Optimization of complex slater‐type functions with analytic derivative methods for describing photoionization differential cross sections

The complex basis function (CBF) method applied to various atomic and molecular photoionization problems can be interpreted as an method to solve the driven‐type (inhomogeneous) Schrödinger equation, whose driven term being dipole operator times the initial state wave function. However, efficient basis functions for representing the solution have not fully been studied. Moreover, the relation between their solution and that of the ordinary Schrödinger equation has been unclear. For these reasons, most previous applications have been limited to total cross sections. To examine the applicability of the CBF method to differential cross sections and asymmetry parameters, we show that the complex valued solution to the driven‐type Schrödinger equation can be variationally obtained by optimizing the complex trial functions for the frequency dependent polarizability. In the test calculations made for the hydrogen photoionization problem with five or six complex Slater‐type orbitals (cSTOs), their complex valued expansion coefficients and the orbital exponents have been optimized with the analytic derivative method. Both the real and imaginary parts of the solution have been obtained accurately in a wide region covering typical molecular regions. Their phase shifts and asymmetry parameters are successfully obtained by extrapolating the CBF solution from the inner matching region to the asymptotic region using WKB method. The distribution of the optimized orbital exponents in the complex plane is explained based on the close connection between the CBF method and the driven‐type equation method. The obtained information is essential to constructing the appropriate basis sets in future molecular applications. © 2017 Wiley Periodicals, Inc.     

Photoionization cross sections with optimized orbital exponents within the complex basis function method

We show a new direction to expand the applicability of the complex basis function method for calculating photoionization cross sections through the imaginary part of the frequency-dependent polarizability. Based on the variational stability of the frequency-dependent polarizability, we made nonlinear optimizations of complex orbital exponents in basis functions representing continuum wave functions, and obtained fairly accurate results for H atom with only one or two complex basis functions particularly with dipole velocity gauge. Results were almost independent of whether Slater-type or Gaussian-type orbitals are used, implying the applicability to general many electron problems. The method was also applied to the (1)S (1s)(2) --> (1)P (1s)(1)(kp)(1) cross section of He atom and the optimized complex orbital exponents were related to those of H atom through the scaling property. The nonlinear optimizations have converged smoothly and the cross sections were in excellent agreement with experiment throughout wide photon energies, which suggest the effectiveness of the approach for many-electron systems.     

Photoionization cross sections of H2(+) and H2 with complex Gaussian-type basis functions optimized for the frequency-dependent polarizabilities

Within the framework of the complex basis function method, the photoionization cross sections of H(2)(+) and H(2) were calculated based on the variational principle for the frequency-dependent polarizabilities. In these calculations, complex orbital exponents of Gaussian-type basis functions for the final state continuum wavefunctions were fully optimized for each photon energy with the numerical Newton-Raphson method. In most cases, the use of only one or two complex Gaussian-type basis functions was enough to obtain excellent agreement with previous high precision calculations and available experimental results. However, there were a few cases, in which the use of complex basis functions having various angular momentum quantum numbers was crucial to obtain the accurate results. The behavior of the complex orbital exponents as a function of photon energy was discussed in relation to the scaling relation and the effective charge for photoelectron. The success of this method implies the effectiveness of the optimization of orbital exponents to reduce the number of basis functions and shows the possibility to calculate photoionization cross sections of general molecules using only Gaussian-type basis functions.     

Absolute photoionization cross sections of two cyclic ketones: cyclopentanone and cyclohexanone

Absolute photoionization cross sections for cyclopentanone and cyclohexanone, as well as partial ionization cross sections for the dissociative ionized fragments, are presented in this investigation. Experiments are performed via a multiplexed photoionization mass spectrometer utilizing vacuum ultraviolet (VUV) synchrotron radiation supplied by the Advanced Light Source of Lawrence Berkeley National Laboratory. These results allow the quantification of these species that is relevant to investigate the kinetics and combustion reactions of potential biofuels. The CBS‐QB3 calculated values for the adiabatic ionization energies agree well with the experimental values, and the identification of possible dissociative fragments is discussed for both systems. Copyright © 2017 John Wiley & Sons, Ltd.     

