Coupling Tensors Research Articles

Primary donor triplet states of Photosystem I and II studied by Q-band pulse ENDOR spectroscopy

The photoexcited triplet state of the “primary donors” in the two photosystems of oxygenic photosynthesis has been investigated by means of electron-nuclear double resonance (ENDOR) at Q-band (34 GHz). The data obtained represent the first set of 1H hyperfine coupling tensors of the 3P700 triplet state in PSI and expand the existing data set for 3P680. We achieved an extensive assignment of the observed electron-nuclear hyperfine coupling constants (hfcs) corresponding to the methine α-protons and the methyl group β-protons of the chlorophyll (Chl) macrocycle. The data clearly confirm that in both photosystems the primary donor triplet is located on one specific monomeric Chl at cryogenic temperature. In comparison to previous transient ENDOR and pulse ENDOR experiments at standard X-band (9–10 GHz), the pulse Q-band ENDOR spectra demonstrate both improved signal-to-noise ratio and increased resolution. The observed ENDOR spectra for 3P700 and 3P680 differ in terms of the intensity loss of lines from specific methyl group protons, which is explained by hindered methyl group rotation produced by binding site effects. Contact analysis of the methyl groups in the PSI crystal structure in combination with the ENDOR analysis of 3P700 suggests that the triplet is located on the Chl aʹ (PA) in PSI. The results also provide additional evidence for the localization of 3P680 on the accessory ChlD1 in PSII.

On variational principles in coupled strain-gradient elasticity

Strain-gradient elasticity is a special case of high-gradient theories in which the potential energy density depends on the first and second gradient of the displacement field. The presence of a coupling term in the material law leads to a non-diagonal quadratic form of the stored energy, which makes it difficult for the derivation of fundamental theorems. In this article, two variational principles of the minimum of potential and complementary energies are argued in the context of the coupled strain-gradient elasticity theory. The basis of the proofs of both variational principles is the equivalent transformation of the stain and strain-gradient energy density that allows to avoid the complication related to the presence of the fifth-rank coupling tensor in the equation for the potential energy density and leads to diagonalization of the quadratic form of the stored energy. This transformation enables to inverse Hook’s law, to determine compliance tensors, and to obtain closed-form relation for the complementary energy. After that the proofs of both principles of a minimum of potential and complementary energies are provided in the usual manner adopted in the classical theory of elasticity.

Geometrically nonlinear formulation of Generalized Beam Theory for the analysis of thin-walled circular pipes

In this paper, the formulation of Generalized Beam Theory (GBT) for the geometrically nonlinear analysis of thin-walled circular pipes is presented. GBT is a computationally efficient numerical method which is especially formulated for thin-walled members with a capacity of determining the cross-sectional deformation through a combination of a set of pre-determined cross-sectional deformation modes. In this study, the current GBT analysis of circular pipes which is limited to buckling analysis is enhanced to a full geometrically nonlinear analysis. This new formulation considers the nonlinear membrane kinematic description based on the Green–Lagrange strain definition. The nonlinear tangent stiffness matrices and the internal force vectors are derived from the variation of the internal energy which results in third and fourth-order GBT deformation mode coupling tensors. These tensors can predetermine the type of GBT deformation modes needed for the nonlinear analysis based on the applied loading conditions. In addition to the classical GBT deformation modes, the non-conventional GBT deformations modes have a vital role since without these modes the coupling tensors and the nonlinear stiffness matrix related to the transverse and the shear membrane energy will be lost. Here, to illustrate the application and capabilities of the developed GBT formulation, two numerical examples involving transverse and longitudinal bending are presented to show the nonlinear relationship between bending, cross-sectional ovalization, and higher local deformation modes. For the purpose of validation, these examples are compared with refined shell finite element models in both displacement and stress fields.

Addressing Single Nuclear Spins Quantum Memories by a Central Electron Spin

Nuclei surrounding single electron spins are valuable resources for quantum technology. For application in this area, one is often interested in weakly coupled nuclei with coupling strength on the order of a few 10–100 kHz. In this paper, we compare methods to address single nuclear spins with this type of hyperfine coupling to a single electron spin. To achieve the required spectral resolution, we specifically focus on two methods, namely dynamical decoupling and correlation spectroscopy. We demonstrate spectroscopy of two single nuclear spins and present a method to derive components of their hyperfine coupling tensor from those measurements.

