The Structure of the Density-Potential Mapping. Part II: Including Magnetic Fields.

The Hohenberg-Kohn theorem of density-functional theory (DFT) is broadly considered the conceptual basis for a full characterization of an electronic system in its ground state by just one-body particle density. In this Part II of a series of two articles, we aim at clarifying the status of this theorem within different extensions of DFT including magnetic fields. We will in particular discuss current-density-functional theory (CDFT) and review the different formulations known in the literature, including the conventional paramagnetic CDFT and some nonstandard alternatives. For the former, it is known that the Hohenberg-Kohn theorem is no longer valid due to counterexamples. Nonetheless, paramagnetic CDFT has the mathematical framework closest to standard DFT and, just like in standard DFT, nondifferentiability of the density functional can be mitigated through Moreau-Yosida regularization. Interesting insights can be drawn from both Maxwell-Schrödinger DFT and quantum-electrodynamic DFT, which are also discussed here.

Unitary coupled-cluster for quantum computation of molecular properties in a strong magnetic field.

In truncated coupled-cluster (CC) theories, non-variational and/or generally complex ground-state energies can occur. This is due to the non-Hermitian nature of the similarity transformed Hamiltonian matrix in combination with CC truncation. For chemical problems that deal with real-valued Hamiltonian matrices, complex CC energies rarely occur. However, for complex-valued Hamiltonian matrices, such as those that arise in the presence of strong magnetic fields, complex CC energies can be regularly observed unless certain symmetry conditions are fulfilled. Therefore, in the presence of magnetic fields, it is desirable to pursue CC methods that are guaranteed to give upper-bound, real-valued energies. In this work, we present the first application of unitary CC to chemical systems in a strong magnetic field. This is achieved utilizing the variational quantum eigensolver algorithm applied to the unitary coupled-cluster singles and doubles (UCCSD) method. We benchmark the method on the H2 molecule in a strong magnetic field and then calculate UCCSD energies for the H4 molecule as a function of both geometry and field angle. We show that while standard CCSD can yield generally complex energies that are not an upper-bound to the true energy, UCCSD always results in variational and real-valued energies. We also show that the imaginary components of the CCSD energy are largest in the strongly correlated region. Last, the UCCSD calculations capture a large percentage of the correlation energy.

Carbon dioxide adsorption to UiO-66: theoretical analysis of binding energy and NMR properties.

UiO-66 is one of the most valuable metal-organic frameworks because of its excellent adsorption capability for gas molecules and its high stability towards water. Herein we investigated adsorption of carbon dioxide (CO2), acetone, and methanol to infinite UiO-66 using DFT calculations on an infinite system under periodic-boundary conditions and post-Hartree-Fock (SCS-MP2 and MP2.5) calculations on cluster models. Three to four molecules are adsorbed at each of four µ-OH groups bridging three Zr atoms in one unit cell (named Site I). Six molecules are adsorbed around three pillar ligands, where the molecule is loosely surrounded by three terephthalate ligands (named Site II). Also, six molecules are adsorbed around the pillar ligand in a different manner from that at Site II, where the molecule is surrounded by three terephthalate ligands (named Site III). Totally fifteen to sixteen CO2 molecules are adsorbed into one unit cell of UiO-66. The binding energy (BE) decreases in the order Site I > Site III > Site II for all three molecules studied here and in the order acetone > methanol ≫ CO2 in the three adsorption sites. At the site I, the protonic H atom of the µ-OH group interacts strongly with the negatively charged O atom of CO2, acetone and methanol, which is the origin of the largest BE value at this site. Although the DFT calculations present these decreasing orders of BE values correctly, the correction by post-Hartree-Fock calculations is not negligibly small and must be added for obtaining better BE values. We explored NMR spectra of UiO-66 with adsorbed CO2 molecules and found that the isotropic shielding constants of the 1H atom significantly differ among no CO2, one CO2 (at Sites I, II, or III), and fifteen CO2 adsorption cases (Sites I to III) but the isotropic 17O and 13C shielding constants change moderately by adsorption of fifteen CO2 molecules. Thus, 1H NMR measurement is a useful experiment for investigating CO2 adsorption.

The Structure of Density-Potential Mapping. Part I: Standard Density-Functional Theory.

