Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “SLOPE”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 865 records · Page 48
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Performance of high-order Godunov-type methods in simulations of astrophysical low Mach number flows

High-order Godunov methods for gas dynamics have become a standard tool for simulating different classes of astrophysical flows. Their accuracy is mostly determined by the spatial interpolant used to reconstruct the pair of Riemann states at cell interfaces and by the Riemann solver that computes the interface fluxes. In most Godunov-type methods, these two steps can be treated independently, so that many different schemes can in principle be built from the same numerical framework. Because astrophysical simulations often test out the limits of what is feasible with the computational resources available, it is essential to find the scheme that produces the numerical solution with the desired accuracy at the lowest computational cost. However, establishing the best combination of numerical options in a Godunov-type method to be used for simulating a complex hydrodynamic problem is a nontrivial task. In fact, formally more accurate schemes do not always outperform simpler and more diffusive methods, especially if sharp gradients are present in the flow. For this work, we used our fully compressible Seven-League Hydro (SLH) code to test the accuracy of six reconstruction methods and three approximate Riemann solvers on two- and three-dimensional (2D and 3D) problems involving subsonic flows only. We considered Mach numbers in the range from 10 −3 to 10 −1 , which are characteristic of many stellar and geophysical flows. In particular, we considered a well-posed, 2D, Kelvin–Helmholtz instability problem and a 3D turbulent convection zone that excites internal gravity waves in an overlying stable layer. Although the different combinations of numerical methods converge to the same solution with increasing grid resolution for most of the quantities analyzed here, we find that (i) there is a spread of almost four orders of magnitude in computational cost per fixed accuracy between the methods tested in this study, with the most performant method being a combination of a low-dissipation Riemann solver and a sextic reconstruction scheme; (ii) the low-dissipation solver always outperforms conventional Riemann solvers on a fixed grid when the reconstruction scheme is kept the same; (iii) in simulations of turbulent flows, increasing the order of spatial reconstruction reduces the characteristic dissipation length scale achieved on a given grid even if the overall scheme is only second order accurate; (iv) reconstruction methods based on slope-limiting techniques tend to generate artificial, high-frequency acoustic waves during the evolution of the flow; and (v) unlimited reconstruction methods introduce oscillations in the thermal stratification near the convective boundary, where the entropy gradient is steep.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

The DEHVILS in the details: Type Ia supernova Hubble residual comparisons and mass step analysis in the near-infrared

Measurements of type Ia supernovae (SNe Ia) in the near-infrared (NIR) have been used both as an alternate path to cosmology compared to optical measurements and as a method of constraining key systematics for the larger optical studies. With the DEHVILS sample, the largest published NIR sample with consistent NIR coverage of maximum light across three NIR bands ( Y, J , and H ), we check three key systematics: (i) the reduction in Hubble residual scatter as compared to the optical, (ii) the measurement of a “mass step” or lack thereof and its implications, and (iii) the ability to distinguish between various dust models by analyzing slopes and correlations between Hubble residuals in the NIR and optical. We produce SN Ia simulations of the DEHVILS sample and find that it is harder to differentiate between various dust models than previously understood. Additionally, we find that fitting with the current SALT3-NIR model does not yield accurate wavelength-dependent stretch-luminosity correlations, and we propose a limited solution for this problem. From the data, we see that (i) the standard deviation of Hubble residual values from NIR bands treated as standard candles are 0.007–0.042 mag smaller than those in the optical, (ii) the NIR mass step is not constrainable with the current sample size of 47 SNe Ia from DEHVILS, and (iii) Hubble residuals in the NIR and optical are correlated in the data. We test a few variations on the number and combinations of filters and data samples, and we observe that none of our findings or conclusions are significantly impacted by these modifications.

