计算数学
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计算数学  2019, Vol. 41 Issue (2): 113-125    DOI:
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类Hartree-Fock方程的数值方法
林霖
加州大学伯克利分校数学系, 劳伦斯伯克利国家实验室, 美国伯克利, 加利福尼亚 94720
NUMERICAL METHODS FOR HARTREE-FOCK-LIKE EQUATIONS
Lin Lin
Department of Mathematics, University of California, Berkeley, and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
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摘要 本文的主要目的是介绍近年来大基组下的类Hartree-Fock方程数值求解的一些进展.类Hartree-Fock方程出现在Hartree-Fock理论和含杂化泛函的Kohn-Sham密度泛函理论中,是电子结构理论中一类重要的方程.该方程在复杂的化学和材料体系的电子结构计算中有广泛地应用.由于计算代价的原因,类Hartree-Fock方程一般只被用在较小规模的量子体系(含几十到几百个电子)的计算.从数学角度上讲,类Hartree-Fock方程是一个非线性积分-微分方程组,其计算代价主要来自于积分算子的部分,也就是Fock交换算子.通过发展和结合自适应压缩交换算子方法(ACE),投影的C-DⅡS方法(PC-DⅡS)方法,以及插值可分密度近似方法(ISDF),我们大大降低了杂化泛函密度泛函理论的计算代价.以含1000个硅原子的体系为例,我们将平面波基组下的杂化泛函的计算代价降至接近不含Fock交换算子的半局域泛函计算的水平.同时,我们发现类Hartree-Fock方程的数学结构也为一类特征值问题的迭代求解提供了新的思路.
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关键词类Hartree-Fock方程   非线性特征值问题   积分-微分算子   量子化学   电子结构理论   密度泛函理论     
Abstract: The main goal of this paper is to introduce some recent developments of numerical methods for solving Hartree-Fock-like equations in the context of large basis sets. HartreeFock-like equations are an important type of equations in electronic structure theory. They appear in the Hartree-Fock theory, as well as the Kohn-Sham density functional theory with hybrid exchange-correlation functionals, and are widely used in electronic structure calculations of complex chemical and materials systems. Because of its high computational cost, Hartree-Fock-like equations are typically only used in systems consisting of tens to hundreds of electrons. From a mathematical perspective, Hartree-Fock-like equations are a system of nonlinear integro-differential equations. The computational cost is mainly due to the integral operator part, namely the Fock exchange operator. Through the development of the adaptive compression method (ACE), the projected commutator-direct inversion in the iterative subspace (PC-DⅡS) method, and the interpolative separable density fitting (ISDF) method, we demonstrate that the cost of Kohn-Sham density functional theory calculations with hybrid functionals can be significantly reduced. Using a silicon system with 1000 atoms for example, we have reduced the cost of hybrid functional calculations with a planewave basis set to a level that is close to the cost of semi-local functional calculations, which do not involve the Fock exchange operator. Meanwhile, we find that the structure of HartreeFock-like equations provides new insights for the iterative solution of one type of eigenvalue problems.
Key wordsHartree-Fock-like equation   nonlinear eigenvalue problem   integro-differential operator   quantum chemistry   electronic structure theory   Density functional theory   
收稿日期: 2019-02-28; 出版日期: 2019-05-18
基金资助:

美国国家自然科学基金DMS-1652330,美国能源部DE-SC0017867资助项目.

作者简介: 林霖,加州大学伯克利分校数学系副教授、劳伦斯伯克利国家实验室计算科学部研究员.2007年和2011年分别在北京大学和普林斯顿大学获学士和博士学位;2011-2013年在劳伦斯伯克利国家实验室从事博士后研究工作.主要研究领域为数值分析、计算量子化学、计算材料科学、多尺度建模和并行计算等方面.曾获DOE Early Career Award (2017-2022)、NSF CAREER Award (2017-2022)、SIAM Computational Science and Engineering (CSE) Early Career Prize (2017)和Alfred P.Sloan fellowship (2015-2017)等.截止目前,在SCI期刊上发表学术论文70余篇.
