Symmetry and reactivity of π-systems in electric and magnetic fields: a perspective from conceptual DFT.

The extension of conceptual density-functional theory (conceptual DFT) to include external electromagnetic fields in chemical systems is utilised to investigate the effects of strong magnetic fields on the electronic charge distribution and its consequences on the reactivity of π-systems. Formaldehyde, H2CO, is considered as a prototypical example and current-density-functional theory (current-DFT) calculations are used to evaluate the electric dipole moment together with two principal local conceptual DFT descriptors, the electron density and the Fukui functions, which provide insight into how H2CO behaves chemically in a magnetic field. In particular, the symmetry properties of these quantities are analysed on the basis of group, representation, and corepresentation theories using a recently developed automatic program for symbolic symmetry analysis, QSYM2. This allows us to leverage the simple symmetry constraints on the macroscopic electric dipole moment components to make profound predictions on the more nuanced symmetry transformation properties of the microscopic frontier molecular orbitals (MOs), electron densities, and Fukui functions. This is especially useful for complex-valued MOs in magnetic fields whose detailed symmetry analyses lead us to define the new concepts of modular and phasal symmetry breaking. Through these concepts, the deep connection between the vanishing constraints on the electric dipole moment components and the symmetry of electron densities and Fukui functions can be formalised, and the inability of the magnetic field in all three principal orientations considered to induce asymmetry with respect to the molecular plane of H2CO can be understood from a molecular perspective. Furthermore, the detailed forms of the Fukui functions reveal a remarkable reversal in the direction of the dipole moment along the CO bond in the presence of a parallel or perpendicular magnetic field, the origin of which can be attributed to the mixing between the frontier MOs due to their subduced symmetries in magnetic fields. The findings in this work are also discussed in the wider context of a long-standing debate on the possibility to create enantioselectivity by external fields.

Modeling interactions between rubidium atom and magnetometer cell wall molecules.

Magnetometer cell wall coat molecules play an important role in preserving the lifetime of pumped alkali metal atoms for use in magnetometers that are capable of measuring very small magnetic fields. The goal of this study is to help rationalize the design of the cell coat molecules. Rubidium-87 is studied in terms of its interaction with three template cell coat molecules: ethane, ethene, and methyltrichlorosilane (MeTS). Ab initio electronic structure methods are applied to investigate the effect that the coat molecules have on the 2S ground state and 2P excited state of 87Rb. We find that, from the ab initio results, the three template molecules have differing effects, with MeTS having the largest effect on the ground state and ethane or ethene having the largest effect on the non-degenerate excited states.

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 n̄xc(r,r'), representing the probability depletion of finding an electron at r' due to an electron at r. Accurate knowledge of n̄xc(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.

Real-time time-dependent self-consistent field methods with dynamic magnetic fields.

The first finite basis set implementation of the real-time time-dependent self-consistent field method in a dynamic (time-dependent) magnetic field using London atomic orbitals (LAOs) is presented. The accuracy of the finite basis approach using LAOs is benchmarked against numerical results from the literature for the hydrogen atom and H2 in the presence of rapidly oscillating magnetic fields. This comparison is used to inform the choice of appropriate basis sets for studies under such conditions. Remarkably, relatively modest compact LAO basis sets are sufficient to obtain accurate results. Analysis of electron dynamics in the hydrogen atom shows that LAO calculations correctly capture the time evolution of orbital occupations. The Fourier transformation of the autocorrelation function yields a power spectrum exhibiting harmonics associated with coherent emission, which closely matches the literature and further confirms the accuracy of this approach. The dynamical response of the electron density in H2 for a magnetic field parallel to the internuclear axis shows similar behavior to benchmark studies. The flexibility of this implementation is then demonstrated by considering how the dynamical response changes as a function of the orientation of the molecule relative to the applied field. At non-parallel orientations, the symmetry of the system is lowered and numerical benchmark data, which exploit cylindrical symmetry, are no-longer readily available. The present study demonstrates the utility of LAO-based calculations for extreme dynamic magnetic fields, providing a stress-test on the choice of basis. Future applications of this approach for less extreme dynamic magnetic fields are briefly discussed.

Magnetic optical rotation from real-time simulations in finite magnetic fields.

We present a numerical approach to magnetic optical rotation based on real-time time-dependent electronic-structure theory. Not relying on perturbation expansions in the magnetic field strength, the formulation allows us to test the range of validity of the linear relation between the rotation angle per unit path length and the magnetic field strength that was established empirically by Verdet 160 years ago. Results obtained from time-dependent coupled-cluster and time-dependent current density-functional theory are presented for the closed-shell molecules H2, HF, and CO in magnetic fields up to 55 kT at standard temperature and pressure conditions. We find that Verdet's linearity remains valid up to roughly 10-20 kT, above which significant deviations from linearity are observed. Among the three current density-functional approximations tested in this work, the current-dependent Tao-Perdew-Staroverov-Scuseria hybrid functional performs the best in comparison with time-dependent coupled-cluster singles and doubles results for the magnetic optical rotation.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Revealing the exotic structure of molecules in strong magnetic fields.

