On the adiabaticity of continum states: tunnelling through a time-dependent barrier

The problem of tunnelling through a time-dependent delta-function barrier,V(x, t) =λτ/tδ(x), has been solved analytically fort≧0. The Massey parameter α=γτk/? controls the tunnel process: For α?1, a continuum electron with momentum ?k will adjust its wavefunction to the variable barrier; for α?1 the change of the barrier is too sudden to allow for adjustment. The characteristic time τ is used to clarify the concept of a tunnelling time.     

Unified treatment of electric multipole transition radial matrix elements using radial parts of STO and complete orthonormal sets of ψα-ETO

A new method is proposed for evaluating electric multipole transition radial (EMTR) matrix elements of the hydrogen-like atoms using the expansion formulae suggested by one of the authors for the radial parts of ψ^α-exponential type orbitals (ψ^α-ETO) in terms of radial parts of Slater type orbitals (STO), where α is the frictional quantum number (?∞<α≤2). A comparative study is carried out between the proposed analytical computations and other numerical simulations. The effectiveness and accuracy of the obtained analytical expressions are demonstrated by calculation of concrete cases.     

New expansion of the Boys function

We propose a new expansion for the Boys function ∫₀¹t^2jexp(−r²t²) dt appearing in the calculation of molecular two-electron matrix elements if Gaussian basis sets are employed. This expansion involves a power series involving the terms C_{i, j}(τ) (r²−R²)ⁱ multiplied by exp(−τr²), where τ is an optimized parameter τ∈[0, 1]. The performances of the introduced expansion are discussed and illustrated by some numerical experiments. It appears that the proposed expansion is considerably shorter than the customary Taylor series, which in turn is the special case for τ=0. This is of some importance, particularly for higher j values. Further, the proposed expansion enables a single expression for calculating erf(x) for the whole range of variable x. The recursive relations for the expansion coefficients are derived and the truncation errors are estimated. A new method for calculating the Boys function by means of asymptotic series is represented too. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 305–315, 1998     

Classical and quantum evaluation in the low temperature limit of the Keesom integral for the interaction between permanent dipoles

In a previous paper on the Keesom effect, we presented a procedure for the evaluation of the average potential energy of interaction between two dipoles of strength μ a distance R apart in the limit of small temperature T. The resulting expansion in terms of the dimensionless T-dependent parameter a = −μ²/R³kT was then proved by us to be an asymptotic one. Here we reformulate our method in a simplified form, extending it to include the quantum effects which must be taken into account in the low-temperature regime. The resulting formulae show explicitly the transition from the quantum regime to the classical one as a function of temperature.     

The graph-like state of matter—VI: Combinatorial approach to the excluded-volume perturbation

The close connection between graphs and matrices, known since the work of Kirchhoff in the 1840's, is reflected in the literature on the perturbation expansion of the excluded-volume theory for polymer chains. The chain molecule is in a graph-like state, and the coefficients of the expansion are related to Petrie matrices. From a purely combinatorial approach to this situation, a number of new insights are derived. The Fixman partition function for an infinite chain is shown to be not only divergent but non-asymptotic. The existence of the asymptotic expansion of the chain-expansion parameter α² in terms of powers of the usual parameter z being assumed, it is extremely likely that these coefficients oscillate and increase rapidly with m. By means of a pre-averaging procedure (primed coefficients c′_m, the first 18 coefficients can be estimated with a desk calculator. The first coefficient c′₁ is thus found to be 1.299 (instead of 1.333) and the higher coefficients then fall away regularly from their ‘correct’ values. As far as these are known (i.e. up to m = 3 inclusive), the plot of α² vs z is only little affected up to z = 0.4. The coefficient c₄ is estimated to be about ?25 by two independent methods. The graph-theoretical approach relates the coefficients c_m either to the number of spanning trees of so-called contracted configuration graphs, or to the sum of determinants of intersection matrices derived from Petrie matrices. The theory of Gordon and Tutte on these sums is summarised, and appropriately extended for application to the pre-averaging technique.     

