Is Quantum Above-Barrier Reflection Important for Molecular Barrier Crossing Rates?

Understanding quantum tunneling and above-barrier reflection effects on unimolecular and bimolecular reaction rate constants remains challenging to this very day. In many applications, especially when considering moderate-to-high temperatures, the "standard" procedure is to use the parabolic barrier approximation. Recent work has shown though that this may be insufficient, and one cannot ignore anharmonicity. In this work, we study the analytic theory, including anharmonicity obtained when expanding the thermal rate up to order ℏ4. Such theories need high-order derivatives of the potential at the barrier top. We show that such derivatives are computed straightforwardly for six different reactions. We suggest a straightforward methodology for assessing whether the parabolic barrier approximation is valid and show that when the reaction asymmetry is large, this may lead to significant quantum above-barrier reflection and transmission coefficients, which are less than unity.

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.

Elucidating Heavy Atom Tunneling Kinetics in the Cope Rearrangement of Semibullvalene.

In this work, we characterize the temperature dependence of kinetic properties in heavy atom tunneling reactions by means of molecular dynamics simulations, including nuclear quantum effects (NQEs) via Path Integral theory. To this end, we consider the prototypical Cope rearrangement of semibullvalene. The reaction was studied in the 25-300K temperature range observing that the inclusion of NQEs modifies the temperature behavior of both free energy barriers and dynamical recrossing factors with respect to classical dynamics. Notably, while in classical simulations the activation free energy shows a very weak temperature dependence, it becomes strongly dependent on temperature when NQEs are included. This temperature behavior shows a transition from a regime where the quantum effects are limited and can mainly be traced back to zero point energy, to a low temperature regime where tunneling plays a dominant role. In this regime, the free energy curve tunnels below the potential energy barrier along the reaction coordinate,  allowing much faster reaction rates. Finally, the temperature dependence of the rate constants obtained from molecular dynamics simulations was compared with available experimental data and with semi-classical transition state theory calculations, showing comparable behaviors and similar transition temperatures from  thermal to (deep) tunneling regime.

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.

Coupled Cluster Semiclassical Estimates of Experimental Reaction Rates: The Interconversion of Glycine Conformer VIp to Ip.

We apply the full-dimensional Semiclassical Transition State Theory (SCTST) to estimate the rate constant of glycine molecule interconversion between the VIp and Ip conformers. We have reached an electronic structure accuracy up to the explicitly correlated Coupled Cluster method (CCSD(T)-F12b/cc-pVDZ-F12) thanks to our parallel implementation. The reaction has been experimentally investigated in the literature and is known to proceed by quantum mechanical tunneling. The SCTST rates improve over other theoretical methods, and our results align with the experimental measurements, thus confirming the accuracy of the fully coupled anharmonic semiclassical tunneling treatment, providing that the level of electronic structure theory gives a reliable estimate of the reaction barrier height and shape. The comparison with experimental half-life times supports the validity of SCTST for glycine VIp-Ip conformer conversion in the cryogenic temperature range, where this theory is usually not considered applicable due to the onset of the deep tunneling regime.

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.

Heavy Atom Tunneling in Organic Reactions at Coupled Cluster Potential Accuracy with a Parallel Implementation of Anharmonic Constant Calculations and Semiclassical Transition State Theory.

We describe and test on some organic reactions a parallel implementation strategy to compute anharmonic constants, which are employed in semiclassical transition state theory reaction rate calculations. Our software can interface with any quantum chemistry code capable of a single point energy estimate, and it is suitable for both minimum and transition state geometry calculations. After testing the accuracy and comparing the efficiency of our implementation against other software, we use it to estimate the semiclassical transition state theory (SCTST) rate constant of three reactions of increasing dimensionality, known as examples of heavy atom tunneling. We show how our method is improved in efficiency with respect to other existing implementations. In conclusion, our approach allows SCTST rates and heavy atom tunneling at a high level of electronic structure theory (up to CCSD(T)) to be evaluated. This work shows how crucial the possibility to perform high level ab initio rate evaluations can be.

Machine learning for vibrational spectroscopy via divide-and-conquer semiclassical initial value representation molecular dynamics with application to N-methylacetamide.

A machine learning algorithm for partitioning the nuclear vibrational space into subspaces is introduced. The subdivision criterion is based on Liouville's theorem, i.e., the best preservation of the unitary of the reduced dimensionality Jacobian determinant within each subspace along a probe full-dimensional classical trajectory. The algorithm is based on the idea of evolutionary selection, and it is implemented through a probability graph representation of the vibrational space partitioning. We interface this customized version of genetic algorithms with our divide-and-conquer semiclassical initial value representation method for the calculation of molecular power spectra. First, we benchmark the algorithm by calculating the vibrational power spectra of two model systems, for which the exact subspace division is known. Then, we apply it to the calculation of the power spectrum of methane. Exact calculations and full-dimensional semiclassical spectra of this small molecule are available and provide an additional test of the accuracy of the new approach. Finally, the algorithm is applied to the divide-and-conquer semiclassical calculation of the power spectrum of 12-atom trans-N-methylacetamide.

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.

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.

Parallel Implementation of Semiclassical Transition State Theory.

This paper presents the parsctst code, an efficient parallel implementation of the semiclassical transition state theory (SCTST) for reaction rate constant calculations. Parsctst is developed starting from a previously presented approach for the computation of the vibrational density of states of fully coupled anharmonic molecules ( Nguyen et al. Chem. Phys. Lett. 2010 , 499 , 915 ). The parallel implementation makes it practical to tackle reactions involving more than 100 fully coupled anharmonic vibrational degrees of freedom and also includes multidimensional tunneling effects. After describing the pseudocode and demonstrating its computational efficiency, we apply the new code for estimating the rate constant of the proton transfer isomerization reaction of the 2,4,6-tri- tert-butylphenyl to 3,5-di- tert-butylneophyl. Comparison with both theoretical and experimental results is presented. Parsctst code is user-friendly and provides a significant computational time saving compared to serial calculations. We believe that parsctst can boost the application of SCTST as an alternative to the basic transition state theory for accurate kinetics modeling not only in combustion or atmospheric chemistry, but also in organic synthesis, where bigger reactive systems are encountered.

A quantum method for thermal rate constant calculations from stationary phase approximation of the thermal flux-flux correlation function integral.

This paper presents a quantum mechanical approximation to the calculation of thermal rate constants. The rate is derived from a suitable stationary phase approximation to the time integral of the thermal flux-flux correlation function. The goal is to obtain an expression that barely depends on the position of the flux operators, i.e., of the dividing surfaces, so that it can be applied also to complex systems by arbitrarily locating the dividing surfaces. The approach is tested on one and two dimensional systems where quantum effects are predominant over a wide range of temperatures. The results are quite accurate, i.e., within a few percent of the exact values for a reasonable range of dividing surface positions.

An Efficient Computational Approach for the Calculation of the Vibrational Density of States.

We present an optimized approach for the calculation of the density of fully coupled vibrational states in high-dimensional systems. This task is of paramount importance, because partition functions and several thermodynamic properties can be accurately estimated once the density of states is known. A new code, called paradensum, based on the implementation of the Wang-Landau Monte Carlo algorithm for parallel architectures is described and applied to real complex systems. We test the accuracy of paradensum on several molecular systems, including some benchmarks for which an exact evaluation of the vibrational density of states is doable by direct counting. In addition, we find a significant computational speedup with respect to standard approaches when applying our code to molecules up to 66 degrees of freedom. The new code can easily handle 150 degrees of freedom. These features make paradensum a very promising tool for future calculations of thermodynamic properties and thermal rate constants of complex systems.