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At least 1,567 records · Page 87
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Thermodynamic stability versus kinetic accessibility: Pareto fronts for programmable self-assembly

A challenge in designing self-assembling building blocks is to ensure the target state is both thermodynamically stable and kinetically accessible. These two objectives are known to be typically in competition, but it is not known how to simultaneously optimize them. We consider this problem through the lens of multi-objective optimization theory: we develop a genetic algorithm to compute the Pareto fronts characterizing the tradeoff between equilibrium probability and folding rate, for a model system of small polymers of colloids with tunable short-ranged interaction energies. We use a coarse-grained model for the particles' dynamics that allows us to efficiently search over parameters, for systems small enough to be enumerated. For most target states there is a tradeoff when the number of types of particles is small, with medium-weak bonds favouring fast folding, and strong bonds favouring high equilibrium probability. Additionally, the tradeoff disappears when the number of particle types reaches a value m *, that is usually much less than the total number of particles. This general approach of computing Pareto fronts allows one to identify the minimum number of design parameters to avoid a thermodynamic–kinetic tradeoff. However, we argue, by contrasting our coarse-grained model's predictions with those of Brownian dynamics simulations, that particles with short-ranged isotropic interactions should generically have a tradeoff, and avoiding it in larger systems will require orientation-dependent interactions.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Identifiability and characterization of transmon qutrits through Bayesian experimental design

Robust control of a quantum system is essential to utilize the current noisy quantum hardware to its full potential, such as quantum algorithms. To achieve such a goal, a systematic search for an optimal control for any given experiment is essential. The design of optimal control pulses requires accurate numerical models and, therefore, accurate characterization of the system parameters. We present an online Bayesian approach for quantum characterization of qutrit systems, which automatically and systematically identifies optimal experiments that provide maximum information on the system parameters, thereby greatly reducing the number of experiments that need to be performed on the quantum testbed. Unlike most characterization protocols that provide point-estimates of the parameters, the proposed approach is able to estimate their probability distribution. The applicability of the Bayesian experimental design technique was demonstrated on test problems, where each experiment was defined by a parameterized control pulse. In addition to this, we also present an approach for iterative pulse extension, which is robust under uncertainties in transition frequencies and coherence times, and shot noise, despite being initialized with wide uninformative priors. Furthermore, we provide a mathematical proof of the theoretical identifiability of the model parameters and present conditions on the quantum state under which the parameters are identifiable. The proof and conditions for identifiability are presented for both closed and open quantum systems using the Schrödinger equation and the Lindblad master equation, respectively.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Grand canonical electronic density-functional theory: Algorithms and applications to electrochemistry

First-principles calculations combining density-functional theory and continuum solvation models enable realistic theoretical modeling and design of electrochemical systems. When a reaction proceeds in such systems, the number of electrons in the portion of the system treated quantum mechanically changes continuously, with a balancing charge appearing in the continuum electrolyte. A grand-canonical ensemble of electrons at a chemical potential set by the electrode potential is therefore the ideal description of such systems that directly mimics the experimental condition. We present two distinct algorithms: a self-consistent field method and a direct variational free energy minimization method using auxiliary Hamiltonians (GC-AuxH), to solve the Kohn-Sham equations of electronic density-functional theory directly in the grand canonical ensemble at fixed potential. Both methods substantially improve performance compared to a sequence of conventional fixed-number calculations targeting the desired potential, with the GC-AuxH method additionally exhibiting reliable and smooth exponential convergence of the grand free energy. Lastly, we apply grand-canonical density-functional theory to the under-potential deposition of copper on platinum from chloride-containing electrolytes and show that chloride desorption, not partial copper monolayer formation, is responsible for the second voltammetric peak.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

Pushing memory bandwidth limitations through efficient implementations of Block-Krylov space solvers on GPUs

