强关联多电子体系的优化模型与算法

doi:10.12286/jssx.j2022-1031

计算数学 ›› 2023, Vol. 45 ›› Issue (2): 141-159.DOI: 10.12286/jssx.j2022-1031

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强关联多电子体系的优化模型与算法

刘歆^1,2

1. 中国科学院数学与系统科学研究院, 计算数学与科学工程计算研究所, 科学与工程计算国家重点实验室, 北京 100190;
2. 中国科学院大学, 北京 100049

收稿日期:2022-11-06 出版日期:2023-05-14 发布日期:2023-05-13
作者简介:刘歆, 中国科学院数学与系统科学研究院研究员, “冯康首席研究员”. 主要研究方向包括流形优化、分布式优化, 及其在材料计算、机器学习与大数据分析等领域中应用. 他于2004年本科毕业于北京大学数学科学学院, 2009年在中国科学院数学与系统科学研究院获得博士学位. 他于2016 年获得国家优秀青年科学基金, 2016年获得中国运筹学会青年科技奖, 2020年获得中国工业与应用数学学会应用数学青年科技奖, 2021年获得国家杰出青年科学基金. 现担任中国青年科技工作者协会理事, 中国运筹学会常务理事, 中国工业与应用数学学会副秘书长, Mathematical Programming Computation, Journal of Computational Mathematics, Journal of Industrial and Management Optimization, Asia-Pacific Journal of Operational Research, 《运筹学学报》等国内外期刊编委.
基金资助:
国家重点研发计划(2020YFA0711900, 2020YFA0 711904), 国家杰出青年基金(12125108) 和国家自然科学基金(11971466, 11991021, 11991020, 12021001, 12288201)资助.

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刘歆. 强关联多电子体系的优化模型与算法[J]. 计算数学, 2023, 45(2): 141-159.

Liu Xin. OPTIMIZATION MODELS AND APPROACHES FOR STRONGLY CORRELATED ELECTRONS SYSTEMS[J]. Mathematica Numerica Sinica, 2023, 45(2): 141-159.

OPTIMIZATION MODELS AND APPROACHES FOR STRONGLY CORRELATED ELECTRONS SYSTEMS

Liu Xin^1,2

1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
2. University of Chinese Academy of Sciences, Beijing 100049, China

Received:2022-11-06 Online:2023-05-14 Published:2023-05-13

在电子结构计算领域, Kohn-Sham方程是最为广泛使用的数学模型之一. 然而, 由于现有的交换关联能近似仍存在缺陷, Kohn-Sham方程无法较好地描述强关联多电子体系. 近年来, 有学者从密度泛函理论的强相关极限出发, 提出了严格关联电子能量的优化模型. 该模型有望弥补Kohn-Sham方程的缺陷, 从而拓宽密度泛函理论的应用面. 由于在该模型中存在维数灾难, 近年来, 它的一些低维转化模型陆续被提出. 在本文中, 我们将介绍严格关联电子能量的优化模型、它的研究重点以及现有的一些低维转化模型. 我们也将介绍这些转化模型的数值求解方法, 并探讨未来的研究方向.

关键词： 强关联多电子体系, 电子结构计算, 维数灾难, 低维转化, 可扩展性

In electronic structure calculations, Kohn-Sham equations rank among the most widely adopted mathematical models. However, due to the deficiency of available approximations for exchange-correlation energy, Kohn-Sham equations cannot well describe strongly correlated electrons systems at present. In recent decades, some researchers have studied the optimization models of strictly-correlated-electrons energy, starting from the strong-interaction limit of density functional theory. These models are hopeful to amend the very deficiency of Kohn-Sham equations and therefore broaden the applicability of the density functional theory. Since the curse of dimensionality resides in these models, some low-dimensional reformulations have been proposed. In this paper, we introduce the optimization models of strictly-correlated-electrons energy, highlight the research focus, and describe some lowdimensional reformulations. We also present the numerical approaches for these reformulations and shed light on the directions of future research.

