Substituent effects on the conformational stability of allyl fluorides

By the B3P86/6‐311G(3d,2p) method, remote substituent effects on trans‐YCH?CHCH₂F were investigated by examining their conformational stabilities, molecular geometries, and stereoelectronic interactions in this paper. The cis conformer is favored for Y?H, Cl, Me, Vinyl, CF₃, CN, CHO, and NO₂, whereas the gauche is favored for Y?OMe, OH. A correlation of ΔH with the substituent constants σ⁺(Y) shows that the increasing electron‐withdrawing ability of the substituent Y increases the relative stability of the cis conformer. It was found that the substituent effect on the molecule stabilization energies (relative to CH₂?CHCH₂F) is more significant in the gauche conformers than in the cis conformers. In agreement, molecular structures of the gauche conformers were also observed to vary more significantly with the substitution than those of the cis conformers. By the second‐order perturbation energy (E(2)) in NBO analysis, it was found that the total C₂–C₃ vicinal hyperconjugation is determinant in the enthalpy difference and consequently controls the conformational stability. Further analysis shows that the substituent effect on the C₂–C₃ vicinal hyperconjugations is much higher in the gauche conformers than in the cis conformers. The highly sensitive π_C?C→σ*_{C? F} interaction to the substitution in the gauche conformers, is the leading factor in variation of molecular stability and geometry. Copyright © 2010 John Wiley & Sons, Ltd.     

Substituent effects on acidity of aryl‐substituted fluoroalkanes: a computational study

The gas‐phase acidities (GA) of various aryl‐substituted fluoroalkanes, XC₆H₄CH(R¹)R², were calculated at the B3LYP/6‐311 + G(d,p)//B3LYP/6‐311 + G(d,p). The acidity values of alkanes having a common substituent X varied significantly with the change of R¹ and R². Their changes in acidity of 1 and 2 having two strong electron‐withdrawing groups (CF₃ or C₂F₅) at the deprotonation site and 8 , 9 , 10 , 11 having no fluorine atom at β‐position were linearly correlated with the corrected number of fluorine atoms contained in the fluorinated alkyl group (R² > 0.999). On the other hand, the GA values of β‐fluorine substituted alkanes ( 3 , 4 , 5 , 6 , 7 ) deviated in a stronger acid direction from the line. The enhanced acidity was attributed to the additional stabilization of the conjugate anion caused by the β‐fluorine negative hyperconjugation. The magnitude of β‐fluorine negative hyperconjugation of the fluorinated alkyl group (ΔG^o_β‐F) given by the deviations from the line decreased with increasing electron‐withdrawing ability of substituent X on the benzene ring, indicating that β‐fluorine negative hyperconjugation competes with the electronic effect of the substituent X. The GA_el values obtained by subtraction ΔG^o_β‐F from the apparent GA value were successfully correlated in terms of the Yukawa–Tsuno equation. The obtained ρ_el and r^?_el values were linearly related to the GA_el value of the respective phenyl‐substituted fluoroalkanes, supporting our previous conclusion that the ρ and r^? values for the substituent effect caused by the electronic effects of the substituent on the acidity are determined by the thermodynamic stability of the parent ion (ring substituent = H). Copyright © 2015 John Wiley & Sons, Ltd.     

Transmission of polar substituent effects across the cubane ring system: 19F substituent chemical shifts (SCS) of 4‐substituted (X) cub‐1‐yl fluorides revisited

¹⁹F NMR shieldings of 4‐substituted (X) cub‐1‐yl fluorides ( 4 ) for a set of substituents (X?H, NO₂, CN, NC, CF₃, COOH, F, Cl, HO, NH₂, CH₃, Si(CH₃)₃, Li, O^? and NH) covering a wide range of electronic substituent effects were calculated using the DFT‐GIAO theoretical model. The level of theory, B3LYP/6‐311+G(2d,p), provided ¹⁹F substituent chemical shifts (SCS) in good agreement with experimental values where known. By means of NBO analysis, various molecular parameters were obtained from the optimized geometries. Linear regression analysis was employed to explore the relationship between the calculated ¹⁹F SCS and polar field, resonance and group electronegativity substituent constants (σ_F, σ_R and σ_x, respectively) and also the NBO derived molecular parameters (fluorine natural charges (Q_n), electron occupancies on fluorine of lone pairs (n_F) and occupation number of the C? F antibonding orbital (σ_CF*)). The key determining parameters appear to be n_F and σ_CF*(occup). Both factors are a function of the electrostatic field influence of the substituent (σ_F effect) but are counteractive in their influence on the shifts. No evidence for a significant resonance effect influence on the shifts could be identified. Copyright © 2009 John Wiley & Sons, Ltd.     

