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At least 541 records · Page 30

Automated discovery of a robust interatomic potential for aluminum

Machine learning, trained on quantum mechanics (QM) calculations, is a powerful tool for modeling potential energy surfaces. A critical factor is the quality and diversity of the training dataset. Here we present a highly automated approach to dataset construction and demonstrate the method by building a potential for elemental aluminum (ANI-Al). In our active learning scheme, the ML potential under development is used to drive non-equilibrium molecular dynamics simulations with time-varying applied temperatures. Whenever a configuration is reached for which the ML uncertainty is large, new QM data is collected. The ML model is periodically retrained on all available QM data. The final ANI-Al potential makes very accurate predictions of radial distribution function in melt, liquid-solid coexistence curve, and crystal properties such as defect energies and barriers. We perform a 1.3M atom shock simulation and show that ANI-Al force predictions shine in their agreement with new reference DFT calculations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Hydrogen ejection from hydrocarbons: Characterization and relevance in soot formation and interstellar chemistry

Polycyclic aromatic hydrocarbons (PAHs) play a major role in the chemistry of combustion, pyrolysis, and the interstellar medium. Production (or activation) of radical PAHs and propagation of their resulting reactions require efficient dehydrogenation, but the preferred method of hydrogen loss is not well understood. Unimolecular hydrogen ejection (i.e., direct C─H bond fission) and bimolecular radical abstraction are two main candidate pathways. We performed a computational study to characterize the role of H ejection, particularly as a driver for radical-centric hydrocarbon-growth mechanisms and particle formation. Electronic structure calculations establish that C─H bond strengths span a broad range of energies, which can be weaker than 30 kcal/mol in some C 9 and C 13 PAH radicals. At T > 1200 K, calculated thermal rates for hydrogen ejection from weak C─H bonds at zigzag sites on PAH radicals are significantly larger than typical H-abstraction rates. These results are highly relevant in the context of chain reactions of radical species and soot inception under fuel-rich combustion conditions. Furthermore, calculated microcanonical rates that include the additional internal energy released by bond formation (e.g., ring closure to yield C 9 H 9 ) yield significantly higher rates than those associated with full thermalization. These microcanonical considerations are relevant to the astrochemical processes associated with hydrocarbon growth and processing in the low-density interstellar environment.

08 HYDROGEN↗

PIONEER: Studies of Rare Pion Decays

A next-generation rare pion decay experiment, PIONEER, is strongly motivated by several inconsistencies between Standard Model (SM) predictions and data pointing towards the potential violation of lepton flavor universality. It will probe non-SM explanations of these anomalies through sensitivity to quantum effects of new particles even if their masses are at very high scales. Measurement of the charged-pion branching ratio to electrons vs. muons $R_{e/\mu}$ is extremely sensitive to new physics effects. At present, the SM prediction for $R_{e/\mu}$ is known to 1 part in $10^4$, which is 15 times more precise than the current experimental result. An experiment reaching the theoretical accuracy will test lepton flavor universality at an unprecedented level, probing mass scales up to the PeV range. Measurement of pion beta decay, $\pi^+\to \pi^0 e^+ \nu (\gamma)$, with 3 to 10-fold improvement in sensitivity, will determine $V_{ud}$ in a theoretically pristine manner and test CKM unitarity, which is very important in light of the recently emerged tensions. In addition, various exotic rare decays involving sterile neutrinos and axions will be searched for with unprecedented sensitivity. The experiment design benefits from experience with the recent PIENU and PEN experiments at TRIUMF and the Paul Scherrer Institut (PSI). Excellent energy and time resolutions, greatly increased calorimeter depth, high-speed detector and electronics response, large solid angle coverage, and complete event reconstruction are all critical aspects of the approach. The PIONEER experiment design includes a 3$\pi$ sr 25 radiation length calorimeter, a segmented low gain avalanche detector stopping target, a positron tracker, and other detectors. Using intense pion beams, and state-of-the-art instrumentation and computational resources, the experiments can be performed at the PSI ring cyclotron.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Modeling Material Interfaces with the Six-Equation Model in FLEXO

