Complete Basis Set Research Articles

Thermochemistry and mechanisms of the Pt+ + SO2 reaction from guided ion beam tandem mass spectrometry and theory.

The kinetic energy dependences of the reactions of Pt+ (2D5/2) with SO2 were studied using a guided ion beam tandem mass spectrometer and theory. The observed cationic products are PtO+ and PtSO+, with small amounts of PtS+, all formed in endothermic reactions. Modeling the kinetic energy dependent product cross sections allows determination of the product bond dissociation energies (BDEs): D0(Pt+-O) = 3.14 ± 0.11eV, D0(Pt+-S) = 3.68 ± 0.31eV, and D0(Pt+-SO) = 3.03 ± 0.12eV. The oxide BDE agrees well with more precise literature values, whereas the latter two results are the first such measurements. Quantum mechanical calculations were performed for PtO+, PtS+, PtO2 +, and PtSO+ at the B3LYP and coupled-cluster with single, double, and perturbative triple [CCSD(T)] levels of theory using the def2-XZVPPD (X = T, Q) and aug-cc-pVXZ (X = T, Q, 5) basis sets and complete basis set extrapolations. These theoretical BDEs agree well with the experimental values. After including empirical spin-orbit corrections, the product ground states are determined as PtO+ (4Σ3/2), PtS+ (4Σ3/2), PtO2 + (2Σg +), and PtSO+ (2A'). Potential energy profiles including intermediates and transition states for each reaction were also calculated at the B3LYP/def2-TZVPPD level. Periodic trends in the thermochemistry of the group 9 metal chalcogenide cations are compared, and the formation of PtO+ from the Pt+ + SO2 reaction is compared with those from the Pt+ + O2, CO2, CO, and NO reactions.

Development and testing of an algorithm for efficient MP2/CCSD(T) energy estimation of molecular clusters with the 2-body approach.

This work reports the development and testing of an automated algorithm for estimating the energies of weakly bound molecular clusters employing correlated theory. Firstly, the monomers and dimers of (homo/hetero) clusters are identified, and the sum of one-body and two-body contributions to correlation energy is calculated. The addition of this contribution to the Hartree-Fock full calculation (FC) energies provides a good estimate of the total energies at Møller-Plesset second-order perturbation theory (MP2)/coupled-cluster method with singles and doubles (CCSD) (T)-level theory using augmented Dunning basis sets. The estimated energies for several test clusters show an excellent agreement with their FC counterparts, with a substantial wall-clock time saving employing off-the-shelf hardware. Furthermore, the complete basis set (CBS) limit for MP2 energy computed using the two-body approach also agrees with its CBS energy with its FC counterpart.

Ab initio potential energy functions, spectroscopy and thermal physics for krypton-contained rare gas dimers

Three state-of-the-art ab initio interatomic potential energy functions (PEFs) for the Kr-Ne, Kr-Ar, Kr-Xe rare gas dimers are presented. These PEFs are based upon extrapolation to complete basis set (CBS) limit results of supermolecular computations at the coupled-cluster level with single, double, and perturbative triple excitations (CCSD(T)) and correlation-consistent basis sets up to sextuple-zeta quality, augmented with mid-bond functions. Core-core and core-valence electron correlation, higher-order coupled-cluster excitations up to perturbative quadruple excitations, CCSDT(Q), and scalar relativistic corrections have also been taken into account. We have employed Morse/Long Range (MLR) potential energy functions (PEFs) to fit the total interaction energies as a function of the internuclear separation, followed by the calculation of spectroscopic parameters, vibrational and rotational energy levels, absorption spectra, and thermophysical properties over an extensive temperature range. Good agreement with experimental data and an empirical potential energy function shows that our ab initio MLR PEFs are superior not only to previous PEFs, but also to most empirical PEFs.

