Recent Advances in Wave Function-Based Methods of Molecular-Property Calculations

Real-Time Coupled Cluster Theory with Approximate Triples

In order to explore the effects of high levels of electron correlation on the real-time coupled cluster formalism and algorithmic behavior, we introduce a time-dependent implementation of the CC3 singles, doubles, and approximate triples method. We demonstrate the validity of our derivation and implementation using specific applications of frequency-dependent properties. Terms with triples are calculated and added to the existing CCSD equations, giving the method a nominal O (N7) scaling. We also use a graphics processing unit accelerated implementation to reduce the computational cost, which we find can speed up the calculation by up to a factor of 13 for test cases of water clusters. In addition, we compare the impact of using single-precision arithmetic compared to conventional double-precision arithmetic. We find no significant difference in polarizabilities and optical-rotation tensor results but a somewhat larger error for first hyperpolarizabilities. Compared to linear response CC3 results, the percentage errors of RT-CC3 polarizabilities and RT-CC3 first hyperpolarizabilities are under 0.1 and 1%, respectively, for a water-molecule test case in a double-ζ basis set. Furthermore, we compare the dynamic polarizabilities obtained using RT-CC3, RT-CCSD, and time-dependent nonorthogonal orbital-optimized coupled cluster doubles (TDNOCCDs) in order to examine the performance of RT-CC3 and the orbital-optimization effect using a set of ten-electron systems.

Toward Explicit Solvation for Simulations of Electrocatalytic Reactions: AIMD for pKa and Redox Potentials of Transition Metal Compounds and Catalyst Models

In this work, we study the possibility to extend electronic structure simulations for electrocatalysis by explicit solvation models. In previous work, we proposed a simulation scheme that explicitly includes the effects of pH and electrochemical potential in density functional theory (DFT) simulations with implicit solvation. Based on calculations of protonation and oxidation reactions, the pH and electrochemical potential can be included given appropriate reference values. In this work, we compute the pKa values and oxidation potentials for a series of transition metal aquo complexes and compare the results including implicit, explicit static and explicit dynamic (AIMD) models for the aqueous solvent and compare vs experimental pKa and redox potential data. This allows the construction of a pKa/redox potential scale that can in principle be extrapolated to the simulation of other transition metal-based materials. An explicit dynamic solvent model is then proposed and applied to a model system for iridium oxide-based catalysts for the oxygen evolution reaction. We outline the advantages and disadvantages of the different approaches and demonstrate that, at the expense of a larger computational effort, the microsolvation environment of a given model can be described in a robust way using a limited amount of solvent molecules and AIMD. Especially for reactions in which water is solvent and reactant like the oxygen evolution reaction (OER) or oxygen reduction reaction (ORR), this model provides a more detailed and complete description that can be exploited in mechanistic studies.

Analysis of the NMR Parameters Shielding and Spin-Spin Coupling Constants of Glycine Conformers

In this study, we worked at the CCSD/aug-cc-pVTZ level to obtain the conformers of glycine in its neutral and zwitterionic forms in the gas and water phases. We then computed the NMR properties (spin-spin coupling constants and nuclear magnetic shieldings) at the SOPPA/aug-cc-pVTZ-J level. We attempt to elucidate the apparent discrepancy arising from two previous works by Valverde et al. [J. Chem. Phys., 2018, 148, 024305] and Caputo and Provasi [Sci, 2021, 3, 41]. Our optimized structures align with previous theoretical predictions, although some geometries were not found. Additionally, our results suggest that in the gas phase the Δ 1J(O,C) = 1J(Ocis,CO) - 1J(Otrans,CO) is positive when the acid group is in the trans conformation and negative in the cis conformation; however, this trend is not well reproduced in water. From magnetic shielding calculations, we cannot distinguish between an O-H in the cis or trans conformation.

Analytical derivative approaches for vibro-polaritonic structures and properties. I. Formalism and implementation

Vibro-polaritons are hybrid light-matter states that arise from the strong coupling between the molecular vibrational transitions and the photons in an optical cavity. Developing theoretical and computational methods to describe and predict the unique properties of vibro-polaritons is of great significance for guiding the design of new materials and experiments. Here, we present the ab initio cavity Born-Oppenheimer density functional theory (CBO-DFT) and formulate the analytic energy gradient and Hessian as well as the nuclear and photonic derivatives of dipole and polarizability within the framework of CBO-DFT to efficiently calculate the harmonic vibrational frequencies, infrared absorption, and Raman scattering spectra of vibro-polaritons as well as to explore the critical points on the cavity potential energy surface. The implementation of analytic derivatives into the electronic structure package is validated by a comparison with the finite-difference method and with other reported computational results. By adopting appropriate exchange-correlation functionals, CBO-DFT can better describe the structure and properties of molecules in the cavity than CBO-Hartree-Fock method. It is expected that CBO-DFT is a useful tool for studying the polaritonic structures and properties.

Understanding and mitigating noise in molecular quantum linear response for spectroscopic properties on quantum computers

The promise of quantum computing to circumvent the exponential scaling of quantum chemistry has sparked a race to develop chemistry algorithms for quantum architecture. However, most works neglect the quantum-inherent shot noise, let alone the effect of current noisy devices. Here, we present a comprehensive study of quantum linear response (qLR) theory obtaining spectroscopic properties on simulated fault-tolerant quantum computers and present-day near-term quantum hardware. This work introduces novel metrics to analyze and predict the origins of noise in the quantum algorithm, proposes an Ansatz-based error mitigation technique, and reveals the significant impact of Pauli saving in reducing measurement costs and noise in subspace methods. Our hardware results using up to cc-pVTZ basis set serve as proof of principle for obtaining absorption spectra on quantum hardware in a general approach with the accuracy of classical multi-configurational methods. Importantly, our results exemplify that substantial improvements in hardware error rates and measurement speed are necessary to lift quantum computational chemistry from proof of concept to an actual impact in the field.