FLEXO is a multiphysics code developed at Sandia National Laboratories for predictive simulation of pulsed power target physics with extended magnetohydrodynamics modeling. Given this application space, FLEXO must be able to perform high-fidelity simulations of pulsed-power systems that include multiple materials. This work presents the augmentation of FLEXO with a six-equation model for multimaterial flow. We design and analyze discretizations of the six-equation model, emphasizing desirable properties for robust multimaterial simulation. We also present solution limiting, bounds preservation, and pressure equilibration/relaxation techniques that enhance the robustness of the overall multimaterial scheme. We present a suite of test problems to document these new capabilities of the FLEXO code.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Critical role of orbital hybridization in the Dzyaloshinskii-Moriya interaction of magnetic interfaces

Dzyaloshinskii-Moriya interaction (DMI), an interfacial spin-orbit coupling (ISOC)-related effect, has become foundational for spintronic research and magnetic memory and computing technologies. However, the underlying mechanism of DMI, including the quantitative role of ISOC, has remained a long-standing unsettled problem due to the great challenge in quantifying and widely tuning ISOC strength in a strong DMI material system. Here, we find that DMI, ISOC, and orbital hybridization at the model magnetic interface Au 1-x Pt x /Co can be quantified and tuned significantly at the same time through the composition of the Au 1-x Pt x , without varying the bulk SOC and the electronegativity. From this ability, we establish that the widespread expectation that DMI should scale in linear proportion to ISOC breaks down at the Au 1-x Pt x /Co interface where degree of orbital hybridization varies with the Au 1-x Pt x composition and that the unexpected DMI behaviors can be understood well by the critical role of orbital hybridization. Our study provides a quantitative frame for comprehensively understanding interfacial DMI of various magnetic interfaces and establishes orbital hybridization as a new degree of freedom for controlling DMI in high-performance chiral domain wall/skyrmion devices and ultrafast magnetic tunnel junctions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Supercomputer-Based Ensemble Docking Drug Discovery Pipeline with Application to Covid-19

In this work, we present a supercomputer-driven pipeline for in silico drug discovery using enhanced sampling molecular dynamics (MD) and ensemble docking. Ensemble docking makes use of MD results by docking compound databases into representative protein binding-site conformations, thus taking into account the dynamic properties of the binding sites. We also describe preliminary results obtained for 24 systems involving eight proteins of the proteome of SARS-CoV-2. The MD involves temperature replica exchange enhanced sampling, making use of massively parallel supercomputing to quickly sample the configurational space of protein drug targets. Using the Summit supercomputer at the Oak Ridge National Laboratory, more than 1 ms of enhanced sampling MD can be generated per day. We have ensemble docked repurposing databases to 10 configurations of each of the 24 SARS-CoV-2 systems using AutoDock Vina. Comparison to experiment demonstrates remarkably high hit rates for the top scoring tranches of compounds identified by our ensemble approach. We also demonstrate that, using Autodock-GPU on Summit, it is possible to perform exhaustive docking of one billion compounds in under 24 h. Finally, we discuss preliminary results and planned improvements to the pipeline, including the use of quantum mechanical (QM), machine learning, and artificial intelligence (AI) methods to cluster MD trajectories and rescore docking poses.

60 APPLIED LIFE SCIENCES↗

Scalable molecular dynamics on CPU and GPU architectures with NAMD

NAMD is a molecular dynamics program designed for high-performance simulations of very large biological objects on CPU- and GPU-based architectures. NAMD offers scalable performance on petascale parallel supercomputers consisting of hundreds of thousands of cores, as well as on inexpensive commodity clusters commonly found in academic environments. It is written in C++ and leans on Charm++ parallel objects for optimal performance on low-latency architectures. NAMD is a versatile, multipurpose code that gathers state-of-the-art algorithms to carry out simulations in apt thermodynamic ensembles, using the widely popular CHARMM, AMBER, OPLS, and GROMOS biomolecular force fields. Here, we review the main features of NAMD that allow both equilibrium and enhanced-sampling molecular dynamics simulations with numerical efficiency. We describe the underlying concepts utilized by NAMD and their implementation, most notably for handling long-range electrostatics; controlling the temperature, pressure, and pH; applying external potentials on tailored grids; leveraging massively parallel resources in multiple-copy simulations; and hybrid quantum-mechanical/molecular-mechanical descriptions. We detail the variety of options offered by NAMD for enhanced-sampling simulations aimed at determining free-energy differences of either alchemical or geometrical transformations and outline their applicability to specific problems. Last, we discuss the roadmap for the development of NAMD and our current efforts toward achieving optimal performance on GPU-based architectures, for pushing back the limitations that have prevented biologically realistic billion-atom objects to be fruitfully simulated, and for making large-scale simulations less expensive and easier to set up, run, and analyze. NAMD is distributed free of charge with its source code at www.ks.uiuc.edu.

