Recent progress in adiabatic and non-adiabatic methods in quantum dynamics
- Basile Curchod (Durham University, United Kingdom)
- Ari Paavo Seitsonen (Ecole Normale Supérieure, France)
- Ivano Tavernelli (IBM-Zurich Research, Switzerland)
- Thomas James Penfold (Newcastle University, United Kingdom)
- Irene BURGHARDT (Goethe University Frankfurt, Germany)
- Felipe Franco de Carvalho (Swiss Federal Institute of Technology of Lausanne (EPFL), Switzerland)
Recently, the non-adiabatic dynamics community has focused not only on the study of model systems and on the improvement of the accuracy of methods, but also on the simulation of more realistic molecular systems with practical applications, for instance, in solar energy conversion and energy storage. The idea is to go beyond the Born-Oppenheimer approximation and beyond the non-relativistic Schrödinger equation (for the inclusion of spin-orbit coupling effects) with the aim at describing complex molecular system in interaction with thermal baths (solvent) and external time-dependent fields (laser fields). This effort will take simulations closer to experiments and will eventually turn them into powerful predictive tools. In addition, advances in femto- and attosecond spectroscopies have allowed for the experimental observation of nuclear and electronic quantum dynamics, which has in turn stimulated the development of more accurate theoretical methods.
At this point in time, we could say that the development of non-adiabatic dynamics has followed two equally challenging and stimulating paths. The first one is mainly based on the simplification of the system Hamiltonian followed by a rigorous (sometimes exact) solution of the quantum dynamics using either wavepackets or trajectory based approaches. This has been for a long time the main focus within the community, and the importance of these calculations is still unquestionable. The second (and newer) path consists in describing the non-adiabatic dynamics of molecular systems keeping the complexity of the molecular Hamiltonian untouched and address the solution of the quantum dynamics in a approximated way for a chosen subset of reaction coordinates or in the full (unconstrained) configuration space. Exploring both these paths and encouraging dialogue between researchers working in both areas will be the focus of this workshop.
In the following we review the most well-known and successful methods in non-adiabatic dynamics and also discuss new developments in the field. We also describe the challenges encountered in treatment of more realistic systems and how they could possibly be tackled.
Since a numerically exact solution of the time-dependent Schroedinger equation is not feasible beyond about 5-6 (nuclear) degrees of freedom - and is thus out of reach for realistic molecular systems - the use of approximate methods is crucial. Two main classes of such approximate methods have emerged, depending on whether the nuclei are treated as quantum or classical particles.
In the first category, the quantum nature of the nuclear dynamics is essentially preserved, and the object that is treated in the numerical simulations is a wavepacket (or superpositions of such wavepackets) describing all quantum effects, such as interference and tunneling. Non-adiabatic effects, involving the splitting and scattering of the wavepacket at potential energy surface (PES) crossings, i.e., avoided crossings or conical intersections, are intrinsically described in a correct way. These are exactly the effects which are decisive for excited-state dynamics, and that are absent in classical simulations. An (incomplete) list of methods based on wavepacket dynamics includes:
i) standard basis set methods (grid/DVR-based, static basis sets) time-dependent Hartree (mean-field),
ii) multiconfigurational methods using moving basis sets: multiconfiguration time-dependent Hartree (MCTDH), multi-layer MCTDH (ML-MCTDH),
iii) non-orthogonal (Gaussian) moving basis sets: Gaussian-based MCTDH (G-MCTDH), Local Coherent State Approximation (LCSA), Coupled Coherent States (CCS), full multiple spawning (FMS).
As mentioned above, standard basis set methods are limited to about 5-6 degrees of freedom, and beyond this, become rapidly unfeasible due to the exponential scaling problem (i.e., exponential scaling of the number of basis functions with the number of degrees of freedom). The multiconfiguration time-dependent Hartree (MCTDH) algorithm [1,2] which corresponds to a multiconfigurational mean-field method does not overcome the exponential scaling, but significantly alleviates the problem through the construction of a variationally optimized moving basis. MCTDH is arguably today's most powerful wavepacket propagation method, and can been applied for systems typically involving 20-50 degrees of freedom.