Lattice quantum chromodynamics simulations in nuclear physics have benefited from a tremendous number of algorithmic advances such as multigrid and eigenvector deflation. These improve the time to solution but do not alleviate the intrinsic memory-bandwidth constraints of the matrix-vector operation dominating iterative solvers. Batching this operation for multiple vectors and exploiting cache and register blocking can yield a super-linear speed up. Block-Krylov solvers can naturally take advantage of such batched matrix-vector operations, further reducing the iterations to solution by sharing the Krylov space between solves. However, practical implementations typically suffer from the quadratic scaling in the number of vector-vector operations. Here, using the QUDA library, we present an implementation of a block-CG solver on NVIDIA GPUs which reduces the memory-bandwidth complexity of vector-vector operations from quadratic to linear. We present results for the HISQ discretization, showing a 5x speedup compared to highly-optimized independent Krylov solves on NVIDIA's SaturnV cluster.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

MICCO: An Enhanced Multi-GPU Scheduling Framework for Many-Body Correlation Functions

Calculation of many-body correlation functions is one of the critical kernels utilized in many scientific computing areas, especially in Lattice Quantum Chromodynamics (Lattice QCD). It is formalized as a sum of a large number of contraction terms each of which can be represented by a graph consisting of vertices describing quarks inside a hadron node and edges designating quark propagations at specific time intervals. Due to its computation- and memory-intensive nature, real-world physics systems (e.g., multi-meson or multi-baryon systems) explored by Lattice QCD prefer to leverage multi-GPUs. Different from general graph processing, many-body correlation function calculations show two specific features: a large number of computation-/data-intensive kernels and frequently repeated appearances of original and intermediate data. The former results in expensive memory operations such as tensor movements and evictions. The latter offers data reuse opportunities to mitigate the data-intensive nature of many-body correlation function calculations. However, existing graph-based multi-GPU schedulers cannot capture these data-centric features, thus resulting in a sub-optimal performance for many-body correlation function calculations. To address this issue, this paper presents a multi-GPU scheduling framework, MICCO, to accelerate contractions for correlation functions particularly by taking the data dimension (e.g., data reuse and data eviction) into account. This work first performs a comprehensive study on the interplay of data reuse and load balance, and designs two new concepts: local reuse pattern and reuse bound to study the opportunity of achieving the optimal trade-off between them. Based on this study, MICCO proposes a heuristic scheduling algorithm and a machine-learning-based regression model to generate the optimal setting of reuse bounds. Specifically, MICCO is integrated into a real-world Lattice QCD system, Redstar, for the first time running on multiple GPUs. The evaluation demonstrates MICCO outperforms other state-of-art works, achieving up to 2.25× speedup in synthesized datasets, and 1.49× speedup in real-world correlation functions.

Wang, Qihan↗
DOE OSTI2022

Coarse-Grained Density Functional Theory Predictions via Deep Kernel Learning

Scalable electronic predictions are critical for soft materials design. Recently, the Electronic Coarse-Graining (ECG) method was introduced to renormalize all-atom quantum chemical (QC) predictions to coarse-grained (CG) resolutions using deep neural networks (DNNs). While DNNs can learn complex representations that prove challenging for kernel-based methods, they are susceptible to overfitting and the overconfidence of uncertainty estimations. Here, we develop ECG within a GPU-accelerated Deep Kernel Learning (DKL) framework to enable CG QC predictions using range-separated hybrid density functional theory (DFT), obtaining a 107 speedup relative to naive all-atom QC. By treating the predicted electronic properties as random Gaussian Processes, DKL incorporates CG mapping degeneracy by learning the distribution of electronic energies as a function of CG configuration. DKL-ECG accurately reproduces molecular orbital energies from range-separated DFT while facilitating efficient training via active learning using the uncertainties provided by DKL. Further, we show that while active learning algorithms enable efficient sampling of a more diverse configurational space relative to random sampling, all explored query methods exhibit comparable performance for the examined system. We attribute this result to the significant overlap of the feature space and output property distributions across multiple temperatures.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Comparison of quantum advantage experiments using random circuit sampling