The Reaction of the Methylidyne Radical (CH X2Π) with the Hydrogen Cyanide (HCN X1∑+) Molecule in Cold Molecular Clouds and Planetary Atmospheres

The reaction of the methylidyne (CH; X2Π) radical with hydrogen cyanide (HCN; X1∑) molecule was studied at a collision energy of 4.0 kJ/mol with ab initio calculations of the potential energy surface (PES). Geometries and potential energies of reactants, products, intermediates and transition states (TS) for the reaction were found by means of ab initio quantum chemical method ωB97xd/cc-pVTZ and the higher-level corrections were evaluated at the CCSD(T)-F12 level of theory with the cc-pVQZ-f12 (E1) and cc-pVTZ-f12 (E2) basis sets. The calculated values then were used for extrapolation to the complete basis set (CBS) limit using the two-point expression E(CBS) = E1 + 0.69377 × (E1 – E2). Analysis of the found energies, structural and kinetic characteristics of the involved compounds allowed us to determine the reaction paths leading to the formation of linear and cyclic intermediates, as well as to the formation of atomic and molecular hydrogen. Those results were utilized in Rice−Ramsperger−Kassel−Marcus calculations of the product branching ratios at the zero pressure limit – common approach in modelling of the cold molecular clouds chemistry. Mechanism identified emphasizes importance of the CH+HCN reaction as an important supplier of the initial bricks for building heterocyclic hydrocarbons in extreme environments.

Generation of Basis Sets for Accurate Molecular Calculations: Application to Helium Atom and Dimer

A new approach for basis set generation is reported and tested in helium atom and dimer. The basis sets thus computed, named sigma, range from DZ to 5Z and consist of the same composition as Dunning basis sets but with a different treatment of contractions. The performance of the sigma sets is analyzed for energy and other properties of He atom and He dimer, and the results are compared with those obtained with Dunning and ANO basis sets. The sigma basis sets and their extended versions up to triple augmented provide better energy values than Dunning basis sets of the same composition, and similar values to those attained with the currently available ANO. Extrapolation to complete basis set of correlation energy is compared between the sigma basis sets and those of Dunning, showing the better performance of the former in this respect.

Early Transition Metals Strengthen the B2 Bond in MB2 Complexes.

The bond dissociation energies of early transition metal diborides (M-B2, M = Sc, Ti, V, Y, Mo) have been measured by observation of the sharp onset of predissociation in a highly congested spectrum. Density functional and CCSD(T) ab initio calculations, extrapolated to the complete basis set limit, have been used to examine the electronic structure of these species. The computations demonstrate the formation of bonding orbitals between the metal d orbitals and the 1πu bonding orbitals of B2, leading to the transfer of metallic electron density into the bonding 1πu orbitals, strengthening both the M-B and B-B bonds in the molecule. This runs counter to most metal-ligand π interactions, where electron density is generally transferred into π antibonding orbitals of the ligand.

Noncanonical Isomers of Nucleoside Cation Radicals: An Ab Initio Study of the Dark Matter of DNA Ionization.

Cation radicals of DNA nucleosides, 2'-deoxyadenosine, 2'-deoxyguanosine, 2'-deoxycytidine, and 2'-deoxythymidine, can exist in standard canonical forms or as noncanonical isomers in which the charge is introduced by protonation of the nucleobase, whereas the radical predominantly resides in the deoxyribose moiety. Density functional theory as well as correlated ab initio calculations with coupled clusters (CCSD(T)) that were extrapolated to the complete basis set limit showed that noncanonical nucleoside ion isomers were thermodynamically more stable than their canonical forms in both the gas phase and as water-solvated ions. This indicated the possibility of exothermic conversion of canonical to noncanonical forms. The noncanonical isomers were calculated to have very low adiabatic ion-electron recombination energies (REad) for the lowest-energy isomers 2'-deoxy-(N-3H)adenos-1'-yl (4.74 eV), 2'-deoxy-(N-7H)guanos-1'-yl (4.66 eV), 2'-deoxy-(N-3H)cytid-1'-yl (5.12 eV), and 2'-deoxy-5-methylene-(O-2H)uridine (5.24 eV). These were substantially lower than the REad value calculated for the canonical 2'-deoxyadenosine, 2'-deoxy guanosine, 2'-deoxy cytidine, and 2'-deoxy thymidine cation radicals, which were 7.82, 7.46, 8.14, and 8.20 eV, respectively, for the lowest-energy ion conformers of each type. Charge and spin distributions in noncovalent cation-radical dA⊂dT and dG⊂dC nucleoside pairs and dAT, dCT, dTC, and dGC dinucleotides were analyzed to elucidate the electronic structure of the cation radicals. Born-Oppenheimer molecular dynamics trajectory calculations of the dinucleotides and nucleoside pairs indicated rapid exothermic proton transfer from noncanonical T+· to A in both dAT+· and dA⊂dT+·, leading to charge and radical separation. Noncanonical T+· in dCT+· and dTC+· initiated rapid proton transfer to cytosine, whereas the canonical dCT+· dinucleotide ion retained the cation radical structure without isomerization. No spontaneous proton transfer was found in dGC+· and dG⊂dC+· containing canonical neutral and noncanonical ionized deoxycytidine.

