G2 Theory Research Articles

Looking for the G2 Higgs branch of 4D rank 1 SCFTs

The Schur index of the Higgs branch of 4-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs is related to the spectrum of non-unitary 2-dimensional CFTs. The rank 1 case has been shown to lead to the non-unitary CFTs with Deligne-Cvitanovic (DC) exceptional sequence of Lie groups. We show that a subsequence (A0,A12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {A}_{\\frac{1}{2}} $$\\end{document}, A1, A2, D4) within the non-unitary sequence is related to a subsequence in the Mathur-Mukhi-Sen (MMS) sequence of unitary theories. We show that 2D non-unitary G2 theory is related to unitary E6 theory, and using this result along with the Galois conjugation, we propose that the G2 Higgs branch is a sub-branch of the E6 Higgs branch.

New insight into the substituent effects on the hydrolytic deamination of saturated and unsaturated cytosine

AbstractAb initio calculations were carried out to understand the effect of electron donating groups (EDG) and electron withdrawing groups (EWG) at the C5 position of cytosine (Cyt) and saturated cytosine (H2Cyt) of the deamination reaction. Geometries of the reactants, transition states, intermediates, and products were fully optimized at the B3LYP/6‐31G(d,p) level in the gas phase as this level of theory has been found to agree very well with G3 theories. Activation energies, enthalpies, and Gibbs energies of activation along with the thermodynamic properties (ΔE, ΔH, and ΔG) of each reaction were calculated. A plot of the Gibbs energies of activation (ΔG‡) for C5 substituted Cyt and H2Cyt against the Hammett σ‐constants reveal a good linear relationship. In general, both EDG and EWG substituents at the C5 position in Cyt results in higher ΔG‡ and lower σ values compared to those of H2Cyt deamination reactions. C5 alkyl substituents (H, CH3, CH2CH3, CH2CH2CH3) increase ΔG‡ values for Cyt, while the same substituents decrease ΔG‡ values for H2Cyt which is likely due to steric effects. However, the Hammett σ‐constants were found to decrease at the C5 position of cytosine (Cyt) and saturated cytosine (H2Cyt) on the deamination reaction. Both ΔG‡ and σ values decrease for the substituents Cl and Br in the Cyt reaction, while ΔG‡ values increase and σ decrease in the H2Cyt reaction. This may be due to high polarizability of bromine which results in a greater stabilization of the transition state in the case of bromine compared to chlorine. Regardless of the substituent at C5, the positive charge on C4 is greater in the TS compared to the reactant complex for both the Cyt and H2Cyt. Moreover, as the charges on C4 in the TS increase compared to reactant, ΔG‡ also increase for the C5 alkyl substituents (H, CH3, CH2CH3, CH2CH2CH3) in Cyt, while ΔG‡ decrease in H2Cyt. In addition, analysis of the frontier MO energies for the transition state structures shows that there is a correlation between the energy of the HOMO–LUMO gap and activation energies.

Explicitly Correlated Double-Hybrid DFT: A Comprehensive Analysis of the Basis Set Convergence on the GMTKN55 Database.

Double-hybrid density functional theory (DHDFT) offers a pathway to accuracy approaching composite wavefunction approaches such as G4 theory. However, the Görling–Levy second-order perturbation theory (GLPT2) term causes them to partially inherit the slow ∝L–3 (with L the maximum angular momentum) basis set convergence of correlated wavefunction methods. This could potentially be remedied by introducing F12 explicit correlation: we investigate the basis set convergence of both DHDFT and DHDFT-F12 (where GLPT2 is replaced by GLPT2-F12) for the large and chemically diverse general main-group thermochemistry, kinetics, and noncovalent interactions (GMTKN55) benchmark suite. The B2GP-PLYP-D3(BJ) and revDSD-PBEP86-D4 DHDFs are investigated as test cases, together with orbital basis sets as large as aug-cc-pV5Z and F12 basis sets as large as cc-pVQZ-F12. We show that F12 greatly accelerates basis set convergence of DHDFs, to the point that even the modest cc-pVDZ-F12 basis set is closer to the basis set limit than cc-pV(Q+d)Z or def2-QZVPPD in orbital-based approaches, and in fact comparable in quality to cc-pV(5+d)Z. Somewhat surprisingly, aug-cc-pVDZ-F12 is not required even for the anionic subsets. In conclusion, DHDF-F12/VDZ-F12 eliminates concerns about basis set convergence in both the development and applications of double-hybrid functionals. Mass storage and I/O bottlenecks for larger systems can be circumvented by localized pair natural orbital approximations, which also exhibit much gentler system size scaling.

