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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 415 records · Page 23

Adaptive spectral solution method for the Landau and Lenard-Balescu equations

In this paper, we present an adaptive spectral method for solving the Landau/Fokker-Planck equation for electron-ion systems. The heart of the algorithm is an expansion in Laguerre polynomials, which has several advantages, including automatic conservation of both energy and particles without the need for any special discretization or time-stepping schemes. One drawback of such an expansion is the O(N 3 ) memory requirement, where N is the number of polynomials used. This can impose an inconvenient limit in cases of practical interest, such as when two particle species have widely separated temperatures. The algorithm we describe here addresses this problem by periodically re-projecting the solution onto a judicious choice of new basis functions that are still Laguerre polynomials but have arguments adapted to the current physical conditions. This results in a reduction in the number of polynomials needed, at the expense of increased solution time. Because the equations are solved with little difficulty, this added time is not of much concern compared to the savings in memory. To demonstrate the algorithm, we solve several relaxation problems that could not be computed with the spectral method without re-projection. Another major advantage of this method is that it can be used for collision operators more complicated than that of the Landau equation, and we demonstrate this here by using it to solve the non-degenerate quantum Lenard-Balescu (QLB) equation for a hydrogen plasma. We conclude with some comparisons of temperature relaxation problems solved with the latter equation and the Landau equation with a Coulomb logarithm inspired by the properties of the QLB operator. We find that with this choice of Coulomb logarithm, there is little difference between using the two equations for these particular systems.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NRC Multiphysics Analysis Capability Deployment (FY2020--Part 3)

This report details progress and activities of Idaho National Laboratory (INL) on the NRC project ”Development and Modeling Support for Advanced Non-Light Water Reactors.” The tasks completed for this report are as follows: First, Task 2c: Explicit modeling of pebble transient temperature response. In this simulation, the PBMR-400 reactor experiences a 20 second power ramp from 100% to 150% power. This is followed by a similar reduction in the power back to 100%. Several multiscale pebble coupling approaches are tested with one pebble per mesh element in the active core region. The results show good conservation behavior and the stability of the coupling. Next, Extended scope part 1: An assessment of the computational efficiency of the DFEM heat transfer solver shows good scalability. The DFEM solver is a factor of 4 more expensive in solution time than the FEM solver for heat transfer problems due to the increased number of degrees of freedom. Nonetheless, the DFEM approach provides the user with the flexibility to model gap heat transfer problems. Then Extended scope part 2: the GapHeatTransferInterfaceMaterial was improved to give the user increased flexibility with the modeling of heat transfer through gaps with the DFEM solver. A number of gap parameters can now be coupled both through functions and variables. Finally, Extended scope part 3: demonstration of how the gap width between hexagonal fuel cells can be calculated during a heat up transient and used in the GapHeat- TransferInterface model. A full domain DFEM model with gap expansion is coupled to a SubApp that models the thermal expansion of the base plate. The results show the expected physical behavior, although have not been fully bench- marked at this point in time.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

SU-F-T-685: Evaluation of Tumor Hypoxic Fraction Using Serial Volumetric Imaging During Radiation Therapy

Purpose: To develop a tumor response model which could be uses to compute tumor hypoxic fraction using serial volumetric tumor imaging. This algorithm may be used for treatment response assessment and also for guidance of more expensive PET imaging of hypoxia. Methods: Previously developed two-level cell population tumor response model was modified to include a third cell level describing hypoxic and necrotic cells. This third level was considered constant value during radiotherapy treatment; therefore, inclusion additional parameter did not compromise stability of model fitting to imaging data. Fitting the model to serial volumetric imaging data was performed using a least squares objective function and simulated annealing algorithm. The problem of reconstruction of radiobiological parameters from serial imaging data was considered as inverse ill-posed problem described by the Fredholm integral equation of the first kind. Variational regularization was used to stabilize solutions. Results: To evaluate performance of the algorithm, we used a set of serial CT imaging data on tumor-volume for 14 head and neck cancer patients. The hypoxic fractions were reconstructed for each patient and the distribution of hypoxic fractions was compared to the distribution of initial hypoxic fractions previously measured using histograph. The measured and reconstructed from imaging data distributions of hypoxic fractions are in good agreement. The reconstructed distribution of cell surviving fraction was also in better agreement with in vitro data than previously obtained using the two-level cell population model. Conclusion: Our results indicate that it is possible to evaluate the initial hypoxic tumor fraction using serial volumetric imaging and a tumor response model. This algorithm can be used for treatment response assessment and guidance of more expensive PET imaging.

