Semiclassical "Divide-and-Conquer" Method for Spectroscopic Calculations of High Dimensional Molecular Systems

Exact factorization method for bound vibrational states: An analytical tool for accurate approximations

The Exact Factorization (XF) method represents an interesting formulation of the Schrödinger equation where subsystem wavefunctions are exactly coupled. Here, I show that the XF method can be employed as an analytical tool to study the quantum vibrational problem of bound systems. In particular, after elaborating suitable XF-based wavefunction Ansätze, the ground-state energy approximated expression for bilinearly and quartically coupled harmonic oscillators is estimated. The XF-based analytical solution is compared with adiabatic and perturbative ones, and it is usually found to be an order of magnitude more accurate than these for estimating the anharmonic and coupling correction part of the ground-state energy. This procedure will possibly increase the numerical stability and accuracy of perturbative or Hartree-product based methods when applied to bound state calculations.

Semiclassical description of nuclear quantum effects in solvated and condensed phase molecular systems

In this perspective we deal with the challenge of investigating nuclear quantum effects in solvated and condensed phase molecular systems in a computationally affordable way. To this end, semiclassical methods are promising theoretical approaches, as we demonstrate through vibrational spectroscopy and reaction kinetics. We show that quantum vibrational features can be found in hydrates of carbonyl compounds and microsolvated amino acids, and we report quantum estimates of the low-temperature reaction rate constant of a unimolecular reaction taking place in a noble-gas matrix. The hallmark of semiclassical methods is their ability to include nuclear quantum effects into classical molecular dynamics simulations. For this reason, unlike other popular methods, semiclassical approaches are able to account also for real-time quantum contributions and are expected to point out the importance of nuclear quantum effects in complex systems for a wider range of chemical properties.

A Perspective on the Investigation of Spectroscopy and Kinetics of Complex Molecular Systems with Semiclassical Approaches

In this Perspective we show that semiclassical methods provide a rigorous hierarchical way to study the vibrational spectroscopy and kinetics of complex molecular systems. The time averaged approach to spectroscopy and the semiclassical transition state theory for kinetics, which have been first adopted and then further developed in our group, provide accurate quantum results on rigorous physical grounds and can be applied even when dealing with a large number of degrees of freedom. In spectroscopy, the multiple coherent, divide-and-conquer, and adiabatically switched semiclassical approaches have practically permitted overcoming issues related to the convergence of results. In this Perspective we demonstrate the possibility of studying the semiclassical vibrational spectroscopy of a molecule adsorbed on an anatase (101) surface, a system made of 51 atoms. In kinetics, the semiclassical transition state theory is able to account for anharmonicity and the coupling between the reactive and bound modes. Our group has developed this technique for practical applications involving the study of phenomena like kinetic isotope effect, heavy atom tunneling, and elusive conformer lifetimes. Here, we show that our multidimensional anharmonic quantum approach is able to tackle on-the-fly the thermal kinetic rate constant of a 135 degree-of-freedom system. Overall, semiclassical methods open up the possibility to describe at the quantum mechanical level systems characterized by hundreds of degrees of freedom leading to the accurate spectroscopic and kinetic description of biomolecules and complex molecular systems.

A time averaged semiclassical approach to IR spectroscopy

We propose a new semiclassical approach to the calculation of molecular IR spectra. The method employs the time averaging technique of Kaledin and Miller upon symmetrization of the quantum dipole-dipole autocorrelation function. Spectra at high and low temperatures are investigated. In the first case, we are able to point out the possible presence of hot bands in the molecular absorption line shape. In the second case, we are able to reproduce accurate IR spectra as demonstrated by a calculation of the IR spectrum of the water molecule, which is within 4% of the exact intensity. Our time averaged IR spectra can be directly compared to time averaged semiclassical power spectra as shown in an application to the CO2 molecule, which points out the differences between IR and power spectra and demonstrates that our new approach can identify active IR transitions correctly. Overall, the method features excellent accuracy in calculating absorption intensities and provides estimates for the frequencies of vibrations in agreement with the corresponding power spectra. In perspective, this work opens up the possibility to interface the new method with the semiclassical techniques developed for power spectra, such as the divide-and-conquer one, to get accurate IR spectra of complex and high-dimensional molecular systems.

Unraveling Water Solvation Effects with Quantum Mechanics/Molecular Mechanics Semiclassical Vibrational Spectroscopy: The Case of Thymidine

We introduce a quantum mechanics/molecular mechanics semiclassical method for studying the solvation process of molecules in water at the nuclear quantum mechanical level with atomistic detail. We employ it in vibrational spectroscopy calculations because this is a tool that is very sensitive to the molecular environment. Specifically, we look at the vibrational spectroscopy of thymidine in liquid water. We find that the C═O frequency red shift and the C═C frequency blue shift, experienced by thymidyne upon solvation, are mainly due to reciprocal polarization effects, that the molecule and the water solvent exert on each other, and nuclear zero-point energy effects. In general, this work provides an accurate and practical tool to study quantum vibrational spectroscopy in solution and condensed phase, incorporating high-level and computationally affordable descriptions of both electronic and nuclear problems.

