A finite-difference solution of the Hartree–Fock equations for diatomic molecules

A more general application of the self-consistent field iteration is coupled with a finite-difference Newton–Raphson algorithm to solve the set of coupled second-order integro-differential equations with split boundary conditions which constitutes the Hartree–Fock problem for diatomic molecules. The N orbitals are assumed to be of the form ψ_α = L_α(λ) M_α (μ)e^imα^? (2π)^?1/2, (α = 1, …?, N), where λ, μ, and ? are the usual confocal elliptical coordinates. Requiring the expectation value of the electronic Hamiltonian to be stationary with respect to independent variations of the functions L_α and M_α, subject to constraints of orthonormality, leads to a set of coupled one-dimensional differential equations for the functions L_α and M_α. In the new method a corresponding set of finite-difference equations including the split boundary conditions for each function, as well as the Lagrange multipliers and associated constraints on normalization and orthogonality, are incorporated into a large system of nonlinear algebraic equations which is solved by means of a coupled self-consistent field-generalized Newton–Raphson iteration. As examples, calculations of the (1)^{2 1}Σ and (1) (2pσ_u) ³Σ states of H₂ are presented. The calculated energy for the ¹Σ state of H₂ is 99.985% of the three-dimensional Hartree–Fock limit. The discrepancy is due to the assumed factored form of the orbitals ψ_α, and a generalization of the finite-difference method is suggested to improve the results.     

Disproportionation-to-combination ratios for cyclohexyl-h11 and -d11 radicals in the gas phase as determined by the radiolysis of water vapor–cyclohexane mixtures

In the radiolysis of water vapor containing small concentrations of cyclohexane, the principal products which account for about 98% of all end products are found to be hydrogen, cyclohexene, and bicyclohexyl. Cyclohexene and bicyclohexyl yields were determined over a range of temperatures (70–200°C), total pressures (50–2400 torr), and total doses (0.15–2.0 Mrad). The disproportionation–combination ratio k/k for c-C₆H₁₁ radicals could be determined as 0.56 ± 0.01 from the ratio of cyclohexene to bicyclohexyl yield. By using c-C₆D₁₂, the ratio k/k for c-C₆D₁₁ radicals is found to be 0.38 ± 0.01. Comparison of the reactivity pattern of C₆H₁₁ and C₆D₁₁ radicals leads to (k)/(k)/(k/k) = 1.47 ± 0.02. The corresponding values for the reactions of c-C₆H₁₁ with c-C₆D₁₁ were also determined.     

Computational analysis of spinodal decomposition dynamics in polymer solutions

In this work a model, composed of the nonlinear Cahn-Hilliard and Flory-Huggins theories, is used to numerically simulate the phase separation and pattern formation phenomena of oligomer and polymer solutions when quenched into the unstable region of their binary phase diagrams. This model takes into account the initial thermal concentration fluctuations. In addition, zero mass flux and natural nonperiodic boundary conditions are enforced to better reflect experimental conditions. The model output is used to characterize the evolution and morphology of the phase separation process. The sensitivity of the time and length scales to processing conditions (initial condition c) and properties (dimensionless diffusion coefficient D) is elucidated. The results replicate frequently reported experimental observations on the morphology of spinodal decomposition (SD) in binary solutions: (1) critical quenches yield interconnected structures, and (2) off-critical quenches yield a droplet-type morphology. As D increases, the dominant dimensionless wave number k increases as well, but the dimensionless transition time t from the early stage to the intermediate stage decreases. In addition, t is shortest when c is at the critical concentration, but increases to infinity when c is at one of the two spinodal concentrations. These results are found when the solute degree of polymerization N₂ is in the range 1 ≤ N₂ ≤ 100. When N₂ > 100, however, a problem of numerical nonconvergence due to diverging relaxation rates occurs because of the very unsymmetric nature of the phase diagram. A novel scaling procedure is introduced to explain the phase separation phenomena due to SD for any value of N₂ during the time range explored in this study.     

Charge distributions and chemical effects. XXXIV. Concepts involved in an approximate,but accurate,description of bond energies

The concepts underlying the definition of bond energies in terms of potentials at the nuclei are outlined. The theory is rooted, first, in a definition of the energy, E_i, of “atom” i in the molecule in terms of the potential energy, V(i, mol), of nucleus Z_i in the field of all the electrons and nuclei of the molecule: E_i = K V(i, mol). The K parameter, which is not required to be a constant in the derivation of the energy expression describing the contribution of an ij bond, turns out to be virtually constant for each atomic species—a situation which is exploited in numerical applications. Second, the Hellmann—Feynman theorem is applied in the calculation of the derivative, δΔE/δZ_i, of the atomization energy, ΔE, using (i) the exact quantum-chemical definition of ΔE and (ii) the view that ΔE is the sum of bond energy contributions, ε_ij, plus a small interaction between nonbonded atoms. The individual bond energies derived in this manner necessarily depend on local charges at the bond-forming atoms. Numerical applications illustrate how this new bond-energy formula provides a simple link between typical saturated, olefinic, acetylenic, and aromatic hydrocarbons.     