Determination of the standard enthalpy of formation of iodine compounds through the G2 and G3(MP2)//B3-SBK theories.

The domain of application of the G3(MP2)//B3-SBK theory was expanded, and its efficiency was evaluated to determinate enthalpies of formation of forty-one iodine compounds. The results were compared to those obtained with the G2 theory for the same set of molecules. The G3(MP2)//B3-SBK theory showed a mean deviation and deviation standard equal to 3.7kcalmol-1 and 6.0kcalmol-1, respectively. The G2 theory (mean deviation = 3.1kcalmol-1 and standard deviation = 4.9kcalmol-1) presented a lower error and standard deviation, but at a significantly higher computational cost. For a more complete evaluation, as a secondary part of the work, it also used different functionals B3LYP, M06-2X, WB97XD, and MP2 method with four different basis sets 6-311G(d,p), LANL2DZ, jorge-ADZP, and CEP-31G(d). The best density functional/basis set combination was obtained with M06-2X/CEP-31G(d) among the three mentioned functionals. However, the produced mean deviation is significant and equal to 17.3kcalmol-1, with a standard deviation equal to 23.0kcalmol-1. The 6-311G(d,p) basis achieved the best performance with the MP2 method, generating an equally significant mean deviation of 12.8kcalmol-1 with a standard deviation equal to 18.7kcalmol-1.

Improving the Accuracy of Composite Methods: A G4MP2 Method with G4-like Accuracy and Implications for Machine Learning.

G4MP2 theory has proven to be a reliable and accurate quantum chemical composite method for the calculation of molecular energies using an approximation based on second-order perturbation theory to lower computational costs compared to G4 theory. However, it has been found to have significantly increased errors when applied to larger organic molecules with 10 or more nonhydrogen atoms. We report here on an investigation of the cause of the failure of G4MP2 theory for such larger molecules. One source of error is found to be the "higher-level correction (HLC)", which is meant to correct for deficiencies in correlation contributions to the calculated energies. This is because the HLC assumes that the contribution is independent of the element and the type of bonding involved, both of which become more important with larger molecules. We address this problem by adding an atom-specific correction, dependent on atom type but not bond type, to the higher-level correction. We find that a G4MP2 method that incorporates this modification of the higher-level correction, referred to as G4MP2A, becomes as accurate as G4 theory (for computing enthalpies of formation) for a test set of molecules with less than 10 nonhydrogen atoms as well as a set with 10-14 such atoms, the set of molecules considered here, with a much lower computational cost. The G4MP2A method is also found to significantly improve ionization potentials and electron affinities. Finally, we implemented the G4MP2A energies in a machine learning method to predict molecular energies.

Three-Dimensional Convolutional Neural Networks Utilizing Molecular Topological Features for Accurate Atomization Energy Predictions.

Deep learning methods provide a novel way to establish a correlation between two quantities. In this context, computer vision techniques such as three-dimensional (3D)-convolutional neural networks become a natural choice to associate a molecular property with its structure due to the inherent 3D nature of a molecule. However, traditional 3D input data structures are intrinsically sparse in nature, which tend to induce instabilities during the learning process, which in turn may lead to underfitted results. To address this deficiency, in this project, we propose to use quantum-chemically derived molecular topological features, namely, localized orbital locator and electron localization function, as molecular descriptors, which provide a relatively denser input representation in a 3D space. Such topological features provide a detailed picture of the atomic and electronic configuration and interatomic interactions in the molecule and hence are ideal for predicting properties that are highly dependent on the physical or electronic structure of the molecule. Herein, we demonstrate the efficacy of our proposed model by applying it to the task of predicting atomization energies for the QM9-G4MP2 data set, which contains ∼134k molecules. Furthermore, we incorporated the Δ-machine learning approach into our model, which enabled us to reach beyond benchmark accuracy levels (∼1.0 kJ mol-1). As a result, we consistently obtain impressive mean absolute errors of the order 0.1 kcal mol-1 (∼0.42 kJ mol-1) versus the G4(MP2) theory using relatively modest models, which could potentially be improved further in a systematic manner using additional compute resources.