The Hohenberg-Kohn theorem of density-functional theory (DFT) is broadly considered the conceptual basis for a full characterization of an electronic system in its ground state by just the one-body particle density. Part I of this review aims at clarifying the status of the Hohenberg-Kohn theorem within DFT and Part II at different extensions of the theory that include magnetic fields. We collect evidence that the Hohenberg-Kohn theorem does not so much form the basis of DFT, but is rather the consequence of a more comprehensive mathematical framework. Such results are especially useful when it comes to the construction of generalized DFTs.

Molecular vibrations in the presence of velocity-dependent forces.

A semiclassical theory of small oscillations is developed for nuclei that are subject to velocity-dependent forces in addition to the usual interatomic forces. When the velocity-dependent forces are due to a strong magnetic field, novel effects arise-for example, the coupling of vibrational, rotational, and translational modes. The theory is first developed using Newtonian mechanics and we provide a simple quantification of the coupling between these types of modes. We also discuss the mathematical structure of the problem, which turns out to be a quadratic eigenvalue problem rather than a standard eigenvalue problem. The theory is then re-derived using the Hamiltonian formalism, which brings additional insight, including a close analogy to the quantum-mechanical treatment of the problem. Finally, we provide numerical examples for the H2, HT, and HCN molecules in a strong magnetic field.

Time-dependent nuclear-electronic orbital Hartree-Fock theory in a strong uniform magnetic field.

In an ultrastrong magnetic field, with field strength B ≈ B0 = 2.35 × 105 T, molecular structure and dynamics differ strongly from that observed on the Earth. Within the Born-Oppenheimer (BO) approximation, for example, frequent (near) crossings of electronic energy surfaces are induced by the field, suggesting that nonadiabatic phenomena and processes may play a more important role in this mixed-field regime than in the weak-field regime on Earth. To understand the chemistry in the mixed regime, it therefore becomes important to explore non-BO methods. In this work, the nuclear-electronic orbital (NEO) method is employed to study protonic vibrational excitation energies in the presence of a strong magnetic field. The NEO generalized Hartree-Fock theory and time-dependent Hartree-Fock (TDHF) theory are derived and implemented, accounting for all terms that result as a consequence of the nonperturbative treatment of molecular systems in a magnetic field. The NEO results for HCN and FHF- with clamped heavy nuclei are compared against the quadratic eigenvalue problem. Each molecule has three semi-classical modes owing to the hydrogen-two precession modes that are degenerate in the absence of a field and one stretching mode. The NEO-TDHF model is found to perform well; in particular, it automatically captures the screening effects of the electrons on the nuclei, which are quantified through the difference in energy of the precession modes.

Berry Population Analysis: Atomic Charges from the Berry Curvature in a Magnetic Field.

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.

Magnetic-translational sum rule and approximate models of the molecular Berry curvature.

The Berry connection and curvature are key components of electronic structure calculations for atoms and molecules in magnetic fields. They ensure the correct translational behavior of the effective nuclear Hamiltonian and the correct center-of-mass motion during molecular dynamics in these environments. In this work, we demonstrate how these properties of the Berry connection and curvature arise from the translational symmetry of the electronic wave function and how they are fully captured by a finite basis set of London orbitals but not by standard Gaussian basis sets. This is illustrated by a series of Hartree-Fock calculations on small molecules in different basis sets. Based on the resulting physical interpretation of the Berry curvature as the shielding of the nuclei by the electrons, we introduce and test a series of approximations using the Mulliken fragmentation scheme of the electron density. These approximations will be particularly useful in ab initio molecular dynamics calculations in a magnetic field since they reduce the computational cost, while recovering the correct physics and up to 95% of the exact Berry curvature.

Molecular dynamics of linear molecules in strong magnetic fields.

Molecular rotations and vibrations have been extensively studied by chemists for decades, both experimentally using spectroscopic methods and theoretically with the help of quantum chemistry. However, the theoretical investigation of molecular rotations and vibrations in strong magnetic fields requires computationally more demanding tools. As such, proper calculations of rotational and vibrational spectra were not feasible up until very recently. In this work, we present rotational and vibrational spectra for two small linear molecules, H2 and LiH, in strong magnetic fields. By treating the nuclei as classical particles, trajectories for rotations and vibrations are simulated from ab initio molecular dynamics. Born-Oppenheimer potential energy surfaces are calculated at the Hartree-Fock and MP2 levels of theory using London atomic orbitals to ensure gauge origin invariance. For the calculation of nuclear trajectories, a highly efficient Tajima propagator is introduced, incorporating the Berry curvature tensor accounting for the screening of nuclear charges.

