Fast evaluation of molecular auxiliary functions <Emphasis Type="Italic">A</Emphasis><Subscript>α</Subscript> and <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> by analytical relations

Analytical relations through the initial values are derived for the molecular auxiliary functions A _α (x) and B _n (x), where α =n+ɛ, 0⩽ ɛ < 1 and n=0,1,2,.... These relations are useful in the fast calculation of multicenter molecular integrals over integer and noninteger n Slater type orbitals. It is shown that the formulas obtained are numerically stable for all values of n,ɛ and x.PACS No: 31.15.+q, 31.20.EjAMS subject classification: 81-V55, 81-V45     

On the Evaluation of Boys Functions Using Downward Recursion Relation

A new, simple, and efficient technique is presented for the accurate evaluation of the Boys functions F _m (x) (BFs) with integer and noninteger values of m appearing for the calculation of multicenter multielectron molecular integrals in a mixed Gaussian and plane-wave basis set. The extensive test calculations show that the proposed in this work algorithm is the most efficient in practical computations.     

Use of analytical relations in evaluation of exponential integral functions

Analytical formulas through the initial values suitable for numerical computation are developed for the exponential integral functions E_n (x). The relationships obtained are numerically stable for all values of n and for x < 1. Numerical results are also given.PACS No: 31.15.+q, 31.20.Ej AMS subject classification:81-V55, 81V45     

Symbolic Calculation of Two‐Center Overlap Integrals Over Slater‐Type Orbitals

Two‐center overlap integrals over Slater type orbitals (STOs) have been expressed in terms of the well‐known Mulliken's integrals B_n(pt) using Rodrigues's formula for normalized associated Legendre functions. A computer program is written in Mathematica 4.0 for the evaluation of two‐center overlap integrals over STOs. Using this computer program, symbolic tables are presented for two‐center overlap integrals up to quantum numbers 1 ≤ n,n′ ≤ 3, 0 ≤ l,l′ ≤ 2, ?2 ≤ m,m′ ≤ 2. Numerical results of this work, for some quantum sets, have also been compared with prior literature and best agreement achieved with recent works of Barnett while some discrepancies were obtained with works of Öztekin et al. and Guseinov et al.     

Evaluation of the auxiliary function $${G_{-ns}^q}$$ using basic overlap integrals over Slater type orbitals

This paper presents a computationally efficient formula in terms of basic overlap integrals over Slater type orbitals (STOs) for the evaluation of auxiliary function which plays a central role in calculations of multicenter molecular integrals. The basic overlap integrals are calculated with the help of recurrence relations. The resulting simple analytical formula for the auxiliary function is completely general for p _a ≤ 1.2 and arbitrary values of parameters p and pt. The efficiency of calculation of auxiliary function is compared with other method.     

On the calculation of arbitrary multielectron molecular integrals over Slater‐type orbitals using recurrence relations for overlap integrals. III. Auxiliary functions \documentclass{article}\pagestyle{empty}\begin{document}$Q_{nn'}^{q}$\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$G_{-nn'}^{q}$\end{document}

The auxiliary functions $Q_{nn'}^{q}(p,pt)$ and $G_{-nn'}^{q}(p_{a},p,pt)$ which are used in our previous paper [Guseinov, I. I.; Mamedov, B. A. Int J Quantum Chem 2001, 81, 117] for the computation of multicenter electron‐repulsion integrals over Slater‐type orbitals (STOs) are discussed in detail, and the method is given for their numerical computation. The present method is suitable for all values of the parameters p_a, p, and pt. Three‐ and four‐center electron‐repulsion integrals are calculated for extremely large quantum numbers using relations for auxiliary functions obtained in this paper. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Use of Auxiliary Functions for Construction of Series Expansion Relations of Noninteger Slater Functions in Standard Convention

Using complete orthonormal sets of ψ ^(α*) ‐self‐frictional exponential type orbitals (ψ ^(α*) ‐SFETOs) and Q^q‐noninteger auxiliary functions (Q^q‐NIAFs) introduced by the author, the combined formulas for the one‐ and two‐center one‐range addition theorems of χ‐noninteger Slater type orbitals (χ‐NISTOs) with arbitrary values of distances between centers R_ab (for R_ab = 0 and R_ab ≠ 0), and of integer (for α* = α, –∞ < α ≤ 2) and noninteger (for α* ≠ α, –∞ < α* < 3) self‐frictional (SF) quantum numbers are suggested. The presented relations for the one‐range addition theorems can be useful tools especially in the electronic structure studies of atoms, molecules and solids when χ‐NISTOs are employed as basis functions.     