high-performance computing↗

Improving Predictions of Spin-Crossover Complex Properties through DFT Calculations with a Local Hybrid Functional

We conducted a study on the performance of the local hybrid exchange-correlation functional PBE0r for a set of 95 experimentally-characterized iron spin crossover (SCO) complexes. The PBE0r functional is a variant of PBE0 where the exchange correction is restricted to on-site terms formulated within the basis of local orbitals. We determine the free parameters of the PBE0r functional against experimental data and other hybrid functionals. With a Hartree-Fock (HF) exchange factor of 4%, the PBE0r functional accurately reproduces the electronic and free energy trends predicted in prior DFT studies for these 95 complexes using the B3LYP functional. Larger values of HF exchange stabilize high-spin states. The PBE0r-predicted bond lengths tend to exceed the experimental bond lengths, and bond lengths are less sensitive to HF exchange. The predicted SCO transition temperatures T 1/2 from PBE0r correlate moderately with the experimental transition temperatures, showing a slight improvement compared to the previous modB3LYP-predicted T 1/2 . Furthermore, this study suggests the PBE0r functional as computationally cost-effective and offers the possibility of simulating larger complexes with accuracy comparable to other global hybrid functionals, provided the HF exchange parameter is carefully optimized.

25 ENERGY STORAGE↗

Predicting Energetics Materials’ Crystalline Density from Chemical Structure by Machine Learning

To expedite new molecular compound development, a long-sought goal within the chemistry community has been to predict molecules’ bulk properties of interest a priori to synthesis from a chemical structure alone. In this work, we demonstrate that machine learning methods can indeed be used to directly learn the relationship between chemical structures and bulk crystalline properties of molecules, even in the absence of any crystal structure information or quantum mechanical calculations. We focus specifically on a class of organic compounds categorized as energetic materials called high explosives (HE) and predicting their crystalline density. An ongoing challenge within the chemistry machine learning community is deciding how best to featurize molecules as inputs into machine learning models—whether expert handcrafted features or learned molecular representations via graph-based neural network models—yield better results and why. We evaluate both types of representations in combination with a number of machine learning models to predict the crystalline densities of HE-like molecules curated from the Cambridge Structural Database, and we report the performance and pros and cons of our methods. Our message passing neural network (MPNN) based models with learned molecular representations generally perform best, outperforming current state-of-the-art methods at predicting crystalline density and performing well even when testing on a data set not representative of the training data. However, these models are traditionally considered black boxes and less easily interpretable. Here, to address this common challenge, we also provide a comparison analysis between our MPNN-based model and models with fixed feature representations that provides insights as to what features are learned by the MPNN to accurately predict density.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Expanded verification and validation studies of hypersonic aerodynamics with multiple physics-fidelity models

Hypersonic aerothermodynamics is an important domain of modern multiphysics simulation. The Multi-Fidelity Toolkit is a simulation tool being developed at Sandia National Laboratories to predict aerodynamic properties for compressible flows from a range of physics fidelities and computational speeds. These models include the Reynolds-averaged Navier–Stokes (RANS) equations, the Euler equations with momentum-energy integral technique (MEIT), and modified Newtonian aerodynamics with flat-plate boundary layer (MNA+FPBL) equations, and they can be invoked independently or coupled with hierarchical Kriging to interpolate between high-fidelity simulations using lower-fidelity data. However, as with any new simulation capability, verification and validation are necessary to gather credibility evidence. This work describes formal code- and solution-verification activities, as well as model validation with uncertainty considerations. Code verification activities on the MNA+FPBL model build on previous work by focusing on the viscous portion of the model. Viscous quantities of interest are compared against those from an analytical solution for flat-plate, inclined-plate, and cone geometries. The code verification methodology for the MEIT model is also presented. Test setup and results of code verification tests on the laminar and turbulent models within MEIT are shown. Solution-verification activities include grid-refinement studies on simulations that model the HIFiRE-1 wind tunnel experiments. These experiments are used for validation of all model fidelities. A thorough validation comparison with prediction error and uncertainty is also presented. Three additional HIFiRE-1 experimental runs are simulated in this study, and the solution verification and validation work examines the effects of the associated parameter changes on model performance. Finally, a study is presented that compares the computational costs and fidelities from each of the different models.