引用本文:   
. 类Hartree-Fock方程的数值方法[J]. 计算数学, 2019, 41(2): 113-125.
. NUMERICAL METHODS FOR HARTREE-FOCK-LIKE EQUATIONS[J]. Mathematica Numerica Sinica, 2019, 41(2): 113-125.
 
[1] Hohenberg P, Kohn W. Inhomogeneous electron gas[J]. Phys. Rev., 1964, 136:B864-B871.
[2] Kohn W, Sham L. Self-consistent equations including exchange and correlation effects[J]. Phys. Rev., 1965, 140:A1133-A1138.
[3] Levy M. Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem[J]. Proc. Natl. Acad. Sci., 1979, 76:6062-6065.
[4] Lieb E H. Density functional for Coulomb systems[J]. Int J. Quantum Chem., 1983, 24:243.
[5] Ceperley D M, Alder B J. Ground state of the electron gas by a stochastic method[J]. Phys. Rev. Lett., 1980, 45:566-569.
[6] Perdew J P, Zunger A. Self-interaction correction to density-functional approximations for manyelectron systems[J]. Phys. Rev. B, 1981, 23:5048-5079.
[7] Lee C, Yang W, Parr R G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[J]. Phys. Rev. B, 1988, 37:785-789.
[8] Becke A D. Density-functional exchange-energy approximation with correct asymptotic behavior[J]. Phys. Rev. A, 1988, 38:3098-3100.
[9] Perdew J P, Burke K, Ernzerhof M. Generalized gradient approximation made simple[J]. Phys. Rev. Lett., 1996, 77:3865-3868.
[10] Staroverov V N, Scuseria G E, Tao J, Perdew J P. Comparative assessment of a new nonempirical density functional:Molecules and hydrogen-bonded complexes[J]. J. Chem. Phys., 2003, 119:12129-12137.
[11] Sun J, Ruzsinszky A, Perdew J P. Strongly Constrained and Appropriately Normed Semilocal Density Functional[J]. Phys. Rev. Lett., 2015, 115:036402.
[12] Becke A D. Density functional thermochemistry. Ⅲ. The role of exact exchange[J]. J. Chem. Phys., 1993, 98:5648.
[13] Perdew J P, Ernzerhof M, Burke K. Rationale for mixing exact exchange with density functional approximations[J]. J. Chem. Phys., 1996, 105:9982-9985.
[14] Heyd J, Scuseria G E, Ernzerhof M. Hybrid functionals based on a screened Coulomb potential[J]. J. Chem. Phys., 2003, 118(18):8207-8215.
[15] Ren X, Rinke P, Joas C, Scheffler M. Random-phase approximation and its applications in computational chemistry and materials science[J]. J. Mater. Sci., 2012, 47:7447-7471.
[16] Zhang I Y, Rinke P, Scheffler M. Wave-function inspired density functional applied to the H2/H2+ challenge[J]. New J. Phys., 2016, 18:073026.
[17] Seidl M, Gori-Giorgi P, Savin A. Strictly correlated electrons in density-functional theory:A general formulation with applications to spherical densities[J]. Phys. Rev. A, 2007, 75:042511.
[18] Malet F, Gori-Giorgi P. Strong Correlation in Kohn-Sham Density Functional Theory[J]. Phys. Rev. Lett., 2012, 109:246402.
[19] Dion M, Rydberg H, Schröder E, Langreth D C, Lundqvist B I. Van der Waals density functional for general geometries[J]. Physical Rev. Lett., 2004, 92:246401.
[20] Tkatchenko A, Scheffler M. Accurate molecular van der Waals interactions from ground-state electron density and free-atom reference data[J]. Phys. Rev. Lett., 2009, 102:073005.
[21] Perdew J P, Schmidt K. Jacob's ladder of density functional approximations for the exchangecorrelation energy[C]. In AIP Conference Proceedings, pages 1-20, 2001.