Ab Initio Composite Approaches for Heavy Element Energetics: Ionization Potentials for the Actinide Series of Elements.

The first, second, and third gas-phase ionization potentials have been determined for the actinide series of elements using an ab initio composite scalar and fully relativistic approach, employing the coupled cluster with single, double, and perturbative triple excitations (CCSD(T)) and Dirac Hartree-Fock (DHF) methods, extrapolated to the complete basis set (CBS) limit. The impact of electron correlation and basis set choice within this framework are examined. Additionally, the first three ionization potentials were obtained using an ab initio heavy element correlation-consistent Composite Approach (here referred to as α-ccCA). This is the first utilization of a ccCA for actinide species.

Small-Basis Set Density-Functional Theory Methods Corrected with Atom-Centered Potentials.

Density functional theory (DFT) is currently the most popular method for modeling noncovalent interactions and thermochemistry. The accurate calculation of noncovalent interaction energies, reaction energies, and barrier heights requires choosing an appropriate functional and, typically, a relatively large basis set. Deficiencies of the density-functional approximation and the use of a limited basis set are the leading sources of error in the calculation of noncovalent and thermochemical properties in molecular systems. In this article, we present three new DFT methods based on the BLYP, M06-2X, and CAM-B3LYP functionals in combination with the 6-31G* basis set and corrected with atom-centered potentials (ACPs). ACPs are one-electron potentials that have the same form as effective-core potentials, except they do not replace any electrons. The ACPs developed in this work are used to generate energy corrections to the underlying DFT/basis-set method such that the errors in predicted chemical properties are minimized while maintaining the low computational cost of the parent methods. ACPs were developed for the elements H, B, C, N, O, F, Si, P, S, and Cl. The ACP parameters were determined using an extensive training set of 118655 data points, mostly of complete basis set coupled-cluster level quality. The target molecular properties for the ACP-corrected methods include noncovalent interaction energies, molecular conformational energies, reaction energies, barrier heights, and bond separation energies. The ACPs were tested first on the training set and then on a validation set of 42567 additional data points. We show that the ACP-corrected methods can predict the target molecular properties with accuracy close to complete basis set wavefunction theory methods, but at a computational cost of double-ζ DFT methods. This makes the new BLYP/6-31G*-ACP, M06-2X/6-31G*-ACP, and CAM-B3LYP/6-31G*-ACP methods uniquely suited to the calculation of noncovalent, thermochemical, and kinetic properties in large molecular systems.

Theoretical investigations on the unimolecular decomposition mechanisms of isopropyl acetate

In this work, a comprehensive theoretical study was performed on the pyrolysis mechanism of Isopropyl acetate (IPA) using density functional methods (M06–2X, ωB97XD) and ab initio complete basis set−quadratic Becke3 (CBS-QB3) methods under combustion conditions over temperature ranges 600−1700 K and pressure ranging from 0.001 to 100 atm. The enthalpies of formation (AE), bond dissociation energies (BDEs), ionization energies (IE), and electron affinities (EA) of IPA and its main dissociation were calculated at the CBS-QB3 method and compared with available experimental results. The IPA pyrolysis mechanism was extended to include fourteen channels, seven complex fission reactions, and seven simple bond fission reactions. Among the investigated reactions, the formation of acetic acid and propene can occur through two different pathways; six-membered ring transition state (TS1SMR) or four-membered ring transition state (TS1FMR), with TS1SMR being 17.40 kcal mol−1 more preferable. The effect of temperature and pressure change was investigated using the transition state theory (TST) in conjunction with Wigner (W) tunneling corrections and the Statistical Rice-Ramsperger-Kassel-Marcus (RRKM) theory, respectively. The branching ratios (BR) of all chemical reactions involved were estimated. The results indicate that acetic acid and propene dominate up to 1400 K, but at high temperatures, simple bond fission reactions lead IPA pyrolysis. The BR analysis results are excellent agreement with the obtained BDEs and previous literature on similar biodiesel esters.

