Force in SCF theories. MC SCF and open-shell RHF theories

A theorem reported for Hartree-Fock SCF theory is shown to be valid for general MC SCF and open-shell RHF theories - a sufficient condition for these wavefunctions to satisfy the Hellmann-Feynman theorem is that the basis set includes the derivative AO ^?X_r/^?X_rfor any basis X_r. The new force approach is applicable to wider fields including electronic processes in chemical reactions. Test calculations are given for some simple systems.     

Calculation of the force constants of molecules by means of the Hellmann-Feynman theorem

The force constants of the molecules LiH, BH, and H₂O have been calculated by means of the Hellmann-Feynman theorem on the basis of the Hartree-Fock-Roothan wave functions in the one-determinant approximation. The results obtained have been compared with the results previously calculated on the basis of the same functions by means of the virial theorem.     

Generic hypersurface singularities

The problem considered here can be viewed as the analogue in higher dimensions of the one variable polynomial interpolation of Lagrange and Newton. Let x₁,...,x _r be closed points in general position in projective spacePn, then the linear subspaceV ofH ⁰ (⨑ⁿ,O(d)) (the space of homogeneous polynomials of degreed on ⨑ⁿ) formed by those polynomials which are singular at eachx _i, is given by r(n + 1) linear equations in the coefficients, expressing the fact that the polynomial vanishes with its first derivatives at x₁,...,x _r. As such, the “expected” value for the dimension ofV is max(0,h ⁰(O(d))−r(n+1)). We prove thatV has the “expected” dimension for d≥5 (theorem A). This theorem was first proven in [A] using a very complicated induction with many initial cases. Here we give a greatly simplified proof using techniques developed by the authors while treating the corresponding problem in lower degrees.     

Effects of the A-site cation number on the properties of Ln5/8M3/8MnO3 manganites

The properties of manganites can be tuned by changing the doping level x in Ln_1−xM_xMnO₃. A second mechanism allows tuning of magnetic and electronic properties, for fixed x values, by varying the average A-cation radius, 〈r_A〉. Moreover, for fixed x and 〈r_A〉 values, the changes in the A-cation size variance, σ², also modify the ferromagnetic and metal-insulator transition temperatures. Here, we investigate the influence of the number of A-site cations on Ln_5/8M_3/8MnO₃ manganites, where x, 〈r_A〉 and σ² values are kept constant, and in the absence of phase separation phenomena. We have found that the number of cation species at the A site (N_A) has a strong influence on the width of the ferromagnetic and metal-insulator transitions, and a small influence on the average transition temperature. This behavior is opposite to that observed for increasing values of the variance σ² in manganites, with the same x and 〈r_A〉 values, where average transition temperatures are strongly reduced.     

The effect of inert additives on the explosive mechanochemical synthesis of some metal chalcogenides

The effect of different inert additives on the mechanochemical synthesis of SnS was studied. A transition from explosive to nonexplosive mechanochemical reaction was observed. It was shown that this change takes place only when a critical volume x_c is occupied by inert material. The critical volume for a transition from an explosive to nonexplosive mechanochemical synthesis were determined for the chalcogenides of Zn, Cd, and Sn. The x_c values obtained are discussed. On the basis of the new data concerning the critical dimensions r_c of the exploding mechanical aggregates and the percolation phenomena, a mechanism for the transition from an explosive to nonexplosive mechanochemical synthesis is suggested.     

Calculations on the helium isoelectronic sequence using Coulomb Green's function variables

Calculations on the ground states of the helium isoelectronic series are carried out using variational wavefunctions of the form ψ (x,y), in which x and y are the combinations r₁ + r₂ ± r₁₂ occurring in the Coulomb Green's function. The results for helium are the most accurate to date using a two-variable wavefunction accounting for 71.5% of the correlation energy.     

Marron-weare variational computation of energy differences for motions of C2H6, B2H6, CH3CH inf2 sup+ , and CH3BH2

The passage of D_3dC₂H₆ and B2H6 toward a D₂ h bridged structure, and the motion of a methyl proton maintaining C symmetry in C₂H _inf5 ^sup+ and CH₃BH₂ are described by integral Hellmann-Feynman computations in a Frost floating spherical Gaussian basis. Marron and Weare's variational corrections to the integral Hellmann-Feynman formula forAE between statesA andB are evaluated with variational functions of the form η(ψ_A/S_AB-ψ_B)) used to refine the stateB. An analogous function ξ(ψ_B/S_AB-ψ_A) refines state A. Both η and ξ are chosen variationally to minimize Marron and Weare's functional. No obvious advantage of the variational method became apparent in this simple application.     