Quantifying the quadrupolar interaction by 45Sc-NMR spectroscopy of single crystals

Single crystals of the compound [{Sc(H2O)5(μ-OH)}2]Cl4 ⋅ 2H2O were studied by 45Sc-NMR, with the effect of the quadrupolar coupling interaction on the spectra of the spin-7/2 nucleus analysed in the hierarchical framework of perturbation theory. Orientation-dependent spectra acquired at B0 = 17.6 T showed strong second-order effects due to the comparatively large coupling constant of χ = |14.613 ± 0.006| MHz, with an associated asymmetry parameter of ηQ = 0.540 9 ± 0.000 4. By analysing the splittings of the ±3/2 satellites, which in good approximation are subjected to first-order effects only, the full quadrupolar coupling tensor could be determined. The second-order effects caused by this tensor were calculated according to theoretical predictions for all orientations, and subtracted from both the centres of gravity of the satellites, and the central transitions. This allowed extraction of the full chemical shift tensor, with the eigenvalues being δ11 = (5.6 ± 0.9) ppm, δ22 = (12.4 ± 0.9) ppm, and δ33 = (38.5 ± 0.9) ppm. In spectra acquired at a lower magnetic field of B0 = 9.4 T, third-order effects could be detected, and similarly quantified using analytical expressions.

Characterization of a Triplet Vinylidene

Singlet vinylidenes (R2C=C:) are proposed as intermediates in a series of organic reactions, and very few have been studied by matrix isolation or gas-phase spectroscopy. Triplet vinylidenes, however, featuring two unpaired electrons at a monosubstituted carbon atom are thus far only predicted as electronically excited-state species and represent an unexplored class of carbon-centered diradicals. We report the photochemical generation and low-temperature EPR/ENDOR characterization of the first ground-state high-spin (triplet) vinylidene. The zero-field splitting parameters (D = 0.377 cm–1 and |E|/D = 0.028) were determined, and the 13C hyperfine coupling tensor was obtained by 13C-ENDOR measurements. Most strikingly, the isotropic 13C hyperfine coupling constant (50 MHz) is far smaller than the characteristic values of triplet carbenes, demonstrating a unique electronic structure which is supported by quantum chemical calculations.

Rotational spectra and conformational analysis of 2-bromobutane

The rotational spectra of 2-bromobutane were obtained by using a chirped-pulse Fourier-transform microwave (CP-FTMW) spectrometer in the frequency range 8–18 GHz. Hyperfine structures of both bromine isotopologues (I = 3/2) for the three conformers (G+, A, and G−) have been investigated to determine the rotational constants, centrifugal distortion constants, and nuclear quadrupole coupling constants of bromine (Br). The conformational compositions for G+, A, and G− in the supersonic jet, estimated by relative intensity measurements, are 55%, 28%, 17%, respectively, for the 79Br species, and 52%, 33%, 15%, respectively, for the 81Br species. However, the small zero-point energy differences of isotopomers can be negligible and can be considered as an average value instead of individual values, which is 54%, 30%, 16% for G+, A, and G− species. The nuclear quadrupole coupling tensors (χxx, χyy, and χzz) of Br and the quadrupolar angles were obtained, and the χzz value of 2-bromobutane is compared with those of other alkyl bromides. The spectroscopic constants, rs coordinates of Br and the dipole moments of the most stable G+ conformer are predicted well from the ab initio calculations at the MP2/6–311++G(2d,2p) level.

Characterization of the negatively charged NV defect through the spin density distribution and the hyperfine coupling constants

The spin density function and hyperfine coupling constants of the NV− defect (a vacancy with one nitrogen atom as a first neighbor) in its triplet ground state, are computed by using hybrid functionals, a Gaussian type all electron basis set, the supercell scheme (216 atoms before the defect formation), and the CRYSTAL code. The charged defect has been simulated by using the charge compensated (CC) scheme, in which a background of positive charge is added to restore the neutrality of the unit cell. The local geometry, charge and spin distribution are also reported and discussed. The EPR constants (Fermi contact, the hyperfine coupling and electric field gradient tensors) of the N (both 14N and 15N) and of two C (for the 13C isotope) atoms around the vacancy are in excellent agreement with the recent experimental results by Felton et al. (Phys. Rev. B, 79,075 203, 2009). The comparison is extended to the many other experimental data collected in the last 30 years, to simpler functionals (LDA and PBE), larger supercells (512 atoms), and to recent simulations. It is shown that the present results are much closer to experiment than previous theoretical studies, with a quantitative, rather than qualitative, agreement.

