SG-0: a small standard grid for DFT quadrature on large systems

We report the development of a new standard quadrature grid for DFT calculations. Standard Grid 0 (SG-0) is designed to be approximately half as large as, and to provide approximately half the accuracy of, the established SG-1 grid. It is based on MultiExp and Lebedev quadrature for radial and angular coordinates, respectively. We find that SG-0 is typically 50% faster than SG-1 for energy, gradient, and hessian calculations for the exchange-correlation energy. This leads to a 35-38% speedup in the total gradient and hessian computations, and we particularly recommend its use for preliminary calculations on moderately large biochemical systems. It has been implemented as the default grid for DFT calculations in the Q-Chem 3.0 package.     

最大熵法分析寡聚核苷酸链内碰撞的荧光相关光谱

准确地由荧光相关光谱(FCS)的实验数据提取动力学信息一直是一个挑战. 本文对比了三种主要的方法: 依赖于模型的多指数函数法, 经验的拓展指数函数法和不依赖于模型的最大熵法. 多指数函数法的物理意义直接但在复杂体系中难以应用和解释. 拓展指数函数法简单易行但其物理意义含混不清. 最大熵法不依赖于具体的物理模型但拟合结果对实验噪音很敏感. 经研究我们发现一个好的选择是将最大熵法和多指数函数法结合在一起使用. 对寡聚核苷酸链内碰撞荧光相关光谱的研究发现, 在单链DNA中可以形成碱基对时, 有两个并行的链内碰撞反应. 以前的拓展指数函数法分析则不能提供这样的信息. 我们建议在荧光相关光谱研究中审慎地使用最大熵法.     

Density functional treatment of water–carbon dioxide van der waals complex

LCGTO–LSD and LCGTO–NLSD methods have been tested for the study of water–carbon dioxide weakly bound binary complex. Different local and nonlocal exchange-correlation energy functionals and many grid radial points have been used. Results show that both nonlocal corrections and a large number of radial points in the grid are mandatory for well reproducing the experimental data. © 1994 John Wiley & Sons, Inc.     

An application of double exponential formula to radial quadrature grid in density functional calculation

We report an application of the double exponential formula to the numerical integration of the radial electron distribution function for atomic and diatomic molecular systems with a quadrature grid. Three types of mapping transformation in the double exponential formula are introduced into the radial quadrature scheme to generate new radial grids. The double exponential grids are examined for the electron-counting integrals of He, Ne, Ar, and Kr atoms which include occupied orbitals up to the 4p shell. The performance of radial grid is compared for the double exponential formula and the formulas proposed in earlier studies. We mainly focus our attention on the behavior of accuracy by the quadrature estimation for each radial grid with varying the mapping parameter and the number of grid points. The convergence behavior of the radial grids with high accuracy for atomic system are also examined for the electron-counting integrals of LiH, NaH, KH, Li₂, Na₂, K₂, HF, HCl, HBr, F₂, Cl₂, Br₂, LiF, NaCl, KBr, [ScH]⁺, [MnH]⁺, and [CuH]⁺ molecules. The results reveal that fast convergence of the integrated values to the exact value is achieved by applying the double exponential formula. It is demonstrated that the double exponential grids show similar or higher accuracies than the other grids particularly for the Kr atom, Br₂ molecule, alkali metal hydrides, alkali metal halogenides, and transition metal hydride cations, suggesting that the double exponential transformations have potential ability to improve the reliability and efficiency of the numerical integration for energy functionals.     