High-order geometric integrators for the local cubic variational Gaussian wavepacket dynamics

Gaussian wavepacket dynamics has proven to be a useful semiclassical approximation for quantum simulations of high-dimensional systems with low anharmonicity. Compared to Heller's original local harmonic method, the variational Gaussian wavepacket dynamics is more accurate, but much more difficult to apply in practice because it requires evaluating the expectation values of the potential energy, gradient, and Hessian. If the variational approach is applied to the local cubic approximation of the potential, these expectation values can be evaluated analytically, but they still require the costly third derivative of the potential. To reduce the cost of the resulting local cubic variational Gaussian wavepacket dynamics, we describe efficient high-order geometric integrators, which are symplectic, time-reversible, and norm-conserving. For small time steps, they also conserve the effective energy. We demonstrate the efficiency and geometric properties of these integrators numerically on a multidimensional, nonseparable coupled Morse potential.

Molecular Hessian matrices from a machine learning random forest regression algorithm

In this article, we present a machine learning model to obtain fast and accurate estimates of the molecular Hessian matrix. In this model, based on a random forest, the second derivatives of the energy with respect to redundant internal coordinates are learned individually. The internal coordinates together with their specific representation guarantee rotational and translational invariance. The model is trained on a subset of the QM7 dataset but is shown to be applicable to larger molecules picked from the QM9 dataset. From the predicted Hessian, it is also possible to obtain reasonable estimates of the vibrational frequencies, normal modes, and zero point energies of the molecules.

Investigating the Spectroscopy of the Gas Phase Guanine-Cytosine Pair: Keto versus Enol Configurations

We report on a vibrational study of the guanine-cytosine dimer tautomers using state-of-the-art quasiclassical trajectory and semiclassical vibrational spectroscopy. The latter includes possible quantum mechanical effects. Through an accurate comparison to the experimental spectra, we are able to shine a light on the hydrogen bond network of one of the main subunits of DNA and put the experimental assignment on a solid footing. Our calculations corroborate the experimental conclusion that the global minimum Watson-and-Crick structure is not detected in the spectra, and there is no evidence of tunnel-effect-based double proton hopping. Our accurate assignment of the spectral features may also serve as a basis for the development of precise force fields to study the guanine-cytosine dimer.

Molecular Dynamics of Artificially Pair-Decoupled Systems: An Accurate Tool for Investigating the Importance of Intramolecular Couplings

We propose a numerical technique to accurately simulate the vibrations of organic molecules in the gas phase, when pairs of atoms (or, in general, groups of degrees of freedom) are artificially decoupled, so that their motion is instantaneously decorrelated. The numerical technique we have developed is a symplectic integration algorithm that never requires computation of the force but requires estimates of the Hessian matrix. The theory we present to support our technique postulates a pair-decoupling Hamiltonian function, which parametrically depends on a decoupling coefficient α ∈ [0, 1]. The closer α is to 0, the more decoupled the selected atoms. We test the correctness of our numerical method on small molecular systems, and we apply it to study the vibrational spectroscopic features of salicylic acid at the Density Functional Theory ab initio level on a fitted potential. Our pair-decoupled simulations of salicylic acid show that decoupling hydrogen-bonded atoms do not significantly influence the frequencies of stretching modes, but enhance enormously the out-of-plane wagging and twisting motions of the hydroxyl and carboxyl groups to the point that the carboxyl and hydroxyl groups may overcome high potential energy barriers and change the salicylic acid conformation after a short simulation time. In addition, we found that the acidity of salicylic acid is more influenced by the dynamical couplings of the proton of the carboxylic group with the carbon ring than with the hydroxyl group.

Vibrational spectroscopy simulation of solvation effects on a G-quadruplex

It is commonly believed that solvation effects on the vibrational properties of a solute are easily accounted for by simple rules of thumbs, that is, solvating a polar molecule in a polar medium has the only effect of red shifting all its spectroscopical features and, similarly, solvating a polar molecule in a nonpolar medium has the opposite effect. In this work, we use theoretical vibrational spectroscopy at quasi-classical and quantum approximate semiclassical level to gain atomistic insights about solvent-solute interactions for 2'-deoxyguanosine and the G-quadruplex. We employ the quasi-classical trajectory method to include full anharmonicity into our calculated spectra, and then introduce quantum nuclear effects by means of divide-and-conquer semiclassical spectroscopy calculations. Solvation is treated explicitly leading to a good reproducibility of the available experimental data and reliable predictions when an experimental reference is missing.Communicated by Ramaswamy H. Sarma.

