Bond Length Dependence Research Articles

Lattice dynamics calculation of infrared active modes of cuprate superconductors

Abstract We have calculated the highest-energy infrared active E u mode for three cuprate compounds with and without apical ions, in order to investigate the experimental energy-difference between T- and T ′ -structures. The calculated energies show the similar Cu–O bond length dependence to the experimental results. The calculated spin densities of the T-structure have more electrons in the d x 2 - y 2 orbital than that of the T ′ -structure. This means that the CuO2 plane in the T-structure is compressed, in comparison with the T ′ -structure even with the same Cu–O bond length. This compression increases the energy of the E u mode in the T-structure. This is the clear evidence that the block layer is important for the electronic state on the CuO2 plane.

Interaction-induced dipole moment of the Ar–H2 dimer: Dependence on the H2 bond length

We present ab initio calculations of the interaction-induced dipole moment of the Ar-H2 van der Waals dimer. The primary focus of our calculations is on the H2 bond length dependence of the dipole moment, which determines the intensities of both the collision-induced H2 upsilon = 1 <-- 0 fundamental band in gaseous Ar-H2 mixtures and the dopant-induced H2 upsilon = 1 <-- 0 absorption feature in Ar-doped solid H2 matrices. Our calculations employ large atom-centered basis sets, diffuse bond functions positioned between the two monomers, and a coupled cluster treatment of valence electron correlation; core-valence correlation effects appear to make negligible contributions to the interaction-induced dipole moment for the Ar-H2 configurations considered here.

Static Dipole Polarizability and Hyperpolarizability of F2 from Density Functional Theory Calculations. Similarities and Dissimilarities with Conventional ab initio Results

We have calculated the bond-length dependence of the static dipole polarizability and hyperpolarizability of F2 relying on finite-field density functional calculations. At the internuclear separation of Re = 2.66816 a0 we obtain α/ e2a0 2Eh-1 = 8.73 (B3LYP) and 8.60 (B3PW91),α./ e2a0 2Eh-1 = 6.42 (B3LYP) and 6.32 (B3PW91), γ / e4a04Eh-3 = 535 (B3LYP) and 503 (B3PW91) with a large, flexible [9s6p4d1f] basis set. For all three properties we observe that both B3LYPand B3PW91 yield values above the more accurate CCSD(T) results: CCSD(T) &lt; B3PW91 &lt; B3LYP. The discrepancy between DFT and conventional ab initio methods is more pronounced for the anisotropy of the (hyper)polarizability.

How H‐Bonding Modifies Molecular Structure and π‐Electron Delocalization in the Ring of Pyridine/Pyridinium Derivatives Involved in H‐Bond Complexation.

Analysis of the experimental geometry of 397 variously substituted pyridine and pyridinium derivatives has shown very little variation of all bond lengths in the ring and substantial changes in the values of α and β angles, being in line with theoretically modeled data (at the B3LYP/6-311+G** level of theory). No dependence of bond lengths and π-electron delocalization in the ring measured by HOMA index on the changes in bond angle at the N-atom is observed. This is at variance with the patterns observed in substituted phenolate/phenol systems involved in H-bonding. Due to significant differences in magnitude of the ipso angle at the nitrogen atom that depend on the location of hydrogen atom in pyridine/pyridinium systems (N, N···H, and N−H), this angle may be used as a valuable indicator of N···H interactions since the X-ray structural analysis does not give a reliable position for the proton.

Influence of Lattice Interactions on the Jahn−Teller Distortion of the [Cu(H2O)6]2+ Ion: Dependence of the Crystal Structure of K2[Cu(H2O)6](SO4)2x(SeO4)2-2x upon the Sulfate/Selenate Ratio