Graphical Approach to Interpreting and Efficiently Evaluating Geminal Wavefunctions

We consider wavefunctions built from antisymmetrized products of two-electron wavefunctions (geminals), which is arguably the simplest extension of the antisymmetrized product of one-electron wavefunctions (orbitals) (i.e., a Slater determinant). Extensive use of geminals in wavefunctions has been limited by their high cost stemming from the many combinations of the two-electron basis functions (orbital pairs) used to build the geminals. When evaluating the overlap of the APG wavefunction with an orthogonal Slater determinant, this cost can be interpreted as the cost of evaluating the permanent, resulting from the symmetry with respect to the interchange of orbital pairs, and the cost of assigning the occupied orbitals to the orbital pairs of the wavefunction. Focusing on the latter, we present a graphical interpretation of the Slater determinant and utilize the maximum weighted matching algorithm to estimate the combination of orbital pairs with the largest contribution to the overlap. Then, the cost due to partitioning the occupied orbitals in the overlap is reduced from 𝒪 ( ( N - 1 ) ! ! ) to 𝒪 ( N 3 log N ) . Computational results show that many of these combinations are not necessary to obtain an accurate solution to the wavefunction. Because the APG wavefunction is the most general of the geminal wavefunctions, this approach can be applied to any of the simpler geminal wavefunction ansätze. In fact, this approach may even be extended to generalized quasiparticle wavefunctions, opening the door to tractable wavefunctions built using components of arbitrary numbers of electrons, not just two electrons.

Solvent effects on the second harmonic responses of donor-acceptor Stenhouse adducts: from implicit to hybrid solvation models

The effect of conformational dynamics and solvent interactions on the second-order nonlinear optical (NLO) responses of the open and closed forms of a donor-acceptor Stenhouse adduct (DASA) are investigated by a mixed quantum/classical computational approach, which couples molecular dynamics (MD) simulations and time-dependent density functional theory (TD-DFT) calculations. The latter are further combined with various solvation schemes, including polarizable continuum models, hybrid QM/MM approaches using either non polarizable or polarizable electrostatic embedding, and QM/QM' schemes with explicit treatment of a few molecules of the first solvation shell. The performances of the different solvation models are discussed in the context of comparisons with experimental data obtained from hyper-Rayleigh scattering measurements.

Synthesis and Optical and Nonlinear Optical Properties of Linear and Two-Dimensional Charge Transfer Chromophores Based on Polyoxometalates

We present the first study of arylimido-polyoxometalate nonlinear optical (NLO) chromophores with two-dimensional (2D) structures, and a comparison with one-dimensional analogues, through the synthesis of a family of arylimido-hexamolybdate derivatives where one or two polyoxometalate (POM) acceptors are connected to a tolyl-amino donor through phenyl bridges. Electronic absorption spectra and TD-DFT calculations reveal significant red-shifts in ligand-to-polyoxometalate charge transfer (LPCT) absorption bands for the 2D species compared to linear, dipolar analogues, consistent with the involvement of a larger conjugated (bridge + POM) system in the transitions. Electrochemical measurements indicate reversible, one-electron processes for the POM acceptors with class II mixed valence behavior observed where the POMs are connected to the same aryl ring and electronic isolation of the acceptors when they are on separate rings. Molecular first hyperpolarizabilities β have been determined using hyper-Rayleigh scattering at 1064 and 1200 nm: for the most active compound, the HRS measurements and depolarization studies reveal a strongly 2D, off-diagonal response (β0,zzz = 190 × 10-30 esu; β0,zyy = -56.5 × 10-30 esu), consistent with the wide A-D-A angle and TD-DFT computed electronic transitions, which show both phenyl bridges and POMs equally involved in the acceptor orbitals.

On the Performance of Second-Order Polarization Propagator Methods in the Calculation of 1JFC and nJFH NMR Spin-Spin Coupling Constants

This study evaluates the performance of doubles-corrected random phase approximation (RPA) and higher random phase approximation (HRPA) approaches in predicting nuclear magnetic resonance (NMR) coupling constants involving fluorine. Their performance is benchmarked against experimental data and compared with that of higher-level theoretical methods, specifically second-order polarization propagator (SOPPA) and SOPPA(CCSD). Additionally, we discuss their performance relative to density functional theory (DFT). We find that RPA(D) is severely constrained by (near) triplet instabilities, while HRPA(D) demonstrates markedly improved stability. Statistical analysis reveals stronger patterns for carbon-fluorine couplings across the methods and systems investigated compared with fluorine-hydrogen couplings. While SOPPA-based methodologies prove to be superior in accuracy, HRPA(D) shows promising performance in reducing the computational burden of these calculations, albeit with a tendency to underestimate the coupling strength. These findings highlight the potential of HRPA(D) as a practical alternative to SOPPA methods, even for such difficult properties as NMR spin-spin coupling constants involving fluorine, emphasizing its role in improving predictive accuracy and efficiency across diverse chemical environments.

Structure, Properties, and Applications of Silica Nanoparticles: Recent Theoretical Modeling Advances, Challenges, and Future Directions

Silica nanoparticles (SNPs), one of the most widely researched materials in modern science, are now commonly exploited in surface coatings, biomedicine, catalysis, and engineering of novel self-assembling materials. Theoretical approaches are invaluable to enhancing fundamental understanding of SNP properties and behavior. Tremendous research attention is dedicated to modeling silica structure, the silica-water interface, and functionalization of silica surfaces for tailored applications. In this review, the range of theoretical methodologies are discussed that have been employed to model bare silica and functionalized silica. The evolution of silica modeling approaches is detailed, including classical, quantum mechanical, and hybrid methods and highlight in particular the last decade of theoretical simulation advances. It is started with discussing investigations of bare silica systems, focusing on the fundamental interactions at the silica-water interface, following with a comprehensively review of the modeling studies that examine the interaction of silica with functional ligands, peptides, ions, surfactants, polymers, and carbonaceous species. The review is concluded with the perspective on existing challenges in the field and promising future directions that will further enhance the utility and importance of the theoretical approaches in guiding the rational design of SNPs for applications in engineering and biomedicine.

An improved DIIS method using a versatile residual matrix to accelerate SCF starting from a crude guess

The minimization of the commutator of the Fock and density matrices as the error matrix in the direct inversion of the iterative subspace (CDIIS) developed by Pulay is a powerful self-consistent field (SCF) acceleration technique for the construction of optimum Fock matrix, if initiated with a fair initial guess. In this work, we present an alternative minimized error matrix to the commutator in the CDIIS, namely the residual or the gradient of the energy-functional for a Slater determinant subject to the orthonormality constraints among orbitals, representing the search for a newly improved Fock matrix in the direction of the residual in the direct inversion of the iterative subspace (RDIIS). Implemented in the computational chemistry package GAMESS, the RDIIS is compared with the standard CDIIS and the second order SCF orbital optimization (SOSCF) for tested molecules started with a crude guess. As a result, the RDIIS stably and efficiently performs the SCF convergence acceleration. Furthermore, the RDIIS is considerably independent on the subspace size with the concentrated linear coefficients accounting proportionally for the Fock matrices close to the current iteration.