Equilibrium molecular structure and intramolecular interactions of picolinic acid hydrazide

Conformational composition and molecule structure of gaseous picolinic acid hydrazide (picolinohydrazide) have been investigated for the first time by means of joint application of gas electron diffraction and quantum chemical calculations. Semi-experimental equilibrium molecular parameters of the prevailing conformer possessing Cs symmetry have been determined. The standard enthalpy of formation of gaseous picolinohydrazide has been estimated using G4 theory. The intermolecular interactions stabilizing major conformation of free picolinohydrazide have been discussed using the results of NBO, AIM and NCI analyses. The structure of gaseous picolinohydrazide was found to be different from that of the solid phase.

6,6′-Dimethyl-1,1′,5,5′-tetraaza-6,6′-bi(bicyclo[3.1.0]hexane): synthesis and investigation of molecular structure by quantum-chemical calculations, NMR spectroscopy and X-ray diffraction analysis

A new diaziridine derivative with two bicyclic diaziridine-containing moieties in one molecule, 6,6′-dimethyl-1,1′,5,5′-tetraaza-6,6′-bi(bicyclo[3.1.0]hexane) (BiDiMDAH), has been synthesized for the first time. Its molecular structure has been investigated in the gas phase by means of quantum-chemical (QC) calculations, in CDCl3 solution by 1D and 2D NMR spectroscopy and in the solid state by the X-ray diffraction (XRD) technique. It was found by QC calculations that BiDiMDAH molecule in a free state presents a single conformer of C2 symmetry with trans orientation of the methyl groups about two carbons connecting diaziridine cycles. The 5-memebered rings in BiDiMDAH molecule has been found to be essentially planar. In order to explain conformational preferences of BiDiMDAH, natural bond orbitals (NBO) and atoms in molecules (AIM) analyses have been performed. According to the XRD data, BiDiMDAH crystallizes in space group P21/c with molecular structure resembling the structure obtained by the means of QC calculations for the lowest energy form. However, the relative arrangement of the methyl groups of the solid BiDiMDAH molecule is similar to that of the one of the transition states. The data of NMR spectroscopy has revealed that only one conformer is present in CDCl3 solution in agreement with the data of QC and XRD studies. The gaseous standard enthalpy of formation of BiDiMDAH has been estimated to be 106.1±1.2 kcal/mol by means of G4 theory.

Effect of ether functionalization and alkyl chain length on the structure and electronic properties of ammonium ionic liquids

The effect of alkyl chain length and ether functionalization on the geometrical and electronic structures of the ammonium-based ionic liquids was studied by density functional theory calculations. The magnitude of the cation–anion interaction energy decreases with increasing side-chain length. The interaction between cations and the chloride anion in the most stable conformers was investigated using the Natural Bond Orbital and Non-Covalent Interaction analyses. The molecular electrostatic potential maps were used to find favorable interaction sites of cations. Moreover, the standard heats of formation were evaluated by combining G3(MP2) theory calculations with the atomization reaction approach.

Accurate Thermochemistry of Complex Lignin Structures via Density Functional Theory, Group Additivity, and Machine Learning

A molecular-level understanding of lignin structures and bond dissociation energies could facilitate depolymerization technologies. Still, this information is currently limited due to the lack of databases and the simplification of surrogate models. Here, substitution effects on seven common linkages in lignin polymers are systematically investigated. An automated reaction network generator is employed to create a database of structures. A new group additivity (GA) model based on principal component analysis (PCA) descriptors is introduced and trained on gas-phase density functional theory data of 4100 species at the M06-2X/6-311++G(d,p) level. Hydrogen bonds, local steric, and nonaromatic ring contributions are also incorporated. Finally, we improve the accuracy of the group additivity model to reach the G4 theory by computing a data set of 770 species at this level and using a data fusion approach.

