Meridional Coordination Research Articles

Mer-[Ni(dien)2]Cl2.H2O and mer-[Ni(dien)2](NO3)2 complexes (dien= diethylenetriamine): Synthesis, physical, computational studies, antioxidant activity, and their use as precursors for Ni/NiO nanoparticles’ preparation

Two new Ni(II) complexes, mer-[Ni(dien)2]Cl2.H2O (I) and mer-[Ni(dien)2](NO3)2 (II), where dien= diethylenetriamine and mer refers to meridional coordination, have been prepared under ambient conditions. Complex (I) crystallizes in the P21/c space group of the monoclinic system, with a = 13.5000(4) Å, b = 8.7170(3) Å, c = 13.9430(4) Å, β = 102.091(2)° and V = 1604.41 Å3, whereas complex (II) crystallizes in the Pbca space group of the orthorhombic system, with a = 8.8117(6) Å, b = 14.0023(8) Å, c = 26.7603(17)Å and V = 3301.79 Å3. Both structures contain the [Ni(C4H13N3)2]2+ cation, together with two chloride anions and a water molecule for (I) and two nitrate anions for (II). The metal centre is coordinated to six nitrogens of the neutral dien ligands in a slightly distorted octahedral geometry. The two compounds have been characterised by FTIR and UV-Vis spectroscopy. Their morphology and size were studied by scanning electron microscopy (SEM). The thermal behaviour of the title compounds was also investigated. Calcination of the two complexes in air yielded Ni/NiO nanoparticles. The structures of both complexes were optimised and their simulated geometric parameters and vibrational modes were discussed and correlated with the experimental data. The antioxidant potential of both complexes was evaluated using DPPH, ferric reducing power tests and phosphomolybdenum assay.

Tuning the Coordination Environment of Ru(II) Complexes with a Tailored Acridine Ligand.

A novel tridentate ligand featuring an acridine core and pyrazole rings, namely 2,7- di-tert-butyl-4,5-di(pyrazol-1-yl)acridine, L, was designed and used to create two ruthenium(II) complexes: [RuL2](PF6)2 and [Ru(tpy)L](PF6)2. Surprisingly, the ligand adopted different coordination modes in the complexes: facial coordination for the homoleptic complex and meridional coordination for the heteroleptic complex. The electronic absorption and electrochemical properties were evaluated. Although both complexes exhibited favorable electronic properties for luminescence, neither emitted light at room temperature nor at 77 K. This study highlights the complex interplay between ligand design, coordination mode, and luminescence in ruthenium(II) complexes.

Synthesis of Asymmetric Variants in a Dinucleating Ligand Family and Application for Dinuclear Copper(II) Complexes

AbstractWe developed a dinucleating ligand system with two identical tetradentate coordination compartments and varying terminal donors. Here, we report the synthesis of two differently asymmetric dinucleating ligands: the ligand selma (=4,7,10‐trimethyl‐1,1,10‐tris(2‐pyridylmethyl)‐1,4,7,10‐tetraazadecane) with one N3‐ and one N4‐ and the ligand susan (=4,7‐dimethyl‐1,1‐bis(6‐methylpyridine‐2‐yl‐methyl)‐10,10‐bis(2‐pyridylmethyl)‐1,4,7,10‐tetraazadecane) with two different N4‐coordination compartments. The initial synthesis of the ligand selma based on extending dipicolylamine with 2‐(methylamino)ethyl units in repeated reductive aminations. Synthesis of 1‐trifluoroacetyl‐4,7‐dimethyl‐1,4,7,10‐tetraazadecane allowed the synthesis of both ligands selma and susan . To study the influence of the terminal donors, the CuII chloride complexes of selma and susan and the two symmetric susan and susan6−Me were synthesized. The introduction of the 6‐methyl substituents on the pyridines increases the bond lengths, reduces the electron donation, and shifts the reduction potentials anodically. The large 6‐methyl substituents enforce a meridional coordination of the dipicolylamine subunits, while it is mainly facial without the 6‐methyl substituents. Analysis of the solid‐state structures and the d‐d transitions in solution demonstrate a flat potential energy surface for the 5‐coordinate polyhedra. Analysis of d‐d transitions and reduction potentials indicate that the properties of the CuIICuII complexes are additive for but cooperative for selma reflecting the difference of these asymmetric dinucleating ligands.