To Be or Not to Be: Demystifying the 2nd-Quantized Picture of Complex Electronic Configuration Patterns in Chemistry with Natural Poly-Electron Population Analysis

We provide a didactic introduction to 2nd-quantized representation of complex electron–hole (e/h) excitation patterns in general configuration interaction wave functions built from orthonormal local orbitals of natural atomic orbital or natural bond orbital (NBO) type. Such local excitation patterns of chemically oriented basis functions can be related to the resonance concepts of valence bond theory, and quantitative evaluation of the associated excitation probabilities then provides an alternative assessment of resonance “weighting” that may be compared with those of NBO-based natural resonance theory. We illustrate the usefulness of anticommutation relations in deriving Pauli-compliant expressions for allowed excitation patterns, showing how the exciton-like promotions φ_λ → φ_ν (creating an e/h excitation with h in φ_λ and e in φ_ν) impose strict constraints on associated e/h-probabilities (requiring, e.g., that the e-probability for an electron “to be” or “not to be” in φ_ν must be rigorously linked to the complementary h-probabilities in φ_λ). Specific examples are presented of the quantum Boolean logic for four or six local spin-orbitals, with emphasis on Natural Poly-Electron Population Analysis (NPEPA) evaluation of VB-type covalent and ionic contributions in conventional 2-center bonding, resonance weightings in 3-center hydrogen bonding, and general characteristics of higher-order m-center bonding motifs for m > 3. Numerical results are presented for methylamine, acrolein, and water dimer to illustrate current NPEPA implementation in the NBO program. © 2019 Wiley Periodicals, Inc.     

Use of Recursion and Analytical Relations in Evaluation of Hypergeometric Functions Arising in Multicenter Integrals with Noninteger <Emphasis Type="Italic">n</Emphasis> Slater Type Orbitals

In this work we present the new recursion and analytical relations for the calculation of hypergeometric functions F(1,b;c;z) occurring in multicenter integrals of noninteger n Slater type orbitals. The formulas obtained are numerically stable for 0 < z < 1 and all integer and noninteger values of parameters b and c The Author cordially congratulates Prof. I.I. Guseinov on his 70th birthday     

Li12Cu16+xAl26−x (x = 3.2): a new intermetallic structure type

The new ternary lithium copper aluminide, Li₁₂Cu_16+xAl_26−x (x = 3.2), dodecalithium nonadecacopper tricosaaluminide, crystallizes in a new structure type with space group P4/mbm. Among nine independent atomic positions, two Al (one of which is statistically disordered with Cu) and three Li atoms have point symmetry m.2m, two statistically disordered Al/Cu atoms are in m.. sites, one Al atom is in a 4/m.. site and one Cu atom occupies a general site. The framework of Li₁₂Cu_16+xAl_26−x consists of pseudo‐Frank–Kasper polyhedra enclosing channels of hexagonal prisms occupied by Li atoms. The crystallochemical peculiarity of this new structure type is discussed in relation to the derivatives from Laves phases (LiCuAl₂ and Li₈Cu_12+xAl_6−x) and to the well known CaCu₅ structure.     

The multi-centre integrals of derivative,spherical GTOs

The multi-centre integrals of the orbital system ⁿ Y _lm () exp (–r ²) are evaluated using the Talmi transformation of nuclear shell theory. The integrals are simpler than those of the systems r ²ⁿ Y lm(r) exp (–r ²), x ^l y ^m z ⁿ exp (–r ²), (/x) ^l (/y) ^m (/z) ⁿ exp (–r ²) and the spherical oscillator functions. The integral types investigated are: overlap, electric dipole transition (momentum operator), kinetic energy, three-centre nuclear attraction, four-centre electronic repulsion, three-centre spin-orbit coupling, and magnetic dipole transition (three-centre integrals of the angular momentum operator).     