Low temperature structural variations and molar heat capacity of stolzite, PbWO4

The crystal structure, thermal expansion and heat capacity of PbWO₄ (mineral name stolzite) scintillator material were comprehensively studied over a wide temperature range. No phase transitions were found down to 2 K (I4₁/a, scheelite structure type). A distinct feature of the temperature induced structural variations in PbWO₄ are the different thermal elongations of shorter and longer Pb-O distances. The low-temperature thermal expansion of PbWO₄ was parameterized on the basis of the 1st order Grüneisen approximation using a Debye function for the internal energy with a Debye temperature of 237 K, a bulk modulus of 67 GPa and a Grüneisen parameter of 1.08. The expansion along the c-axis is about 2.5-3 times higher in the range 23-290 K than along the a-direction. This pronounced anisotropy of the thermal expansion arises from the arrangements of rigid tetrahedral WO₄²⁻ units along 〈100〉-directions while Pb²⁺ cations occupy the sites between WO₄²⁻ in 〈001〉-directions.     

Lanczos cluster expansion for non-extensive systems

A cluster expansion of the Lanczos recursion for non-extensive systems is developed based on the plaquette expansion for extensive systems, in which an auxiliary scaling parameter, Ω, plays the role of volume and introduces extensivity into the problem. Connected Hamiltonian moments of the non-extensive system are computed and introduced into the plaquette expansion in the usual way with Ω. The extensive energy is calculated for increasing orders of the expansion in 1/Ω and the ground state and mass gap of the finite few body problem recovered in the limit Ω → ∞. This new non-perturbative method is applied to the case of N bosons interacting harmonically in one dimension and the ground state energy and mass gap in the vacuum sector are calculated exactly.     

An application of the 1/Z expansion method to reorganization corrections of the Koopmans ionization energy of atomic valence state

Using the 1/Z expansion method coupled with the random spin vector model, the reorganization corrections of the Koopmans ionization energies of the atomic valence state whose spatial quantization is referred to the cartesian coordinate system were formulated up to the second order. The theoretical formulations were applied to the valence electron ionizations of s²x², s²xy and sxyz isoelectronic series. The reorganization corrections calculated with the resolvent expansion were in fairly good agreement with the corresponding terms calculated with the direct ΔSCF method using the analytical approximations to the Hartree—Fock atomic orbitals.     

Substitution–dilution method to correct the matrix effect in multi-element quantitative analysis by X-ray fluorescence

A mathematical model based on the dilution–addition method (DAM) for multi-elemental analysis using an X-ray fluorescence technique is proposed. The conditions for sample preparation do not require both the unknown and standard samples to be similar in composition and mineralogy, and the unknown sample is replaced quantitatively by the standard sample, hence the denomination substitution–dilution method (SDM). This method makes it possible to correct the matrix effect in multi-elemental quantitative analysis by X-ray fluorescence for each analyte. The proposed model presents hyperbolic behaviour of the experimental data when the X-ray fluorescence intensities are represented versus the substitution factor (h) for each analyte. After calculating the A/B parameter relations, which depend on the X-ray fluorescence intensity of each analyte (I_i^ns) and the substitution factor (h) and determining the analyte concentration in the multi-element standard sample (C_i^p), it is possible to calculate the analyte concentration in the multi-element unknown using an algorithm suggested for this purpose. This work studies the substitution–dilution phase proposed in the method, and the factors arising from incorporation of the standard and diluent are established according to the nature of the samples and the modifications. These factors make it possible to establish the experimental interval of analyte concentration, generally narrow, which corresponds to a section of the hyperbolic function which is so short that it can be accepted as linear. This linear model can be accepted for a wide variety of samples with a diluent/sample ratio greater than 10. The proposed linear method provides satisfactory results which are comparable to those calculated by applying the hyperbolic method. The proposed method (SDM) has been applied to two different types of matrices, a binary alloy (without diluent, using the hyperbolic model) and a geological sample (with diluent, using both hyperbolic and linear models). In all cases the results were satisfactory.     