Random circuit sampling, the task of sampling bit strings from a random unitary operator, has been implemented to demonstrate quantum advantage on the Sycamore quantum processor with 53 qubits and on the Zuchongzhi quantum processor with 56 and 61 qubits. Recently, it was claimed that classical computers using tensor network simulation could catch on to current noisy quantum processors for random circuit sampling. While the linear cross-entropy benchmark fidelity was used to certify all these claims, it may not capture statistical properties of outputs in detail. Here, we compare the bit strings sampled from classical computers using tensor network simulation by Pan et al. [F. Pan, K. Chen, and P. Zhang, Phys. Rev. Lett. 129, 090502 (2022)] and by Kalachev et al. [G. Kalachev, P. Panteleev, P. Zhou, and M.-H. Yung, arXiv:2112.15083] with the bit strings from the Sycamore quantum processor. It is shown that all of Kalachev et al.'s samples passed the NIST random number tests. The heat maps of bit strings show that Pan et al.'s and Kalachev et al.'s samples are quite different from the Sycamore or Zuchongzhi samples. The analysis with the Marchenko-Pastur distribution and the Wasssertein distances demonstrates that Kalachev et al.'s samples are statistically closer to the Sycamore samples than Pan et al.'s while the three datasets have similar values for the linear cross-entropy fidelity. In conclusion, our finding implies that further study is needed to certify or beat the claims of quantum advantage using random circuit sampling.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Dynamical Complexity of Non-Gaussian Many-Body Systems with Dissipation

We characterize the dynamical state of many-body bosonic and fermionic many-body models with intersite Gaussian couplings, on-site non-Gaussian interactions, and local dissipation comprising incoherent particle loss, particle gain, and dephasing. We first establish that, for fermionic systems, if the dephasing noise is larger than the non-Gaussian interactions, irrespective of the Gaussian coupling strength, the system state is a convex combination of Gaussian states at all times. Furthermore, for bosonic systems, we show that if the particle loss and particle gain rates are larger than the Gaussian intersite couplings, the system remains in a separable state at all times. Building on this characterization, we establish that at noise rates above a threshold, there exists a classical algorithm that can efficiently sample from the system state of both the fermionic and bosonic models. Finally, we show that, unlike fermionic systems, bosonic systems can evolve into states that are not convex Gaussian even when the dissipation is much higher than the on-site non-Gaussianity. Similarly, unlike bosonic systems, fermionic systems can generate entanglement even with noise rates much larger than the intersite couplings.

Computational complexity↗
DOE OSTI2018

Possibilities of laser amplification and measurement of the field structure of ultrashort pulses in the range of 2.7 – 3μm in tellurite glass fibres doped with erbium ions

The amplification of ultrashort signals in the range 2.7 – 3 μm in active tellurite glass fibres with high Kerr nonlinearity and high concentration of erbium ions in the core is theoretically investigated under diode pumping at a wavelength of 0.975 μm. The possibility of producing pulses with the energy ∼10 nJ and the duration smaller than 100 fs with ∼20 dB gain is shown. Moreover, using a specially developed iterative computer algorithm, the possibility of determining the intensity and phase of the generated pulses by measuring their fundamental spectra and two additional spectra obtained by means of nonlinear-optical conversion in short pieces of these optical fibres is demonstrated. Optical fibre samples are made of high-purity TeO{sub 2} – ZnO – La{sub 2}O{sub 3} – Na{sub 2}O glasses with 10{sup 21} cm{sup −3} concentration of erbium ions in the core and a low content of hydroxyl groups that causes small losses in the spectral range under consideration. The group velocity dispersions, effective mode fields, nonlinear coefficients and field overlap integrals of the LP{sub 01} fundamental modes with a doped core for optical fibres with different core diameters are calculated, and the emission and absorption cross sections are calculated basing on the experimental luminescence spectrum. The optimal parameters of the amplifier are determined by numerically modelling the dynamics of ultrashort pulses using the measured and calculated characteristics. (paper)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Importance of kernel bandwidth in quantum machine learning