[22] Dunning T H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen[J]. J. Chem. Phys., 1989, 90:1007-1023.
[23] Soler J M, Artacho E, Gale J D, García A, Junquera J, Ordejón P, Sánchez-Portal D. The SIESTA method for ab initio order-N materials simulation[J]. J. Phys.:Condens. Matter, 2002, 14:2745-2779.
[24] Blum V, Gehrke R, Hanke F, Havu P, Havu V, Ren X, Reuter K, Scheffler M. Ab initio molecular simulations with numeric atom-centered orbitals[J]. Comput. Phys. Commun., 2009, 180:2175-2196.
[25] Kresse G, Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[J]. Phys. Rev. B, 1996, 54:11169-11186.
[26] Tsuchida E, Tsukada M. Electronic-structure calculations based on the finite-element method[J]. Phys. Rev. B, 1995, 52:5573-5578.
[27] Suryanarayana P, Gavani V, Blesgen T, Bhattacharya K, Ortiz M. Non-periodic finite-element formulation of Kohn-Sham density functional theory[J]. J. Mech. Phys. Solids, 2010, 58:258-280.
[28] Bao G, Hu G, Liu D. An h-adaptive finite element solver for the calculations of the electronic structures[J]. J. Comput. Phys., 2012, 231:4967-4979.
[29] Chen H, Dai X, Gong X, He L, Zhou A. Adaptive Finite Element Approximations for Kohn-Sham Models[J]. Multiscale Model. Simul., 2014, 12:1828-1869.
[30] Genovese L, Neelov A, Goedecker S, Deutsch T, Ghasemi S A, Willand A, Caliste D, Zilberberg O, Rayson M, Bergman A, Schneider R. Daubechies wavelets as a basis set for density functional pseudopotential calculations[J]. J. Chem. Phys., 2008, 129:014109.
[31] Payne M C, Teter M P, Allen D C, Arias T A, Joannopoulos J D. Iterative minimization techniques for ab initio total energy calculation:molecular dynamics and conjugate gradients[J]. Rev. Mod. Phys., 1992, 64:1045-1097.
[32] Duchemin I, Gygi F. A scalable and accurate algorithm for the computation of Hartree-Fock exchange[J]. Comput. Phys. Commun., 2010, 181:855-860.
[33] Valiev M, Bylaska E J, Govind N, Kowalski K, Straatsma T P, Dam H J J V, Wang D, Nieplocha J, Apra E, Windus T L, Jong W D. NWChem:a comprehensive and scalable open-source solution for large scale molecular simulations[J]. Comput. Phys. Commun., 2010, 181:1477-1489.
[34] Kohn W. Density Functional and Density Matrix Method Scaling Linearly with the Number of Atoms[J]. Phys. Rev. Lett., 1996, 76:3168-3171.
[35] Goedecker S. Linear scaling electronic structure methods[J]. Rev. Mod. Phys., 1999, 71:1085-1123.
[36] Wu X, Selloni A, Car R. Order-N implementation of exact exchange in extended insulating systems[J]. Phys. Rev. B, 2009, 79(8):085102.
[37] Dawson W, Gygi F. Performance and Accuracy of Recursive Subspace Bisection for Hybrid DFT Calculations in Inhomogeneous Systems[J]. J. Chem. Theory Comput., 2015, 11:4655-4663.
[38] Hu W, Lin L, Yang C. DGDFT:A massively parallel method for large scale density functional theory calculations[J]. J. Chem. Phys., 2015, 143:124110.
[39] Davidson E. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices[J]. J. Comput. Phys., 1975, 17:87-94.
[40] Lin L. Adaptively Compressed Exchange Operator[J]. J. Chem. Theory Comput., 2016, 12:2242.