The temperature dependence of the structure of the mixed-anion Tutton salt K2[Cu(H2O)6](SO4)(2x)(SeO4)(2-2x) has been determined for crystals with 0, 17, 25, 68, 78, and 100% sulfate over the temperature range of 85-320 K. In every case, the [Cu(H2O)6]2+ ion adopts a tetragonally elongated coordination geometry with an orthorhombic distortion. However, for the compounds with 0, 17, and 25% sulfate, the long and intermediate bonds occur on a different pair of water molecules from those with 68, 78, and 100% sulfate. A thermal equilibrium between the two forms is observed for each crystal, with this developing more readily as the proportions of the two counterions become more similar. Attempts to prepare a crystal with approximately equal amounts of sulfate and selenate were unsuccessful. The temperature dependence of the bond lengths has been analyzed using a model in which the Jahn-Teller potential surface of the [Cu(H2O)6]2+ ion is perturbed by a lattice-strain interaction. The magnitude and sign of the orthorhombic component of this strain interaction depends on the proportion of sulfate to selenate. Significant deviations from Boltzmann statistics are observed for those crystals exhibiting a large temperature dependence of the average bond lengths, and this may be explained by cooperative interactions between neighboring complexes.

How H-bonding Modifies Molecular Structure and π-Electron Delocalization in the Ring of Pyridine/Pyridinium Derivatives Involved in H-Bond Complexation

[structure: see text] Analysis of the experimental geometry of 397 variously substituted pyridine and pyridinium derivatives has shown very little variation of all bond lengths in the ring and substantial changes in the values of alpha and beta angles, being in line with theoretically modeled data (at the B3LYP/6-311+G level of theory). No dependence of bond lengths and pi-electron delocalization in the ring measured by HOMA index on the changes in bond angle at the N-atom is observed. This is at variance with the patterns observed in substituted phenolate/phenol systems involved in H-bonding. Due to significant differences in magnitude of the ipso angle at the nitrogen atom that depend on the location of hydrogen atom in pyridine/pyridinium systems (N, N...H, and N-H), this angle may be used as a valuable indicator of N...H interactions since the X-ray structural analysis does not give a reliable position for the proton.

Relaxing the graphite lattice along critical directions: The effect of the electron–phonon coupling on the π electron band structure

Abstract We investigate the effects on the electronic structure induced by a relaxation of the graphite lattice along the direction of the phonons associated to the G and to the D peak characteristic of the Raman spectra of graphitic materials. To this aim the bond length dependence of the hopping integral β is introduced in a Huckel (tight-binding) hamiltonian, giving rise to two different β1 and β2 parameters. A Peierls-like gap opening is found for a relaxation along the D peak phonon and not for the G peak phonon. The electronic density of states is discussed and a link is shown between the electronic structure of benzene and the electronic structure of graphene at the K point in reciprocal space.

Structural variation of cubic and hexagonal MgxZn1−xO layers grown on MgO(111)∕c-sapphire

We report on the structure study of MgxZn1−xO films and, in particular, we will focus on MgxZn1−xO layers with x=0.28 and 0.41 MgxZn1−xO layers with different crystal structures of cubic and wurtzite that have been grown by plasma-assisted molecular-beam epitaxy on MgO∕c-sapphire with Mg∕Zn flux ratio control. The MgxZn1−xO films have been characterized by high-resolution transmission electron microscopy (HRTEM) and high-resolution x-ray diffraction. The dependence of the cation-anion bond length to Mg content has been studied. A virtual crystal model of MgZnO has been applied to interpret the bond-length variation. HRTEM results indicate that the initial stage of the MgZnO growth on a MgO buffer layer starts with a cubic structure even in the case of a wurtzite structure at the end of growth.

Anatomy of bond formation. Bond length dependence of the extent of electron sharing in chemical bonds

Summary This paper reports the application of an index, which we propose to call the shared-electron distribution index, for quantifying the changes to the extent of electron sharing during the making or breaking of chemical bonds. For this purpose we use this index, which is based on the correlated pair density, to monitor the effects of increasing the bond length in four representative diatomic molecules. We do of course employ wave functions which describe correctly the dissociation process. It is found for the nonpolar bonds in H 2 and N 2 that the degree of electron sharing increases monotonically with shortening of the bond length, but that there appears to be no molecule-specific significance to the location of the point of inflexion. The variation of the extent of electron sharing with internuclear distance is somewhat more complex for the polar bonds in LiH and BeH, but it reflects directly the known variation of the nature of the wave functions.

