解析-数值积分混合的Fock-Like矩阵算法及在BDF程序中的应用

In the integral-direct quantum chemistry methods such as Hartree-Fock, DFT, TDDFT and MCSCF, the key step is to calculate the Fock-like matrix, in which the two-electron repulsive integrals (ERIs) and first and second order derivatives of ERIs are computed and contracted with the density matrix on-the-fly to construct the Coulomb (J) and exchange (K) operators. To improve the computational efficiency of Beijing Density Functional Package (BDF), 1) This project will develop a new analytic-numerical hybrid integration algorithm to calculate the Fock-Like matrix. In this approach, the analytic integral of the Gaussian basis functions and the numeric integration scheme based on the multipole expansion will be used to calculate the Coulomb operator while a semi-numerical integration scheme will be used to calculate the exchange operator. 2) The analytic-numerical hybrid integration approach will be extended to calculate the first and second order derivatives of HF, DFT, TDDFT and MCSCF, as well as molecular properties such as the nuclear magnetic resonance (NMR) and the dipole polarizability. 3) Some basic functions in BDF such as the molecular geometry optimization of the ground and excited states, the vibrational analyze，the transition dipole and QM/MM will be enhanced. To demonstrate the capability of the BDF package, the luminance mechanism of some typical metal-organic and organic conjugated molecules will be studied. These works will make BDF not only a more friendly theoretical and algorithmic research platform, but also a practical tool for the broader users.

双电子排斥积分及其一阶、二阶导数与密度矩阵收缩构造Fock-Like矩阵是常用量子化学方法，如Hartree-Fock、DFT、TDDFT及MCSCF等的核心计算任务。本项目拟发展：1）高效的高斯解析积分—多极展开数值积分混合驱动的Fock-Like矩阵算法，利用解析积分和多极展开计算库伦算符，半数值方法计算交换算符，用于HF、DFT、TDDFT和MCSCF等计算；2）高斯解析积分和多极展开数值积分混合的一阶、二阶导数算法，用于能量梯度、二阶导数及NMR、极化率等分子性质计算；3）以金属有机和有机共轭分子的发光机理为示范应用，完善BDF（Beijing Density Functional Package）的分子基态、激发态几何结构优化、频率分析、跃迁矩及QM/MM等计算功能。本项目的研究将提高BDF核心模块的计算效率，使BDF完成从量子化学理论与算法研究平台到更普遍的应用研究工具的跨越。

大多数量子化学方法，如Hartree-Fock，DFT，TDDFT等的主要瓶颈在于Fock-Like矩阵，即库伦和交换矩阵的计算。本项目在BDF(Beijing Density Functional)程序中发展了解析和多级展开库伦势(analytical and Multipole expension for Coulomb potential - aMECP)计算库伦矩阵，改进的半数值COSX方法(Chain-of-Sphere Exchange - COSX)计算交换矩阵。其中，aMPEC基于将库伦相互作用“分而治之”的思想，通过密度矩阵的分解，将与原子占据壳层的库伦相互作用用解析积分处理，其余的用半数值或纯数值方式计算，在保证计算精度的同时，提高了计算效率。对于COSX方法，我们改进了积分格点、积分格点对基组的筛选等计算细节。测试表明，aMEPC+COSX方法的精度对不同的基组是稳定的。对一个包含20个分子(Mole20)的测试集，Hartree-Fock能量的平均绝对偏差(Mean absolute error - MAE)大约为4μΗ/atom，其中aMPEC方法的精度约为2μΗ/atom，COSX方法精度约为4μΗ/atom。对化学反应能垒，分子异构化能计算误差在0.1Kcal/mol之内；对TDDFT激发能误差在0.005eV之内。aMPEC+COSX相比与精确的解析积分精确算法，在cc-pVTZ级别的基组下，计算Mole20分子DFT能量及梯度平均加速8.25和5.01倍，MPEC+COSX方法计算TDDFT能量平均加速20.63倍。此外，我们还基于紧束缚近似的半经验哈密顿，发展了加速SCF迭代收敛的算法。测试表明，新方法对传统的DIIS+Leve Shift难以收敛的体系可以加速40%。

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