Elucidating NO x Surface Chemistry at the Anatase (101) Surface in TiO2 Nanoparticles

Understanding NO _x chemistry at titania nanoparticle surfaces is important for photocatalytic environmental remediation processes. We focus on this problem and put forward an experimental-computational approach based on vibrational spectroscopy grounds. Temperature-dependent IR experiments of NO _x adsorption on shape-engineered nanoparticle (101) anatase surfaces are paired with power spectra obtained from Born-Oppenheimer trajectories. Then, the harmonic versus anharmonic vibrational frequencies of several adsorption scenarios are directly compared with the IR experiments. We conclude that molecules are adsorbed mainly by the N-end side and both the intermolecular interactions between adsorbed molecules and (NO)₂ dimer formation are responsible for the main NO adsorption spectroscopic features. We also investigate the spectroscopy and the mechanism of formation on defective anatase surfaces of the long-lived greenhouse gas N₂O.

A Semiclassical Framework for Mixed Quantum Classical Dynamics

Semiclassical (SC) approximations for quantum dynamic simulations in complex chemical systems range from rigorously accurate methods that are computationally expensive to methods that exhibit near-classical scaling with system size but are limited in their ability to describe quantum effects. In practical studies of high-dimensional reactions, neither extreme is the best choice: frequently a high-level quantum mechanical description is only required for a handful of modes, while the majority of environment modes that do not play a key role in the reactive event of interest are well served with a lower level of theory. In this feature, we introduce modified Filinov filtration as a powerful tool to construct mixed quantum-classical SC theories where different subsystems can be quantized to different extents without introducing ad hoc intersubsystem interaction terms. We demonstrate that these Filinov-based SC methods can systematically tune between quantum and classical limit SC behavior, offering a practical way forward to accurate and computationally efficient simulations of high-dimensional quantum processes.

Quantum Anharmonic Calculations of Vibrational Spectra for Water Adsorbed on Titania Anatase(101) Surface: Dissociative versus Molecular Adsorption

The interaction of water molecules and hydroxyl groups with titanium dioxide (TiO₂) surfaces is ubiquitous and very important in anatase nanoparticle photocatalytic processes. Infrared spectroscopy, assisted by ab initio calculations of vibrational frequencies, can be a powerful tool to elucidate the mechanisms behind water adsorption. However, a straightforward comparison between measurements and calculations remains a challenging task because of the complexity of the physical phenomena occurring on nanoparticle surfaces. Consequently, severe computational approximations, such as harmonic vibrational ones, are usually employed. In the present work we partially address this complexity issue by overcoming some of the standard approximations used in theoretical simulations and employ the Divide and Conquer Semiclassical Initial Value Representation (DC-SCIVR) molecular dynamics. This method allows to perform simulations of vibrational spectra of large dimensional systems accounting not only for anharmonicities, but also for nuclear quantum effects. We apply this computational method to water and deuterated water adsorbed on the ideal TiO₂ anatase(101) surface, contemplating both the molecular and the dissociated adsorption processes. The results highlight not only the presence of an anharmonic shift of the frequencies in agreement with the experiments, but also complex quantum mechanical spectral signatures induced by the coupling of molecular vibrational modes with the surface ones, which are different in the hydrogenated case from the deuterated one. These couplings are further analyzed by exploiting the mode subdivision performed during the divide and conquer procedure.

Applicability of the Thawed Gaussian Wavepacket Dynamics to the Calculation of Vibronic Spectra of Molecules with Double-Well Potential Energy Surfaces

Simulating vibrationally resolved electronic spectra of anharmonic systems, especially those involving double-well potential energy surfaces, often requires expensive quantum dynamics methods. Here, we explore the applicability and limitations of the recently proposed single-Hessian thawed Gaussian approximation for the simulation of spectra of systems with double-well potentials, including 1,2,4,5-tetrafluorobenzene, ammonia, phosphine, and arsine. This semiclassical wavepacket approach is shown to be more robust and to provide more accurate spectra than the conventional harmonic approximation. Specifically, we identify two cases in which the Gaussian wavepacket method is especially useful due to the breakdown of the harmonic approximation: (i) when the nuclear wavepacket is initially at the top of the potential barrier but delocalized over both wells, e.g., along a low-frequency mode, and (ii) when the wavepacket has enough energy to classically go over the low potential energy barrier connecting the two wells. The method is efficient and requires only a single classical ab initio molecular dynamics trajectory, in addition to the data required to compute the harmonic spectra. We also present an improved algorithm for computing the wavepacket autocorrelation function, which guarantees that the evaluated correlation function is continuous for arbitrary size of the time step.

Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths

This paper reviews methods that aim at simulating nuclear quantum effects (NQEs) using generalized thermal baths. Generalized (or quantum) baths simulate statistical quantum features, and in particular zero-point energy effects, through non-Markovian stochastic dynamics. They make use of generalized Langevin Equations (GLEs), in which the quantum Bose–Einstein energy distribution is enforced by tuning the random and friction forces, while the system degrees of freedom remain classical. Although these baths have been formally justified only for harmonic oscillators, they perform well for several systems, while keeping the cost of the simulations comparable to the classical ones. We review the formal properties and main characteristics of classical and quantum GLEs, in relation with the fluctuation–dissipation theorems. Then, we describe the quantum thermostat and quantum thermal bath, the two generalized baths currently most used, providing several examples of applications for condensed matter systems, including the calculation of vibrational spectra. The most important drawback of these methods, zero-point energy leakage, is discussed in detail with the help of model systems, and a recently proposed scheme to monitor and mitigate or eliminate it—the adaptive quantum thermal bath—is summarised. This approach considerably extends the domain of application of generalized baths, leading, for instance, to the successful simulation of liquid water, where a subtle interplay of NQEs is at play. The paper concludes by overviewing further development opportunities and open challenges of generalized baths.