Gas‐phase Acidities of 2‐Aryl‐2‐chloro‐1,1,1‐trifluoroethanes and Related Compounds. Experimental and Computational Studies

The gas‐phase acidities (GA) of 2‐aryl‐2‐chloro‐1,1,1‐trifluoroethanes ( 1a ), 2‐aryl‐2‐fluoro‐1,1,1‐trifluoroethanes ( 2a ), and related compounds, XC₆H₄CH(Z)R where Z = Cl ( 1 ) or F ( 2 ) and R = C₂F₅ ( b ), t‐C₄F₉ ( c ), C(CF₃)₂C₂F₅ ( d ), C(CF₃)₂Me ( e ), Me ( f ), H ( g ), were investigated experimentally and computationally. On the basis of an excellent linear correlation (R² > 0.99) of acidities of 1c , 1d , 1e , 1f and 2c , 2d , 2e , 2f where there is no fluorine atom at β‐position to the deprotonation site with the corrected number of fluorine atoms contained in the fluorinated alkyl group, the extent of β‐fluorine negative hyperconjugation of the CF₃ and C₂F₅ groups (ΔG^o_β‐F) was evaluated. The GA_el values given by subtraction ΔG^o_β‐F from the apparent GA value were considered to represent the electronic effect of the substituent X. The substituent effects on the GA_el values and GA values for 1c , 1d , 1e , 1f and 2c , 2d , 2e , 2f were successfully analyzed in terms of the Yukawa–Tsuno equation. The variation of resonance demand parameter r^? with the R group observed for various XC₆H₄CH(Z)R was linearly related to the GA (GA_el) value of the respective phenyl‐substituted fluorinated alkanes. On the other hand, the corresponding correlation for the ρ values provided three lines for ArCH(Cl)R, ArCH(F)R and ArCH₂R, respectively. These results supported our previous conclusion that the r^? and ρ values are governed by the thermodynamic stability of the parent ion (ring substituent = H). Other factors arising from an atom bonded to the acidic center also influence the ρ value. Copyright © 2016 John Wiley & Sons, Ltd.     

Solvent polarity scales: determination of new ET(30) values for 84 organic solvents

The solvent polarity parameter E_T(30) is newly measured from the solvatochromism of the betaine dye 30 for 84 solvents and re‐measured for 186 additional ones. The results are organized in a database. It is shown that the validity of linear solvation energy relationships used for the determination of secondary E_T(30) values is limited to non‐hydrogen‐bond donor solvents. Relationships with the chain length n are given for the determination of tertiary E_T(30) values of the homologous H(CH₂)_nY solvent series. The parameter E_T(30) is orthogonal to the function of the refractive index (n² ? 1) / (2n² + 1). For non hydrogen‐bond donor solvents, this allows to enter E_T(30) as an almost pure electrostatic parameter in a new linear solvation energy relationship. Copyright © 2014 John Wiley & Sons, Ltd.     

Evolution of excess electron binding motifs under both internal‐push (from exo C–F bonds) and external‐push (from endo C–F bonds) electron effects in endohedral metallofullerenes with endo C–F bonds

How does the endo C–F bond influence the excess electron binding motif? For lithium‐doped endohedral perfluorofullerenes with endo C–F bonds, under both internal‐push (from exo C–F bonds) and external‐push (from endo C–F bonds) electron effects, the singly occupied molecular orbital electron cloud of the sphere‐like Li···F₈@C₆₀F₅₂ (D₂) is partially dispersed within the σ_p–s antibonding orbital of endo C–F bonds and the space between C^δ+–F_endo^δ– double electric layers, which makes Li···F₈@C₆₀F₅₂ have partial excess electron (electride characteristics) and partial lithium salt characteristics, while in the tube‐like Li···F₂@C₆₀F₅₈ (C_s), as the Li is changing from approaching F to keeping away from F and to approaching another one, the singly occupied molecular orbital electron cloud is mainly dispersed from within the p orbital of the short endo C–F bond to within the middle of the two F atoms and again to within the p orbital of the short endo C–F bond, which indicates an evolution from lithium salt characteristic to excess electron characteristic, and again to lithium salt characteristic. Copyright © 2012 John Wiley & Sons, Ltd.     