An important related class of multiconfigurational methods do not use a fully flexible moving basis like MCTDH, but are based on time-evolving Gaussian wavepackets. This is the case for the G-MCTDH , CCS , LCSA , and FMS  methods mentioned above. These approaches employ superpositions of variational (G-MCTDH, LCSA) or non-variational (CCS, FMS) Gaussian wavepackets. They interpolate between quantum basis set methods and semiclassical methods. In the case of G-MCTDH, it has been shown that results can be converged to the “exact” MCTDH results, e.g., for the standard test case of non-adiabatic dynamics of pyrazine at the S2/S1 conical intersection . Most of these methods are, however, applied in a regime that is relatively far from convergence. Still, they arguably have significant advantages over trajectory-based propagation methods. When compared to full quantum methods, they further have the advantage that the Gaussian wavepackets follow classical-like paths, and can thus be used in the context of on-the-fly simulations. This is illustrated, e.g., by the great success of the full multiple spawning (FMS) method . This type of method can be very powerful in gaining insight into the complicated dynamics at conical intersections as well as for the investigation of light-driven reactions.
Quantum trajectory methods [7-10] have the particularity of being able to describe all nuclear quantum effects (just as wavepacket methods) and being on-the-fly. Furthermore, we also note that an exact factorization of the molecular wavefunction has recently been proposed. It brings about a new and illuminating perspective of the concept of potential energy surface .
In the second category of methods, the wavepacket is approximated by an ensemble of particles that follow classical trajectories. These methods are well suited for the study of the nuclear dynamics in the full phase space (without the need of introducing constraints or reaction coordinates) and can easily be implemented in software packages that allow for the on-the-fly calculation of energies and forces. An (incomplete) list of such trajectory-based methods includes:
i) classical trajectories (classical-path, mean-field approaches) semiclassical (WKB and related approaches),
ii) Gaussian wavepacket dynamics (Heller wavepackets, spawning, CCS, G-MCTDH) mixed quantum-classical surface-hopping type methods mixed quantum-classical Liouville dynamics,
iii) path integrals,
iv) real-time RPMD (ring polymer molecular dynamics).
Among these methods, the classical path, or Ehrenfest approximation is the most straightforward one. Here, the classical subsystem evolves under the mean field generated by the electrons, and the electronic dynamics is evaluated along the classical path of the nuclei. The intuitively appealing picture of trajectories hopping between coupled potential-energy surfaces gave rise to a number of quasiclassical implementations [12,13]. The most well-known method is Tully's “fewest switches” surface hopping method , which has evolved into a widely used and successful technique. Finally, semiclassical  and quasi-classical methods such as quantum-classical Liouville approach  and semiclassical WKB-type approaches can be seen as being the middle ground between full-quantum (wavepacket- or trajectory-based) methods and the classical trajectory methods.
Moving towards the description of more realistic systems, we first acknowledge that this is still a work-in-progress research area, which presents many challenges (this makes the matter all the more interesting). Examples are the simulation of complex molecular systems describing photoinduced charge transfer reactions, which require the accurate description of the molecular electronic structure and of the environment (solvent). Possible solutions to these problems in the context of MCTDH might come from the ML-MCTDH  (for larger systems) and G-MCTDH (for the inclusion of the environment and for direct dynamics) methods. When it comes to TSH, there has been increased interest in combining it with QM/MM schemes in order to take the environment into account. This idea has been used to describe reactive molecules in solvents , and noble gas matrices .
It is also worth noting that, besides the development of new methods that can deal with quantum dynamics of larger systems, the use of GPUs in quantum chemistry has vastly increased the computational power at the disposal of researchers , which means that even relatively computationally expensive (but accurate) methods can be used to study large molecules. Moreover, improved algorithms -with improved parallelization- have also been recently developed and applied to describe the quantum dynamics of complex molecular systems.
As previously mentioned, describing molecules in a more realistic way does not simply implies moving towards the simulation of larger systems, but also requires the inclusion of effects that used to be neglected. One of example of this is spin-orbit coupling, which is essential for the description of photophysical and photochemical processes in which inter-system crossing plays a role (including phosphorescence and the development OLEDs). In this domain we can cite a modified version of MCTDH  and methods combining TSH with the computation of spin-orbit coupling [21-23].
Another important aspect usually neglected in quantum dynamics is the direct coupling between the molecular system and the external time-dependent electromagnetic field (including strong laser fields). Once again, the accurate description of such effects is important in bridging the gap between theory and modern experiments. There have been propositions to implement coupling with an external field within TSH [22,23,25], despite the limitations of the method, because of the possibility to calculate everything on the fly. MCTDH can also be used for this purpose [26,27] (with the disadvantage that quantities cannot be computed on the fly).
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