arXiv subjects

Trygve Helgaker

Publications and source records attributed to Trygve Helgaker.

At least 19 recordsLinked to original sources

arXiv2025

Perspective on Moreau-Yosida Regularization in Density-Functional Theory

Within density-functional theory, Moreau-Yosida regularization enables both a reformulation of the theory and a mathematically well-defined definition of the Kohn-Sham approach. It is further employed in density-potential inversion schemes and, through the choice of topology for the density and potential space, can be directly linked to classical field theories. This perspective collects various appearances of the regularization technique within density-functional theory alongside possibilities for their future development.

cond-mat.mtrl-sci↗

arXiv2024

Non-adiabatic coupling matrix elements in a magnetic field: geometric gauge dependence and Berry phase

Non-adiabatic coupling matrix elements (NACMEs) are important in quantum chemistry, particularly for molecular dynamics methods such as surface hopping. However, NACMEs are gauge dependent. This presents a difficulty for their calculation in general, where there are no restrictions on the gauge function except that it be differentiable. These cases are relevant for complex-valued electronic wave functions, such as those that arise in the presence of a magnetic field or spin-orbit coupling. Additionally, the Berry curvature and Berry force play an important role in molecular dynamics in a magnetic field, and are also relevant in the context of spin-orbit coupling. For methods such as surface hopping, excited-state Berry curvatures will also be of interest. With this in mind, we have developed a scheme for the calculation of continuous, differentiable NACMEs as a function of the molecular geometry for complex-valued wave functions. We demonstrate the efficacy of the method by using the H$_2$ molecule at the full configuration-interaction (FCI) level of theory. Additionally, ground- and excited- state Berry curvatures are computed for the first time using FCI theory. Finally, Berry phases are computed directly in terms of diagonal NACMEs.

physics.chem-ph↗

arXiv2023

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 $\lambda$ at $\lambda=0$. 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 ($\lambda \searrow 0$) involving the exchange-correlation potential $v_\mathrm{xc}^\lambda$, which exists by assumption of $v$-representability. On the other hand, a local-exchange potential $v_\mathrm{x}$ is not warranted to exist as such a limit.

physics.chem-ph↗

arXiv2023

Propagators for molecular dynamics in a magnetic field

Ab initio molecular dynamics in a magnetic field requires solving equations of motion with velocity-dependent forces -- namely, the Lorentz force arising from the nuclear charges moving in a magnetic field and the Berry force arising from the shielding of these charges from the magnetic field by the surrounding electrons. In this work, we revisit two existing propagators for these equations of motion, the auxiliary-coordinates-and-momenta (ACM) propagator and the Tajima propagator (TAJ), and compare them with a new exponential (EXP) propagator based on the Magnus expansion. Additionally, we explore limits (for example, the zero-shielding limit), the implementation of higher-order integration schemes, and series truncation to reduce computational cost by carrying out simulations of a HeH$^+$ model system for a wide range of field strengths. While being as efficient as the TAJ propagator, the EXP propagator is the only propagator that converges to both the schemes of Spreiter and Walter (derived for systems without shielding of the Lorentz force) and to the exact cyclotronic motion of a charged particle. Since it also performs best in our model simulations, we conclude that the EXP propagator is the recommended propagator for molecules in magnetic fields.

physics.chem-ph↗

arXiv2022

Time-Dependent Nuclear-Electronic Orbital Hartree-Fock Theory in a Strong Uniform Magnetic Field

The Born-Oppenheimer (BO) approximation is less accurate in the presence of a strong magnetic field than in the absence of a field. This is due to the complicated and unpredictable response of electronic structure to the field, especially in the mixed regime $B \approx B_0 = 2.35 \times 10^5\,$T. Therefore, it is desirable to explore non-BO methods in magnetic fields. 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. NEO Generalized Hartree-Fock theory and time-dependent Hartree-Fock 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 (QEP). 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.

physics.chem-ph↗

arXiv2022

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 H$_2$, HT, and HCN molecules in a strong magnetic field.

physics.chem-ph↗

arXiv2022

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 dependence 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 structure of molecules.

physics.chem-ph↗

arXiv2022

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.

physics.chem-ph↗

arXiv2022

Lieb variation principle in density-functional theory

Lieb's convex formulation of density-functional theory is presented in a pedagogical manner, emphasizing its connection to Hohenberg-Kohn theory and to Levy's constrained-search theory. The Hohenberg-Kohn and Lieb variation principles are discussed, highlighting the dual relationship between the ground-state energy and the universal density functional. Applications of the Lieb variation principle are reviewed, demonstrating how it may be utilized to calculate the Kohn-Sham potential of atoms and molecules, to study the exchange-correlation functional and the adiabatic connection by high-precision many-body methods, and to calculate the exchange-correlation hole and energy densities of atoms and molecules.

physics.chem-ph↗

arXiv2022

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 H$_2$, 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 upon the magnetic field and varies between molecules. We also identify instances of broken time-reversal symmetry for the dissociation curves of BH and CH$^+$.

physics.chem-ph↗

arXiv2021

Ab-Initio Molecular Dynamics with Screened Lorentz Forces. Part 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 of 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 H$_2$ at two field strengths characteristic of magnetic white dwarfs (0.1 $B_0 = 2.35 \times 10^4$ T and $B_0 = 2.35 \times 10^5$ T). While the He simulations clearly demonstrate the importance of electron screening of the Lorentz force in the dynamics, the extracted rovibrational spectra of H$_2$ 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.

physics.chem-ph↗

arXiv2021

Ab-Initio Molecular Dynamics with Screened Lorentz Forces. Part I. Calculation and Atomic Charge Interpretation of Berry Curvature

The dynamics of a molecule in a magnetic field is significantly different form 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, by 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 H$_2$ 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 the 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. Calculation of the Berry curvature allows for its inclusion in \textit{ab initio} molecular dynamics simulations in a magnetic field.

physics.chem-ph↗

arXiv2020

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 semi-local 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 analysing the performance of approximate kinetic-energy functionals in finite systems.

physics.chem-ph↗

arXiv2020

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.

physics.chem-ph↗

arXiv2019

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 $L^1\cap L^{3/2}$ 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.

physics.chem-ph↗

arXiv2018

Generalized Kohn-Sham iteration on Banach spaces

A detailed account of the Kohn-Sham 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 to rigorously introduce, in contrast to the common unregularized approach, a well-defined Kohn-Sham 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.

physics.chem-ph↗

arXiv2017

Excitation energies from G{ö}rling-Levy perturbation theory along the range-separated adiabatic connection

A G{ö}rling-Levy (GL)-based perturbation theory along the range-separated adiabatic connection is assessed for the calculation of electronic excitation energies. In comparison with the Rayleigh-Schr{ö}dinger (RS)-based perturbation theory introduced in a previous work [E. Rebolini, J. Toulouse, A. M. Teale, T. Helgaker, A. Savin, Mol. Phys. 113, 1740 (2015)], this GL-based perturbation theory keeps the ground-state density constant at each order and thus gives the correct ionization energy at each order. Excitation energies up to first order in the perturbation have been calculated numerically for the helium and beryllium atoms and the hydrogen molecule without introducing any density-functional approximations. In comparison with the RS-based perturbation theory, the present GL-based perturbation theory gives much more accurate excitation energies for Rydberg states but similar excitation energies for valence states.

physics.chem-ph↗

arXiv2017

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 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 analogues in LDFT. We prove results concerning $N$-representability, Hohenberg--Kohn-like mappings, existence of minimizers in the constrained-search expression, and a restricted analogue 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.

physics.chem-ph↗