A new method of determination of radii of ions with arbitrary effective charges

An equation for calculation of the radii of ions with an arbitrary effective charge has been derived. The ionic radii of halogens, chalcogens, and other atoms with different charges have been calculated. Under the assumption of pure ionic bond character, these values have been used for calculating the R ₁₂ distances for a large group of different molecules—halogens, interhalides, oxygen, chalcogens, and nitrogen and their compounds—by the previously derived equation for the a priori determination of internuclear distances. The error of calculation of the internuclear distance for halogens and interhalides is no more than 0.07 Å (3–4%). The internuclear distance in dihalogen cations Hal ₂ ⁺ and binary ionic molecules of p elements and their oxides and sulfides have been calculated. It has been demonstrated that the coordination (environment) of atoms should be taken into account and that there is a possibility of estimating the bond multiplicity and the character of bonding electron pairs in a molecule.     

Computation of overlap integrals over Slater-type orbitals using auxiliary functions

Analytical expressions through the binomial coefficients and recursive relations are derived for the expansion coefficients of overlap integrals in terms of a product of well-known auxiliary functions A_k and B_k. These formulas are especially useful for the calculation of overlap integrals for large quantum numbers. Accuracy of the computer results is satisfactory for the values of quantum numbers up to 50 and for the arbitrary values of screening constants of atomic orbitals and internuclear distances. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 199–204, 1998     

Origin of the variation with vibrational and rotational state of the fine-structure constants in O2, SO and S2

Calculations of the spin—orbit part λ_SO of the fine-structure parameter λ have been performed at several internuclear distances for the molecules O₂, SO and S₂ in their ground (X ³Σ^?) states. Only the interaction with the lowest ¹Σ⁺ state was considered and two-centre integrals were neglected. λ_SO is found to make the dominant contribution to λ. The variation of the matrix elements in λ_SO with internuclear distance has been derived and when combined with expressions for the variation of the energy denominator in λ_SO enables the coefficients for the vibrational and rotational dependence of λ to be estimated. Excellent agreement with experiment is found.     

Ab initio calculations of relativistic and electron correlation effects in polyatomics using the universal Gaussian basis set: XeF2

Ab initio accurate all-electron relativistic molecular orbital Dirac–Fock self-consistent field calculations are reported for the linear symmetric XeF₂ molecule at various internuclear distances with our recently developed relativistic universal Gaussian basis set. The nonrelativistic limit Hartree–Fock calculations were also performed for XeF₂ at various internuclear distances. The relativistic correction to the electronic energy of XeF₂ was calculated as ～ ?215 hartrees (?5850 eV) by using the Dirac–Fock method. The dominant magnetic part of the Breit interaction correction to the nonrelativistic interelectron Coulomb repulsion was included in our calculations by both the Dirac–Fock–Breit self-consistent field and perturbation methods. The calculated Breit correction is ～6.5 hartrees (177 eV) for XeF₂. The relativistic Dirac–Fock as well as the nonrelativistic HF wave functions predict XeF₂ to be unbound, due to neglect of electron correlation effects. These effects were incorporated for XeF₂ by using various ab initio post Hartree–Fock methods. The calculated dissociation energy obtained using the MP 2(full) method with our extensive basis set of 313 primitive Gaussians that included d and f polarization functions on Xe and F is 2.77 eV, whereas the experimental dissociation energy is 2.78 eV. The calculated correlation energy is ～ ?2 hartrees (?54 eV) at the predicted internuclear distance of 1.986 Å, which is in excellent agreement with the experimental Xe—F distance of 1.979 Å in XeF₂. In summary, electron correlation effects must be included in accurate ab initio calculations since it has been shown here that their inclusion is crucial for obtaining theoretical dissociation energy (D_e) close to experimental value for XeF₂. Furthermore, relativistic effects have been shown to make an extremely significant contribution to the total energy and orbital binding energies of XeF₂. © 1995 John Wiley & Sons, Inc.     