Deep Zoom-In Simulation of a Fuzzy Dark Matter Galactic Halo

Fuzzy dark matter (FDM) made of ultralight bosonic particles is a viable alternative to cold dark matter with clearly distinguishable small-scale features in collapsed structures. On large scales, it behaves gravitationally like cold dark matter deviating only by a cutoff in the initial power spectrum and can be studied using N-body methods. In contrast, wave interference effects near the de Broglie scale result in new phenomena unique to FDM. Interfering modes in filaments and halos yield a stochastically oscillating granular structure which condenses into solitonic cores during halo formation. Investigating these highly nonlinear wave phenomena requires the spatially resolved numerical integration of the Schrödinger equation. In previous papers we introduced a hybrid zoom-in scheme that combines N-body methods to model the large-scale gravitational potential around and the mass accretion onto pre-selected halos with simulations of the Schrödinger-Poisson equation to capture wave-like effects inside these halos. In this work, we present a new, substantially improved reconstruction method for the wave function inside of previously collapsed structures. We demonstrate its capabilities with a deep zoom-in simulation of a well-studied sub-L_{*}-sized galactic halo from cosmological initial conditions. With a particle mass of m=2.5×10^{-22}  eV and halo mass M_{vir}=1.7×10^{11}  M_{⊙} in a (60  h^{-1} comoving Mpc)^{3} cosmological box, it reaches an effective resolution of 20 comoving pc. This pushes the values of m and M accessible to simulations significantly closer to those relevant for studying galaxy evolution in the allowed range of FDM masses.

The complex vibrational spectrum of proline explained through the adiabatically switched semiclassical initial value representation

Proline, a 17-atom amino acid with a closed-ring side chain, has a complex potential energy surface characterized by several minima. Its IR experimental spectrum, reported in the literature, is of difficult and controversial assignment. In particular, the experimental signal at 3559 cm−1 associated with the OH stretch is interesting because it is inconsistent with the global minimum, trans-proline conformer. This suggests the possibility that multiple conformers may contribute to the IR spectrum. The same conclusion is obtained by investigating the splitting of the CO stretch at 1766 and 1789 cm−1 and other, more complex spectroscopic features involving CH stretches and COH/CNH bendings. In this work, we perform full-dimensional, on-the-fly adiabatically switched semiclassical initial value representation simulations employing the ab initio dft-d3-B3LYP level of theory with aug-cc-pVDZ basis set. We reconstruct the experimental spectrum of proline in its main features by studying the vibrational features of trans-proline and cis1-proline, and provide a new assignment for the OH stretch of trans-proline.

Quantum Vibrational Spectroscopy of Explicitly Solvated Thymidine in Semiclassical Approximation

In this paper, we demonstrate the possibility to perform spectroscopy simulations of solvated biological species taking into consideration quantum effects and explicit solvation. We achieve this goal by interfacing our recently developed divide-and-conquer approach for semiclassical initial value representation molecular dynamics with the polarizable AMOEBABIO18 force field. The method is applied to the study of solvation of the thymidine nucleoside in two different polar solvents, water and N,N-dimethylformamide. Such systems are made of up to 2476 atoms. Experimental evidence concerning the different behavior of thymidine in the two solvents is well reproduced by our study, even though quantitative estimates are hampered by the limited accuracy of the classical force field employed. Overall, this study shows that semiclassically approximate quantum dynamical studies of explicitly solvated biological systems are both computationally affordable and insightful.

Zwitter Ionization of Glycine at Outer Space Conditions due to Microhydration by Six Water Molecules

We investigate glycine microsolvation with water molecules, mimicking astrophysical conditions, in our laboratory by embedding these clusters in helium nanodroplets at 0.37 K. We recorded mass selective infrared spectra in the frequency range 1500-1800  cm^{-1} where two bands centered at 1630 and 1724  cm^{-1} were observed. By comparison with the extensive accompanying calculations, the band at 1630  cm^{-1} was assigned to the COO^{-} asymmetric stretching mode of the zwitter ion and the band at 1724  cm^{-1} was assigned to redshifted C=O stretch within neutral clusters. We show that zwitter ion formation of amino acids readily occurs with only few water molecules available even under extreme conditions.

Singular value decomposition analysis of the electron density changes occurring upon electrostatic polarization of water

In-depth elucidation of how molecules are electrically polarized would be one key factor for understanding the properties of those molecules under various thermodynamic and/or spatial conditions. Here this problem is tackled for the case of hydrogen-bonded water by conducting singular value decomposition of the electron density changes that occur upon electrostatic polarization. It is shown that all those electron density changes are approximately described as linear combinations of ten orthonormal basis “vectors”. One main component is the interatomic charge transfer through each OH bond, while some others are characterized as the atomic dipolar polarizations, meaning that both of these components are important for the electrostatic polarization of water. The interaction parameters that reasonably well reproduce the induced dipole moments are derived, which indicate the extent of mixing of the two components in electrostatic polarization.

