The Quantum-Electrodynamical Time-Dependent Density Functional Theory (QED-TDDFT)equations are solved by time propagating the wave function on a tensor product of a Fock-space andreal-space grid. Applications for molecules in cavities show the accuracy of the approach. Examplesinclude the coupling strength and light frequency dependence of the energies, wave functions, op-tical absorption spectra, and Rabi splitting magnitudes in cavities, as well as a description of highharmonic generation in cavities.

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In this work, we present a stochastic variational calculation (SVM) of energies and wave functions of few particle systems coupled to quantum fields in cavity QED. The spatial wave function and the photon spaces are optimized by a random selection process. Using correlated basis functions, the SVM approach solves the problem accurately and opens the way to the same precision that is reached the non light coupled quantum systems.Examples for a two-dimensional trion and confined electrons as well as for the He atom and the H2 molecule are presented showing that the light-matter coupling drastically changes the electronic states.

Deformed explicitly correlated Gaussian (DECG) basis functions are introduced, and their matrix elements are calculated. All matrix elements can be calculated analytically in a closed form, except the Coulomb one, which has to be approximated by a Gaussian expansion. The DECG basis functions can be used to solve problems with nonspherical potentials. One example of such potential is the dipole self-interaction term in the Pauli–Fierz Hamiltonian. Examples are presented showing the accuracy and necessity of deformed Gaussian basis functions to accurately solve light–matter coupled systems in cavity QED.

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