42 ENGINEERING↗

Nanoscale Tracking of the High-Temperature Spin-State Transition in LaCoO 3

The high-temperature spin and electronic transitions in LaCoO 3 have recently been leveraged to create neuromorphic (brain-inspired) devices. While these devices have shown the potential for impactful functionality in next-generation computing systems, the nanoscale dynamics of the spin and electronic transitions that underlie their operation are not well understood. Inhomogeneities related to interfaces, electrode contacts, strain, and crystal defects can all affect device performance, making nanoscale characterization of the transitions essential for producing consistent and reliable devices. Here, we demonstrate the first nanoscale in situ measurement of the spin transition in LaCoO 3 at device-relevant temperatures (25–325 °C) over length scales of tens of nanometers using STEM-EELS. This measurement is enabled by an Al 2 O 3 coating, which prevents unwanted reduction of the LaCoO 3 specimen at high temperature and vacuum. The detailed understanding of LaCoO 3 transition dynamics enabled by such measurements will be crucial for optimizing LaCoO 3 -based neuromorphic devices and increasing reliability for real-world application.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Impact of particle loading and phase coupling on gas–solid flow dynamics: A case study of a two-phase, gas–solid flow in an annular pipe

The present study is devoted to a two-phase, gas–solid flow in an annular pipe (hollow cylinder) at an elevated pressure of 15 bars and moderate Reynolds number of circa 6000. The influence of the particle loading, the interaction between the phases, and turbulence dispersion on the flow dynamics is systematically studied by means of computational fluid dynamics simulations, employing the Ansys FLUENT commercial package. The cases with a particle volumetric fraction of 1.2% are referred to as “high particle loading,” and those with 0.13% are denoted as “low particle loading.” The following cases are investigated: (1) pure gas flow; (2) low particle loading two-phase flow with one-way coupling and with turbulence dispersion; (3) low particle loading two-phase flow with two-way coupling but without turbulence dispersion; (4) low particle loading two-phase flow with two-way coupling and with turbulence dispersion; (5) high particle loading two-phase flow with one-way coupling and with turbulence dispersion; (6) high particle loading two-phase flow with two-way coupling but without turbulence dispersion; and (7) high particle loading two-phase flow with two-way coupling and with turbulence dispersion. Additionally, the boundary layer is found to grow without fluctuations of the turbulent kinetic energy (TKE) for cases 1, 2, and 5. For case 4, the TKE fluctuations have been identified, although they appear to be less substantial than those in cases 6 and 7. The authors attribute the semi-chaotic nature of the TKE fluctuations to the particle loading and two-way coupling. In addition, the onset and development of the flow instability have been observed at a random axial distance in cases 4, 6, and 7. Such instability is also attributed to the two-way coupling with turbulence dispersion in the flow. It is concluded that the particle loading, one-way, or two-way coupling between the phases, and the turbulence dispersion models significantly influence the development of the flow dynamics with the same inlet and boundary conditions. Consequently, it is not a trivial question, which result a user should trust. The present computational results inspire to perform verification as well as experimental validation of the simulations, so the simulation results can subsequently be used with confidence for design analysis.

42 ENGINEERING↗

End-to-end protocol for high-quality quantum approximate optimization algorithm parameters with few shots

The quantum approximate optimization algorithm (QAOA) is a quantum heuristic for combinatorial optimization that has been demonstrated to scale better than state-of-the-art classical solvers for some problems. For a given problem instance, QAOA performance depends crucially on the choice of the parameters. While average-case optimal parameters are available in many cases, meaningful performance gains can be obtained by fine-tuning these parameters for a given instance. This task is especially challenging, however, when the number of circuit executions (shots) is limited. In this work, we develop an end-to-end protocol that combines multiple parameter settings and fine-tuning techniques. We use large-scale numerical experiments to optimize the protocol for the shot-limited setting and observe that optimizers with the simplest internal model (linear) perform best. We implement the optimized pipeline on a trapped-ion processor using up to 32 qubits and 5 QAOA layers, and we demonstrate that the pipeline is robust to small amounts of hardware noise. To the best of our knowledge, these are the largest demonstrations of QAOA parameter fine-tuning on a trapped-ion processor in terms of two-qubit gate count.