Investigation on the absolute and relative photoionization cross sections of 3 potential propargylic fuels

Absolute photoionization cross sections for 2 potential propargylic fuels (propargylamine and dipropargyl ether) along with the partial ionization cross sections for their dissociative fragments are measured and presented for the first time via synchrotron photoionization mass spectrometry. The experimental setup consists of a multiplexed orthogonal time‐of‐flight mass spectrometer and is located at the Advanced Light Source facility of the Lawrence Berkeley National Laboratory in Berkeley, California. Data for a third propargylic compound (propargyl alcohol) were taken; however, because of its low signal, due to its weakly bound cation, only the dissociative ionization fragment from the H‐loss channel is observed and presented. Suggested pathways leading to formation of dissociative photoionization fragments along with CBS‐QB3 calculated adiabatic ionization energies and appearance energies for the dissociative fragments are also presented.     

Evaluation of expansion coefficients for translation of Slater‐type orbitals using complete orthonormal sets of exponential‐type functions

By the use of exponential‐type functions (ETFs) the simpler formulas for the expansion of Slater‐type orbitals (STOs) in terms of STOs at a displaced center are derived. The expansion coefficients for translation of STOs are presented by a linear combination of overlap integrals. The final results are of a simple structure and are, therefore, especially useful for machine computations of arbitrary multielectron multicenter molecular integrals over STOs that arise in the Hartree–Fock–Roothaan approximation and also in the Hylleraas correlated wave function method for the determination of arbitrary multielectron properties of atoms and molecules. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 126–129, 2001     

Canonical two‐range addition theorem for slater‐type orbitals

The radial Slater‐type orbitals (STO) ${r^\mu }{e^{ - \alpha r}}$ can be simply obtained by repeated parametric differentiation of the Yukawa Potential $({e^{ - \alpha r}}/r)$ with respect to α. A new compact two‐range addition theorem (AdT) for the STO is herein derived by explicit parametric differentiation of the well‐known Yukawa AdT. The resulting addition formula is combined with the single‐range AdT for solid spherical harmonics $({r^l}Y_l^m(\hat r))$ to present a new AdT for three‐dimensional spherical coordinate STOs. We advance the proposition that this formula is “canonical” in the same sense that the Laplace expansion of the Coulomb potential is considered canonical. We demonstrate how this procedure can be employed for all exponential‐type orbitals. © 2012 Wiley Periodicals, Inc.     

Absolute photoionization cross sections of furanic fuels: 2‐ethylfuran, 2‐acetylfuran and furfural

Absolute photoionization cross sections of the molecules 2‐ethylfuran, 2‐acetylfuran and furfural, including partial ionization cross sections for the dissociative ionized fragments, are measured for the first time. These measurements are important because they allow fuel quantification via photoionization mass spectrometry and the development of quantitative kinetic modeling for the complex combustion of potential fuels. The experiments are carried out using synchrotron photoionization mass spectrometry with an orthogonal time‐of‐flight spectrometer used for mass analysis at the Advanced Light Source of Lawrence Berkeley National Laboratory. The CBS‐QB3 calculations of adiabatic ionization energies and appearance energies agree well with the experimental results. Several bond dissociation energies are also derived and presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Role of resonances in building cross sections: Comparison between the Mittag–Leffler and the T‐matrix Green function expansion approaches

Peaks in collision cross sections are often interpreted as resonances. The complex dilation method, as well as other methods relying on analytic continuation of the scattering formalism, can be used to clarify whether these structures are true resonances in the sense that they are poles of the S‐matrix and the associated Green function. The performance of the Mittag–Leffler expansion and T‐matrix Green function expansion methods are formally and computationally compared. The two methods are applied to two model potentials. Eigenenergies, s‐wave residues, and cross sections are computed with both methods. The resonance contributions to the cross sections are further analyzed by removing the residue contributions from the Mittag–Leffler and Green function expansion sums, respectively. It is suggested that the contribution of a resonance to a cross section should be defined through its S‐matrix residue. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

The role of resonances in building cross sections: The Mittag‐Leffler expansion in a two‐channel scattering

The Mittag‐Leffler formalism for identifying resonance contributions to a two‐body scattering cross section is extended here from the one channel case to the two channel case. The reduced partial wave cross sections based on subtracting S‐matrix residues are compared with expressions which can be derived from the Breit‐Wigner approximation. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Representation of Kohn–Sham free atom eigenfunctions by Slater‐type orbitals