Quantum chemistry study on the hyper-Rayleigh scattering optical activity of R-carvone and (1R,5R)-α-pinene

In this work, the HRS-OA circular differential scattering ratios of medium-size chiral molecules, R-carvone and (1R,5R)-α-pinene, has been computed by means of TD-DFT calculations. The results have been discussed as a function of the basis set size. For R-carvone a conformational analysis has been taken into account. In particular, the three most stable conformations of R-carvone present chiral terms that contribute to the HRS-OA circular differential scattering ratio, with opposite signs. Finally, the modified unit sphere representations (mUSR) have been produced for the molecules under study.

Electronic structure simulations in the cloud computing environment

The transformative impact of modern computational paradigms and technologies, such as high-performance computing (HPC), quantum computing, and cloud computing, has opened up profound new opportunities for scientific simulations. Scalable computational chemistry is one beneficiary of this technological progress. The main focus of this paper is on the performance of various quantum chemical formulations, ranging from low-order methods to high-accuracy approaches, implemented in different computational chemistry packages and libraries, such as NWChem, NWChemEx, Scalable Predictive Methods for Excitations and Correlated Phenomena, ExaChem, and Fermi-Löwdin orbital self-interaction correction on Azure Quantum Elements, Microsoft's cloud services platform for scientific discovery. We pay particular attention to the intricate workflows for performing complex chemistry simulations, associated data curation, and mechanisms for accuracy assessment, which is demonstrated with the Arrows automated workflow for high throughput simulations. Finally, we provide a perspective on the role of cloud computing in supporting the mission of leadership computational facilities.

Experimental and theoretical aspects of magnetic circular dichroism and magnetic circularly polarized luminescence in the UV, visible and IR ranges: A review

A historical sketch of the MCD (magnetic circular dichroism) spectroscopy is reported in its experimental and theoretical aspects. MCPL (magnetic circularly polarized luminescence) is also considered. The main studies are presented encompassing porphyrinoid systems, aggregates and materials, as well as simple organic molecules useful for the advancement of the interpretation. The MCD of chiral systems is discussed with special attention to new studies of natural products with potential pharmaceutical valence, including Amaryllidaceae alkaloids and related isocarbostyrils. Finally, the vibrational form of MCD, called MVCD, which is recorded in the IR part of the spectrum is also discussed. A final brief note on perspectives is given.

Why are information-theoretic descriptors powerful predictors of atomic and molecular polarizabilities

CONTEXT

We rationalize the excellent performance of information-theoretic descriptors for predicting atomic and molecular polarizabilities. It seems that descriptors which capture information about the change in valence-shell structure, especially the relative Fisher information measures, are particularly useful. Using this, we can rationalize why the G3 form of the relative Fisher information, which measures the deviation of effective nuclear charge between an atom-in-a-molecule and the reference pro-atom, is especially effective as a predictor of molecular polarizability.

METHODS

There are no methods used in this paper, which relies on mathematical derivation and analysis.

Tinned: A symbolic library for response theory and high-order derivatives

A symbolic C++ library-Tinned-has been developed for symbolic differentiation and manipulation in response theory. By recognizing different key building blocks in the density matrix-based (Thorvaldsen et al., J. Chem. Phys. 2008, 129, 214108) and coupled-cluster response theories, we have implemented their corresponding C++ symbolic classes, including but not limited to one- and two-electron operators, exchange-correlation energy and potential, and coupled-cluster operator. Formulas of response theory can be well expressed in terms of the symbolic classes in the library Tinned. Their high-order perturbation-strength derivatives can be straightforwardly computed and extracted afterwards for numerical evaluation. The library Tinned will greatly facilitate the development work of response theory and may lead to a unified framework for response theory at different levels of electronic structure theory.

Theory of Magnetic Properties in Quantum Electrodynamics Environments: Application to Molecular Aromaticity

In this work, we present ab initio cavity quantum electrodynamics (QED) methods which include interactions with a static magnetic field and nuclear spin degrees of freedom using different treatments of the quantum electromagnetic field. We derive explicit expressions for QED-HF magnetizability, nuclear shielding, and spin-spin coupling tensors. We apply this theory to explore the influence of the cavity field on the magnetizability of saturated, unsaturated, and aromatic hydrocarbons, showing the effects of different polarization orientations and coupling strengths. We also examine how the cavity affects aromaticity descriptors, such as the nucleus-independent chemical shift and magnetizability exaltation. We employ these descriptors to study the trimerization reaction of acetylene to benzene. We show how the optical cavity induces modifications in the aromatic character of the transition state leading to variations in the activation energy of the reaction. Our findings shed light on the effects induced by the cavity on magnetic properties, especially in the context of aromatic molecules, providing valuable insights into understanding the interplay between the quantum electromagnetic field and molecules.

Time-Resolved Circular Dichroism in Molecules: Experimental and Theoretical Advances

Following changes in chirality can give access to relevant information on the function or reactivity of molecular systems. Time-resolved circular dichroism (TRCD) spectroscopy proves to be a valid tool to achieve this goal. Depending on the class of molecules, different temporal ranges, spanning from seconds to femtoseconds, need to be investigated to observe such chiroptical changes. Therefore, over the years, several approaches have been adopted to cover the timescale of interest, especially based on pump-probe schemes. Moreover, various theoretical approaches have been proposed to simulate and explain TRCD spectra, including linear and non-linear response methods as well as non-adiabatic molecular dynamics. In this review, an overview on both experimental and theoretical advances in the TRCD field is provided, together with selected applications. A discussion on future theoretical developments for TRCD is also given.

Reduced Density Matrix Formulation of Quantum Linear Response

The prediction of spectral properties via linear response (LR) theory is an important tool in quantum chemistry for understanding photoinduced processes in molecular systems. With the advances of quantum computing, we recently adapted this method for near-term quantum hardware using a truncated active space approximation with orbital rotation, named quantum linear response (qLR). In an effort to reduce the classic cost of this hybrid approach, we here derive and implement a reduced density matrix (RDM) driven approach of qLR. This allows for the calculation of spectral properties of moderately sized molecules with much larger basis sets than so far possible. We report qLR results for benzene and R-methyloxirane with a cc-pVTZ basis set and study the effect of shot noise on the valence and oxygen K-edge absorption spectra of H2O in the cc-pVTZ basis.

