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The interstellar medium (ISM) and planetary atmospheres are the place of myriad of molecular processes [1] such as bimolecular reactions, photodissociations or detachments, recombinations, inelastic collisions, dissociative adsorptions on grain surfaces, the reverse associative desorptions, etc... These processes play a key role in the composition, temperature, and motions of the previous media. Detailed information on these features are revealed by patterns in spectra measurable by means of telescopes even for interstellar clouds or exoplanets millions to billions of miles away [2,3].

Besides, molecular beam experiments provide fascinating data on how atoms move in the course of chemical reactions through the measurement of differential and integral cross sections [4,5], or via femtosecond pump-probe spectroscopy [6]. In addition to that, kinetics experiments, for instance of the CRESU type, provide rate constants down to a few tens of Kelvin [7]. In turn, these data are used as parameters in models of the ISM and planetary atmospheres. It should also be noted that this research has technological spin-offs in areas as diverse as combustion (radical-radical reactions), catalysis or thermonuclear fusion (gas-surface processes) to cite but a few.

In order to ensure the crucial synergy between experiment and theory, the previous measurements of ever increasing precision require theoretical predictions of comparable accuracy. For triatomic or tetratomic reactions, or for diatom-surface reactions within the rigid surface approximation, exact quantum calculations of the dynamics and kinetics are often feasible [8]. This is generally also the case for the spectra of small molecular species. On the other hand, the exponential scaling of quantum mechanics with dimensionality makes exact quantum calculations very difficult if not unfeasible for systems involving five atoms or more. For reaction rate constants, ring polymer molecular dynamics (RPMD) is a very promising alternative as it scales linearly with dimensionality [9]. However, this approach is in principle a theory for molecular processes in thermodynamic equilibrium and does not address spectra. In contrast with quantum methods, classical trajectory simulations are very friendly (free of tricky parametrization issues) and applicable to any system, whatever the dimensionality and the state of equilibrium [10]. Their main drawback, however, is that they are meaningful only far from the quantum regime, limiting thereby their applicability. A question naturally arises: can we combine the realism of quantum mechanics with the simplicity and versatility of classical trajectory calculations to fight the curse of dimensionality in the quantum regime ?

The answer is (nearly) yes in principle, not always in practice. As illustrated in the drawing on the first page, there are two avenues of approach, each presenting different strengths and weaknesses. The first and most rigorous one, pioneered by Miller, Heller and others, consists in expressing S-matrix elements, rate constants or absorption spectra in terms of the space-time propagator [11,12]. The latter is then replaced by one of its semiclassical limits, in general the van Vleck-Gutzwiller or Herman-Kluk expressions [11,13]. Passage to initial value representations allows to express the quantities of interest as multidimensional integrals over initial conditions. Consequently, the resulting methods replace the difficult wave propagation step by a simple run of intuitively appealing classical paths while accounting for quantum interferences (left framework in the drawing). These approaches are expected to be accurate provided that de Broglie wavelength is small enough and the classical dynamics sufficiently regular. Note that if they have been successfully applied to spectra or rate constants [14], their ability to reproduce S-matrix elements for three- dimensional collisions is still questionable, in particular for processes involving trapped trajectories.

The second avenue of approach, especially for chemical dynamics, amounts to introduce ad hoc quantum corrections in classical trajectory simulations to deal with major quantum effects such as tunneling [15], non adiabatic transitions [16,17], the quantization of initial and final internal motions [18], etc... (right framework in the drawing). This introduction is performed in part on the basis of qualitative knowledge and intuition (zero point energy leakage corrections, Gaussian binning,...) and in part on pre-existing semiclassical models (WKB, Landau-Zener, surface hopping algorithm...). Eventhough these approaches do not take into account the superposition principle, they take advantage of the fact that quantum interferences are more often quenched that not. Hence, ignoring them weakly affects the quality of the predictions in many cases. These methods are extremely popular as they can be applied to high-dimensional systems and do not necessarily require potential energy surfaces, replacing them whenever necessary by on-the-fly calculations [19,20]. Attempts to put such methods on firm theoretical grounds within the rigorous semiclassical theories previously discussed are regularly published in the literature [21]. 