Reaching for [6]n SiC‐cyclacenes and [6]n SiC‐acenes: A DFT approach

Density functional theory (DFT) calculations introduced triplet ground states for [6]_n SiC‐cyclacenes and ‐acenes with alternate silabenzene rings including silicon atoms in 2 opposite edges (n = 6, 8, 10, 12). The singlet‐triplet energy gap (ΔE_(S‐T)), binding energy per atom (BE/n), and NBO calculation with very small band gap (ΔE_LUMO‐HOMO) confirmed the triplet ground states. In contrast to polyacenes, the singlet [6]_n SiC‐cyclacenes displayed more stability improvement than triplets, through n increasing. This may open the way for synthesis of larger stable [6]_n SiC‐cyclacenes. The ΔE_(S‐T), BE/n, and the strain energy through homodesmic equations indicated more stability for larger [6]_n SiC‐cyclacenes, which was more noticeable in singlet states. Cyclacenes and acenes with high conductivity and full point charge were introduced as suitable candidates for hydrogen storage.     

Hydrogen bond and internal rotations barrier: DFT study on heavier group‐14 analogues of formamide

A theoretical study on heavier group‐14 substituting effect on the essential property of formamide, strong hydrogen bond with water and internal rotational barrier was performed within the framework of natural bond orbital (NBO) analysis and based on the density functional theory calculation. For heavier group‐14 analogues of formamide (YHONH₂, Y = Si, Ge and Sn), the n_N–π_Y=O conjugation strength does not always reduce as Y becomes heavier, for example, silaformamide and germaformamide have similar strength of delocalization. Heavier formamides prefer being H‐bond donors to form FYO–H₂O complexes to being H‐bond acceptors to form FYH–H₂O complexes. The NEDA analysis indicates that H‐bond energies of FYO–H₂O complexes increase as moving down group 14 due to concurrently stronger charge transfer (CT) and electrostatic attraction and for the FYH–H₂O complexes H‐bond strengths are similar. The model of CTs from FYO to H₂O differs from that at FYH–H₂O complexes, which are contributed not only by aligning lone‐pair orbital of O but also by another lone‐pair orbital. At two lowest lying excited states (the triplet and S₁ excited states), formamide and its heavier analogues form double H‐bonds with H₂O molecule at the same time. The barrier heights of internal rotation become gradually low from C to Sn, formamide (15.73 kcal/mol) > silaformamide (11.73 kcal/mol) > germaformamide (9.45 kcal/mol) > stannaformamide (7.50 kcal/mol) at the CCSD(T)/aug‐cc‐pVTZ//B3LYP/cc‐pVTZ level. NBO analysis indicates that the barrier does not only come from the n_N→π*_YO conjugation, and for heavier analogues of formamide, the n_N→σ*_YO hyperconjugation effect and steric effect considerably contribute to the overall rotational barrier. Copyright © 2013 John Wiley & Sons, Ltd.     

Chemically triggered C–ON bond homolysis in alkoxyamines. Part 7. Remote polar effect

In a recent work (Org. Lett. 2012, 14, 358–361), we showed that the activation by benzylation of alkoxyamine 1 (diethyl (1‐(tert‐butyl(1‐(pyridin‐4‐yl)ethoxy)amino)‐2,2‐dimethylpropyl)phosphonate) afforded a surprisingly large C–ON bond homolysis rate constant k_d. Taking advantage of the easy preparation of para‐X‐benzyl‐activated alkoxyamines 2 and of the presence of a shielding methylene group between the two aromatic moieties, we investigated the long range (10 bonds between the X group and the C–ON bond) polar effect for X = H, F, OMe, CN, NO₂, NMe₂, ⁺NHMe₂,Br^?. It was observed that the effect was weak (4‐fold) and mainly due to the zwiterionic mesomeric forms generated by the presence of group X on the para position, i.e. k_d increased for CN and NO₂ and decreased for OMe, NMe₂ and ⁺NMe₂H,Br^?. DFT calculations at the B3LYP/6‐31G(d,p) level were performed to determine orbital interactions (natural bond orbital (NBO) analysis), Mulliken and NBO charges which support the reactivity described. Copyright © 2014 John Wiley & Sons, Ltd.     

