Born-Oppenheimer Research Articles

Non-adiabatic Couplings in Surface Hopping with Tight Binding Density Functional Theory: The Case of Molecular Motors.

Nonadiabatic molecular dynamics (NAMD) has become an essential computational technique for studying the photophysical relaxation of molecular systems after light absorption. These phenomena require approximations that go beyond the Born-Oppenheimer approximation, and the accuracy of the results heavily depends on the electronic structure theory employed. Sophisticated electronic methods, however, make these techniques computationally expensive, even for medium size systems. Consequently, simulations are often performed on simplified models to interpret the experimental results. In this context, a variety of techniques have been developed to perform NAMD using approximate methods, particularly density functional tight binding (DFTB). Despite the use of these techniques on large systems, where ab initio methods are computationally prohibitive, a comprehensive validation has been lacking. In this work, we present a new implementation of trajectory surface hopping combined with DFTB, utilizing nonadiabatic coupling vectors. We selected the methaniminium cation and furan systems for validation, providing an exhaustive comparison with the higher-level electronic structure methods. As a case study, we simulated a system from the class of molecular motors, which has been extensively studied experimentally but remains challenging to simulate with ab initio methods due to its inherent complexity. Our approach effectively captures the key photophysical mechanism of dihedral rotation after the absorption of light. Additionally, we successfully reproduced the transition from the bright to dark states observed in the time-dependent fluorescence experiments, providing valuable insights into this critical part of the photophysical behavior in molecular motors.

Hybrids, tetraquarks, pentaquarks, doubly heavy baryons, and quarkonia in Born-Oppenheimer effective theory

The discovery of XYZ exotic states in the hadronic sector with two heavy quarks represents a significant challenge in particle theory. Understanding and predicting their nature remains an open problem. In this work, we demonstrate how the Born-Oppenheimer (BO) effective field theory (BOEFT), derived from quantum chromodynamics (QCD) on the basis of scale separation and symmetries, can address XYZ exotics of any composition. We derive the Schrödinger coupled equations that describe hybrids, tetraquarks, pentaquarks, doubly heavy baryons, and quarkonia at leading order, incorporating nonadiabatic terms, and present the predicted multiplets. We define the static potentials in terms of the QCD static energies for all relevant cases. We provide the precise form of the nonperturbative low-energy gauge-invariant correlators required for the BOEFT: static energies, generalized Wilson loops, gluelumps, and adjoint mesons. These are to be calculated on the lattice, and we calculate here their short-distance behavior. Furthermore, we outline how spin-dependent corrections and mixing terms can be incorporated using matching computations. Lastly, we discuss how static energies with the same BO quantum numbers mix at large distances leading to the phenomenon of avoided level crossing. This effect is crucial to understand the emergence of exotics with molecular characteristics, such as the χc1(3872). With BOEFT both the tetraquark and the molecular picture appear as part of the same description. Published by the American Physical Society 2024

Modeling of warm dense hydrogen via explicit real-time electron dynamics: Dynamic structure factors

We present two methods for computing the dynamic structure factor for warm dense hydrogen without invoking either the Born-Oppenheimer approximation or the Chihara decomposition, by employing a wave-packet description that resolves the electron dynamics during ion evolution. First, a semiclassical method is discussed, which is corrected based on known quantum constraints, and second, a direct computation of the density response function within the molecular dynamics. The wave-packet models are compared to PIMC and DFT-MD for the static and low-frequency behavior. For the high-frequency behavior the models recover the expected behavior in the limits of small and large momentum transfers and show the characteristic flattening of the plasmon dispersion for intermediate momentum transfers due to interactions, in agreement with commonly used models for x-ray Thomson scattering. By modeling the electrons and ions on an equal footing, both the ion and free electron part of the spectrum can now be treated within a single framework where we simultaneously resolve the ion-acoustic and plasmon mode, with a self-consistent description of collisions and screening. Published by the American Physical Society 2024

Tetraquarks at large M and large N

We study tetraquarks in large N QCD with heavy quarks, in the domain where non-relativistic quantum mechanics offers an adequate approximation. Within the regime of validity of the Born-Oppenheimer approximation, we systematically study and explicitly construct tetraquark states. At leading order in the 1/N expansion, the bound spectrum consists of free mesons, while the 1/N corrections give rise to a Born-Oppenheimer potential that can bind the mesons into tetraquarks. We find two different types of tetraquarks, each endowed with distinct color-spatial wavefunctions. These states arise in the presence of an O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(N) mass hierarchy between the quarks and the antiquarks. We provide a quantitative argument indicating that only for such a hierarchy is the ground state of the system a tetraquark. We discuss what the extrapolation of our results to realistic values of the parameters may imply for the QCD tetraquark states.