19F NMR studies on ionic conduction pathways in tysonite-CeF3

There has been a long-standing controversy over ion conducting pathways in tysonite. To unravel the pathways in CeF3, high-resolution MAS NMR spectra of 19F in CeF3 at −40–240 °C are analyzed on the bases of the calculated dipolar coupling tensor between 19F and paramagnetic electron spins at Ce3+ to assign the three spinning-sideband manifolds for the three crystallographically inequivalent sites (F1, F2, and F3). Clearly resolved sideband manifolds for F1∼F3 at 240 °C indicate that ion exchange among the three sites is not fast enough to affect the 19F spectrum, whereas F1 motion is suggested by its broader linewidth. Further, we show that addition of only 0.1% Sr2+ in CeF3 (Ce1-xSrxF3-x (x = 0.001)) brings significant effects on F-ion mobility. While some of the F2 ions remain at the F2 site, most of the F3 ions undergo exchange motion between F1 at 240 °C. The preference of the exchange pathways in CeF3 is thus consistent with that postulated for LaF3, that is, F1–F1 exchange occurs first followed by F1–F3, and F1–F2 exchange is most restricted.

Solid-State Nuclear Magnetic Resonance Techniques for the Structural Characterization of Geminal Alane-Phosphane Frustrated Lewis Pairs and Secondary Adducts.

The first comprehensive solid‐state nuclear magnetic resonance (NMR) characterization of geminal alane‐phosphane frustrated Lewis pairs (Al/P FLPs) is reported. Their relevant NMR parameters (isotropic chemical shifts, direct and indirect 27Al‐31P spin‐spin coupling constants, and 27Al nuclear electric quadrupole coupling tensor components) have been determined by numerical analysis of the experimental NMR line shapes and compared with values computed from the known crystal structures by using density functional theory (DFT) methods. Our work demonstrates that the 31P NMR chemical shifts for the studied Al/P FLPs are very sensitive to slight structural inequivalences. The 27Al NMR central transition signals are spread out over a broad frequency range (>200 kHz), owing to the presence of strong nuclear electric quadrupolar interactions that can be well‐reproduced by the static 27Al wideband uniform rate smooth truncation (WURST) Carr‐Purcell‐Meiboom‐Gill (WCPMG) NMR experiment. 27Al chemical shifts and quadrupole tensor components offer a facile and clear distinction between three‐ and four‐coordinate aluminum environments. For measuring internuclear Al⋅⋅⋅P distances a new resonance‐echo saturation‐pulse double‐resonance (RESPDOR) experiment was developed by using efficient saturation via frequency‐swept WURST pulses. The successful implementation of this widely applicable technique indicates that internuclear Al⋅⋅⋅P distances in these compounds can be measured within a precision of ±0.1 Å.

The NV0 defects in diamond: A quantum mechanical characterization through its vibrational and Electron Paramagnetic Resonance spectroscopies

The NV0 defect in diamond has been investigated quantum-mechanically, by analyzing its structural, electronic, vibrational and magnetic properties. The possible spin states for NV0 are a quartet NVq0 (4A2 symmetry) and a doublet (2E symmetry). In the latter state a single electron occupies a double degenerate level producing a Jahn Teller distortion that removes the degeneracy and lowers the total energy. The symmetry reduces from C3v to CS and two local minima are indeed identified, to be indicated as NVd10 (2A′ symmetry) and NVd20 (2A″ symmetry). NVd10 is the ground state, and is more stable than NVq0 by ~ 0.5 eV, and than NVd20 by ~ 0.2 eV, irrespective of the functional or basis set adopted.The EPR hyperfine coupling tensor has been computed for NVq0 and has been found to be in excellent agreement with available experimental data. The IR spectra of the NV defects (the three neutral cases and also the negatively charged one, for comparison) show specific peaks and shape that characterize each system and differentiate the spectra according to the spin and charge state. The Raman spectra of the NV0 defects shows a single peak, redshifted with respect to the single peak of pristine diamond by 2 cm−1 only, but in turn well separated from the negatively charged form of the defect.