The main features of the electron density changes that occur upon electrostatic polarization of water are elucidated by conducting singular value decomposition analysis of those changes.

On-the-fly adiabatically switched semiclassical initial value representation molecular dynamics for vibrational spectroscopy of biomolecules

Semiclassical (SC) vibrational spectroscopy is a technique capable of reproducing quantum effects (such as zero-point energies, quantum resonances, and anharmonic overtones) from classical dynamics runs even in the case of very large dimensional systems. In a previous study [Conte et al. J. Chem. Phys. 151, 214107 (2019)], a preliminary sampling based on adiabatic switching has been shown to be able to improve the precision and accuracy of semiclassical results for challenging model potentials and small molecular systems. In this paper, we investigate the possibility to extend the technique to larger (bio)molecular systems whose dynamics must be integrated by means of ab initio "on-the-fly" calculations. After some preliminary tests on small molecules, we obtain the vibrational frequencies of glycine improving on pre-existing SC calculations. Finally, the new approach is applied to 17-atom proline, an amino acid characterized by a strong intramolecular hydrogen bond.

Unsupervised Machine Learning Neural Gas Algorithm for Accurate Evaluations of the Hessian Matrix in Molecular Dynamics

The Hessian matrix of the potential energy of molecular systems is employed not only in geometry optimizations or high-order molecular dynamics integrators but also in many other molecular procedures, such as instantaneous normal mode analysis, force field construction, instanton calculations, and semiclassical initial value representation molecular dynamics, to name a few. Here, we present an algorithm for the calculation of the approximated Hessian in molecular dynamics. The algorithm belongs to the family of unsupervised machine learning methods, and it is based on the neural gas idea, where neurons are molecular configurations whose Hessians are adopted for groups of molecular dynamics configurations with similar geometries. The method is tested on several molecular systems of different dimensionalities both in terms of accuracy and computational time versus calculating the Hessian matrix at each time-step, that is, without any approximation, and other Hessian approximation schemes. Finally, the method is applied to the on-the-fly, full-dimensional simulation of a small synthetic peptide (the 46 atom N-acetyl-l-phenylalaninyl-l-methionine amide) at the level of DFT-B3LYP-D/6-31G* theory, from which the semiclassical vibrational power spectrum is calculated.

Nuclear Quantum Effects in Liquid Water at Near Classical Computational Cost Using the Adaptive Quantum Thermal Bath

We demonstrate the accuracy and efficiency of a recently introduced approach to account for nuclear quantum effects (NQEs) in molecular simulations: the adaptive quantum thermal bath (adQTB). In this method, zero-point energy is introduced through a generalized Langevin thermostat designed to precisely enforce the quantum fluctuation-dissipation theorem. We propose a refined adQTB algorithm with improved accuracy and report adQTB simulations of liquid water. Through extensive comparison with reference path integral calculations, we demonstrate that it provides excellent accuracy for a broad range of structural and thermodynamic observables as well as infrared vibrational spectra. The adQTB has a computational cost comparable to that of classical molecular dynamics, enabling simulations of up to millions of degrees of freedom.

Caldeira-Leggett model vs ab initio potential: A vibrational spectroscopy test of water solvation

We present a semiclassically approximate quantum treatment of solvation with the purpose of investigating the accuracy of the Caldeira-Leggett model. We do that by simulating the vibrational features of water solvation by means of two different approaches. One is entirely based on the adoption of an accurate ab initio potential to describe water clusters of increasing dimensionality. The other one consists of a model made of a central water molecule coupled to a high-dimensional Caldeira-Leggett harmonic bath. We demonstrate the role of quantum effects in the detection of water solvation and show that the computationally cheap approach based on the Caldeira-Leggett bath is only partially effective. The main conclusion of the study is that quantum methods associated with high-level potential energy surfaces are necessary to correctly study solvation features, while simplified models, even if attractive owing to their reduced computational cost, can provide some useful insights but are not able to come up with a comprehensive description of the solvation phenomenon.

How many water molecules are needed to solvate one?

Many efforts undertaken to study the solvation process have led to general theories that may describe mean properties, but are unable to provide a detailed understanding at the molecular level. Remarkably, the basic question of how many solvent molecules are necessary to solvate one solute molecule is still open. By exploring several water aggregates of increasing complexity, in this contribution we employ semiclassical spectroscopy to determine on quantum dynamical grounds the minimal network of surrounding water molecules to make the central one display the same vibrational features of liquid water. We find out that double-acceptor double-donor tetrahedral coordination constituting the standard picture is necessary but not sufficient, and that particular care must be reserved for the quantum description of the combination band due to the coupling of the central monomer bending mode with network librations. It is actually our ability to investigate the combination band with a quantum-derived approach that allows us to answer the titular question. The minimal structure eventually responsible for proper solvation is made of a total of 21 water molecules and includes two complete solvation shells, of which the whole first one is tetrahedrally coordinated to the central molecule.