Molecular integrals by numerical quadrature. I. Radial integration

 This article presents a numerical quadrature intended primarily for evaluating integrals in quantum chemistry programs based on molecular orbital theory, in particular density functional methods. Typically, many integrals must be computed. They are divided up into different classes, on the basis of the required accuracy and spatial extent. Ideally, each batch should be integrated using the minimal set of integration points that at the same time guarantees the required precision. Currently used quadrature schemes are far from optimal in this sense, and we are now developing new algorithms. They are designed to be flexible, such that given the range of functions to be integrated, and the required precision, the integration is performed as economically as possible with error bounds within specification. A standard approach is to partition space into a set of regions, where each region is integrated using a spherically polar grid. This article presents a radial quadrature which allows error control, uniform error distribution and uniform error reduction with increased number of radial grid points. A relative error less than 10⁻¹⁴ for all s-type Gaussian integrands with an exponent range of 14 orders of magnitude is achieved with about 200 grid points. Higher angular l quantum numbers, lower precision or narrower exponent ranges require fewer points. The quadrature also allows controlled pruning of the angular grid in the vicinity of the nuclei. Received: 30 August 2000 / Accepted: 21 December 2000 / Published online: 3 April 2001     

Numerical integration of atomic electron density with double exponential formula for density functional calculation

In the preceding study, we reported an application of the double exponential formula to the radial quadrature grid for numerical integration of the radial electron distribution function. Three-type new radial grids with the double exponential transformation were introduced. The performance of radial grids was compared between the double exponential grids and the grids proposed in earlier studies by applying to the electron-counting integrals of noble gas atoms and diatomic molecules including alkali metals, halogens, and transition metals. It was confirmed that the change in accuracy of the quadrature approximation depending on atomic or molecular species is not significant for the double exponential integration schemes rather than the other integration schemes. In the present study, we further investigate the accuracy of the double exponential formula for the electron-counting integrals of all the atoms from H to Kr in the periodic table to elucidate the stable performance of the double exponential radial grids. The electron densities of the atoms are calculated with the Gauss-type orbital basis functions at the B3LYP level. The quadrature accuracy and convergence behavior of numerical integration are compared among the double exponential formula and the formulas proposed by Treutler et al. and by Mura et al. The results reveal that the double exponential radial grids remarkably improve the convergence rate toward high accuracy compared with the previous radial grids, particularly for heavy elements in the 4th period, without fine tuning of the radial grids for each atom.     

Picoliter water contact angle measurement on polymers

Water contact angle measurement is the most common method for determining a material's wettability, and the sessile drop approach is the most frequently used. However, the method is generally limited to macroscopic measurements because the base diameter of the droplet is usually greater than 1 mm. Here we report for the first time on a dosing system to dispense smaller individual droplets with control of the position and investigate whether water contact angles determined from picoliter volume water droplets are comparable with those obtained from the conventional microliter volume water droplets. This investigation was conducted on a group of commonly used polymers. To demonstrate the higher spatial resolution of wettability that can be achieved using picoliter volume water droplets, the wettability of a radial plasma polymer gradient was mapped using a 250 microm interval grid.     

基于高光谱和化学计量学方法的除草剂残留分类识别研究

本研究基于高光谱技术和化学计量学方法,对薄荷叶上的异丙甲草胺、烟嘧(莠去津、敌草胺和砜嘧(精喹4类除草剂残留进行种类判别.高光谱成像光谱范围为450nm～950nm的可见-近红外区域.为降低噪音对光谱数据的干扰、提升判别精度,采用SG平滑和多元散射校正对高光谱曲线进行处理.利用主成分分析算法(PCA)对原始数据进行降维...     

A divide and conquer real-space approach for all-electron molecular electrostatic potentials and interaction energies

A computational scheme to perform accurate numerical calculations of electrostatic potentials and interaction energies for molecular systems has been developed and implemented. Molecular electron and energy densities are divided into overlapping atom-centered atomic contributions and a three-dimensional molecular remainder. The steep nuclear cusps are included in the atom-centered functions making the three-dimensional remainder smooth enough to be accurately represented with a tractable amount of grid points. The one-dimensional radial functions of the atom-centered contributions as well as the three-dimensional remainder are expanded using finite element functions. The electrostatic potential is calculated by integrating the Coulomb potential for each separate density contribution, using our tensorial finite element method for the three-dimensional remainder. We also provide algorithms to compute accurate electron-electron and electron-nuclear interactions numerically using the proposed partitioning. The methods have been tested on all-electron densities of 18 reasonable large molecules containing elements up to Zn. The accuracy of the calculated Coulomb interaction energies is in the range of 10(-3) to 10(-6) E(h) when using an equidistant grid with a step length of 0.05 a(0).     