Breathing viability into cyclonona‐3,5,7‐trienylidenes via α‐dimethyl and ά‐moieties at DFT

Cyclonona‐3,5,7‐trienylidene ( 1 ) changes from being a transition state (TS) to minimum states when substituted by α‐methyl groups and ?‐X, where X = CMe₂, NMe, PMe, O, S, cyclopropyl, and SiMe₂ ( 2 , 3 , 4 , 5 , 6 , 7 , 8 , respectively) at density functional theory. Specifically, the parent carbene 1 exhibits a negative vibrational force constant and proves to be an unreachable electrophilic TS while shows C_s symmetry with an NBO atomic charge of +0.70 on its carbenic center. It has a triplet ground state with a rather small singlet‐triplet energy gap (ΔE_s–t = ?4.1 kcal/mol). In contrast, all of its seven scrutinized derivatives enjoy reachable global minima, with C₁ symmetry, desired nucleophilicity, and singlet closed shell (S_cs) ground states (for all but 8 which remains triplet). Stability is indicated by relative ΔE_s–t values: 2 > 3 > 4 > 5 > 6 > 7 > 1 > 8 . The highest ΔE_s–t as well as NBO carbenic atomic negative charge (?0.74) are displayed by 2 . Our carbenes ( 2 , 3 , 4 , 5 , 6 , 7 ) appear more nucleophilic than the synthesized N‐heterocyclic carbenes (imidazol‐2‐ylidenes). Copyright © 2013 John Wiley & Sons, Ltd.     

Quantum chemical studies on a series of transition metal carbon dioxide complexes: Metal–carbon bonding and electronic structures

The bonding features and electronic structures of a series of transition metal carbon dioxide complexes have been studied by density functional theory (DFT) calculations combined with natural bond orbital (NBO) analysis and energy-decomposition analysis (EDA). NBO analysis shows that the interaction between the metal center and the carbon atom of the carbon dioxide ligand (M–C) is stronger than the other interaction between the metal center and the carbon dioxide ligand. Natural hybrid orbital (NHO) analysis gives the detailed bonding features of the M–C bond for each complex. The NBO charge distribution on the carbon dioxide unit in all studied complexes is negative, which indicates charge transfer from the metal center to the carbon dioxide ligand for all studied complexes. The hyperconjugation effect of the metal center and the two C–O bonds of the carbon dioxide ligand has been estimated using the NBO second-order perturbation stabilization energy. It was found that the NBO second-order stabilization energy of C–O?→?nM* is sensitive to the coordinated sphere and the metal center. Frontier molecular orbital (FMO) analysis shows that complexes 1 and 4 may be good nucleophilic reagents for activation of the carbon dioxide molecule. However, the EDAs show that the M–CO₂ bond interaction energy of complex 4 is about two times as large as that of complex 1. The high M–CO₂ bond interaction energy of complex 4 may limit its practical application.     

A DFT‐GIAO and DFT‐NBO study of polar substituent effects on NMR 17O chemical shifts in some rigid polycyclic alkanes

¹⁷O NMR shieldings of 3‐substituted(X)bicyclo[1.1.1]pentan‐1‐ols ( 1 , Y = OH), 4‐substituted(X)bicyclo[2.2.2]octan‐1‐ols ( 2 , Y = OH), 4‐substituted(X)‐bicyclo[2.2.1]heptan‐1‐ols ( 3 , Y = OH), 4‐substituted(X)‐cuban‐1‐ols ( 4 , Y = OH) and exo‐ and endo‐ 6‐substituted(X)exo‐bicyclo[2.2.1]heptan‐2‐ols ( 5 and 6 , Y = OH, respectively), as well as their conjugate bases ( 1 – 6 , Y = O^?), for a set of substituents (X = H, NO₂, CN, NC, CF₃, COOH, F, Cl, OH, NH₂, CH₃, SiMe₃, Li, O^?, and NH) covering a wide range of electronic substituent effects were calculated using the DFT‐GIAO theoretical model at the B3LYP/6‐311 + G(2d, p) level of theory. By means of natural bond orbital (NBO) analysis various molecular parameters were obtained from the optimized geometries. Linear regression analysis was employed to explore the relationship between the calculated ¹⁷O SCS and polar field and group electronegativity substituent constants (σ_F and σ_χ, respectively) and also the NBO derived molecular parameters (oxygen natural charge, Q_n, occupation numbers of the oxygen lone pairs, n_o, and occupancy of the C? O antibonding orbital, σ*_CO(occup)). In the case of the alcohols ( 1 – 6 , Y = OH) the ¹⁷O SCS appear to be governed predominantly by the σ_χ effect of the substituent. Furthermore, the key determining NBO parameters appear to be n_o and σ*_CO(occup). Unlike the alcohols, the calculated ¹⁷O SCS of the conjugate bases ( 1 – 6 , Y = O^?), except for system 1 , do not respond systematically to the electronic effects of the substituents. An analysis of the SCS of 1 (Y = O^?) raises a significant conundrum with respect to their origin. Copyright © 2010 John Wiley & Sons, Ltd.     