Improvement of the effective nuclear charge model for triatomic molecules—VII

The effective nuclear charge model previously proposed for predicting the quadratic valence force constants of polyatomic molecules is further developed to allow the prediction of their anharmonic force constants. Especially, the correction term included in the effective intramolecular potential function which is assumed in the model is expanded in the analytic form of an inverse power series of the bonded and non-bonded internuclear distances. The improved model is tested using some triatomic molecules (CO₂, CS₂, OCS, N₂O) whose experimental anharmonic force constants are presently available, and it is shown that the model is useful in discussing anharmonic force fields of more complicated polyatomic molecules.     

Structure and real composition of undoped and Cr- and Ni-doped Sr<Subscript>0.61</Subscript>Ba<Subscript>0.39</Subscript>Nb<Subscript>2</Subscript>O<Subscript>6</Subscript> single crystals

Strontium barium niobate crystals with congruent melting composition Sr_0.61Ba_0.39Nb₂O₆ (SBN-61), both nominally pure and doped with Cr³⁺ и Ni³⁺ ions, have been investigated by neutron diffraction. Different strontium and barium contents as well as their different distribution over the Sr1, of Sr2 and Ba2 crystallographic sites of SBN-61 structure, caused by introduction of dopants, have been revealed. Coordination polyhedra of cations have been established based on the analysis of cation–anion internuclear distances together with the calculation of bond-valence sums for cations, which are equal to their formal charge. It was found that the Nb1 and Nb2 atoms are located in distorted octahedra with quadfurcated (the Nb1O₆ polyhedron) or bifurcated (the Nb2O₆ polyhedron) vertices, and the Sr1 atoms are located in a cuboctahedron with bifurcated vertices in the base plane. Different polyhedra have been revealed for the Sr2 and Ba2 atoms: Sr2 atoms are coordinated by 15 oxygen atoms to form a highly distorted five-capped pentagonal prism, whereas Ba2 atoms are located in a highly distorted three-capped trigonal prism with a coordination number 9. Comparison of interatomic and internuclear distances, determined by X-ray and neutron diffraction analyses, respectively, allowed to reveal a highly pronounced shift of electron density in Nb1 and Sr2 polyhedra, responsible for the covalent bond and properties of crystals. Location of Cr³⁺ и Ni³⁺ dopant ions in the SBN-61 structure as well as their formal charges has been discussed.     

Crystal Structure and Electron-Density Distribution of Two [1.1.1] Propellane Derivatives at 81 K

The molecular structure and electron-density distribution of two [1.1.1]propellane derivatives have been determined from accurate single-crystal X-ray diffraction measurements at 81 K. The crystals of these highly reactive compounds (both are liquid at room temperature) were grown directly on the X-ray diffractometer at ca. 208 and 228 K, respectively. Both compounds crystallize in the space group P2₁/c with one molecule in the asymmetric unit. The symmetry of the propellane C-atom skeleton is close to D_3h for both molecules. The distances between the two bridgehead C-atoms are 1.587 and 1.585 Å, and the mean lengths of the propellane side bonds are 1.525 and 1.528 Å, respectively. The deformation density peaks of the propellane side bonds (ca. 0.25 e/ų) lie somewhat outside the internuclear connection lines and so correspond to ‘bent bonds’ as expected. On the other hand, the difference electron density between the bridgehead nuclei is slightly negative in both molecules. An interesting feature, observed in these difference maps, is presence of a diffuse, positive difference density at each inverted C-atom, outside the bridgehead bond, a probable site of electrophilic attack.     

Supplementary d and f functions in molecular wave functions at large and small internuclear separations

The CO, CO₂, CS, CIF, and SO₂ molecules were used to test the dependence of supplementary d and f function exponents to changes in bond lengths and bond angles in MO calculations utilizing Gaussian basis sets in Hartree–Fock and Moller–Plesset calculations. Using Dunning–Hay double zeta basis sets, optimizations were performed at internuclear separations from 100–200 pm and beyond. The energy cost of not reoptimizing d function exponents when bonds are stretched or compressed is much smaller for correlated calculations than for those at the Hartree–Fock level and is greatest at the lower end of the range of internuclear distances. The problem is much less serious at all levels when multiple sets of d functions are used. © 1993 John Wiley & Sons, Inc.     