On a priori bounding the growth of thermal instability waves

We have previously shown that the nonlinear growth of a finite-amplitude perturbation to a basic state given by a baroclinic zonal flow on the β-plane in a thermal quasigeostrophic reduced-gravity model can be a priori bounded. In this note, we show that, unlike we stated earlier, Lyapunov stability can be proved even when buoyancy varies linearly with the meridional coordinate. In addition to rectifying our earlier results, we expand them by deriving an instability saturation bound by making use of the existence of such a class of Lyapunov-stable basic states. This bound can be smaller than that one we estimated before, reinforcing our previous conclusions. We also present a numerical test of the accuracy of the derived bound.

Influence of triphosphine ligand coordination geometry in Mn(I) hydride complexes [(P∩P∩P)(CO)2MnH] on their kinetic hydricity.

Octahedral Mn(I) complexes bearing tridentate donor ligands [(L∩L'∩L'')(CO)2MnX] have recently emerged as major players in catalytic (de)hydrogenation processes. While most of these systems are still based on structurally rigid pincer scaffolds imposing a meridional coordination mode, for some more flexible tridentate ligands a facial arrangement of donor moieties becomes possible. Accordingly, the geometry of the corresponding Mn(I) hydrides [(L∩L'∩L'')(CO)2MnH] directly involved in the catalytic processes, namely the nature of the donor extremity located in the trans-position of the hydride (CO and L for mer- and fac-configurations, respectively) may influence their hydride transfer ability. Herein, low-temperature IR and NMR spectroscopy studies of two model Mn(I) complexes, mer-[(L1)(CO)2MnH] and fac-[(L2)(CO)2MnH], bearing similar triphosphine ligands (L1 = PhP(CH2CH2PPh2)2; L2 = MeC(CH2PPh2)3) in the presence of B(C6F5)3 as the H- abstractor revealed for the first time a higher kinetic hydricity of the tripodal system. Even for the pincer complex, hydride transfer proceeds from the non-covalent adduct fac-[(L1)(CO)2MnH]⋯B(C6F5)3 with the facial geometry arising from the mer-to-fac isomerization of the initial mer-[(L1)(CO)2MnH]. The higher reactivity of the fac-hydride derivatives was found to be consistent with the catalytic performance of the corresponding Mn(I) bromide complexes in the benchmark ester hydrosilylation.

Five-Spin Copper(II) Nitroxide Complex with Apparently Compressed Octahedral Geometry: Design, Synthesis, and Magnetostructural Studies

Multispin Cu(II)-nitroxide compounds have attracted significant interest in the field of molecular magnetism; however, complexes involving more than three spins in a single molecule have rarely been reported. In this work we describe the synthesis and detailed physicochemical study of the solvated six-coordinated five-spin complex [CuL2] (L = 2,2′-(1H-pyrrole-2,5-diyl)bis(4,4,5,5-tetramethyl-4,5-dihydro-1H-imidazole-3-oxide-1-oxyl)). This complex features a tridentate meridional coordination of the biradical L, resulting in an exchange-coupled cluster of four radical and one Cu(II) spin. The structural and magnetic properties of [CuL2] have been thoroughly investigated using single-crystal X-ray diffraction (XRD), SQUID magnetometry, electron paramagnetic resonance (EPR), Fourier-transform infrared (FTIR) spectroscopy, and quantum chemical calculations (QCC). An apparently compressed octahedral coordination geometry has been found by XRD, which, however, stems from the superposition of two isomeric structures of elongated octahedral CuN2O4 units. Neither X-ray nor FTIR has revealed any structural transitions in [CuL2] at 8–300 K, whereas the magnetic behavior studied by magnetometry and QCC is governed by temperature-dependent populations of high-spin multiplets. EPR spectroscopy revealed a drastic temperature dependence of the line width in [CuL2], which changes by almost 2 orders of magnitude within 4–300 K. A comparison with EPR and QCC data for close-structure four- and three-spin analogues has revealed an interplay of the temperature-dependent population of high-spin multiplets and intermolecular exchange interactions, which govern the line width in such systems.

Nitrogen Reduction to Ammonia by a Phosphorus-Nitrogen PN 3 P-Mo(V) Nitride Complex: Significant Enhancement via Ligand Postmodification

Nitrogen Reduction to Ammonia by a Phosphorus-Nitrogen PN ³ P-Mo(V) Nitride Complex: Significant Enhancement via Ligand Postmodification

3d Metal Complexes in T-shaped Geometry as a Gateway to Metalloradical Reactivity.

