Exchange-only virial relation from the adiabatic connection.

The exchange-only virial relation due to Levy and Perdew is revisited. Invoking the adiabatic connection, we introduce the exchange energy in terms of the right-derivative of the universal density functional w.r.t. the coupling strength λ at λ = 0. This agrees with the Levy-Perdew definition of the exchange energy as a high-density limit of the full exchange-correlation energy. By relying on v-representability for a fixed density at varying coupling strength, we prove an exchange-only virial relation without an explicit local-exchange potential. Instead, the relation is in terms of a limit (λ ↘ 0) involving the exchange-correlation potential vxcλ, which exists by assumption of v-representability. On the other hand, a local-exchange potential vx is not warranted to exist as such a limit.

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.

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.

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.

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.

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+.

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.

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.

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.

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.

First-Principles Calculation of 1H NMR Chemical Shifts of Complex Metal Polyhydrides: The Essential Inclusion of Relativity and Dynamics.

1H NMR spectroscopy has become an important technique for the characterization of transition-metal hydride complexes, whose metal-bound hydrides are often difficult to locate by X-ray diffraction. In this regard, the accurate prediction of 1H NMR chemical shifts provides a useful, but challenging, strategy to help in the interpretation of the experimental spectra. In this work, we establish a density-functional-theory protocol that includes relativistic, solvent, and dynamic effects at a high level of theory, allowing us to report an accurate and reliable interpretation of 1H NMR hydride chemical shifts of iridium polyhydride complexes. In particular, we have studied in detail the hydride chemical shifts of the [Ir6(IMe)8(CO)2H14]2+ complex in order to validate previous assignments. The computed 1H NMR chemical shifts are strongly dependent on the relativistic treatment, the choice of the DFT exchange-correlation functional, and the conformational dynamics. By combining a fully relativistic four-component electronic-structure treatment with ab initio molecular dynamics, we were able to reliably model both the terminal and bridging hydride chemical shifts and to show that two NMR hydride signals were inversely assigned in the experiment.

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.

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.

Dalton Project: A Python platform for molecular- and electronic-structure simulations of complex systems.

The Dalton Project provides a uniform platform access to the underlying full-fledged quantum chemistry codes Dalton and LSDalton as well as the PyFraME package for automatized fragmentation and parameterization of complex molecular environments. The platform is written in Python and defines a means for library communication and interaction. Intermediate data such as integrals are exposed to the platform and made accessible to the user in the form of NumPy arrays, and the resulting data are extracted, analyzed, and visualized. Complex computational protocols that may, for instance, arise due to a need for environment fragmentation and configuration-space sampling of biochemical systems are readily assisted by the platform. The platform is designed to host additional software libraries and will serve as a hub for future modular software development efforts in the distributed Dalton community.

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.

A computational quantum-mechanical model of a molecular magnetic trap.

A model for describing the states of a molecular system trapped in a cavity created by a fast-rotating strong magnetic field is proposed and implemented. All-particle explicitly correlated Gaussian functions with shifted centers are employed in the model to expand the wave functions of the system. Both "internal" states associated with the system's rovibrational and electronic motions and the "external" states associated with translational motion of the center of mass of the system in the cavity are calculated. The states are visualized by density plots. The model is applied to a trapped HD molecule.

Bethe-Salpeter correlation energies of atoms and molecules.

A variety of approaches are presented for the computation of atomic and molecular correlation energies based on the Bethe-Salpeter equation in the framework of the adiabatic-connection fluctuation-dissipation theorem. The performance of the approaches is assessed by computing the total energies of the atoms H-Ne and the atomization energies of the high-accuracy extrapolated ab initio thermochemistry set of small molecules as well as by determining the bond lengths and harmonic vibrational frequencies of the metal monoxides MO with M=Ca-Zn.

Generalized Kohn-Sham iteration on Banach spaces.

A detailed account of the Kohn-Sham (KS) algorithm from quantum chemistry, formulated rigorously in the very general setting of convex analysis on Banach spaces, is given here. Starting from a Levy-Lieb-type functional, its convex and lower semi-continuous extension is regularized to obtain differentiability. This extra layer allows us to rigorously introduce, in contrast to the common unregularized approach, a well-defined KS iteration scheme. Convergence in a weak sense is then proven. This generalized formulation is applicable to a wide range of different density-functional theories and possibly even to models outside of quantum mechanics.