## Periodic Reporting for period 3 - MOLEQULE (Unraveling molecular quantum dynamics with accelerated ab initio algorithms)

Reporting period: 2019-09-01 to 2021-02-28

Many physical and chemical processes in nature as well as an increasing number of man-made devices exploit the quantum properties of electrons, nuclei, and the quantum signatures of the coupling between nuclear and electronic motions. To optimize the design of novel devices and to correctly interpret physical processes studied, e.g. by experiments probing the molecular dynamics induced by interactions with ultrafast laser pulses, quantitative simulations are required. The first goal of this project is to combine accurate ab initio electronic structure calculations with accurate quantum or semiclassical treatment of the nuclear dynamics. Although ninety years have passed since the discovery of the Schrödinger equation, these simulations remain extremely difficult for systems with more than a few degrees of freedom. Since the exact solution of the time-dependent Schrödinger equation scales exponentially with the number of atoms, accelerating computers even by orders of magnitude will not break the exponential barrier to simulating molecular quantum dynamics. The second goal of this project is, therefore, developing and implementing both exact and approximate computationally efficient quantum dynamics methods applicable to polyatomic molecules. The last goal of the project is developing systematic methods for interpreting spectra of complex systems in terms of the underlying nuclear and electronic dynamics. To summarize in simple terms, the overall objective is developing theoretical methods that will allow replacing the popular classical molecular dynamics movies by their quantum analogues.

"Since the beginning of the project, we have developed several efficient methods that combine accurate ab initio quantum-mechanical treatment of electrons with the semiclassical treatment of nuclei. These methods enabled us to evaluate vibrationally resolved electronic absorption and emission spectra of various molecules, including the electronically ""forbidden"" spectrum of benzene, which would be simply zero if computed using the standard Condon approximation and which would contain completely wrong peaks if evaluated with standard harmonic models (see Figure). The method approximates the nuclear wavefunction by a many-dimensional Gaussian multiplied with a polynomial and, therefore, makes a simple interpretation and visualization possible (see Figure). We have extended this method in two ways: First, we generalized it in order to compute so-called pump-probe spectra, which allow the study of the time dependence of molecular spectra with an extraordinary time resolution of several femtoseconds, i.e. 10^(-15) s, and used it to compute time-resolved stimulated emission spectrum of the phenyl radical. Second, we accelerated the method by reducing the number of evaluations of the curvature of the potential energy surface on which the molecule is propagated, and thus reduced the cost of this semiclassical method to the cost of now almost standard ab initio classical molecular dynamics methods, in which the nuclear quantum effects are ignored completely.

In addition, we have developed several highly accurate [with relative errors below 10^(-10), e.g.] quantum methods for the coupled motion of nuclei and electrons. These so-called geometric integrators do not rely on a semiclassical approximation, and, as their name suggests, preserve exactly [i.e. within machine accuracy, of about 10^(-14)] various geometric properties of the exact quantum evolution. This distinguishes them from standard numerical methods for solving the time-dependent Schrödinger equation, which may not preserve the energy, norm, or time reversibility of the exact solution. These geometric quantum integrators will provide a benchmark for other, approximate methods, which will be developed in the second phase of the project.

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In addition, we have developed several highly accurate [with relative errors below 10^(-10), e.g.] quantum methods for the coupled motion of nuclei and electrons. These so-called geometric integrators do not rely on a semiclassical approximation, and, as their name suggests, preserve exactly [i.e. within machine accuracy, of about 10^(-14)] various geometric properties of the exact quantum evolution. This distinguishes them from standard numerical methods for solving the time-dependent Schrödinger equation, which may not preserve the energy, norm, or time reversibility of the exact solution. These geometric quantum integrators will provide a benchmark for other, approximate methods, which will be developed in the second phase of the project.

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The state-of-the art molecular simulations are either ab initio molecular dynamics calculations, in which the electrons are treated quantum mechanically, but the nuclei are purely classical, or quantum simulations of nuclear dynamics, but on potential energy surfaces of a simple, approximate form. In the first phase of the project, we have gone beyond these standards by treating the quantum effects of both electrons and nuclei simultaneously. In the ab initio semiclassical approaches, this allowed us to treat tens of nuclear degrees of freedom; as the name suggests, the nuclear quantum effects were included only approximately. In the second phase of the project, we will increase the accuracy of the semiclassical treatment of nuclei in several different ways.

As for quantum simulations of nuclear dynamics, the state-of-the-art algorithms are either very accurate but do not preserve the geometric properties, or preserve the geometric properties but at the cost of lower accuracy. Our geometric integrators preserve the geometric properties of the exact solution within machine accuracy, and, moreover, achieve high accuracy of the propagated wavefunction. For convergence errors below 10^(-10), we observed speedups by three orders of magnitude compared to standard second-order methods (split-operator algorithm and the Crank-Nicolson method). In the second phase of the project, we will increase the number of nuclear degrees of freedom accessible by these highly accurate geometric integrators.

As for quantum simulations of nuclear dynamics, the state-of-the-art algorithms are either very accurate but do not preserve the geometric properties, or preserve the geometric properties but at the cost of lower accuracy. Our geometric integrators preserve the geometric properties of the exact solution within machine accuracy, and, moreover, achieve high accuracy of the propagated wavefunction. For convergence errors below 10^(-10), we observed speedups by three orders of magnitude compared to standard second-order methods (split-operator algorithm and the Crank-Nicolson method). In the second phase of the project, we will increase the number of nuclear degrees of freedom accessible by these highly accurate geometric integrators.