Li8Cu12+xAl6−x (x = 1.16): a new structure type related to Laves phases

The new ternary lithium copper aluminide Li₈Cu_12+xAl_6−x (x = 1.16) crystallizes in the P6₃/mmc space group with six independent atom positions of site symmetries m. (Al/Cu mixture), m2 (Li atoms), 3m. (Al/Cu mixture and Li atoms) and .m. (Cu atoms). The compound is a derivative of the K₇Cs₆ binary structure type and is related to the binary MgZn₂ Laves phase and the LiCuAl₂, MgCu_1.07Al_0.93 and Mg(Cu_1−xAl_x)₂ (x = 0.465) ternary Laves phases. The coordination polyhedra of the atoms in this structure are icosahedra (Cu atoms), slightly distorted icosahedra and bicapped hexagonal antiprisms (Al/Cu statistical mixture), and Frank–Kasper and distorted Frank–Kasper polyhedra (Li atoms). All interatomic distances indicate metallic type bonding.     

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).     

Estimation of the lattice heat capacity of rare-earth compounds

On the basis of the experimental data reported in literature, the contributions of cation mass (m) and molar volume (V) to lattice heat capacity (C) were analyzed. The volumetric-mass formula, C_x=(l —f)·C₁+f·C₂+C_m·(m_x — m_x′), was presented for estimating the heat capacities of rare-earth compounds. In the formula C₁ and C₂ represent the lattice heat capacities of two reference substances respectively, f = V_x—V₁/V₂—V₁ and C_m represents the lattice heat capacity variation with the variation 1 g of cation mass. The equation relating the C_m with temperatures was derived as follows: C_m = 0.084 e ?0.0074T ?0.27 e ?0.045T, and m_x and m_x′ (= (1 - f) m₁₊f m₂) represent the practical and “assumed” cation masses of the substance in question respectively.     

Study of the 1A1?5T2 spin transition in mix-ligand complexes of iron(II) hexafluorosilicate with 1,2,4-triazole and 4-amino-1,2,4-triazole

Synthesis procedures for mix-ligand complexes with a composition of Fe(Htrz)_3(1-x)(NH₂trz)_3x SiF₆ · mH₂O, 0 ≤ x ≤ 1, m=1,2 are developed. Solid phases are isolated from water-ethanol solutions and studied by X-ray phase analysis, the method of static magnetic susceptibility (temperature range of 100 K to 400 K), IR and M?ssbauer spectroscopy (at 200 K and 298 K). The temperatures of direct and inverse transitions are shown to depend on the complex composition.     

Die Kristallstruktur von TlFe3Te3

TlFe₃Te₃ is hexagonal, space groupP6₃/m–C _2h ⁶ ,a=9.350(2) Å,c=4.2230 (7) Å,Z=2. Iron and tellurium atoms occupy the positions 6 (h) withx=0.170,y=0.149 andx=0.046,y=0.357 respectively. Thallium atoms are situated in 2 (d). The structure was determined on the basis of single crystal data obtained form a four circle diffractometer. Refinement yielded andR-value of 4.8% for an asymmetric set of 267 reflections. TlFe₃Te₃ is a new structure type. The structure and its relations to the Mn₅Si_3–, the Nb₃Te₄-and the Tl _x V₆S₈-type are discussed.

     

Relative Rates for Plasma Homo- and Copolymerizations of a Homologous Series of Fluorinated Ethylenes

The relative rates of plasma homo- and Copolymerizations of ethylene, vinyl fluoride, vinylidene fluoride, trifluoroethylene and tetrafluoroethylene (VF _x , x = 0–4, respectively) were determined in an rf, capacitively coupled, tubular reactor with external electrodes using identical plasma parameters. The deposition rates for VF _x (x = 1–3) and 20 different monomer blends, when plotted versus the F/C ratios of the monomers or monomer blends, followed a concave-downward curve situated above the straight-line joining the rates for VF₀ and VF₄. The deposition rates for VF _m /VF _n blends (m = 3 or 4; n = 0–2) likewise yielded concave-downward curves situated above the straight lines joining the rates for the respective monomers; the rates for VF _m /VF _n blends (m = 0 or 1; n = 1 or 2) yielded concave-upward curves situated below the straight lines joining the rates for the respective monomers; while the rates for VF₃/VF₄ blends fell along the straight line joining the rates for VF₃ and VF₄. The mechanisms for plasma (co) polymerizations of VF _x monomers responsible for the wide range of relative deposition rates remain to be elucidated.     