How quantum spectroscopic simulations can explain water solvation by comparison with experimental spectra.

Representing molecular ground and excited vibrational eigenstates with nuclear densities obtained from semiclassical initial value representation molecular dynamics

We present in detail and validate an effective Monte Carlo approach for the calculation of the nuclear vibrational densities via integration of molecular eigenfunctions that we have preliminary employed to calculate the densities of the ground and the excited OH stretch vibrational states in the protonated glycine molecule [Aieta et al., Nat Commun 11, 4348 (2020)]. Here, we first validate and discuss in detail the features of the method on a benchmark water molecule. Then, we apply it to calculate on-the-fly the ab initio anharmonic nuclear densities in the correspondence of the fundamental transitions of NH and CH stretches in protonated glycine. We show how we can gain both qualitative and quantitative physical insight by inspection of different one-nucleus densities and assign a character to spectroscopic absorption peaks using the expansion of vibrational states in terms of harmonic basis functions. The visualization of the nuclear vibrations in a purely quantum picture allows us to observe and quantify the effects of anharmonicity on the molecular structure, also to exploit the effect of IR excitations on specific bonds or functional groups, beyond the harmonic approximation. We also calculate the quantum probability distribution of bond lengths, angles, and dihedrals of the molecule. Notably, we observe how in the case of one type of fundamental NH stretching, the typical harmonic nodal pattern is absent in the anharmonic distribution.

Anharmonic quantum nuclear densities from full dimensional vibrational eigenfunctions with application to protonated glycine

The interpretation of molecular vibrational spectroscopic signals in terms of atomic motion is essential to understand molecular mechanisms and for chemical characterization. The signals are usually assigned after harmonic normal mode analysis, even if molecular vibrations are known to be anharmonic. Here we obtain the quantum anharmonic vibrational eigenfunctions of the 11-atom protonated glycine molecule and we calculate the density distribution of its nuclei and its geometry parameters, for both the ground and the O-H stretch excited states, using our semiclassical method based on ab initio molecular dynamics trajectories. Our quantum mechanical results describe a molecule elongated and more flexible with respect to what previously thought. More importantly, our method is able to assign each spectral peak in vibrational spectroscopy by showing quantitatively how normal modes involving different functional groups cooperate to originate that spectroscopic signal. The method will possibly allow for a better rationalization of experimental spectroscopy.

Plumbing Potentials for Molecules with Up To Tens of Atoms: How to Find Saddle Points and Fix Leaky Holes

Potential energy surfaces (PESs) play an indispensable role in molecular dynamics but are notoriously difficult to flesh out properly in large-dimensional spaces. In particular, the undetected presence of PES holes, i.e., unphysical saddle points beyond which the potential energy drops arbitrarily, can have devastating effects on both classical and quantum dynamics calculations. In this study, the Crystal algorithm is developed as a tool for efficiently and accurately finding PES holes, as well as legitimate saddle points, even in very large-dimensional configuration spaces. The approach is applied to three large-dimensional PESs for molecular systems of current interest: uracil, naphthalene, and formic acid dimer. Low-lying PES holes are discovered and located for the first two systems-including naphthalene, for which no holes were previously suspected, to the best of our knowledge. Likewise, the double-well, double-proton-transfer isomerization saddle point for formic acid dimer is also located.

Semiclassical Vibrational Spectroscopy of Biological Molecules Using Force Fields

Semiclassical spectroscopy is a practical way to get an accurately approximate quantum description of spectral features starting from ab initio molecular dynamics simulations. The computational bottleneck for the method is represented by the cost of ab initio potential, gradient, and Hessian matrix estimates. This drawback is particularly severe for biological systems due to their unique complexity and large dimensionality. The main goal of this manuscript is to demonstrate that quantum dynamics and spectroscopy, at the level of semiclassical approximation, are doable even for sizable biological systems. To this end, we investigate the possibility of performing semiclassical spectroscopy simulations when ab initio calculations are replaced by computationally cheaper force field evaluations. Both polarizable (AMOEBABIO18) and nonpolarizable (AMBER14SB) force fields are tested. Calculations of some particular vibrational frequencies of four nucleosides, i.e., uridine, thymidine, deoxyguanosine, and adenosine, show that ab initio simulations are accurate and widely applicable. Conversely, simulations based on AMBER14SB are limited to harmonic approximations, but those relying on AMOEBABIO18 yield acceptable semiclassical values if the investigated conformation has been included in the force field parametrization. The main conclusion is that AMOEBABIO18 may provide a viable route to assist semiclassical spectroscopy in the study of large biological molecules for which an ab initio approach is not computationally affordable.