The linear and nonlinear inverse Compton scattering between microwaves and electrons in a resonant cavity

The new scheme of the energy measurement of the extremely high energy electron beam with the inverse Compton scattering between electrons and microwave photons requires the precise calculation of the interaction cross section of electrons and microwave photons in a resonant cavity. In the local space of the cavity, the electromagnetic field is expressed by Bessel functions. Although Bessel functions can form a complete set of orthogonal basis, it is difficult to quantify them directly as fundamental wave functions. Fortunately, with the Fourier expansion of Bessel functions, the local electromagnetic field can be considered as the superposition of a series of plane waves. Therefore, with corresponding corrections of the cross section formula of the classical Compton scattering, the cross section of the linear or nonlinear microwave Compton scattering in the local space can be described accurately. As an important application of our results in astrophysics, corresponding ground verification devices can be designed to perform experimental verifications on the prediction of the Sunyaev–Zeldovich (SZ) effect of the cosmic microwave background radiation. Our results could also provide a new way to generate wave sources with strong practical value, such as the terahertz waves, the ultra-violet (EUV) waves, or the mid-infrared beams.

Ab initio Morse/long-range potential energy functions plus spectroscopic and thermophysical properties of heteronuclear diatomic complexes of xenon with the rare gases

New interatomic potential energy curves of xenon-helium (Xe-He), xenon-neon (Xe-Ne), and xenon-argon (Xe-Ar) atom pairs are developed by state-of-the-art ab initio calculations. Coupled cluster theory with single, double, and perturbative triple excitations [CCSD(T)] is used to calculate the majority of the interaction between atoms, on (multiple) augmented correlation-consistent basis sets for valence orbitals, extrapolated to the complete basis set (CBS) limit. Contributions from inner nuclear electronic correlation, relativistic effects, and higher-order excitations beyond CCSD(T) are also considered as corrections. For each system, an analytical Morse/long-range (MLR) potential functions, is fitted to 32 ab initio data using only 13 parameters: the poorest-quality fit yielded a dimensionless-root-mean-square-deviation (dd¯) of only 1.51. The ro-vibrational energy levels and wave functions, resonance states, effective radial potential energy curves, absorption spectra and thermophysical properties have also been computed.

Fast and Accurate Quantum Mechanical Modeling of Large Molecular Systems Using Small Basis Set Hartree-Fock Methods Corrected with Atom-Centered Potentials.

There has been significant interest in developing fast and accurate quantum mechanical methods for modeling large molecular systems. In this work, by utilizing a machine learning regression technique, we have developed new low-cost quantum mechanical approaches to model large molecular systems. The developed approaches rely on using one-electron Gaussian-type functions called atom-centered potentials (ACPs) to correct for the basis set incompleteness and the lack of correlation effects in the underlying minimal or small basis set Hartree-Fock (HF) methods. In particular, ACPs are proposed for ten elements common in organic and bioorganic chemistry (H, B, C, N, O, F, Si, P, S, and Cl) and four different base methods: two minimal basis sets (MINIs and MINIX) plus a double-ζ basis set (6-31G*) in combination with dispersion-corrected HF (HF-D3/MINIs, HF-D3/MINIX, HF-D3/6-31G*) and the HF-3c method. The new ACPs are trained on a very large set (73 832 data points) of noncovalent properties (interaction and conformational energies) and validated additionally on a set of 32 048 data points. All reference data are of complete basis set coupled-cluster quality, mostly CCSD(T)/CBS. The proposed ACP-corrected methods are shown to give errors in the tenths of a kcal/mol range for noncovalent interaction energies and up to 2 kcal/mol for molecular conformational energies. More importantly, the average errors are similar in the training and validation sets, confirming the robustness and applicability of these methods outside the boundaries of the training set. In addition, the performance of the new ACP-corrected methods is similar to complete basis set density functional theory (DFT) but at a cost that is orders of magnitude lower, and the proposed ACPs can be used in any computational chemistry program that supports effective-core potentials without modification. It is also shown that ACPs improve the description of covalent and noncovalent bond geometries of the underlying methods and that the improvement brought about by the application of the ACPs is directly related to the number of atoms to which they are applied, allowing the treatment of systems containing some atoms for which ACPs are not available. Overall, the ACP-corrected methods proposed in this work constitute an alternative accurate, economical, and reliable quantum mechanical approach to describe the geometries, interaction energies, and conformational energies of systems with hundreds to thousands of atoms.