Characterization of Mixed Crystals in the System Cu2MnxCo1−xGeS4 and Investigations of the Tetrahedra Volumes

In a series of investigations on normal tetrahedral compounds we present mixed crystals in the system Cu₂Mn_xCo_1?xGeS₄ (0 < x < 1) and an inspection of their tetrahedra volumes. Cu₂CoGeS₄ crystallizes tetragonal in a stannite type structure, Cu₂MnGeS₄ crystallizes orthorhombic in the wurtzstannite structure type. The crystal structures of Cu₂CoGeS₄ and Cu₂Mn_0.68Co_0.32GeS₄ were refined from single crystal diffraction data. The refinement of Cu₂CoGeS₄ converged to R = 0.0547 and wR2 = 0.0847 for 299 unique reflections. The refinement of Cu₂Mn_0.68Co_0.32GeS₄ converged to R = 0.0481 and wR2 = 0.0877 for 1556 unique reflections. From these data the tetrahedra volumes of the end members and of Cu₂Mn_0.68Co_0.32GeS₄ are calculated. In Cu₂CoGeS₄ tetrahedra [MS₄] are similar in size. In contrast, the differences of the volumes of the polyhedra [MS₄] in the orthorhombic wurtzite superstructure type compounds Cu₂MnGeS₄ and Cu₂Mn_0.68Co_0.32GeS₄ are significant (M = Cu, Mn, (Mn_0.68Co_0.32), Co, Ge). From x = 0 to x = 0.5 the tetragonal structure type dominates while from x = 0.7 to the Cu₂MnGeS₄ end member the products crystallize in the orthorhombic structure type. Melting points of the mixed crystals decrease linearly with increasing manganese content.     

Atomically Precise Alloy Nanoclusters

Metal nanoclusters (NCs) have a particle size of about one nanometer, which makes them the smallest unit that can give a function to a substance. In addition, metal NCs possess physical and chemical properties that are different from those of the corresponding bulk metals. Metal NCs with such characteristics are expected to be important for use in nanotechnology. Research on the precise synthesis of metal NCs and elucidation of their physical/chemical properties and functions is being actively conducted. When metal NCs are alloyed, it is possible to obtain further various electronic and geometrical structures and functions. Thus, research on alloy NCs has become a hot topic in the study of metal NCs and the number of publications on alloy NCs has increased explosively in recent years. Such publications have provided much insight into the effects of alloying on the electronic structure and function of metal NCs. However, the rapid increase in knowledge has made it difficult for researchers (especially those new to the field) to grasp all of it. Therefore, in this review, we summarize the reported chemical composition, geometrical structure, electronic structure, and physical and chemical properties of Au_n−xM_x(SR)_m, Ag_n−xM_x(SR)_m, Au_n−xM_x(PR₃)_l(SR)_m, and Ag_n−xM_x(PR₃)_l(SR)_m (Au=gold, Ag=silver, M=heteroatom, PR₃=phosphine, and SR=thiolate) NCs. This review is expected to help researchers understand the characteristics of alloy NCs and lead to clear design guidelines to develop new alloy NCs with intended functions.     

Tb2–xNdxZn17–yNiy (x = 0.5, y = 4.83): a new intermetallic with a maximum disordered structure and its hydrogen storage properties

The ternary Tb_2–xNd_xZn_17–yNi_y (x = 0.5, y = 4.83) disordered phase belongs to the structural family based on the rhombohedral Th₂Zn₁₇ structure type. The structure is maximally disordered since all the sites are occupied by statistical mixtures of atoms. The Tb/Nd mixture of atoms occupies the 6c site (site symmetry 3m). The statistical mixtures Ni/Zn consisting of more Ni atoms are located in the 6c and 9d (symmetry .2/m) sites. In the following 18f (site symmetry .2) and 18h (site symmetry .m) sites are located Zn/Ni statistical mixtures which consist of more Zn atoms. Zn/Ni atoms form three-dimensional networks with hexagonal channels that fill statistical mixtures of Tb/Nd and Ni/Zn. The Tb_2–xNd_xZn_17–yNi_y compound belongs to the family of intermetallic phases capable of absorbing hydrogen. In the structure, there are three types of voids, namely, 9e (site symmetry .2/m), 3b (site symmetry m) and 36i (site symmetry 1), in which hydrogen can be inserted, and the maximum total absorption capacity can reach 1.21 wt% H₂. Electrochemical hydrogenation shows that the phase absorbs 1.03% of H₂, which indicates partial filling of the voids with H atoms.