Dynamics Of Molecular Systems Research Articles

Attosecond emission delay from atoms and molecules using multi-dimensional XUV interferometry.

Abstract Multi-dimensional interferometry of high harmonic generation is
demonstrated using an inline Gouy interferometer. The rich data sets acquired with this technique, coupled with its inherent stability, enable robust phase analysis to be performed, allowing the delay in harmonic emission from a range of atomic and molecular gases to be measured relative to harmonic emission from argon with single-digit attosecond precision. Delays of −78 ± 6 as, −25 ± 6 as, 30 ± 3 as and 60 ± 3 as relative to emission from argon were measured for Xe, Kr, CO2 and N2, respectively. The scheme can easily be incorporated in other high harmonic experiments, such as tomographic imaging of molecular orbitals, and provides an experimentally simple route towards probing ultrafast dynamics in molecular systems.

Impact of Loss Mechanisms on Linear Spectra of Excitonic and Polaritonic Aggregates.

The presence of loss mechanisms governed by empirical timescales can profoundly affect the dynamics in molecular systems, leading to changes in their spectra. However, incorporation of these effects along with the system's interaction with the thermal dissipative environments proves to be challenging. In this work, we demonstrate the possibility of utilizing the recently developed path integral Lindblad dynamics (PILD) method to study the linear spectra of molecular aggregates. PILD presents a uniquely powerful simulation technique for retaining the effects of the vibrations in a numerically exact manner through the Feynman-Vernon influence functional while incorporating the effects of losses in an empirical manner using the Lindbald master equation. As illustrations of this technique, we provide examples taken from chiral excitonic and polaritonic aggregates for which we simulate both the absorption spectra and circular dichroism (CD) spectra. We demonstrate that the effect of loss on particular states can differ not just on the basis of the symmetries of the state but also on the basis of complicated "interactions" of the structured dissipative environments with the system and its loss mechanisms. Due to the different selection rules between the absorption and CD spectra and the relative intensities and broadening of the peaks, the two linear spectra together give an interesting insight into the contributions of the various eigenstates to the correlation functions. While the focus here is on linear spectroscopy, it should be possible in the future to use PILD to study multidimensional spectra under loss mechanisms as well.

Investigating fracture mechanisms in glassy polymers using coupled particle-continuum simulations

We study the fracture behavior of glassy polymers using a multiscale simulation method that integrates a molecular dynamics (MD) system within a continuum domain. By employing a nonlinear viscoelastic constitutive model in the continuum domain, the MD system undergoes non-uniform deformation with flexible boundaries through interaction with the surrounding continuum. Systems with pre-defined double cracks are subjected to tensile stretch under various geometric constraints and bond breakage criteria. The simulation results show that geometric constraints primarily affect deformation behavior at small strains, but their influence diminishes at larger strains. In the stage of fracture, increased deformation correlates with a narrowing distribution of microscopic structures at molecular scales, such as bond length. This distribution converges across different systems just before fracture, irrespective of ultimate stress, fracture strain, bond breakage criteria, and geometric constraints. This convergence suggests that the distribution of microscopic structures is a fundamental property linked to the fracture behavior of glassy polymers.

Large-scale molecular dynamics simulation and aggregate behavior research on asphalt

In this comprehensive study, a molecular dynamics (MD) model consisting of approximately 150,000 atoms was developed within the Strategic Highway Research Program (SHRP) framework to investigate the behavior of AAA asphalt, focusing on asphaltene aggregate dynamics behavior over a simulation time span of 0.282 microseconds. The findings reveal that the model achieves a stable state in terms of both energy and microstructure without significant changes or the formation of densely clustered molecules, despite the extensive simulation period. Notably, the patterns of asphaltene aggregation were found to significantly influence the MD system energy, with variations ranging from loosely organized structures due to dispersed asphaltenes to more structured, energy-efficient clusters formed by encapsulated asphaltene pentamers. The size of asphaltene aggregates emerged as a pivotal factor, affecting their encapsulation and leading to distinct formation patterns. Furthermore, the research highlighted that increasing shear rates prompt the depolymerization of asphaltene aggregates, shifting from large-scale structures to smaller ones and adapting dynamically to the flow conditions by aligning parallel to the shear direction. This study underscores the critical impact of asphaltene aggregate behavior and shear rates on the structural integrity and distribution within the asphalt model, offering profound insights into the MD of asphalt materials.

