[1] Adams R A. Sobolev Spaces, Academic Press, New York, 1975.[2] Anderson D G. Iterative procedures for nonlinear integral equations[J]. J. Assoc. Comput. Mach., 1965, 12:547-560.[3] Bai Z, Demmel J, Dongarra J, Ruhe A and van der Vorst H, editors. Templates for the Solution of Algebraic Eigenvalue Problems:A Practical Guide. SIAM, Philadelphia, 2000.[4] Bao G, Hu G and Liu D. Numerical solution of the Kohn-Sham equation by finite element methods with an adaptive mesh redistribution technique[J]. J. Sci. Comput., 2013, 55:372-391.[5] Beck T L. Real-space mesh techniques in density-function theory[J]. Rev. Mod. Phys., 2000, 72:1041-1080.[6] Becke A D. A new mixing of Hartree-Fock and local density-functional theories[J]. J. Chem. Phys., 1993, 98:1372-1377.[7] Becke A D. Density-functional thermochemistry. III. The role of exact exchange[J]. J. Chem. Phys., 1993, 98:5648-5652.[8] Blöchl P E. Generalized separable potentials for electronic-structure calculations[J]. Phys. Rev. B, 1990, 41:5414-5416.[9] Born M and Oppenheimer J R. Zur quantentheorie der Molekeln[J]. Ann. Physik, 1927, 84:457-484.[10] Broyden C G. A class of methods for solving nonlinear simultaneous equations[J]. Math. Comput., 1965, 19:577-593.[11] Cai Y, Zhang L, Bai Z, and Li R. On an eigenvector-dependent nonlinear eigenvalue problem[J]. SIAM J. Matrix Anal. Appl., 2018, 39:1360-1382.[12] Calvetti D, Reichel L, Sorensen D C. An implicitly restarted Lanczos method for large symmetric eigenvalue problems[J]. Electron T. Numer. Ana., 1994, 2:1-21.[13] Cancès E, Chakir R, and Maday Y. Numerical analysis of nonlinear eigenvalue problems[J]. J. Sci. Comput., 2010, 45:90-117.[14] Cancès E, Chakir R, and Maday Y. Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models[J]. M2AN, 2012, 46:341-388.[15] Cascon J M, Kreuzer C, Nochetto R H, and Siebert K G. Quasi-optimal convergence rate for an adaptive finite element method[J]. SIAM J. Numer. Anal., 2008, 46:2524-2550.[16] 陈华杰. 密度泛函理论的有限维逼近[D]. 中国科学院研究生院博士论文, 2010.[17] Chen H, Dai X, Gong X, He L, and Zhou A. Adaptive finite element approximations for Kohn-Sham models[J]. Multiscale Model. Simul., 2014, 12:1828-1869.[18] Chen H, Gong X, He L, Yang Z, and Zhou A. Numerical analysis of finite dimensional approximations of Kohn-Sham equations[J]. Adv. Comput. Math., 2013, 38:225-256.[19] Chen H, Gong X, He L, and Zhou A. Adaptive finite element approximations for a class of nonlinear eigenvalue problems in quantum physics[J]. Adv. Appl. Math. Mech., 2011, 3:493-518.[20] Chen H, He L, and Zhou A. Finite element approximations of nonlinear eigenvalue problems in quantum physics[J]. Comput. Methods Appl. Mech. Engrg., 2011, 200:1846-1865.[21] Ciarlet P G. The Finite Element Method for Elliptic Problems. North-Holland, 1978.[22] Cramer C J. Essentials of Computational Chemistry:Theories and Models. Wiley, 2002.[23] 戴小英. 第一原理电子结构计算的有限元自适应及局部算法研究[D]. 中国科学院研究生院博士论文, 2008.[24] 戴小英, 高兴誉, 周爱辉. 特征值问题的Davidson型方法及其实现技术[J]. 数值计算与计算机应用, 2006, 27:218-240.[25] Dai X, Gong X, Yang Z, Zhang D, and Zhou A. Finite volume discretizations for eigenvalue problems with applications to electronic structure calculations[J]. Multiscale Model. Simul., 2011, 9:208-240.[26] Dai X, Gong X, Zhou A, and Zhu J. A parallel orbital-updating approach for electronic structure calculations[J]. Multiscale Model. Simul., 2014, 12:182-1869.[27] Dai X, He L, and Zhou A. Convergence rate and quasi-optimal complexity of adaptive finite element computations for multiple eigenvalues[J]. IMA J Numer. Anal., 2015, 35:1934-1977.