We develop a formalism to calculate the quasi-particle energy within the GW many-body perturbation correction to the density functional theory (DFT). The occupied and virtual orbitals of the Kohn-Sham (KS) Hamiltonian are replaced by stochastic orbitals used to evaluate the Green function, the polarization potential, and thereby the GW self-energy. The stochastic GW (sGW) relies on novel theoretical concepts such as stochastic time-dependent Hartree propagation, stochastic matrix compression and spatial/temporal stochastic decoupling techniques. Beyond the theoretical interest, the formalism enables linear scaling GW calculations breaking the theoretical scaling limit for GW as well as circumventing the need for energy cutoff approximations. We illustrate the method for silicon nanocrystals of varying sizes with Ne > 3000 electrons.The GW approximation [1, 2] to many-body perturbation theory (MBPT) [3] offers a reliable and accessible theory for quasi-particles (QPs) and their energies [2,[4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], enabling estimation of electronic excitations [19][20][21][22][23][24][25] quantum conductance [26][27][28][29][30] and level alignment in hybrid systems [31,32]. Practical use of GW for large systems is severely limited because of the steep CPU and memory requirements as system size increases. The most computationally intensive element in the GW method, the calculation of the polarization potential (screen Coulomb interaction), involves an algorithmic complexity that scales as the fourth power of the system size [33,34]. Various approaches have been developed to reduce the computational bottlenecks of the GW approach [8,18,23,[33][34][35][36][37]. Despite these advances, GW calculations are still quite expensive for many of the intended applications in the fields of materials science, surface science and nanoscience.In this letter we develop a stochastic, orbital-less, formalism for the GW theory, unique in that it does not reference occupied or virtual orbitals and orbital energies of the KS Hamiltonian. While the approach is inspired by recent developments in electronic structure theory using stochastic orbitals [38][39][40][41][42] it introduces three powerful and basic notions: Stochastic decoupling, stochastic matrix compression and stochastic time-dependent Hartree (sTDH) propagation. The result is a stochastic formulation of GW, where the QP energies become random variables sampled from a distribution with a mean equal to the exact GW energies and a statistical error proportional to the inverse number of stochastic orbitals (iterations, I sGW ).We illustrate the sGW formalism for silicon nanocrystals (NCs) with varying sizes and band gaps [43,44] and demonstrate that the CPU time and memory required by sGW scales nearly linearly with system size, thereby providing means to study QPs excitations in large systems of experimental and technological interest.In the reformulation of the GW approach, we treat the QP energy (ε QP = ω QP ) as a perturbative correction to th...

We report a global analysis of spin-orbit coupling in the mono-halocarbenes, CH(D)X, where X = Cl, Br, and I. These are model systems for examining carbene singlet-triplet energy gaps and spin-orbit coupling. Over the past decade, rich data sets collected using single vibronic level emission spectroscopy and stimulated emission pumping spectroscopy have yielded much information on the ground vibrational level structure and clearly demonstrated the presence of perturbations involving the low-lying triplet state. To model these interactions globally, we compare two approaches. First, we employ a diabatic treatment of the spin-orbit coupling, where the coupling matrix elements are written in terms of a purely electronic spin-orbit matrix element which is independent of nuclear coordinates, and an integral representing the overlap of the singlet and triplet vibrational wavefunctions. In this way, the structures, harmonic frequencies, and normal mode displacements from ab initio calculations were used to calculate the vibrational overlaps of the singlet and triplet state levels, including the full effects of Duschinsky mixing. These calculations have allowed many new assignments to be made, particularly for CHI, and provided spin-orbit coupling parameters and values for the singlet-triplet gaps. In a second approach, we have computed and fit full geometry dependent spin-orbit coupling surfaces and used them to compute matrix elements without the product form approximation. Those matrix elements were used in similar fits varying the anharmonic constants and singlet-triplet gap to reproduce the experimental levels. The derived spin-orbit parameters for carbenes CHX (X = Cl, Br, and I) show an excellent linear correlation with the atomic spin-orbit constant of the corresponding halogen, indicating that the spin-orbit coupling in the carbenes is consistently around 14% of the atomic value.

Building upon our recent studies of noncovalent interactions in chlorobenzene and bromobenzene clusters, in this work we focus on interactions of chlorobenzene (PhCl) with a prototypical N atom donor, ammonia (NH3). Thus, we have obtained electronic spectra of PhCl···(NH3)n (n = 1-3) complexes in the region of the PhCl monomer S0 -S1 (ππ*) transition using resonant 2-photon ionization (R2PI) methods combined with time-of-flight mass analysis. Consistent with previous studies, we find that upon ionization the PhCl···NH3 dimer cation radical reacts primarily via Cl atom loss. A second channel, HCl loss, is identified for the first time in R2PI studies of the 1:1 complex, and a third channel, H atom loss, is identified for the first time. While prior studies have assumed the dominance of a π-type complex, we find that the reactive complex corresponds instead to an in-plane σ-type complex. This is supported by electronic structure calculations using density functional theory and post-Hartree-Fock methods and Franck-Condon analysis. The reactive pathways in this system were extensively characterized computationally, and consistent with results from previous calculations, we find two nearly isoenergetic arenium ions (Wheland intermediates; denoted WH1, WH2), which lie energetically below the initially formed dimer cation radical complex. At the energy of our experiment, intermediate WH1, produced from ipso-addition, is not stable with respect to Cl or HCl loss, and the relative branching between these channels observed in our experiment is well reproduced by microcanonical transition state theory calculations based upon the calculated parameters. Intermediate WH2, where NH3 adds ortho to the halogen, decomposes over a large barrier via H atom loss to form protonated o-chloroaniline. This channel is not open at the (2-photon) energy of our experiments, and it is suggested that photodissociation of a long-lived (i.e., several ns) WH2 intermediate leads to the observed products.