Density functional theory calculation and vibrational spectral analysis of 4‐hydroxy‐3‐(3‐oxo‐1‐phenylbutyl)‐2H‐1‐benzopyran‐2‐one

The Fourier transform infrared (4000–400 cm⁻¹) and Fourier transform Raman (3500–500 cm⁻¹) spectra of 4‐hydroxy‐3‐(3‐oxo‐1‐phenylbutyl)‐2H‐1‐benzopyran‐2‐one (Warfarin) have been measured and calculated. The structure optimization has been made using density functional theory (DFT) calculations. Complete vibrational assignments of the observed spectra have been compared with theoretical wavenumbers. The wavenumber increasing in the methyl group shows the electronic hyperconjugation effect. The natural bond orbital (NBO) analysis reveals the hyperconjugation interaction and the intramolecular hydrogen bonding. The first‐order hyperpolarizability has been calculated. Copyright © 2009 John Wiley & Sons, Ltd.     

Long distance structural consequences of H‐bonding: the case of complexes of para‐substituted phenol derivatives

H‐bonded complexes of p‐X‐PhOH/p‐X‐PhO^? with fluoride and hydrofluoric acid (X = OH, H, NO₂) were subject of optimization (by means of B3LYP/6‐311+G**) for gradually changed O···F distance from d_O···F = 4.0 Å down to (i) the distance of the proton transfer from the hydroxyl group to fluoride leading to O^?···HF interaction and (ii) fully optimized system (O^?···HF type). In this way, we simulate gradual changes of H‐bond strength estimating simultaneously the energy of interaction, E_int, energy of deformation, E_def, and the binding energy, E_tot. The obtained geometrical parameters allow us to show that H‐bond formation causes substantial changes in geometry, even at so distant parts of the system as the ring and bond length in para‐substituents (OH and NO₂). All these changes are monotonically dependent on interaction and deformation energies. Copyright © 2009 John Wiley & Sons, Ltd.     

Basicity of the polydentate captodative aminoenones. Ab initio,DFT, and FTIR study

Electron donating ability of the oxygen, nitrogen, and carbon atoms of captodative aminoenones R'CH = C(NR₂)EWG, EWG = CHO, C(O)Me, C(O)CF₃, C(O)Ph was investigated using ab initio and Density Functional Theory (DFT) calculations, Natural Bond Analysis (NBO) analysis, and Fourier Transform InfraRed (FTIR) spectroscopy. The influence of both electron withdrawing group (EWG) and double bond substituents on the proton affinity of the basic centers, orbital interaction, as well as resonance stabilization energies between heteroatoms and double bonds are discussed. The results obtained are critically compared with the push–pull aminoenones. The comparison of experimental values ΔνOH and theoretical values of H‐bonding energy was used to determine the H‐complex type of aminoenone with phenol and the H‐bond strength. Copyright © 2016 John Wiley & Sons, Ltd.     

Effect of proton donors on the intramolecular coordination C=O→Si–F in (acyloxymethyl)trifluorosilanes. Ab initio,DFT and FTIR study,QTAIM analysis

The intramolecular С=O→Si coordination in H‐complexes of (acetoxymethyl)trifluorosilane and (benzoyloxymethyl)trifluorosilane with proton donors HCl, PhOH, MeOH, and CHCl₃ was investigated by density functional theory and second‐order Møller‐Plesset perturbation theory (MP2) methods. Interrelation and mutual influence of the intramolecular coordination bond С=O→Si and intermolecular hydrogen bonds C=O···H and Si–F···H in H‐complexes was established using the AIM and NBO analyses. The С=O→Si coordination is weakened by the C=O···H hydrogen bonding but enhanced by the Si–F_ax···H hydrogen bonding. The structure of H‐complexes of (acetoxymethyl)trifluorosilane with proton donors in solution was determined by comparing the ν(C=O) and ν(Si–F) frequencies calculated using the conductor‐like polarizable continuum model and their experimental Fourier transform infrared values. Copyright © 2014 John Wiley & Sons, Ltd.     