[28] Dai X, Liu Z, Zhang X and Zhou A. A parallel orbital-updating based optimization method for electronic structure calculations. arXiv:1510.07230(2015).[29] Dai X, Liu Z, Zhang L and Zhou A. A conjugate gradient method for electronic structure calculations[J]. SIAM J. Sci. Comput., 2017, 39:A2702-A2740.[30] Dai X, Xu J, and Zhou A. Convergence and optimal complexity of adaptive finite element eigenvalue computations[J]. Numer. Math., 2008, 110:313-355.[31] Dai X, Zhang L, and Zhou A. An adaptive step size strategy for orthogonality constrained line search methods. arXiv:1906.02883(2019).[32] Dai X, Zhang L and Zhou A. Pactical Newton methods for electronic structure calculations. arXiv:2001.09285(2020).[33] 戴小英, 周爱辉. 电子结构计算的有限元方法[J]. 中国科学:化学, 2015, 45:800-811.[34] 戴小英, 周爱辉. 第一原理实空间并行自适应计算程序设计原理[J]. 中国科学:信息科学, 2016, 46:1421-1441.[35] Dai Y and Yuan Y. A nonlinear conjugate gradient method with a strong global convergence property[J]. SIAM J. Optim., 1999, 10:177-182.[36] Dirac P A M. The Principles of Quantum Mechanics. Oxford University Press, Oxford, 4th edition, 1988.[37] Dörfler W. A convergent adaptive algorithm for Poisson's equation[J]. SIAM J. Numer. Anal., 1996, 33:1106-1124.[38] Ernzerhof M and Perdew J P. Generalized gradient approximation to the angle- and system-averaged exchange hole[J]. J. Chem. Phys., 1998, 109:3313-3320.[39] 方俊. 第一原理计算的若干实空间算法研究[D]. 中国科学院大学博士论文, 2013.[40] Fang J, Gao X, Zhou A. A Symmetry-based decomposition approach to eigenvalue problems[J]. J. Sci. Comput., 2013, 57:638-669.[41] Fermi E. Un metodo statistics per la determinazione di alcune proprieta dell'atomoi[J]. Rend. Accad. Lincei, 1927, 6:602-607.[42] Fermi E. A statistical method for the determination of some atomic properties and the application of this method to the theory of the periodic system of elements[J]. Zeit. Fur Physik, 1928, 48:73-79.[43] Francisco J B, Martinez J M, and Martinez L. Globally convergent trust-region methods for self-consistent field electronic structure calculations[J]. J. Chem. Phys., 2004, 121:10863-10878.[44] Fock V. Näherungsmethode zur lösung des quantenmechanischen mehrkörper problems[J]. Z. Phys., 1930, 61:126-148.[45] 高兴誉. 第一原理电子结构计算的六面体有限元方法[D]. 中国科学院研究生院博士论文, 2009.[46] Gao B, Liu X, Chen X, and Yuan Y. A new first-order framework for orthogonal constrained optimization problems[J]. SIAM J. Optim., 2018, 28:302-332.[47] Gao B, Liu X and Yuan Y. Parallelizable algorithms for optimization problems with orthogonality constraints[J]. SIAM J. Sci. Comput., 2019, 41:A1949-A1983.[48] Garau E M. and Morin P. Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems[J]. IMA J. Numer. Anal., 2011, 31:914-946.[49] Garau E M, Morin P, and Zuppa C. Convergence of adaptive finite element methods for eigenvalue problems[J]. M3AS, 2009, 19:721-747.[50] Gong X, Shen L, Zhang D, and Zhou A. Finite element approximations for Schrödinger equations with applications to electronic structure computations[J]. J. Comput. Math., 2008, 23:310-327.[51] Gunnarsson O, Jonson M, and Lundqvist B I. Descriptions of exchange and correlation effects in inhomogeneous electron systems[J]. Phys. Rev. B, 1979, 20:3136-3164.[52] Hamann D R, Schlüter M and Chiang C, Norm-conserving pseudopotentials[J]. Phys. Rev. Lett., 1979, 43:1494-1497.[53] Hartree D R. The wave mechanics of an atom with non-coulombic central field:parts I, II, III[J]. in Proc. Cambridge Phil. Soc, 1928, 24:89-110, 111-132, 426-437.[54] 何连花. 第一原理电子结构计算研究:数值分析与数值模拟[D]. 中国科学院研究生院博士论文, 2012.