Astronomy & Astrophysics↗
DOE OSTI2025

Stability analysis of the Eulerian–Lagrangian finite volume methods for nonlinear hyperbolic equations in one space dimension

In this paper, we construct a novel Eulerian–Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine–Hugoniot junmp condition. Unfortunately, to avoid the intersection of the partition lines, the time step sizes are still limited. To fix this gap, we detect effective troubled cells (ETCs) and carefully design the influence region of each ETC, within which the partitioned space-time regions are merged together to form a new one. Then with the new partition of the space-time domain, we theoretically prove that the proposed first-order scheme with Euler forward time discretization is total-variation-diminishing and maximum-principle-preserving with at least twice larger time step constraints than the classical first order Eulerian method for Burgers’ equation. Numerical experiments verify the optimality of the designed time step sizes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Dynamic experiments to study the α - ε phase transition in cerium

The ability to understand and predict the response of matter at extreme conditions requires knowledge of a material’s equation-of-state including the location of phase boundaries, transition kinetics, and the evolution of material strength. Cerium is a material with a complex phase diagram that continues to attract significant scientific interest. Recent dynamic experiments have provided information on the low-pressure γ–α phase transition, sound speed, and Hugoniot data for the higher-pressure α phase, as well as the incipient shock melt transition. Despite these efforts, there are still regions of the phase diagram that are largely unexplored dynamically, including the high-pressure region below the melt boundary. Along a room temperature isotherm, diamond anvil cell data report a transition to the ϵ phase between 13 and 17 GPa. At higher temperatures, similar diamond anvil cell data show significant disagreement regarding the existence, location, and slope of the ϵ-phase boundary. In this work, double-shock loading was used to access the α–ϵ region of the phase diagram to obtain equation-of-state information and to determine the location of the ϵ-phase boundary for shock loading.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2020

High power edge-cum-surface emitting terahertz laser arrays phased locked by vacuum guided plasmon waves

Terahertz semiconductor quantum-cascade lasers (QCLs) are widely implemented with metallic cavities that support low-loss plasmonic optical modes at long wavelengths. However, resonant optical modes in such cavities suffer from poor radiative characteristics due to their subwavelength transverse dimensions. Consequently, single-mode terahertz QCLs with metallic cavities and large (>100 mW) output power have only been realized in the surface-emitting configuration that affords a large radiating surface. Here, we demonstrate a method to enhance radiative outcoupling from such plasmonic lasers for high-power emission in the edge-emitting (end-fire or longitudinal) direction. Single-sided plasmon waves propagating in vacuum are resonantly excited in surrounding medium of metallic cavities with the QCL semiconductor medium. The vacuum guided plasmon waves with a large wavefront phase-lock multiple metallic cavities longitudinally, which leads to intense radiation in multiple directions, including that in the longitudinal direction in a narrow single-lobed beam. The multicavity array radiates predominantly in a single spectral mode. A peak-power output of 260 mW and a slope efficiency of 303 mW/A are measured for the end-fire beam from a 3.3 THz QCL operating at 54 K in a Stirling cooler. Finally, single-mode operation and lithographic tuning across a bandwidth of ~150 GHz are demonstrated.

42 ENGINEERING↗
DOE OSTI2020

First-principles equations of state and structures of liquid metals in multi-megabar conditions

Liquid metals at extreme pressures and temperatures are widely interested in the high-pressure community. Based on density functional theory molecular dynamics, we conduct first-principles investigations on the equation of state ( EOS) and structures of four metals (Cu, Fe, Pb, and Sn) at 1.5–5 megabar conditions and 5 × 10 3 –4 × 10 4 K. Our first-principles EOS data enable evaluating the performance of four EOS models in predicting Hugoniot densities and temperatures of the four systems. We find the melting temperature of Cu is 1000–2000 K higher and shows a similar Clapeyron slope, in comparison to those of Fe. Our structure, coordination number, and diffusivity analysis indicates all the four liquid metals form similar simple close-packed structures. Our results set theoretical benchmarks for EOS development and structures of metals in their liquid states and under dynamic compression.