Slater‐type orbitals are applied to represent the numerically obtained Kohn–Sham eigenfunction of free atom. The algorithm evaluating the nonlinear expansion coefficients of this approximation is described. Standard iterative solution of Kohn–Sham equation to obtain the nonlinear expansion coefficients is avoided and replaced by the projection method. First, the eigenfunction is obtained in the B‐spline space based on the Galerkin formulation of the finite element method. Then, based on the density functional theory, the conditions are formulated, which leads to the set of nonlinear equations. The proposed algorithm is general and can be applied for any atomic Kohn–Sham eigenfunction. As an examplary application of the proposed algorithm, the set of nonlinear equations is derived for occupied states of N, Al, Ga, and In atoms. The expansion coefficients, obtained for these atoms, are evaluated numerically by Newton procedure and listed in the tables. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Auxiliary functions for molecular integrals with Slater‐type orbitals. I. Translation methods

Many types of molecular integrals involving Slater functions can be expressed, with the ζ‐function method in terms of sets of one‐dimensional auxiliary integrals whose integrands contain two‐range functions. After reviewing the properties of these functions (including recurrence relations, derivatives, integral representations, and series expansions), we carry out a detailed study of the auxiliary integrals aimed to facilitate both the formal and computational applications of the ζ‐function method. The usefulness of this study in formal applications is illustrated with an example. The high performance in numerical applications is proved by the development of a very efficient program for the calculation of two‐center integrals with Slater functions corresponding to electrostatic potential, electric field, and electric field gradient. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Use of noninteger n‐generalized exponential type orbitals with hyperbolic cosine in atomic calculations

The efficiency of noninteger n‐generalized exponential type orbitals (NGETO) r^n*?1 e with hyperbolic cosine (HC) cosh (βr^μ) as radial basis functions in atomic ground state total energy calculations is studied. By the use of these functions, the combined Hartree‐Fock‐Roothaan calculations have been performed for some closed and open shell neutral atoms and their anions and cations with Z ≤ 21. The performance of new basis functions within the minimal basis framework has been compared with numerical Hartree‐Fock (NHF) results. Our total energy values are significantly close to NHF results. The presented minimal basis total energies obtained from the noninteger NGETO with HC are notably better than minimal basis functions total energies previously reported in the literature. It is found that the accuracy of new noninteger NGETO with HC almost correspond to the accuracy of the conventional double‐zeta functions. All the nonlinear parameters are fully optimized. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Auxiliary functions for molecular integrals with Slater‐type orbitals. II. Gauss transform methods

The Gauss transform of Slater‐type orbitals is used to express several types of molecular integrals involving these functions in terms of simple auxiliary functions. After reviewing this transform and the way it can be combined with the shift operator technique, a master formula for overlap integrals is derived and used to obtain multipolar moments associated to fragments of two‐center distributions and overlaps of derivatives of Slater functions. Moreover, it is proved that integrals involving two‐center distributions and irregular harmonics placed at arbitrary points (which determine the electrostatic potential, field and field gradient, as well as higher order derivatives of the potential) can be expressed in terms of auxiliary functions of the same type as those appearing in the overlap. The recurrence relations and series expansions of these functions are thoroughly studied, and algorithms for their calculation are presented. The usefulness and efficiency of this procedure are tested by developing two independent codes: one for the derivatives of the overlap integrals with respect to the centers of the functions, and another for derivatives of the potential (electrostatic field, field gradient, and so forth) at arbitrary points. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Comment on “Symbolic Calculation of Two‐Center Overlap Integrals Over Slater‐type Orbitals”

Published by Gümü?. and Özdo?an formulas for the evaluation of two‐center overlap integrals (Gümü?, S.; Özdo?an, T. J. Chin. Chem. Soc. 2004, 51, 243) are critically analyzed. It is demonstrated that the formulas presented in this work are not original and they can easily be derived from the relationships contained in our papers (Guseinov, I. I. J. Phys. B 1970 , 3, 1399; Phys. Rev. A 1985 , 32, 1864; J. Mol Struct. (Theochem) 1995 , 336, 17) by changing the summation indices and application of a simple algebra. It should be noted that the symbolic results of overlap integrals between different combinations of quantum numbers given in Table 1 and 2 can also be obtained from the use of established in above mentioned our papers general formulas or presented in the literature relations for overlap integrals in terms of the products of molecular auxiliary functions A_n(p) and B_n(pt) (see, e.g., Lofthus, A. Mol. Phys. 1962 , 5, 105).     

