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     

分子轨道成分的计算

量子化学文献中经常涉及到“分子轨道成分”概念, 然而相应的计算方法却没有普遍得到重视, 甚至存在严重误区. 专门对分子轨道中基函数的成分、原子轨道的成分以及原子的成分这些基本概念和计算方法进行了详细讨论, 通过实例分析比较了不同方法间的差异, 同时指出了计算和分析时需要注意的问题, 给出了在方法选择上的建议.     

On the importance of intramolecular hydrogen bond cooperativity in d‐glucose – an NMR and QTAIM approach

The idea that hydrogen bond cooperativity is responsible for the structure and reactivity of carbohydrates is examined. Density functional theory and gauge‐including atomic orbital calculations on the known conformers of the α and β anomers of d ‐glucopyranose in the gas phase are used to compute proton NMR chemical shifts and interatomic distances, which are taken as criteria for probing intramolecular interactions. Atom–atom interaction energies are calculated by the interacting quantum atoms approach in the framework of the quantum theory of atoms in molecules. Association of OH1 in the counterclockwise conformers with a strong acceptor, pyridine, is accompanied by cooperative participation from OH2, but there is no significant change in the bonding of the two following 1,2‐diol motifs. The OH6 ^... O5 (G?g+/cc/t and G+g?/cc/t conformers) or OH6 ^... O4 (Tg+/cc/t conformer) distance is reduced, and the OH6 proton is slightly deshielded. In the latter case, this shortening and the associated increase in the OH6–O4 interaction energy may be interpreted as a small cooperative effect, but intermolecular interaction energies are practically the same for all three conformers. In most of the pyridine complexes, one ortho proton interacts with the endocyclic oxygen O5. Analogous results are obtained when the clockwise conformer, G?g+/cl/g?, detected for the α anomer, and a hypothetical conformer, Tt/cl/g?, are complexed with pyridine through OH6. Generally, the cooperative effect does not go beyond the first two OH groups of a chain. Copyright © 2017 John Wiley & Sons, Ltd.     

Overlap Integrals Between Irregular Solid Harmonics and STOs Via the Fourier Transform Methods

In this paper, a unified analytical and numerical treatment of overlap integrals between Slater type orbitals (STOs) and irregular Solid Harmonics (ISH) with different screening parameters is presented via the Fourier transform method. Fourier transform of STOs is probably the simplest to express of overlap integrals. Consequently, it is relatively easy to express the Fourier integral representations of the overlap integrals as finite sums and infinite series of STOs, ISHs, Gegenbauer, and Gaunt coefficients. The another mathematical tools except for Fourier transform have used partial-fraction decomposition and Taylor expansions of rational functions. Our approach leads to considerable simplification of the derivation of the previously known analytical representations for the overlap integrals between STOs and ISHs with different screening parameters. These overlap integrals have also been calculated for extremely large quantum numbers using Gegenbauer, Clebsch-Gordan and Binomial coefficients. The accuracy of the numerical results is quite high for the quantum numbers of Slater functions, irregular solid harmonic functions and for arbitrary values of internuclear distances and screening parameters of atomic orbitals.     

Calculation of molecular integrals for systems confined by hard spherical walls: Use of the single‐center expansion of floating spherical gaussians

Assuming a gaussian basis set representation of atomic and molecular wave functions, the single‐center expansion of off‐centered spherical gaussian orbitals is exploited to calculate the one and two‐electron integrals for multielectronic atoms and molecules confined within hard spherical walls. As a validating test, the ground‐state energy of a helium atom positioned off‐center in a spherical box is calculated by applying the simplest form of the floating spherical gaussian orbital (FSGO) scheme, i.e., the use of a primitive basis set consisting of a single FSGO per electron pair. Comparison with corresponding recent accurate calculations gives supporting evidence of the adequacy of the method for its application to more elaborate gaussian‐type basis set representations for confined atoms and molecules. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 271–278, 2001     

Atomic orbital basis sets for use with effective core potentials

Basis sets developed for use with effective core potentials describe pseudo‐orbitals rather than orbitals. The primitive Gaussian functions and the contraction coefficients in the basis set must therefore both describe the valence region effectively and allow the pseudo‐orbital to be small in the core region. The latter is particularly difficult using 1s primitive functions, which have their maxima at the nucleus. Several methods of choosing contraction coefficients are tried, and it is found that natural orbitals give the best results. The number and optimization of primitive functions are done following Dunning's correlation‐consistent procedure. Optimization of orbital exponents for larger atoms frequently results in coalescence of adjacent exponents; use of orbitals with higher principal quantum number is one alternative. Actinide atoms or ions provide the most difficult cases in that basis sets must be optimized for valence shells of different radial size simultaneously considering correlation energy and spin‐orbit energy. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 516–520, 2000     

Use of Cartesian coordinates in evaluation of multicenter multielectron integrals over Slater type orbitals and their derivatives

Using addition theorems for interaction potentials and Slater type orbitals (STOs) obtained by the author, and the Cartesian expressions through the binomial coefficients for complex and real regular solid spherical harmonics (RSSH) and their derivatives presented in this study, the series expansion formulas for multicenter multielectron integrals of arbitrary Coulomb and Yukawa like central and noncentral interaction potentials and their first and second derivatives in Cartesian coordinates were established. These relations are useful for the study of electronic structure and electron-nuclei interaction properties of atoms, molecules, and solids by Hartree–Fock–Roothaan and correlated theories. The formulas obtained are valid for arbitrary principal quantum numbers, screening constants and locations of STOs.     

Evaluation of two-center overlap integrals using slater type orbitals in terms of bessel type orbitals

Using the definition of STOs in terms of BTOs, we have presented analytical formula for two-center overlap integrals. The obtained formula contains generalized binomial coefficients and Mulliken integrals A_k and B_k. Taking into account the recent advances on the efficient calculation of Mulliken integrals (Harris, Int. J. Quantum Chem., 100 (2004) 142), we have obtained many more satisfactory results for two-center overlap integrals, for arbitrary quantum numbers, scaling parameters, and location of atomic orbitals.PACS No: 31.15.+qAMS Subject Classification: 81V55, 81–08     

Molecular integrals for Gaussian and exponential‐type functions: Shift operators

Basis functions with arbitrary quantum numbers can be attained from those with the lowest numbers by applying shift operators. We derive the general expressions and the recurrence relations of these operators for Cartesian basis sets with Gaussian and exponential radial factors. In correspondence, the expressions of molecular integrals involving functions with arbitrary quantum numbers can be obtained by applying these operators on the integrals with the lowest quantum numbers. Since the original form of the shift operators is not appropriate to deal with integrals, we give their representation in terms of derivatives with respect to the parameters on which these integrals explicitly depend. Moreover, we translate the recurrence relations to the new representation and, finally, we analyze the general expressions ot the molecular integrals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 137–145, 2000