Methods for Classical-Mechanical Molecular Simulation in Chemistry: Achievements, Limitations, Perspectives.

More than a half century ago it became feasible to simulate, using classical-mechanical equations of motion, the dynamics of molecular systems on a computer. Since then classical-physical molecular simulation has become an integral part of chemical research. It is widely applied in a variety of branches of chemistry and has significantly contributed to the development of chemical knowledge. It offers understanding and interpretation of experimental results, semiquantitative predictions for measurable and nonmeasurable properties of substances, and allows the calculation of properties of molecular systems under conditions that are experimentally inaccessible. Yet, molecular simulation is built on a number of assumptions, approximations, and simplifications which limit its range of applicability and its accuracy. These concern the potential-energy function used, adequate sampling of the vast statistical-mechanical configurational space of a molecular system and the methods used to compute particular properties of chemical systems from statistical-mechanical ensembles. During the past half century various methodological ideas to improve the efficiency and accuracy of classical-physical molecular simulation have been proposed, investigated, evaluated, implemented in general simulation software or were abandoned. The latter because of fundamental flaws or, while being physically sound, computational inefficiency. Some of these methodological ideas are briefly reviewed and the most effective methods are highlighted. Limitations of classical-physical simulation are discussed and perspectives are sketched.

Using the Haken-Strobl-Reineker Model to Determine the Temperature Dependence of the Diffusion Coefficient.

Stochastic quantum Liouville equations (SQLE) are widely used to model energy and charge dynamics in molecular systems. The Haken-Strobl-Reineker (HSR) SQLE is a particular paradigm in which the dynamical noise that destroys quantum coherences arises from a white noise (i.e., constant-frequency) spectrum. A system subject to the HSR SQLE thus evolves to its "high-temperature" limit, whereby all the eigenstates are equally populated. This result would seem to imply that the predictions of the HSR model, e.g., the temperature dependence of the diffusion coefficient, have no validity for temperatures lower than the particle bandwidth. The purpose of this paper is to show that this assumption is incorrect for translationally invariant systems. In particular, provided that the diffusion coefficient is determined via the mean-squared-displacement, considerations about detailed-balance are irrelevant. Consequently, the high-temperature prediction for the temperature dependence of the diffusion coefficient may be extrapolated to lower temperatures, provided that the bath remains classical. Thus, for diagonal dynamical disorder the long-time diffusion coefficient, D∞(T) = c1/T, while for both diagonal and off-diagonal disorder, D∞(T) = c1/T + c2T, where c2 ≪ c1. An appendix discusses an alternative interpretation from the HSR model of the "quantum to classical" dynamics transition, whereby the dynamics is described as stochastically punctuated coherent motion.

#2263 Computational drug repurposing screen targeting PLA2R antibody binding

Abstract Background and Aims Approximately 60% of patients with membranous nephropathy have antibodies against the phospholipase A2 receptor (PLA2R). The cysteine rich (CysR) epitope of PLA2R is a likely site of protein-antibody interaction. Current treatment regimens for membranous nephropathy involve maximal conservative management or systemic immunosuppression in severe cases. We have developed protocols and programs for an in silico high-throughput repurposing screen using structural modelling, docking, and molecular dynamics (MD). Uniquely, our approach uses a bespoke interface on Google Colab for graphic processing units to run the MD simulations. We describe their application to a repurposing screen aiming to identify larger licensed drug molecules capable of impeding antibody-protein interactions at the site of the CysR epitope. Method A complete model of PLA2R was obtained from I-TASSER using the canonical sequence on the UNIPROT database. This structure was equilibrated using a five nanosecond molecular dynamics (MD) simulation. The simulation was prepared using CHARMM-GUI and run on the GROMACS-ON-COLAB platform. The centroid of the MD system between three and five nanoseconds was used for docking experiments using PLANTS PLP with a study area that included the CysR epitope. A library of compounds listed in the British National Formulary was used for the docking studies. The results would be analysed by PLANTS energy and by key residue overlap score. Results Viable docking results were obtained for 891 of the 1510 compounds tested. 60 of these compounds were in contact with at least 10% of the residues in the CysR region. By restricting these results to compounds in the lowest 20% of the total sample for calculated binding energy, 22 compounds remained. 16 of these compounds have been assessed for their safety for oral or intravenous use. These sixteen compounds were then identified for further investigation by MD to assess the stability of the identified interactions. The sixteen compounds include Lymecycline, Trazodone, Mirabegron, Sulfasalazine, Methotrexate, Rosuvastatin and Montelukast. Conclusion Structural bioinformatics methods provide a relatively fast and economical option for identifying protein-drug interactions that could result in new treatment strategies. The compounds identified for investigation are used for an array of different health conditions. However, they all share the chemical properties of large molecular weight and have the potential to form multiple hydrogen bonds. If validated experimentally, these compounds could provide new treatment options for PLA2R antibody positive membranous nephropathy.