Analytic calculation of the Berry curvature and diagonal Born-Oppenheimer correction for molecular systems in uniform magnetic fields.

The diagonal nonadiabatic term arising from the Born-Oppenheimer wave function ansatz contains contributions from a vector and scalar potential. The former is provably zero when the wave function can be taken to be real valued, and the latter, known as the diagonal Born-Oppenheimer correction (DBOC), is typically small in magnitude. Therefore, unless high accuracy is sought, the diagonal nonadiabatic term is usually neglected when calculating molecular properties. In the presence of a magnetic field, the wave function is generally complex, and the geometric vector potential gives rise to a screening force that is qualitatively important for molecular dynamics. This screening force is written in terms of the Berry curvature and is added to the bare Lorentz force acting on the nuclei in the presence of the field. In this work, we derive analytic expressions for the Berry curvature and DBOC using both first- and second-quantization formalisms for the case of generalized and restricted Hartree-Fock theories in a uniform magnetic field. The Berry curvature and DBOC are calculated as a function of the magnetic field strength and the bond distance for the ground-state singlets of H2, LiH, BH, and CH+. We also examine the stability and time-reversal symmetry of the underlying self-consistent field solutions. The character of the DBOC and Berry curvature is found to depend on the magnetic field and varies between molecules. We also identify instances of broken time-reversal symmetry for the dissociation curves of BH and CH+.

Ab Initio molecular dynamics with screened Lorentz forces. II. Efficient propagators and rovibrational spectra in strong magnetic fields.

Strong magnetic fields have a large impact on the dynamics of molecules. In addition to the changes in the electronic structure, the nuclei are exposed to the Lorentz force with the magnetic field being screened by the electrons. In this work, we explore these effects using ab initio molecular dynamics simulations based on an effective Hamiltonian calculated at the Hartree-Fock level of theory. To correctly include these non-conservative forces in the dynamics, we have designed a series of novel propagators that show both good efficiency and stability in test cases. As a first application, we analyze simulations of He and H2 at two field strengths characteristic of magnetic white dwarfs (0.1 B0 = 2.35 × 104 T and B0 = 2.35 × 105 T). While the He simulations clearly demonstrate the importance of electron screening of the Lorentz force in the dynamics, the extracted rovibrational spectra of H2 reveal a number of fascinating features not observed in the field-free case: couplings of rotations/vibrations with the cyclotron rotation, overtones with unusual selection rules, and hindered rotations that transmute into librations with increasing field strength. We conclude that our presented framework is a powerful tool to investigate molecules in these extreme environments.

Ab initio molecular dynamics with screened Lorentz forces. I. Calculation and atomic charge interpretation of Berry curvature.

The dynamics of a molecule in a magnetic field is significantly different from its zero-field counterpart. One important difference in the presence of a field is the Lorentz force acting on the nuclei, which can be decomposed as the sum of the bare nuclear Lorentz force and a screening force due to the electrons. This screening force is calculated from the Berry curvature and can change the dynamics qualitatively. It is therefore important to include the contributions from the Berry curvature in molecular dynamics simulations in a magnetic field. In this work, we present a scheme for calculating the Berry curvature numerically using a finite-difference technique, addressing challenges related to the arbitrary global phase of the wave function. The Berry curvature is calculated as a function of bond distance for H2 at the restricted and unrestricted Hartree-Fock levels of theory and for CH+ as a function of the magnetic field strength at the restricted Hartree-Fock level of theory. The calculations are carried out using basis sets of contracted Gaussian functions equipped with London phase factors (London orbitals) to ensure gauge-origin invariance. In this paper, we also interpret the Berry curvature in terms of atomic charges and discuss its convergence in basis sets with and without London phase factors. The calculation of the Berry curvature allows for its inclusion in ab initio molecular dynamics simulations in a magnetic field.

Revisiting density-functional theory of the total current density.