[55] Hedin L. New method for calculating the one-particle Green's function with application to the electron-gas problem[J]. Phys. Rev., 1965, 139:A796-A823.[56] Hohenberg P and Kohn W. Inhomogeneous electron gas[J]. Phys. Rev., 1964, 136:B864-B871.[57] Hu J, Jiang B, Lin L, Wen Z and Yuan Y. Structured quasi-Newton methods for optimization with orthogonality constraints[J]. SIAM J. Sci. Comput., 2019, 41:A2239-A2269.[58] Hu G, Xie H, and Xu F. A multilevel correction adaptive finite element method for Kohn-Sham equation[J]. J. Comput. Phys., 2018, 355:436-449.[59] Johnson D D, Modified Broyden's method for accelerating convergence in self-consistent calculations[J]. Phys. Rev. B, 1988, 38:12807-12813[60] Knyazev A V. Toward the optimal preconditioned eigensolver:Locally optimal block preconditioned conjugate gradient method[J]. SIAM J. Sci. Comput., 2001, 23:517-541.[61] Kohn W and Sham L J. Self-consistent equations including exchange and correlation effects[J]. Phys. Rev. A, 1965, 140:1133-1138.[62] Lee C, Yang W, and 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.[63] Lieb E H. Density functionals for Coulomb systems[J]. Inter. J. Quantum Chem., 1983, 24:243-277.[64] 林霖. 类Hartree-Fock方程的数值方法[J]. 计算数学, 2019, 41(2):113-125.[65] Lin L, Lu J, Ying L. Numerical methods for Kohn-Sham density functional theory[J]. Acta Numerica, 2019, 28:405-539.[66] Lin L and Yang C. Elliptic preconditioner for accelerating the self-consistent field iteration in Kohn-Sham density functional theory[J]. SIAM J. Sci. Comput., 2013, 35:S277-S298.[67] 刘壮. 第一原理电子结构计算的优化算法若干研究[D]. 中国科学院大学博士学位论文, 2016.[68] Liu X, Wang X, Wen Z, Yuan Y. On the convergence of the self-consistent field iteration in Kohn-Sham density functional theory[J]. SIAM J. Matrix Anal. A., 2014, 35:546-558.[69] Liu X, Wang X, Wen Z, Ulbrich M, and Yuan Y. On the analysis of the discretized Kohn-Sham density functional theory[J]. SIAM J. Numer. Anal., 2015, 53:1758-1785[70] Martin R M. Electronic Structure:Basic Theory and Practical Methods. Cambridge University Press. 2004.[71] Pan Y, Dai X, S de Gironcoli, Gong X, Rignanese G M, Zhou A. A parallel orbital-updating based plane-wave basis method for electronic structure calculations[J]. J. Comput. Phys., 2017, 348:482-492.[72] Parr R G and Yang W T. Density-Functional Theory of Atoms and Molecules. Oxford University Press, New York, Oxford, 1994.[73] Perdew J P. Climbing the ladder of density functional approximations[J]. MRS Bull., 2013, 28:743-750.[74] Perdew J P, Burke K and Ernzerhof M. Generalized gradient approximation made simple[J]. Phys. Rev. Lett., 1996, 77:3865-3868.[75] Perdew J P and Schmidt K. Jacob's ladder of density functional approx-imations for the exchange-correlation energy[J]. InAIP Conference Proceedings, 2001, 577:1-20.[76] Pulay P. Convergence acceleration of iterative sequences. the case of scf iteration[J]. Chem. Phys. Lett., 1980, 73:393-398.[77] Pulay P. Improved SCF convergence acceleration[J]. J. Comput. Chem., 1982, 3:556-560.[78] Saad Y. Numerical Methods for Large Eigenvalue Problems[M]. New York:Halstead Press, 1992.[79] Saad Y. Numerical methods for electronic structure calculations of materials[J]. SIAM Rev., 2010, 52:3-54.[80] Sleijpen G L G and van der Vorst H A. A Jacobi-Davidson iteration method for linear eigenvalue problems[J]. SIAM J. Matrix Anal. Appl., 1996, 17:401-425.[81] Sherrill C D and Schaefer III H F. The configuration interaction method:Advances in highly correlated approaches[J]. Adv. Quantum Chem., 1999, 34:143-269.