Complete basis set,hybrid‐DFT study and NBO interpretations of conformational behaviors of trans‐2,3‐ and trans‐2,5‐dihalo‐1,4‐dithianes

The conformational behaviors of trans‐2,3‐dihalo‐1,4‐dithiane [halo = F ( 1 ), Cl ( 2 ), Br ( 3 )] and trans‐2,5‐dihalo‐1,4‐dithiane [halo = F ( 4 ), Cl ( 5 ), Br ( 6 )] have been analyzed by means of complete basis set CBS‐4, hybrid‐density functional theory (B3LYP/6‐311 + G**//B3LYP/6‐311 + G**) based methods, and natural bond orbital (NBO) interpretation. Both methods showed that the axial conformations of compounds 1–5 are more stable than their equatorial conformations but CBS‐4 resulted in an equatorial preference for compound 6 . The Gibbs free energy difference (G_eq?G_ax) values (i.e., ΔG_eq–ax) at 298.15 K and 1 atm between the axial and equatorial conformations decrease from compound 1 to compound 2 but increase from compound 2 to compound 3 . Also, the calculated ΔG_eq–ax values decrease from compound 4 to compound 6 . The NBO analysis of donor–acceptor (LP → σ*) interactions showed that the anomeric effect (AE) increase from compound 1 to compound 3 and also from compound 4 to compound 6 . On the other hand, the calculated dipole moment values between the axial and equatorial conformations [Δ(µ_eq?µ_ax)] decrease from compound 1 to compound 3 . The conflict between the increase of AE and the decrease of Δ(µ_eq?µ_ax) values could explain the variation of the calculated ΔG_eq–ax for compounds 1–3 . The Gibbs free energy difference values between the axial and equatorial conformations (i.e., ΔG_ax–ax and ΔG_eq–eq) of compounds 1 and 4 , 2 and 5 and also 3 and 6 have been calculated. The correlations between the AE, bond orders, pairwise steric exchange energies (PSEE), ΔG_eq–ax, ΔG_ax–ax, ΔG_eq–eq, dipole–dipole interactions, structural parameters, and conformational behaviors of compounds 1–6 have been investigated. Copyright © 2010 John Wiley & Sons, Ltd.     

Raman studies and optical properties of some (PbO)x(Bi2O3)0.2(B2O3)0.8−x glasses

Five (PbO)_x(Bi₂O₃)_0.2(B₂O₃)_0.8−x glasses, where x = 0, 0.2, 0.3, 0.4 and 0.6, were prepared. The dilatometric glass transition temperature (T_g) was found in the region 470 (x = 0)≥ T_g ( °C) ≥ 347 (x = 0.6), and the density (ρ) varied within 4.57 (x = 0) ≤ ρ (g/cm³) ≤ 8.31 (x = 0.6). Raman spectra indicated the conversion of BO₃ to BO₄ entities for low x values but for x > 0.3, namely, for x → 0.6, back‐conversion occurred, most probably. From the measurements of the optical transmission on very thin bulk samples, the room temperature optical gap values (E_g) were determined to be in the range 4.03 (x = 0)≥ E_g (eV) ≥ 3.08 (x = 0.6). The temperature (T) dependence of the optical gap (E_g(T)) in the region 300 ≤ T(K) ≤ 600 was examined and approximated by a linear relationship of the form of E_g(T) = E_g(0)− γT, where γ × 10⁻⁴(eV/K) varied from 5.1 to 6.8. The non‐linear refractive index (n₂) was estimated from the optical gap values and it was found to correspond to the n₂ values calculated from the experimental third‐order non‐linear optical susceptibility taken from the literature. Copyright © 2008 John Wiley & Sons, Ltd.     

Superconformal Indices of $${{\mathcal N}=4}$$ SYM Field Theories

Superconformal indices (SCIs) of 4d N = 4{{\mathcal N} = 4} SYM theories with simple gauge groups are described in terms of elliptic hypergeometric integrals. For F ₄, E ₆, E ₇, E ₈ gauge groups this yields first examples of integrals of such type. S-duality transformation for G ₂ and F ₄ SCIs is equivalent to a change of integration variables. Equality of SCIs for SP(2N) and SO(2N + 1) group theories is proved in several important special cases. Reduction of SCIs to partition functions of 3d N=2{{\mathcal N}=2} SYM theories with one matter field in the adjoint representation is investigated, corresponding 3d dual partners are found, and some new related hyperbolic beta integrals are conjectured.