Evaluation of the Coulomb energy in relativistic self-consistent-field theory

 Computational schemes are presented with which to evaluate the electrostatic Coulomb energy in relativistic molecular electronic structure calculations using a basis of four-component Dirac spinor amplitudes. We demonstrate that algorithms may be constructed and implemented which differ only in minor details from those in common use in nonrelativistic quantum chemistry, and that the four-component formalism is neither as complicated nor as expensive as has been suggested recently in the literature. Spherically symmetrical atomic basis sets are presented which indicate that accurate representations of the Coulomb energy may be obtained using modest expansions of the electronic density in a scalar auxiliary basis set of spherical harmonic Gaussian-type functions. Received: 15 April 2002 / Accepted: 15 May 2002 / Published online: 29 July 2002     

Small-angle X-ray study of air-dried elastoidin using three- and one- dimensional correlation functions obtained from slit-smeared intensity

In the present paper an isotropic sample of air-dried elastoidin has been studied considering it to be a non-ideal two-phase densely packed system after the theories by Vonk [1]. The relevant important physical parameters obtained for the sample areE, the width of the transition layer,D, the average periodicity transverse to the layers, S/V, the specific inner surface ₁ and ₂, the volume fraction of two phases, i. e. matter and void,l ₁ andl ₂ the transversal lengths,l _r, the range of inhomogeneity,l _c, the length of coherence and 2E/D, the volume fraction of the transition layer. The values ofE obtained by two approches as given by Vonk [1] and Ruland [2] show but a small difference indicating the correctness of the analysis.     

Closed‐form rectangular atomic wave functions of an N‐dimensional atom

A closed‐form expression has been derived to obtain the N‐dimensional rectangular atomic wave functions. The wave functions can be obtained straightforwardly with the knowledge of the angular momentum quantum numbers and the dimension and there is no need to employ recursive methods as in previous works. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem 88: 263–274, 2002     

Optimisation of a flow injection system with electrochemical detection using the desirability function: Application to the determination of hydroquinone in cosmetics

A procedure is proposed in which the determination of hydroquinone using a flow injection system with electrochemical detection is described. Size and coefficient of variation of the signal are optimised by a desirability function and a central composite design. The robustness of the optimum reached in the optimisation step is evaluated by means a Plackett-Burman design. The optimised FIA system is able to determine hydroquinone with a minimum detectable net concentration of 10 μg l⁻¹ with a false positive probability of 0.05 and a false negative probability less than 0.05. In samples of bleaching cream, the proposed procedure has a recovery of 102.2% with standard deviation of 4.4% and a relative error of 6.2%.     

The W algorithm and the D transformation for the numerical evaluation of three‐center nuclear attraction integrals

Three‐center nuclear attraction integrals over exponential‐type functions are required for ab initio molecular structure calculations and density functional theory (DFT). These integrals occur in many millions of terms, even for small molecules, and they require rapid and accurate numerical evaluation. The use of a basis set of B functions to represent atomic orbitals, combined with the Fourier transform method, led to the development of analytic expressions for these molecular integrals. Unfortunately, the numerical evaluation of the analytic expressions obtained turned out to be extremely difficult due to the presence of two‐dimensional integral representations, involving spherical Bessel integral functions. % The present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of these spherical Bessel integrals. This algorithm, which is based on the nonlinear D transformation and the W algorithm of Sidi, can be computed recursively, allowing the control of the degree of accuracy. Numerical analysis tests were performed to further improve the efficiency of our algorithm. The numerical results section demonstrates the efficiency of this new algorithm for the numerical evaluation of three‐center nuclear attraction integrals. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Thermodynamic excess properties of alkanol + amine mixtures and application of the ERAS model

New experimental data of the molar excess volume V ^E of the mixtures ethanol/n-butylamine, heptanol/n-butylamine, n-propanol/dibutylamine have been obtained using the technique of the vibrating tube densitometer. Together with the data for the molar excess enthalpy H ^E from the literature, the V ^E data have been used for testing the applicability of the socalled ERAS model which accounts for hydrogen bonding effects as well as for free volume effects in associating mixtures. The results obtained by adjusting the model parameters reveal a strong cross association between the unlike molecules in the mixture resulting from strong negative values for the hydrogen bonding energy and the hydrogen bonding volume.Communicated at the Festsymposium celebrating Dr. Henry V. Kehiaian's 60^th birthday, Clermont-Ferrand, France, 17–18 May 1990.     

Application of coset representations to the construction of symmetry adapted functions

Summary A coset representation (G(/G _i )), which is defined algebraically by a coset decomposition of a finite groupG by its subgroupG _i , is shown to be a method for the decomposition of a regular body into its point group orbits. This proof also shows that each member of theG(/G _i ) orbit belongs to theG _i site-symmetry. In addition, a general equation concerning the multiplicities of such coset representations is derived and shown to involve Brester's equations and thek-value equations of framework groups as special cases. The relationship of the coset representation and the site-symmetry affords a general procedure for obtaining symmetry adapted functions.     

A modified steepest descent method with applications to maximizing likelihood functions

In maximizing a non-linear function G(), it is well known that the steepest descent method has a slow convergence rate. Here we propose a systematic procedure to obtain a 1–1 transformation on the variables , so that in the space of the transformed variables, the steepest descent method produces the solution faster. The final solution in the original space is obtained by taking the inverse transformation. We apply the procedure in maximizing the likelihood functions of some generalized distributions which are widely used in modeling count data. It was shown that for these distributions, the steepest descent method via transformations produced the solutions very fast. It is also observed that the proposed procedure can be used to expedite the convergence rate of the first derivative based algorithms, such as Polak-Ribiere, Fletcher and Reeves conjugate gradient methods as well.     

Lipschitzian inverse functions,directional derivatives,and applications inC 1,1 optimization

The paper shows that Thibault's limit sets allow an iff-characterization of local Lipschitzian invertibility in finite dimension. We consider these sets as directional derivatives and extend the calculus in a way that can be used to clarify whether critical points are strongly stable inC ^1,1 optimization problems.Many fruitful discussions with colleagues D. Klatte and K. Tammer as well as with H. Th. Jongen and F. Nozicka have influenced the present investigations in a very constructive manner. For the original papers concerning the sets f(x; u), the author is indebted to Prof. L. Thibault.     

Metastable systems driven by colored noise: The stationary state

The influence of the Bardeen-Herring back-jump correlations on the Fermi-Dirac statistics of the one-dimensional nonhomogeneous fermionic lattice gas is studied by the Monte Carlo simulation technique and semianalytically. The resulting distribution is obtained, exhibiting increased population of the lower levels in comparison to the Fermi-Dirac statistics.