Theoretical Investigations on the Hydroxyl-Initialized Oxidation of Hexafluoro-2-butyne in the Presence of Oxygen.

Hexafluoro-2-butyne (C4F6) is a potential eco-friendly alternative gas in plasma, refrigerants, and electrical insulation applications. Mechanisms for the reactions of C4F6 with OH/O2 have been revealed in detail using various theoretical methods including ROCBS-Q, RCCSD(T), multireference RS2C, and extrapolations to the complete basis-set limit with Aug-cc-pVnZ (n = T, Q, 5) basis sets. Rate coefficients and product branching ratios were predicted for a wide range of temperatures and pressures using the solution of master equations. The vibrationally adiabatic ground-state barrier for the initial C4F6 + OH association was best estimated to be 1.53-2.26 kcal/mol. Energetically preferable decomposition paths for the conformation-dependent C4F6OH adducts include six-center HF elimination, four-center proton migration, and C-C bond cleavage, but the collisional deactivation is dominant under ambient conditions. The subsequent oxidation of C4F6OH by O2 bifurcates in two orientations and proceeds without any well-defined barrier followed by the successive isomerization/elimination steps, forming perfluorobiacetyl to regenerate OH radicals or trifluoroacetic acid with trifluoroacetyl radicals. The OH-recycling path accounts for a branching ratio of 70% under ambient conditions. Theoretical rate coefficients are in good agreement with the available experimental results. The effect of fluorination on the reactivity of alkynes toward OH/O2 is discussed.

Rotational excitation of CO2 induced by He: New potential energy surface and scattering calculations.

The CO2 molecule is of great interest for astrophysical studies since it can be found in a large variety of astrophysical media where it interacts with the dominant neutral species, such as He, H2, or H2O. The CO2-He collisional system was intensively studied over the last two decades. However, collisional data appear to be very sensitive to the potential energy surface (PES) quality. Thus, we provide, in this study, a new PES of the CO2-He van der Waals complex calculated with the coupled-cluster method and a complete basis set extrapolation in order to provide rotational rate coefficients that are as accurate as possible. The PES accuracy was tested through the calculations of bound state transition frequencies and pressure broadening coefficients that were compared to experimental data. An excellent agreement was globally found. Then, revised collisional data were provided for the 10-300K temperature range. Rate coefficients were compared to previously computed ones and are found to be up to 50% greater than previously provided ones. These differences can induce non-negligible consequences for the modeling of CO2 abundance in astrophysical media.

Basis Set Dependence of Optical Rotation Calculations with Different Choices of Gauge.

In this work, the basis set dependence of optical rotation (OR) calculations is examined for various choices of gauge/level of theory. The OR is calculated for a set of 50 molecules using B3LYP and CAM-B3LYP and 17 molecules using coupled cluster with single and double excitations (CCSD). The calculations employ the correlation-consistent basis sets, aug-cc-pVζZ with ζ = D, T, Q. An inverse-power extrapolation formula is then utilized to obtain OR values at the complete basis set (CBS) limit. We investigate the basis set convergence for these methods and three choices of gauge: length gauge (with gauge-including atomic orbitals, LG(GIAOs), for DFT), the origin-invariant length gauge [LG(OI)], and the modified velocity gauge (MVG). The results show that all methods converge smoothly to the CBS limit and that the LG(OI) approach has a slightly faster convergence rate than the other choices of gauge. While the DFT methods reach gauge invariance at the CBS limit, CCSD does not. The significant difference between the MVG and LG(OI) results at the CBS limit, 26%, indicates that CCSD is not quite at convergence in the description of electron correlation for this property. On the other hand, gauge invariance at the CBS limit for DFT does not lead to the same OR values for the two density functionals, which is also due to electron correlation incompleteness. A limited comparison to gas-phase experimental OR values for the DFT methods shows that CAM-B3LYP seems more accurate than B3LYP. Overall, this study shows that the LG(OI) approach with the aug-cc-pVTZ basis set for DFT, and with the CBS(DT) extrapolation for CCSD, provides a good cost/accuracy balance.

Theoretical Investigations for Kinetics of the Chemical Reactions: H + SiClx (x = 1, 2, 3).