Density Functional Waltz: Molecular Theory of Spin Dynamics

Spin dynamics in molecular systems lies at the heart of contemporary research, promising revolutionary applications in fields like spintronics and quantum information processing. The paper provides insights into green techniques to improve spin dynamics for utilization in quantum computing and magnetic materials. This dynamic approach describes the temperature-dependent evolution of spin-lattice interactions. Snapshots acquired from various simulations enable additional DFT investigations, providing a more sophisticated understanding of spin dynamics under temperature variations. By seamlessly integrating theoretical insights with computational approaches, the Density Functional Waltz emerges as a potent paradigm for unraveling the mesmerizing choreography of spin dynamics at the molecular scale. This review analyzes the evolution of machine-learning approaches in the field of chemical sciences. The proposed review will highlight the significance of collaboration between quantum catalysis and the chemical physics community. There is an outline of the unique features of electron spin alongside the ingenious integration of macromolecules to control and fine-tune electron spin-based innovation

Unraveling motion in proteins by combining NMR relaxometry and molecular dynamics simulations: A case study on ubiquitin.

Nuclear magnetic resonance (NMR) relaxation experiments shine light onto the dynamics of molecular systems in the picosecond to millisecond timescales. As these methods cannot provide an atomically resolved view of the motion of atoms, functional groups, or domains giving rise to such signals, relaxation techniques have been combined with molecular dynamics (MD) simulations to obtain mechanistic descriptions and gain insights into the functional role of side chain or domain motion. In this work, we present a comparison of five computational methods that permit the joint analysis of MD simulations and NMR relaxation experiments. We discuss their relative strengths and areas of applicability and demonstrate how they may be utilized to interpret the dynamics in MD simulations with the small protein ubiquitin as a test system. We focus on the aliphatic side chains given the rigidity of the backbone of this protein. We find encouraging agreement between experiment, Markov state models built in the χ1/χ2 rotamer space of isoleucine residues, explicit rotamer jump models, and a decomposition of the motion using ROMANCE. These methods allow us to ascribe the dynamics to specific rotamer jumps. Simulations with eight different combinations of force field and water model highlight how the different metrics may be employed to pinpoint force field deficiencies. Furthermore, the presented comparison offers a perspective on the utility of NMR relaxation to serve as validation data for the prediction of kinetics by state-of-the-art biomolecular force fields.

Elevated temperature effects (T > 100 °C) on the interfacial water and microstructure swelling of Na-montmorillonite

Montmorillonite-based barriers are key elements of the engineered barrier systems (EBS) in geological disposal facilities (GDF). Their performance at temperatures above 100 °C is not sufficiently understood to assess the possibility of raising the temperature limits in GDF designs that could reduce construction costs and CO2 footprint. The present work provides new fundamental insights through molecular dynamics (MD) simulations of Na-montmorillonite's water-clay interactions and swelling pressure at temperatures 298–500 K and basal spacings of 1.5–3.5 nm. At temperatures above 100 °C, the swelling behaviour is governed by the attractive van der Waals force and the repulsive hydration force instead of the repulsive electrostatic (double layer) force. The swelling pressure reduction with increasing temperature is related to the weakened hydration repulsion and electric double layer repulsion, which result from the deterioration of the interlayer water layer structure and the shrinkage of the electric double layer. The applicability and breakdown of the classic Derjaguin-Landau-Verwey-Overbeek (DLVO) theory at elevated temperatures are examined. By excluding the osmotic contribution in the DLVO theory, the summation of the van der Waals interaction in DLVO and an additional non-DLVO hydration interaction can predict our MD system's swelling under high temperatures. The findings of this study provide a fundamental understanding of the swelling behaviour and the underlying molecular-level mechanisms of the clay microstructure under extreme conditions.