Ab initio Hartree-Fock calculations of electronic wave functions for the c 3πu state of H2

Theoretical electronic wave functions, potential curves, and expectation values of some one-electron properties are given for the c³II_u state of the hydrogen molecule. The calculations are carried out by the matrix Hartree-Fock method and use a 2-center basis of Slater-type orbitals. A total energy of ?0.7292 a.u. is obtained in the best calculation. Our potential curve is reasonably consistent with that calculated by Browne, but we have examined the region of small internuclear distances (those at and below R_e for the ground state) more extensively than any previous calculation. At R ≦ 1.6 a.u. our calculated potential curve is in excellent agreement with experiment.     

Effects of anharmonicity of molecular vibrations on the diffraction of electrons: Part III. Predictive models

It has long been known how to relate anharmonic vibrational distribution functions to scattered electron intensities when deriving molecular parameters from gas-phase electron diffraction patterns. What has been lacking is a convenient procedure for estimating the characteristic asymmetry parameters of radial distribution functions for polyatomic molecules, particularly in the case of non-bonded internuclear distances. In the present work alternative models of bond distribution functions are discussed briefly and a plausible model is proposed for geminal non-bonded distances. Numerical examples are worked out for the ground-states of CO₂, CS₂, H₂O, and D₂O. Variations of the asymmetry parameters of SO₂ and SF₆ with temperature are examined. It is shown that the effect of asymmetry can become quite large at elevated temperatures.     

Regularity and asymptotic properties of the discrete spectrum of electronic hamiltonians

We analyze the electronic part of diatomic molecular Hamiltonians as a function of the internuclear distance, x. We prove analyticity outside x ≠ 0 of eigenvalues in the discrete spectrum and differentiability up to second order for the associated projection operators. Furthermore, it is shown that for large internuclear distances they converge to eigenvalues and eignprojectors of the separated atoms.     

On energy bounds derived from the conjugate eigenvalue problem

An upper bound for E₀, which has been derived from the conjugate eigenvalue problem by Hall, is discussed. It is emphasized that the bound is only guaranteed when V is negative-definite. An alternative bound is presented which is free from this restriction, and the underlying iterative procedure is given. Hall's result is generalized to admit internuclear distances, and the theory is illustrated by a one-dimensional system with delta-function potentials. Some disadvantages of the approach are mentioned.     

Broken symmetry in valence molecular region within Hartree-Fock calculations

Symmetry restricted and unrestricted Hartree-Fock calculations at theab initio LCAO-MO-SCF level have been carried out on the ground, core and valence hole states of N₂ at various N-N distances. A one-particle criterion for symmetry breaking is discussed. Strong broken-symmetry effects in the inner valence molecular region of N₂ have been found at larger N-N distances. This breaking of symmetry accompanying the symmetry unrestricted Hartree-Fock calculations of the inner valence hole states at large internuclear separations can be considered to be a common phenomenon with all highly symmetric molecules. The outer valence broken-symmetry effects with N₂ have showed some deviations as compared with these effects in the inner valence and core molecular regions.     

Correlation of the Stretching Vibration Frequencies of EX n Diatomic Compounds with Each Other and with Structural and Thermodynamic Characteristics

A correlation was revealed between stretching vibration frequncies, internuclear distances, bondangles, thermodynamic parameters for EX _n compounds in which elements E belong to the same group ofthe periodic table.     

Theoretical electronic structure including spin–orbit effects of the alkali dimer cation

A theoretical study of the electronic structure of has been performed, including or not spin–orbit coupling. Potential energy curves for all the molecular states dissociating up to the limit Cs⁺ + Cs (8s ²S_1/2), i.e 26 states in the representation and 38 states in the representation Ω_g,u, are displayed. Equilibrium distances, transition energies and depths for the wells predicted at short and large range of internuclear separation R are reported. The existence of some of the long-range wells are confirmed by a long range model. Extensive tables of energy values versus internuclear distances are available at the following address: http://lasim.univ-lyon1.fr/allouche/cs2plus.htm.     

