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.

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.

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.

A Fermi smearing variant of the Tamm-Dancoff approximation for nonadiabatic dynamics involving S1-S0 transitions: Validation and application to azobenzene.

The main shortcoming of time-dependent density functional theory (TDDFT) regarding its use for nonadiabatic molecular dynamics (NAMD) is its incapability to describe conical intersections involving the ground state. To overcome this problem, we combine Fermi smearing (FS) DFT with a fractional-occupation variant of the Tamm-Dancoff approximation (TDA) of TDDFT in the generalized gradient approximation. The resulting method (which we denote as FS-TDA) gives access to ground- and excited-state energies, gradients, and nonadiabatic coupling vectors, which are physically correct even in the vicinity of S1-S0 conical intersections. This is shown for azobenzene, a widely used photoswitch, via single point calculations and NAMD simulations of its cis-trans photoisomerization. We conclude that FS-TDA may be used as an efficient alternative to investigate these processes.

Identifying Free Energy Hot-Spots in Molecular Transformations.

The free energy is one of the central quantities in material and natural sciences. While being well-established, e.g., in drug design or catalyst optimization, computational methods lack a straightforward way to gain deeper insights into the calculated free energy, and thus the underlying chemical or physical processes. Here, we present a generally applicable, spectrum-based ansatz that tackles this shortcoming by identifying contributions from specific atoms or groups to the vibrational free energy. We illustrate this in studies of the bromodomain-inhibitor binding and the anomeric effect in glucose providing quantitative evidence in line with chemical intuition in both cases. For the latter example we also report an experimental infrared spectrum and find excellent agreement with our simulated spectra.

Calculating free energies from the vibrational density of states function: Validation and critical assessment.

We explore and show the usefulness of the density of states function for computing vibrational free energies and free energy differences between small systems. Therefore, we compare this density of states integration method (DSI) to more established schemes such as Bennett's Acceptance Ratio method (BAR), the Normal Mode Analysis (NMA), and the Quasiharmonic Analysis (QHA). The strengths and shortcomings of all methods are highlighted with three numerical examples. Furthermore, the free energy of the ionization of ammonia and the mutation from serine to cysteine are computed using extensive ab initio molecular dynamics simulations. We conclude that DSI improves upon the other frequency-based methods (NMA and QHA) regarding the treatment of anharmonicity and yielding results comparable to BAR in all cases without the need for alchemical transformations. Low-frequency modes lead to larger errors indicating that long simulation times might be required for larger systems. In addition, we introduce the use of DSI for the localization of the vibrational free energy to specific atoms or residues, leading to insights into the underlying process, a unique feature that is only offered by this method.

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.

Quantitative Comparison of Experimental and Computed IR-Spectra Extracted from Ab Initio Molecular Dynamics.

Experimentally measured infrared spectra are often compared to their computed equivalents. However, the accordance is typically characterized by visual inspection, which is prone to subjective judgment. The primary challenge for a similarity-based analysis is that the artifacts introduced by each approach are very different and, therefore, may require preprocessing steps to determine and correct impeding irregularities. To allow for automated objective assessment, we propose a practical and comprehensive workflow involving scaling factors, a novel baseline correction scheme, and peak smoothing. The resulting spectra can then easily be compared quantitatively using similarity measures, for which we found the Pearson correlation coefficient to be the most suitable. The proposed procedure is then applied to compare the agreement of the experimental infrared spectra from the NIST Chemistry Web book with the calculated spectra using standard harmonic frequency analysis and spectra extracted from ab initio molecular dynamics simulations at different levels of theory. We conclude that the direct, quantitative comparison of calculated and measured IR spectra might become a novel, sophisticated approach to benchmark quantum-chemical methods. In the present benchmark, simulated spectra based on ab initio molecular dynamics show in general better agreement with the experiment than static calculations.

Combining Graphics Processing Units, Simplified Time-Dependent Density Functional Theory, and Finite-Difference Couplings to Accelerate Nonadiabatic Molecular Dynamics.