The effect of a cation radii on structural, magnetic and electrical properties of doped manganites La0.6−xPrxSr0.4MnO3

Structural, magnetic and transport properties of La_0.6−xPr_xSr_0.4MnO₃ with x=0.0, 0.03, 0.06, 0.18, 0.3, 0.42, 0.54 and 0.6 are studied. The system exhibits a rhombohedrally distorted perovskite structure for x?0.3. A rhombohedral-orthorhombic (Pnma) structure transition is detected in the doping range from x=0.42 to 0.6. The structure refinement by Rietveld analysis of the X-ray powder diffraction data shows that the average distance Mn-O increases in the rhombohedral phases and decreases in the orthorhombic phases. Results show that the Curie temperature decreases from 374 to 310 K when 〈r_A〉 varies from 1.254 to 1.231 Å. Electrical measurements show that all samples exhibit a metallic to semiconducting transition with increasing temperature. Meanwhile, the size of the resistivity ρ increases near T_C. This phenomenon is interpreted as a gradual bending of the Mn-O-Mn bond angle, with decreasing 〈r_A〉, which causes the narrowing of the electronic bandwidth and the effect of the A-site variance σ².     

Role of aluminum ions on the dielectric and conducting properties of multiferroic Tb1−xAlxMnO3: Study at high temperatures

Dielectric and conducting properties of Tb_1−xAl_xMnO₃ (x = 0, 0.05) synthesized by the solid–state reaction method have been investigated. The Al ion has the same valence as substituted Tb but is nonmagnetic and its small size gives rise to microstructural strain which affects the multiferroic properties of the parent compound. Samples were characterized by means of complex impedance spectroscopy (CIS) in the frequency range from 40 Hz to 5 MHz, at temperatures above room temperature. The conductivity curves for the two samples are well fitted by the Jonscher law σ(ω) = σ_dc + Aωⁿ. Results of the fitting procedure indicate that for the studied samples, the hopping motion involves localized hopping without the species leaving the neighbors. Frequency dependence of the dielectric constant (ε″) and tangent loss (tan δ) display a dispersive behavior at low frequencies that can be explained on the basis of the Maxwell–Wagner model and Koop's theory. The relaxation dynamics of charge carriers has been studied by means of the electric modulus formalism. In turn, the variation of the imaginary part of the complex impedance, Z″, shows a peak at a relaxation angular frequency (ω_r) related to the relaxation time (τ) by τ = 1/ω_r. The complex impedance spectra (Nyquist plots) show well-defined semicircles which are strongly dependent on the Al-doping level and temperature. The complex impedance data have been modeled using electrical equivalent circuits.     

A new insight on the role of Mg in calcite

Magnesium is a key determinant in calcite (CaCO₃) mineralization having a notable significance upon the carbonate growth. Magnesium is also present in CaCO₃ mineral scale formation which is a persistent and expensive problem in oil and gas production. In the present work, we attempt to treat CaCO₃ calcite mineral bulk inducing different concentrations of Mg impurity. The study is done using a quantum‐chemical computer code based on the Hartree–Fock method. It is found that small Mg impurity concentrations produce a significant distortion of the defect‐closest O atoms thus developing local microstructures being distinct to those of the original calcite structure. This structural alteration is accompanied by the electron density redistribution from the O 2p_z atomic orbitals (AOs) toward the O 2p_x and 2p_y AOs for impurity nearest oxygens. On the other hand, the high Mg impurity concentrations make the chemical bonding in calcite more ionic. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