Direct deperturbation analysis of the A 2Pi approximately B 2Sigma+ complex of 7,6LiAr isotopomers

Direct deperturbation analysis of the highly accurate experimental rovibronic term values of the A (2)Pi approximately B (2)Sigma(+) complex of LiAr [R. Bruhl and D. Zimmermann, J. Chem. Phys. 114, 3035 (2001)] has been performed in the framework of inverted close-coupling approach implicitly adjusted to the unified treatment of the overall A approximately B coupling effect without reducing the rovibrational dimensionality. The nonlinear fitting procedure was supported by the ab initio calculations on the spin-orbit and angular coupling matrix elements between the lowest X (2)Sigma(+), A (2)Pi, and B (2)Sigma(+) states. The analytical grid mapping based on the reduced variable representation of the radial coordinate r was used to improve the efficiency of the solution of the close-coupling radial equations near the dissociation limit. The mutual A approximately X perturbation effect on the A (2)Pi term values and spin-rotation splitting of the ground state were evaluated for both (7,6)LiAr isotopomers. The resulting empirical potential-energy curves for the adiabatic A (2)Pi and B (2)Sigma(+) states, along with the refined r-dependent nonadiabatic matrix elements, reproduce the total rovibronic structure of the (7)LiAr complex with the standard deviation of 0.003 cm(-1). The mass invariance of the deperturbed electronic parameters was confirmed by the calculation of the rovibronic term values of the (6)LiAr isotopomer which coincided with their experimental counterparts within 0.004 cm(-1).     

An efficient approach to calculation of zero-flux atomic surfaces and generation of atomic integration data

A new robust method for variational determination of atomic zero-flux surfaces is presented. The zero-flux surface sheets are expressed in terms of variational trial functions in prolate spheroidal coordinates. The trial functions are optimized with a Newton procedure to satisfy the zero-flux condition on a grid. The data required for radial integrations are generated by an adaptive quadrature procedure that employs model electron densities and utilizes an original third-order algorithm for linear search. Results of test calculations involving variational determination of atomic surfaces are presented for a representative set of 20 molecules. The new approach is both less time consuming and substantially more accurate than the previously published algorithms. © 1995 John Wiley & Sons, Inc.     

A Laser Reduced Graphene Oxide Grid Electrode for the Voltammetric Determination of Carbaryl

Laser-reduced graphene oxide (LRGO) on a polyethylene terephthalate (PET) substrate was prepared in one step to obtain the LRGO grid electrode for sensitive carbaryl determination. The grid form results in a grid distribution of different electrochemically active zones affecting the electroactive substance diffusion towards the electrode surface and increasing the electrochemical sensitivity for carbaryl determination. Carbaryl is electrochemically irreversibly oxidized at the secondary amine moiety of the molecule with the loss of one proton and one electron in the pH range from 5 to 7 by linear scan voltammetry (LSV) on the LRGO grid electrode with a scan rate of 300 mV/s. Some interference of the juice matrix molecules does not significantly affect the LSV oxidation current of carbaryl on the LRGO grid electrode after adsorptive accumulation without applied potential. The LRGO grid electrode can be used for LSV determination of carbaryl in fruit juices in the concentration range from 0.25 to 128 mg/L with LOD of 0.1 mg/L. The fabrication of the LRGO grid electrode opens up possibilities for further inexpensive monitoring of carbaryl in other fruit juices and fruits     

Numerical solution of Dalgarno-Lewis equations by a mapped Fourier grid method

Inhomogeneous radial differential equations emerging in applications of standard perturbation theory are numerically solved by a novel approach making use of Fourier grid methods in conjunction with a simple mapping scheme. The proposed algorithm is applied along the lines of the Dalgarno-Lewis method [Proc. R. Soc. London, Ser. A 223, 70 (1955)] to the calculation of the static dipole polarizabilities and hyperpolarizabilities of 1s, 2s, and 2p states of hydrogen atom and their frequency dependent dynamic dipole polarizabilities. The high efficiency and accuracy of the algorithm are demonstrated for the above test cases, where exact values are available. Then, the frequency dependent dipole polarizability of the ground state of lithium atom is computed by a variationally stable method combining an effective local potential approach with a second-order energy correction. The obtained results are in perfect agreement with other elaborate theoretical approaches.     