Simple bond length dependence: A correspondence between reactive fluid theories

Two elementary models of reactive fluids are examined, the first being a standard construction assuming molecular dissociation at infinite separation; the second is an open mixture of nondissociative molecules and free atoms in which the densities of free atoms and molecules are coupled. An approximation to the density of molecules, to low order in site density, is derived in terms of the classical associating fluid theory variously described by Wertheim [J. Chem. Phys. 87, 7323 (1987)] and Stell [Physica A 231, 1 (1996)]. The results are derived for a fluid of dimerizing hard spheres, and predict dependence of the molecular density on the total site density, the hard sphere diameter, and the bond length of the dimer. The results for the two reactive models are shown to be qualitatively similar, and lead to equivalent predictions of the molecular density for the infinitely short and infinitely long bond lengths.

Magnetic structure, magnetostriction, and magnetic transitions of the Laves-phase compoundNdCo2

The crystal and the magnetic structures as well as magnetostriction of the Laves-phase compound $\mathrm{Nd}{\mathrm{Co}}_{2}$ are investigated by means of temperature-dependent high-resolution neutron-powder diffraction. The compound crystallizes in the cubic Laves phase $C15$ structure at high temperature, undergoes a tetragonal distortion (space group $I{4}_{1}∕amd$) around ${T}_{C}\ensuremath{\approx}100\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and an orthorhombic distortion (space group $Fddd$) at $T\ensuremath{\approx}42\phantom{\rule{0.3em}{0ex}}K$. The temperature dependence of lattice constants, magnetostriction constant, and magnetic moment indicate that the magnetic and structural transitions are second order in the vicinity of ${T}_{C}$ and are first-order around 42 K. Refinements of magnetic structure reveal that the Nd moment distinctly exhibits an abrupt increase at the first-order transition and the easy magnetization direction of the compound changes from [001] in the tetragonal lattice to [011] in the orthorhombic lattice, indicating a strong coupling between crystal structure and magnetic properties at zero field. Analysis of the temperature dependence of bondlength suggests a strong magnetic exchange striction in the tetragonal structure and that the abrupt increase of the Nd moment is attributed essentially to a change in crystal electric field. Field-dependent neutron diffraction reveals a decoupling of the magnetic and structural transitions under relatively modest magnetic fields.

Bond Length Dependence on Quantum States as Shown by Spectroscopy

Undergraduate students often have the misconception that molecules have fixed, unchanging bond lengths. This article discusses how linear-molecule rotational band spacings in infrared spectroscopy can be used as a qualitative, visual demonstration of the elongation of average bond lengths on vibrational excitation. The method does not depend on a detailed mathematical analysis of the spectra. In UV–vis spectroscopy, the rotational band spacings give rise to distinctive linear-molecule rotational contours, which easily show whether the average bond length has increased or decreased. The method is based on a spreadsheet simulation of the vibration–rotation or rovibronic (electronic–vibration–rotation) spectrum and is applied to hydrogen chloride IR, iodine UV–vis, and nitrogen UV–vis spectra in this article.

Molecular orbital excitations in cuprates: Resonant inelastic x-ray scattering studies

We report resonant inelastic x-ray scattering studies of electronic excitations in a wide variety of cuprate compounds. Specifically, we focus on the charge-transfer type excitation of an electron from a bonding molecular orbital to an antibonding molecular orbital in a copper oxygen plaquette. Both the excitation energy and the amount of dispersion are found to increase significantly as the copper oxygen bond-length is reduced. We also find that the estimated bond-length dependence of the hopping integral t_pd is much stronger than that expected from tight-binding theory.