36 MATERIALS SCIENCE↗
DOE OSTI2020

A simple model to scope out parameter space for indirect drive designs on NIF

In this work, we present a simple model to scope out parameter space for indirect-drive, inertial confinement fusion designs for the National Ignition Facility laser. Because the parameter space is large, simple models can be used to identify regions of parameter space for further study with more sophisticated models and experiments. We include a model for Hohlraum radiation drive and symmetry—both based on empirical scalings from the data. The model for radiation drive is based on assuming that the high atomic number (Z) Hohlraum wall dominates the energy balance during the high power, peak of the pulse (≳ 300 TW). We find that the time-dependent radiation drive flux can be described by the running integral of the laser energy divided by the Hohlraum area multiplied by constant slopes in two distinct time periods. The first period is when the laser power rises rapidly, so the radiation temperature increases due to changes in laser power and wall albedo. The second period is during peak power—here, the laser power is typically held constant—so, the radiation temperature increases only due to changes in the wall albedo. This model is applied to several NIF designs with different Hohlraum sizes, laser pulse length durations, and peak powers and energies. Drive and symmetry models can be combined to find regions of parameter space that have high capsule absorbed energy while maintaining a symmetric implosion. We propose a new metric for evaluating designs based on minimizing the radius at which the maximum implosion kinetic energy is achieved.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Determination of acoustic nonlinearity parameters using thermal modulation of ultrasonic waves

This study presents a test method and its theoretical framework to determine the acoustic nonlinearity parameters (α, β, δ) of material using thermal modulation of ultrasonic waves. Temperature change-induced thermal strain excites the nonlinear response of the material and modulates the ultrasonic wave propagating in it. Experimental results showed a strong correlation between the relative wave velocity change and the temperature change. With a quadratic polynomial model, the acoustic nonlinearity parameters were obtained from the polynomial coefficients by curve fitting the experimental data. Their effects on thermal-induced velocity change were discussed. The parameters α, β, and δ govern the hysteretic gap, average slope, and curvature of the correlation curve, respectively. The proposed theory was validated on aluminum, steel, intact and damaged concrete samples. The obtained nonlinear parameters show reasonable agreement with values reported in the literature. Compared to other nonlinear acoustic methods using vibration or acoustic excitation, the thermal modulation method generates more uniform, slow changing, and larger strain field in the test sample. Employing the thermal effect as the driving force for nonlinearity instead of an undesired influencing factor, this method can measure the absolute values of α, β, and δ with good accuracy using a simple ultrasonic test setup.

Physics↗
DOE OSTI2020

Tri-gate GaN junction HEMT

This work presents a tri-gate GaN junction high-electron-mobility transistor (JHEMT) concept in which the p–n junction wraps around the AlGaN/GaN fins in the gate region. This tri-gate JHEMT differs from all existing GaN FinFETs and tri-gate HEMTs, as they employ a Schottky or a metal-insulator-semiconductor (MIS) gate stack. A tri-gate GaN JHEMT is fabricated using p-type NiO with the gate metal forming an Ohmic contact to NiO. The device shows minimal hysteresis and a subthreshold slope of 63 ± 2 mV/decade with an on-off current ratio of 10 8 . Compared to the tri-gate MISHEMTs fabricated on the same wafer, the tri-gate JHEMTs exhibit higher threshold voltage (V TH ) and achieve positive V TH without the need for additional AlGaN recess. In addition, this tri-gate JHEMT with a fin width of 60 nm achieves a breakdown voltage (BV) > 1500 V (defined at the drain current of 1 μA/mm at zero gate bias) and maintains the high BV with the fin length scaled down to 200 nm. In comparison, the tri-gate MISHEMTs with narrower and longer fins show punch-through at high voltages. Moreover, when compared to planar enhancement mode HEMTs, tri-gate JHEMTs show significantly lower channel sheet resistance in the gate region. Finally, these results illustrate a stronger channel depletion and electrostatic control in the junction tri-gate compared to the MIS tri-gate and suggest great promise of the tri-gate GaN JHEMTs for both high-voltage power and low-voltage power/digital applications.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Multi-energy reconstructions, central electron temperature measurements, and early detection of the birth and growth of runaway electrons using a versatile soft x-ray pinhole camera at MST

