Comparison study of several numerical integration schemes used in calculations of density functional theory

Several numerical integration schemes for the evaluation of matrix elements in density functional theory calculations have been studied and compared by computational practice. The best scheme was found to be the combination of the atomic partition function proposed by Becke with the scaled generalized Gauss-Laguerre quadrature formula for radial integration suggested by Yang, which achieve the highest convergence rate to the numerical integration. With the same number of integration points, the accuracy of the calculated results by this scheme is higher by 1 to 2 orders of magnitudes than that by other schemes. The reason for achieving higher accuracy by this scheme has been proposed preliminarily.     

Radial quadrature for multiexponential integrands

We introduce a Gaussian quadrature, based on the polynomials that are orthogonal with respect to the weight function ln(2)x on the interval [0, 1], which is suitable for the evaluation of radial integrals. The quadrature is exact if the non-Jacobian part of the integrand is a linear combination of a geometric sequence of exponential functions. We find that the new scheme is a useful alternative to existing approaches, particularly for integrands that exhibit multiexponential behavior.     

An accurate numerical multicenter integration for molecular orbital theory

A powerful and accurate numerical three‐dimensional integration scheme was developed especially for molecular orbital calculations. A multicenter integral is decomposed into the sum of single‐center integrals using nuclear weight functions and calculated using Gaussian quadrature rules. The decomposed single‐center integrands show strong anisotropy. With a careful selection of the Gaussian quadrature rule according to the anisotropy, it is possible to obtain an accuracy of 13 digits with a small number of integration points for the overlap integrals, normalization integrals, and molecular integrals for the hydrogen molecule. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 509–523, 1999     

Study of chemical bonding in H2 and HF molecules: Wave function and density functional theory (DFT) parameters approach

Balint Kurti's Fourier grid Hamiltonian method is employed to obtain the molecular wave function and equilibrium bond length for H₂ and HF molecules. The density functional theory parameter, namely, the chemical hardness (η) value, was determined for some diatomic hydride molecules using this wave function and the results are found to be in good agreement with the values obtained from the ab initio HF–SCF method. A new formula for chemical hardness (η=1/2Dr, where D is the proportionality constant and r is the internuclear distance) is introduced in binding energy and change of hardness equations to determine the chemical hardness and chemical potential values for different bond lengths. The binding energy and change of hardness values are calculated for H₂, H, H, HF, HF⁺, and HF⁻ molecules and the bond stability is discussed. Finally, the concept of an atom in a molecule is examined in the context of DFT parameters and comparison is made between an atom in a molecule and the isolated atom. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 662–669, 2000     

分析实验数据处理中的测量不确定度

在现行的分析标准中,大多未对测量不确定度提出相应的要求．本文通过对测量不确定度概念的探讨及对几种化验数据的测量不确定度分析,对检测数据出具时的规范化、标准化有重要意义．     

Quantum-Chemical Calculation of Matrix Elements in a Basis of Functions with Polynomial Tails

A numerical integration method is suggested for calculating Hamiltonian matrix elements in a basis of functions with polynomial tails with allowance for discontinuities of higher-order derivatives of the basis function within the domain of integration. The method is tested by calculating matrix elements for a copper crystal. The results for the overlap matrix elements are presented demonstrating efficiency of the method.     

An Alternate [2×2] Grid Constructed Around TiO4N2 Units

The formation of a tetranuclear self-assembled species constructed around a TiO₄N₂ motif is reported. This aggregate is generated from Ti(OⁱPr)₄, 2,2’-bipyrimidine (bpym) and a bis-biphenol strand (L²H₄) where two 2,2’-biphenol units are connected with a biphenyl spacer. The solid-state structure of the [Ti₄(L²)₄(bpym)₄] architecture reveals the formation of an unprecedented chiral alternate [2×2] grid. In addition to the structural characterization of the [Ti₄(L²)₄(bpym)₄] architecture, geometry optimisation on various possible isomeric tetrameric assemblies ([2×2] grid, alternate [2×2] grid, circular helicate or cyclic hemihelicate) is performed using DFT calculations. These results confirm the higher stability of the alternate [2×2] grid over the other possible tetranuclear isomers and allow examining the replacement of the bpym ligands by two novel diimine chelates within the tetranuclear assembly (2,2’-bipyridine=bipy and 2,2’-bipyrazine=bipyraz). From this initial theoretical investigation, the competition between these three nitrogen ligands in the course of the self-assembly process is next evaluated. Overall, this investigation shows that the exclusive formation of the alternate [2×2] grid is driven by CH⋅⋅⋅N interactions.     

离子选择性电极标准加入法的测量误差研究

研究了离子选择性电极标准加入法的测量误差与标准溶液加入量的关系。从理论上导出了离子选择性电极一次标准加入法测量结果的相对误差计算公式。当加入标准溶液使试液增加的浓度等于试液原来的浓度,测量结果的相对误差最小。     

红外光谱定量分析中标准添加法的新计算方法

对红外光谱定量分析中的标准添加法进行了误差分析,提出了一种新的数学处理方法,使标准添加法的误差大为降低,准确度得到提高。用简单的苯－氯仿二元体系对此法进行了验证。