[41] Giannozzi P, Baroni S, Bonini N, Calandra M, Car R, Cavazzoni C, Ceresoli D, Chiarotti G L, Cococcioni M, Dabo I, Corso A D, de Gironcoli S, Fabris S, Fratesi G, Gebauer R, Gerstmann U, Gougoussis C, Kokalj A, Lazzeri M, Martin-Samos L, Marzari N, Mauri F, Mazzarello R, Paolini S, Pasquarello A, Paulatto L, Sbraccia C, Scandolo S, Sclauzero G, Seitsonen A P, Smogunov A, Umari P, Wentzcovitch R M. QUANTUM ESPRESSO:a modular and open-source software project for quantum simulations of materials[J]. J. Phys.:Condens. Matter, 2009, 21:395502-395520.
[42] Carnimeo I, Baroni S, Giannozzi P. Fast hybrid density-functional computations using plane-wave basis sets[J]. Electronic Structure, 2018.
[43] Jia W, Fu J, Cao Z, Wang L, Chi X, Gao W, Wang L W. Fast plane wave density functional theory molecular dynamics calculations on multi-GPU machines[J]. J. Comput. Phys., 2013, 251:102-115.
[44] Lin L, Lindsey M. Convergence of adaptive compression methods for Hartree-Fock-like equations[J]. Commun. Pure Appl. Math., 2019, 72:0451.
[45] Pulay P. Improved SCF Convergence Acceleration[J]. J. Comput. Chem., 1982, 3:54-69.
[46] Saad Y, Schultz M H. GMRES:A generalized minimal residual algorithm for solving nonsymmetric linear systems[J]. SIAM J. Sci. Stat. Comput., 1986, 7:856-869.
[47] Fang H R, Saad Y. Two classes of multisecant methods for nonlinear acceleration[J]. Numer. Linear Algebra Appl., 2009, 16:197-221.
[48] Hu W, Lin L, Yang C. Projected Commutator DⅡS Method for Accelerating Hybrid Functional Electronic Structure Calculations[J]. J. Chem. Theory Comput., 2017, 13:5458.
[49] Lu J, Sogge C D, Steinerberger S. Approximating pointwise products of Laplacian eigenfunctions[M], 2018. preprint, arXiv:1811.10447.
[50] Aquilante F, Pedersen T B, Lindh R. Low-cost evaluation of the exchange Fock matrix from Cholesky and density fitting representations of the electron repulsion integrals[J]. J. Chem. Phys., 2007, 126:194106.
[51] Ren X, Rinke P, Blum V, Wieferink J, Tkatchenko A, Sanfilippo A, Reuter K, Scheffler M. Resolution-of-identity approach to Hartree-Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions[J]. New J. Phys., 2012, 14:053020.
[52] Lu J, Ying L. Compression of the electron repulsion integral tensor in tensor hypercontraction format with cubic scaling cost[J]. J. Comput. Phys., 2015, 302:329.
[53] Lu J, Ying L. Fast algorithm for periodic density fitting for Bloch waves[J]. Ann. Math. Sci. Appl., 2016, 1:321-339.
[54] Hu W, Lin L, Yang C. Interpolative separable density fitting decomposition for accelerating hybrid density functional calculations with applications to defects in silicon[J]. J. Chem. Theory Comput., 2017, 13:5420.
[55] Dong K, Hu W, Lin L. Interpolative separable density fitting through centroidal Voronoi tessellation with applications to hybrid functional electronic structure calculations[J]. J. Chem. Theory Comput., 2018, 14:1311.
[56] Ratcliff L E, Degomme A, Flores-Livas J A, Goedecker S, Genovese L. Affordable and accurate large-scale hybrid-functional calculations on GPU-accelerated supercomputers[J]. J. Phys.:Condens. Matter, 2018, 30:095901.
[57] Hu J, Jiang B, Lin L, Wen Z, Yuan Y. Structured Quasi-Newton Methods for Optimization with Orthogonality Constraints[J]. arXiv:1809.00452, 2018.
[1] 曹阳, 戴华. 非线性特征值问题的二次近似方法[J]. 计算数学, 2014, 36(4): 381-392.

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