Calculation of mean-square vibrational amplitudes and shrinkage for linear triatomic molecules of AB2 type

Formulae for the calculation of shrinkage and mean-square amplitudes for linear molecules of AB₂ type have been derived without the use of small harmonic vibration theory. The probability density of the internuclear distance distribution being calculated with allowance for the rotation—vibration interaction. The formulae obtained agree, within experimental error, with electron diffraction data for molecules characterized by large displacements of the nuclei from their equilibrium positions. These data cannot be described by the relations usually employed in gas electron-diffraction investigations.     

The AMO method at small internuclear distances

The AMO function of the hydrogen molecule ψ = ψ_c + η ψ_i, where ψ_c is the covalent part and ψ_i the ionic part, is investigated for small internuclear distances R. We found η → ?1 as R →,?1 as R → 0, contrary to the intuitively expected limit η → 1. However, near R = 0 an analytical expression of ψ is derived, showing that ψ reduces to the helium ground state as R → 0. We have proved that the empirical concept ?covalent and ionic character”? should be replaced by the symmetry argument in the case of small R.     

Structure of SmCl3, DyCl3, and HoCl3 molecules based on the data of synchronous electron diffraction and mass-spectrometric experiment

Saturated vapors of SmCl₃, DyCl₃, and HoCl₃ have been studied in the framework of a synchronous electron diffraction and mass-spectrometric experiment at temperatures 1205 K, 1160 K, and 1148 K, respectively. In vapors of all compounds, along with monomer molecular forms, an insignificant (up to 2 mol.%) amount of dimers was detected. Parameters of the effective configuration of monomer molecules were determined. For molecules SmCl₃, DyCl₃, and HoCl₃ values of internuclear distances r _g(Ln-Cl) were 2.511(5) Å, 2.453(5) Å, and 2.444(5) Å, values of valence angles ∠_g(Cl-Ln-Cl) were 115.6(11)°, 116.8(10)°, and 116.6(10)°, respectively. It is shown that parameters of the r _g-structure are not incompatible with the notion of a planar equilibrium geometrical configuration of molecules SmCl₃, DyCl₃, and HoCl₃. Main tendencies in the change of structural and vibration characteristics in the series of lanthanide trichlorides are considered.     

Energetics and structure of the hydrated gaseous halide anions

Energetics and geometries for the hydrated gaseous halide anions have been computed from a simple model in which the molecular dipole of water was composed of two parts, one due to a lone pair on oxygen (60%) and the rest to formal charges on the nuclei. The calculations were made for both the symmetric and nonsymmetric structures. A variety of structures were used to compute potential energies and distances with up to six water molecules. The total energy consisted of a sum of electrostatic, polarization, dispersion, and repulsion terms. Various sets of repulsive potential parameters, ranging from those determined from molecular beam experiments to those determined using experimental ion–water distances or energies, have been employed to compute repulsive interaction energies. It was found that the range parameters play a significant role in deciding the magnitudes of the distances and energies, as the latter are most sensitive to them. It was also shown that with a simple correlation scheme the consistency of the experimental energies and distances can be tested separately without using repulsive potential parameters from other sources. It also suggests that a range of parameters can be used to compute repulsion energies. Despite the fact that the model is greatly simplified, the agreement of both the predicted ion-oxygen distances and energies with both experiment and other calculations is excellent. A detailed analysis of our calculation suggests that the negative ion clusters with one to three water molecules contain symmetric orientation of water molecules, while those with more than three may contain asymmetric orientations of water molecules or a mixture of both. From the log–log plots of hydration energies versus (R + radius of water molecule), we have proposed empirical expressions of the type ΔE_n?1,n = 10·0^x (R + 1.38)^?y with both Pauling's and Ladd's radii for univalent ions with which stepwise hydration energies of the latter can be predicted if we know thier radii. The values predicted for the alkali cations are in excellent agreement with the experimental and theoretical values, indicating the consistency of the simple model.