[1] The Physics and Chemistry of the Interstellar Medium, A.G.G.M Tielens, Cambridge University Press, 2005.
[2] B. T. Soifer, G. Helou, and M. Werner. The Spitzer view of the extragalactic universe. Annual Review of Astronomy and Astrophysics, 46:2010 (2008).
[3] B. A. McGuire, O. Asvany, S. Brünken et al. Laboratory spectroscopy techniques to enable observations of interstellar ion chemistry. Nat. Rev. Phys., (2020). Doi: 10.1038/s42254-020-0198-0.
[4] J. Werdecker, M. E. van Reijzen, B.-J. Chen, and R. D. Beck, Vibrational energy redistribution in a gas-surface encounter: state-to-state scattering of CH4 from Ni(111). Phys. Rev. Lett., 120, 053402 (2018).
[5] Y. Xie, H. Zhao, Y. Wang, Y. Huang, T. Wang, X. Xu, C. Xiao, Z. Sun, D. H. Zhang, X. Yang. Quantum interference in H + HD → H  Dbetweendirectabstractionand H + D between direct abstraction and
roaming insertion pathways. Science, 368: 767 (2020).
[6] W. Li, A. A. Jaroń-Becker,C. W.Hogle, V.Sharma,X.Zhou, A.Becker,H.C.- ecker, C. W. Hogle, V. Sharma, X. Zhou, A. Becker, H. C. Kapteyn, and M. M. Murnane, Visualizing electron rearrangement in space and time during the transition from a molecule to atom. PNAS, 107, 20219 (2010).
[7] Y. Wu, J. Cao, H. Ma, C. Zhang, W. Bian, D. Nunez-Reyes, K. M. Hickson, Conical intersection–regulated intermediates in bimolecular reactions: Insights from C( 1D) + HD dynamics. Science Advances, 26 Apr 2019: Vol. 5, no. 4, eaaw0446, DOI: 10.1126/sciadv.aaw0446.
[8] B. Fu, X. Shan, D.-H. Zhang, and D. Clary, Recent advances in quantum scattering calculations on polyatomic bimolecular reactions. Chemical Society Reviews, 46, 7625 (2017).
[9] J. E. Lawrence and D. E. Manolopoulos, Path integral methods for reaction rates in complex systems. Faraday Discuss., 221, 9 (2020).
[10] J. F. Aoiz, L. Banares, and V. Herrero, Recent results from quasiclassical
trajectory computations of elementary chemical reactions. J. Chem. Soc. Faraday Trans. 94, 2483 (1998).
[11] W. H. Miller, The semiclassical initial value representation: A potentially practical way for adding quantum effects to classical molecular dynamics simulations. J. Phys. Chem. A, 105, 2942 (2001); W. H. Miller, Quantum dynamics of complex molecular systems. PNAS, 102, 6660 (2005).
[12] E. J. Heller, Quantum corrections to classical photodissociation models. J. Chem. Phys., 68, 2066 (1978); E. J. Heller, Cellular dynamics: A new semiclassical approach to time-dependent quantum mechanics. J. Chem. Phys., 94, 2723 (1991). [13] E. Kluk, M. F. Herman, and H. L. Davis, Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited anharmonic oscillator. J. Chem. Phys., 84, 326 (1986).
[14] G. Di Liberto and M. Ceotto, The importance of the pre-exponential factor in semiclassical molecular dynamics. J. Chem. Phys., 145, 144107 (2016).
[15] J. Zheng, X. Xu, R. Meana-Paneda, and D. G. Truhlar, Army ants tunneling for classical simulations. Chem. Sci., 5, 2091 (2014).
[16] J. E. Subotnik, A. Jain, B. Landry, A. Petit, W. Ouyang, and N. Bellonzi, Understanding the Surface Hopping View of Electronic Transitions and Decoherence. Annu. Rev. Phys. Chem., 67, 387 (2016).
[17] S. J. Cotton and W. H. Miller, A symmetrical quasi-classical windowing model for the molecular dynamics treatment of non-adiabatic processes involving many electronic states. J. Chem. Phys., 150, 104101 (2019).
[18] A. Rodríguez-Fernandez, L. Bonnet, C. Crespos, P. Larregaray, and R. Díez Muiño, When Classical Trajectories Get to Quantum Accuracy: The Scattering of H 2on Pd(111). J. Phys. Chem. Lett., 10, 7629 (2019)
[19] F. Kossoski, M. T. do N. Varella, and M. Barbatti, On-the-fly dynamics simulations of transient anions, J. Chem. Phys., 151, 224104 (2019).
[20] M. Richter, P. Marquetand, J. González-Vázquez, I. Sola, L. González, SHARC: ab Initio Molecular Dynamics with Surface Hopping in the Adiabatic Representation Including Arbitrary Couplings, J. Chem. Theory Comput., 7, 1253 (2011).
[21] Jeremy O. Richardson, Derivation of instanton rate theory from first principles, J. Chem. Phys., 144, 114106 (2016).