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研究激发态的多体格林函数理论
- 马玉臣*, 刘成卜
作者信息
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Many-Body Green's Function Theory for the Study of Excited States
- Ma Yuchen, Liu Chengbu
Author information
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文章历史
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摘要
本文介绍的多体格林函数理论是一种建立在一套格林函数(包括单粒子格林函数和双粒子格林函数)方程基础上的,用以研究物质激发态性质的第一性原理方法。该理论包括计算准粒子性质的GW方法和描述电子-空穴对运动的Bethe-Salpeter方程。GW方法可以以很高的精度计算轨道能量、能带结构、准粒子寿命等物理量;Bethe-Salpeter方程则在研究激发能、光吸收谱、激发态动力学等光学性质上有广泛的应用前景。多体格林函数理论通过自能算符描述电子之间以及电子与空穴之间的交换关联作用。本文将详细阐述该理论的基本概念和原理,并对其在各种材料中的应用做简要介绍。
Abstract
Many-body Green's function theory is a first-principle method used to investigate excited states, which is based on a set of Green's function equations. This theory includes GW method, which is used to calculate the properties of quasiparticles, and Bethe-Salpeter equation, which describes the motion of the electron-hole pair. GW method predicts orbital energies, band structures and quasiparticle lifetimes with high accuracy, while Bethe-Salpeter equation is a promising approach to study excitation energy, optical absorption spectrum and excited-state dynamics. Many-body Green's function theory uses self-energy operator to describe the exchange and correlation interactions among electrons and those between electron and hole. Here we give an overview of the fundamental concepts and principles of many-body Green's function theory, and a discussion on its applications in various materials.
Contents
1 Introduction
2 GW method
2.1 One-particle Greens function
2.2 Hedins equations
2.3 GW approximation
2.4 Dielectric function
2.5 Quasiparticle correction
2.6 Applications
3 Bethe-Salpeter equation
3.1 Two-particle Greens function
3.2 Bethe-Salpeter equation
3.3 Tamm-Dancoff approximation and Random-phase approximation
3.4 Dynamical screening effect
3.5 Applications
4 Excited-state dynamics
4.1 BSE force
4.2 Combination of constrained densityfunctional theory and many-body Greensfunction theory
5 Conclusions and Outlook
1 Introduction
2 GW method
2.1 One-particle Greens function
2.2 Hedins equations
2.3 GW approximation
2.4 Dielectric function
2.5 Quasiparticle correction
2.6 Applications
3 Bethe-Salpeter equation
3.1 Two-particle Greens function
3.2 Bethe-Salpeter equation
3.3 Tamm-Dancoff approximation and Random-phase approximation
3.4 Dynamical screening effect
3.5 Applications
4 Excited-state dynamics
4.1 BSE force
4.2 Combination of constrained densityfunctional theory and many-body Greensfunction theory
5 Conclusions and Outlook
关键词
关键词 / 多体格林函数理论 / GW方法 / Bethe-Salpeter方程 / 激发态 {{custom_keyword}} /
Key words
Many-body Green's function theory / GW method / Bethe-Salpeter equation / excited states {{custom_keyword}} /
中图分类号：
O641.1
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参考文献
[1] Hohenberg P, Kohn W. Phys. Rev., 1964, 136: B864-B871
[2] Kohn W, Sham L J. Phys. Rev., 1965, 140: A1133-A1138
[3] Perdew J P, Burke K, Ernzerhof M. Phys. Rev. Lett., 1996, 77: 3865-3868