Structure of liquid water at ambient temperature from ab initio molecular dynamics performed in the complete basis set limit

Structural properties of liquid water at ambient temperature were studied using Car-Parrinello [Phys. Rev. Lett. 55, 2471 (1985)] ab initio molecular dynamics (CPAIMD) simulations combined with the Kohn-Sham (KS) density functional theory and the BLYP exchange-correlation functional for the electronic structure. Unlike other recent work on the same subject, where plane-wave (PW) or hybrid Gaussian/plane-wave basis sets were employed, in the present paper, a discrete variable representation (DVR) basis set is used to expand the KS orbitals, so that with the real-space grid adapted in the present work, the properties of liquid water could be obtained very near the complete basis set limit. Structural properties of liquid water were extracted from a 30 ps CPAIMD-BLYP/DVR trajectory at 300 K. The radial distribution functions (RDFs), spatial distribution functions, and hydrogen bond geometry obtained from the CPAIMD-BLYP/DVR simulation are generally in good agreement with the most up to date experimental measurements. Compared to recent ab initio MD simulations based on PW basis sets, less significant overstructuring was found in the RDFs and the distributions of hydrogen bond angles, suggesting that previous plane-wave and Gaussian basis set calculations have exaggerated the tendency toward overstructuring.     

Modelling the thermal behaviour of the low-thermal mass liquid chromatography system

We report upon the experimental investigation of the heat transfer in low thermal mass LC (LTMLC) systems, used under temperature gradient conditions. The influence of the temperature ramp, the capillary dimensions, the material selection and the chromatographic conditions on the radial temperature gradients formed when applying a temperature ramp were investigated by a numerical model and verified with experimental temperature measurements. It was found that the radial temperature gradients scale linearly with the heating rate, quadratically with the radius of the capillary and inversely to the thermal diffusivity. Because of the thermal radial gradients in the liquid zone inside the capillary lead to radial viscosity and velocity gradients, they form an additional source of dispersion for the solutes. For a temperature ramp of 1 K/s and a strong temperature dependence of the retention of small molecules, the model predicts that narrow-bore columns (i.d. 2.1 mm) can be used. For a temperature ramp of 10 K/s, the maximal inner diameter is of the order of 1 mm before a substantial increase in dispersion occurs.     

Moving boundary truncated grid method: Multidimensional quantum dynamics

The moving boundary truncated grid (TG) method is used to study wave packet dynamics of multidimensional quantum systems. As time evolves, appropriate Eulerian grid points required for propagating a wave packet are activated and deactivated with no advance information about the dynamics. This method is applied to the Henon-Heiles potential and wave packet barrier scattering in two, three, and four dimensions. Computational results demonstrate that the TG method not only leads to a great reduction in the number of grid points needed to perform accurate calculations but also is computationally more efficient than the full grid calculations.     

Wavepacket dynamics on dynamically adapting grids: application of the equidistribution principle

A moving grid approach to wavepacket dynamics is described that enables grid points to be used efficiently in regions where high resolution of the wavepacket is required. The grid movement is based on the principle of equidistribution and by using a grid smoothing technique the grid points trace a path that continuously adapt to reflect the dynamics of the wavepacket. The technique is robust and allows accurate computations to be obtained for long wavepacket propagation times. Results are presented for two systems: tunnelling dynamics in a double well potential and scattering of a wavepacket from a repulsive Eckart barrier.