quantum algorithms & computation↗

Interpretable, extensible linear and symbolic regression models for charge density prediction using a hierarchy of many-body correlation descriptors

Here, density functional theory (DFT) is routinely used to make electronic structure predictions for high-throughput screening of materials and molecules for technologically relevant areas, like the identification of better catalysts, electronic materials, and drug discovery. However, the DFT formalism is limited by (a) its poor (quadratic-to-quartic) scaling, and (b) the need to perform repeated eigenvalue computations of the electronic Hamiltonian as part of its self-consistent field (SCF) iteration procedure to obtain the converged ground state electron density, ρ (r). Approaches that directly predict ρ (r) of a structure with high accuracy can accelerate conventional SCF calculations and can also be used in linearly scaling methods such as orbital-free DFT. To this end, we present a procedure to predict the ground state electron density of molecular and periodic three-dimensional systems directly from the atomic structure with a particular emphasis on physical interpretability. In our framework, ρ (r) is modeled using many-body correlation descriptors that accurately capture the effects of local atomic arrangements in the neighborhood of a grid point. Our use of a linear regression scheme to fit to charge density data enables transparent analysis of the relative contributions of various types of local atomic correlations. By systematically including increasingly complex correlations, our model is shown to accurately predict ρ (r) for a variety of chemically and electronically diverse systems — amorphous Ge, Al(001) slab, crystalline Ga 2 O 3 , molecular benzene, and polyethylene. We then demonstrate a symbolic regression-based protocol to construct easily computable, interpretable features from lower-order correlations that significantly improves our electron density predictions with effectively no increase in the computational cost.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Rapid Initial-State Preparation for the Quantum Simulation of Strongly Correlated Molecules

Studies on quantum algorithms for ground-state energy estimation often assume perfect ground-state preparation; however, in reality the initial state will have imperfect overlap with the true ground state. Here, we address that problem in two ways: by faster preparation of matrix-product-state (MPS) approximations and by more efficient filtering of the prepared state to find the ground-state energy. We show how to achieve unitary synthesis with a Toffoli complexity about 7 × lower than that in prior work and use that to derive a more efficient MPS-preparation method. For filtering, we present two different approaches: sampling and binary search. For both, we use the theory of window functions to avoid large phase errors and minimize the complexity. We find that the binary-search approach provides better scaling with the overlap at the cost of a larger constant factor, such that it will be preferred for overlaps less than about 0.003 . Finally, we estimate the total resources to perform ground-state energy estimation of Fe - S cluster systems, including the Fe Mo cofactor by estimating the overlap of different MPS initial states with potential ground states of the Fe Mo cofactor using an extrapolation procedure. With a modest MPS bond dimension of 4000 , our procedure produces an estimate of approximately 0.9 overlap squared with a candidate ground state of the Fe Mo cofactor, producing a total resource estimate of 7.3 × 10 10 Toffoli gates; neglecting the search over candidates and assuming the accuracy of the extrapolation, this validates prior estimates that have used perfect ground-state overlap. This presents an example of a practical path to prepare states of high overlap in a challenging-to-compute chemical system. Published by the American Physical Society 2025

Berry, Dominic W. (ORCID:0000000334461449)↗

Dynamic load balancing with enhanced shared-memory parallelism for particle-in-cell codes

Furthering our understanding of many of today’s interesting problems in plasma physics – including plasma based acceleration and magnetic reconnection with pair production due to quantum electrodynamic effects – requires large-scale kinetic simulations using particle-in-cell (PIC) codes. However, these simulations are extremely demanding, requiring that contemporary PIC codes be designed to efficiently use a new fleet of exascale computing architectures. To this end, the key issue of parallel load balance across computational nodes must be addressed. We discuss the implementation of dynamic load balancing by dividing the simulation space into many small, self-contained regions or ‘‘tiles,’’ along with shared-memory (e.g., OpenMP) parallelism both over many tiles and within single tiles. The load balancing algorithm can be used with three different topologies, including two space-filling curves. Here, we tested this implementation in the code Osiris and show low overhead and improved scalability with OpenMP thread number on simulations with both uniform load and severe load imbalance. Compared to other load-balancing techniques, our algorithm gives order-of-magnitude improvement in parallel scalability for simulations with severe load imbalance issues.