Spin-Spin Coupling Constant Based on Reference Interaction Site Model Self-Consistent Field with Constrained Spatial Electron Density

A method for computing spin-spin coupling constants (SSCCs) using the reference interaction site model self-consistent field with constrained spatial electron density (RISM-SCF-cSED) is proposed for the first time. Describing solvents using integral equation theory allows us to reflect solvent effects at atomic resolution in SSCCs while accounting for thermal fluctuations at a low computational cost. Applying the method to water, 1,1-difluoroethylene, and 1-methylaminomethylene-2-naphthalenone revealed that the solvent shift was evaluated to a greater extent than in the continuum solvent model. The origin of this phenomenon was analyzed in terms of the physical mechanisms underlying SSCCs.

Nonadiabatic Molecular Dynamics with Fermionic Subspace-Expansion Algorithms on Quantum Computers

We introduce a novel computational framework for excited-state molecular quantum dynamics simulations driven by quantum-computing-based electronic-structure calculations. This framework leverages the fewest-switches surface-hopping method for simulating the nuclear dynamics and calculates the required excited-state transition properties with different flavors of the quantum subspace expansion and quantum equation-of-motion algorithms. We apply our method to simulate the collision reaction between a hydrogen atom and a hydrogen molecule. For this system, we critically compare the accuracy and efficiency of different quantum subspace expansion and equation-of-motion algorithms and show that only methods that can capture both weak and strong electron correlation effects can properly describe the nonadiabatic effects that tune the reactive event.

On the third-order nonlinear optical responses of cis and trans stilbenes - a quantum chemistry investigation

The second hyperpolarizabilities (γ) of the stilbene molecular switch in its trans and cis forms have been calculated using quantum chemistry methods to address their third-order nonlinear optical contrasts, to assess the reliability of lower-cost DFT methods, and to make comparisons with experiments. First, the reference CCSD(T) method shows that trans-stilbene presents a γ‖ value twice larger than its cis isomer (its γTHS value is 2.7 times larger). Among more cost-effective methods, reliable results are obtained at MP2 as well as with DFT, provided the CAM-B3LYP or ωB97X-D XCFs are employed. Supplementary DFT calculations have investigated the relationships between the accuracy of the exchange-correlation functionals, the fulfillment of Koopmans' theorem, and the delocalization error, and they demonstrated that satisfying Koopmans' theorem is not the condition for the best accuracy but that functionals with small delocalization errors are generally efficient. Using the selected CAM-B3LYP, large γ enhancements by about 70% (trans-stilbene) and 50% (cis-stilbene) have been evidenced when accounting for solvent effects using an implicit solvation model (IEFPCM), even for apolar solvents. Then, the frequency dispersion of the γ responses has been described using Bishop polynomial expansions, allowing comparisons with a broad set of experimental data. To a certain extent, no systematic agreement between the calculations and the measured values was found. On the one hand, the agreement is satisfactory for the γ(-ω;ω,-ω,ω) quantities, provided that the dominant vibrational contribution is taken into account. On the other hand, the agreement is poor for the γ(-2ω;ω,ω,0) and γ(-3ω;ω,ω,ω) quantities, while some inconsistencies between experimental values are also highlighted.

Quantum Equation of Motion with Orbital Optimization for Computing Molecular Properties in Near-Term Quantum Computing

Determining the properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is how to use imperfect near-term quantum computers to solve problems of practical value. Inspired by the recently developed variants of the quantum counterpart of the equation-of-motion (qEOM) approach and the orbital-optimized variational quantum eigensolver (oo-VQE), we present a quantum algorithm (oo-VQE-qEOM) for the calculation of molecular properties by computing expectation values on a quantum computer. We perform noise-free quantum simulations of BeH2 in the series of STO-3G/6-31G/6-31G* basis sets and of H4 and H2O in 6-31G using an active space of four electrons and four spatial orbitals (8 qubits) to evaluate excitation energies, electronic absorption, and, for twisted H4, circular dichroism spectra. We demonstrate that the proposed algorithm can reproduce the results of conventional classical CASSCF calculations for these molecular systems.

Which Options Exist for NISQ-Friendly Linear Response Formulations?

Linear response (LR) theory is a powerful tool in classic quantum chemistry crucial to understanding photoinduced processes in chemistry and biology. However, performing simulations for large systems and in the case of strong electron correlation remains challenging. Quantum computers are poised to facilitate the simulation of such systems, and recently, a quantum linear response formulation (qLR) was introduced [Kumar et al., J. Chem. Theory Comput. 2023, 19, 9136-9150]. To apply qLR to near-term quantum computers beyond a minimal basis set, we here introduce a resource-efficient qLR theory, using a truncated active-space version of the multiconfigurational self-consistent field LR ansatz. Therein, we investigate eight different near-term qLR formalisms that utilize novel operator transformations that allow the qLR equations to be performed on near-term hardware. Simulating excited state potential energy curves and absorption spectra for various test cases, we identify two promising candidates, dubbed "proj LRSD" and "all-proj LRSD".

Subspace Methods for the Simulation of Molecular Response Properties on a Quantum Computer

We explore Davidson methods for obtaining excitation energies and other linear response properties within the recently developed quantum self-consistent linear response (q-sc-LR) method. Davidson-type methods allow for obtaining only a few selected excitation energies without explicitly constructing the electronic Hessian since they only require the ability to perform Hessian-vector multiplications. We apply the Davidson method to calculate the excitation energies of hydrogen chains (up to H10) and analyze aspects of statistical noise for computing excitation energies on quantum simulators. Additionally, we apply Davidson methods for computing linear response properties such as static polarizabilities for H2, LiH, H2O, OH-, and NH3, and show that unitary coupled cluster outperforms classical projected coupled cluster for molecular systems with strong correlation. Finally, we formulate the Davidson method for damped (complex) linear response, with application to the nitrogen K-edge X-ray absorption of ammonia, and the C6 coefficients of H2, LiH, H2O, OH-, and NH3.