Density-functional theory (DFT) requires an extra variable besides the electron density in order to properly incorporate magnetic-field effects. In a time-dependent setting, the gauge-invariant, total current density takes that role. A peculiar feature of the static ground-state setting is, however, that the gauge-dependent paramagnetic current density appears as the additional variable instead. An alternative, exact reformulation in terms of the total current density has long been sought but to date a work by Diener is the only available candidate. In that work, an unorthodox variational principle was used to establish a ground-state DFT of the total current density as well as an accompanying Hohenberg-Kohn-like result. We here reinterpret and clarify Diener's formulation based on a maximin variational principle. Using simple facts about convexity implied by the resulting variational expressions, we prove that Diener's formulation is unfortunately not capable of reproducing the correct ground-state energy and, furthermore, that the suggested construction of a Hohenberg-Kohn map contains an irreparable mistake.

Lower Semicontinuity of the Universal Functional in Paramagnetic Current-Density Functional Theory.

A cornerstone of current-density functional theory (CDFT) in its paramagnetic formulation is proven. After a brief outline of the mathematical structure of CDFT, the lower semicontinuity and expectation-valuedness of the CDFT constrained-search functional is proven, meaning that there is always a minimizing density matrix in the CDFT constrained-search universal density functional. These results place the mathematical framework of CDFT on the same footing as that of standard DFT.

Benchmarking Density Functional Approximations for Diamagnetic and Paramagnetic Molecules in Nonuniform Magnetic Fields.

In this article, correlated studies on a test set of 36 small molecules are carried out with both wavefunction (HF, MP2, CCSD) and density functional (LDA, KT3, cTPSS, cM06-L) methods. The effect of correlation on exotic response properties such as molecular electronic anapole susceptibilities is studied and the performance of the various density functional approximations are benchmarked against CCSD and/or MP2. Atoms and molecules are traditionally classified into "diamagnetic" and "paramagnetic" based on their isotropic response to uniform magnetic fields. However, in this article, we propose a more fine-grained classification of molecular systems on the basis of their response to generally nonuniform magnetic fields. The relation of orientation to different qualitative responses is also considered.

Bonding in the helium dimer in strong magnetic fields: the role of spin and angular momentum.

We investigate the helium dimer in strong magnetic fields, focusing on the spectrum of low-lying electronic states and their dissociation curves, at the full configuration-interaction level of theory. To address the loss of cylindrical symmetry and angular momentum as a good quantum number for nontrivial angles between the bond axis and magnetic field, we introduce the almost quantized angular momentum (AQAM) and show that it provides useful information about states in arbitrary orientations. In general, strong magnetic fields dramatically rearrange the spectrum, with the orbital Zeeman effect bringing down states of higher angular momentum below the states with pure σ character as the field strength increases. In addition, the spin Zeeman effect pushes triplet states below the lowest singlet; in particular, a field of one atomic unit is strong enough to push a quintet state below the triplets. In general, the angle between the bond axis and the magnetic field also continuously modulates the degree of σ, π, and δ character of bonds and the previously identified perpendicular paramagnetic bonding mechanism is found to be common among excited states. Electronic states with preferred skew field orientations are identified and rationalized in terms of permanent and induced electronic currents.

Erratum: Guaranteed Convergence of a Regularized Kohn-Sham Iteration in Finite Dimensions [Phys. Rev. Lett. 123, 037401 (2019)].

This corrects the article DOI: 10.1103/PhysRevLett.123.037401.

Guaranteed Convergence of a Regularized Kohn-Sham Iteration in Finite Dimensions.

The exact Kohn-Sham iteration of generalized density-functional theory in finite dimensions with a Moreau-Yosida regularized universal Lieb functional and an adaptive damping step is shown to converge to the correct ground-state density.

Excited States of Molecules in Strong Uniform and Nonuniform Magnetic Fields.

This paper reports an implementation of Hartree-Fock linear response with complex orbitals for computing electronic spectra of molecules in strong external magnetic fields. The implementation is completely general, allowing for spin-restricted, spin-unrestricted, and general two-component reference states. The method is applied to small molecules placed in strong uniform and nonuniform magnetic fields of astrochemical importance at the Random Phase Approximation level of theory. For uniform fields, where comparison is possible, the spectra are found to be qualitatively similar to those recently obtained with equation of motion coupled cluster theory. We also study the behavior of spin-forbidden excitations with progressive loss of spin symmetry induced by nonuniform magnetic fields. Finally, the equivalence of length and velocity gauges for oscillator strengths when using complex orbitals is investigated and found to hold numerically.

Kohn-Sham Theory with Paramagnetic Currents: Compatibility and Functional Differentiability.

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 L1 ∩ L3/2 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.