Stochastic orbital techniques offer reduced computational scaling and memory requirements to describe ground and excited states at the cost of introducing controlled statistical errors. Such techniques often rely on two basic operations, stochastic trace estimation and stochastic resolution of identity, both of which lead to statistical errors that scale with the number of stochastic realizations (\$N\_\\textbackslashxi\\$) as \$\textbackslashsqrt\N\_\\textbackslashxi\ˆ\-1\\\$. Reducing the statistical errors without significantly increasing \$N\_\\textbackslashxi\\$ has been challenging and is central to the development of efficient and accurate stochastic algorithms. In this work, we build upon recent progress made to improve stochastic trace estimation based on the ubiquitous Hutchinson's algorithm and propose a two-step approach for the stochastic resolution of identity, in the spirit of the Hutch++ method. Our approach is based on employing a randomized low-rank approximation followed by a residual calculation, resulting in statistical errors that scale much better than \$\textbackslashsqrt\N\_\\textbackslashxi\ˆ\-1\\\$. We implement the approach within the second-order Born approximation for the self-energy in the computation of neutral excitations and discuss three different low-rank approximations for the two-body Coulomb integrals. Tests on a series of hydrogen dimer chains with varying lengths demonstrate that the Hutch++-like approximations are computationally more efficient than both deterministic and purely stochastic (Hutchinson) approaches for low error thresholds and intermediate system sizes. Notably, for arbitrarily large systems, the Hutchinson-like approximation outperforms both deterministic and Hutch++-like methods.

The evolution of open quantum systems is a fundamental topic in various scientific fields. During time propagation, the environment occasionally makes measurements, forcing the system's wave function to collapse randomly. The von Neumann density matrix incorporates the statistics involved in these random processes, and its time development is often described by Markovian quantum master equations that incorporate a dissipator. For large systems, the complexity of the dissipator grows with the increasing number of possible measurements, posing conceptual and severe computational challenges. This article introduces a stochastic representation of the dissipator, using bundled measurement operators to address this complexity. Taking the Morse oscillator as an example, we demonstrate that small samples of bundled operators capture the system's dynamics. This stochastic bundling is different from the stochastic unraveling and the jump operator formalism and offers a new way of understanding quantum dissipation and decoherence.

Engineered metallic nanoparticles, which are found in numerous applications, are usually stabilized by organic ligands influencing their interfacial properties. We found that the ligands affect tremendously the electrochemical peak oxidation potentials of the nanoparticles. In this work, identical gold nanoparticles were ligand-exchanged and carefully analyzed to enable a precise and highly reproducible comparison. The peak potential difference between gold nanoparticles stabilized by various ligands, such as 2and 4-mercaptobenzoic acid, can be as high as 71 mV, which is substantial in energetic terms. A detailed study supported by density functional theory (DFT) calculations aimed to determine the source of this interesting effect. The DFT simulations of the ligand adsorption modes on Au surfaces were used to calculate the redox potentials through the thermodynamic cycle method. The DFT results of the peak potential shift were in good agreement with the experimental results for a few ligands, but showed some discrepancy, which was attributed to kinetic effects. The kinetic rate constant of the oxidation of Au nanoparticles stabilized by 4mercaptobenzoic acid was found to be twice as large as that of the Au nanoparticles stabilized by citrate, as calculated from Laviron’s theory and the Tafel equation. Finally, these findings could be applied to some novel applications such as determining the distribution of nanoparticle population in a dispersion as well as monitoring the ligand exchange between nanoparticles.

We report high-level calculations of the excited states of [2,2]-paracyclophane (PCP), which was recently investigated experimentally by ultrafast pump-probe experiments on oriented single crystals [Haggag et al., ChemPhotoChem 6 e202200181 (2022)]. PCP, in which the orientation of the two benzene rings and their range of motion are constrained, serves as a model for studying benzene exciplex formation. The character of the excimer state and the state responsible for the brightest transition are similar to those in benzene dimer. The constrained structure of PCP allows one to focus on the most important degree of freedom, the inter-ring distance. The calculations explain the main features of the transient absorption spectral evolution. This brightest transition of the excimer is polarized along the inter-fragment axis. The absorption of light polarized in the plane of the rings reveals the presence of other absorbing states of Rydberg character, with much weaker intensities. We also report new transient absorption data obtained by a broadband 8 fs pump, which time-resolve strong modulations of the excimer absorption. The combination of theory and experiment provides a detailed picture of the evolution of the electronic structure of the PCP excimer in the course of a single molecular vibration.