Practical phase-space electronic Hamiltonians for ab initio dynamics

Modern electronic structure theory is built around the Born-Oppenheimer approximation and the construction of an electronic Hamiltonian Ĥel(X) that depends on the nuclear position X (and not the nuclear momentum P). In this article, using the well-known theory of electron translation (Γ') and rotational (Γ″) factors to couple electronic transitions to nuclear motion, we construct a practical phase-space electronic Hamiltonian that depends on both nuclear position and momentum, ĤPS(X,P). While classical Born-Oppenheimer dynamics that run along the eigensurfaces of the operator Ĥel(X) can recover many nuclear properties correctly, we present some evidence that motion along the eigensurfaces of ĤPS(X,P) can better capture both nuclear and electronic properties (including the elusive electronic momentum studied by Nafie). Moreover, only the latter (as opposed to the former) conserves the total linear and angular momentum in general.

Harvesting Chemical Understanding with Machine Learning and Quantum Computers

It is tenable to argue that nobody can predict the future with certainty, yet one can learn from the past and make informed projections for the years ahead. In this Perspective, we overview the status of how theory and computation can be exploited to obtain chemical understanding from wave function theory and density functional theory, and then outlook the likely impact of machine learning (ML) and quantum computers (QC) to appreciate traditional chemical concepts in decades to come. It is maintained that the development and maturation of ML and QC methods in theoretical and computational chemistry represent two paradigm shifts about how the Schrödinger equation can be solved. New chemical understanding can be harnessed in these two new paradigms by making respective use of ML features and QC qubits. Before that happens, however, we still have hurdles to face and obstacles to overcome in both ML and QC arenas. Possible pathways to tackle these challenges are proposed. We anticipate that hierarchical modeling, in contrast to multiscale modeling, will emerge and thrive, becoming the workhorse of in silico simulations in the next few decades.

Sonification of molecular electronic energy density and its dynamics

A method is proposed for sonification of the molecular electronic energy density. The characteristic energetic structures of the individual complicated electronic wavefunctions are extracted in terms of the Energy Natural Orbitals (ENO), which are the eigenfunctions of the electronic energy density operator [K. Takatsuka and Y. Arasaki, J. Chem. Phys., 2021, 154, 094103]. Then, the frequency corresponding to each ENO energy is linearly transformed to the audible range. The time-variation of the population of the ENO serves as the volume (amplitude) of the sound. We demonstrate the sonification and associated voiceprints for a couple of very basic chemical bondings, from across an avoided crossing, and from the bond dissociation of a cluster.

Scalar Relativistic Effects with Multiwavelets: Implementation and Benchmark

The importance of relativistic effects in quantum chemistry is widely recognized, not only for heavier elements but throughout the periodic table. At the same time, relativistic effects are strongest in the nuclear region, where the description of electrons through a linear combination of atomic orbitals becomes more challenging. Furthermore, the choice of basis sets for heavier elements is limited compared with lighter elements where precise basis sets are available. Thanks to the framework of multiresolution analysis, multiwavelets provide an appealing alternative to overcoming this challenge: they lead to robust error control and adaptive algorithms that automatically refine the basis set description until the desired precision is reached. This allows one to achieve a proper description of the nuclear region. In this work, we extended the multiwavelet-based code MRChem to the scalar zero-order regular approximation framework. We validated our implementation by comparing the total energies for a small set of elements and molecules. To confirm the validity of our implementation, we compared both against a radial numerical code for atoms and the plane-wave-based code EXCITING.

Formulation and Implementation of Frequency-Dependent Linear Response Properties with Relativistic Coupled Cluster Theory for GPU-Accelerated Computer Architectures

We present the development and implementation of relativistic coupled cluster linear response theory (CC-LR), which allows the determination of molecular properties arising from time-dependent or time-independent electric, magnetic, or mixed electric-magnetic perturbations (within a common gauge origin for the magnetic properties) as well as taking into account the finite lifetime of excited states in the framework of damped response theory. We showcase our implementation, which is capable to offload the computationally intensive tensor contractions characteristic of coupled cluster theory onto graphical processing units, in the calculation of (a) frequency-(in)dependent dipole-dipole polarizabilities of IIB atoms and selected diatomic molecules, with a particular emphasis on the calculation of valence absorption cross sections for the I2 molecule; (b) indirect spin-spin coupling constants for benchmark systems such as the hydrogen halides (HX, X = F-I) as well the H2Se-H2O dimer as a prototypical system containing hydrogen bonds; and (c) optical rotations at the sodium D line for hydrogen peroxide analogues (H2Y2, Y = O, S, Se, Te). Thanks to this implementation, we are able to show the similarities in performance, but often the significant discrepancies, between CC-LR and approximate methods such as density functional theory. Comparing standard CC response theory with the flavor based upon the equation of motion formalism, we find that for valence properties such as polarizabilities, the two frameworks yield very similar results across the periodic table as found elsewhere in the literature; for properties that probe the core region, such as spin-spin couplings, on the other hand, we show a progressive differentiation between the two as relativistic effects become more important. Our results also suggest that as one goes down the periodic table, it may become increasingly difficult to measure pure optical rotation at the sodium D line due to the appearance of absorbing states.

Full Breit Hamiltonian in the Multiwavelets Framework

New techniques in core-electron spectroscopy are necessary to resolve the structures of oxides of f-elements and other strongly correlated materials that are present only as powders and not as single crystals. Thus, accurate quantum chemical methods must be developed to calculate core spectroscopic properties in such materials. In this contribution, we present an important development in this direction, extending our fully adaptive real-space multiwavelet basis framework to tackle the four-component Dirac-Coulomb-Breit Hamiltonian. We show that multiwavelets can reproduce one-dimensional grid-based approaches. They are however a fully three-dimensional approach which can later be extended to molecules and materials. Our multiwavelet implementation attained precise results irrespective of the chosen nuclear model, provided that the error threshold is tight enough and that the chosen polynomial basis is sufficiently large. Furthermore, our results confirmed that in two-electron species, the magnetic and Gauge contributions from s-orbitals are identical in magnitude and can account for the experimental evidence from K and L edges.

Second-order nonlinear optical properties of X-shaped pyrazine derivatives

This work reports an investigation of the second-order NLO properties of two isomer series of X-shaped pyrazine derivatives, by means of HRS measurements and DFT calculations. The systems differ in the relative position of the donor and acceptor substituents with respect to the axis formed by the nitrogen atoms of the central pyrazine ring. Although the magnitude of the second harmonic signal is similar, HRS measurements revealed that the anisotropy of the NLO response strongly differs in the two chromophore series, the one of the 2,3-isomers being strikingly dipolar, while the one of the 2,6-isomers is mostly octupolar. The experimental observations are well supported by DFT calculations. In particular, the sum-over-states approach allows us to rationalize the different NLO anisotropies observed in the two isomer series through a detailed analysis of the symmetry of the low-lying excited states.

