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## Non-adiabatic quantum dynamics: From Theory to Experiments

#### Location: CECAM-HQ-EPFL, Lausanne, Switzerland

#### Organisers

The breakdown of the Born-Oppenheimer approximation and the non-adiabatic phenomena that ensue is the basis of significant research interest in both fundamental and applied fields. Importantly, the increasing complexity of the systems being studied necessitates a strong synergy between experiment and theory [1]. This promotes understanding of the respective approaches, their advantages and limitations and is essential in order to draw reliable conclusions from the many exciting new research in the field.

Over the past decade, there have been significant developments in experimental methodologies. Nonlinear spectroscopies, such as two-dimensional electronic spectroscopy (2D-ES) [2, 3], have been able to probe quantum coherences and the electronic couplings between different states. Femtosecond impulsive stimulated Raman [4] and Broadband impulsive vibrational spectroscopies [5] have made it possible to follow molecular vibrations during a non-equilibrium dynamical process. Tools such as femtosecond electron diffraction [6], X-ray Free Electron Lasers (X-FELs) [7–9] and High Harmonic Generation (HHG) [10, 11], have led to a paradigm shift in the capability of short wavelength techniques to deliver ultrafast dynamics with direct structural information.

From a theoretical perspective, the description of nonadiabatic phenomena calls for the treatment of coupled electron-nuclear dynamics, and not just nuclear quantum effects. This can be achieved by solving the time-dependent Schrödinger equation. Here the nuclear wavefunction is most commonly represented on a grid of points. However, even with the development of sophisticated algorithms, such as the multi-configurational time-dependent Hartree (MCTDH) approach, [12] these grid-based solutions are limited to a small number of nuclear degrees of freedom owing to the exponential scaling brought about by the direct product basis. While this problem can be reduced using the Multi-Layer variant of MCTDH [13] or by adopting approximations such as Gaussian-based MCTDH (G-MCTDH) [14], the very challenging problem of obtaining an accurate multi-dimensional potential remains. In addition, in each case this potential must be computed a priori limiting the dynamics to models or very few (<10) degrees of freedom.

To overcome this difficulty, many approximate methods based, to a greater or lesser extent, on trajectories have been developed. Among these trajectory surface hopping [15–17] remains the most widely used method. However, path integral based methods such as ring polymer molecular dynamics (RPMD) [18,19] have recently been extended to non-adiabatic systems using mean-field approximations and mapping variables [20, 21], which builds upon the early work of Meyer and Miller [22] and Stock and Thoss [23]. In addition, (partially) linearized and semiclassical methods have been developed for multiple electronic surfaces. These include partially linearized density matrix approaches [24], quantum master equation formalism [25, 26], quantum-classical Liouville dynamics [27–30], quantum-classical path integral methods [31–33] and the symmetrical quasi-classical (SQC) windowing model [34, 35].

Wavefunction approaches based upon Exact Factorisation [36, 37] and Gaussian basis functions (GBF) [38] are also receiving much attention. The former has provided new perspectives on established problems such as dynamics around a conical intersection [39], novel approaches to mixed quantum-classical dynamics [40,41] and the simulatation of experimental observables [42]. The latter GBF approaches arise from the work of Heller [43, 44] and include the delocalised nature of quantum mechanics, while conveniently maintaining the localised trajectory representation of classical mechanics. A number of sophisticated on-the-fly approaches based upon GBF have been developed. These include multiple spawning [45, 46], coupled-coherent states (CCS) [47], multi-configurational Ehrenfest (MCE) [48], the multiple cloning method [49] and the variational multi-configruational Gaussian (vMCG) method [50]. Alongside this, there is also much work addressing the problems with GBF, such as their non-orthogonality [51–53].

Despite considerable progress, there is still an unmet need for widely-applicable and computationally inexpensive methods [54], which is able to address fundamental questions as well as enhance the synergy between theory and experiment. Theoretical challenges include (i) developing methods with predictive power by avoiding uncontrolled approximations, (ii) identifying regimes of applicability for each method and (iii) developing implementations that allow for computationally efficient simulations of complex systems. Implementation of these methods will require the development of stable and inexpensive algorithms for time-propagation [55, 56]

To address this need we propose a workshop which will bring together a select number of experimentalists and theoreticians to actively discuss state-of-the-art research and to foster collaboration. There will be particular emphasis on ‘bridging the gap’ between theoretical calculation and experimental measurement. This workshop has the potential to increase interdisciplinary understanding, enhance research productivity and kickstart collaboration.

The field of molecular quantum dynamics is undergoing rapid development due to an increase in computational power and the emergence of new methodologies (see state of the art). This is offering exciting new opportunities to calculate the quantum properties of matter. At present a key challenge for the molecular quantum dynamics community is to coordinate the efforts between a number of subgroups, each developing their specific theoretical methods. Although many methods have the same foundation, they differ in their approximations and algorithmic implementations, bringing their own particular advantages and drawbacks. It is important that these groups come together to exchange ideas and advance research in this area. Concurrently, the rapid development in time-resolved spectroscopies across a broad range of wavelengths has increased the information content available from experiment, but also the complexity. Now more than ever, we must seek a strong synergy between theory and experiment. Indeed detailed theoretical studies are often essential to provide a firm link between the spectroscopic observables and the underlying molecular structure and dynamics. Developing such synergy is the objective of the present workshop which will address the following key questions:

• What are the methods’ advantages and limitations? For example, how do they scale with dimensionality/temperature/electronic states? How rapidly does the calculation converge bf for different system parameters?

• Do the methods provide a consistent description of dynamics and statistics?

• Which research areas are presently most poorly described by theory/experiment?

• How can experimentalists and theoreticians improve collaboration?

To achieve these objectives we propose a format where invited speakers (theoreticians and experimentalists) will be asked to present lectures on the state-of-the-art in their specific area of expertise. Each lecture will be followed by discussion. At the end of each day, we will hold a detailed discussion on the synergy between the different theories and experimental techniques. This will be used to identify shortfalls and propose new objectives. A small number of contributed talks will be scheduled, covering topics related to the invited lectures.

## References

**Canada**

Raymond Kapral (University of Toronto) - Organiser

**United Kingdom**

Thomas James Penfold (Newcastle University) - Organiser

**United States**

Nandini Ananth (Cornell University) - Organiser