Key words: Strongly correlated electrons, Electronic structure calculations, Curse of dimensionality, Low-dimensional reformulations, Scalability

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[1] Hohenberg P and Kohn W. Inhomogeneous electron gas[J]. Phys Rev, 1964, 136(3B):B864.
[2] Kohn W and Sham L J. Self-consistent equations including exchange and correlation effects[J]. Phys Rev, 1965, 140(4A):A1133.
[3] Cohen A J, Mori-Sánchez P and Yang W. Challenges for density functional theory[J]. Chem Rev, 2012, 112(1):289-320.
[4] Gori-Giorgi P, Seidl M and Vignale G. Density-functional theory for strongly interacting electrons[J]. Phys Rev Lett, 2009, 103(16):166402.
[5] Gori-Giorgi P and Seidl M. Density functional theory for strongly-interacting electrons:perspectives for physics and chemistry[J]. Phys Chem Chem Phys, 2010, 12(43):14405-14419.
[6] Liu Z and Burke K. Adiabatic connection for strictly correlated electrons[J]. J Chem Phys, 2009, 131(12):124124.
[7] Vuckovic S, Gerolin A, Daas T J, Bahmann H, Friesecke G and Gori-Giorgi P. Density functionals based on the mathematical structure of the strong-interaction limit of DFT[J]. Wiley Interdiscip Rev Comput Mol Sci, 2022:e1634.
[8] Born M and Oppenheimer R. Zur quantentheorie der molekeln[J]. Ann Phys, 1927, 389(20):457-484.
[9] Becke A D. Perspective:fifty years of density-functional theory in chemical physics[J]. J Chem Phys, 2014, 140(18):18A301.
[10] Burke K. Perspective on density functional theory[J]. J Chem Phys, 2012, 136(15):150901.
[11] Lin L, Lu J and Ying L. Numerical methods for Kohn-Sham density functional theory[J]. Acta Numer, 2019, 28:405-539.
[12] Saad Y, Chelikowsky J R and Shontz S M. Numerical methods for electronic structure calculations of materials[J]. SIAM Rev, 2010, 52(1):3-54.
[13] 戴小英. 电子结构计算的数值方法与理论[J]. 计算数学, 2020, 42(2):131-158.
[14] Dreizler R M and Gross E K U. Density Functional Theory:an Approach to the Quantum ManyBody Problem[M]. Springer Science & Business Media, 2012.
[15] Parr R G and Yang W. Density Functional Theory of Atoms and Molecules[M]. Oxford University Press, 1989.
[16] Zhou A. Hohenberg-Kohn theorem for Coulomb type systems and its generalization[J]. J Math Chem, 2012, 50(10):2746-2754.
[17] Zhou A. A mathematical aspect of Hohenberg-Kohn theorem[J]. Sci China Math, 2019, 62(1):63-68.
[18] Levy M. Electron densities in search of Hamiltonians[J]. Phys Rev A, 1982, 26(3):1200.
[19] Lieb E H. Density functionals for Coulomb systems[J]. Int J Quantum Chem, 1983, 24:243-277.
[20] Slater J C. The theory of complex spectra[J]. Phys Rev, 1929, 34(10):1293.
[21] Anisimov V I, Zaanen J and Andersen O K. Band theory and Mott insulators:Hubbard U instead of Stoner I[J]. Phys Rev B, 1991, 44(3):943.
[22] Chen H, Friesecke G and Mendl C B. Numerical methods for a Kohn-Sham density functional model based on optimal transport[J]. J Chem Theory Comput, 2014, 10(10):4360-4368.
[23] Cohen A J, Mori-Sánchez P and Yang W. Insights into current limitations of density functional theory[J]. Science, 2008, 321(5890):792-794.
[24] Grüning M, Gritsenko O V and Baerends E J. Exchange-correlation energy and potential as approximate functionals of occupied and virtual Kohn-Sham orbitals:application to dissociating H₂[J]. J Chem Phys, 2003, 118(16):7183-7192.
[25] Ghosal A, Güçlü A D, Umrigar C J, Ullmo D and Baranger H U. Incipient Wigner localization in circular quantum dots[J]. Phys Rev B, 2007, 76(8):085341.
[26] Seidl M, Perdew J P and Levy M. Strictly correlated electrons in density-functional theory[J]. Phys Rev A, 1999, 59(1):51.
[27] Seidl M. Strong-interaction limit of density-functional theory[J]. Phys Rev A, 1999, 60(6):4387.
[28] Seidl M, Perdew J P and Kurth S. Simulation of all-order density-functional perturbation theory, using the second order and the strong-correlation limit[J]. Phys Rev Lett, 2000, 84(22):5070.
[29] Bindini U and De Pascale L. Optimal transport with Coulomb cost and the semiclassical limit of density functional theory[J]. J Éc polytech Math, 2017, 4:909-934.
[30] Buttazzo G, De Pascale L and Gori-Giorgi P. Optimal-transport formulation of electronic densityfunctional theory[J]. Phys Rev A, 2012, 85(6):062502.