Revisiting Gauge-Independent Kinetic Energy Densities in Meta-GGAs and Local Hybrid Calculations of Magnetizabilities

In a recent study [J. Chem. Theory Comput. 2021, 17, 1457-1468], some of us examined the accuracy of magnetizabilities calculated with density functionals representing the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA (mGGA), as well as global hybrid (GH) and range-separated (RS) hybrid functionals by assessment against accurate reference values obtained with coupled-cluster theory with singles, doubles, and perturbative triples [CCSD(T)]. Our study was later extended to local hybrid (LH) functionals by Holzer et al. [J. Chem. Theory Comput. 2021, 17, 2928-2947]; in this work, we examine a larger selection of LH functionals, also including range-separated LH (RSLH) functionals and strong-correlation LH (scLH) functionals. Holzer et al. also studied the importance of the physically correct handling of the magnetic gauge dependence of the kinetic energy density (τ) in mGGA calculations by comparing the Maximoff-Scuseria formulation of τ used in our aforementioned study to the more physical current-density extension derived by Dobson. In this work, we also revisit this comparison with a larger selection of mGGA functionals. We find that the newly tested LH, RSLH, and scLH functionals outperform all of the functionals considered in the previous studies. The various LH functionals afford the seven lowest mean absolute errors while also showing remarkably small standard deviations and mean errors. Most strikingly, the best two functionals are scLHs that also perform remarkably well in cases with significant multiconfigurational character, such as the ozone molecule, which is traditionally excluded from statistical error evaluations due to its large errors with common density functionals.

Frequency-Dependent Quadratic Response Properties and Two-Photon Absorption from Relativistic Equation-of-Motion Coupled Cluster Theory

We present the implementation of quadratic response theory based upon the relativistic equation-of-motion coupled cluster method. We showcase our implementation, whose generality allows us to consider both time-dependent and time-independent electric and magnetic perturbations, by considering the static and frequency-dependent hyperpolarizability of hydrogen halides (HX, X = F-At), providing comprehensive insights into their electronic response characteristics. Additionally, we evaluated the Verdet constant for noble gases Xe and Rn and discussed the relative importance of relativistic and electron correlation effects for these magneto-optical properties. Finally, we calculate the two-photon absorption cross sections of transition [ns1S0 → (n + 1)s1S0] of Ga+ and In+, which are suggested as candidates for new ion clocks. As our implementation allows for the use of nonrelativistic Hamiltonians as well, we have compared our EOM-QRCC results to the QR-CC implementation in the DALTON code and show that the differences between CC and EOMCC response are in general smaller than 5% for the properties considered. Collectively, the results underscore the versatility of our implementation and its potential as a benchmark tool for other approximated models, such as density functional theory for higher-order properties.

Linear Response Equations Revisited: A Simple and Efficient Iterative Algorithm

We present an algorithm to solve the linear response equations for Hartree-Fock, Density Functional Theory, and the Multiconfigurational Self-Consistent Field method that is both simple and efficient. The algorithm makes use of the well-established symmetric and antisymmetric combinations of trial vectors but further orthogonalizes them with respect to the scalar product induced by the response matrix. This leads to a standard, symmetric block eigenvalue problem in the expansion subspace that can be solved by diagonalizing a symmetric, positive definite matrix half the size of the expansion space. Numerical tests showed that the algorithm is robust and stable.

Magnetic optical rotation from real-time simulations in finite magnetic fields

We present a numerical approach to magnetic optical rotation based on real-time time-dependent electronic-structure theory. Not relying on perturbation expansions in the magnetic field strength, the formulation allows us to test the range of validity of the linear relation between the rotation angle per unit path length and the magnetic field strength that was established empirically by Verdet 160 years ago. Results obtained from time-dependent coupled-cluster and time-dependent current density-functional theory are presented for the closed-shell molecules H2, HF, and CO in magnetic fields up to 55 kT at standard temperature and pressure conditions. We find that Verdet's linearity remains valid up to roughly 10-20 kT, above which significant deviations from linearity are observed. Among the three current density-functional approximations tested in this work, the current-dependent Tao-Perdew-Staroverov-Scuseria hybrid functional performs the best in comparison with time-dependent coupled-cluster singles and doubles results for the magnetic optical rotation.

Tailoring light-induced charge transfer and intersystem crossing in FeCO using time-dependent spin-orbit configuration interaction

Real-time (RT) electronic structure methods provide a natural framework for describing light-matter interactions in arbitrary time-dependent electromagnetic fields (EMF). Optically induced excited state transitions are of particular interest, which require tuned EMF to drive population transfer to and from the specific state(s) of interest. Intersystem crossing, or spin-flip, may be driven through shaped EMF or laser pulses. These transitions can result in long-lived "spin-trapped" excited states, which are especially useful for materials requiring charge separation or protracted excited state lifetimes. Time-dependent configuration interaction (TDCI) is unique among RT methods in that it may be implemented in a basis of eigenstates, allowing for rapid propagation of the time-dependent Schrödinger equation. The recent spin-orbit TDCI (TD-SOCI) enables a real-time description of spin-flip dynamics in an arbitrary EMF and, therefore, provides an ideal framework for rational pulse design. The present study explores the mechanism of multiple spin-flip pathways for a model transition metal complex, FeCO, using shaped pulses designed to drive controlled intersystem crossing and charge transfer. These results show that extremely tunable excited state dynamics can be achieved by considering the dipole transition matrix elements between the states of interest.

Response properties in phaseless auxiliary field quantum Monte Carlo

We present a method for calculating first-order response properties in phaseless auxiliary field quantum Monte Carlo by applying automatic differentiation (AD). Biases and statistical efficiency of the resulting estimators are discussed. Our approach demonstrates that AD enables the calculation of reduced density matrices with the same computational cost scaling per sample as energy calculations, accompanied by a cost prefactor of less than four in our numerical calculations. We investigate the role of self-consistency and trial orbital choice in property calculations. We find that orbitals obtained using density functional theory perform well for the dipole moments of selected molecules compared to those optimized self-consistently.

