Researchers from Nanyang Technological University in Singapore presented results from simulations achieved with ArrayFire in the Faraday Discussions journal of The Royal Society of Chemistry. The simulations model the effects of a dissipative environment on the ultrafast vibronic couplings at conical intersections.

In this post, we first define these terms to gain understanding. Subsequently, we provide a summary of this research and the utility provided by ArrayFire in the simulation framework.

## Defining Terms

**Dissipative Environment** (ref1, ref2)

A dissipative system is a thermodynamically open system that is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter. A tornado may be thought of as a dissipative system. Dissipative systems stand in contrast to conservative systems.

In physics, an open quantum system is a quantum-mechanical system that interacts with an external quantum system, which is known as the environment or a bath. In general, these interactions significantly change the dynamics of the system and result in quantum dissipation, such that the information contained in the system is lost to its environment. Because no quantum system is completely isolated from its surroundings, it is important to develop a theoretical framework for treating these interactions in order to obtain an accurate understanding of quantum systems.

**Nonadiabatic Coupling** (ref)

Vibronic coupling (also called nonadiabatic coupling or derivative coupling) in a molecule involves the interaction between electronic and nuclear vibrational motion. The term “vibronic” originates from the combination of the terms “vibrational” and “electronic”, denoting the idea that in a molecule, vibrational, and electronic interactions are interrelated and influence each other. The magnitude of vibronic coupling reflects the degree of such interrelation.

In theoretical chemistry, the vibronic coupling is neglected within the Born–Oppenheimer approximation. Vibronic couplings are crucial to the understanding of nonadiabatic processes, especially near points of conical intersections. The direct calculation of vibronic couplings is not common due to difficulties associated with its evaluation.

**Conical Intersections** (ref)

In quantum chemistry, a conical intersection of two or more potential energy surfaces is the set of molecular geometry points where the potential energy surfaces are degenerate (intersect) and the non-adiabatic couplings between these states are non-vanishing. In the vicinity of conical intersections, the Born–Oppenheimer approximation breaks down and the coupling between electronic and nuclear motion becomes important, allowing non-adiabatic processes to take place. The location and characterization of conical intersections are therefore essential to the understanding of a wide range of important phenomena governed by non-adiabatic events, such as photoisomerization, photosynthesis, vision, and the photostability of DNA.

Conical intersections are also called molecular funnels or diabolic points as they have become an established paradigm for understanding reaction mechanisms in photochemistry as important as transitions states in thermal chemistry. This comes from the very important role they play in non-radiative de-excitation transitions from excited electronic states to the ground electronic state of molecules. For example, the stability of DNA with respect to UV irradiation is due to such conical intersection. The molecular wave packet excited to some electronically excited state by the UV photon follows the slope of the potential energy surface and reaches the conical intersection from above. At this point, the very large vibronic coupling induces a non-radiative transition (surface-hopping) which leads the molecule back to its electronic ground state. The singularity of vibronic coupling at conical intersections is responsible for the existence of the Geometric phase, which was discovered by Longuet-Higgins in this context.

## Research Summary

**Simulations with the Hierarchy Equations of Motion Method**

The effect of a dissipative environment on the ultrafast nonadiabatic dynamics at conical intersections is analyzed for a two-state two-mode model chosen to represent the conical intersection in pyrazine (the system) which is bilinearly coupled to infinitely many harmonic oscillators in thermal equilibrium (the bath). The system-bath coupling is modeled by the Drude spectral function. The equation of motion for the reduced density matrix of the system is solved numerically exactly with the hierarchy equation of motion method using graphics-processor-unit (GPU) technology. The simulations are valid for arbitrary strength of the system-bath coupling and arbitrary bath memory relaxation time. The present computational studies overcome the limitations of weak system-bath coupling and short memory relaxation time inherent in previous simulations based on multi-level Redfield theory.

**Using ArrayFire to Make a Simulation Tractable**

The numerical evaluation requires the propagation of a large number of coupled equations of motion for auxiliary matrices of large dimensions, which is computationally demanding. The researchers overcame the computational challenge by using a GPU (Graphics Processing Unit) implementation of the ArrayFire package which provides an easy-to-use API (application programming interface) and an array-based function set that facilitates GPU programming. The numerical integration of the equations of motion for the auxiliary density matrices is performed on an NVIDIA Tesla M2050 GPU with 448 processors and 2GB ECC-protected onboard memory.

## Conclusion

With this efficient algorithm implemented on GPU hardware, the propagation of these equations of motion for up to 1000 fs on a single GPU is accomplished in a tractable amount of time, from several hours to 10 hours, depending on the truncation number N. Without ArrayFire and GPU computing, the simulation time was so long it was not achievable in a realistic amount of time and was not pursued.

Thanks to these researchers for publishing their excellent work!