ConspectusLow-valent, low-coordinate 3d metal complexes represent a class of extraordinarily reactive compounds that can act as reagents and catalysts for challenging bond-activation reactions. The pursuit of these electron-deficient metal complexes in low oxidation states demands ancillary ligands capable of providing not only energetic stabilization but also sufficiently high steric bulk at the metal center. From this perspective, pincer ligands are particularly advantageous, as their prearranged, meridional coordination mode scaffolds the active center while the substituents of the peripheral donor atoms provide effective steric shielding for the coordination sphere. In a T-shaped geometry, the transition metal complexes possess a precisely defined vacant coordination site, which, combined with the often observed high-spin electron configuration, exhibits unusually high selectivity of these compounds with respect to one-electron redox chemistry. In light of the intractable reaction pathways typically observed with related electronically unsaturated 3d transition metal complexes, the pincer coordination mode enables the isolation of low-valent compounds with more controlled and unique reactivity. We have thus investigated a series of T-shaped metal(I) complexes using three different types of pincer ligands, which may be regarded as "metalloradicals" due to their selectively exposed unpaired electrons.These compounds display remarkably high thermal stability and represent rarely observed "naked" monovalent metal species featuring both monomeric and dimeric structures. Extensive reactivity studies using various organic substrates highlight a strong tendency of these paramagnetic compounds to undergo one-electron oxidation, leading to the isolation of a plethora of metal(II) species with reduced organic ligands as unusual structural elements. The exploration of C2 symmetric T-shaped Ni(I) complexes as asymmetric catalysts also shows success in enantioselective hydrodehalogenation of geminal dihalogenides. In addition, this specific class of low-valent, low-coordinate complexes can be further diversified by introducing redox-active pincer ligands or building homobimetallic systems with two T-shaped units.This Account focuses on the discussion of selected examples of iron, cobalt, and nickel pincer complexes bearing a [P,N,P] or [N,N,N] donor set; however, their electronic structure and radical-type reactivity can be broadly extended to other pincer systems. The availability of various types of pincer ligands should allow fine-tuning of the reactivity of the T-shaped complexes. Given the unprecedented reactivity observed with these compounds, we expect the studies of T-shaped 3d metal complexes to be a fertile field for advancing base metal catalysis.

Facial vs. meridional coordination in gaseous Ni(II)-hexacyclen complexes revealed with infrared ion spectroscopy.

We report fingerprint infrared multiple-photon dissociation (IRMPD) spectra of the isolated gaseous hexa-coordinated complex of the macrocycle hexa-aza-18-crown-6 (hexacyclen, 1,4,7,10,13,16-hexaazacyclooctadecane, 18-azacrown-6) with Ni2+. The metal-ligand complexes are generated using electrospray ionization (ESI) and IR action spectra are recorded in a Fourier transform ion cyclotron resonance mass spectrometer (FTICR) MS coupled to the infrared free-electron laser FELIX. We investigate geometric structure of the complexes and in particular the chelation motif, by comparison with computed vibrational spectra, obtained using density functional theory (DFT) at the B3LYP/6-31++G(d,p) level. The quasi-octahedral chelation motif of the complex has been well documented in condensed-phase studies, and we focus here on the gas-phase structure, addressing in particular the question of a facial (fac) versus a meridional (mer) octahedral chelation geometry. Based on the good agreement between calculated linear IR spectra and experimental IRMPD spectra, we conclude that the gas-phase complex adopts a mer chelation geometry and we exclude significant contribution of the fac isomer, which is computed to lie about 10 kJ mol-1 higher in energy. We also address the possible presence of both meridional diastereomers and of higher energy conformers of meridional isomers. Finally, as expected for the d8 Ni2+-ion in an octahedral ligand environment, the IR spectrum also shows that the complexes are in a high-spin electron configuration.