[31] Cotar C, Friesecke G and Klüppelberg C. Density functional theory and optimal transportation with Coulomb cost[J]. Commun Pure Appl Math, 2013, 66(4):548-599.
[32] Cotar C, Friesecke G and Klüppelberg C. Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional[J]. Arch Ration Mech Anal, 2018, 228(3):891-922.
[33] Malet F and Gori-Giorgi P. Strong correlation in Kohn-Sham density functional theory[J]. Phys Rev Lett, 2012, 109(24):246402.
[34] Malet F, Mirtschink A, Cremon J C, Reimann S M and Gori-Giorgi P. Kohn-Sham density functional theory for quantum wires in arbitrary correlation regimes[J]. Phys Rev B, 2013, 87(11):115146.
[35] Peyré G and Cuturi M. Computational optimal transport:with applications to data science[J]. Found Trends Mach Learn, 2019, 11(5-6):355-607.
[36] Villani C. Topics in Optimal Transportation[M]. American Mathematical Society, 2021.
[37] Lin T, Ho N, Cuturi M and Jordan M I. On the complexity of approximating multimarginal optimal transport[J]. J Mach Learn Res, 2022, 23:1-43.
[38] Hu Y, Chen H and Liu X. A global optimization approach for multi-marginal optimal transport problems with Coulomb cost[J]. SIAM J Sci Comput, 2023, in press.
[39] Friesecke G and Vögler D. Breaking the curse of dimension in multi-marginal Kantorovich optimal transport on finite state spaces[J]. SIAM J Math Anal, 2018, 50(4):3996-4019.
[40] Mendl C B and Lin L. Kantorovich dual solution for strictly correlated electrons in atoms and molecules[J]. Phys Rev B, 2013, 87(12):125106.
[41] Friesecke G, Mendl C B, Pass B, Cotar C and Klüppelberg C. N-density representability and the optimal transport limit of the Hohenberg-Kohn functional[J]. J Chem Phys, 2013, 139(16):164109.
[42] Friesecke G, Schulz A S and Vögler D. Genetic column generation:fast computation of highdimensional multimarginal optimal transport problems[J]. SIAM J Sci Comput, 2022, 44(3):A1632-A1654.
[43] Friesecke G, Gerolin A and Gori-Giorgi P. The strong-interaction limit of density functional theory. arXiv preprint arXiv:2202.09760, 2022.
[44] Seidl M, Gori-Giorgi P and Savin A. Strictly correlated electrons in density-functional theory:a general formulation with applications to spherical densities[J]. Phys Rev A, 2007, 75(4):042511.
[45] Seidl M, Di Marino S, Gerolin A, Nenna L, Giesbertz K J H and Gori-Giorgi P. The strictlycorrelated electron functional for spherically symmetric systems revisited. arXiv preprint arXiv:1702.05022, 2017.
[46] De los Reyes J C. Numerical PDE-Constrained Optimization[M]. Springer, 2015.
[47] Flegel M L and Kanzow C. On the Guignard constraint qualification for mathematical programs with equilibrium constraints[J]. Optimization, 2005, 54(6):517-534.
[48] Sun W and Yuan Y. Optimization Theory and Methods:Nonlinear Programming[M]. Springer Science & Business Media, 2006.
[49] Bednarek S, Szafran B, Chwiej T and Adamowski J. Effective interaction for charge carriers confined in quasi-one-dimensional nanostructures[J]. Phys Rev B, 2003, 68(4):045328.
[50] Grossi J, Kooi D P, Giesbertz K J H, Seidl M, Cohen A J, Mori-Sánchez P and Gori-Giorgi P. Fermionic statistics in the strongly correlated limit of density functional theory[J]. J Chem Theory Comput, 2017, 13(12):6089-6100.
[51] Zeldovich Y B and Popov V S. Electronic structure of superheavy atoms[J]. Sov Phys Usp, 1972, 14(6):673.
[52] Hu Y and Liu X. The exactness of the ȴ1 penalty function for a class of mathematical programs with generalized complementarity constraints. arXiv preprint arXiv:2206.04844, 2022.
[53] Liu G, Han J and Zhang J. Exact penalty functions for convex bilevel programming problems[J]. J Optim Theory Appl, 2001, 110(3):621-643.
[54] Luo Z-Q, Pang J-S, Ralph D and Wu S. Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints[J]. Math Program, 1996, 75(1):19-76.
[55] Pardalos P M and Vavasis S A. Quadratic programming with one negative eigenvalue is NPhard[J]. J Global Optim, 1991, 1(1):15-22.
[56] Attouch H, Bolte J, Redont P and Soubeyran A. Proximal alternating minimization and projection methods for nonconvex problems:an approach based on the Kurdyka-Ƚojasiewicz inequality[J]. Math Oper Res, 2010, 35(2):438-457.