Molecular Optimization for Nuclear Spin State Control via a Single Electron Spin Qubit by Optimal Microwave Pulses: Quantum Control of Molecular Spin Qubits

Quantum state control is one of the most important concepts in advanced quantum technology, emerging quantum cybernetics and related fields. Molecular open shell entities can be a testing ground for implementing quantum control technology enabling us to manipulate molecular spin quantum bits (molecular spin qubits). In well-designed molecular spins consisting of unpaired electron and nuclear spins, the electrons and nuclear spins can be bus and client qubits, respectively. Full control of molecular spin qubits, in which client spins interact via hyperfine coupling, is a key issue for implementing quantum computers (QCs). In solid-state QCs, there are two approaches to the control of nuclear client qubits, namely, direct control of nuclear spins by radio-wave (RF) pulses and indirect control via hyperfine interactions by microwave pulses applied to electron spin qubits. Although the latter is less popular in the literature, the indirectness has advantage of greatly reducing unnecessary interactions between a qubit system and its environment. In this work, we investigate molecular spin optimization to find optimal experimental conditions which can afford to achieve the high fidelity of quantum gates by the indirect control scheme. In the present quantum systems, one electron is directly controlled by pulsed ESR techniques without manipulating individual hyperfine resonance, but the states of two nuclear client spins are indirectly steered via hyperfine interactions. Single crystals of potassium hydrogen maleate (KHM) radical and 13C-labeled malonyl radical are chosen as typical molecular spin qubits which exemplify the importance of the symmetry of hyperfine tensors and their collinear properties. We have found that both the non-collinearity of the principal axes of hyperfine coupling tensors and the non-distinguishability/non-equivalency between nuclear spins are key issues which extremely reduce the gate fidelity.

Neighborhood matters: Steric effects on methyl internal rotation and chlorine nuclear quadrupole coupling in 2-fluoro-4-chlorotoluene

The rotational spectrum of 2-fluoro-4-chlorotoluene was studied in the frequency range 4 to 24 GHz by Fourier transform microwave spectroscopy. Fine torsional splittings into doublets were found for all rotational transitions due to methyl internal rotation hindered with a barrier of about 230 cm−1. In combination with the quadrupole coupling hyperfine splittings on the same order of magnitude originating from the chlorine nucleus, rich spectral features were observed. Rotational and quadrupole coupling constants for the 35Cl and 37Cl isotopologues studied in natural abundance were precisely determined. Results from the experiment were supported by quantum chemical calculations at several levels of theory for the rotational constants and by Bailey's semi-experimental method for the quadrupole coupling constants. The χcc quadrupole coupling tensor element and the barrier to methyl internal rotation were compared with those of other halogen-substituted toluene derivatives.

Analysis of the Complex Quadrupole Hyperfine Patterns for Two Chlorine Nuclei in the Rotational Spectrum of 2,5-Dichlorothiophene.

The rotational spectrum of 2,5-dichlorothiophene (DCT) was measured for the first time using Fourier transform microwave spectroscopy from 5.5-19 GHz. Dense hyperfine splitting patterns due to the two quadrupolar chlorine nuclei (I = 3/2) were resolved and assigned for the 35Cl-35Cl, 37Cl-35Cl, and 37Cl-37Cl isotopologues and for the two 13C and one 34S analogues with two 35Cl atoms, allowing derivation of their respective nuclear quadrupole coupling tensors. The rotational constants obtained from fitting the spectra of the six isotopic species allowed derivation of the experimental geometry of DCT for comparison with the equilibrium structure computed at the MP2/aug-cc-pVTZ level. This revealed that the electron-withdrawing effect of chlorine causes small distortions in the ring geometry relative to thiophene, including a 1.1° increase in the two S-C-C angles and a 0.012 Å increase in the two S-C bond lengths.

