Complete Fourth Order Research Articles

On the regularity of Hamiltonian stationary Lagrangian submanifolds

We prove a Morrey-type theorem for Hamiltonian stationary Lagrangian submanifolds of Cn: If a C1 Lagrangian submanifold is a critical point of the volume functional under Hamiltonian variations, then it must be real analytic. Locally, a Hamiltonian stationary Lagrangian submanifold is determined geometrically by harmonicity of its Lagrangian phase function, or variationally by a nonlinear fourth order elliptic equation of the potential function whose gradient graph defines the Hamiltonian stationary Lagrangian submanifolds locally. Our result shows that Morrey's theorem for minimal submanifolds admits a complete fourth order analogue. We establish full regularity and removability of singular sets of capacity zero for weak solutions to the fourth order equation with C1,1 norm below a dimensional constant, and to C1,1 potential functions, under certain convexity conditions, whose Lagrangian phase functions are weakly harmonic.

Isomeric structures and energies of Hn+ clusters (n=13, 15, and 17)

Ab initio calculations have been performed for the Hn+ clusters (n=3–17; odd) at Møller–Plesset second order (MP2)/6-311G(mp), Møller–Plesset complete fourth order (MP4)/6-311G(mp), and coupled-cluster single-double-triple [CCSD(T)/6-311G(1p)] levels of calculations. Such hydrogen clusters are constituted by an H3+ core in which H2 units are bound. In order to understand the features of these bindings, enthalpy and entropy variations upon cluster formation, binding energies, and charge distributions have been computed, and a molecular orbital analysis, based on localized orbital, was performed. Our results show that the way the first three H2 units bind to the H3+ core is fundamentally different from the others, providing an explanation for the binding energies observed for these molecules. For the H13+, H15+, and H17+ clusters, the way in which the external H2 units are distributed around the H3+ plane leads to the formation of different isomers with very close energies, but with a rotational barrier large enough to inhibit the interconversions.

Upper miocene carbonate ramp deposits from the southernmost part of Maiella Nountain (Abruzzo, Central Italy)

The development of carbonate ramp depositional systems in the Neogene of the Mediterranean Region represents a widespread feature so far analysed in several papers. It is striking to note that the evolution of upper Miocene carbonate ramps, characterized by the presence of coralgal bioherms, highlights the events leading to the Messinian salinity crisis. The coralgal bioherms of preevaporite Messinian age exhibit fossil assemblages indicating marine waters with normal salinity, whereas stromatolitic and microbial encrustations underline the deterioration of the environment during the Messinian salinity crisis. Maiella Mountain is a broad carbonate massif located in Abruzzo (Central Italy). The late lower Oligocene-Messinian part of its stratigraphic succession consists of stacked non-tropical carbonate ramp deposits related to third and higher order sequences. The investigations performed in the southernmost portion of the massif allowed to recognize a complete fourth order carbonate depositional sequence on a homoclinal ramp of pre-evaporite Messinian age. The presence of small coralgal patch reefs and overlaying microbial encrustations is significant. A transect exhibits the stratigraphic framework of the area. The data show how local parameters play a notable role in the development of these deposits.

Cooperative effects in hydrogen bonding: Fourth-order many-body perturbation theory studies of water oligomers and of an infinite water chain as a model for ice