Anharmonic calculations of vibrational spectra for molecular adsorbates: A divide-and-conquer semiclassical molecular dynamics approach

The vibrational spectroscopy of adsorbates is becoming an important investigation tool for catalysis and material science. This paper presents a semiclassical molecular dynamics method able to reproduce the vibrational energy levels of systems composed by molecules adsorbed on solid surfaces. Specifically, we extend our divide-and-conquer semiclassical method for power spectra calculations to gas-surface systems and interface it with plane-wave electronic structure codes. The Born-Oppenheimer classical dynamics underlying the semiclassical calculation is full dimensional, and our method includes not only the motion of the adsorbate but also those of the surface and the bulk. The vibrational spectroscopic peaks related to the adsorbate are accounted together with the most coupled phonon modes to obtain spectra amenable to physical interpretations. We apply the method to the adsorption of CO, NO, and H₂O on the anatase-TiO₂ (101) surface. We compare our semiclassical results with the single-point harmonic estimates and the classical power spectra obtained from the same trajectory employed in the semiclassical calculation. We find that CO and NO anharmonic effects of fundamental vibrations are similarly reproduced by the classical and semiclassical dynamics and that H₂O adsorption is fully and properly described in its overtone and combination band relevant components only by the semiclassical approach.

Improved semiclassical dynamics through adiabatic switching trajectory sampling

We introduce an improved semiclassical dynamics approach to quantum vibrational spectroscopy. In this method, a harmonic-based phase space sampling is preliminarily driven toward non-harmonic quantization by slowly switching on the actual potential. The new coordinates and momenta serve as initial conditions for the semiclassical dynamics calculation, leading to a substantial decrease in the number of chaotic trajectories to deal with. Applications are presented for model and molecular systems of increasing dimensionality characterized by moderate or high chaoticity. They include a bidimensional Henon-Heiles potential, water, formaldehyde, and methane. The method improves accuracy and precision of semiclassical results and it can be easily interfaced with all pre-existing semiclassical theories.

Semiclassical dynamics in the mixed quantum-classical limit

The semiclassical double Herman-Kluk initial value representation is an accurate approach to computing quantum real time correlation functions, but its applications are limited by the need to evaluate an oscillatory integral. In previous work, we have shown that this "sign problem" can be mitigated using the modified Filinov filtration technique to control the extent to which individual modes of the system contribute to the overall phase of the integrand. Here, we follow this idea to a logical conclusion: we analytically derive a general expression for the mixed quantum-classical limit of the semiclassical correlation function-analytical mixed quantum-classical-initial value representation (AMQC-IVR), where the phase contributions from the "classical" modes of the system are filtered while the "quantum" modes are treated in the full semiclassical limit. We numerically demonstrate the accuracy and efficiency of the AMQC-IVR formulation in calculations of quantum correlation functions and reaction rates using three model systems with varied coupling strengths between the classical and quantum subsystems. We also introduce a separable prefactor approximation that further reduces computational cost but is only accurate in the limit of weak coupling between the quantum and classical subsystems.

Reduced rovibrational coupling Cartesian dynamics for semiclassical calculations: Application to the spectrum of the Zundel cation

We study the vibrational spectrum of the protonated water dimer, by means of a divide-and-conquer semiclassical initial value representation of the quantum propagator, as a first step in the study of larger protonated water clusters. We use the potential energy surface from the work of Huang et al. [J. Chem. Phys. 122, 044308 (2005)]. To tackle such an anharmonic and floppy molecule, we employ fully Cartesian dynamics and carefully reduce the coupling to global rotations in the definition of normal modes. We apply the time-averaging filter and obtain clean power spectra relative to suitable reference states that highlight the spectral peaks corresponding to the fundamental excitations of the system. Our trajectory-based approach allows for the physical interpretation of the very challenging proton transfer modes. We find that it is important, for such a floppy molecule, to selectively avoid initially exciting lower energy modes, in order to obtain cleaner spectra. The estimated vibrational energies display a mean absolute error (MAE) of ∼29 cm^-1 with respect to available multiconfiguration time-dependent Hartree calculations and MAE ∼ 14 cm^-1 when compared to the optically active experimental excitations of the Ne-tagged Zundel cation. The reasonable scaling in the number of trajectories for Monte Carlo convergence is promising for applications to higher dimensional protonated cluster systems.

Vibrational investigation of nucleobases by means of divide and conquer semiclassical dynamics

In this work, we report a computational study of the vibrational features of four different nucleobases employing the divide-and-conquer semiclassical initial value representation molecular dynamics method. Calculations are performed on uracil, cytosine, thymine, and adenine. Results show that the overall accuracy with respect to experiments is within 20 wavenumbers, regardless of the dimensionality of the nucleobase. Vibrational estimates are accurate even in the complex case of cytosine, where two relevant conformers are taken into account. These results are promising in the perspective of future studies on more complex systems, such as nucleotides or nucleobase pairs.