Synthesis, structural characterisation and antibacterial activities of lead(II) and some transition metal complexes derived from quinoline-2-carboxaldehyde 4-methyl-3-thiosemicarbazone

Some metal complexes of formulae, [PbHNNS(NO3)2], [Cu(NNS)], [Zn(NNS)2], [Ni(NNS)2] and [Cd(NNS)2] (NNS = anionic form of the Schiff base of quinoline-2-carboxaldehyde 4-methyl-3-thiosemicarbazone) have been synthesised and structurally characterised by a variety of physico-chemical techniques. The crystal structure of [PbHNNS(NO3)2] and [Zn(NNS)2] have also been determined by X-ray crystallography. The mono-ligated nitrato-complex adopts a distorted square pyramidal geometry where the thiosemicarbazone ligand coordinates to the lead(II) ion as a uninegatively charged bidentate NS chelating agent via the azomethine nitrogen atom and the thiolate sulphur atom. The third and fourth coordinations are occupied by two nitrato ligands. In the bis-ligand zinc(II) complex [Zn(NNS)2], the zinc atom adopts a distorted octahedral geometry around the zinc(II) ion, with two uninegatively charged tridentate NNS thiosemicarbazone ligands coordinating via one of the quinoline nitrogen atom, the azomethine nitrogen atom and the mercaptide sulphur atom displaying meridional coordination. Some of the metal complexes exhibit greater antibacterial activity against the Bacillus subtilis, Staphylococcus aureus, Pseudomonas aeruginosa and Escherichia coli, as compared to the parent Schiff base ligand.

Cover Feature: Facial vs. Meridional Coordination Modes in ReI Tricarbonyl Complexes with a Carbodiphosphorane‐based Tridentate Ligand (14/2021)

Cover Feature: Facial vs. Meridional Coordination Modes in Re^I Tricarbonyl Complexes with a Carbodiphosphorane‐based Tridentate Ligand (14/2021)

Facial vs. Meridional Coordination Modes in ReI Tricarbonyl Complexes with a Carbodiphosphorane‐based Tridentate Ligand

AbstractThe reactivity of the novel cationic ligand precursor [(dppm)2CH](PF6) (1) with a weakly coordinating anion and [ReBr(CO)5] is reported. The initial coordination results in the formation of the facially coordinated dicationic complex fac‐[({dppm}2CH)Re(CO)3]X2 (fac‐2, X=Br, PF6). The facial coordination mode is retained in fac‐[({dppm}2C)Re(CO)3](PF6) (fac‐3) upon deprotonation of the central cationic donor group. Quantum chemical investigations indicate that for both complexes, 2 and 3, the meridional coordination mode is thermodynamically favored. In line with these findings, the isomerization of the facially coordinated complex fac‐3 to the meridionally coordinated complex mer‐[({dppm}2C)Re(CO)3](PF6) (mer‐3) is observed under irradiation with UV‐light.

Rapid atmospheric carbon dioxide fixation by nickel(II) complexes: meridionally coordinated diazepane-based 3N ligands facilitate fixation.

Octahedral complexes of the type [Ni(L)(H2O)3](ClO4)2 (1 and 2), where L is the tridentate 3N ligand 4-methyl-1-(pyrid-2-ylmethyl)-1,4-diazacycloheptane (L1, 1), or 4-methyl-1-(N-methylimidazolyl)-1,4-diazacycloheptane (L2, 2), have been isolated and characterized using elemental analysis, ESI-MS and electronic absorption spectroscopy. The DFT optimized structures of 1 and 2 reveal that the tridentate 3N ligands are coordinated meridionally constituting a distorted octahedral coordination geometry around nickel(ii). In methanol solution, the complexes, upon treatment with triethylamine, generate the reactive red colored low-spin square planar Ni-OH intermediate [Ni(L1/L2)(OH)]+ (1a and 2a), as characterized by ESI-MS and electronic absorption spectroscopy, and energy minimized structures. The latter when exposed to the atmosphere rapidly absorbs atmospheric CO2 to produce the carbonate bridged dinickel(ii) complexes [Ni2(L1/L2)2(μ-CO3)(H2O)2](ClO4)2 (3 and 4), as characterized by elemental analysis and the IR spectral feature (∼1608 cm-1) characteristic of bridging carbonate. The single crystal X-ray structure of 3 reveals the presence of a dinickel(ii) core bridged by a carbonate anion in a symmetric mode. Both the Ni(ii) centers are identical to each other with each Ni(ii) possessing a distorted octahedral coordination geometry constituted by a meridionally coordinated 3N ligand, a carbonate ion and a water molecule. The decay kinetics of the red intermediates generated by 1 (kobs, 7.7 ± 0.1 × 10-5 s-1) and 2 (kobs, 5.8 ± 0.3 × 10-4 s-1) in basic methanol solution with atmospheric CO2 has been determined by absorption spectroscopy. DFT studies illustrate that meridional coordination of the 3N ligand and the electron-releasing imidazole ring as in 2 facilitate fixation of CO2. The carbonate complex 3 efficiently catalyzes the conversion of styrene oxide into cyclic carbonate by absorbing atmospheric and pure CO2 with excellent selectivity.