Time-dependent rovibronic wave function of H2+ by multiconfiguration theory

Abstract We simulate the time-dependent rovibronic wave function of H 2 + in an intense, few-cycle 400 nm laser field using a time-dependent multiconfiguration method. By calculating the expectation value of the total angular momentum squared, the time-dependent expectation value of the internuclear distance, the induced dipole moment, and the probability of electron ejection, we show that rotational, vibrational and electronic excitation as well as ionization can be simulated. We compare the results obtained using the time-dependent multiconfiguration method with the results obtained using the standard two-state Born-Oppenheimer approximation and the close-coupling expansion. We find that the multiconfiguration results for a wave function constructed using 21 time-dependent orbitals are in good agreement with the close-coupling results, but that the two-state Born-Oppenheimer approximation leads to an underestimation of the induced dipole moment by a factor of two and an overestimation the expectation value of the total angular momentum squared by a factor of 2.3.

Cavity-modified local and non-local electronic interactions in molecular ensembles under vibrational strong coupling.

Resonant vibrational strong coupling (VSC) between molecular vibrations and quantized field modes of low-frequency optical cavities constitutes the conceptual cornerstone of vibro-polaritonic chemistry. In this work, we theoretically investigate the role of complementary nonresonant electron-photon interactions in the cavity Born-Oppenheimer (CBO) approximation. In particular, we study cavity-induced modifications of local and non-local electronic interactions in dipole-coupled molecular ensembles under VSC. Methodologically, we combine CBO perturbation theory (CBO-PT) [E. W. Fischer and P. Saalfrank, J. Chem. Theory Comput. 19, 7215 (2023)] with non-perturbative CBO Hartree-Fock (HF) and coupled cluster (CC) theories. In a first step, we derive up to second-order CBO-PT cavity potential energy surfaces, which reveal non-trivial intra- and inter-molecular corrections induced by the cavity. We then introduce the concept of a cavity reaction potential (CRP), minimizing the electronic energy in the cavity subspace to discuss vibro-polaritonic reaction mechanisms. We present reformulations of CBO-HF and CBO-CC approaches for CRPs and derive second-order approximate CRPs from CBO-PT for unimolecular and bimolecular scenarios. In the unimolecular case, we find small local modifications of molecular potential energy surfaces for selected isomerization reactions dominantly captured by the first-order dipole fluctuation correction. Excellent agreement between CBO-PT and non-perturbative wave function results indicates minor VSC-induced state relaxation effects in the single-molecule limit. In the bimolecular scenario, CBO-PT reveals an explicit coupling of interacting dimers to cavity modes besides cavity-polarization dependent dipole-induced dipole and van der Waals interactions with enhanced long-range character. An illustrative CBO-coupled cluster theory with singles and doubles-based numerical analysis of selected molecular dimer models provides a complementary non-perturbative perspective on cavity-modified intermolecular interactions under VSC.

Cavity Born-Oppenheimer approximation for molecules and materials via electric field response.

We present an abinitio method for computing vibro-polariton and phonon-polariton spectra of molecules and solids coupled to the photon modes of optical cavities. We demonstrate that if interactions of cavity photon modes with both nuclear and electronic degrees of freedom are treated on the level of the cavity Born-Oppenheimer approximation, spectra can be expressed in terms of the matter response to electric fields and nuclear displacements, which are readily available in standard density functional perturbation theory implementations. In this framework, results over a range of cavity parameters can be obtained without the need for additional electronic structure calculations, enabling efficient calculations on a wide range of parameters. Furthermore, this approach enables results to be more readily interpreted in terms of the more familiar cavity-independent molecular electric field response properties, such as polarizability and Born effective charges, which enter into the vibro-polariton calculation. Using corresponding electric field response properties of bulk insulating systems, we are also able to obtain the Γ point phonon-polariton spectra of two dimensional (2D) insulators. Results for a selection of cavity-coupled molecular and 2D crystal systems are presented to demonstrate the method.

Molecular Structure Theory without the Born-Oppenheimer Approximation: Rotationless Vibrational States of LiH.