Characterization of Biological Samples Using Ultra-Short and Ultra-Bright XFEL Pulses.

The advent of X-ray Free Electron Lasers (XFELs) has ushered in a transformative era in the field of structural biology, materials science, and ultrafast physics. These state-of-the-art facilities generate ultra-bright, femtosecond-long X-ray pulses, allowing researchers to delve into the structure and dynamics of molecular systems with unprecedented temporal and spatial resolutions. The unique properties of XFEL pulses have opened new avenues for scientific exploration that were previously considered unattainable. One of the most notable applications of XFELs is in structural biology. Traditional X-ray crystallography, while instrumental in determining the structures of countless biomolecules, often requires large, high-quality crystals and may not capture highly transient states of proteins. XFELs, with their ability to produce diffraction patterns from nanocrystals or even single particles, have provided solutions to these challenges. XFEL has expanded the toolbox of structural biologists by enabling structural determination approaches such as Single Particle Imaging (SPI) and Serial X-ray Crystallography (SFX). Despite their remarkable capabilities, the journey of XFELs is still in its nascent stages, with ongoing advancements aimed at improving their coherence, pulse duration, and wavelength tunability.

A fully dynamical description of time-resolved resonant inelastic X-ray scattering of pyrazine.

Recent advancements in ultrashort and intense X-ray sources have enabled the utilisation of resonant inelastic X-ray scattering (RIXS) as a probing technique for monitoring photoinduced dynamics in molecular systems. To account for dynamic phenomena like non-adiabatic transitions across the relevant electronic state manifold, a time-dependent framework is crucial. Here, we introduce a fully time-dependent approach for calculating transient RIXS spectra using wavepacket dynamics simulations, alongside an explicit treatment of the X-ray probe pulse that surpasses Kramers-Heisenberg-Dirac constraints. Our analysis of pyrazine at the nitrogen K-edge underscores the importance of considering nuclear motion effects in all electronic states involved in the transient RIXS process. As a result, we propose a numerically exact approach to computationally support and predict cutting-edge time-resolved RIXS experiments.

Simulations of photoinduced processes with the exact factorization: state of the art and perspectives.

This perspective offers an overview of the applications of the exact factorization of the electron-nuclear wavefunction to the domain of theoretical photochemistry, where the aim is to gain insights into the ultrafast dynamics of molecular systems via simulations of their excited-state dynamics beyond the Born-Oppenheimer approximation. The exact factorization offers an alternative viewpoint to the Born-Huang representation for the interpretation of dynamical processes involving the electronic ground and excited states as well as their coupling through the nuclear motion. Therefore, the formalism has been used to derive algorithms for quantum molecular-dynamics simulations where the nuclear motion is treated using trajectories and the electrons are treated quantum mechanically. These algorithms have the characteristic features of being based on coupled and on auxiliary trajectories, and have shown excellent performance in describing a variety of excited-state processes, as this perspective illustrates. We conclude with a discussion on the authors' point of view on the future of the exact factorization.

Enhancing gel behavior of yellow croaker surimi by fruit extracts: Physicochemical properties and molecular mechanism.

The aim of this study was to investigate the effects of grape seed extract (GSE), acerola cherry extract (ACE), and blueberry extract (BBE) on the physicochemical properties and structure of the yellow croaker surimi gel. In addition, molecular docking and molecular dynamics (MD) simulation were utilized to study the binding mechanism of yellow croaker's fibrillin and fruit extracts. Surimi gel with 1.5% GSE, ACE, and BBE had the highest water holding capacity, hardness, chewability, cohesion, breaking force, breaking distance, gel strength, and densest 3D network structure, according to the experiment's findings. Nevertheless, the cross-linking of proteins in surimi was blocked with the further increase of fruit extract (1.5%-2.0%), and the existing network of surimi was weakened or even destroyed. Three fruit extracts had little effect on the secondary structure of the surimi gel. Besides, hydrophobic and disulfide bonds are the main chemical bonds of croaker surimi. Molecular docking showed that B-type procyanidine (BP) interacted with ASN-183, SER-571, ASP-525, ARG-350, LYS-188, GLU-349, CYS-353, and other active amino acids in croaker protein. Moreover, it can form strong hydrogen bond interaction with ASN-183, SER-571, ASP-525, and ARG-350 at the active sites of protein. The BP-Larimichthys crocea protein system's MD simulation was carried out, and calculations for the simulation's root mean square deviation, root mean square fluctuation, radius of gyration, solvent accessible surface area, and hydrogen bonds were made. It was found that these indices can demonstrate that the BP binding contributes to the stability of the yellow croaker structure.