A different approach to X-ray stress analysis

A different approach to X-ray stress analysis has been developed. At the outset, it must be noted that the material to be analyzed is assumed homogeneous and isotropic. If a sphere with radius r within a specimen is subjected to a state of stress, the sphere is deformed into an ellipsoid. The semi-axes of the ellipsoid have the values of (r + ε_x), (r + ε_y), and (r + ε_z), which are replaced by d_x, d_y, and d_z, or for the cubic case, a_x, a_y, and a_z. In this technique, at a particular ϕ angle (see Fig. 1), the two-theta position of a high angle (hkl) peak is determined at ψ angles of 0, 15, 30, and 45°. These measurements are repeated for 3 to 6 ϕ angles in steps of 30°. The d_ϕψ or a_ϕψ values are then determined from the peak positions. The data is then fitted to the general quadratic equation for an ellipsoid by the method of least squares. From the coefficients of the quadratic equation, the angle between the laboratory and the specimen coordinates (direction of the principle stress) can be determined. Applying the general rotation of axes equations to the quadratic, the equation of the ellipse in the x–y plane is determined. The a_x, a_y, and a_z values for the principal axes of the lattice parameter ellipsoid are then evaluated. It is then possible to determine the unstressed a₀ value from Hooke's Law using a_x, a_y, and a_z. The magnitude of the principal strains/stresses is then determined.     

Ferroelectric properties and ac conductivity studies of yttrium modified Pb(Fe0.5Nb0.5)O3 ceramics

Using a high-temperature standard solid-state reaction (i.e., mixed oxide) method several yttrium modified ferroelectric ceramics of type Pb_1−xY_x(Fe_0.5Nb_0.5)_1−x/4O₃ (PYFN)with different concentration (x = 0.00, 0.02, 0.04, 0.06, 0.08) were prepared. Preliminary X-ray structural analysis reveals that all the samples were crystallized in monoclinic crystal system. The crystal structure is almost invariant with small Y-concentration. Microstructure of the samples exhibits grain growth on increasing Y content. Curie temperature of Y-substituted samples decreases slightly as compared to that of PFN (Pb(Fe_1/2Nb_1/2)O₃). However, Y substitution results in a notable enlargement of remnant polarization (2P_r). The 2P_r of PYFN (x = 0.02) reaches to a large value (23 μC/cm²) which is nearly 5 times greater than that of x = 0.00 (4.6 μC/cm²). A phenomenological model (i.e., ferroelectric capacitor model) used to simulate hysteresis loops was found to be suitable to explain experimental results. Jonscher's single power law (σ_ac(ω) = σ(0) + Aωⁿ) was well fitted to the frequency dependence of conductivity for all the samples.     

Adhesive and electrical properties of the interface between Pb1 − x Mn x Te Crystals and In-Ag-Au alloy

The effect of annealing of Pb_{1 ? x} Mn _x Te crystals at ～690 K and structures on their basis at ～383 K on the adhesive and electric properties of the interface in the Pb_{1 ? x} Mn _x Te-(In-Ag-Au) structure was studied over the temperature range ～77–300 K. The contacts possessed high adhesive strength. The effect of annealing on contact resistance r _c was determined by a change in the specific resistance of crystals, diffusion of Ag atoms into the near-contact area of crystals, and the formation of intermediate phases of the Ag₂Te type at the interface.     

Ternary phases MPtSi (M = Ca,Eu, Sr,Ba) with the LalrSi-type structure

Four ternary phases MPtSi (M = Ca, Eu, Sr, Ba) have been shown to crystallize in the LaIrSi-type structure (space group P2₁3). This ternary structure is a derivative structure of the binary SrSi₂-type structure (space group P4₃32 or P4₁32). In the MPtSi series the LaIrSi-type structure has a stability range for metals with radii from r_Ca = 1.973 Å to r_Ba = 2.243 Å in contrast to MSi₂ compounds which exist with the SrSi₂-type structure only from r_Sr = 2.151Å to r_Ba 2.243 Å. From a single-crystal investigation on CaPtSi remarkably short PtSi distances of 2.30 Å (3x) are obtained. Structural relations are discussed.     

Bond dissociation energies and bond orders for diatomic alkali halides

The bond dissociation energies for Alkali halides have been estimated based on the derived relations: $$\begin{gathered} D_{AB} = \bar D_{AB} + 31.973{\text{ e}}^{0.363\Delta x} {\text{ and}} \hfill \\ D_{AB} = \bar D_{AB} (1 - 0.2075\Delta xr_e ) + 52.29\Delta x, \hfill \\ \end{gathered} $$ where \(\bar D_{AB} = (D_{AA} \cdot D_{BB} )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} \) , Δx represents Pauling electronegativity differences x(_A ?x_B) and r _e is the internuclear distance. A simplified formula relating bond orders, q, to spectroscopic constants is suggested. The formula has the form q = 1.5783 × 10^?3 (ω _e ² r_e/ B_e)^1/2. The ambiguity arising from the Parr and Borkman relation is discussed. The present study supports the view of Politzer that q/(0.5r _e)² is the correct definition of bond order. The estimated bond energies and bond orders are in reasonably good agreement with the literature values. The bond energies estimated with the relations we suggested, for alkali halides give an error of 4.5% and 5.3%, respectively. The corresponding error associated with Pauling's equation is 40.2%.     