[82] Simon B. Schrödinger operators in the twentieth century[J]. J. Math. Phys., 2000, 41:3523-3555.[83] 沈丽华. 基于密度泛函理论的电子结构有限元并行自适应算法[D]. 中国科学院研究生院博士论文, 2005.[84] Suryanarayana P, Gavini V, Blesgen T, Bhattacharya K, and Ortiz M. Non-periodic finite-element formulation of Kohn-Sham density functional theory[J]. J. Mech. Phys. Solids, 2010, 58:256-280.[85] Thøgersen L, Olsen J, Yeager D, Jørgensen P, Sa?ek P, and Helgaker T. The trustregion self-consistent field method:Towards a black-box optimization in Hartree-Fock and Kohn-Sham theories[J]. J. Chem. Phys., 2004, 121:16-27.[86] Thomas L H. The calculation of atomic fields[J]. Proc. Cambridge Phil. Soc., 1927, 23:542-548. Reprinted in March(1975).[87] Troullier N and Martins J L. Efficient pseudopotentials for plane-wave calculations[J]. Phys. Rev. B, 1991, 43:1993-2006.[88] Tsuchida E and Tsukada M. Adaptive finite-element method for electronic-structure calculations[J]. Phys. Rev. B, 1996, 54:7602-7605.[89] Tsuchida E and Tesukada M. Large-scale electronic-structure calculations based on the adaptive finite element method[J]. J. Phy. Soc. Jpn., 1998, 67:3844-3858.[90] Ulbrich M, Wen Z, Yang C, Klöckner D, and Lu Z. A proximal gradient method for ensemble density functional theory[J]. SIAM J. Sci. Comput., 2015, 37:A1975-A2002.[91] Vanderbilt D. Soft self-consistent pseudopotentials in a generalized eigenvalue formalism[J]. Phys. Rev. B, 1990, 41:7892-7895.[92] Wen Z and Yin W. A feasible method for optimization with orthogonality constraints[J]. Math. Program. Ser. A., 2013, 142:397-434.[93] 杨章. 基于有限体积离散的第一原理电子结构计算[D]. 中国科学院研究生院博士论文, 2011.[94] Yang C, Meza J C, and Wang L. A trust region direct constrained minimization algorithm for the Kohn-Sham equation[J]. SIAM J. Sci. Comput., 2007, 29:1854-1875.[95] 谢希德, 陆栋. 固体能带理论[M]. 复旦大学出版设, 上海, 1998.[96] Yang C, Gao W, Meza J C. On the convergence of the self-consistent field iteration for a class of nonlinear eigenvalue problems[J]. SIAM J. Matrix Anal. Appl., 2009, 30:1773-1788.[97] Yang B and Zhou A. Eigenfunction behaviors and adaptive finite element approximations of nonlinear eigenvalue problems in quantum physics. arXiv:1907.03968(2019).[98] 张笛儿. 有限元方法在电子结构计算中的应用[D]. 复旦大学博士论文, 2007.[99] Zhang X, Zhu J, Wen Z and Zhou A. Gradient type optimization methods for electronic structure calculations[J]. SIAM J. Sci. Comput., 2014, 36:265-289.[100] Zhou A. An analysis of finite-dimensional approximations for the ground state solution of Bose-Einstein condensates[J]. Nonlinearity, 2004, 17:541-550.[101] Zhou A. Finite dimensional approximations for the electronic ground state solution of a molecular system[J]. Math. Meth. Appl. Sci., 2007, 30:429-447.[102] 周爱辉. 电子结构模型的数学基础. 中国科学院研究生院讲义, 2010.[103] Zhou A. Hohenberg-Kohn theorem for Coulomb type systems and its generalization[J]. J. Math. Chem., 2012, 50:2746-2754.[104] 周爱辉. 电子结构模型与计算的数学问题[J]. 中国科学:数学, 2015, 45(6):929-938.[105] Zhou A. A mathematical aspect of Hohenberg-Kohn theorem[J]. Science China Mathematics, 2019, 62:63-68[106] Zhou Y, Wang H, Liu Y, Gao X, Song H. Applicability of Kerker preconditioning scheme to the self-consistent density functional theory calculations of inhomogeneous systems[J]. Phys. Rev. E, 2018, 97:033305.[107] 朱金伟. 基于平均场模型与强关联理论的第一原理计算[D]. 中国科学院大学博士论文, 2014.[108] ABACUS, http://abacus.ustc.edu.cn/.[109] ABINIT, http://www.abinit.org.[110] Gaussian, http://www.gaussian.com/.[111] PHG, http://lsec.cc.ac.cn/phg/.[112] Quantum Espresso, http://www.quantum-espresso.org.[113] VASP, http://www.vasp.at. |