A multi-energy soft x-ray pinhole camera has been designed, built, and deployed at the Madison Symmetric Torus to aid the study of particle and thermal transport, as well as MHD stability physics. This novel imaging diagnostic technique employs a pixelated x-ray detector in which the lower energy threshold for photon detection can be adjusted independently on each pixel. The detector of choice is a PILATUS3 100 K with a 450 μm thick silicon sensor and nearly 100 000 pixels sensitive to photon energies between 1.6 and 30 keV. An ensemble of cubic spline smoothing functions has been applied to the line-integrated data for each time-frame and energy-range, obtaining a reduced standard-deviation when compared to that dominated by photon-noise. The multi-energy local emissivity profiles are obtained from a 1D matrix-based Abel-inversion procedure. Central values of Te can be obtained by modeling the slope of the continuum radiation from ratios of the inverted radial emissivity profiles over multiple energy ranges with no a priori assumptions of plasma profiles, magnetic field reconstruction constraints, high-density limitations, or need of shot-to-shot reproducibility. In tokamak plasmas, a novel application has recently been tested for early detection, 1D imaging, and study of the birth, exponential growth, and saturation of runaway electrons at energies comparable to 100 × Te,0; thus, early results are also presented.

Delgado-Aparicio, L. F. (ORCID:0000000167394380)↗
DOE OSTI2021

Development of a deep learning based automated data analysis for step-filter x-ray spectrometers in support of high-repetition rate short-pulse laser-driven acceleration experiments

We present a deep learning based framework for real-time analysis of a differential filter based x-ray spectrometer that is common on short-pulse laser experiments. The analysis framework was trained with a large repository of synthetic data to retrieve key experimental metrics, such as slope temperature. With traditional analysis methods, these quantities would have to be extracted from data using a time-intensive and manual analysis. Furthermore, this framework was developed for a specific diagnostic, but may be applicable to a wide variety of diagnostics common to laser experiments and thus will be especially crucial to the development of high-repetition rate (HRR) diagnostics for HRR laser systems that are coming online.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Neutron irradiation induced defects in oxides and their impact on the oxide properties

Understanding the irradiation-induced defects in oxides is of interest for a wide range of applications. ZnO is an interesting oxide with mixed ionic and covalent bonding that contains a variety of point defect structures—making it an excellent model for studying irradiation-induced defects and their impact on properties. Here, we investigate the effects of neutron irradiation on the formation of defects and on the structural, optical, and electrical properties of ZnO single crystals. We observe the formation of vacancies and voids via positron annihilation spectroscopy. Neutron irradiation led to a significant deterioration of the ZnO structure and formed a high concentration of point defects, vacancy clusters, and voids with large disparities in their structure across variable irradiation times. It also led to significant changes in the optical properties and sample color. Irradiation for 444 h induced a high concentration of Cu acceptors as well as a high concentration of Ga donors. Additionally, temperature-dependent Hall effect measurements revealed the competing production of donors and acceptors and showed an increase in the slope of the carrier freeze-out curve with increasing irradiation dose. This work demonstrates the combined effects of neutron irradiation in producing a wide range of structural defects, impurities, and dopants in oxides and their enormous impact on modifying the oxide structure and both the optical and electronic properties. It particularly emphasizes the importance of considering the production of new impurities and dopants during the neutron irradiation of oxides.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Two-dimensional electronic–vibrational spectroscopy: Exploring the interplay of electrons and nuclei in excited state molecular dynamics