Magnetic Order and Sign of the Dzyaloshinskii–Moriya Interaction in 2-D Antiferromagnet Ba2CoGe2O7 Under Applied Magnetic Field

The Dzyaloshinskii–Moriya interaction (DMI), i.e., the antisymmetric part of the exchange coupling tensor, favors the perpendicular arrangement of magnetic moment, thus inducing canting in otherwise collinear structures. The DMI is the prerequisite for the emergence of weak ferromagnetism in antiferromagnets, but can stabilize twisted magnetic textures, such as spin spirals, soliton lattices, and magnetic skyrmions. While the magnitude of the DMI determines the canting angle of adjacent spins, its sign dictates the sense of the spin rotation. Based on focused polarized neutron diffraction (PND) study, combined with symmetry analysis, we determine the sign of the DMI in the unconventional multiferroic Ba <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> CoGe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> O <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7</sub> and reveal its detailed magnetic structure in magnetic fields applied in the tetragonal plane. As PND gives unique access to the scattering contribution from the phase-sensitive nuclear-magnetic interference, it is a valuable tool for a straightforward DMI sign determination in bulk materials and allows to disclose even very weak magnetic moments. Remarkably, the sign of the DMI could be determined from the PND measurement of a single reflection, which is demonstrated to be reliable for a large range of applied magnetic field directions and values.

Experimental and Theoretical Evidence for an Unusual Almost Triply Degenerate Electronic Ground State of Ferrous Tetraphenylporphyrin.

Iron porphyrins exhibit unrivalled catalytic activity for electrochemical CO2-to-CO conversion. Despite intensive experimental and computational studies in the last 4 decades, the exact nature of the prototypical square-planar [FeII(TPP)] complex (1; TPP2- = tetraphenylporphyrinate dianion) remained highly debated. Specifically, its intermediate-spin (S = 1) ground state was contradictorily assigned to either a nondegenerate 3A2g state with a (dxy)2(dz2)2(dxz,yz)2 configuration or a degenerate 3Egθ state with a (dxy)2(dxz,yz)3(dz2)1/(dz2)2(dxy)1(dxz,yz)3 configuration. To address this question, we present herein a comprehensive, spectroscopy-based theoretical and experimental electronic-structure investigation on complex 1. Highly correlated wave-function-based computations predicted that 3A2g and 3Egθ are well-isolated from other triplet states by ca. 4000 cm-1, whereas their splitting ΔA-E is on par with the effective spin-orbit coupling (SOC) constant of iron(II) (≈400 cm-1). Therfore, we invoked an effective Hamiltonian (EH) operating on the nine magnetic sublevels arising from SOC between the 3A2g and 3Egθ states. This approach enabled us to successfully simulate all spectroscopic data of 1 obtained by variable-temperature and variable-field magnetization, applied-field 57Fe Mössbauer, and terahertz electron paramagnetic resonance measurements. Remarkably, the EH contains only three adjustable parameters, namely, the energy gap without SOC, ΔA-E, an angle θ that describes the mixing of (dxy)2(dxz,yz)3(dz2)1 and (dz2)2(dxy)1(dxz,yz)3 configurations, and the ⟨rd-3⟩ expectation value of the iron d orbitals that is necessary to estimate the 57Fe magnetic hyperfine coupling tensor. The EH simulations revealed that the triplet ground state of 1 is genuinely multiconfigurational with substantial parentages of both 3A2g (<88%) and 3Eg (>12%), owing to their accidental near-triple degeneracy with ΔA-E = +950 cm-1. As a consequence of this peculiar electronic structure, 1 exhibits a huge effective magnetic moment (4.2 μB at 300 K), large temperature-independent paramagnetism, a large and positive axial zero-field splitting, strong easy-plane magnetization (g⊥ ≈ 3 and g∥ ≈ 1.7) and a large and positive internal field at the 57Fe nucleus aligned in the xy plane. Further in-depth analyses suggested that g⊥ ≫ g∥ is a general spectroscopic signature of near-triple orbital degeneracy with more than half-filled pseudodegenerate orbital sets. Implications of the unusual electronic structure of 1 for CO2 reduction are discussed.

Determination of the g-, hyperfine coupling- and zero-field splitting tensors in EPR and ENDOR using extended Matlab codes

The analysis of single crystal electron magnetic resonance (EMR) data has traditionally been performed using software in programming languages that are difficult to update, are not easily available, or are obsolete. By using a modern script-language with tools for the analysis and graphical display of the data, three MatLab® codes were prepared to compute the g, zero-field splitting (zfs) and hyperfine coupling (hfc) tensors from roadmaps obtained by EPR or ENDOR measurements in three crystal planes. Schonland’s original method was used to compute the g- and hfc -tensors by a least-squares fit to the experimental data in each plane. The modifications required for the analysis of the zfs of radical pairs with S = 1 were accounted for. A non-linear fit was employed in a second code to obtain the hfc -tensor from EPR measurements, taking the nuclear Zeeman interaction of an I = ½ nucleus into account. A previously developed method to calculate the g- and hfc -tensors by a simultaneous linear fit to all data was used in the third code. The validity of the methods was examined by comparison with results obtained experimentally, and by roadmaps computed by exact diagonalization. The probable errors were estimated using functions for regression analysis available in MatLab. The software will be published at https://doi.org/10.17632/ps24sw95gz.1, Input and output examples presented in this work can also be downloaded from https://old.liu.se/simarc/downloads?l=en.