Recent progress in quantifying substituent effects

This paper summarizes significant progress in quantifying organic substituent effects in the last 20 years. The main content is as follows: (1) The principle of electronegativity equalization has gained wide acceptance, and has been used to calculate the intramolecular charge distribution and inductive effect of groups. A valence electrons equalization method was proposed to compute the molecular electronegativity on the basis of geometric mean method, harmonic mean method, and weighted mean method. This new calculation method further extended the application of the principle of electronegativity equalization. (2) A scale method was established for experimentally determining the electrophilic and nucleophilic ability of reagents, in which benzhydryliumions and quinone methides were taken as the reference compounds, and the research field was extended to the gas phase conditions, organometallic reaction and radicals system. Moreover, the nucleophilicity parameters N and electrophilicity parameters E for a series of reagents were obtained. The definition and quantitative expression of electrophilicity index and nucleophilicity index were proposed theoretically, and the correlation between the parameters from experimental determination and the indexes from theoretical calculation was also investigated. (3) The polarizability effect parameter was initially calculated by empirical method and further developed by quantum chemistry method. Recently, the polarizability effect index of alkyl (PEI) and groups (PEI X ) were proposed by statistical method, and got wide applications in explaining and estimating gas-phase acidity and basicity, ionization energy, enthalpy of formation, bond energy, reaction rate, water solubility and chromatographic retention for organic compounds. (4) The excited-state substituent constant ex CC obtained directly from the UV absorption energy data of substituted benzenes, is different from the polar constants in molecular ground state and the radical spin-delocalization effects constants in molecular radical state. The proposed constant ex CC correlated well with the UV absorption energy of many kinds of organic compounds, such as 1,4-disubstituted benzenes, substituted stilbenes, and disubstituted N-benzylidenebenzenamine. (5) The establishment of the steric shielding effect distinguished the three traditional steric effects. The stereoselectivity index C i was proposed to quantify the stereoselectivity of the addition reaction of carbonyl with nucleophilic reagent. The shielding parameter S R was defined to quantitatively express the specific surface of the reaction center screened by a group. Further, the Topological Steric Effect Index (TSEI) of a group was proposed on the basis of the relative specific volume of reaction center screened by the atoms of substituents. These parameters can be applied in estimating the intramolecular dihedral angles, stereoselectivity of reaction, enthalpies of formation of alkenes and alkylbenzene, acidity of substitutedimidazolium ionic liquid, and the reaction rate of alkane and hydroxyl radical. In addition, some suggestions and prospects for further studies on quantifying the organic substituent effects were presented in this paper.     

The use of scaled moments of inertia in experimental structure determinations of polyatomic molecules

Recently developed procedures for determining near-r _e structures (r _m ^p )from scaled ground-state moment of inertia data have been extended to general polyatomic molecules. A simple and general empirical method for correctingl _m ^p moments for the large vibration-rotation effects of deuterium substitution has been proposed and illustrated for a variety of polyatomic molecules. The correction requires utilization of a bond elongation parameter,δr _D ,whose experimental value is found to be approximately 0.003 Å. A theoretical justification of the approximation has been provided for linear triatomic molecules.     

On the structure and stability of Ar n and Ar n + clusters at finite temperature

For Ar_2–29 and Ar _2–29 ⁺ clusters at 20 K in the polarization model presented here the electrodynamical dipole-dipole many-body problem is solved selfconsistently with the Monte-Carlo method (MC) at 20 K, i.e. the instantaneous dipole-dipole interaction is solved to infinite perturbation order and in cluster expansion to the order of the cluster size. The long range many-body dipole-dipole interaction is coupled to exchange interaction by a modified effective dipole polarizability. This model will be compared to the dimer model and classical MC simulation of Ar _n . The resulting different magic numbers in the binding energies are discussed in this connection with different experimental techniques of cluster ionization. By the mean square cluster diameter a shape parameter is introduced and it is found that with this parameter structural form transition in cluster growth can be resolved, and surprisingly do not correlate with the magic numbers.     