A novel implementation for the calculation of molecular gradients under strong magnetic fields is employed at the current-density functional theory level to optimize the geometries of molecular structures, which change significantly under these conditions. An analog of the ab initio random structure search is utilized to determine the ground-state equilibrium geometries for Hen and CHn systems at high magnetic field strengths, revealing the most stable structures to be those in high-spin states with a planar geometry aligned perpendicular to the field. The electron and current densities for these systems have also been investigated to develop an explanation of chemical bonding in the strong field regime, providing an insight into the exotic chemistry present in these extreme environments.

Exact constraints and appropriate norms in machine-learned exchange-correlation functionals.

Machine learning techniques have received growing attention as an alternative strategy for developing general-purpose density functional approximations, augmenting the historically successful approach of human-designed functionals derived to obey mathematical constraints known for the exact exchange-correlation functional. More recently, efforts have been made to reconcile the two techniques, integrating machine learning and exact-constraint satisfaction. We continue this integrated approach, designing a deep neural network that exploits the exact constraint and appropriate norm philosophy to de-orbitalize the strongly constrained and appropriately normed (SCAN) functional. The deep neural network is trained to replicate the SCAN functional from only electron density and local derivative information, avoiding the use of the orbital-dependent kinetic energy density. The performance and transferability of the machine-learned functional are demonstrated for molecular and periodic systems.

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.

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.

GW quasiparticle energies of atoms in strong magnetic fields.

Quasiparticle energies of the atoms H-Ne have been computed in the GW approximation in the presence of strong magnetic fields with field strengths varying from 0 to 0.25 atomic units (0.25 B0=0.25 ℏe-1a0 -2≈58 763 T). The GW quasiparticle energies are compared with equation-of-motion ionization-potential (EOM-IP) coupled-cluster singles-and-doubles (CCSD) calculations of the first ionization energies. The best results are obtained with the evGW@PBE0 method, which agrees with the EOM-IP-CCSD model to within about 0.20 eV. Ionization potentials have been calculated for all atoms in the series, representing the first systematic study of ionization potentials for the first-row atoms at field strengths characteristic of magnetic white dwarf stars. Under these conditions, the ionization potentials increase in a near-linear fashion with the field strength, reflecting the linear field dependence of the Landau energy of the ionized electron. The calculated ionization potentials agree well with the best available literature data for He, Li, and Be.

Uniform magnetic fields in density-functional theory.

We construct a density-functional formalism adapted to uniform external magnetic fields that is intermediate between conventional density functional theory and Current-Density Functional Theory (CDFT). In the intermediate theory, which we term linear vector potential-DFT (LDFT), the basic variables are the density, the canonical momentum, and the paramagnetic contribution to the magnetic moment. Both a constrained-search formulation and a convex formulation in terms of Legendre-Fenchel transformations are constructed. Many theoretical issues in CDFT find simplified analogs in LDFT. We prove results concerning N-representability, Hohenberg-Kohn-like mappings, existence of minimizers in the constrained-search expression, and a restricted analog to gauge invariance. The issue of additivity of the energy over non-interacting subsystems, which is qualitatively different in LDFT and CDFT, is also discussed.

Interpolated energy densities, correlation indicators and lower bounds from approximations to the strong coupling limit of DFT.

We investigate the construction of approximated exchange-correlation functionals by interpolating locally along the adiabatic connection between the weak- and the strong-coupling regimes, focussing on the effect of using approximate functionals for the strong-coupling energy densities. The gauge problem is avoided by dealing with quantities that are all locally defined in the same way. Using exact ingredients at weak coupling we are able to isolate the error coming from the approximations at strong coupling only. We find that the nonlocal radius model, which retains some of the non-locality of the exact strong-coupling regime, yields very satisfactory results. We also use interpolation models and quantities from the weak- and strong-coupling regimes to define a correlation-type indicator and a lower bound to the exact exchange-correlation energy. Open problems, related to the nature of the local and global slope of the adiabatic connection at weak coupling, are also discussed.

Connections between variation principles at the interface of wave-function and density-functional theories.

A recently proposed variation principle [N. I. Gidopoulos, Phys. Rev. A 83, 040502(R) (2011)] for the determination of Kohn-Sham effective potentials is examined and extended to arbitrary electron-interaction strengths and to mixed states. Comparisons are drawn with Lieb's convex-conjugate functional, which allows for the determination of a potential associated with a given electron density by maximization, yielding the Kohn-Sham potential for a non-interacting system. The mathematical structure of the two functionals is shown to be intrinsically related; the variation principle put forward by Gidopoulos may be expressed in terms of the Lieb functional. The equivalence between the information obtained from the two approaches is illustrated numerically by their implementation in a common framework.

The importance of current contributions to shielding constants in density-functional theory.