Dissociative Photoionization of Chloro-, Bromo-, and Iodocyclohexane: Thermochemistry and the Weak C-Br Bond in the Cation.

Double-imaging photoelectron photoion coincidence spectroscopy (i2PEPICO) with tunable synchrotron vacuum ultraviolet radiation was used to record threshold ionization mass spectra of the halocyclohexanes C6H11X (X = Cl, Br, and I). Calculations show that experimental dissociative ionization thresholds correspond to thermochemical limits. Among the processes observed (X loss, followed by C2H4 or C3H6 loss; C2H3Cl loss; HCl loss, followed by CH3 or C2H4 loss), halogen atom loss can be used to derive enthalpies of formation and C-X bond energies in the cation. As an ancillary value, we propose a new proton affinity for cyclohexene at PA298K(c-C6H10) = 771.5 ± 1.7 kJ mol-1. The halogen loss onsets 10.74 ± 0.06 eV, 10.125 ± 0.005, and 9.474 ± 0.005 eV thus yield ΔfHo298K(C6H11X (g)) = -164.4 ± 6.2, -114.4 ± 2.3, and -56.3 ± 2.3 kJ mol-1 for X = Cl, Br, and I, respectively. The last two agree with DFT-calculated isodesmic reaction energies very well, as opposed to G4 theory for X = Br. The C-X bond energy in the cation is the lowest for X = Br. This is the sum result of the weakening C-X bond in the neutral and the increasing stabilization of the parent ion with increasing halogen size.

Potential main group amine borane-based chemical hydrogen storage molecular systems

The thermodynamics as a function of the temperature for the decomposition of the substituted amine boranes X(NH2BH3)3 and X(BH2NH3)3 for X = N and P and E(NH2BH3)4 and E(BH2NH3)4 for E = C and Si were predicted using composite correlated molecular orbital G3(MP2) theory. The relative stabilities of isomers were explained using bond dissociation energies. Equilibrium concentrations of products with respect to temperature were determined using Gibbs free energy minimization. The activation energy for loss of the first H2 was calculated at the G3(MP2)B3 level and is larger than the respective bond dissociation energy to lose terminal BH3 or NH3 via dative bond cleavage. Dipole moments and polarizabilities were calculated at the DFT/B3LYP level and boiling points were calculated at the DFT/BP/COSMO-RS level. Planar fits were developed for the boiling points of N(BH2NH3)3, P(BH2NH3)3, C(BH2NH3)4, and Si(BH2NH3)4 dehydrogenation products using dipole moment, polarizability, and solvent cavity surface area.

Exploring enthalpies of formation of imidazolium-, pyridinium-, and pyrrolidinium-based ionic liquids with dicyanamide anion using quantum chemical methods

Abstract In this work, quantum chemical calculations have been performed on the ionic liquids based on imidazolium, pyridinium, and pyrrolidinium cations with the dicyanamide anion. The dispersion-corrected B3LYP exchange-correlation functional was used to optimize geometries. Enthalpies of formation were calculated at the high-level G3(MP2) theory by employing both the atomization energy and reaction enthalpy approaches. There exists a very good agreement between the predicted enthalpies of formation and available experimental values. For the first time, the enthalpies of formation have been predicted for 21 ion pairs with no available experimental data. The calculated enthalpies of formation are decreased with increasing alkyl chain length. Linear correlations were observed between enthalpy of formation and number of carbon atoms in the alkyl side-chain for all of the studied series of cations. Regardless of the cation type, the contribution of each CH2 group in the alkyl chain to the enthalpy of formation was also determined.

Structural considerations for charge‐enhanced Brønsted acid catalysts

AbstractAll three N‐methylated and N‐protonated hydroxypyridinium BArF4– salt isomers were synthesized and their hydrogen bond donating abilities were investigated. DFT and G4 theory computations along with IR spectroscopic measurements were found to be effective methods for predicting the catalytic activities of these O–H and N–H Brønsted acids. A UV‐vis titration approach for rapidly quantifying hydrogen bond donating ability revealed that carbon‐hydrogen bonds also can participate in electrostatic interactions, but the presence of multiple equilibrium complexes results in a limitation of this method. In the methylated series of hydroxypyridines, the ortho and para isomers displayed modest rate enhancements relative to the meta derivative. Protonation introduces a new acidic site and the ortho hydroxypyridinium ion salt is a significantly more active catalyst than all of the other species examined. This is indicative of bidentate activation by the N–H and O–H acidic sites, and suggests a new design strategy for improving charge‐enhanced catalysts.