CH bond dissociation energies,isolated stretching frequencies,and radical stabilization energy

An earlier correlation between isolated CH stretching frequencies, v, and experimental CH bond dissociation energies, in hydrocarbons, fluorocarbons, and CHO compounds, is updated. A stabilization energy, E, which reflects only the properties of the radical, is defined by the deviation of a point from the above correlation. E values for a variety of radicals are listed and discussed. In H? C? N and H? C? O compounds E is low or negligible, due to the low v found in these compounds. The conventional definition of E_S then represents a serious misnomer, which distracts attention from the probable source of discrepancies between experimental and ab initio values of DH°(C? H), namely, the parent molecules. Stereo electronic effects concerned with the breaking of CH bonds are predicted in a variety of situations. Some experimental determinations of DH°(C? H), viz., in C₂H₄, HCOOH, CH₃CHO, CH₃NH₂, are considered to be probably in error. Schemes for partitioning energies of atomization into ‘standard’ or ‘intrinsic’ bond energies are criticized.     

Control of Oxidation States of Transition Elements (Cr,Mn) in Nitridometalates and the Solid Solution Series Li6(Ca1‐xSrx)2[Mn2N6]

The formation of ternary nitridometalates from the elements in the case of the systems Li—Cr, V, Mn—N leads to compounds which contain the transition metals in the highest (V^V, Cr^VI) or a comparably high (Mn^V) oxidation state. In the corresponding calcium and strontium systems, the transition metals show a lower oxidation state (V^III, Cr^III, Mn^III). Transition metals with intermediate oxidation states (Cr^V, Mn^IV) are present in the quaternary (mixed cation) compounds Li₄Sr₂[CrN₆], Li₆Ca₂[MnN₆], and Li₆Sr₂[MnN₆] (R3¯(#148), a = 585.9(3) pm, c = 1908.6(4) pm, Z = 3), as well as in the solid solution series Li₆(Ca_1—xSr_x)₂[MnN₆].     

Hartree Fock instabilities of sulfur-nitrogen ring systems: S2N2

The Hartree-Fock instablities of S₂N₂ are reported and compared with those of S₃N and S₄N. These unsaturated sulfur nitrogen planar rings are π electron rich and although the symmetry adapted HF solutions are singlet stable at the experimental bond lengths they become unstable with only a very modest increase in bond length. The broken symmetry solutions for S₂N₃, S₃N, and S₄N are of planar C_2v type with one of the nitrogens stripped of its π electrons, producing a π hole.     

SCF and MC–SCF Study of CO2 with bond angle variation

Multiconfiguration (MC ) SCF calculations are reported for CO₂ for bond angles between 60° and 180°. The ground state configuration is found to be …?5a4b∣b∣a for small bending angles and …?6a3b∣b∣a for large bending angles, the change in ground state character occurring at a bond angle of about 100°. The force constant for bending obtained from the MC –SCF function is about 8.0% lower than the corresponding SCF value, and in considerably better agreement with experiment.     

Bis[N-(trimethylsilyl)iminobenzoyl]phosphanide des Lithiums und Zinks – Synthese sowie NMR-spektroskopische,strukturelle und quantenchemische Untersuchungen