[4] Van Schilfgaarde M, Kotani T, Faleev S. Phys. Rev. Lett., 2006, 96: art. no. 226402
[5] Onida G, Reining L, Rubio A. Rev. Mod. Phys., 2002, 74: 601-659
[6] Koopmans T. Physica, 1934, 1: 104-113
[7] Hedin L, Lundqvist S. Solid State Physics: Advances in Research and Application. NY: Academic Press, 1969. 23: 1-181
[8] Hedin L. Phys. Rev., 1965, 139: A796-A823
[9] Hybertsen M S, Louie S G. Phys. Rev. B, 1986, 34: 5390-5413
[10] Hybertsen M S, Louie S G. Phys. Rev. B, 1987, 35: 5585-5601
[11] Hybertsen M S, Louie S G. Phys. Rev. B, 1987, 35: 5602-5610
[12] Aulbur W G, Jönsson L, Wilkins J W. Solid State Physics: Advances in Research and Application. NY: Academic Press, 2000. 54: 1-218
[13] Aryasetiawan F, Gunnarsson O. Rep. Prog. Phys., 1998, 61: 237-312
[14] Rohlfing M, Krüger P, Pollmann J. Phys. Rev. B, 1993, 48: 17791-17805
[15] Rohlfing M, Krüger P, Pollmann J. Phys. Rev. B, 1995, 52: 1905-1917
[16] Spataru C D, Cazalilla M A, Rubio A, Benedict L X, Echenique P M, Louie S G. Phys. Rev. Lett., 2001, 87: art. no. 246405
[17] Godby R W, Schlüter M, Sham L J. Phys. Rev. B, 1988, 37: 10159-10175
[18] Shishkin M, Kresse G. Phys. Rev. B, 2006, 74: art. no. 035101
[19] Strinati G. Riv. Nuovo Cimento, 1988, 11: 1-86
[20] Strinati G. Phys. Rev. Lett., 1982, 49: 1519-1522
[21] Spataru C D, Ismail-Beigi S, Capaz R B, Louie S G. Phys. Rev. Lett., 2005, 95: art. no. 247402
[22] Strinati G. Phys. Rev. B, 1984, 29: 5718-5726
[23] Puschnig P, Ambrosch-Draxl C. Phys. Rev. B, 2002, 66: art. no. 165105
[24] Rohlfing M, Louie S G. Phys. Rev. B, 2000, 62: 4927-4944
[25] Tiago M L, Chelikowsky J R. Phys. Rev. B, 2006, 73: art. no. 205334
[26] Rödl C, Fuchs F, Fruthmüller J, Bechstedt F. Phys. Rev. B, 2008, 77: art. no. 184408
[27] Marini A, Hogan C, Grüning M, Varsano D. Comput. Phys. Commun., 2009, 180: 1392-1403
[28] Albrecht S, Reining L, Del Sole R, Onida G. Phys. Rev. Lett., 1998, 80: 4510-4513
[29] Shirley E L. Phys. Rev. Lett., 1998, 80: 794-797
[30] Benedict L X, Shirley E L, Bohn R B. Phys. Rev. B, 1998, 80: 4514-4517
[31] Rohlfing M, Louie S G. Phys. Rev. Lett., 1999, 82: 1959-1962
[32] Rohlfing M, Louie S G. Phys. Rev. Lett., 1998, 81: 2312-2315
[33] Rohlfing M, Louie S G. Phys. Rev. Lett., 1998, 80: 3320-3323
[34] van der Horst J W, Bobbert P A, Michels M A J, Brocks G, Kelly P J. Phys. Rev. Lett., 1999, 83: 4413-4416
[35] Kaczmarski M S, Ma Y C, Rohlfing M. Phys. Rev. B, 2010, 81: art. no. 115433
[36] Ismail-Beigi S, Louie S G. Phys. Rev. Lett., 2003, 90: art. no. 076401
[37] Ismail-Beigi S, Louie S G. Phys. Rev. Lett., 2005, 95: art. no. 156401
[38] Linderberg J, Öhrn Y. Propagator in Quantum Chemistry, Second Edition. Hoboken-New Jersey: John Wiley &Sons, Inc., 2004
[39] Von Niessen W, Schirmer J, Cederbaum L S. Comp. Phys. Rep., 1984, 1: 57-125
[40] Ortiz J V. J. Chem. Phys., 1988, 89: 6348-6352
[41] Zakrzewski V G, Ortiz J V. Int. J. Quant. Chem., 1995, 53: 583-590
[42] 徐光宪(Xu G X),黎乐民(Li L M),王德民(Wang D M),陈敏伯(Chen M B). 量子化学-基本原理和从头计算法(Quantum Chemistry-Basic Principles and Ib Inition Calcuation)(下册)(2nd ed.). 北京: 科学出版社(Science Press), 2008
