From vertical excitations towards excited-state relaxation: a journey with quantum Monte Carlo

**Alice Cuzzocrea is a PhD student in the department Computational Chemical Physics. Supervisor is prof.dr. C. Filippi from the faculty of Science & Technology.**

With this dissertation, we contribute to the study of molecular excitations by assessing and expanding a specific electronic structure method: quantum Monte Carlo (QMC). The excited states of molecules play a central role in activating many natural processes such as human vision, and in the functioning of new technologies like solar panels. Their theoretical study is a very active field of research but, due to their complex nature, we still lack a standard procedure to analyze them.

On one side, in many photo-induced processes, the effects of the environment on the molecule become decisive and have to be taken into account, for example, by using mixed quantum/classical approaches.

On the other, outside the ground state equilibrium, the potential energy surfaces to be described is very complex, and especially when multiple states interact, the available quantum chemistry methods often fail.

In this thesis, we do not consider any effect of the environment and work on improving the quantum mechanical description of excited states by focusing on QMC methods, a class of techniques for solving the Schrödinger equation in a stochastic manner. Lately, they are attracting increasing interest in the electronic structure community thanks to their favorable scaling with the number of electrons and the natural ease in parallelization. In the context of excited states, they are affirming as a valid alternative to other methods, especially for those complex cases where the cheapest options (such as time-dependent density functional theory) fail in the description. Moreover, recent algorithmic advancements in the QMC community have made it possible to extend the description from small to medium and relatively large (about 100 non-hydrogen atoms) molecules and to compute accurate geometries in both ground and excited states. Encouraged by this work, in this dissertation, we further investigate the use of QMC for the study of molecular excited states. In particular, in the first half of the manuscript (Chapters 3 and 4), we try to build robust protocols to compute vertical excitations. In the second (Chapters 5 and 6), we explore and try to address the problems connected to using QMC methods to describe excited-state relaxation.

*Building robust strategies to study vertical excitations.*

The use of QMC methods in the analysis of vertical excitation is a quite recent and fastly developing field. We here contribute to it by investigating two crucial aspects: the construction of the starting trial wave functions and their optimization. For the optimization of the wave function, there are different quantities that one can minimize. In Chapter 3, we discuss which variational principle is more effective between variance and energy minimization in the context of excited states. In particular, we obtain accurate excitation energies for two prototypical molecules by minimizing the energy, but we encounter severe difficulties following the variance. By analyzing a simple model, we infer that the variance landscape has little or no barriers between its minima. For this reason, while minimizing the variance, the optimization leads to the,* a priori *unknown, global minimum of the variance, making it hard or impossible to target a specific state.

Using energy minimization, we then discuss in Chapter 3 and in more depth in Chapter 4 the possible ways of constructing the trial wave function when the aim is a balanced description of multiple states. We demonstrate how, by using a selected configuration interaction (sCI) scheme, we can build compact trial wave functions that, after being fully optimized in QMC, give vertical excitations energies in line with other accurate quantum chemistry methods. Differently than the scheme previously used (based on complete active space (CAS) calculations), with sCI, we have an automatic way of selecting important contributions to the wave function, removing any dangerous bias that could be introduced by the user's understanding of the problem. Moreover, with the sCI scheme, we obtain better accuracy with fewer parameters (less computationally demanding). Therefore, we can describe relatively large molecules (in Chapter 4 we go from 3 to 19 atoms, excluding hydrogens) that are not accessible by other accurate methods such us full CI (FCI) or approximate coupled cluster singles, doubles, and triples (CC3).

*Exploring the use of QMC methods for describing excited-state relaxations.*

Establishing systematic protocols for accurately calculating vertical excitations opens the way for the study of excited-state relaxations. At the moment, few electronic structure methods can achieve the task; with this work, we try to asses if we can use QMC for such studies by investigating its strengths and limitations. In Chapter 5, we focus on the consequences of having a statistical error (typical of any quantity estimated with an MC integration) on the forces while performing a molecular dynamics (MD) calculation. The error in the forces creates a random walk in the velocities resulting in an increase in the total energy in time. Of course, the error would go to zero in the limit of an infinitely long MC run (infinite points for the integral), and the MD total energy would be stable. However, such long MC runs are currently too expensive, especially if we want to perform picoseconds long MD runs (as is needed to follow excited state relaxation processes). For this reason, we develop new strategies to obtain as stable as possible total energies without increasing the computational cost. In particular, we constrain the molecule's center of mass motion, removing unphysical rotations and translations, and we develop an on-the-fly fit procedure that uses the information of the past MD forces to improve the estimate of the present one. In this way, we obtain relatively stable total energies by tuning few fit parameters. To further improve on it, we also attempt to develop a Langevin-like scheme to thermalize the excess noise without corrupting the dynamical path.

In Chapter 6, we examine two additional technical problems relative to QMC-driven MD calculations: the infinite estimator of the variance of the forces and the effects of the optimization of the QMC wave function at every MD step. The first is a known problem that we solve in a standard way by introducing a guiding wave function designed to remove the infinity. Regarding the second problem, we find that a partial optimization of the wave function results in a decrease of the total energy in time, so at each MD, it is crucial to perform a strict optimization (although expensive). Combining all these findings and strategies, we show how we converge towards performing stable MD simulations; moreover, quite encouragingly, our preliminary results in the excited states align with previous findings.

To summarize, this dissertation contributes to establishing and enhancing QMC methods for the study of photo-excitations. Parallel to this work, new efficient procedures for performing wave function optimization have been proposed, and they have successfully been used for computationally challenging cases such as the excited states with a strong double excitation character. Moreover, with the rise of machine learning methods, an increasing effort is being devoted to creating wave function structures based on neural networks, and into borrowing and integrating efficient machine learning optimization algorithms into QMC. Additionally, big-scale research projects, such as the European center of excellence TREX project, are improving the efficiency and user-friendliness of the main QMC codes, enabling more users to experiment with these methods. In this very active research picture, this work has the advantage of not only contributing to the creation of robust backbones for QMC calculations in excited states, but also of exploring the new territory of molecular dynamics simulation with QMC forces. This direction is quite exciting since it allows us to move from the static description of vertical excitations toward following actual photo-excitation processes. Of course, more work is still necessary, for example in understanding how to build efficient wave functions that are suitable for molecular dynamic simulations. Furthermore, creating a more effective optimizer would reduce the cost of the optimization (currently this being the bottleneck of QMC-driven MD simulations).

In summary, this dissertation helps setting up the pillars needed to extend the description to more complex scenarios. For example, together with the aforementioned advancements, a possibility is now to incorporate non-adiabatic effects, for example, by including hopping probabilities among energy surfaces.