Acyl- and Alkylidenephosphanes. XXXV. Bis[ N -(trimethylsilyl)iminobenzoyl]phosphanides of Lithium and Zinc – Syntheses as well as NMR Spectroscopic, Structural, and Quantumchemical Studies From the reaction of bis(tetrahydrofuran)lithium bis(trimethylsilyl)phosphanide with two equivalents of benzonitrile in 1,2-dimethoxyethane, the yellow dme complex ( 2 a ) of lithium bis[N-(trimethylsilyl)iminobenzoyl]phosphanide ( 2 ) was obtained in 69% yield. However, the intermediate {1-[N-lithium-N-(trimethylsilyl)amido]benzylidene}trimethylsilylphosphane ( 1 ), formed by an analogous 1 : 1 addition in diethyl ether, turned out to be unstable and as a consequence could be characterized by nmr spectroscopic methods only; attempts to isolate the compound failed, but small amounts of the neutral complex 2 b , with the ligands benzonitrile and tetrahydrofuran coordinated to lithium, precipitated. The reaction of compound 2 with zinc(II) chloride in diethyl ether gives the orange-red spiro-complex zinc bis{bis[N-(trimethylsilyl)iminobenzoyl]phosphanide} ( 3 ); this complex is also formed from bis[N-(trimethylsilyl)iminobenzoyl]phosphane ( 4 ), easily amenable by a lithium hydrogen exchange of 2 a with trifluoroacetic acid [18], and zinc bis[bis(trimethylsilyl)amide]. As derived from nmr spectroscopic studies and x-ray structure determinations, compounds 2 a {δ³¹P +63.3 ppm; P2₁/n; Z = 4; R₁ = 0.067}, 2 b {δ³¹P +63.3 ppm; P2₁/c; Z = 4; R₁ = 0.063}, 3 {δ³¹P +58.2 ppm; C2/c; Z = 4; R₁ = 0.037} and 4 {δ³¹P +58.1 ppm [18]} exist as cyclic 3-imino-2λ³σ²-phosphapropenylamides and -propenylamine, respectively, in solution as well as in the solid state. Unlike hydrogen derivative 4 the bis[N-(trimethylsilyl)iminobenzoyl]phosphanide fragments N,N′-coordinating either a lithium or a zinc cation are characterized by almost completely equalized bond lengths; typical mean distances and angles are: PC 180.3 and 178.7; CN 130.5 and 131.8; N–Si 175.3 and 179.3; N–Li 202.3; N–Zn 203.5 pm; CPC 108.8° and 110.5°; PCN 130.9° and 132.9°; CN–Li 113.0°, CN–Zn 117.4°; N–Li–N 104.6°; N–Zn–N 108.8°. Alterations in the shape of the six membered chelate rings, caused by an exchange of the 3-imino-2λ³σ²-phosphapropenylamide or related 2λ³σ²-phospha-1,3-dionate units for the corresponding phosphorus free ligands, are discussed in detail. The results of quantumchemical DFT-B3LYP calculations coincide very well with the experimentally obtained findings.     

Structure and Bonding in Cyclic Sulfur-Nitrogen Compounds—Molecular Orbital Considerations

The molecular structures of monocyclic sulfur-nitrogen ring systems, such as S₂N₂, S₃N, S₄N and S₅N, can be considered as examples of electron rich (4n + 2)π systems. The structures of S₄N₄, S₄N, P₄S₄, As₄S₄ and the bicyclic structures S₄N, S₄N as well as S₅N₆ can be rationalized on the basis of a planar tetrasulfur tetranitride with 12π electrons.     

Length dependence of the optical transition energies of the exactly solvable Hubbard model for linear polyenes

Some accurate results on the length dependence of the excitation energies from the ground state to ionic excited states in the Hubbard model of linear polyenes are obtained based on the method of Lieb and Wu. To this end, it is first shown that singly ionic excited states with “plus” alternancy symmetry in the Hubbard model are described by the wave functions in which the two electron operator [∑(?)ⁿCC] is acted on (N ? 2)-electron covalent eigenstates. Then by solving the Lieb-Wu equations the exact excitation energies of the lowest ionic state, which corresponds to the E state in this model, are calculated for systems with up to 50 electrons. The result, together with a correction for the end effect, indicates that the excitation energies do not decrease as 1/N but converge to the limiting value more rapidly when the number of electrons N becomes large.     

Lithiumtriamidostannat(II), Li[Sn(NH2)3] – Synthese und Kristallstruktur

Lithium Triamidostannate(II), Li[Sn(NH₂)₃] – Synthesis and Crystal Structure Rusty-red glistening, transparent crystals of Li[Sn(NH₂)₃] were obtained by reaction of metallic lithium with tetraphenyl tin in liquid ammonia at 110 °C. The structure was determined from X-ray single-crystal diffractometer data: Space group P 2₁/n, Z = 4, a = 8.0419(9) Å, b = 7.1718(8) Å, c = 8.5085(7) Å, β = 90.763(8)°, R₁ (F _o ≥ 4σ(F _o)) = 2.8%, wR₂ (F ≥ 2σ(F )) = 5.3%, N(F ≥ 2σ(F )) = 1932, N(Var.) = 65. The crystal structure contains trigonal pyramidal complex anions [Sn(NH₂)₃]^– with tin at the apex, which are connected to layers of sequence A B A B … by lithium in tetrahedra-double units [Li(NH₂)_2/2(NH₂)₂]₂.     