Cost-Efficient High-Resolution Linear Absorption Spectra through Extrapolating the Dipole Moment from Real-Time Time-Dependent Electronic-Structure Theory

We present a novel function fitting method for approximating the propagation of the time-dependent electric dipole moment from real-time electronic structure calculations. Real-time calculations of the electronic absorption spectrum require discrete Fourier transforms of the electric dipole moment. The spectral resolution is determined by the total propagation time, i.e., the trajectory length of the dipole moment, causing a high computational cost. Our developed method uses function fitting on shorter trajectories of the dipole moment, achieving arbitrary spectral resolution through extrapolation. Numerical testing shows that the fitting method can reproduce high-resolution spectra by using short dipole trajectories. The method converges with as little as 100 a.u. dipole trajectories for some systems, though the difficulty converging increases with the spectral density. We also introduce an error estimate of the fit, reliably assessing its convergence and hence the quality of the approximated spectrum.

TURBOMOLE: Today and Tomorrow

TURBOMOLE is a highly optimized software suite for large-scale quantum-chemical and materials science simulations of molecules, clusters, extended systems, and periodic solids. TURBOMOLE uses Gaussian basis sets and has been designed with robust and fast quantum-chemical applications in mind, ranging from homogeneous and heterogeneous catalysis to inorganic and organic chemistry and various types of spectroscopy, light-matter interactions, and biochemistry. This Perspective briefly surveys TURBOMOLE's functionality and highlights recent developments that have taken place between 2020 and 2023, comprising new electronic structure methods for molecules and solids, previously unavailable molecular properties, embedding, and molecular dynamics approaches. Select features under development are reviewed to illustrate the continuous growth of the program suite, including nuclear electronic orbital methods, Hartree-Fock-based adiabatic connection models, simplified time-dependent density functional theory, relativistic effects and magnetic properties, and multiscale modeling of optical properties.

Reduced Scaling Real-Time Coupled Cluster Theory

Real-time coupled cluster (CC) methods have several advantages over their frequency-domain counterparts, namely, response and equation of motion CC theories. Broadband spectra, strong fields, and pulse manipulation allow for the simulation of complex spectroscopies that are unreachable using frequency-domain approaches. Due to the high-order polynomial scaling, the required numerical time propagation of the CC residual expressions is a computationally demanding process. This scaling may be reduced by local correlation schemes, which aim to reduce the size of the (virtual) orbital space by truncation according to user-defined parameters. We present the first application of local correlation to real-time CC. As in previous studies of locally correlated frequency-domain CC, traditional local correlation schemes are of limited utility for field-dependent properties; however, a perturbation-aware scheme proves promising. A detailed analysis of the amplitude dynamics suggests that the main challenge is a strong time dependence of the wave function sparsity.

State-Interaction Approach for Evaluating g-Tensors within EOM-CC and RAS-CI Frameworks: Theory and Benchmarks

Among various techniques designed for studying open-shell species, electron paramagnetic resonance (EPR) spectroscopy plays an important role. The key quantity measured by EPR is the g-tensor, describing the coupling between an external magnetic field and molecular electronic spin. One theoretical framework for quantum chemistry calculations of g-tensors is based on response theory, which involves substantial developments that are specific to the underlying electronic structure models. A simplified and easier-to-implement approach is based on the state-interaction scheme, in which perturbation is included by considering a small number of states. We describe and benchmark the state-interaction approach using equation-of-motion coupled-cluster and restricted-active-space configuration interaction wave functions. The analysis confirms that this approach can deliver accurate results and highlights caveats of applying it, such as a choice of the reference state, convergence with respect to the number of states used in calculations, etc. The analysis also contributes toward a better understanding of challenges in calculations of higher-order properties using approximate wave functions.

Automated, Consistent, and Even-Handed Selection of Active Orbital Spaces for Quantum Embedding

A widely used strategy to reduce the computational cost of quantum-chemical calculations is to partition the system into an active subsystem, which is the focus of the computational efforts, and an environment that is treated at a lower computational level. The system partitioning is mostly based on localized molecular orbitals. When reaction paths or energy differences are to be calculated, it is crucial to keep the orbital space consistent for all structures. Inconsistencies in orbital space can lead to unpredictable errors on the potential energy surface. While successful strategies to ensure this consistency have been established for organic and even metal-organic systems, these methods often fail for metal clusters or nanoparticles with a high density of near-degenerate and delocalized molecular orbitals. However, such systems are highly relevant for catalysis. Accurate yet feasible quantum-mechanical ab initio calculations are therefore highly desired. In this work, we present an approach based on the subsystem projected atomic orbital decomposition algorithm that allows us to ensure automated and consistent partitioning even for systems with delocalized and near-degenerate molecular orbitals and demonstrate the validity of this method for the binding energies of small molecules on transition-metal clusters.

Extreme NMR shielding in fluoro-nitrogen cations

The structure and NMR shielding of a set of N-F containing cations is reported to near-quantitative accuracy from extensive ab initio calculations. Currently, the shortest experimentally confirmed N-F bond is 1.2461(10) Å in NNF+, however CCSD(T)-F12b/cc-pVQZ-F12 optimised geometries suggest that even shorter N-F bonds are possible for both monocations (1.236 Å, HNF+) and dications (1.098 Å, NF2+). NMR shielding constants have been calculated in a composite manner with individual components from coupled-cluster expansions up to CCSDTQP and basis sets up to aug-cc-pCV8Z, together with vibrational and relativistic corrections. 15N and 19F NMR chemical shifts correlate well with available experimental data. Extreme 19F chemical shifts are predicted for HNF+ (1628.9 ppm) and NH2F2+ (1298.0 ppm), which are by far the largest 19F chemical shifts ever reported and well outside the known range of +865 ppm (F2O2) to -448 ppm (ClF). The 15N chemical shift of -1283.07 ppm in HNF+ is similarly extreme, being well outside the known range of 15N chemical shifts of -730 to 260 ppm (CH3NO2 reference). This work highlights the application of state-of-the-art theoretical techniques, and provides accurate NMR properties of both isolated and yet unknown N-F cations, which can serve to guide and supplement NMR experimentation.

responsefun: Fun with Response Functions in the Algebraic Diagrammatic Construction Framework

We present the open-source responsefun package, which implements a universally applicable procedure for computing molecular response properties within the algebraic diagrammatic construction (ADC) framework, exploiting the intermediate state representation (ISR) approach. With symbolic mathematics, the user can simply enter textbook sum-over-states (SOS) expressions from time-dependent perturbation theory, which are then automatically translated into the corresponding symbolic ADC/ISR formulations. Using the data structures provided by the hybrid Python/C++ module adcc for calculating excited states with ADC, the specified response property is directly evaluated, and the result is returned to the user. Employing the novel responsefun package, we present the first ADC/ISR calculations of second-order hyperpolarizability tensors and three-photon-absorption matrix elements.