Bonding and (hyper)polarizability in the sodium dimer

We report a conventional ab initio and density functional theory study of the polarizability (alpha(alphabeta)/e(2)a(0) (2)E(h) (-1)) and hyperpolarizability (gamma(alphabetagammadelta)/e(4)a(0) (4)E(h) (-3)) of the sodium dimer. A large [18s14p9d2f1g] basis set is thought to yield near-Hartree-Fock values for both properties: alpha=272.28, Deltaalpha=127.22 and gamma=2157.6 x 10(3) at R(e)=3.078 87 A. Electron correlation has a remarkable effect on the Cartesian components of gamma(alphabetagammadelta). Our best value for the mean is gamma=1460.1 x 10(3). The (hyper)polarizability shows very strong bond-length dependence. The effect is drastically different for the longitudinal and transverse components of the hyperpolarizability. The following first derivatives were extracted from high-level coupled cluster calculations: (dalpha/dR)(e)=54.1, (dDeltaalpha/dR)(e)=88.1e(2)a(0)E(h) (-1), and (dgamma/dR)(e)=210 x 10(3)e(4)a(0) (3)E(h) (-3). We associate the (hyper)polarizability to bonding effects between the two sodium atoms by introducing the differential property per atom Q(diff)/2 identical with (Q[Na(2)(X (1)Sigma(g) (+))]/2-Q[Na((2)S)]). The differential (hyper)polarizability per atom is predicted to be strongly negative for the dimer at R(e), as [alpha(Na(2))/2-alpha(Na)]=-33.8 and [gamma(Na(2))/2-gamma(Na)]=-226.3 x 10(3). The properties calculated with the widely used B3LYP and B3PW91 density functional methods differ significantly. The B3PW91 results are in reasonable agreement with the conventional ab initio values. Last, we observe that low-level ab initio and density functional theory methods underestimate the dipole polarizability anisotropy. Experimental data on this important property are highly desirable.

Bond Length Dependence of Cluster Expansions of Pi-Electron Energy

There are two long established methods for representing the delocalized portion of the pi-electron energy of conjugated molecules in a simple local fashion. One, introduced by Hess and Schaad, can be used to provide a very accurate estimate of the energy of acyclic polyenes. Another, introduced originally by Herndon, has been successfully applied to the energy of aromatic ring systems. Both methods can be shown to be forms of cluster expansions, in which the energy of a molecule is expressed exactly as a sum of (successively smaller) contributions from successively larger molecular fragments. As such, it should in principle be possible to combine the methods to provide a consistent description of molecules of almost any structure. However past attempts to do so have yielded physically unreasonable results. It is shown that the key missing ingredient in these attempts is a failure to account for bond stretching and compression. When the energy changes associated with bond length changes are accounted for properly, the combined cluster expansion is quite successful in accounting for molecular pi electron energies in a very compact form.

Single (C–C) and triple (CC) bond-length dependence of the static electric polarizability and hyperpolarizability of H–CC–CC–H

Abstract We report an ab initio study of the static electric (hyper)polarizability of diacetylene and its dependence on the single (C–C) and triple (CC) bond length. At the CCSD(T) level of theory we find for the mean dipole polarizability and its derivatives α =49.10 e 2 a 0 2 E h −1 ,( ∂ α / ∂ R C − C ) e =−4.41 and ( ∂ α / ∂ R C  C ) e =34.57 e 2 a 0 E h −1 . For the anisotropy Δ α=54.45 e 2 a 0 2 E h −1 , (∂Δα/∂RC–C)e=−20.42 and ( ∂Δ α/ ∂ R C  C ) e =64.56 e 2 a 0 E h −1 . The dependence of the mean hyperpolarizability on RC–C and RCC around the equilibrium is quite distinct. Varying the single bond by ΔR/a0 around the equilibrium entails changes of [ γ (R C – C )− γ (R e )]/ e 4 a 0 4 E h −3 =−3643 Δ R−230 Δ R 2 −184 Δ R 3 +453 Δ R 4 The mean second hyperpolarizability increases strongly with RC≡C around the equilibrium [ γ (R C  C )− γ (R e )]/ e 4 a 0 4 E h −3 =22 259 Δ R+11 293 Δ R 2 +2384 Δ R 3 +6445 Δ R 4

Electric (hyper)polarizability derivatives for the symmetric stretching of carbon dioxide