The sources of error in the calculation of nuclear-magnetic-resonance shielding constants determined by density-functional theory are examined. Highly accurate Kohn-Sham wave functions are obtained from coupled-cluster electron density functions and used to define accurate-but current independent-density-functional shielding constants. These new reference values, in tandem with high-accuracy coupled-cluster shielding constants, provide a benchmark for the assessment of errors in common density-functional approximations. In particular the role of errors arising in the diamagnetic and paramagnetic terms is investigated, with particular emphasis on the role of current-dependence in the latter. For carbon and nitrogen the current correction is found to be, in some cases, larger than 10 ppm. This indicates that the absence of this correction in general purpose exchange-correlation functionals is one of the main sources of error in shielding calculations using density functional theory. It is shown that the current correction improves the shielding performance of many popular approximate DFT functionals.

Revisiting the density scaling of the non-interacting kinetic energy.

Scaling relations play an important role in the understanding and development of approximate functionals in density functional theory. Recently, a number of these relationships have been redefined in terms of the Kohn-Sham orbitals [Calderín, Phys. Rev. A: At., Mol., Opt. Phys., 2013, 86, 032510]. For density scaling the author proposed a procedure involving a multiplicative scaling of the Kohn-Sham orbitals whilst keeping their occupation numbers fixed. In the present work, the differences between this scaling with fixed occupation numbers and that of previous studies, where the particle number change implied by the scaling was accommodated through the use of the grand canonical ensemble, are examined. We introduce the terms orbital and ensemble density scaling for these approaches, respectively. The natural ambiguity of the density scaling of the non-interacting kinetic energy functional is examined and the ancillary definitions implicit in each approach are highlighted and compared. As a consequence of these differences, Calderín recovered a homogeneity of degree 1 for the non-interacting kinetic energy functional under orbital scaling, contrasting recent work by the present authors [J. Chem. Phys., 2012, 136, 034101] where the functional was found to be inhomogeneous under ensemble density scaling. Furthermore, we show that the orbital scaling result follows directly from the linearity and the single-particle nature of the kinetic energy operator. The inhomogeneity of the non-interacting kinetic energy functional under ensemble density scaling can be quantified by defining an effective homogeneity. This quantity is shown to recover the homogeneity values for important approximate forms that are exact for limiting cases such as the uniform electron gas and one-electron systems. We argue that the ensemble density scaling provides more insight into the development of new functional forms.

Differentiable but exact formulation of density-functional theory.

The universal density functional F of density-functional theory is a complicated and ill-behaved function of the density-in particular, F is not differentiable, making many formal manipulations more complicated. While F has been well characterized in terms of convex analysis as forming a conjugate pair (E, F) with the ground-state energy E via the Hohenberg-Kohn and Lieb variation principles, F is nondifferentiable and subdifferentiable only on a small (but dense) subset of its domain. In this article, we apply a tool from convex analysis, Moreau-Yosida regularization, to construct, for any ε > 0, pairs of conjugate functionals ((ε)E, (ε)F) that converge to (E, F) pointwise everywhere as ε → 0(+), and such that (ε)F is (Fréchet) differentiable. For technical reasons, we limit our attention to molecular electronic systems in a finite but large box. It is noteworthy that no information is lost in the Moreau-Yosida regularization: the physical ground-state energy E(v) is exactly recoverable from the regularized ground-state energy (ε)E(v) in a simple way. All concepts and results pertaining to the original (E, F) pair have direct counterparts in results for ((ε)E, (ε)F). The Moreau-Yosida regularization therefore allows for an exact, differentiable formulation of density-functional theory. In particular, taking advantage of the differentiability of (ε)F, a rigorous formulation of Kohn-Sham theory is presented that does not suffer from the noninteracting representability problem in standard Kohn-Sham theory.

Excitation energies along a range-separated adiabatic connection.

We present a study of the variation of total energies and excitation energies along a range-separated adiabatic connection. This connection links the non-interacting Kohn-Sham electronic system to the physical interacting system by progressively switching on the electron-electron interactions whilst simultaneously adjusting a one-electron effective potential so as to keep the ground-state density constant. The interactions are introduced in a range-dependent manner, first introducing predominantly long-range, and then all-range, interactions as the physical system is approached, as opposed to the conventional adiabatic connection where the interactions are introduced by globally scaling the standard Coulomb interaction. Reference data are reported for the He and Be atoms and the H2 molecule, obtained by calculating the short-range effective potential at the full configuration-interaction level using Lieb's Legendre-transform approach. As the strength of the electron-electron interactions increases, the excitation energies, calculated for the partially interacting systems along the adiabatic connection, offer increasingly accurate approximations to the exact excitation energies. Importantly, the excitation energies calculated at an intermediate point of the adiabatic connection are much better approximations to the exact excitation energies than are the corresponding Kohn-Sham excitation energies. This is particularly evident in situations involving strong static correlation effects and states with multiple excitation character, such as the dissociating H2 molecule. These results highlight the utility of long-range interacting reference systems as a starting point for the calculation of excitation energies and are of interest for developing and analyzing practical approximate range-separated density-functional methodologies.