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We develop a stochastic formulation of the optimally-tuned range-separated hybrid density functional theory which enables significant reduction of the computational effort and scaling of the non-local exchange operator at the price of introducing a controllable statistical error. Our method is based on stochastic representations of the Coulomb convolution integral and of the generalized Kohn-Sham density matrix. The computational cost of the approach is similar to that of usual Kohn-Sham density functional theory, yet it provides much more accurate description of the quasiparticle energies for the frontier orbitals. This is illustrated for a series of silicon nanocrystals up to sizes exceeding 3000 electrons. Comparison with the stochastic GW many-body perturbation technique indicates excellent agreement for the fundamental band gap energies, good agreement for the band-edge quasiparticle excitations, and very low statistical errors in the total energy for large systems. The present approach has a major advantage over one-shot GW by providing a self-consistent Hamiltonian which is central for additional post-processing, for example in the stochastic Bethe-Salpeter approach., 7 pages, 3 figures

Interference dips in electronic absorption spectra caused by coupling between excited states are calculated and interpreted using three methods: an analytical expression, an integratable expression, and wave packet propagation. All of the analyses give similar results. The dips (distantly related to Fano’s antiresonance) are caused by spin-orbit coupling between the states involved in spin-forbidden transitions with narrow bandwidths and the state involved in a spin-allowed transition with a large bandwidth caused by progressions in normal vibrational modes that are displaced between the excited and ground electronic states. The analytical expression involves the assumptions that only one vibrational eigenstate of the “forbidden” electronic state is involved and that the broad background is represented by a Lorentzian-type function. The integratable expression replaces the Lorentzian function by one that represents the band shape caused by a progression of unresolved vibronic peaks. The wave packet propagation method is exact for the model. Analytical expressions for multiple interfering states and integratable expressions for multiple electronic states and multiple vibrational modes are derived. The spectra of Ni(H2O)6 2+ and octahedral CrO6 9- units in ZrO2‚33%Y2O3 doped with chromium(III) are analyzed. Physical meanings and mathematical origins of the interference dips are interpreted.

Filter diagonalization (Wall, M. R.; Neuhauser, D. J. Chem. Phys. 1995, 102, 8011) is a new and efficient method for extracting frequencies and damping constants from a short-time segment of any time-dependent signal, whether of quantum nature or not. In this letter, we will demonstrate that filter diagonalization can be used to follow the anharmonic vibrational dynamics of Si6. Using the technique, we can locate (resolve) the normal modes (including the widths) from even a very short-time correlation signal generated by ab initio molecular dynamics calculations. Our results show that filter diagonalization can reduce significantly the necessary sampling time with an ab initio molecular simulation and aid in understanding normal-mode dynamics for systems that are affected by anharmonicity.