Computer modeling of thermotransport in a uniform binary liquid solution with equimolar n-alkane mixtures

By means of molecular dynamics (MD), two novel methods, a thermal mean-path that may outline temperature profiles effectively in the MD system and a modified coarse-grained force field potential (the MCG-FFP) that may depict inter/intra-molecular interactions fairly well among n-alkane species, are employed to simulate a thermotransport process in a uniform liquid solution with two equimolar n-pentane (nC-5) and n-decane (nC-10) mixtures. In addition, all the MD simulations are running under two constraints: a weak thermal gradient exerting on the MD system from its hot through cold boundary sides and the standard-state acting on the MD system from its outer environment. During the whole MD simulations, coefficients of thermal diffusion and mass mutual diffusion, and the Soret coefficient (SC) for the MD system are calculated by using the MCG-FFP. As a result, the MD simulations indicate that nC-5 species with light molar-mass would migrate toward the hot boundary region, while nC-10 species with heavy molar-mass would migrate toward the cold one. Coefficients calculated from the MCG-FFP are found to meet relevant experimental outputs fairly well. Furthermore, an empirical formula developed by means of relevant continuum methods is used for calculating coefficients of mass mutual diffusion in solutions mixing with multimolar nC-5 and nC-10 species. Its one output is found to corroborate pretty well with that from the MD simulations. This may expect that such the formula would perform universally when characterizing properties of mass mutual diffusion in binary liquid solutions with other multimolar alkane mixtures in the petroleum engineering.

DETECTION OF VOID NUCLEATION, COALESCENCE AND COLLAPSE IN ATOMISTIC SIMULATIONS

Extraction of required information from atomic configurations is crucial for efficient use of molecular dynamics (MD) simulations, and automating this process allows the collection of statistical data to develop and validate physical models and data-driven approaches. Here we update the previously developed pore searching algorithm to automatically detect the events of void nucleation, collapse and coalescence by comparing pairs of atomic configurations. We calculate an intersection matrix, which shows common regions between voids of current and previous time frames, and analysis of this intersection matrix gives a numerical criterion for determination of the considered events. The updated algorithm is verified in the case of small MD system and further used to analyze a large MD system with many hundreds of voids. The predominance of void collapse is confirmed in the case of uniform triaxial tension of solid aluminum, while void coalescence instead of collapse is detected in the case of a more complex loading with axial compression and lateral tension. This dependence of the predominant mechanism on the loading path correlates with previous finite element analysis in the literature. Another interesting finding is a continuing nucleation and collapse of voids far beyond the fracture beginning at a level of negative pressure much lower than the spall strength, which can be attributed to a developed defect structure of material produced by plastic growth of voids.

Adsorption of fatty acid potassium salts on diamond-like carbon: Effects of alkyl chain lengths and adsorptive conditions

Organic monolayers on solid substrates have received considerable attention due to their excellent versatilities in many fields. In this paper, molecular dynamics (MD) simulations were performed to study the adsorption of fatty acid potassium salts (FAPS) on diamond-like carbon (DLC) substrates. The results indicated that adsorbed molecules tend to aggregate on DLC surfaces due to the intense attractions between FAPS molecules. In order to obtain evenly distributed molecular films, the external energy input is needed to facilitate molecular adsorption and overcome the potential barriers. Also, the effects of alkyl chain lengths on molecular adsorption were investigated. A positive correlation was observed between the alkyl length and the adsorptive energy exerted by DLC substrates. Furthermore, the Coulomb interaction between adsorbed molecules is the most powerful interaction within the MD system. Affected by the Coulomb interactions, adsorbed molecules would undergo severe deformation. And a net-like structure composed of polar ends is formed in molecular films. The hollow area within the network is full of alkyl chains of molecules. It can be concluded that the powerful Coulomb interactions between adsorbed molecules play a dominant role in determining the adsorption behavior and structural properties of potassium salts on DLC substrates.

