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.