Abstract Simulating mixed-state evolution in open quantum systems is crucial for various chemical physics, quantum optics, and computer science applications. These simulations typically follow the Lindblad master equation dynamics. An alternative approach known as quantum state diffusion unraveling is based on the trajectories of pure states generated by random wave functions, which evolve according to a nonlinear Itô-Schrödinger equation (ISE). This study introduces weak first- and second-order solvers for the ISE based on directly applying the Itô-Taylor expansion with exact derivatives in the interaction picture. We tested the method on free and driven Morse oscillators coupled to a thermal environment and found that both orders allowed practical estimation with a few dozen iterations. The variance was relatively small compared to the linear unraveling and did not grow with time. The second-order solver delivers much higher accuracy and stability with bigger time steps than the first-order scheme, with a small additional workload. However, the second-order algorithm has quadratic complexity with the number of Lindblad operators as opposed to the linear complexity of the first-order algorithm.

Electron-transfer reactions are ubiquitous in chemistry and biology. The electrons quantum nature allows its transfer across long distances. In the well-known harpoon mechanism, electron-transfer results in Coulombic attraction between initially neutral reactants that leads to dramatic increase in the reaction rate. Here we present a different mechanism, in which electron-transfer from a neutral reactant to a multiply charged cation results in strong repulsion that encodes the electron-transfer distance in the kinetic energy release. 3D coincidence-imaging allows to identify such “inverse” harpoon products, predicted by non adiabatic molecular dynamics simulations to occur between H2 and HCOH2+ following double-ionization of isolated methanol molecules. Detailed comparison of measured and simulated data indicates that while the relative probability of long-range electron-transfer events is correctly predicted, theory overestimates the electron-transfer distance.

Linear scaling density functional theory (DFT) approaches to the electronic structure of materials are often based on the tendency of electrons to localize in large atomic and molecular systems. However, in many cases of actual interest, such as semiconductor nanocrystals, system sizes can reach a substantial extension before significant electron localization sets in, causing a considerable deviation from linear scaling. Herein, we address this class of systems by developing a massively parallel DFT approach which does not rely on electron localization and is formally quadratic scaling yet enables highly efficient linear wall-time complexity in the weak scalability regime. The method extends from the stochastic DFT approach described in Fabian et al. (WIRES: Comp. Mol. Sci. 2019, e1412) but is entirely deterministic. It uses standard quantum chemical atomcentered Gaussian basis sets to represent the electronic wave functions combined with Cartesian real-space grids for some operators and enables a fast solver for the Poisson equation. Our main conclusion is that when a processor-abundant high-performance computing (HPC) infrastructure is available, this type of approach has the potential to allow the study of large systems in regimes where quantum confinement or electron delocalization prevents linear scaling.

We review a suite of stochastic vector computational approaches for studying the electronic structure of extended condensed matter systems. These techniques help reduce algorithmic complexity, facilitate efficient parallelization, simplify computational tasks, accelerate calculations, and diminish memory requirements. While their scope is vast, we limit our study to ground-state and finite temperature density functional theory (DFT) and second-order perturbation theory. More advanced topics, such as quasiparticle (charge) and optical (neutral) excitations and higher-order processes, are covered elsewhere. We start by explaining how to use stochastic vectors in computations, characterizing the associated statistical errors. Next, we show how to estimate the electron density in DFT and discuss highly effective techniques to reduce statistical errors. Finally, we review the use of stochastic vector techniques for calculating correlation energies within the secondorder Møller-Plesset perturbation theory and its finite temperature variational form. Example calculation results are presented and used to demonstrate the efficacy of the methods.

We develop a formalism for calculating forces on the nuclei within the linear-scaling stochastic density functional theory (sDFT) in a nonorthogonal atomcentered basis set representation (Fabian et al. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2019, 9, e1412, 10.1002/wcms.1412) and apply it to the Tryptophan Zipper 2 (Trpzip2) peptide solvated in water. We use an embedded-fragment approach to reduce the statistical errors (fluctuation and systematic bias), where the entire peptide is the main fragment and the remaining 425 water molecules are grouped into small fragments. We analyze the magnitude of the statistical errors in the forces and find that the systematic bias is of the order of 0.065 eV/Å (∼1.2 × 10−3Eh/a0) when 120 stochastic orbitals are used, independently of system size. This magnitude of bias is sufficiently small to ensure that the bond lengths estimated by stochastic DFT (within a Langevin molecular dynamics simulation) will deviate by less than 1% from those predicted by a deterministic calculation.