Dynamic x-ray diffraction studies of high-density polyethylene

Dynamic x-ray diffraction is conducted to explore the structural origin of the α and β mechanical dispersions of a melt-crystallized high-density polyethylene. It is shown that the real component of the strain orientation coefficient for the crystal c axis C decreases with increasing frequency at a rate which decreases with decreasing temperature. Values of C for the c axis are positive, C for the a axis negative, and C for the b axis close to zero, suggesting that the predominant relaxation process is crystal rotation about the b axis. The activation energy found from Arrhenius plots of C corresponds to that of the α₁ mechanical dispersion. The dynamic birefringence in this region is dominated by the contribution from crystal orientation changes. At low temperatures, the imaginary component KC of the strain-optical coefficient of the crystal phase approaches zero, while KC of the amorphous phase exhibits a somewhat broad dispersion peak corresponding to the β birefringence dispersion. This suggests that the principal contribution to the β birefringence dispersion arises from the amorphous phase, probably owing to the amorphous orientation process. Contrary to the case of low-density polyethylene, the dynamic crystal lattice deformation and compliance functions reveal distinct frequency dispersions corresponding to the α₁ and α₂ mechanical processes. The α₁ lattice dispersion is thought to be associated with the α₁ crystal orientation dispersion, while the α₂ lattice dispersion is believed to be the inherent one arising from the onset of intracrystalline chain motions.     

Goldreiche Auride mit Caesium: Cs1,34Rb0,66RbAu7 und Cs1,60Rb0,40RbAu7

Gold-rich Aurides with Caesium: Cs_1.34Rb_0.66RbAu₇ and Cs_1.60Rb_0.40RbAu₇ Cs_1,60Rb_0,40RbAu₇, Raumgruppe Cmmm, Z = 2, a = 5,677(1) Å, b = 13,273(3) Å, c = 7,288(1) Å, R1/wR2 = 0,0392/0,0892, Z(F) ≥ 2σ(F) = 700 and Z(Var.) = 23. Silver coloured, brittle single crystals of Cs_1.34Rb_0.66RbAu₇ and Cs_1.60Rb_0.40RbAu₇ were obtained by the reaction of CsN₃, RbN₃ and gold sponge at 903 K. The structures were determined from X-ray single-crystal diffractometry data: Cs_1.34Rb_0.66RbAu₇, space group Cmmm, Z = 2, a = 5.657(1) Å, b = 13.265(4) Å, c = 7.281(2) Å, R1/wR2 = 0.0373/0,0628, N(F) ≥ 2σ(F) = 818 and N(var.) = 23.     

Determination of molecular weight and its distribution of rigid‐rod polymers determined by phase‐modulated flow birefringence technique

The phase‐modulated flow birefringence (PMFB) method is widely accepted as one of the most sensitive and accurate techniques suitable for experimental tests on the molecular theory of polymer solutions. The objective of this study is to develop a systematic method to determine molecular weight and distribution of rigid‐rod polymers by the PMFB technique. Using molecular theory for rigid polymers, birefringence Δn and orientation angle χ have been expressed as a function of molecular weight and distribution. Δn has been shown to be proportional to Σc_iM, and cot 2χ turned out to have a linear relationship with Σc_iM/Σc_iM. From the experimental results for PBLG solutions, birefringence and orientation angle data were in some degree matched with the theory presented. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 509–515, 2000     

Dielectric anisotropy in oriented poly(chlorotrifluoroethylene)

Dielectric measurements between ?50 and 60°C have been made on isotropic and oriented samples of poly(chlorotrifluoroethylene) with draw ratios λ of 1 to 3.5 at frequencies ranging from 30 Hz to 1 MHz. For the oriented samples, the dielectric loss has been measured with the electric field normal (ε) and parallel (ε) to the draw direction. At low frequency (say 60 Hz) the loss data for the oriented samples reveal two peaks at 25 and ?5°C, which are associated with the amorphous (γ_a) and the crystalline (γ_c) relaxations, respectively. Analysis of these data using a two-phase model yields values for the amorphous orientation function f_a which are only about 25 to 60% of those for the crystalline orientation function f_c. Upon annealing, the anisotropy ε/ε at the γ_a peak decreases significantly while that at the γ_c peak remains largely unchanged. This implies a roughly unaltered f_c and a large decrease in f_a, which is consistent with the results of wide-angle x-ray diffraction and birefringence measurements.     