60 APPLIED LIFE SCIENCES↗

Isolating Chemical Reaction Mechanism as a Variable with Reactive Coarse-Grained Molecular Dynamics: Step-Growth versus Chain-Growth Polymerization

Here we present a general approach to isolate chemical reaction mechanism as an independently controllable variable across chemically distinct systems. Modern approaches to reduce the computational expense of molecular dynamics simulations often group multiple atoms into a single “coarse-grained” interaction site, which leads to a loss of chemical resolution. In this work we convert this shortcoming into a feature and use identical coarse-grained models to represent molecules that share nonreactive characteristics but react by different mechanisms. As a proof of concept, we use this approach to simulate and investigate distinct, yet similar, trifunctional isocyanurate resin formulations that polymerize by either chain- or step-growth. Because the underlying molecular mechanics of these models are identical, all emergent differences are a function of the reaction mechanism only. We find that the microscopic morphologies resemble related all-atom simulations and that simulated mechanical testing reasonably agrees with experiment.

36 MATERIALS SCIENCE↗

Optimizers for stabilizing likelihood-free inference

A growing number of applications in particle physics and beyond use neural networks as unbinned likelihood ratio estimators applied to real or simulated data. Precision requirements on the inference tasks demand a high-level of stability from these networks, which are affected by the stochastic nature of training. We show how physics concepts can be used to stabilize network training through a physics-inspired optimizer. In particular, the energy conserving descent (ECD) optimization framework uses classical Hamiltonian dynamics on the space of network parameters to reduce the dependence on the initial conditions while also stabilizing the result near the minimum of the loss function. We develop a version of this optimizer known as , which has few free hyperparameters with limited ranges guided by physical reasoning. We apply to representative likelihood-ratio estimation tasks in particle physics and find on average that it out-performs the widely used Adam optimizer. We expect that ECD will be a useful tool for wide array of data-limited problems, where it is computationally expensive to exhaustively optimize hyperparameters and mitigate fluctuations with ensembling.

Monte Carlo methods↗

Model for the electrical conductivity in dense plasma mixtures

A new density functional theory, average atom based model for the electrical conductivity of dense plasmas with a mixture of ion species, containing no adjustable parameters, is presented herein. The model takes the temperature, mass density and relative abundances of the species as input. It takes into account partial ionization, ionic structure, and core-valence orthogonality, and uses quantum mechanical calculations of cross sections. Comparison to an existing high fidelity but computationally expensive method reveals good agreement. The new model is computationally efficient and can reach high temperatures. A new mixing rule is also presented that gives reasonably accurate conductivities for high temperature plasma mixtures.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Nonadiabatic Molecular Dynamics Simulations Based on Time-Dependent Density Functional Tight-Binding Method

Nonadiabatic excited-state molecular dynamics underpin many photophysical and photochemical phenomena, such as exciton dynamics, charge separation and transport. In this work, we present an efficient nonadiabatic molecular dynamic (NAMD) simulation method based on time-dependent density functional tight-binding (TDDFTB) theory. Specifically, the adiabatic electronic structure, an essential NAMD input, is described at the TDDFTB level. Here, the nonadiabatic effects originating from the coupled motions of electrons and nuclei are treated by the trajectory surface hopping algorithm. To improve the computational efficiency, nonadiabatic couplings between excited states within the TDDFTB method are derived and implemented using an analytical approach. Further, the time-dependent nonadiabatic coupling scalars are calculated based on the overlap between molecular orbitals rather than the Slater determinants to speed up the simulations. In addition, the electronic decoherence scheme and a state reassigned unavoided crossings algorithm, which has been implemented in the NEXMD software, are used to improve the accuracy of the simulated dynamics and handle trivial unavoided crossings. Finally, the photoinduced nonadiabatic dynamics of a benzene molecule are simulated to demonstrate our implementation. The results for excited state NAMD simulations of benzene molecule based on TDDFTB method compare well that obtained with numerically expensive time-dependent density functional theory. The proposed methodology provides an attractive theoretical simulation tool for predicting the photophysical and photochemical properties of complex materials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