Membrane Deformation Modulated by Hydration and Cholesterol Unveiled by Solid-State 2H NMR Spectroscopy

Phospholipid membranes are biological liquid crystals in which intermolecular interactions at mesoscopic length scales play key roles in the emergence of membrane physical properties like structure and rigidity. Such interactions modulated by membrane hydration [1] and cholesterol [2] have been investigated using solid-state 2H NMR spectroscopy. The average structure is manifested by the lineshapes through principal values of static or residual coupling tensors due to quadrupolar interactions. The residual quadrupolar couplings (RQCs) yield segmental order parameters for the individual acyl segments (i), providing average membrane properties such as bilayer thickness, area/lipid, and spontaneous curvature [3]. We show that membrane dehydration and increased cholesterol content result in greater RQCs, indicative of membrane thickening. Elastic membrane deformations are exhibited as collective order-director fluctuations (ODF) contributing to the nuclear magnetic relaxation. At mesoscopic length scales the thermal fluctuations connect to membrane mechanical properties via the fluctuation-dissipation theorem. Model-free interpretation of the functional dependence of spin-lattice relaxation rates, R1Z, on segmental order parameters (square-law) reveals emergent material properties in the liquid-crystalline state. The R1Z relaxation dispersion and temperature dependence of square-law plots indicate a nematic-like membrane hydrocarbon core. Solid-state 2H NMR relaxation studies demonstrate that both dehydration and increased cholesterol content in the membrane lead to decreased slopes of square-law plots, indicating an increase in membrane bending rigidity, consistent with neutron spin-echo (NSE) measurements. Analogous 2H NMR studies with various membrane lipid compositions and peptide-reconstituted membranes have addressed lipid-mediated cell signaling and lipid-protein interactions. Molecular dynamics simulations provide further insights on the correspondence of the solid-state NMR and NSE results. [1] K.J. Mallikarjunaiah et al. (2019) Phys. Chem. Chem. Phys. 21, 18422. [2] S. Chakraborty et al. (2020) Proc. Natl. Acad. Sci. 117, 21896. [3] Mallikarjunaiah et al. (2011) Biophys. J. 100, 98.

Positive definiteness for 4th order symmetric tensors and applications

In particle physics, the vacuum stability of scalar potentials is to check positive definiteness (or copositivity) of its coupling tensors, and such a coupling tensor is a 4th order and symmetric tensor. In this paper, we mainly discuss precise expressions of positive definiteness of 4th order tensors. More specifically, two analytically sufficient conditions of positive definiteness for 4th order 2 dimensional symmetric tensors are given by reducing orders of tensors, and applying these conclusions, some sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors are derived. We also present several other sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors. Finally, we test and verify the vacuum stability of general scalar potentials of two real singlet scalar fields and the Higgs boson by using these results.

Modeling flexoelectricity in soft dielectrics at finite deformation

This paper develops the equilibrium equations describing the flexoelectric effect in soft dielectrics under large deformations. Previous works have developed related theories using a flexoelectric coupling tensor of mixed material–spatial character. Here, we formulate the model in terms of a flexoelectric tensor completely defined in the material frame, with the same symmetries of the small-strain flexocoupling tensor and leading naturally to objective flexoelectric polarization fields. The energy potential and equilibrium equations are first expressed in terms of deformation and polarization, and then rewritten in terms of deformation and electric potential, yielding an unconstrained system of fourth order partial differential equations (PDE). We further develop a theory of geometrically nonlinear extensible flexoelectric rods under open and closed circuit conditions, with which we examine analytically cantilever bending and buckling under mechanical and electrical actuation. Besides being a simple and explicit model pertinent to slender structures, this rod theory also allows us to test our general theory and its numerical implementation using B-splines. This numerical implementation is robust as it handles the electromechanical instabilities in soft flexoelectric materials.