Semiclassical vibrational spectroscopy with Hessian databases

We report on a new approach to ease the computational overhead of ab initio "on-the-fly" semiclassical dynamics simulations for vibrational spectroscopy. The well known bottleneck of such computations lies in the necessity to estimate the Hessian matrix for propagating the semiclassical pre-exponential factor at each step along the dynamics. The procedure proposed here is based on the creation of a dynamical database of Hessians and associated molecular geometries able to speed up calculations while preserving the accuracy of results at a satisfactory level. This new approach can be interfaced to both analytical potential energy surfaces and on-the-fly dynamics, allowing one to study even large systems previously not achievable. We present results obtained for semiclassical vibrational power spectra of methane, glycine, and N-acetyl-L-phenylalaninyl-L-methionine-amide, a molecule of biological interest made of 46 atoms.

Complex saddle trajectories for multidimensional quantum wave packet and coherent state propagation: Application to a many-body system

A practical search technique for finding the complex saddle points used in wave packet and coherent state propagation is developed which works for a large class of Hamiltonian dynamical systems with many degrees of freedom. The method can be applied to problems in atomic, molecular, and optical physics and other domains. A Bose-Hubbard model is used to illustrate the application to a many-body system where discrete symmetries play an important and fascinating role. For multidimensional wave packet propagation, locating the necessary saddles involves the seemingly insurmountable difficulty of solving a boundary value problem in a high-dimensional complex space, followed by determining whether each particular saddle found actually contributes. In principle, this must be done for each propagation time considered. The method derived here identifies a real search space of minimal dimension, which leads to a complete set of contributing saddles up to intermediate times much longer than the Ehrenfest timescale for the system. The analysis also gives a powerful tool for rapidly identifying the various dynamical regimes of the system.

Protonated glycine supramolecular systems: the need for quantum dynamics

Quantum mechanical simulations unequivocally explain experimental IR spectra of protonated supramolecular systems.

IR spectroscopy is one of the most commonly employed techniques to study molecular vibrations and interactions. However, characterization of experimental IR spectra is not always straightforward. This is the case of protonated glycine supramolecular systems like Gly₂H⁺ and (GlyH + nH₂), whose IR spectra raise questions which have still to find definitive answers even after theoretical spectroscopy investigations. Specifically, the assignment of the conformer responsible for the spectrum of the protonated glycine dimer (Gly₂H⁺) has led to much controversy and it is still debated, while structural hypotheses formulated to explain the main experimental spectral features of (GlyH + nH₂) systems have not been theoretically confirmed. We demonstrate that simulations must account for quantum dynamical effects in order to resolve these open issues. This is achieved by means of our divide-and-conquer semiclassical initial value representation technique, which approximates the quantum dynamics of high dimensional systems with remarkable accuracy and outperforms not only the commonly employed but unfit scaled-harmonic approaches, but also pure classical dynamics simulations. Besides the specific insights concerning the two particular cases presented here, the general conclusion is that, due to the widespread presence of protonated systems in chemistry, quantum dynamics may play a prominent role and should not be totally overlooked even when dealing with large systems including biological structures.

"Divide and conquer" semiclassical molecular dynamics: A practical method for spectroscopic calculations of high dimensional molecular systems

We extensively describe our recently established "divide-and-conquer" semiclassical method [M. Ceotto, G. Di Liberto, and R. Conte, Phys. Rev. Lett. 119, 010401 (2017)] and propose a new implementation of it to increase the accuracy of results. The technique permits us to perform spectroscopic calculations of high-dimensional systems by dividing the full-dimensional problem into a set of smaller dimensional ones. The partition procedure, originally based on a dynamical analysis of the Hessian matrix, is here more rigorously achieved through a hierarchical subspace-separation criterion based on Liouville's theorem. Comparisons of calculated vibrational frequencies to exact quantum ones for a set of molecules including benzene show that the new implementation performs better than the original one and that, on average, the loss in accuracy with respect to full-dimensional semiclassical calculations is reduced to only 10 wavenumbers. Furthermore, by investigating the challenging Zundel cation, we also demonstrate that the "divide-and-conquer" approach allows us to deal with complex strongly anharmonic molecular systems. Overall the method very much helps the assignment and physical interpretation of experimental IR spectra by providing accurate vibrational fundamentals and overtones decomposed into reduced dimensionality spectra.

Atomistic Explanation for Interlayer Charge Transfer in Metal-Semiconductor Nanocomposites: The Case of Silver and Anatase

A concerted theoretical and experimental investigation of the silver/anatase hybrid nanocomposite, a very promising material for advanced sensing applications, is presented. We measure its exceptional electrochemical virtues in terms of current densities and reproducibility, providing their explanation at the atomic-scale level and demonstrating how and why silver acts as a positive electrode. Using periodic plane-wave DFT calculations, we estimate the overall amount of electron transfer toward the semiconductor side of the interface at equilibrium. Suitably designed (photo)electrochemical experiments strictly agree, both qualitatively and quantitatively, with the theoretical charge transfer estimates. The unique permanent charge separation occurring in the device is possible because of the favorable synergy of Ag and TiO₂, which exploits in a favorable band alignment, while the electron-hole recombination rate and carrier mobility decrease when electrons cross the metal-semiconductor interface. Finally, the hybrid material is proven to be extremely robust against aging, showing complete regeneration, even after 1 year.