Direct fluorination of tetrafluoroethylene at low temperatures

Mechanisms for the processes of direct fluorination of tetrafluoroethylene (TFE) in matrices of TFE, hexafluoropropylene (HFP), and dimers and trimers of HFP ((HFP)2 and (HFP)3) from 77 K to 300 K have been developed. Electronic structure calculations at the composite correlated G3(MP2) and G4 molecular orbital theory levels of the energetics of a range of reactions involving TFE and fluorine are presented to aid in the development of these mechanisms. The equilibrium products of the direct fluorination of TFE in the gas phase at varied temperature and initial composition was determined by Gibbs free energy minimization. Spontaneous reactions (explosions) were observed for the fluorination of pure crystalline TFE and HFP. Fluorination of TFE can be performed without explosion in glassy matrixes of (HFP)2 or (HFP)3 at low temperatures. The explosive nature of the reaction decreases in the matrix order TFE > (HFP)2 > (HFP)3. The fluorination of TFE begins at the phase transition temperature of the matrix, i.e., after the transition of devitrified (HFP)2 and (HPF)3 into the supercooled liquid state at 110 K and 150 K, respectively. The experiments show that either the presence of a branched structure (C9F20, the saturated analog of (HFP)3) or the presence of unsaturated bonds (perfluorotoluene) separately cannot provide a medium for the safe fluorination of TFE as the direct fluorination of TFE in these matrices led to an explosion. HFP oligomers provide an effective environment for TFE fluorination because of the presence of double bonds surrounded by the branched perfluorinated groups. The unsaturated bonds of the HFP oligomers are an active participant in the chemical processes involved in the safe, direct fluorination of TFE.

Theoretical study of the internal rotational barriers of fluorine, chlorine, bromine, and iodine-substituted ethanes

Physical and chemical characteristics of flexible compounds are extremely dependent of internal rotation barriers. On the other hand, halides of organic compounds are extremely common, and the nature of their rotation barriers are still poorly understood. As a simple example, the experimental height of the internal rotational barrier decreases with the number of fluorine-substituted in ethane, C2H5F, C2H4F2, and C2H3F3. For chlorine-substituted ethane C2H5Cl, C2H4Cl2 and C2H3Cl3 the internal rotational barrier converges in an opposite manner. No experimental results are available for the bromine- and iodine-substituted compounds C2H5Br, C2H4Br2, C2H3Br3, C2H5I, C2H4I2, and C2H3I3. In light of lack of both an adequate explanation for this phenomenon and experimental results for Br and I, the present work studied these compounds using different levels of theory: MP2, MP3, MP4, QCISD(T) and CCSD(T) methods, and the G3 and G3CEP composite theories. The results showed that the G3 and G3CEP theories were the most accurate calculations for the F- and Cl-substituted compounds as a result of the additive contributions of the energy values. From Natural Bond Orbitals (NBO), Quantum Theory of Atoms in Molecules (QTAIM) and Energy Decomposition Analysis (EDA) it was verified that the internal rotational barriers decrease with the addition of Fluorine atoms for fluorine-containing ethanes and can be explained by: π character in the CF bond and delocalization indexes (DI(A)) by QTAIM analysis; the interaction of the ligand and anti-ligand orbitals by NBO analysis and for the descriptors by EDA. On the other hand, the enlargement of the barriers with the addition of Cl, Br and I atoms for substituted ethanes, are dominated by the large electronic cloud of these halogens, which generates significant steric repulsion.