A quantum chemical investigation of the second hyperpolarizability of p-nitroaniline

Recent measurements of the third harmonic scattering responses of molecules have given a new impetus for computing molecular second hyperpolarizabilities (γ) and for deducing structure-property relationships. This paper has employed a variety of wavefunction and density functional theory methods to evaluate the second hyperpolarizability of the p-nitroaniline prototypical push-pull π-conjugated molecule, addressing also numerical aspects, such as the selection of an integration grid and the impact of the order of differentiation vs the achievable accuracy by using the Romberg quadrature. The reliability of the different methods has been assessed by comparison to reference Coupled-Cluster Singles and Doubles with perturbative treatment of the Triples results. On the one hand, among wavefunction methods, the MP2 scheme offers the best accuracy/cost ratio for computing the static γ. On the other hand, using density functional theory, γ remains a challenging property to compute because all conventional, global hybrid or range-separated hybrid, exchange-correlation functionals underestimate static γ values by at least 15%. Even tuning the range-separating parameter to minimize the delocalization errors does not enable to improve the γ values. Nevertheless, the original double-hybrid B2-PLYP functional, which benefits from 27% of PT2 correlation and 53% Hartree-Fock exchange, provides accurate estimates of static γ values. Unfortunately, the best performing exchange-correlation functionals for γ are not necessarily reliable for the first hyperpolarizability, β, and vice versa. In fact, the β of p-nitroaniline (pNA) could be predicted, with a good accuracy, with several hybrid exchange-correlation functionals (including by tuning the range-separating parameter), but these systematically underestimate γ. As for γ, the MP2 wavefunction method remains the best compromise to evaluate the first hyperpolarizability of pNA at low computational cost.

Third-harmonic scattering optical activity: QED theory, symmetry considerations, and quantum chemistry applications in the framework of response theory

In this work, expressions for the third-harmonic scattering optical activity (THS-OA) spectroscopic responses are derived by combining molecular quantum electrodynamics (QED) and response theory, allowing their computational implementation. The QED theory of THS-OA presented here is meant to be an extension of a previous study by Andrews [Symmetry 12, 1466 (2020)]. In particular, the THS-OA phenomena are described within the Power-Zienau-Woolley multipolar Hamiltonian by including the electric-dipole, magnetic-dipole, and electric-quadrupole interactions for the absorption as well as the emission processes between the dynamic electromagnetic field (the photons) and matter. Moreover, we derive the expressions for the differential scattering ratios as a function of the scattering angle defined by the wavevectors of the incident and scattered photons. We show how the pure and mixed second hyperpolarizabilities can be obtained in the framework of response theory as specific cases of a generic cubic response function, thus enabling the computational implementation of THS-OA spectroscopy. We prove the origin-independence of the theory for exact wavefunctions. Preliminary computations on a prototype chiral molecule (methyloxirane) are considered together with an analysis of the basis set convergence and of the origin-dependence.

Efficient Protocol for Computing MCD Spectra in a Broad Frequency Range Combining Resonant and Damped CC2 Quadratic Response Theory

Coupled cluster response theory offers a path to high-accuracy calculations of spectroscopic properties, such as magnetic circular dichroism (MCD). However, divergence or slow convergence issues are often encountered for electronic transitions in high-energy regions with a high density of states. This is here addressed for MCD by an implementation of damped quadratic response theory for resolution-of-identity coupled cluster singles-and-approximate-doubles (RI-CC2), along with an implementation of the MCD A term from resonant response theory. Combined, damped and resonant response theory calculations provide an efficient strategy to obtain MCD spectra over a broad frequency range and for systems that include highly symmetric molecules with degenerate excited states. The protocol is illustrated by application to zinc tetrabenzoporphyrin in the energy region of 2-8 eV and comparison to experimental data. Timings are reported for the resonant and damped approaches, showing that a greater part of the calculation time is consumed by the construction of the building blocks for the final MCD ellipticity. A recommendation on how to use the procedure is outlined.

Electronic-state chaos, intramolecular electronic energy redistribution, and chemical bonding in persisting multidimensional nonadiabatic systems

We study the chaotic, huge fluctuation of electronic state, resultant intramolecular energy redistribution, and strong chemical bonding surviving the fluctuation with exceedingly long lifetimes of highly excited boron clusters. Those excited states constitute densely quasi-degenerate state manifolds. The huge fluctuation is induced by persisting multidimensional nonadiabatic transitions among the states in the manifold. We clarify the mechanism of their coexistence and its physical significance. In doing so, we concentrate on two theoretical aspects. One is quantum chaos and energy randomization, which are to be directly extracted from the properties of the total electronic wavefunctions. The present dynamical chaos takes place through frequent transitions from adiabatic states to others, thereby making it very rare for the system to find dissociation channels. This phenomenon leads to the concept of what we call intramolecular nonadiabatic electronic-energy redistribution, which is an electronic-state generaliztion of the notion of intramolecular vibrational energy redistribution. The other aspect is about the peculiar chemical bonding. We investigate it with the energy natural orbitals (ENOs) to see what kind of theoretical structures lie behind the huge fluctuation. The ENO energy levels representing the highly excited states under study appear to have four robust layers. We show that the energy layers responsible for chaotic dynamics and those for chemical bonding are widely separated from each other, and only when an event of what we call "inter-layer crossing" happens to burst can the destruction of these robust energy layers occur, resulting in molecular dissociation. This crossing event happens only rarely because of the large energy gaps between the ENO layers. It is shown that the layers of high energy composed of complex-valued ENOs induce the turbulent flow of electrons and electronic-energy in the cluster. In addition, the random and fast time-oscillations of those high energy ENOs serve as a random force on the nuclear dynamics, which can work to prevent a concentration of high nuclear kinetic energy in the dissociation channels.

Corrigendum: Coupled cluster theory on modern heterogeneous supercomputers

[This corrects the article DOI: 10.3389/fchem.2023.1154526.].