A practical extension of the recursive multi-fidelity model for the emulation of hole closure experiments

We report in regimes of high strain rate, the strength of materials often cannot be measured directly in experiments. Instead, the strength is inferred based on an experimental observable, such as a change in shape, that is matched by simulations supported by a known strength model. In hole closure experiments, the rate and degree to which a central hole in a plate of material closes during a dynamic loading event are used to infer material strength parameters. Due to the complexity of the experiment, many computationally expensive, three-dimensional simulations are necessary to train an emulator for calibration or other analyses. These simulations can be run at multiple grid resolutions, where dense grids are slower but more accurate. In an effort to reduce the computational cost, a combination of simulations with different resolutions can be combined to develop an accurate emulator within a limited training time. We explore the novel design and construction of an appropriate functional recursive multi-fidelity emulator of a strength model for tantalum in hole closure experiments that can be applied to arbitrarily large training data. Hence, by formulating a multi-fidelity model to employ low-fidelity simulations, we were able to reduce the error of our emulator by approximately 81% with only an approximately 1.6% increase in computing resource utilization.

36 MATERIALS SCIENCE↗

End-to-end orientation estimation from 2D cryo-EM1images

Cryo-electron microscopy (cryo-EM) is a Nobel Prize-winning technique for deter-mining high-resolution 3D structures of biological macromolecules. A 3D structure is reconstructed from hundreds of thousands of noisy 2D projection images. However, existing 3D reconstruction methods are still time-consuming, and one of the major computational bottlenecks is to recover the unknown orientation of the particle in16each 2D image. The dominant methods typically exploit expensive global search on each image to estimate the missing orientations. Here, a novel end-to-end supervised learning method is introduced to directly recover the missing orientations from 2D cryo-EM images. A neural network is used to approximate the mapping from images to orientations. Furthermore, a robust loss function is proposed for optimizing the parameters of the network, which can handle both asymmetric and symmetric 3D structures. Experiments on synthetic datasets with various symmetry types confirm that the neural network is capable of recovering orientations from 2D cryo-EM images, and the results on one real cryo-EM dataset further demonstrate its potential in more challenging imaging conditions.

3D reconstruction↗

NRC Multiphysics Analysis Capability Deployment FY2020: Part 3

This report details progress and activities of Idaho National Laboratory (INL) on the Nuclear Regulatory Commission (NRC) project “Development and Modeling Sup- port for Advanced Non-Light Water Reactors.” The tasks completed for this report are: Task2c: Explicit modeling of pebble transient temperature response. In this simulation, the 400 MWth Pebble-Bed Modular Reactor (PBMR) design, PBMR- 400, experiences a 20-second power ramp from 100% to 150% power. This is followed by a similar reduction in the power back to 100%. Several multiscale pebble coupling approaches are tested with one pebble per mesh element in the active core region. The results show good conservation behavior and the stability of the coupling.; Extended scope part 1: An assessment of the computational efficiency of the Discontinuous Finite Element Method (DFEM) heat transfer solver shows good scalability. The DFEM solver is a factor of 4 more expensive in solution time than the Finite Element Method (FEM) solver for heat transfer problems due to the increased number of degrees of freedom. Nonetheless, the DFEM approach provides the user with the flexibility to model gap heat transfer problems.; Extended scope part 2: The GapHeatTransferInterfaceMaterial was improved to give the user increased flexibility with the modeling of heat transfer through gaps with the DFEM solver. A number of gap parameters can now be coupled both through functions and variables.; Extended scope part 3: Demonstration of how the gap width between hexagonal fuel cells can be calculated during a heat-up transient and used in the GapHeatTransferInterface model. A full-domain DFEM model with gap expansion is coupled to a SubApp that models the thermal expansion of the base plate. The results show the expected physical behavior, although have not been fully bench-marked at this point in time.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Coalescence Prevention Algorithm for Level Set Method