As a step toward the first principles quantum mechanical modeling of the structural and electronic properties of ice, hydrogen-bonded periodic infinite chains of water molecules have been investigated by the ab initio crystal orbital method at the Hartree–Fock (HF) level and by including electron correlation up to the complete fourth order of Mo/ller–Plesset perturbation theory (MP4). The Bloch functions of the crystal have been expanded in a series of high quality atomic orbital basis sets complemented by extended sets of polarization functions, up to TZ(3d2f,3p2d). Basis set superposition errors have been (partly) eliminated by the counterpoise method and the infinite lattice sums have been computed using the multipole expansion technique. The systematically increasing size of the basis sets has allowed the extrapolation of structural and electronic indices of this ice model to the limit of an infinite atomic basis at both the HF and various correlated levels, respectively. For each theoretical model, detailed comparisons have been made with the corresponding physical properties of water monomers, dimers, and some larger linear oligomers. The results convincingly prove that hydrogen bonding in ice is a highly cooperative phenomenon, both from the structural and energetic points of view. The cohesive energy per hydrogen bond of the crystal is −5.30 kcal/mol at the HF level (with RHFO,O=2.88 Å) as compared with the dimer value of −3.60 kcal/mol (at the optimized distance of 3.03 Å). At the MP2 level of theory, the crystalline binding energy decreases to −6.60 kcal/mol and the lattice contracts to RMP2O,O=2.73 Å (compared with −4.50 kcal/mol at 2.88 Å for the dimer). The correlation corrections at third and fourth order slightly expand the crystal lattice (to RMP4O,O=2.75 Å) and reduce the cohesion by 0.15 kcal/mol. A decomposition of the intermolecular interactions according to different terms of MP4 theory suggests that the cohesive energy of ice results from a delicate balance between different repulsive and attractive terms in third and fourth order, which exhibit different long-range behaviors. The detailed study of the role of high-energy virtual energy bands in computing electron correlation effects in ice provides further insight into the important role that basis set flexibility plays in such investigations. The resulting cohesive energy of −6.83 kcal/mol at the MP4 level is in reasonable agreement with the experimental energy per hydrogen bond in ice I, −6.7 kcal/mol.

Electron correlation in extended systems: Fourth-order many-body perturbation theory and density-functional methods applied to an infinite chain of hydrogen atoms.

Linear equidistant and bond-alternating infinite chains of hydrogen atoms have been investigated by the ab initio crystal-orbital method at the Hartree-Fock (HF) level, by including electron correlation up to the complete fourth order of the Mo/ller-Plesset perturbation theory (MP4-PT), and by using different versions of density-functional theory (DFT). The Bloch functions have been expanded in all cases in a series of high-quality atomic-orbital basis sets and complemented by extended sets of polarization functions up to 6s3p2d1f per H atom. In order to compare the performance of the PT and DFT methods, several physical properties have been computed at all theoretical levels including lattice geometry, cohesive energy, mechanisms of bond alternation (Peierls instability), and energetic features of nonequilibrium configurations (dissociation). For these latter quantities, both spin-restricted (RHF) and unrestricted (UHF) wave functions have been employed in all orders of PT. The methods described have been used parallel to infinite chains and to the ${\mathit{H}}_{2}$ molecule, to be able to check their accuracy on experiments. In the case of the DFT, six different functionals (combining Slater and Becke exchange with local and gradient-corrected correlation potentials) have been utilized to test their accuracy in comparison with the MP4 results.

Electron correlation effects in the cohesive properties of ice

An infinite cooperative network of hydrogen bonded water molecules has been studied at the Hartree—Fock (HF) level and by including correlation up to the complete fourth order of Møller—Plesser perturbation theory (MP4) using Bloch transformed atomic basis functions of the type TZ (2df, 2pd). The HF cohesive energy per hydrogen bond of the crystal is −5.30 kcal/mol at RHFO,O = 2.8804 Å as compared with the dimer values of −3.60 kcal/mol and 3.0284 Å. For MP2, the binding energy decreases to −6.60 kcal/mol and the lattice contracts to RMP2O,O=2.7416 Å (compared with −4.52 kcal/mol at 2.8928 Å for the dimer). Further correlation slightly expands the lattice (to RMP4O,O = 2.7548 Å) and leads to an estimate of the cohesive energy of −6.825 kcal/mol per water molecule in ice.

Mo/ller–Plesset perturbation investigation of the He2 potential and the role of midbond basis functions

The Mo/ller–Plesset perturbation theory up to the complete fourth order is applied to calculate the ab initio van der Waals energy potential of He2. A scheme of constructing a very efficient basis set is proposed based on theoretical considerations. Such a basis set contains a set of Gaussian functions (usually of low angular momentum) centered at the midbond of the van der Waals molecule, which have been proven very efficient to reproduce the intersystem correlation interaction energy normally to be achieved by use of the nucleus-centered high angular momentum polarization functions. A nuclear-centered moderate set [5s4p2d] augmented by a set of midbond functions such as {3s3p2d} produces a value of 10.04 K for the well depth of He2 at the complete fourth-order (MP4) theory. Results from a series of other basis sets suggest that this value is nearly saturated with the further increase of basis, which is in agreement with the estimate of the complete basis-set limit. Calculations of the potential at the minimum and the other distances (3.0–10.0 a0) confirm that the MP4 method, together with the use of midbond functions, is a reliable and convenient choice for the accurate calculations.