Quantum kernel methods are considered a promising avenue for applying quantum computers to machine learning problems. Identifying hyperparameters controlling the inductive bias of quantum machine learning models is expected to be crucial given the central role hyperparameters play in determining the performance of classical machine learning methods. In this work we introduce the hyperparameter controlling the bandwidth of a quantum kernel and show that it controls the expressivity of the resulting model. We use extensive numerical experiments with multiple quantum kernels and classical data sets to show consistent change in the model behavior from underfitting (bandwidth too large) to overfitting (bandwidth too small), with optimal generalization in between. We draw a connection between the bandwidth of classical and quantum kernels and show analogous behavior in both cases. Furthermore, we show that optimizing the bandwidth can help mitigate the exponential decay of kernel values with qubit count, which is the cause behind recent observations that the performance of quantum kernel methods decreases with qubit count. Here, we reproduce these negative results and show that if the kernel bandwidth is optimized, the performance instead improves with growing qubit count and becomes competitive with the best classical methods.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Orbits and Masses of Binaries from Speckle Interferometry at SOAR

We present results from Speckle inteferometric observations of 15 visual binaries and one double-line spectroscopic binary, carried out with the HRCam Speckle camera of the SOAR 4.1 m telescope. These systems were observed as a part of an on-going survey to characterize the binary population in the solar vicinity, out to a distance of 250 pc. We obtained orbital elements and mass sums for our sample of visual binaries. The orbits were computed using a Markov Chain Monte Carlo algorithm that delivers maximum likelihood estimates of the parameters, as well as posterior probability density functions that allow us to evaluate their uncertainty. Their periods cover a range from 5 yr to more than 500 yr; and their spectral types go from early A to mid M, implying total system masses from slightly more than 4M{sub ⊙} down to 0.2M {sub ⊙}. They are located at distances between approximately 12 and 200 pc, mostly at low Galactic latitude. For the double-line spectroscopic binary YSC8, we present the first combined astrometric/radial-velocity orbit resulting from a self-consistent fit, leading to individual component masses of 0.897 ± 0.027 M {sub ⊙} and 0.857 ± 0.026 M {sub ⊙}; and an orbital parallax of 26.61 ± 0.29 mas, which compares very well with the Gaia DR2 trigonometric parallax (26.55 ± 0.27 mas). In combination with published photometry and trigonometric parallaxes, we place our objects on an H-R diagram and discuss their evolutionary status. We also present a thorough analysis of the precision and consistency of the photometry available for them.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2024

Universal framework for simultaneous tomography of quantum states and SPAM noise

We present a general denoising algorithm for performing simultaneous tomography of quantum states and measurement noise. This algorithm allows us to fully characterize state preparation and measurement (SPAM) errors present in any quantum system. Our method is based on the analysis of the properties of the linear operator space induced by unitary operations. Given any quantum system with a noisy measurement apparatus, our method can output the quantum state and the noise matrix of the detector up to a single gauge degree of freedom. We show that this gauge freedom is unavoidable in the general case, but this degeneracy can be generally broken using prior knowledge on the state or noise properties, thus fixing the gauge for several types of state-noise combinations with no assumptions about noise strength. Such combinations include pure quantum states with arbitrarily correlated errors, and arbitrary states with block independent errors. This framework can further use available prior information about the setting to systematically reduce the number of observations and measurements required for state and noise detection. Our method effectively generalizes existing approaches to the problem, and includes as special cases common settings considered in the literature requiring an uncorrelated or invertible noise matrix, or specific probe states.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Python and HPC for High Energy Physics Data Analyses

High level abstractions in Python that can utilize computing hardware well seem to be an attractive option for writing data reduction and analysis tasks. In this paper, we explore the features available in Python which are useful and efficient for end user analysis in High Energy Physics (HEP). A typical vertical slice of an HEP data analysis is somewhat fragmented: the state of the reduction/analysis process must be saved at certain stages to allow for selective reprocessing of only parts of a generally time-consuming workflow. Also, algorithms tend to to be modular because of the heterogeneous nature of most detectors and the need to analyze different parts of the detector separately before combining the information. This fragmentation causes difficulties for interactive data analysis, and as data sets increase in size and complexity (O10 TiB for a “small” neutrino experiment to the O10 PiB currently held by the CMS experiment at the LHC), data analysis methods traditional to the field must evolve to make optimum use of emerging HPC technologies and platforms. Mainstream big data tools, while suggesting a direction in terms of what can be done if an entire data set can be available across a system and analysed with high-level programming abstractions, are not designed with either scientific computing generally, or modern HPC platform features in particular, such as data caching levels, in mind. Our example HPC use case is a search for a new elementary particle which might explain the phenomenon known as “Dark Matter”. Here, using data from the CMS detector, we will use HDF5 as our input data format, and MPI with Python to implement our use case.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Polynomial-time preparation of low-temperature Gibbs states for two-dimensional toric code