Two-dimensional electronic-vibrational spectroscopy (2DEVS) is an emerging spectroscopic technique which exploits two different frequency ranges for the excitation (visible) and detection (infrared) axes of a 2D spectrum. In contrast to degenerate 2D techniques, such as 2D electronic or 2D infrared spectroscopy, the spectral features of a 2DEV spectrum report cross correlations between fluctuating electronic and vibrational energy gaps rather than autocorrelations as in the degenerate spectroscopies. The center line slope of the spectral features reports on this cross correlation function directly and can reveal specific electronic-vibrational couplings and rapid changes in the electronic structure, for example. The involvement of the two types of transition moments, visible and infrared, makes 2DEVS very sensitive to electronic and vibronic mixing. 2DEV spectra also feature improved spectral resolution, making the method valuable for unraveling the highly congested spectra of molecular complexes. The unique features of 2DEVS are illustrated in this paper with specific examples and their origin described at an intuitive level with references to formal derivations provided. Although early in its development and far from fully explored, 2DEVS has already proven to be a valuable addition to the tool box of ultrafast nonlinear optical spectroscopy and is of promising potential in future efforts to explore the intricate connection between electronic and vibrational nuclear degrees of freedom in energy and charge transport applications.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Diverse simulations of time-resolved photoluminescence in thin-film solar cells: A SnO 2 /CdSe y Te 1-y case study

Time-resolved photoluminescence (TRPL) is widely used to measure carrier lifetime in thin-film solar cell absorbers. However, the injection dependence of data and frequent non-exponential decay shapes complicate the interpretation. Here, we develop a numerical model to simulate injection-dependent TRPL measurements in a SnO 2 /CdSe y Te 1-y case study solar cell structure, considering parameters of interest to researchers in industry and academia. Previous simulations have shown that, in low injection, excess electrons and holes injected by the laser pulse are rapidly separated in the electric field formed by the pn junction. As a result, at early times, the PL signal can decay faster than the Shockley-Read-Hall lifetime in the absorber bulk (t_bulk). Prior simulations have shown that the charge stored in the junction can slowly leak out to affect decays at late times. However, it has not been clear if and to what degree charge storage can affect the slopes extracted from TRPL decays - t_2 - commonly cited as the TRPL-measured lifetime. Here, we show that charge storage can, in some cases, result in t_2 values that substantially overestimate t_bulk. Previous simulations indicate that high injection conditions can screen the junction field and minimize charge separation. Here, we show that continued injection increases can drive down t_2 below t_bulk as radiative recombination becomes dominant. We catalog charge storage and radiative recombination impacts for a diverse set of material parameters and compare results to double-heterostructure models.

CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS,S↗
DOE OSTI2022

Nanoscale laser flash measurements of diffuson transport in amorphous Ge and Si

The thermal properties of amorphous materials have attracted significant attention due to their technological importance in electronic devices. In addition, the disorder-induced breakdown of the phonon gas model makes vibrational transport in amorphous materials a topic of fundamental interest. In the past few decades, theoretical concepts, such as propagons, diffusons, and locons, have emerged to describe different types of vibrational modes in disordered solids. However, experiments can struggle to accurately determine which types of vibrational states carry the majority of the heat. In this study, we use nanoscale laser flash measurements (front/back time-domain thermoreflectance) to investigate thermal transport mechanisms in amorphous Ge and amorphous Si thin-films. We observe a nearly linear relationship between the amorphous film’s thermal resistance and the film’s thickness. The slope of the film’s thermal resistance vs thickness corresponds to a thickness-independent thermal conductivity of 0.4 and 0.6 W/(m K) for a-Ge and a-Si, respectively. This result reveals that the majority of heat currents in amorphous Si and Ge thin films prepared via RF sputtering at room temperature are carried by diffusons and/or propagons with mean free paths less than a few nanometers.

36 MATERIALS SCIENCE↗
DOE OSTI2022

The role of breakup and coalescence in fine-scale bubble-induced turbulence. I. Dynamics

