Adiabatic and non-adiabatic methods in quantum dynamics
- Ivano Tavernelli (IBM-Zurich Research, Switzerland)
- Irene BURGHARDT (Goethe University Frankfurt, Germany)
- Basile Curchod (Durham University, United Kingdom)
A wide variety of methods have been developed to simulate the quantum dynamics of molecular systems . In the following, we focus on nuclear quantum dynamics , but promising recent developments of electronic quantum dynamics are underway [2,3]. 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:
- standard basis set methods (grid/DVR-based, static basis sets)
- time-dependent Hartree (mean-field)
- multiconfigurational methods using moving basis sets: multiconfiguration time-dependent Hartree (MCTDH), multi-layer MCTDH (ML-MCTDH)
- 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 [4,5] which corresponds to a multiconfigurational mean-field method does not overcome the exponential scaling, but significantly alleviates the problem due to 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. The multi-layer variant of MCTDH (ML-MCTDH)  adds to the potential of this family of methods. Further, this method has recently been extended to fermionic systems (electrons) , similarly to the MCTDH-F approach .
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. These approaches interpolate between quantum basis set methods and the semiclassical methods addressed below. 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. As compared with 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 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.
In the second category of methods, the wavepacket is approximated by an ensemble of particles that follow classical trajectories. An (incomplete) list of such trajectory-based methods includes:
- classical trajectories (classical-path, mean-field approaches)
- semiclassical (WKB and related approaches)
- Gaussian wavepacket dynamics (Heller wavepackets, spawning, CCS, G-MCTDH)
- mixed quantum-classical surface-hopping type methods
- mixed quantum-classical Liouville dynamics
- full quantum trajectory dynamics (Bohmian & Wigner trajectories)
- path integrals
- 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. An important limitation of the classical path approach is the absence of a "back-reaction" of the classical DoF to the dynamics of the quantum DoF. On the other hand, 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. One way is to employ Ehrenfest's theorem and calculate the effective force on the classical trajectory through a mean potential that is averaged over the quantum DoF . The quantum nature of the nuclear wavefunction is represented by a swarm of trajectories evolving in time according to classical equations and initiated at phase space configurations sampled from a quasiclassical distribution (mainly a Wigner distribution).
Beyond such quasi-classical methods, the semiclassical WKB-type approach has a long tradition in adding part of the missing quantum effects to the classical simulations. Semiclassical methods  take into account the phase exp(iS(t)/hbar) evaluated along a classical trajectory and are therefore capable - at least in principle - of describing quantum nuclear effects including tunneling, interference effects, and zero-point energies. The most common semiclassical methods have been reviewed in recent articles [13-15]. To treat non-adiabatic transitions beyond the simplest Landau-Zener approximation, the "connection approach" can be used which was proposed independently by Landau, Zener, and Stueckelberg  and has later been adopted and generalized by many authors . In this formulation, non-adiabatic transitions of classical trajectories are described in terms of a connection formula of the semiclassical WKB wave functions associated with two or more coupled electronic states.
The intuitively appealing picture of trajectories hopping between coupled potential-energy surfaces gave rise to a number of quasiclassical implementations [18,19]. The most well-known method is Tully's "fewest switches" surface hopping method , which has evolved into a widely used and successful technique. In recent years, the term "surface hopping" and its underlying ideas have also been used in the stochastic modeling of a given deterministic differential equation, for example, the quantum-classical Liouville equation (see below).
The quantum-classical Liouville approach  constructs a well-defined quantum-classical approximation based upon the classical Liouville limit (i.e., the limit by which the Liouville equation is obtained from the Lie bracket for the quantum phase space distribution). This method has the great advantage of preserving quantum coherence in the quantum-classical limit. More approximate schemes like surface hopping can be derived as limiting cases where the rapid loss of coherence is assumed.
Finally, using the Bohmian (or hydrodynamical) interpretation of quantum mechanics it is possible to derive formally exact equations of motion for quantum trajectories (or fluid elements). At the end of the 90ies, there was a strong hope to turn this approach into efficient computational methods for the description of nuclear quantum dynamics [1,21]. While this has not been entirely borne out in the meantime, the concept remains without any doubt interesting.
Given this enormous variety of developments, the aim of the workshop is to assess the use of these methods in view of applications to large systems, including, in particular excited-state dynamics situations and on-the-fly calculations, as well as the generalization to electronic and combined electronic-nuclear dynamics.
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