Finite-field fourth-order MBPT calculations of dipole moments, field, and field-gradient polarizabilities of diatomic molecules

The dipole moments and the independent components of the field polarizability α for H 2, CO, F 2,CH + and OH − and of the field-gradient polarizability A for CH +, CO and FH molecules were calculated at the self consistent field Hartree-Fock (SCF-HF) and at the complete fourth order many body perturbation theory (MBPT (4) ) level of approximation using the POL1 basis sets described recently by Sadlej. Excellent overall agreement with respect to the best values available is obtained for all electric properties and for all the species examined in this work. The only exceptions are the relatively low values obtained for the α xx component of the dipole polarizability of the OH − ion and for the A x,xz component of the FH molecule, owing to the absence of f-type GTOs in the POL1 basis.

Basis set and electron correlation effects on the electron affinities of first row atoms

The electron affinities of boron, carbon, oxygen, and fluorine atoms have been evaluated by Mo/ller–Plesset perturbation calculations through complete fourth order using several large basis sets. The convergence of the perturbation series has been evaluated carefully by means of modified coupled cluster methods. Large basis sets and electron correlation effects involving higher substitutions are found to be important in the accurate calculation of electron affinities. Triple substitutions are particularly important in the correlation treatment, contributing more than 0.2 eV to the electron affinities of O and F. The electron affinities of all the systems are calculated to be within 0.1 eV of the corresponding experimental values.

Finite-field many-body perturbation theory

The results of complete fourth order many-body perturbation theory (MBPT) calculations of the quadrupole moment tensor for the FH and H2O molecules are presented. The finite-field method employed in this study is based on the coupled Hartree-Fock (CHF) solutions for the externally perturbed system. The results are very close to the available experimental data. Of interesting features of the correlation perturbation series for the quadrupole moment components one should mention that the fourth order correlation correction is dominated by contributions involving triple and single substitutions in the reference Hartree-Fock function. Moreover, the correlation perturbation series for quadrupole moments appears to be less regular than the corresponding series for molecular dipole moments. Additionally complete fourth order MBPT calculations are performed for the quadrupole polarizability tensor of FH and H2O. The correlation corrections to the corresponding CHF values are large and exhibit similar regularities...

Prediction of temperature distribution along a thin heated wire of finite length

An exact elliptic-integral solution is given for the temperature distribution of a thin, heated wire between two supports. The solution includes the variation of thermal conductivity, electrical resistivity, convective coefficient, and emissivity of the wire with temperature. The complete fourth-order (and higher) temperature effects of thermal radiation can be included in the general solution. The hot-wire-anemometer case is evaluated numerically and is successfully compared with measured wire-temperature distributions. The exact theoretical temperature distribution is also compared with a first-order hyperbolic solution.

Finite element analysis of translational shells

A finite element procedure is presented for solving thin shell problems of arbitrary geometry and boundary conditions. The actual smoothly curved shell surface is approximated by the assemblage of flat triangular plate elements. A quadratic displacement function for the tangential displacements in the triangular element is assumed and the membrane stiffness is derived accordingly. A complete fourth order displacement function is used to derive the flexural stiffness of the triangular element. These higher order displacement functions, resulting in 27 generalized displacements, are used for the purpose of obtaining a more accurate solution to the stress conditions especially in the regions near supporting edge members for various doubly curved shells. The paper includes shells which are supported by edge beams or is stiffened by ribs. A beam type member is idealized as an assemblage of segments of straight beam elements. The twisting and axial stiffnesses, the eccentricity and the bending stiffness of the beam element are all considered.