Cu2Sc2Ge4O13, a novel germanate isotypic with the quasi-1D compound Cu2Fe2Ge4O13 between 100 and 298 K

The germanate compound Cu₂Sc₂Ge₄O₁₃ has been synthesized by solid-state ceramic sintering techniques between 1173 and 1423 K. The structure was solved from single-crystal data by Patterson methods. The title compound is monoclinic, a=12.336(2) Å, b=8.7034(9) Å, c=4.8883(8) Å, β=95.74(2), space group P2₁/m, Z=4. The compound is isotypic with Cu₂Fe₂Ge₄O₁₃, described very recently. The structure consists of crankshaft-like chains of edge-sharing ScO₆ octahedra running parallel to the crystallographic b-axis. These chains are linked laterally by [Cu₂O₆]⁸⁻ dimers forming a sheet of metal-oxygen-polyhedra within the a-b plane. These sheets are separated along the c-axis by [Ge₄O₁₃]¹⁰⁻ units. Cooling to 100 K does not alter the crystallographic symmetry of Cu₂Sc₂Ge₄O₁₃. While the b, c lattice parameter and the unit cell volume show a positive linear thermal expansion (α=6.4(2)×10⁻⁶, 5.0(2)×10⁻⁶ and 8.3(2)×10⁻⁶ K⁻¹ respectively), the a lattice parameter exhibits a negative thermal expansion (α=−3.0(2)×10⁻⁶ K⁻¹) for the complete T-range investigated. This negative thermal expansion of a is mainly due to the increase of the Cu-Cu interatomic distance, which is along the a-axis. Average bond lengths remain almost constant between 100 and 298 K, whereas individual ones partly show both significant shortages and lengthening.     

A new mixing rule—modified conventional mixing rule

The asymmetry of the corrective cohesion parameter a^c against the mole fraction is found out to be a controlling factor of the vapour—liquid equilibrium(VLE) calculations.A new mixing rule which satisfies the asymmetry of a^c and has two adjustable parameters is proposed for systems containing strongly polar substances. Furthermore, it still keeps the functional form of the conventional mixing rule.The parameters of the proposed mixing rule for the MVDW equation of state are correlated by the system temperature and well applied to the ternary systems containing strongly polar substances.     

Integration of the rate constant and linearization of the kinetic equations in non-isothermal reaction kinetics

The theory of reaction rates gives approximations of the type A T^b e^?E/RT for rate constants. In thermal analysis the case where b = 0 is used in kinetic calculations. However, if b is regarded as a variable, treatment of the non-isothermal kinetic equations is not more complicated than in the special case of b = 0. In this paper Padé approximations, described in the literature of the special functions, and Legendre's continued fractions are proposed for evaluation of the A∫T^b e^?E/RT dT integrals. The Coats—Redfern types methods for the determination of the kinetic parameters are discussed analysing the errors of approximation involved in their deduction. On this basis a modified parameter estimation scheme is proposed.     

Temperature dependence of limiting viscosity number and radius of gyration of cellulose diacetate in acetone

Acetone solutions of a cellulose diacetate fraction were studied by viscosity and light scattering methods over the range 12.6–50.3². The temperature dependences of the limiting viscosity number [η], the mean-square radius of gyration 〈s²〉, and the second virial coefficient A₂ were determined. The unperturbed mean-square radius of gyration 〈s²〉_o and the expansion factor α, were estimated by using theoretical relations to the interpenetration function. It was found that dln 〈s²〉_o/dT is ?6.4 × 10^?3 deg^?1, while α, is close to unity over the whole temperature range studied. The viscosity results are interpreted to show that the draining effect is not negligible and the Flory viscosity parameter Φ slightly increases with increasing temperature. It is finally concluded that the value of ?6.9 × 10^?3 deg^?1 for dln [η]/dT can be ascribable to the rapid decrease in 〈s²〉_o.     