Hydrogen atoms and SiClx (x = 1, 2, 3) radicals coexist during the hydrogenation of silicon tetrachloride (STC, SiCl4), an important process in the fabrication of industrial polysilicon. In this work, the mechanisms and kinetics of the reactions between H and SiClx (x = 1, 2, 3) were studied by theory. The structures and vibrational frequencies of reactants, products, intermediates, and transition states (TSs) were determined at the B2PLYP/may-cc-pVTZ level. The single-point energies of minima and saddle points were refined using the coupled-cluster single-double with triple perturbative (CCSD(T)) with the complete basis set extrapolation method. Some special treatments were designed to obtain reliable wave functions for unimolecular reactions without tight TSs by the density functional theory. Subsequently, Lennard-Jones (L-J) parameters between each intermediate (SiHClx) and bath gas (He) were obtained at the MP2/jul-cc-pVTZ level to derive reliable temperature- and pressure-dependent rate coefficients for unimolecular reactions according to the variational Rice-Ramsperger-Kassel-Marcus theory. For bimolecular reactions, rate coefficients were determined by the variational transition-state theory. The rate coefficients of barrierless reactions were derived based on the loose TSs with the maximum free energy. Finally, the master equation analysis was used to investigate the variation of the rate coefficients with pressure and temperature in the activated paths.

Coupled cluster theory for the ground and excited states of two-dimensional quantum dots

We present a study of the two-dimensional circular quantum dot model Hamiltonian using a range of quantum chemical ab initio methods. Ground and excited state energies are computed on different levels of perturbation theories, including the coupled cluster method. We outline a scheme to compute the required Coulomb integrals in real space and utilize a semianalytic solution to the integral over the Coulomb kernel in the vicinity of the singularity. Furthermore, we show that the remaining basis set incompleteness error for two-dimensional quantum dots scales with the inverse number of virtual orbitals, allowing us to extrapolate to the complete basis set limit energy. By varying the harmonic potential parameter we tune the correlation strength and investigate the predicted ground and excited state energies.

Correlation-Consistent Gaussian Basis Sets for Solids Made Simple.

The rapidly growing interest in simulating condensed-phase materials using quantum chemistry methods calls for a library of high-quality Gaussian basis sets suitable for periodic calculations. Unfortunately, most standard Gaussian basis sets commonly used in molecular simulation show significant linear dependencies when used in close-packed solids, leading to severe numerical issues that hamper the convergence to the complete basis set (CBS) limit, especially in correlated calculations. In this work, we revisit Dunning's strategy for construction of correlation-consistent basis sets and examine the relationship between accuracy and numerical stability in periodic settings. We find that limiting the number of primitive functions avoids the appearance of problematic small exponents while still providing smooth convergence to the CBS limit. As an example, we generate double-, triple-, and quadruple-ζ correlation-consistent Gaussian basis sets for periodic calculations with Goedecker-Teter-Hutter (GTH) pseudopotentials. Our basis sets cover the main-group elements from the first three rows of the periodic table. Especially for atoms on the left side of the periodic table, our basis sets are less diffuse than those used in molecular calculations. We verify the fast and reliable convergence to the CBS limit in both Hartree-Fock and post-Hartree-Fock (MP2) calculations, using a diverse test set of 19 semiconductors and insulators.

Open-Shell Variant of the London Dispersion-Corrected Hartree-Fock Method (HFLD) for the Quantification and Analysis of Noncovalent Interaction Energies.

The London dispersion (LD)-corrected Hartree–Fock (HF) method (HFLD) is an ab initio approach for the quantification and analysis of noncovalent interactions (NCIs) in large systems that is based on the domain-based local pair natural orbital coupled-cluster (DLPNO-CC) theory. In the original HFLD paper, we discussed the implementation, accuracy, and efficiency of its closed-shell variant. Herein, an extension of this method to open-shell molecular systems is presented. Its accuracy is tested on challenging benchmark sets for NCIs, using CCSD(T) energies at the estimated complete basis set limit as reference. The HFLD scheme was found to be as accurate as the best-performing dispersion-corrected exchange-correlation functionals, while being nonempirical and equally efficient. In addition, it can be combined with the well-established local energy decomposition (LED) for the analysis of NCIs, thus yielding additional physical insights.