An algorithm to prevent or delay bubble coalescence for the level set (LS) method is presented. This novel algorithm uses the LS method field to detect when bubbles are in close proximity, indicating a potential coalescence event, and applies a repellent force to simulate the unresolved liquid drainage force. The model is introduced by locally modifying the surface tension force near the liquid film drainage area. The algorithm can also simulate the liquid drainage time of the thin film by controlling the length of time the increased surface tension has been applied. Thus, a new method of modeling bubble coalescence has been developed. Several test cases were designed to demonstrate the capabilities of the algorithm. The simulations, including a mesh study, confirmed the abilities to identify and prevent coalescence as well as implement the time tracking portion, with an additional 10–25% computational cost. Ongoing tests aim to verify the algorithm's functionality for simulations with different flow conditions, a ranging number of bubbles, and both structured and unstructured computational mesh types. Specifically, a bubble rising toward a free surface provides a test of performance and demonstrates the ability to consistently prevent coalescence. In addition, a two bubble case and a seven bubble case provide a more complex demonstration of how the algorithm performs for larger simulations. These cases are compared to much more expensive simulations capable of resolving the liquid film drainage (through very high local mesh resolution) to investigate how the algorithm replicates the liquid film drainage process.

Engineering↗

Verifying quantitative reliability for programs that execute on unreliable hardware

Emerging high-performance architectures are expected to contain unreliable components that may exhibit soft errors, which silently corrupt the results of computations. Full detection and masking of soft errors is challenging, expensive, and, for some applications, unnecessary. For example, approximate computing applications (such as multimedia processing, machine learning, and big data analytics) can often naturally tolerate soft errors. We introduce Rely, a programming language that enables developers to reason about the quantitative reliability of an application—namely, the probability that it produces the correct result when executed on unreliable hardware. Rely allows developers to specify the reliability requirements for each value that a function produces. We perform a static quantitative reliability analysis that verifies quantitative requirements on the reliability of an application, enabling a developer to perform sound and verified reliability engineering. The analysis takes a Rely program with a reliability specification and a hardware specification that characterizes the reliability of the underlying hardware components and verifies that the program satisfies its reliability specification when executed on the underlying unreliable hardware platform. We demonstrate the application of quantitative reliability analysis on six computations implemented in Rely.

97 MATHEMATICS AND COMPUTING↗

Multidisciplinary Design Optimization and Cruise Mach Number Study of Truss-Braced Wing Aircraft

The Subsonic Ultra Green Aircraft Research (SUGAR) Phase III was led by Dr. Rakesh K. Kapania and Dr. Joseph A. Schetz at the Multidisciplinary Analysis and Design Center for Advanced Vehicles, Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg VA. The research was performed from December 2014 to December 2015. Three major areas were investigated: Multidisciplinary Design Optimization (MDO) studies of truss braced wing (TBW) and strut braced wing (SBW) vehicles at cruise Mach numbers of 0.7 and 0.8 for a flight mission similar to current market single aisle configurations. The performance and the characteristics of the optimized vehicles were compared to the SUGAR Phase II TBW vehicle. These results were obtained without applying any of the extended transonic aerodynamic and aeroelastic tools that will be discussed later. It was found that the cruise Mach number has a large effect on the best “truss” configuration. At Mach 0.7, an SBW has a better fuel consumption and better take-off gross weight (TOGW). However, at Mach 0.8, the TBW is superior because the jury strut aids in satisfying the flutter constraint; Two-dimensional, steady, transonic aerodynamic analysis of the Boeing Airfoil J (BACJ) airfoil was performed for a range of thickness ratios, Mach numbers and lift coefficients. Reynolds-averaged Navier-Stokes (RANS) equations were solved to obtain the lift-curve slope, wave drag coefficient, the location of the center of pressure and to predict the separation at the trailing edge, which may lead to buffeting. One of the goals was to develop a database of lift-curve slope and the location of center of pressure, which could be used in a transonic aeroelastic analysis. Another goal was to compare the wave drag coefficients to those predicted by Lock’s fourth-power law and also to compare the transonic effects obtained from RANS simulations to those predicted by the Korn equations. A third goal was to develop a buffet boundary, which can be integrated into the MDO framework to prevent the optimized designs from probable buffeting; A state-space transonic aeroelastic analysis tool was developed, which can incorporate the nonlinear transonic effects in the unsteady aerodynamics but is yet computationally cheap when used within the VT MDO framework. The aeroelastic analysis uses Leishman- Beddoes (LB) indicial functions, which generated a state-space representation of the aeroelastic system. The indicial functions allow the incorporation of data for steady lift-curve slope and location of the center of pressure. Thus, the steady transonic effects are included, and the unsteady aerodynamic responses are a linearization about the steady results. The aeroelastic approach discretizes the wing into numerous strips, which results in a large eigenvalue problem as each strip has eight augmented aerodynamic states as per the LB theory. Thus, to reduce the computation expense, a reduced order model (ROM) was developed. The approach was validated using a few examples.