Low-lying rotationless states of the lithium hydride molecule are studied in the framework of the variational method without assuming the Born-Oppenheimer (BO) approximation. Highly accurate solutions to the six-particle (two nuclei and four electrons) Schrödinger equation are obtained by means of expanding the wave functions of the considered states in terms of many thousands of all-particle explicitly correlated Gausssians. The basis functions are optimized independently for each state using the analytic energy gradient with respect to the nonlinear parameters. The non-BO wave functions obtained in the calculations are used to evaluate the leading-order relativistic and quantum electrodynamics energy corrections in the framework of the perturbation theory. The geometric structure of the molecule in the ground and excited states is discussed based on the analysis of the nucleus-nucleus correlation functions. The non-BO energies and structural parameters obtained of this work are compared with the most accurate BO results currently available.

Second-Order Mass-Weighting Scheme for Atom-Centered Density Matrix Propagation Molecular Dynamics.

The atom-centered density matrix propagation (ADMP) method is an extended Lagrangian approach to ab initio molecular dynamics, which includes the density matrix in an orthonormalized atom-centered Gaussian basis as additional, fictitious, electronic degrees of freedom, classically propagated along with the nuclear ones. A high adiabaticity between the nuclear and electronic subsystems is mandatory in order to keep the trajectory close to the Born-Oppenheimer (BO) surface. In this regard, the fictitious electronic mass μ, being a symmetric, nondiagonal matrix in its most general form, represents a free parameter, exploitable to optimize the propagation of the electronic density. Although mass-weighting schemes in ADMP exist, a systematic procedure to define an optimal value of the fictitious masses is not available yet. In this work, in order to rationally evaluate the electronic mass, fictitious electronic normal modes are defined through the diagonalization of the Hessian of the electronic density matrix. If the same frequency is imposed on all such modes (compatible with the chosen integration time step), then the corresponding μ matrix can be calculated and then employed for the following propagation. Analysis of several ADMP test simulations reveals that such Hessian-based mass-weighting approach is able to ensure, together with a 0.1/0.2 fs time steps, a high separation between the (real) nuclear and the (fictitious) electronic frequencies, which determines a high adiabaticity. This high, unprecedented, accuracy in the propagation leads, in turn, to low errors in the estimated nuclear vibrational frequencies, making the ADMP method totally comparable to a fully converged BO molecular dynamics simulation but more computationally efficient. This work, therefore, contributes to a further development of the ADMP ab initio molecular dynamics method, aimed at improving its accuracy through a more rational evaluation of the fictitious electronic mass parameter.

Corrections of Electron–Phonon Coupling for Second‐Order Structural Phase Transitions

Structural phase transitions are accompanied by a movement of one nucleus (or a few) in the crystallographic unit cell. If the nucleus movement is continuous, a second‐order phase transition without latent heat results, whereas an abrupt nucleus displacement indicates a first‐order phase transition with accompanying latent heat. Herein, a Hamiltonian including electron–phonon coupling (EPC) as proposed by Kristoffel and Konsel is taken. Contrary to their treatment, both the kinetic energy of the nucleus and its position are treated. The interaction of the many‐electron system with the single nucleus is taken into account by the Born–Oppenheimer approximation and perturbative expressions for the free energies are derived. The nuclei corrections due to the entangled electrons are found to be minor, but highlight the importance of the symmetry breaking at low temperature. Furthermore the free energy for a canonical ensemble is computed, whereas Kristoffel and Konsel use a grand canonical ensemble, which allows to derive more stringent bounds on the free energy. For the zero‐order nucleus correction, the shift of the phase transition temperature by evaluating the free energy is deduced.

Doubly-charged Tcc++ states in the dynamical diquark model

One of the celebrated tools in explaining the hydrogen atom is Born-Oppenheimer approximation. The resemblance of QQq¯q¯ tetraquarks to hydrogen atom within quantum chromodynamics implies usage of Born-Oppenheimer approximation for these multiquark states. In this work, we use dynamical diquark model to calculate mass spectra and sizes of doubly charmed and charged tetraquark states denoted as Tcc++. Our results for mass spectra indicate some bound-state candidates with respect to corresponding two-meson thresholds. Calculation of expectation values of ⟨r2⟩ reflects that doubly charmed and charged tetraquark states are compact. Published by the American Physical Society 2024

Angular Momentum Transfer between a Molecular System and a Continuous Circularly Polarized Light Field within a Semiclassical Born-Oppenheimer Surface Hopping Framework.

