, and prospects for future approximations based on the local adiabatic connection are discussed

Robust All-Electron Optimization in Orbital-Free Density-Functional Theory Using the Trust-Region Image Method

We present a Gaussian-basis implementation of orbital-free density-functional theory (OF-DFT) in which the trust-region image method (TRIM) is used for optimization. This second-order optimization scheme has been constructed to provide benchmark all-electron results with very tight convergence of the particle-number constraint, associated chemical potential, and electron density. It is demonstrated that, by preserving the saddle-point nature of the optimization and simultaneously optimizing the density and chemical potential, an order of magnitude reduction in the number of iterations required for convergence is obtained. The approach is compared and contrasted with a new implementation of the nested optimization scheme put forward by Chan, Cohen, and Handy. Our implementation allows for semilocal kinetic-energy (and exchange-correlation) functionals to be handled self-consistently in all-electron calculations. The all-electron Gaussian-basis setting for these calculations will enable direct comparison with a wide range of standard high-accuracy quantum-chemical methods as well as with Kohn-Sham density-functional theory. We expect that the present implementation will provide a useful tool for analyzing the performance of approximate kinetic-energy functionals in finite systems

Extending conceptual DFT to include external variables: the influence of magnetic fields

An extension of conceptual DFT to include the influence of an external magnetic field is proposed in the context of a program set up to cope with the ever increasing variability of reaction conditions and concomitant reactivity. The two simplest global reactivity descriptors, the electronic chemical potential (μ) and the hardness (η), are considered for the main group atoms H-Kr using current density-functional theory. The magnetic field strength, |B|, is varied between 0.0 and 1.0 B0 = ħe−1a0−2 ≈ 2.3505 × 105 T, encompassing the Coulomb and intermediate regimes. The carbon atom is studied as an exemplar system to gain insight into the behaviour of the neutral, cationic and anionic species under these conditions. Their electronic configurations change with increasing |B|, leading to a piecewise behaviour of the ionization energy (I) and electron affinity (A) values as a function of |B|. This results in complex behaviour of properties such as the electronegativity χ = −1/2(I + A) = −μ and hardness η = 1/2(I − A). This raises an interesting question: to what extent are atomic properties periodic in the presence of a magnetic field? In the Coulomb regime, close to |B| = 0, we find the familiar periodicity of the atomic properties, and make the connections to response functions central to conceptual DFT. However, as the field increases in the intermediate regime configurational changes of the atomic species lead to discontinuous changes in their properties; fundamentally changing their behaviour, which is illustrated by constructing a periodic table of χ and η values at |B| = 0.5 B0. These values tend to increase for groups 1-2 and decrease for groups 16-18, leading to a narrower range overall and suggesting substantial changes in the chemistry of the main group elements. Changes within each group are also examined as a function of |B|. These are more complex to interpret due to the larger number of configurations accessible to heavier elements at high field. This is illustrated for group 17 where Cl and Br have qualitatively different configurations to their lighter cogener at |B| = 0.5 B0. The insight into periodic trends in strong magnetic fields may provide a crucial starting point for predicting chemical reactivity under these exotic conditions

Molecular charge distributions in strong magnetic fields: a conceptual and current DFT study

The effect of strong magnetic fields on the charge distribution of the hydrogen halides, H2O and NH3 is studied in the context of recent extensions of conceptual density functional theory to include additional variables such as external magnetic fields. From conceptual DFT studies on atoms in strong magnetic fields, changes in electronegativity and hardness suggest a reversal in polarity for all three diatomic molecules under these conditions. This is confirmed by current DFT calculations on these molecules in the presence of strong magnetic fields parallel and perpendicular to the internuclear axis; in the former case the electric dipole moment only undergoes small changes whereas in the latter case it changes significantly and also reverses in direction, doing so at lower field strength if the geometry is relaxed. The absence of a dipole moment induced perpendicular to the bond when a magnetic field is applied in this direction is understood by consideration of time reversal symmetry. Similar results are obtained for H2O and NH3; this may be an important point to consider in future studies focused on the unresolved question on the behaviour of hydrogen bonding in applied magnetic fields

Understanding ground and excited-state molecular structure in strong magnetic fields using the maximum overlap method

The maximum overlap method (MOM) provides a simple but powerful approach for performing calculations on excited states by targeting solutions with non-Aufbau occupations from a reference set of molecular orbitals. In this work, the MOM is used to access excited states of (Formula presented.) and (Formula presented.) in strong magnetic fields. The lowest (Formula presented.), (Formula presented.) and (Formula presented.) states of (Formula presented.) in the absence of a field are compared with the corresponding states in strong magnetic fields. The changes in molecular structure in the presence of the field are examined by performing excited state geometry optimisations using the MOM. The (Formula presented.) state is significantly stabilised by the field, becoming the ground state in strong fields with a preferred orientation perpendicular to the applied field. Its potential energy surface evolves from being repulsive to bound, with an equilateral equilibrium geometry. In contrast, the (Formula presented.) state is destabilised and its structure distorts to an isosceles form with the longest H−H bond parallel to the applied field. Comparisons are made with the (Formula presented.) state of H3, which also becomes bound with an equilateral geometry at high fields. The structures of the high-spin ground states are rationalised by orbital correlation diagrams constructed using constrained geometry optimisations

Analyzing Magnetically Induced Currents in Molecular Systems Using Current-Density-Functional Theory

A suite of tools for the analysis of magnetically induced currents is introduced. These are applicable to both the weak-field regime, well described by linear response perturbation theory, and to the strong-field regime, which is inaccessible to such methods. A disc-based quadrature scheme is proposed for the analysis of magnetically induced current susceptibilities, providing quadratures that are consistently defined between different molecular systems and applicable to both planar 2D and general 3D molecular systems in a black-box manner. The applicability of the approach is demonstrated for a range of planar ring systems, the ground and excited states of the benzene molecule, and the ring, bowl, and cage isomers of the C20 molecule in the presence of a weak magnetic field. In the presence of a strong magnetic field, the para- to diamagnetic transition of the BH molecule is studied, demonstrating that magnetically induced currents present a visual interpretation of this phenomenon, providing insight beyond that accessible using linear response methods

Capturing the electron–electron cusp with the coupling-constant averaged exchange–correlation hole: A case study for Hooke’s atoms

In density-functional theory, the exchange–correlation (XC) energy can be defined exactly through the coupling-constant (λ) averaged XC hole n¯xc(r, r′), representing the probability depletion of finding an electron at r′ due to an electron at r. Accurate knowledge of n¯xc(r, r′) has been crucial for developing XC energy density-functional approximations and understanding their performance for molecules and materials. However, there are very few systems for which accurate XC holes have been calculated since this requires evaluating the one- and two-particle reduced density matrices for a reference wave function over a range of λ while the electron density remains fixed at the physical (λ = 1) density. Although the coupled-cluster singles and doubles (CCSD) method can yield exact results for a two electron system in the complete basis set limit, it cannot capture the electron–electron cusp using finite basis sets. Focusing on Hooke’s atom as a two-electron model system for which certain analytic solutions are known, we examine the effect of this cusp error on the XC hole calculated using CCSD. The Lieb functional is calculated at a range of coupling constants to determine the λ-integrated XC hole. Our results indicate that, for Hooke’s atoms, the error introduced by the description of the electron–electron cusp using Gaussian basis sets at the CCSD level is negligible compared to the basis set incompleteness error. The system-, angle-, and coupling-constant averaged XC holes are also calculated and provide a benchmark against which the Perdew–Burke–Ernzerhof and local density approximation XC hole models are assessed