Kapania, Rakesh K.↗

Automated Work Package: Initial Wireless Communication Platform Design, Development, and Evaluation

The Department of Energy’s Light Water Reactor Sustainability Program is developing the scientific basis to ensure long-term reliability, productivity, safety, and security of the nuclear power industry in the United States. The Instrumentation, Information, and Control (II&C) pathway of the program aims to increase the role of advanced II&C technologies to achieve this objective. One of the pathway efforts at Idaho National Laboratory (INL) is to improve the work packages execution process by replacing the expensive, inefficient, bulky, complex, and error-prone paper-based work orders with automated work packages (AWPs). An AWP is an automated and dynamic presentation of the work package designed to guide the user through the work process. It is loaded on a mobile device, such as a tablet, and is capable of communicating with plant equipment and systems to acquire plant and procedure states. The AWP replaces those functions where a computer is more efficient and reliable than a human. To enable the automatic acquisition of plant data, it is necessary to design and develop a prototype platform for data exchange between the field instruments and the AWP mobile devices. The development of the platform aims to reveal issues and solutions generalizable to large-scale implementation of a similar system. Topics such as bandwidth, robustness, response time, interference, and security are usually associated with wireless communication. These concerns, along with other requirements, are listed in an earlier INL report. Specifically, the targeted issues and performance aspects in this work are relevant to the communication infrastructure from the perspective of promptness, robustness, expandability, and interoperability with different technologies.

Automated Work Packages↗

Automated Work Package: Initial Wireless Communication Platform Design, Development, and Evaluation

The Department of Energy's Light Water Reactor Sustainability Program is developing the scientific basis to ensure long-term reliability, productivity, safety, and security of the nuclear power industry in the United States. The Instrumentation, Information, and Control (II and C) pathway of the program aims to increase the role of advanced II and C technologies to achieve this objective. One of the pathway efforts at Idaho National Laboratory (INL) is to improve the work packages execution process by replacing the expensive, inefficient, bulky, complex, and error-prone paper-based work orders with automated work packages (AWPs). An AWP is an automated and dynamic presentation of the work package designed to guide the user through the work process. It is loaded on a mobile device, such as a tablet, and is capable of communicating with plant equipment and systems to acquire plant and procedure states. The AWP replaces those functions where a computer is more efficient and reliable than a human. To enable the automatic acquisition of plant data, it is necessary to design and develop a prototype platform for data exchange between the field instruments and the AWP mobile devices. The development of the platform aims to reveal issues and solutions generalizable to large-scale implementation of a similar system. Topics such as bandwidth, robustness, response time, interference, and security are usually associated with wireless communication. These concerns, along with other requirements, are listed in an earlier INL report. Specifically, the targeted issues and performance aspects in this work are relevant to the communication infrastructure from the perspective of promptness, robustness, expandability, and interoperability with different technologies. The development of a prompt system entailed significant performance optimization, including the use of parallel processing, parallel communication, modularity, noise filtering, and frequency tuning. A robust system was achieved by enforcing a persistent continuous data exchange that automatically detects and conveys a failure of any link or component in any part of the system, and that automatically restores the communication once the link or component is restored. An expandable topology was enabled by modularity and load sharing. The number of servers, mobile devices, and field instruments is not limited from a system structure point of view. The system interoperability was achieved by enforcing the independence of the field communication application from the mobile device application. (authors)

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

A computationally efficient approach for calculating galaxytwo-point correlations

We developed a modification to the calculation of the two-point correlation function commonly used in the analysis of large-scale structure in cosmology. An estimator of the two-point correlation function is constructed by contrasting the observed distribution of galaxies with that of a uniformly populated random catalogue. The assumption that the distribution of random galaxies in redshift is independent of angular position allows us to replace pair-wise combinatorics with fast integration over probability maps. The new method significantly reduces the computation time while simultaneously increasing the precision of the calculation. Also, it only requires cosmological parameters at the last and least computationally expensive stage, which allows for faster exploration of the parameter space.