Effective Model of Load-Bearing Layers for Layer-by-Layer Analysis of the Stress-Strain State of Three-Layer Cylindrical Irregular Shells of Revolution

The use of effective approximations that increase the rate of convergence of numerical results when constructing a finite element model of bearing layers for a more accurate layer-by-layer analysis of the stress-strain state of three-layer irregular cylindrical shells is considered. It is believed that the carrier layers are sufficiently thin and rigid, and two-dimensional finite elements of natural curvature, constructed on the basis of the classical theory of moment shells, are used to simulate the stress-strain state. It is assumed that the aggregate layer can be modeled in thickness by the required number of three-dimensional finite elements, which allows one to take into account the change in geometric and physical-mechanical characteristics, as well as the parameters of the stress-strain state, not only along the meridional and circumferential coordinates, but also along the thickness of the shell and the aggregate layer. The finite element approximations considered allow one to reduce the order of systems of equations, i.e. to reduce the dimensionality of the problems being solved in comparison with the traditionally used approximations, which is especially important for layer-by-layer analysis of layered-heterogeneous structures. The high convergence rate of the numerical results obtained using the considered finite element model of the bearing layers is confirmed by a comparison with other known finite elements.

Binding a meridional ligand in a facial geometry: A square peg in a round hole

The bis(pyridylimino)isoindoline (BPI) ligand is a tridentate chelate that binds to metals via a meridional coordination mode. However, when this ligand forms a complex with Re(CO)3, an almost exclusively facial moiety, the BPI ligand deforms to coordinate in a facial mode. We have investigated this deformation via structural and theoretical means, and the non-planar binding mode of the ligand bathochromically shifts the metal to ligand charge transfer (MLCT) transition.

Model for Layer-by-Layer Analysis of the Stress-Strain State of Three-Layer Irregular Shells of Revolution of Double Curvature

Based on the layer-by-layer analysis approach, the construction of a model of two types of finite elements (FE) of natural curvature (two-dimensional FE of the momentary bearing layers and three-dimensional FE of the filler) is considered for refined investigation of the stress-strain state (SSS) in layers of three-layer generally irregular shells of revolution of double curvature. The presented model-building algorithm allows one to avoid errors due to the discontinuity of generalized displacements on the docking surfaces of the FE of the bearing layers and the aggregate, to analyze the SSS for all three coordinates to which the shell is assigned, and to obtain a solution in an updated statement for various shell shapes and boundary conditions of the layers, as well as violations of their continuity. At the same time, the aggregate layer can be modeled in thickness by the required number of finite elements, which allows one to take into account the change in geometric characteristics, physical and mechanical properties, and SSS parameters not only in the meridional and circumferential coordinates, but also in the thickness of the shell and aggregate layer. The proposed approach significantly expands the range of problems solved in the refined formulation and allows us to carry out the calculation with a high degree of accuracy and detail. As an example, the calculation of the SSS of a three-layer animated casing with cut-outs is given. The influence of the size of the cut-outs on the stress – strain state in the layers of the three-layer shell of rotation of double curvature is studied.

Alkyl yttrium complexes of doubly cyclometallated xanthene- and naphthalene-backbone bis(phosphinimine) ligands