Bound state energies for the exponential cosine screened Coulomb potential

The energy eigenvalues of bound states of an electron in the general exponential cosine screened Coulomb potential are obtained using the shifted 1/N expansion method. The energies for the states from 1s to 8k are calculated from six to eight significant figures. The energy eigenvalues for the 1s, 2s – 2p, 3s – 3d, and 4s – 4f states are also presented as a function of the screening parameter λ. Results are compared with the ones obtained by other workers. The agreement reduces roughly for large λ. It is also observed that the convergence of the expansion series increases remarkably asl increases.     

A numerical method for the solution of the one-dimensional Schrödinger equation

A numerical method for the calculation of bound states in a one-dimensional potential is suggested, the application of which is very simple and economic. The wavefunctions as well as the potential are approximated by Lagrange interpolation polynomials of the order 2N. The application of the variational principle yields a (2N?1)^th order eigenvalue problem of the symmetric matrix H derived from the universal matrices' T and V ^(l) (l = 0,1,..., N) and (2N + 1) discrete potential values of the actual problem by means of simple matrix algebra. The accuracy of the obtained energies and wavefunctions depends on the parameter N. The proposed method has been applied to different types of potentials. Using polynomials of the order twenty (N=10), generally the results are quite satisfactory for the states of quantum numbers n≦4.     

Application of thermal blob theory to intrinsic viscosity measurements of poly(4-chlorostyrene) above the theta point

Macromolecular chains obey purely Gaussian statistics close to the theta temperature. An asymptotic regime is also reached far above the theta point. Thermal blob theory permits an approximate calculation of thermal expansion factor. In this theory, an adjustable parameter is used as a prefactor (A*N₁) in order to calculate the reduced blob parameter (N/N_c). In this work, we proposed a different approach to evaluate (A*N₁) by plotting τM_ω^1/2/α_η⁵ against τM_ω^1/2. Chain expansion data of poly(4-chlorostyrene) in various solvents is used to evaluate viscosity expansion factors at various temperatures. It seems that predictions of thermal blob theory provide a better fit to experimental points rather than those evaluations based on Flory-type approaches. © 1996 John Wiley & Sons, Inc.     

DATINK pilot study: An effective methodology for ballpoint pen ink dating in questioned documents

Establishing the approximate age of an ink entry from a questioned document is often a complicated task and a controversial issue in forensic sciences. Among the existing approaches, the analysis of solvents in ballpoint inks may be a useful parameter for determining the age of ink on paper. In recent years, several ink dating methods have been proposed. These methods have been based on the analysis of common ink solvents using gas chromatography/mass spectrometry (GC/MS) as the analytical platform. Despite these recent methods, several questions remain. The aim of this work was to develop an ink dating methodology (DATINK) for documents written by ballpoint pens based on the disappearance of volatile solvents from the ink entry. Multiple solid-phase microextraction (MHS-SPME) coupled to GC/MS was used to measure the solvents from ink entries made with four BIC^® ballpoint pens. The β parameter, the remaining fraction of the analyte in the system after one equilibration, corresponding to the successive extractions was considered for modelling a mathematical equation for later ink age dating. Preliminary tests of DATINK method showed that it was possible to detect the presence of ink solvents on documents up to the studied five years. The analyses of different real samples of known age were analyzed in terms of β values, which provided a mean relative error of 21%. The proposed use of β parameter for estimating the absolute age of ballpoint ink entries has shown promising results with a standard deviation of β ranging from 0.002 to 0.004.