We study the effect of bubble breakup and coalescence on fine-scale dynamics of bubbly turbulent flows using direct numerical simulations. We perform two different simulations of dilute bubbly flows of void fraction 0.5%: one with bubbles breaking up and coalescing and the other without these physical processes. The volume of the fluid method is used for simulating bubbles undergoing breakup and coalescence while the bubbles are treated as rigid spheres in the immersed boundary method simulation. The energy spectrum in both types of simulation, consistent with previous studies, exhibits a -3 slope. We follow a single infinitesimal fluid element as it evolves to understand velocity gradient dynamics using conditional mean trajectories. We note finite-time divergence when the fluid element evolves under the action of inertial and pressure dynamics. The inertial, pressure, and viscous velocity gradient dynamics, when considered individually, produce the same results for bubble-induced turbulence (BIT) as with the classical homogeneous isotropic turbulence (HIT). Yet when the overall velocity gradient dynamics is considered, BIT results in non-cyclic trajectories moving toward stable node and unstable saddle while classical HIT shows cyclic behavior in their trajectories that move toward the origin. Interestingly, both the volume of fluid and immersed boundary simulations produce similar results. Therefore, there are two main takeaways from this research. First, new velocity gradient models are needed for BIT as their velocity gradients behavior is entirely different from the HIT. Second, we can neglect the bubble topology, breakup, and coalescence while studying or modeling the fine-scale dynamics of BIT.

42 ENGINEERING↗
DOE OSTI2022

Computational studies of structural, energetic, and electronic properties of pure Pt and Mo and mixed Pt/Mo clusters: Comparative analysis of characteristics and trends

The added technological potential of bimetallic clusters and nanoparticles, as compared to their pure (i.e., one-component) counterparts, stems from the ability to further fine-tune their properties and, consequently, functionalities through a simultaneous use of the “knobs” of size and composition. The practical realization of this potential can be greatly advanced by the knowledge of the correlations and relationships between the various characteristics of bimetallic nanosystems on the one hand and those of their pure counterparts as well as pure constituent components on the other hand. Here, in this study, we present results of a density functional theory based study of pure Pt n and Mo n clusters aimed at revisiting and exploring further their structural, electronic, and energetic properties. These are then used as a basis for analysis and characterization of the results of calculations on two-component Pt n-m Mo m clusters. The analysis also includes establishing relationships between the properties of the Pt n-m Mo m clusters and those of their Pt n-m and Mo m components. One of the particularly intriguing findings suggested by the calculated data is a linear dependence of the average binding energy per atom in sets of Pt n-m Mo m clusters that have the same fixed number m of Mo atoms and different number n-m of Pt atoms on the fractional content (n-m)/n of Pt atoms. We derive an analytical model that establishes the fundamental basis for this linearity and expresses its parameters—the m-dependent slope and intercept—in terms of characteristic properties of the constituent components, such as the average binding energy per atom of Mo m and the average per-atom adsorption energy of the Pt atoms on Mo m . The conditions of validity and degree of robustness of this model and of the linear relationship predicted by it are discussed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Dynamical electron correlation and the chemical bond. II. Recoupled pair bonds in the $a$ 4 Σ - states of CH and CF

We extended our studies of the effect of dynamical electron correlation on the covalent bonds in the AH and AF series (A = B–F) to the recoupled pair bonds in the excited $a$ 4 Σ - states of the CH and CF molecules. Dynamical correlation is energetically less important in the $a$ 4 Σ - states than in the corresponding X 2 Π states for both molecules, which is reflected in smaller changes in bond energies ($D$ e ). Changes in the equilibrium bond distance ($R$ e ) and vibrational frequency ($ω$ e ), on the other hand, are influenced by the changes in the slope and curvature of the dynamical electron correlation energy as a function of the internuclear distance $R$, $E$ DEC ($R$). In the CH($a$ 4 Σ - ) state, these changes are much smaller than in the CH(X 2 Π) state, but in the CF($a$ 4 Σ - ) state, they are larger, reflecting a significant difference in the shapes of $E$ DEC ($R$) curves. At large $R$, the shape of $E$ DEC ($R$) curves for covalent and recoupled pair bonds is similar although different in magnitude. For the CH($a$ 4 Σ - ) state, $E$ DEC ($R$) has a minimum at $R$ = $R$ e + 0.72 Å as the orbitals associated with the formation of the recoupled pair bond switch places. $E$ DEC (R) for the CF($a$ 4 Σ - ) state decreases continuously throughout the bound region of the potential energy curve because the dynamical correlation energy associated with the electrons in the lone pair orbitals is increasing. In conclusion, these results support our earlier conclusion that we still have much to learn about the nature of dynamical electron correlation in molecules.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