The Effect of the Capping Agents of Nanoparticles on Their Redox Potential.

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 2- and 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 4-mercaptobenzoic 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.

Stochastic density functional theory combined with Langevin dynamics for warm dense matter.

This study overviews and extends a recently developed stochastic finite-temperature Kohn-Sham density functional theory to study warm dense matter using Langevin dynamics, specifically under periodic boundary conditions. The method's algorithmic complexity exhibits nearly linear scaling with system size and is inversely proportional to the temperature. Additionally, a linear-scaling stochastic approach is introduced to assess the Kubo-Greenwood conductivity, demonstrating exceptional stability for dc conductivity. Utilizing the developed tools, we investigate the equation of state, radial distribution, and electronic conductivity of hydrogen at a temperature of 30 000 K. As for the radial distribution functions, we reveal a transition of hydrogen from gaslike to liquidlike behavior as its density exceeds 4g/cm^{3}. As for the electronic conductivity as a function of the density, we identified a remarkable isosbestic point at frequencies around 7 eV, which may be an additional signature of a gas-liquid transition in hydrogen at 30 000 K.

Revisiting the benzene excimer using [2,2] paracyclophane model system: Experiment and theory.

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 excimer formation. The character of the excimer state and the state responsible for the brightest transition are similar to those of the 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 the 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.

Weak second-order quantum state diffusion unraveling of the Lindblad master equation.

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-order 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 a 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.

Stochastic Real-Time Second-Order Green's Function Theory for Neutral Excitations in Molecules and Nanostructures.

We present a real-time second-order Green's function (GF) method for computing excited states in molecules and nanostructures, with a computational scaling of O(Ne3), where Ne is the number of electrons. The cubic scaling is achieved by adopting the stochastic resolution of the identity to decouple the 4-index electron repulsion integrals. To improve the time propagation and the spectral resolution, we adopt the dynamic mode decomposition technique and assess the accuracy and efficiency of the combined approach for a chain of hydrogen dimer molecules of different lengths. We find that the stochastic implementation accurately reproduces the deterministic results for the electronic dynamics and excitation energies. Furthermore, we provide a detailed analysis of the statistical errors, bias, and long-time extrapolation. Overall, the approach offers an efficient route to investigate excited states in extended systems with open or closed boundary conditions.

Structure optimization with stochastic density functional theory.

Linear-scaling techniques for Kohn-Sham density functional theory are essential to describe the ground state properties of extended systems. Still, these techniques often rely on the localization of the density matrix or accurate embedding approaches, limiting their applicability. In contrast, stochastic density functional theory (sDFT) achieves linear- and sub-linear scaling by statistically sampling the ground state density without relying on embedding or imposing localization. In return, ground state observables, such as the forces on the nuclei, fluctuate in sDFT, making optimizing the nuclear structure a highly non-trivial problem. In this work, we combine the most recent noise-reduction schemes for sDFT with stochastic optimization algorithms to perform structure optimization within sDFT. We compare the performance of the stochastic gradient descent approach and its variations (stochastic gradient descent with momentum) with stochastic optimization techniques that rely on the Hessian, such as the stochastic Broyden-Fletcher-Goldfarb-Shanno algorithm. We further provide a detailed assessment of the computational efficiency and its dependence on the optimization parameters of each method for determining the ground state structure of bulk silicon with varying supercell dimensions.

High-frequency limit of spectroscopy.

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 ω0 -1, 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.

An "inverse" harpoon mechanism.

Electron-transfer reactions are ubiquitous in chemistry and biology. The electrons' quantum nature allows their transfer across long distances. For example, in the well-known harpoon mechanism, electron transfer results in Coulombic attraction between initially neutral reactants, leading to a marked 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. Three-dimensional coincidence imaging allows to identify such "inverse" harpoon products, predicted by nonadiabatic molecular dynamics simulations to occur between H2 and HCOH2+ following double ionization of isolated methanol molecules. These dynamics are experimentally initiated by single-photon double ionization with ultrafast extreme ultraviolet pulses, produced by high-order harmonic generation. A 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.

Time-Dependent Second-Order Green's Function Theory for Neutral Excitations.

We develop a time-dependent second-order Green's function theory (GF2) for calculating neutral excited states in molecules. The equation of motion for the lesser Green's function (GF) is derived within the adiabatic approximation to the Kadanoff-Baym (KB) equation, using the second-order Born approximation for the self-energy. In the linear response regime, we recast the time-dependent KB equation into a Bethe-Salpeter-like equation (GF2-BSE), with a kernel approximated by the second-order Coulomb self-energy. We then apply our GF2-BSE to a set of molecules and atoms and find that GF2-BSE is superior to configuration interaction with singles (CIS) and/or time-dependent Hartree-Fock (TDHF), particularly for charge-transfer excitations, and is comparable to CIS with perturbative doubles (CIS(D)) in most cases.

Sequential and concerted C-C and C-O bond dissociation in the Coulomb explosion of 2-propanol.

We study the competing mechanisms involved in the Coulomb explosion of 2-propanol CH3 2CHOH2+ dication, formed by an ultrafast extreme ultraviolet pulse. Over 20 product channels are identified and characterized using 3D coincidence imaging of the ionic fragments. The momentum correlations in the three-body fragmentation channels provide evidence for a dominant sequential mechanism, starting with the cleavage of a C-C bond, ejecting CH3 + and CH3CHOH+ cations, followed by a secondary fragmentation of the hydroxyethyl cation that can be delayed for up to a microsecond after ionization. The C-O bond dissociation channels are less frequent, involving proton transfer and double proton transfer, forming H2O+ and H3O+ products, respectively, and exhibiting mixed sequential and concerted character. These results can be explained by the high potential barrier for the C-O bond dissociation seen in our ab initio quantum chemical calculations. We also observe coincident COH+ + C2Hn + ions, suggesting exotic structural rearrangements, starting from the Frank-Condon geometry of the neutral 2-propanol system. Remarkably, the relative yield of the H3 + product is suppressed compared with methanol and alkene dications. Ab initio potentials and ground state molecular dynamics simulations show that a rapid and direct C-C bond cleavage dominates the Coulomb explosion process, leaving no time for H2 roaming, which is a necessary precursor to the H3 + formation.