In this work, we propose a polynomial-time algorithm for preparing the Gibbs state of the two-dimensional toric code Hamiltonian at any temperature, starting from any initial state, significantly improving upon prior estimates that suggested exponential scaling with inverse temperature. We prove that fast mixing at low temperature for the two-dimensional toric code can be achieved by augmenting local jump operators with simple global jump operators, which enable efficient transitions between logical sectors. To establish tight lower bounds on the spectral gap, we introduce a new reduction method that eventually maps the problem to estimating the spectral gap of a perturbed graph Laplacian on a stair graph. Our proof also shows that the Lindblad dynamics with a digitally implemented low-temperature local Davies generator is able to efficiently drive the quantum state toward the ground state manifold.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Optimal Orbital Selection for Full Configuration Interaction (OptOrbFCI): Pursuing the Basis Set Limit under a Budget

Full configuration interaction (FCI) solvers are limited to small basis sets due to their expensive computational costs. An optimal orbital selection for FCI (OptOrbFCI) is proposed to boost the power of existing FCI solvers to pursue the basis set limit under a computational budget. The optimization problem coincides with that of the complete active space SCF method (CASSCF), while OptOrbFCI is algorithmically quite different. OptOrbFCI effectively finds an optimal rotation matrix via solving a constrained optimization problem directly to compress the orbitals of large basis sets to one with a manageable size, conducts FCI calculations only on rotated orbital sets, and produces a variational ground-state energy and its wave function. Coupled with coordinate descent full configuration interaction (CDFCI), we demonstrate the efficiency and accuracy of the method on the carbon dimer and nitrogen dimer under basis sets up to cc-pV5Z. We also benchmark the binding curve of the nitrogen dimer under the cc-pVQZ basis set with 28 selected orbitals, which provide consistently lower ground-state energies than the FCI results under the cc-pVDZ basis set. Furthermore, the dissociation energy in this case is found to be of higher accuracy.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

Neutrino Oscillations in Supernovae

A (core-collapse) supernova is the final death call of a massive star during which a gigantic amount of a special kind of elementary particles known as neutrinos are released for a brief period of time. An earlier work by the PI and his collaborators has shown that the dense neutrino gas deep inside the supernova can experience "flavor transformation" or "oscillations" in a collective fashion, a macroscopic quantum phenomenon spanning over distances of 100-1000 kilometers. During this process neutrinos of different energies and propagation directions change their identities simultaneously. This phenomenon may be a key to understanding the mechanism of supernova explosion and the origin of many of the chemical elements found on the earth which were produced in the supernovae exploded in the past. The previous understanding of neutrino oscillations in supernovae was based on a stationary 3-dimensional model with 1 spatial dimension and 2 momentum dimensions. In this "neutrino bulb model" neutrinos are emitted from a spherically symmetric surface and they are decoupled from the motion of matter and the production of chemical elements. In order to accurately model the neutrino oscillations in supernovae and determine its physical consequences, sophisticated 7-dimensional (1 temporal dimension + 3 spatial dimensions + 3 momentum dimensions) modelling of neutrino transport must be performed self-consistently together with the hydrodynamic simulation of the flow of energy and matter inside the supernova and with the calculation of the production and destruction of chemical elements through a large nucleosynthesis network. Although it will be long journey to achieve the lofty goal, our project has made important progress in three different fronts towards this goal. In the first front we explored new computing hardware on which the next generation simulation codes may be run. We ported the neutrino bulb code to the platform of Graphics Processing Unit (GPU) and then later developed a new bulb code that can be run on supercomputers equipped with the Intel Xeon CPUs and/or the Xeon Phi coprocessors. In the second front we developed new algorithms that can be used towards computing neutrino oscillations in multi-dimensional models. We have developed a moment expansion method which is approximately 10 times more efficient than the old algorithm. We have also developed numerical codes to solve neutrino oscillations in toy models with two spatial dimensions. In the last front we used analytic methods to study neutrino oscillations in more sophisticated models, e.g. of multiple spatial dimensions and/or with time dependence. By analyzing the neutrino oscillations in toy models with two spacial dimensions we found that the approximate spacial translation symmetry, which is similar to the spherical symmetry in the neutrino bulb model, can be broken spontaneously as neutrino oscillations begin. By analyzing a time-dependent neutrino bulb model we found that certain modes of neutrino oscillations that are suppressed in the stationary model can become active in the time-dependent model. We have also developed theories which give insights to how ordinary matter may affect neutrino oscillations in supernovae and other astrophysical environments.

