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A time-dependent vibrational electronic coupled-cluster (VECC) approach is proposed to simulate photoelectron/ UV-VIS absorption spectra, as well as time-dependent properties for non-adiabatic vibronic models, going beyond the Born-Oppenheimer approximation. A detailed derivation of the equations of motion and a motivation of the ansatz are presented. The VECC method employs second-quantized bosonic construction operators and a mixed linear and exponential ansatz to form a compact representation of the time-dependent wave-function. Importantly, the method does not require a basis set, has only few user-defined inputs, and has a classical (polynomial) scaling with respect to the number of degrees of freedom (of the vibronic model), resulting in a favourable computational cost. In benchmark applications to small models and molecules the VECC method provides accurate results, compared to Multi-Configurational Time-dependent Hartree (MCTDH) calculations when predicting short-time dynamical properties (i.e. photo-elecron / UV-VIS absorption spectra) for non-adiabatic vibronic models. To illustrate the capabilities the VECC method is also applied successfully to a large vibronic model for hexahelicene with 14 electronic states and 63 normal modes, developed in the group by Santoro.

A normal ordered exponential parametrization is used to obtain equations for thermal one-and two-particle reduced density matrices, as well as free energies, partition functions and entropy for both Fermionic (electronic) and Bosonic (vibrational) Hamiltonians. A first principles derivation of the equations, relying only on a simple Wick's theorem and starting from the differential equation $\frac{d \hat{D}}{d \beta}= - (\hat{H}-\mu \hat{N})\hat{D}$, is presented that yields a differential equation for the amplitudes representing density cumulants, as well as the grand potential. In contrast to other approaches reported in the literature the theory does not use perturbation theory in the interaction picture and an integral formulation as a starting point, but rather requires a propagation of the resulting differential equation for the amplitudes. While the theory is applicable to general classes of many-body problems in principle, here, the theory is illustrated using simple model systems. For one-body Fermionic Hamiltonians, Fermi-Dirac one-body reduced density matrices are recovered for the grand-canonical formulation. For multidimensional harmonic oscillators numerically exact results are obtained using the thermal normal ordered exponential (TNOE) approach. As an application of the related time-dependent formulation numerically exact time-autocorrelation functions and absorption spectra are obtained for harmonic Franck Condon problems. These examples illustrate the basic soundness of the scheme and are used for pedagogical purposes. Other approaches in the literature are only discussed briefly and no detailed comparative discussion is attempted.