A multi-GPU implementation of Real-Time Time-Dependent Auxiliary Density Functional Theory for the investigation of nanosystems irradiations

We report a new Multi-GPU (Graphical Processor Unit) implementation of real-time time-dependent Auxiliary Density Functional Theory (DFT) for simulations of attosecond electronic dynamics in molecular systems subjected to strong perturbations. Our code relies on the Kohn-Sham formalism of DFT and has been implemented in the deMon2k Fortran code. We expand single-particle wave functions (i.e molecular orbitals) as linear combinations of Gaussian-type-orbitals centered on atoms. The density matrix propagation is carried out on GPU while the Kohn-Sham potential is operated on CPUs (Central Processor Unit) with the help of variationally fitted densities. We propose a parallelization strategy using the MAGMA/CUDA libraries to calculate the exponential of dense Hermitian matrices entering the mathematical definition of the propagator, here using Taylor expansions. We report performance benchmarks on water droplets and on fullerenes (C50 to C540). They show a clear advantage of GPU over CPU (using the Scalapack library). The benchmarks also show the benefit of using more than one GPU for systems comprised of up to more than 10,000 basis functions. There, a speed-up of almost 40 between pure 40 CPU and four 4 GPU is obtained. Attosecond electron dynamics simulation in molecular systems comprised of several thousands of electrons becomes amenable to routine simulations in our code. We assess the accuracy of the GPU implementation considering various applications, namely, the calculation of extreme UV absorption spectra with non-Hermitian dynamics, the response of C180 to an electric perturbation, and finally the irradiation of a DNA/protein complex by a 0.4 MeV proton. The results demonstrate the robustness of the implementation. This work also paves the way for future even more efficient implementations. Program SummaryProgram title: deMon2k-RT-TDDFTCPC Library link to program files: https://doi.org/10.17632/whwc5zkst5.1Developer's repository link: DOI: 10.5281/zenodo.8301468Licensing provisions: CeCILL Free Software License Agreement v2.1Programming language: FortranExternal routines/libraries: MPI, BLAS/LAPACK, CUDA, MAGMASupplementary material: User manual, installation instructions, test cases. Nature of problem: Electron dynamics within molecular systems is amenable to simulation with Real-Time Time-Dependent Density Functional Theory (RT-TD-DFT). The methodology consists in propagating on the attosecond time scale the single particle electronic wave functions on the underlying Kohn-Sham potential [1,2]. In deMon2k-RT-TDDFT, the propagation is coupled to the Auxiliary DFT framework for fast evaluation of the potential [3]. A remaining computational bottleneck is the matrix exponential calculation of the Hermitian Hamiltonian matrix. This task is achieved in deMon2k via diagolanization or expansions formulas (Taylor, Chebychev or Baker-Campbell-Hausdorff). The operation is cumbersome on central-processor-unit machines for large matrices (e.g. > 5000 × 5000), which corresponds to molecular systems comprising around 2,000 electrons. Alternatives are needed to overcome this bottleneck.Solution method: We propose to carry out the matrix exponentiation on graphical-processor-units thanks to the MAGMA[4]/CUDA libraries. The code permits a drastic reduction of the computational cost, opening the attosecond dynamic simulations of large molecular systems comprised thousands of electrons with controlled accuracy. The method is tested here for Taylor expansions. Additional comments including restrictions and unusual features: The code described in this article opens the realization of RT-TD-ADFT simulations. The code is linked to the Master version 6.1.6 of deMon2k (http://www.demon-software.com), which must be pre-installed by the user.

Quantum dynamics in molecular systems

Quantum dynamics in molecular systems

Quantum simulation of non-Born–Oppenheimer dynamics in molecular systems by path integrals

A numerical algorithm based on the probabilistic path integral approach for solving Schrödinger equation has been devised to treat molecular systems without Born–Oppenheimer approximation in the nonrelativistic limit at zero temperature as an alternative to conventional variational and perturbation methods. Using high-quality variational trial functions and path integral method based on generalized Feynman–Kac method, we have been able to calculate the non-Born–Oppenheimer energy for hydrogen molecule for the [Formula: see text] state and hydrogen molecular ion. Combining these values and the value for ionization potential for atomic hydrogen, the dissociation energy and ionization potential for hydrogen molecules have been determined to be 36 113.672(3)[Formula: see text]cm[Formula: see text] and 124 446.066(10)[Formula: see text]cm[Formula: see text], respectively. Our results favorably compare with other theoretical and experimental results and thus show the promise of being a nonperturbative alternative for testing fundamental physical theories.