Duan, Huaiyu↗
DOE OSTI2021

Parallel physics-informed neural networks via domain decomposition

Here we develop a distributed framework for the physics-informed neural networks (PINNs) based on two recent extensions, namely conservative PINNs (cPINNs) and extended PINNs (XPINNs), which employ domain decomposition in space and in time-space, respectively. This domain decomposition endows cPINNs and XPINNs with several advantages over the vanilla PINNs, such as parallelization capacity, large representation capacity, efficient hyperparameter tuning, and is particularly effective for multi-scale and multi-physics problems. Here, we present a parallel algorithm for cPINNs and XPINNs constructed with a hybrid programming model described by MPI + X, where X ∈ {CPUs, GPUs}. The main advantage of cPINN and XPINN over the more classical data and model parallel approaches is the flexibility of optimizing all hyperparameters of each neural network separately in each subdomain. We compare the performance of distributed cPINNs and XPINNs for various forward problems, using both weak and strong scalings. Our results indicate that for space domain decomposition, cPINNs are more efficient in terms of communication cost but XPINNs provide greater flexibility as they can also handle time-domain decomposition for any differential equations, and can deal with any arbitrarily shaped complex subdomains. To this end, we also present an application of the parallel XPINN method for solving an inverse diffusion problem with variable conductivity on the United States map, using ten regions as subdomains.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Monte Carlo Explicitly Correlated Second-Order Many-Body Green’s Function Calculations of Semiconductor Band Gaps

A systematically converging series of ab initio, post-density-functional, size-consistent, electron-correlated approximations is desired for predictive computing of felectronic band structures of insulating, semiconducting, and metallic solids. A series that meets all of these desiderata (except the applicability to metals) is ab initio many-body Green's function theory based on Gaussian-type-orbital (GTO) basis sets. Here, its leading-order approximation, the second-order Green's function (GF2) method in the diagonal and frequency-independent approximations with the aug-cc-pVDZ basis set, is applied to the fundamental band gaps of three semiconductors (diamond, silicon, and silicon carbide in the zincblende structure) using cluster models. Corrections are made to the basis-set-incompleteness errors by the explicit-correlation (F12) ansatz (GF2-F12) for the valence band edges. The crystals are modeled as surface-passivated clusters of increasing sizes, whose wave functions are expanded by up to 2709 GTO basis functions. Immense computational costs of these calculations are overcome by the highly scalable stochastic algorithm of the Monte Carlo GF2-F12 method, whose operation cost per state increases only as a cubic power of system size, which has a tiny memory footprint and easily achieves near-perfect parallel efficiency on thousands of CPUs or on hundreds of GPUs. The correlated, F12-corrected highest-occupied and lowest-unoccupied molecular-orbital energy (HOMO-LUMO) gap is 5.78 ± 0.07 eV for C 87 H 76 as compared with the experimental value of the fundamental (indirect) band gap of bulk diamond at 5.48 eV. The correlated, F12-corrected HOMO-LUMO gaps for Si 75 H 76 and Si 32 C 43 H 76 are 2.56 ± 0.15 eV and 3.50 ± 0.12 eV, respectively, which are expected to decrease further with increasing cluster sizes. As a result, the experimental fundamental (indirect) band gaps of bulk silicon and silicon carbide are 1.17 eV and 2.42 eV, respectively.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