Starting from our recently published implementation of nonadiabatic molecular dynamics (NAMD) on graphics processing units (GPUs), we explore further approaches to accelerate ab initio NAMD calculations at the time-dependent density functional theory (TDDFT) level of theory. We employ (1) the simplified TDDFT schemes of Grimme et al. and (2) the Hammes-Schiffer-Tully approach to obtain nonadiabatic couplings from finite-difference calculations. The resulting scheme delivers an accurate physical picture while virtually eliminating the two computationally most demanding steps of the algorithm. Combined with our GPU-based integral routines for SCF, TDDFT, and TDDFT derivative calculations, NAMD simulations of systems of a few hundreds of atoms at a reasonable time scale become accessible on a single compute node. To demonstrate this and to present a first, illustrative example, we perform TDDFT/MM-NAMD simulations of the rhodopsin protein.

Nonadiabatic Molecular Dynamics on Graphics Processing Units: Performance and Application to Rotary Molecular Motors.

Nonadiabatic molecular dynamics (NAMD) simulations of molecular systems require the efficient evaluation of excited-state properties, such as energies, gradients, and nonadiabatic coupling vectors. Here, we investigate the use of graphics processing units (GPUs) in addition to central processing units (CPUs) to efficiently calculate these properties at the time-dependent density functional theory (TDDFT) level of theory. Our implementation in the FermiONs++ program package uses the J-engine and a preselective screening procedure for the calculation of Coulomb and exchange kernels, respectively. We observe good speed-ups for small and large molecular systems (comparable to those observed in ground-state calculations) and reduced (down to sublinear) scaling behavior with respect to the system size (depending on the spatial locality of the investigated excitation). As a first illustrative application, we present efficient NAMD simulations of a series of newly designed light-driven rotary molecular motors and compare their S1 lifetimes. Although all four rotors show different S1 excitation energies, their ability to rotate upon excitation is conserved, making the series an interesting starting point for rotary molecular motors with tunable excitation energies.

Finding Reactive Configurations: A Machine Learning Approach for Estimating Energy Barriers Applied to Sirtuin 5.

Sirtuin 5 is a class III histone deacetylase that, unlike its classification, mainly catalyzes desuccinylation and demanoylation reactions. It is an interesting drug target that we use here to test new ideas for calculating reaction pathways of large molecular systems such as enzymes. A major issue with most schemes (e.g., adiabatic mapping) is that the resulting activation barrier height heavily depends on the chosen educt conformation. This makes the selection of the initial structure decisive for the success of the characterization. Here, we apply machine learning to a large number of molecular dynamics frames and potential energy barriers obtained by quantum mechanics/molecular mechanics calculations in order to identify (1) suitable start-conformations for reaction path calculations and (2) structural features relevant for the first step of the desuccinylation reaction catalyzed by Sirtuin 5. The latter generally aids the understanding of reaction mechanisms and important interactions in active centers. Using our novel approach, we found eleven key features that govern the reactivity. We were able to estimate reaction barriers with a mean absolute error of 3.6 kcal/mol and identified reactive configurations.

Efficient and Accurate Born-Oppenheimer Molecular Dynamics for Large Molecular Systems.

An efficient scheme for the calculation of Born-Oppenheimer molecular dynamics (BOMD) simulations is introduced. It combines the corrected small basis set Hartree-Fock (HF-3c) method by Sure and Grimme [J. Comput. Chem. 2013, 43, 1672], extended Lagrangian BOMD (XL-BOMD) by Niklasson et al. [J. Chem. Phys. 2009, 130, 214109], and the calculation of the two electron integrals on graphics processing units (GPUs) [J. Chem. Phys. 2013, 138, 134114; J. Chem. Theory Comput. 2015, 11, 918]. To explore the parallel performance of our strong scaling implementation of the method, we present timings and extract, as its validation and first illustrative application, high-quality vibrational spectra from simulated trajectories of ß-carotene, paclitaxel, and liquid water (up to 500 atoms). We conclude that the presented BOMD scheme may be used as a cost-efficient and reliable tool for computing vibrational spectra and thermodynamics of large molecular systems including explicit solvent molecules containing 500 atoms and more. Simulating 50 ps of maitotoxin (nearly 500 atoms) employing time steps of 0.5 fs requires ∼3 weeks on 12 CPUs (Intel Xeon E5 2620 v3) with 24 GPUs (AMD FirePro 3D W8100).