Linear Weak Scalability of Density Functional Theory Calculations without Imposing Electron Localization.

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 atom-centered 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.

Forces from Stochastic Density Functional Theory under Nonorthogonal Atom-Centered Basis Sets.

We develop a formalism for calculating forces on the nuclei within the linear-scaling stochastic density functional theory (sDFT) in a nonorthogonal atom-centered 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 (Trp-zip2) 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.

Stochastic Vector Techniques in Ground-State Electronic Structure.

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 many-body 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 effective techniques to reduce statistical errors. Finally, we review the use of stochastic vectors for calculating correlation energies within the second-order Møller-Plesset perturbation theory and its finite temperature variational form. Example calculation results are presented and used to demonstrate the efficacy of the methods.

Tempering stochastic density functional theory.

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 in the statistical fluctuations and systematic deviation compared to sDFT for the same computational effort. We demonstrate the method's performance on large hydrogen-passivated silicon nanocrystals, finding a reduction in the systematic deviation in the energy by more than an order of magnitude, while the systematic deviation in the forces is also quenched. Similarly, the statistical fluctuations are reduced by factors of ≈4-5 for the total energy and ≈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.

Stochastic density functional theory: Real- and energy-space fragmentation for noise reduction.

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.

Stochastically Realized Observables for Excitonic Molecular Aggregates.

We show that a stochastic approach enables calculations of the optical properties of large 2-dimensional and nanotubular excitonic molecular aggregates. Previous studies of such systems relied on numerically diagonalizing the dense and disordered Frenkel Hamiltonian, which scales approximately as O(N3) for N dye molecules. Our approach scales much more efficiently as O(Nlog(N)), enabling quick study of systems with a million of coupled molecules on the micrometer size scale. We calculate several important experimental observables, including the optical absorption spectrum and density of states, and develop a stochastic formalism for the participation ratio. Quantitative agreement with traditional matrix diagonalization methods is demonstrated for both small- and intermediate-size systems. The stochastic methodology enables the study of the effects of spatial-correlation in site energies on the optical signatures of large 2D aggregates. Our results demonstrate that stochastic methods present a path forward for screening structural parameters and validating experiments and theoretical predictions in large excitonic aggregates.

Absence of Triplets in Single-Photon Double Ionization of Methanol.

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 >3 order of magnitude enhancement of the singlet:triplet ratio compared with their respective 1:3 multiplicities.

Range-separated stochastic resolution of identity: Formulation and application to second-order Green's function theory.

We develop a range-separated stochastic resolution of identity (RS-SRI) approach for the four-index electron repulsion integrals, where the larger terms (above a predefined threshold) are treated using a deterministic RI and the remaining terms are treated using a SRI. The approach is implemented within a second-order Green's function formalism with an improved O(N3) scaling with the size of the basis set, N. Moreover, the RS approach greatly reduces the statistical error compared to the full stochastic version [T. Y. Takeshita et al., J. Chem. Phys. 151, 044114 (2019)], resulting in computational speedups of ground and excited state energies of nearly two orders of magnitude, as demonstrated for hydrogen dimer chains and water clusters.

Efficient Langevin dynamics for "noisy" forces.

Efficient Boltzmann-sampling using first-principles methods is challenging for extended systems due to the steep scaling of electronic structure methods with the system size. Stochastic approaches provide a gentler system-size dependency at the cost of introducing "noisy" forces, which could limit the efficiency of the sampling. When the forces are deterministic, the first-order Langevin dynamics (FOLD) offers efficient sampling by combining a well-chosen preconditioning matrix S with a time-step-bias-mitigating propagator [G. Mazzola and S. Sorella, Phys. Rev. Lett. 118, 015703 (2017)]. However, when forces are noisy, S is set equal to the force-covariance matrix, a procedure that severely limits the efficiency and the stability of the sampling. Here, we develop a new, general, optimal, and stable sampling approach for FOLD under noisy forces. We apply it for silicon nanocrystals treated with stochastic density functional theory and show efficiency improvements by an order-of-magnitude.

Linear-Response Time-Dependent Density Functional Theory with Stochastic Range-Separated Hybrids.

Generalized Kohn-Sham density functional theory is a popular computational tool for the ground state of extended systems, particularly within range-separated hybrid (RSH) functionals that capture the long-range electronic interaction. Unfortunately, the heavy computational cost of the nonlocal exchange operator in RSH-DFT usually confines the approach to systems with at most a few hundred electrons. A significant reduction in the computational cost is achieved by representing the density matrix with stochastic orbitals and a stochastic decomposition of the Coulomb convolution (J. Phys. Chem. A 2016, 120, 3071). Here, we extend the stochastic RSH approach to excited states within the framework of linear-response generalized Kohn-Sham time-dependent density functional theory (GKS-TDDFT) based on the plane-wave basis. As a validation of the stochastic GKS-TDDFT method, the excitation energies of small molecules N2 and CO are calculated and compared to the deterministic results. The computational efficiency of the stochastic method is demonstrated with a two-dimensional MoS2 sheet (∼1500 electrons), whose excitation energy, exciton charge density, and (excited state) geometric relaxation are determined in the absence and presence of a point defect.