Abstract We have relied on finite-field Coupled Cluster calculations with carefully designed basis sets to obtain the bond length dependence of the static electric (hyper)polarizability for the symmetric stretching of carbon dioxide. We find that around the experimental bond length of R e / a 0 the mean ( α ) and the anisotropy (Δ α ) of the dipole polarizability and the mean hyperpolarizability ( γ ) vary at the CCSD(T) level of theory as [ α (R)− α (R e )]/e 2 a 0 2 E h −1 =13.81(R−R e )+3.78(R−R e ) 2 −0.83(R−R e ) 3 +0.44(R−R e ) 4 , [ Δ α(R)− Δ α(R e )]/e 2 a 0 2 E h −1 =19.35(R−R e )+11.23(R−R e ) 2 +1.06(R−R e ) 3 +0.17(R−R e ) 4 , [ γ (R)− γ (R e )]/e 4 a 0 4 E h −3 =1493(R−R e )+1041(R−R e ) 2 −433(R−R e ) 3 −350(R−R e ) 4 . . Our values for the derivatives ( d α / d R) e , ( d 2 α / d R 2 ) e and the anisotropy Δ α agree quite well with the recent experimental findings.

Microscopic theory of vibronic dynamics in linear polyenes

We propose an approach to calculate dynamical processes at an ultrafast time scale in molecules in which vibrational and electronic motions are strongly mixed. The relevant electronic orbitals and their interactions are described by a Hubbard model, while electron-phonon interaction terms account for the dependence of the hopping on bond length and the dependence of the equilibrium bond length (atomic radii) on the local charge. The latter term plays a crucial role in the nonadiabatic internal-conversion process of the molecule. The time-resolved photoelectron spectra are in good qualitative agreement with experiments.

Structural properties of magnesium and aluminium co-substituted lithium ferrite

The structural properties of Mg2+ and Al3+ co-substituted Li0.5Fe2.5O4 are studied by synthesizing the spinel solid solution series MgxAl2xLi0.5(1−x)Fe2.5(1−x)O4. Polycrystalline samples of this series with x=0.0, 0.1, 0.2, 0.3, 0.4 and 0.5 have been prepared by double-sintering ceramic method. The structural details like: lattice constant and distribution of cations in the tetrahedral and octahedral interstitial voids have been deduced through X-ray diffraction (XRD) data analysis. The x dependence of bond length, oxygen positional parameter, site ionic radii, bulk density, porosity and shrinkages have also been determined.

Electric Quadrupole and Hexadecapole Moment, Dipole and Quadrupole Polarizability, Second Electric Dipole Hyperpolarizability for P2, and a Comparative Study of Molecular Polarization in N2, P2, and As2

We have obtained electric properties of P⋮P from finite-field Møller−Plesset perturbation theory, density functional theory and coupled cluster techniques. Reference, near-Hartree−Fock values have been obtained with a very large (20s15p10d5f) uncontracted basis set consisting of 300 Gaussian-type functions. At the experimental equilibrium bond length of Re = 1.8934 Å we obtain self-consistent field values of 1.0682 ea02 for the quadrupole moment (ϑ), −41.68 ea04 for the hexadecapole moment (Φ), 51.16 for the mean ( ) and 28.58 e2a02Eh-1 for the anisotropy of the dipole polarizability, and 16.5 × 103 e4a04Eh-3 for the mean second dipole hyperpolarizability ( ). Electron correlation reduces strongly the magnitude of the electric moments. Both components of the dipole polarizability are reduced by electron correlation, but a small increase is observed for the dipole hyperpolarizability. Our best post-Hartree−Fock values have been obtained with a [9s7p5d3f] basis set at the CCSD(T) level of theory: ϑ = 0.4850 ea02, Φ = −31.25 ea04, = 49.20 and Δα = 28.02 e2a02Eh-1, = 16.8 × 103 e4a04Eh-3. The bond-length dependence around Re has been obtained for all properties. Conventional density functional theory methods predict dipole polarizabilities close enough to the most accurate CCSD(T) values but overestimate the second dipole hyperpolarizability. The mean dipole polarizability changes as (NaK) > (AlCl) > (SiS) > (P2) > (Zn) for some isoelectronic, 30-electron systems. It is seen that the electric properties in the sequence N2→P2→As2 display regular changes.