We simulate semiclassically angular momentum transfer for a molecular system subject to a circularly polarized light (CPL) field either moving along a single Born-Oppenheimer (BO) surface or moving along multiple BO surfaces. Both sets of simulations are able to conserve the total angular momentum around the propagation direction of the CPL field, the former requiring a Berry force and the latter requiring a surface parametrized by both nuclear position and momentum (a so-called phase-space approach). Our results provide new insight into the nature of semiclassical nonadiabatic dynamics methods and further demonstrate the power of such methods to capture angular momentum transfer between different media, highlighting the need for accurate algorithms that conserve the total angular momentum.

On accurate calculations of equilibrium H/D fractionations at super-cold conditions (≤200 K) I: Full Partition Function Ratio (FPFR) vs. Path Integral Monte Carlo (PIMC)

Accurate estimations of Hydrogen/Deuterium (H/D) isotope effects have been a long-standing problem since the beginning of stable isotope geochemistry, primarily due to their strong nuclear quantum effects (NQEs). Currently, the D/H ratio plays a key role in deciphering the origin and evolution of water ice and organics in the solar system. Additionally, the Big-bang origin of Deuterium necessitates that the observed D/H variations be linked to isotope exchange reactions, such as HD + H2O ↔ H2 + HDO. Therefore, an accurate benchmark dataset describing the equilibrium H/D isotope fractionations at low temperatures (≤ 200 K) is essential for interpreting rapidly accumulating astrochemical observation data. In this study, we implemented and compared two methods developed for evaluating the NQEs of H/D isotopes: 1) the Full Partition Function Ratio (FPFR) method and 2) the Path Integral Monte Carlo (PIMC) method. Both were assessed for their accuracy and efficiency in estimating equilibrium isotope effects (EIEs) using 10 common interstellar small molecules (H2, HF, HCl, HCN, HNC, H2O, H2S, NH3, HCHO, and CH4) as examples. Most of them are calculated at high precision for the first time. For the FPFR method, several higher-order corrections to the Urey-Bigeleisen-Mayer model were considered, including anharmonicity, quantum mechanical rotation with nuclear-spin statistics, and corrections to the Born-Oppenheimer approximation. For the PIMC method, four different actions related to the thermal density matrix are used. These include the primitive and three fourth-order actions (Takahashi and Imada, Suzuki-Chin and Chin approximations), based on the direct scaled-coordinates transformation. Three main conclusions are obtained: 1) Under the framework of Born-Oppenheimer approximation, the PIMC method holds the same and even slightly better accuracy (more comprehensive descriptions of quantum anharmonicities) in estimating EIEs for H/D isotopes as the FPFR method. However, both methods suffer from the neglection of the DBOC when using the Born-Oppenheimer potential energy surfaces. For the PIMC method, additional corrections for particle distinguishability must be considered when nuclear spin statistics exhibit its effects at low temperatures (e.g., < 160 K for H2). 2) Among the four actions, three fourth-order actions show better convergence and efficiency (2.2 to 26.5 times speedups at 50 K) than the primitive action, making them superior candidates for PIMC simulations at super-cold (≤ 200 K) conditions. 3) A combination of accurate potential energy surface with the PIMC method serves as an alternative way for theoretical estimations of H/D isotope effects.

Electronic Vector Potential from the Exact Factorization of a Complex Wavefunction.

We generalize the definitions of local scalar potentials named and , which are relevant to properly describe phenomena such as molecular dissociation with density-functional theory, to the case in which the electronic wavefunction corresponds to a complex current-carrying state. In such a case, an extra term in the form of a vector potential appears which cannot be gauged away. Both scalar and vector potentials are introduced via the exact factorization formalism which allows us to express the given Schrödinger equation as two coupled equations, one for the marginal and one for the conditional amplitude. The electronic vector potential is directly related to the paramagnetic current density carried by the total wavefunction and to the diamagnetic current density in the equation for the marginal amplitude. An explicit example of this vector potential in a triplet state of two non-interacting electrons is showcased together with its associated circulation, giving rise to a non-vanishing geometric phase. Some connections with the exact factorization for the full molecular wavefunction beyond the Born-Oppenheimer approximation are also discussed.

Reliability of the Born-Oppenheimer Approximation in Noninteger Dimensions

Reliability of the Born-Oppenheimer Approximation in Noninteger Dimensions

Time evolution as an optimization problem: The hydrogen atom in strong laser fields in a basis of time-dependent Gaussian wave packets.