[57] Attouch H, Bolte J and Svaiter B F. Convergence of descent methods for semi-algebraic and tame problems:proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods[J]. Math Program, 2013, 137(1):91-129.
[58] Auslender A. Asymptotic properties of the Fenchel dual functional and applications to decomposition problems[J]. J Optim Theory Appl, 1992, 73(3):427-449.
[59] Bolte J, Sabach S and Teboulle M. Proximal alternating linearized minimization for nonconvex and nonsmooth problems[J]. Math Program, 2014, 146(1):459-494.
[60] Tseng P. Convergence of a block coordinate descent method for nondifferentiable minimization[J]. J Optim Theory Appl, 2001, 109(3):475-494.
[61] Xu Y and Yin W. A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion[J]. SIAM J Imaging Sci, 2013, 6(3):1758-1789.
[62] Li X, Sun D and Toh K-C. On the efficient computation of a generalized Jacobian of the projector over the Birkhoff polytope[J]. Math Program, 2020, 179(1):419-446.
[63] Frankel P, Garrigos G and Peypouquet J. Splitting methods with variable metric for Kurdyka- Lojasiewicz functions and general convergence rates[J]. J Optim Theory Appl, 2015, 165(3):874-900.
[64] Hu Y and Liu X. The convergence properties of infeasible inexact proximal alternating linearized minimization[J]. Sci China Math, 2023, in press.
[65] Colombo M and Di Marino S. Equality between Monge and Kantorovich multimarginal problems with Coulomb cost[J]. Ann Mat Pura Appl (1923-), 2015, 194(2):307-320.
[66] Colombo M, De Pascale L and Di Marino S. Multimarginal optimal transport maps for onedimensional repulsive costs[J]. Canad J Math, 2015, 67(2):350-368.
[67] Bindini U, De Pascale L and Kausamo A. On Seidl-type maps for multi-marginal optimal transport with Coulomb cost. arXiv preprint arXiv:2011.05063, 2020.
[68] Papadimitriou C H and Steiglitz K. Combinatorial Optimization:Algorithms and Complexity (2nd Edition)[M]. Dover Publications, 1998.
[69] Di Marino S and Gerolin A. Optimal transport losses and Sinkhorn algorithm with general convex regularization. arXiv preprint arXiv:2007.00976, 2020.
[70] De Pascale L. Optimal transport with Coulomb cost. Approximation and duality[J]. ESAIM Math Model Numer Anal, 2015, 49(6):1643-1657.
[71] Buttazzo G, Champion T and De Pascale L. Continuity and estimates for multimarginal optimal transportation problems with singular costs[J]. Appl Math Optim, 2018, 78(1):185-200.
[72] Di Marino S, Gerolin A and Nenna L. Optimal transportation theory with repulsive costs[G]//Topological Optimization and Optimal Transport, 2017, 17:204-256.
[73] Conn A R, Scheinberg K and Vicente L N. Introduction to Derivative-Free Optimization[M]. Society for Industrial and Applied Mathematics, 2009.
[74] Larson J, Menickelly M and Wild S M. Derivative-free optimization methods[J]. Acta Numer, 2019, 28:287-404.
[75] Powell M J D. The NEWUOA software for unconstrained optimization without derivatives[G]//Large-Scale Nonlinear Optimization. Springer, Boston, MA, 2006:255-297.
[76] Lelotte R. Asymptotic of the Kantorovich potential for the optimal transport with Coulomb cost. arXiv preprint arXiv:2210.07830, 2022.
[77] Khoo Y and Ying L. Convex relaxation approaches for strictly correlated density functional theory[J]. SIAM J Sci Comput, 2019, 41(4):B773-B795.
[78] Yang L, Sun D and Toh K-C. SDPNAL+:a majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints[J]. Math Program Comput, 2015, 7(3):331-366.
[79] Khoo Y, Lin L, Lindsey M and Ying L. Semidefinite relaxation of multimarginal optimal transport for strictly correlated electrons in second quantization[J]. SIAM J Sci Comput, 2020, 42(6):B1462-B1489.
[80] Holland J H. Adaptation in Natural and Artificial Systems:an Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence[M]. MIT press, 1992.
[81] Desaulniers G, Desrosiers J and Solomon M M. Column Generation[M]. Springer Science & Business Media, 2006.
[82] Arjovsky M, Chintala S and Bottou L. Wasserstein generative adversarial networks[C]//International Conference on Machine Learning. PMLR, 2017:214-223.

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强关联多电子体系的优化模型与算法

OPTIMIZATION MODELS AND APPROACHES FOR STRONGLY CORRELATED ELECTRONS SYSTEMS

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