Application of Löwdin's canonical orthogonalization method to the Slater‐type orbital configuration‐interaction basis set

We apply Löwdin's canonical orthogonalization method to investigate the linearly dependent problem arising from the variational calculation of atomic systems using Slater‐type orbital configuration‐interaction (STO‐CI) basis functions. With a specific arithmetic precision used in numerical computations, the nonorthogonal STO‐CI basis is easily linearly dependent when the number of basis functions is sufficiently large. We show that Löwdin's canonical orthogonalization method can successfully overcome such problem and simultaneously reduce the dimension of basis set. This is illustrated first through an S‐wave model He atom, and then the real two‐electron atoms in both the ground and excited states. In all of these calculations, the variational bound state energies of the two‐electron systems are obtained in reasonably high accuracy using over‐redundant STO‐CI bases, however, without using extended high‐precision technique. © 2015 Wiley Periodicals, Inc.     

Evaluation of elastic‐scattering cross sections for electrons and positrons over a wide energy range

Quantification of surface‐ and bulk‐analytical methods, e.g. Auger‐electron spectroscopy (AES), X‐ray photoelectron spectroscopy (XPS), electron‐probe microanalysis (EPMA), and analytical electron microscopy (AEM), requires knowledge of reliable elastic‐scattering cross sections for describing electron transport in solids. Cross sections for elastic scattering of electrons and positrons by atoms, ions, and molecules can be calculated with the recently developed code ELSEPA (Elastic Scattering of Electrons and Positrons by Atoms) for kinetic energies of the projectile from 10 eV to 50 eV. These calculations can be made after appropriate selection of the basic input parameters: electron‐density distribution, a model for the nuclear‐charge distribution, and a model for the electron‐exchange potential (the latter option applies only to scattering of electrons). Additionally, the correlation‐polarization potential and an imaginary absorption potential can be considered in the calculations. We report comparisons of calculated differential elastic‐scattering cross sections (DCSs) for silicon and gold at selected energies (500 eV, 5 keV, 30 keV) relevant to AES, XPS, EPMA, and AEM, and at 100 MeV as a limiting case. The DCSs for electrons and positrons differ considerably, particularly for medium‐ and high‐atomic‐number elements and for kinetic energies below about 5 keV. The DCSs for positrons are always monotonically decreasing functions of the scattering angle, while the DCSs for electrons have a diffraction‐like structure with several minima and maxima. A significant influence of the electron‐exchange correction is observed at 500 eV. The correlation‐polarization correction is significant for small scattering angles at 500 eV, while the absorption correction is important at energies below about 10 keV. Copyright © 2005 John Wiley & Sons, Ltd.     

Evaluation of two‐center,three‐ and four‐electron integrals over Slater‐type orbitals in elliptical coordinates

An algorithm for evaluation of two‐center, three‐electron integrals with the correlation factors of the type rr and rrr as well as four‐electron integrals with the correlation factors rrr and rrr in the Slater basis is presented. This problem has been solved here in elliptical coordinates, using the generalized and modified form of the Neumann expansion of the interelectronic distance function r for k ≥ ?1. Some numerical results are also included. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Synthesis of a new salphen‐type titanium complex and its application in ethylene polymerization and copolymerization

TiCl₂[salphen(di‐tBu)] was synthesized, characterized and employed as pre‐catalyst in ethylene homo‐ and copolymerization with propylene, 1‐octene and 10‐undecen‐1‐ol. X‐ray diffraction study on the titanium complex revealed a distorted octahedral coordination of the central metal with a trans‐Cl, cis‐O, cis‐N arrangement. The complex combined with MAO afforded moderate catalytic activities in ethylene polymerization. Furthermore the catalyst not only copolymerized ethylene with apolar monomer (propylene and 1‐octene), but also possessed significant capability of incorporation with polar monomer (10‐undecen‐1‐ol). Only single insertion of 1‐octene unit in ethylene‐co‐1‐octene polymer was detected by ¹³C NMR spectrum. Copyright © 2010 John Wiley & Sons, Ltd.     

Computation of two‐center Coulomb integrals over Slater‐type orbitals using elliptical coordinates

A general analytic formula is obtained for the two‐center Coulomb integrals over Slater‐type orbitals in elliptical coordinates. Finite series expansions are used in the evaluation of the radial part of the integrals. The analytic formula is expressed in terms of a product of the well‐known auxiliary functions A_k(p) and B_k(p) and incomplete gamma functions. Recursive relations for the computer evaluation of these functions are given as well. The recursive relations are stable and our computer results are in good agreement with the benchmark values given in the literature. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003