Effect of cation size and size-disorder on the magnetic and electron transport properties of La0.7−xLnxSr0.3MnO3 (Ln=Gd and Dy)

Magnetic and electron transport properties of La_0.7−xLn_xSr_0.3MnO₃ with Ln=Gd and Dy have been investigated over a wide range of compositions to explore the effects of the radius of the A-site cation, 〈r_A〉, and the size-disorder as measured by σ². These manganates exhibit ferromagnetism and insulator–metal transitions up to a critical value of x (x_c), with the T_c (T_IM) values decreasing with 〈r_A〉. When x⩾x_c, however, the manganates cease to be ferromagnetic and instead they show an abrupt decrease in magnetization and increase in resistivity, suggesting the likely occurrence of electronic phase separation.     

Energy-density relations in momentum space

The integrated Hellmann-Feynman theorem is used to derive a rigorous relation between the energy and the electron density in momentum space. Choosing the electron mass as a differential parameter, we obtain a formula corresponding to the Wilson-Frost formula in coordinate space. Analysing the mass-dependence of momentum density, we then show that the present formula is equivalent to one of the previous results deduced from the virial theorem. Use of the integral Hellmann-Feynman theorem is also discussed. Several illustrative examples are given for the calculation of energy from momentum density.     

Rechnergestützte DTA- und DMC-messungen: Apparatur und methodische aspekte

The evaluation of data from DTA and calorimetric measurements is still, in most cases, made by hand. This is not only a highly time-consuming procedure, but also includes subjective errors. This is especially true for the determination of starting points and end points of DTA peaks. Via integration between these limits, errors are introduced into the calculation of peak areas, i.e. thermal energies. Since many research groups possess high-performance microcomputers or desktop computers, which are able to perform measurements with subsequent data evaluation by suitable software, we developed a computer-controlled DTA/DMC equipment.In the present paper, we first describe the hardware coupling of a desktop computer to a DTA/DMC apparatus. We proceed with a discussion of the capabilities of both the apparatus and the data evaluation methods. A calorimetric measurement of the melting point of tin is the basis for a rating of the proposed numeric procedure for finding the peak limits and for peak integration in contrast to the evaluation of data by hand. With the argyrodite—cuprodite system. Ag_8?xCu_xGeS₆, O?x?4, as an example, we present a proposed method for obtaining phase diagrams.     

GPS patch antenna performance by modification of Zn(1−x)CaxAl2O4-based microwave dielectric ceramics

This study reports the characterization, fabrication, and performance of global positioning systems (GPS) patch antennas as a function of calcium (Ca) concentration and dielectric constant (? _r ). Zn_(1?x)Ca_xAl₂O₄ (x = 0.00, 0.05, 0.10, 0.20, 0.25, and 0.30) thin films were prepared through a sol–gel method. The effects of added Ca on the nanostructures and dielectric properties of ZnAl₂O₄ ceramics were investigated. The addition of Ca increased the crystallite size, grain size, and surface morphology, thereby increasing the density and dielectric constant. As the Ca content increased, the ? _r values linearly increased. However, the Q _u values decreased (at x = 0.25 to x = 0.25) after achieving the optimum values at x = 0.20. Finally, GPS patch antennas were successfully fabricated using the Zn_(1?x)Ca_xAl₂O₄ material. The patch antenna sizes decreased as ? _r increased from 2.88  × 4.37 cm (? _r  ≈ 8.52) to 2.88  × 4.37 cm (? _r  ≈ 10.16). The performance (return loss analysis) and operating frequencies of the GPS patch antennas were measured using the PNA series network analyzer. Results show that the patch antenna resonates at frequency of 1.570 GHz and produces a return loss bandwidth between ?16.6 and ?27.5 dB. The optimal performance of GPS patch antenna with ? _r  ≈ 9.95, Q _u  ≈ 6,186, and return loss = ?27.5 dB was obtained from specimen using Zn_0.80Ca_0.20Al₂O₄ (x = 0.20) ceramics.