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Unraveling molecular quantum dynamics with accelerated ab initio algorithms

Periodic Reporting for period 4 - MOLEQULE (Unraveling molecular quantum dynamics with accelerated ab initio algorithms)

Reporting period: 2021-03-01 to 2022-02-28

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 was 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 was, therefore, developing and implementing both exact and approximate computationally efficient quantum dynamics methods applicable to polyatomic molecules. The last goal of the project was 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 was developing theoretical methods that would allow replacing the popular classical molecular dynamics movies by their quantum analogues.
Over the course of this project, we have developed a family of efficient methods that combine accurate ab initio quantum-mechanical treatment of electrons with the semiclassical treatment of nuclei. By including quantum effects on both nuclear and electronic motions efficiently, these methods enabled us to compute vibrationally resolved electronic absorption and emission spectra of various molecules, such as benzene, phenyl radical, azulene, and quinquethiophene. In the first half of the project we implemented methods that can describe anharmonicity, Duschinsky, and non-Condon effects on linear spectra. This is nicely demonstrated on 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). As an additional benefit, this 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).

In the second part of the project, we made many further improvements: First, we accelerated this semiclassical method by reducing its cost to the cost of now standard ab initio classical molecular dynamics methods, in which the nuclear quantum effects are, however, ignored completely. Second, we included finite-temperature effects by combining the method with so-called thermofield dynamics, which allowed us to capture even the so-called “hot bands” (the small side peaks in the Figure). Third, we generalized this semiclassical method in order to compute ultrafast time-resolved spectra, specifically the pump-probe and two-dimensional spectra, by which experimentalists study the time dependence of molecular spectra with an extraordinary time resolution of several femtoseconds, i.e. 10^(-15) s. Again, we were able to capture anharmonicity, Duschinsky, non-Condon, and finite-temperature effects on such spectra.

In addition to these efficient on-the-fly ab initio semiclassical methods, we have developed a family of 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. Specifically, we developed geometric integrators for nonadiabatic quantum dynamics in adiabatic, diabatic, and exact quasidiabatic representations, which allowed us to answer questions “Which form of the molecular Hamiltonian is the most suitable for simulating the nonadiabatic quantum dynamics at a conical intersection?” and “How important are the residual nonadiabatic couplings for an accurate simulation of nonadiabatic quantum dynamics in a quasidiabatic representation?” We have extended applicability of these accurate methods to higher-dimensional systems by employing an adaptive grid that moves with the molecular wavepacket and generalized them to treat the so-called nonlinear Schrödinger equation, in which the Hamiltonian depends on the state that is propagated. These methods can be used both for high-accuracy calculations of molecular quantum dynamics and as benchmarks for developing more efficient approximate methods in the future.
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 this 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.

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 for both linear and nonlinear time-dependent Schrödinger equation 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).
Ab initio semiclassical calculation of the electronically forbidden absorption spectrum of benzene.