The synthesis of a rigid 4,5-bis(triphenylphosphinimino)-2,7-di-tert-butyl-9,9-dimethylxanthene {(Ph3PN)2XT} ligand is outlined, along with a modified synthesis for previously reported 1,8-bis(triphenylphosphinimino)naphthalene {(Ph3PN)2NAP}. Reaction of neutral (Ph3PN)2XT with [Y(CH2SiMe3)3(THF)2] resulted in double cyclometallation, yielding the base-free monoalkyl complex, [({Ph2(C6H4)PN2XT)Y(CH2SiMe3)] (1). Layering a concentrated THF solution of 1 with hexanes at −28 °C afforded THF-coordinated [({Ph2(C6H4)PN}2XT)Y(CH2SiMe3)(THF)]·2THF (1-THF·2THF), with a distorted pentagonal bipyramidal geometry and approximately meridional coordination of the pentadentate {Ph2(C6H4)PN}2XT dianion. Similarly, (Ph3PN)2NAP reacted with [Y(CH2SiMe3)3(THF)2] to afford a THF-coordinated monoalkyl complex, [{(Ph2(C6H4)PN)2NAP}Y(CH2SiMe3)(THF)] (2-THF). Layering a DME solution of 2-THF with hexanes at −28 °C afforded X-ray quality crystals of [[{(Ph2(C6H4)PN)2NAP}Y(CH2SiMe3)(κ2-DME)]·hexane (2-DME·hexane), with a highly distorted pentagonal bipyramidal geometry and a facial coordination mode of the tetradentate {Ph2(C6H4)PN}2NAP dianion.

Numerical Algorithm for Investigating the Stress-Strain State of Cylindrical Shells of Railway Tanks

On the basis of the synthesis of the grid and Godunov orthogonal sweep methods, an algorithm is proposed for solving a boundary value problem in partial derivatives describing the stress-strain state of a shell of revolution of a copper of a railway tank. A cylindrical shell arbitrarily loaded by inertial forces and pressure is considered under the combined fastening conditions at its ends. The grid method according to the explicit scheme has made it possible to transform the equation system of the shell theory to eight differential equations of the first order with respect to the meridional coordinate and the fourth order with respect to the circumferential coordinate to the system of algebraic equations with a five-diagonal matrix, whose non-zero elements are eight-order matrices. To solve the system of algebraic equations with a rare matrix of non-zero elements, the sweep method is applied in which the Gram-Schmidt orthogonalization of “sweep” vectors is used to eliminate the accumulation of computational errors, which allows us to exclude the formation of a singular matrix from the “sweep” vectors when calculating the coefficients of the solution to the boundary value problem. An example is considered in which the stress-strain state of the shell of the boiler of a railway tank undergoes the internal forces variable along the meridional and circumferential coordinates and has a rigid fastening at each end of one part of the circle and a free state at another part.

Chiral palladium pincer complexes for asymmetric catalytic reactions.

Palladium pincer complexes, containing a monoanionic terdentate ligand composed of an anionic aryl carbon atom and two mutually compatible donor sites, have aroused considerable interest since their first reports in the late 1970s. The high stability of the Pd pincer complexes and particularly their high modularity make these species ideal candidates for catalysis. Furthermore, the nature of the meridional coordination of the pincer ligands, and along with this their ability to enforce a stereo-specific environment around the Pd center, provide a good opportunity for developing chiral Pd pincer catalysts. Thus, a broad variety of chiral Pd pincer complexes have been prepared by the introduction of various stereochemical centers in the pincer skeletons. These chiral Pd pincer complexes have been successfully applied to many asymmetric catalytic reactions such as hydrophosphination reactions, allylation of aldehydes and imines, Michael and aldol reactions, Suzuki-Miyaura reactions as well as reactions of nitrile compounds with imines. This review focuses on the synthetic methods and the applications of chiral Pd pincer complexes in asymmetric catalysis.

A Finite Strip for the Vibration Analysis of Rotating Toroidal Shell Under Internal Pressure

In this paper, a finite strip for vibration analysis of rotating toroidal shells subjected to internal pressure is developed. The expressions for strain and kinetic energies are formulated in a previous paper in which vibrations of a toroidal shell with a closed cross section are analyzed using the Rayleigh–Ritz method (RRM) and Fourier series. In this paper, however, the variation of displacements u, v, and w with the meridional coordinate is modeled through a discretization with a number of finite strips. The variation of the displacements with the circumferential coordinate is taken into account exactly by using simple sine and cosine functions of the circumferential coordinate. A unique argument nφ+ω t is used in order to be able to capture traveling modes due to the shell rotation. The finite strip properties, i.e., the stiffness matrix, the geometric stiffness matrix, and the mass matrices, are defined by employing bar and beam shape functions, and by minimizing the strain and kinetic energies. In order to improve the convergence of the results, also a strip of a higher-order is developed. The application of the finite strip method is illustrated in cases of toroidal shells with closed and open cross sections. The obtained results are compared with those determined by the RRM and the finite element method (FEM).