97 MATHEMATICS AND COMPUTING↗

Silicon nitride stress-optic microresonator modulator for optical control applications

Modulation-based control and locking of lasers, filters and other photonic components is a ubiquitous function across many applications that span the visible to infrared (IR), including atomic, molecular and optical (AMO), quantum sciences, fiber communications, metrology, and microwave photonics. Today, modulators used to realize these control functions consist of high-power bulk-optic components for tuning, sideband modulation, and phase and frequency shifting, while providing low optical insertion loss and operation from DC to 10s of MHz. In order to reduce the size, weight and cost of these applications and improve their scalability and reliability, modulation control functions need to be implemented in a low loss, wafer-scale CMOS-compatible photonic integration platform. The silicon nitride integration platform has been successful at realizing extremely low waveguide losses across the visible to infrared and components including high performance lasers, filters, resonators, stabilization cavities, and optical frequency combs. Yet, progress towards implementing low loss, low power modulators in the silicon nitride platform, while maintaining wafer-scale process compatibility has been limited. Here we report a significant advance in integration of a piezo-electric (PZT, lead zirconate titanate) actuated micro-ring modulation in a fully-planar, wafer-scale silicon nitride platform, that maintains low optical loss (0.03 dB/cm in a 625 µm resonator) at 1550 nm, with an order of magnitude increase in bandwidth (DC - 15 MHz 3-dB and DC - 25 MHz 6-dB) and order of magnitude lower power consumption of 20 nW improvement over prior PZT modulators. The modulator provides a >14 dB extinction ratio (ER) and 7.1 million quality-factor (Q) over the entire 4 GHz tuning range, a tuning efficiency of 162 MHz/V, and delivers the linearity required for control applications with 65.1 dB·Hz 2/3 and 73.8 dB·Hz 2/3 third-order intermodulation distortion (IMD3) spurious free dynamic range (SFDR) at 1 MHz and 10 MHz respectively. We demonstrate two control applications, laser stabilization in a Pound-Drever Hall (PDH) lock loop, reducing laser frequency noise by 40 dB, and as a laser carrier tracking filter. This PZT modulator design can be extended to the visible in the ultra-low loss silicon nitride platform with minor waveguide design changes. This integration of PZT modulation in the ultra-low loss silicon nitride waveguide platform enables modulator control functions in a wide range of visible to IR applications such as atomic and molecular transition locking for cooling, trapping and probing, controllable optical frequency combs, low-power external cavity tunable lasers, quantum computers, sensors and communications, atomic clocks, and tunable ultra-low linewidth lasers and ultra-low phase noise microwave synthesizers.

Wang, Jiawei (ORCID:0000000257965220)↗

Finding the order in complexity: The electronic structure of 14-1-11 zintl compounds

Yb 14 MnSb 11 and Yb 14 MgSb 11 have rapidly risen to prominence as high-performing p-type thermoelectric materials. However, the fairly complex crystal structure of A 14 MX 11 Zintl compounds renders the interpretation of the electronic band structure obscure, making it difficult to chemically guide band engineering and optimization efforts. In this work, we delineate the valence-balanced Zintl chemistry of A 14 MX 11 compounds using the molecular orbital theory. By analyzing the electronic band structures of Yb 14 MgSb 11 and Yb 14 AlSb 11 , we show that the conduction band minimum is composed of either an antibonding molecular orbital originating from the (Sb 3 ) 7– trimer or a mix of atomic orbitals of A, M, and X. The singly degenerate valence band is comprised of non-bonding Sb pz orbitals primarily from the Sb atoms in the (MSb 4 ) m– tetrahedra and of isolated Sb atoms distributed throughout the unit cell. Such a chemical understanding of the electronic structure enables strategies to engineer electronic properties (e.g., the bandgap) of A 14 MX 11 compounds.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