Enthalpies of formation of diamantanes in the gas and crystalline phase: comparison of theory and experiment

Gas-phase enthalpies of formation of diamantane and some of its derivatives were calculated from isodesmic reactions using adamantane as reference species. The G4 theory was applied to calculate the energies of all molecules involved in the isodesmic reactions. The sublimation enthalpies were predicted by the molecular electrostatic potential (MEP) model. Their values were computed from an equation that describes the relationship between the sublimation enthalpy and computed crystal density, and molecular surface area plus four quantities obtained from the surface electrostatic potential. The coefficients in this equation were determined earlier from least squares fitting to reliable values of sublimation enthalpy of 23 adamantanes. The enthalpy formation of diamantanes in the crystalline phase was estimated on the basis of calculated gas-phase enthalpies of formation and sublimation enthalpies. The difference between experimental and theoretical data and inconsistencies in available experimental data are discussed.

Accurate heats of formation of polycyclic saturated hydrocarbons predicted by using the XYG3 type of doubly hybrid functionals.

Polycyclic saturated hydrocarbons (PSHs) are attractive candidates as hydrocarbon propellants. To assess their potential values, one of the key factors is to determine their energy contents, such as to calculate their heats of formation (HOF). In this work, we have calculated HOFs for a set of 36 PSHs including exo-Tricyclo[5.2.1.0(2,6) ] decane, the principal component of the high-energy density hydrocarbon fuel commonly identified as JP-10. The results from B3LYP, B3LYP-D3BJ, M06-2X, B2PLYP, B2PLYP-D3BJ, and the XYG3 type of doubly hybrid (xDH) functionals are presented. It is demonstrated here that the xDH functionals yield accurate HOFs in good agreement with those from experiments or the G4 theory. In particular, XYGJ-OS, a low scaling xDH functional, is shown to hold the promise for accurate prediction of HOFs for PSHs of larger sizes. © 2018 Wiley Periodicals, Inc.

Effects of Positive and Negative Ionization on Prototropy in Pyrimidine Bases: An Unusual Case of Isocytosine.

Intramolecular proton-transfers (prototropic conversions) have been studied for the guanine building block isocytosine (iC), and effects of positive ionization, called one-electron oxidation (iC - e → iC+•), and negative ionization, called one-electron reduction (iC + e → iC-•), on tautomeric conversions when proceeding from neutral to ionized isocytosine have been discussed. Although radical cations and radical anions are very short-lived species, the ionization effects could be investigated by quantum-chemical methods. Such kind of studies gives some information about the labile protons and the most basic positions in the neutral and radical forms of the tautomeric system. For investigations, the complete isomeric mixture of isocytosine has been considered and calculations performed in two extreme environments, apolar {DFT(B3LYP)/6-311+G(d,p)} and polar {PCM(water)//DFT(B3LYP)/6-311+G(d,p)}. For selected isomers, the G4 theory has also been applied. There are no good relations for energetic parameters of neutral and ionized forms. Ionization energies depend on localization of labile protons. Tautomeric equilibria for neutral and ionized isocytosine, favored sites of protonation and deprotonation, and favored structures of protonated and deprotonated forms strongly depend on environment. Acidity of iC+• is close to that of the iC conjugate acid, and basicity of iC-• is close to that of the iC conjugate base. This increase of acid-base properties of charged radicals explains the proton-transfer in ionized pairs of nucleobases. When compared to other pyrimidine bases such as uracil (U) and cytosine (C), which exhibit analogous tautomeric equilibria between nine prototropic tautomers as isocytosine, the tautomeric preferences for iC, iC+•, iC-•, U, U+•, U-•, C, C+•, and C-• are completely different. The differences suggest that acid-base properties of functional groups, their stabilities, and ionization energies play a principal role in proton-transfers for pyrimidine bases and influence compositions of tautomeric mixtures.

Exact results in 3d mathcal{N} = 2 Spin(7) gauge theories with vector and spinor matters

We study three-dimensional mathcal{N} = 2 Spin(7) gauge theories with NS spinorial matters and with Nf vectorial matters. The quantum Coulomb branch on the moduli space of vacua is one- or two-dimensional depending on the matter contents. For particular values of (Nf, NS ), we find s-confinement phases and derive exact superpotentials. The 3d dynamics of Spin(7) is connected to the 4d dynamics via KK-monopoles. Along the Higgs branch of the Spin(7) theories, we obtain 3d mathcal{N} = 2 G2 or SU(4) theories and some of them lead to new s-confinement phases. As a check of our analysis we compute superconformal indices for these theories.