We consider an arbitrary quantum mechanical system, initially in its ground-state, exposed to a time-dependent electromagnetic pulse with a carrier frequency ω₀ and a slowly varying envelope of finite duration. By working out a solution to the time-dependent Schrödinger equation in the high-ω₀ limit, we find that, to the leading order in ω−10, a perfect self-cancellation of the system’s linear response occurs as the pulse switches off. Surprisingly, the system’s observables are, nonetheless, describable in terms of a combination of its linear density response function and nonlinear functions of the electric field. An analysis of a jellium slab and jellium sphere models reveals a very high surface sensitivity of the considered setup, producing a richer excitation spectrum than accessible within the conventional linear response regime. On this basis, we propose a new spectroscopic technique, which we provisionally name the Nonlinear High-Frequency Pulsed Spectroscopy (NLHFPS). Combining the advantages of the extraordinary surface sensitivity, the absence of constraints by the traditional dipole selection rules, and the clarity of theoretical interpretation utilizing the linear response time-dependent density functional theory, NLHFPS has a potential to evolve into a powerful characterization method for nanoscience and nanotechnology

We introduce a tempering approach with stochastic density functional theory (sDFT), labeled t-sDFT, which reduces the statistical errors in the estimates of observable expectation values. This is achieved by rewriting the electronic density as a sum of a "warm" component complemented by "colder" correction(s). Since the "warm" component is larger in magnitude but faster to evaluate, we use many more stochastic orbitals for its evaluation than for the smaller-sized colder correction(s). This results in a significant reduction of the statistical fluctuations and the bias compared to sDFT for the same computational effort. We the method's performance on large hydrogen-passivated silicon nanocrystals (NCs), finding a reduction in the systematic error in the energy by more than an order of magnitude, while the systematic errors in the forces are also quenched. Similarly, the statistical fluctuations are reduced by factors of around 4-5 for the total energy and around 1.5-2 for the forces on the atoms. Since the embedding in t-sDFT is fully stochastic, it is possible to combine t-sDFT with other variants of sDFT such as energy-window sDFT and embedded-fragmented sDFT.

The trihydrogen ion has a central role in creating complex molecules in the interstellar medium. Therefore, its formation and destruction mechanisms in high photon energy environments involving organic molecules are drawing significant experimental and theoretical attention. Here, we employ a combination of time-resolved ultrafast extreme-ultraviolet pump and near-infrared probe spectroscopy applied to the deuterated CH3OD methanol molecule. Similar to other double-ionization studies, the isotopic labeling reveals two competing pathways for forming trihydrogen: A) H+3 + COH+ and B) H+3 + HCO+. We validate our high-level ab initio nonadiabatic molecular dynamic simulations by showing that it closely reproduces the essential features of the measured kinetic energy release distribution and branching ratios of the two pathways of the deuterated system. The success of ab initio simulation in describing single photon double-ionization allows for an unprecedented peek into the formation pathways for the undeuterated species, beyond present experimental reach. For this case, we find that the kinetic energy release of pathway B shifts to lower energies by more than 0.6 eV due to a dynamical isotope effect. We also determine the mechanism for trihydrogen formation from excited states of the dication and elucidate the isotope effect’s role in the observed dynamics.

Stochastic density functional theory (sDFT) is becoming a valuable tool for studying ground-state properties of extended materials. The computational complexity of describing the Kohn–Sham orbitals is replaced by introducing a set of random (stochastic) orbitals leading to linear and often sub-linear scaling of certain ground-state observables at the account of introducing a statistical error. Schemes to reduce the noise are essential, for example, for determining the structure using the forces obtained from sDFT. Recently, we have introduced two embedding schemes to mitigate the statistical fluctuations in the electron density and resultant forces on the nuclei. Both techniques were based on fragmenting the system either in real space or slicing the occupied space into energy windows, allowing for a significant reduction in the statistical fluctuations. For chemical accuracy, further reduction of the noise is required, which could√be achieved by increasing the number of stochastic orbitals. However, the convergence is relatively slow as the statistical error scales as 1/ Nχ according to the central limit theorem, where Nχ is the number of random orbitals. In this paper, we combined the embedding schemes mentioned above and introduced a new approach that builds on overlapped fragments and energy windows. The new approach significantly lowers the noise for ground-state properties, such as the electron density, total energy, and forces on the nuclei, as demonstrated for a G-center in bulk silicon.

Despite the abundance of data concerning single-photon double ionization of methanol, the spin state of the emitted electron pair has never been determined. Here we present the first evidence that identifies the emitted electron pair spin as overwhelmingly singlet when the dication forms in low-energy configurations. The experimental data show that while the yield of the CH2O+ + H3+ Coulomb explosion channel is abundant, the metastable methanol dication is largely absent. According to high-level ab initio simulations, these facts indicate that photoionization promptly forms singlet dication states, where they quickly decompose through various channels, with significant H3+ yields on the low-lying states. In contrast, if we assume that the initial dication is formed in one of the low-lying triplet states, the ab initio simulations exhibit a metastable dication, contradicting the experimental findings. Comparing the average simulated branching ratios with the experimental data suggests a \textgreater3 order of magnitude enhancement of the singlet:triplet ratio compared with their respective 1:3 multiplicities.