Exact results on homopolar singlet excitation energies of the hubbard model for linear polyenes

The Lieb–Wu equations for homopolar singlet excited states of the Hubbard model of linear polyenes are formulated in a form tractable by numerical calculation. They are solved to obtain exact excitation energies of the lowest two excited states, corresponding to the ¹B and ¹E states of the benzene molecule, for systems with up to 50 and 30 electrons, respectively. These results may be particularly useful for checking accuracy of approximate methods for the homopolar states for which the conventional CI method is very difficult to apply. Calculation is also carried out for the ¹B⁺ ionic excited state involving a spin excitation, which corresponds to the ¹B state of the benzene molecule. The length dependence of the excitation energies is examined and shown to have no N^?1 dependence for large N, where N is the number of electrons. This supplements the result on the ¹E ionic state obtained in our previous paper.     

Simple perturbation and perturbation-variation treatments of the 1sσg and 2pσu states of H

First-order wave functions and binding energies of the 1sσ_g and 2pσ_u states of H are calculated by simple methods for internuclear separations between 0.2 and 10_a0. An exact perturbation treatment of the lsσ_g state with a zeroth-order function of the form N exp (-sRλ/2) exp (sR μ2) yields only fair results. An alternative method starts with zeroth-order functions of the form N exp (-sRλ/2)[exp (-sRμ/2) ± exp (sRμ/2)] for the 1 and 2 states, respectively. An approximate first-order trial function is set up and the energies are determined variationally. For both states the results are comparable to those obtained by exact perturbation treatments of the same order.     

Darstellung,Kristallstruktur und Schwingungsspektren von (n-Bu4N)2[(Mo6I)(NCS)]

Synthesis, Crystal Structure, and Vibrational Spectra of (n-Bu₄N)₂[(Mo₆I)(NCS)] By treatment of [(Mo₆I)I]^2– with (SCN)₂ in dichloromethane at –20 °C the hexaisothiocyanato cluster anion [(Mo₆I)(NCS)]^2– is formed. The X-ray structure determination of (n-Bu₄N)₂[(Mo₆I)(NCS)] · 2 Me₂CO (monoclinic, space group P2₁/c, a = 13.168(5), b = 11.964(5), c = 24.636(5) Å, β = 104.960(5)°, Z = 2) shows, that the thiocyanate groups are coordinated exclusively via N atoms with Mo–N bond lengths of 2.141–2.150 Å, Mo–N–C angles of 166–178° and N–C–S-angles of 174–180°. The vibrational spectra exhibit characteristic innerligand vibrations at 2073–2054 (ν_CN), 846–844 (ν_CS) and 480–462 cm^–1 (δ_NCS).     

Nd4N2Se3 und Tb4N2Se3: Zwei nicht‐isotype Lanthanoid(III)‐Nitridselenide

Nd₄N₂Se₃ and Tb₄N₂Se₃: Two non‐isotypical Lanthanide(III) Nitride Selenides The non‐isotypical nitride selenides M₄N₂Se₃ of neodymium (Nd₄N₂Se₃) and terbium (Tb₄N₂Se₃) are formed by the reaction of the respective rare‐earth metal with sodium azide (NaN₃), selenium and the corresponding rare‐earth tribromide (MBr₃) at 900 °C in evacuated silica ampoules after seven days. Each of them crystallizes monoclinically in the space group C2/c with Z = 4 for Nd₄N₂Se₃ (a = 1300.47(4), b = 1009.90(3), c = 643.33(2) pm, β = 90.039(2)°) and in the space group C2/m with Z = 2 for Tb₄N₂Se₃ (a = 1333.56(5), b = 394.30(2), c = 1034.37(4) pm, β = 130.377(2)°), respectively. The crystal structures differ fundamentally in the linkage of the structure dominating N^3‐ centred (M³⁺)₄ tetrahedra. In Nd₄N₂Se₃, the [NNd₄] units are edge‐linked to bitetrahedra which are cross‐connected to [N(Nd1)(Nd2)]³⁺ layers via their remaining four corners, whereas the [NTb₄] tetrahedra in Tb₄N₂Se₃ share cis‐oriented edges to form strands [N(Tb1)(Tb2)]³⁺. Both structures contain two crystallographically different M³⁺ cations, that show coordination numbers of six and seven (Nd₄N₂Se₃) or twice six (Tb₄N₂Se₃), respectively, relative to the anions (N^3‐ und Se^2‐). Each of the two independent kinds of Se^2‐ anions provide the three‐dimensional linkage as well as the charge balance. The particular axial ratio a/c and the monoclinic reflex angle offer two choices for fixing the unit cell of Tb₄N₂Se₃.