Recent advances in attosecond science have made it increasingly important to develop stable, reliable, and accurate algorithms and methods to model the time evolution of atoms and molecules in intense laser fields. A key process in attosecond science is high-harmonic generation, which is challenging to model with fixed Gaussian basis sets, as it produces high-energy electrons, with a resulting rapidly varying and highly oscillatory wave function that extends over dozens of ångström. Recently, Rothe's method, where time evolution is rephrased as an optimization problem, has been applied to the one-dimensional Schrödinger equation. Here, we apply Rothe's method to the hydrogen wave function and demonstrate that thawed, complex-valued Gaussian wave packets with time-dependent width, center, and momentum parameters are able to reproduce spectra obtained from essentially exact grid calculations for high-harmonic generation with only 50-181 Gaussians for field strengths up to 5 × 1014 W/cm2. This paves the way for the inclusion of continuum contributions into real-time, time-dependent electronic-structure theory with Gaussian basis sets for strong fields and eventually accurate simulations of the time evolution of molecules without the Born-Oppenheimer approximation.

Periodic orbits, pair nucleation, and unbinding of active nematic defects on cones.

Geometric confinement and topological constraints present promising means of controlling active materials. By combining analytical arguments derived from the Born-Oppenheimer approximation with numerical simulations, we investigate the simultaneous impact of confinement together with curvature singularity by characterizing the dynamics of an active nematic on a cone. Here, the Born-Oppenheimer approximation means that textures can follow defect positions rapidly on the timescales of interest. Upon imposing strong anchoring boundary conditions at the base of a cone, we find a rich phase diagram of multidefect dynamics, including exotic periodic orbits of one or two +1/2 flank defects, depending on activity and nonquantized geometric charge at the cone apex. By characterizing the transitions between these ordered dynamical states, we present detailed understanding of (i) defect unbinding, (ii) defect absorption, and (iii) defect pair nucleation at the apex. Numerical simulations confirm theoretical predictions of not only the nature of the circular orbits but also defect unbinding from the apex.

Anomalous diffusion, prethermalization, and particle binding in an interacting flat band system

We study the broadening of initially localized wave packets in a quasi one-dimensional diamond ladder with interacting, spinless fermions. The lattice possesses a flat band causing localization. We place special focus on the transition away from the flat band many-body localized case by adding very weak dispersion. By doing so, we allow propagation of the wave packet on significantly different timescales which causes anomalous diffusion. Due to the temporal separation of dynamic processes, an interaction-induced, prethermal equilibrium becomes apparent. A physical picture of light and heavy modes for this prethermal behavior can be obtained within Born–Oppenheimer approximation via basis transformation of the original Hamiltonian. This reveals a detachment between light, symmetric and heavy, anti-symmetric particle species. We show that the prethermal state is characterized by heavy particles binding together mediated by the light particles.

A potential approach to the X(3872) thermal behavior

We study the potential of X(3872) at finite temperature in the Born-Oppenheimer approximation under the assumption that it is a tetraquark. We argue that, at large number of colors, it is a good approximation to assume that the potential consists in a real part plus a constant imaginary term. The real part is then computed adapting an approach by Rothkopf and Lafferty and using as input lattice QCD determinations of the potential for hybrids. This model allows us to qualitatively estimate at which temperature range the formation of a heavy tetraquark is possible, and to propose a qualitative picture for the dissociation of the state in a medium. Our approach can be applied to other suggested internal structures for the X(3872) and to other exotic states.

Density functional theory beyond the Born–Oppenheimer approximation: exact mapping onto an electronically non-interacting Kohn–Sham molecule

This work presents an alternative, general, and in-principle exact extension of electronic Kohn–Sham density functional theory (KS-DFT) to the fully quantum-mechanical molecular problem. Unlike in existing multi-component or exact-factorization-based DFTs of electrons and nuclei, both nuclear and electronic densities are mapped onto a fictitious electronically non-interacting molecule (referred to as KS molecule), where the electrons still interact with the nuclei. Moreover, in the present molecular KS-DFT, no assumption is made about the mathematical form (exactly factorized or not) of the molecular wavefunction. By expanding the KS molecular wavefunction à la Born–Huang, we obtain a self-consistent set of ‘KS beyond Born–Oppenheimer’ electronic equations coupled to nuclear equations that describe nuclei interacting among themselves and with non-interacting electrons. An exact adiabatic connection formula is derived for the Hartree-exchange-correlation energy of the electrons within the molecule and, on that basis, a practical adiabatic density-functional approximation is proposed and discussed.