79 ASTRONOMY AND ASTROPHYSICS↗

A high-order multiscale finite-element method for time-domain acoustic-wave modeling

Accurate and efficient wave equation modeling is vital for many applications in such as acoustics, electromagnetics, and seismology. However, solving the wave equation in large-scale and highly heterogeneous models is usually computationally expensive because the computational cost is directly proportional to the number of grids in the model. We develop a novel high-order multiscale finite-element method to reduce the computational cost of time-domain acoustic-wave equation numerical modeling by solving the wave equation on a coarse mesh based on the multiscale finite-element theory. In contrast to existing multiscale finite-element methods that use only first-order multiscale basis functions, our new method constructs high-order multiscale basis functions from local elliptic problems which are closely related to the Gauss–Lobatto–Legendre quadrature points in a coarse element. Essentially, these basis functions are not only determined by the order of Legendre polynomials, but also by local medium properties, and therefore can effectively convey the fine-scale information to the coarse-scale solution with high-order accuracy. Numerical tests show that our method can significantly reduce the computation time while maintain high accuracy for wave equation modeling in highly heterogeneous media by solving the corresponding discrete system only on the coarse mesh with the new high-order multiscale basis functions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

A Massively Parallel Implementation of the CCSD(T) Method Using the Resolution-of-the-Identity Approximation and a Hybrid Distributed/Shared Memory Parallelization Model

In this work, a parallel algorithm is described for the coupled-cluster singles and doubles method augmented with a perturbative correction for triple excitations [CCSD(T)] using the resolution-of-the-identity (RI) approximation for two-electron repulsion integrals (ERIs). The algorithm bypasses the storage of four-center ERIs by adopting an integral-direct strategy. The CCSD amplitude equations are given in a compact quasi-linear form by factorizing them in terms of amplitude-dressed three-center intermediates. A hybrid MPI/OpenMP parallelization scheme is employed, which uses the OpenMP-based shared memory model for intranode parallelization and the MPI-based distributed memory model for internode parallelization. Parallel efficiency has been optimized for all terms in the CCSD amplitude equations. Two different algorithms have been implemented for the rate-limiting terms in the CCSD amplitude equations that entail and -scaling computational costs, where N O and N V denote the number of correlated occupied and virtual orbitals, respectively. One of the algorithms assembles the four-center ERIs requiring N V 4 and N O 2 N V 2 -scaling memory costs in a distributed manner on a number of MPI ranks, while the other algorithm completely bypasses the assembling of quartic memory-scaling ERIs and thus largely reduces the memory demand. It is demonstrated that the former memory-expensive algorithm is faster on a few hundred cores, while the latter memory-economic algorithm shows a better strong scaling in the limit of a few thousand cores. The program is shown to exhibit a near-linear scaling, in particular for the compute-intensive triples correction step, on up to 8000 cores. The performance of the program is demonstrated via calculations involving molecules with 24–51 atoms and up to 1624 atomic basis functions. As the first application, the complete basis set (CBS) limit for the interaction energy of the π-stacked uracil dimer from the S66 data set has been investigated. This work reports the first calculation of the interaction energy at the CCSD(T)/aug-cc-pVQZ level without local orbital approximation. The CBS limit for the CCSD correlation contribution to the interaction energy was found to be -8.01 kcal/mol, which agrees very well with the value -7.99 kcal/mol reported by Schmitz, Hättig, and Tew [ Phys. Chem. Chem. Phys. 2014 , 16 , 22167-22178]. The CBS limit for the total interaction energy was estimated to be -9.64 kcal/mol.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