[43] Schindlmayr A, Godby R W. Phys. Rev. Lett., 1998, 80: 1702-1705
[44] Marini A, Onida G, Del Sole R. Phys. Rev. Lett., 2001, 88: art. no. 016403
[45] Marini A, Del Sole R, Onida G. Phys. Rev. B, 2002, 66: art. no. 115101
[46] Marini A, Del Sole R, Rubio A, Onida G. Phys. Rev. B, 2002, 66: art. no. 161104(R)
[47] Yi Z J, Ma Y C, Rohlfing M, Silkin V M, Chulkov E V. Phys. Rev. B, 2010, 81: art. no. 125125
[48] Johnson D L. Phys. Rev. B, 1974, 9: 4475-4484
[49] von der Linden W, Horsch P. Phys. Rev. B, 1988, 37: 8351-8362
[50] Hott R. Phys. Rev. B, 1991, 44: 1057-1065
[51] Hybertsen M S, Louie S G. Phys. Rev. B, 1988, 37: 2733-2736
[52] Bechstedt F, Del Sole R, Cappellini G, Reining L. Solid State Commun., 1992, 84: 765-770
[53] Ma Y C, Rohlfing M. Phys. Rev. B, 2007, 75: art. no. 205114
[54] Rohlfing M, Krüger P, Pollmann J. Phys. Rev. Lett., 1995, 75: 3489-3492
[55] Rohlfing M, Krüger P, Pollmann J. Phys. Rev. B, 1998, 57: 6485-6492
[56] Hamann D R, Schlüter M, Chiang C. Phys. Rev. Lett., 1979, 43: 1494-1497
[57] Louie S G, Froyen S, Cohen M L. Phys. Rev. B, 1982, 26: 1738-1742
[58] Grossman J C, Rohlfing M, Mitas L, Louie S G, Cohen M L. Phys. Rev. Lett., 2001, 86: 472-475
[59] Faber C, Attaccalite C, Olevano V, Runge E, Blase X. Phys. Rev. B, 2011, 83: art. no. 115123
[60] Blase X, Attaccalite C, Olevano V. Phys. Rev. B, 2011, 83: art. no. 115103
[61] Ma Y C, Rohlfing M, Molteni C. Phys. Rev. B, 2009, 80: art. no. 241405(R)
[62] Papakonstantinou P. EPL, 2007, 78: art. no. 12001
[63] De Groot M, Buma W J, Gromov E V, Burghardt I, Köppel H, Cederbaum L S. J. Chem. Phys., 2006, 125: art. no. 204303
[64] González E M, Guidoni L, Molteni C. Phys. Chem. Chem. Phys., 2009, 11: 4556-4563
[65] De Groot M, Gromov E V, Köppel H, Buma W J. J. Phys. Chem. B, 2008, 112: 4427-4434
[66] Putschügl M, Zirak P, Penzkofer A. Chem. Phys., 2008, 343: 107-120
[67] Sergi A, Grüning M, Ferrario M, Buda F. J. Phys. Chem. B, 2001, 105: 4386-4391
[68] Ko C, Levine B, Toniolo A, Manohar L, Olsen S, Werner H J, Martínez T J. J. Am. Chem. Soc., 2003, 125: 12710-12711
[69] Nielsen I B, Boyé-Péronne S, El Ghazaly M O A, Kristensen M B, Nielsen S B, Andersen L H. Biophy. J., 2005, 89: 2597-2604
[70] Rocha-Rinza T, Christiansen O, Rajput J, Gopalan A, Rahbek D B, Andersen L H, Bochenkova A V, Granovsky A A, Bravaya K B, Nemukhin A V, Christiansen K L, Nielsen M B. J. Phys. Chem. A, 2009, 113: 9442-9449
[71] Nemukhin A V, Bochenkova A V, Bravaya K B, Granovsky A A. Proc. SPIE, 2007, 6449: art. no. 64490N
[72] Andersen L H, Bochenkova A V. Eur. Phys. J. D, 2009, 51: 5-14
[73] Molina V, Merchán M. Proc. Natl. Acad. Sci. USA, 2001, 98: 4299-4304
[74] Gromov E V, Burghardt I, Köppel H, Cederbaum L S. J. Am. Chem. Soc., 2007, 129: 6798-6806
[75] Nielsen S B, Lapierre A, Andersen J U, Pedersen U V, Tomita S, Andersen L H. Phys. Rev. Lett., 2001, 87: art. no. 228102
[76] Toniolo A, Olsen S, Manohar L, Martínez T J. Faraday Discuss., 2004, 127: 149-163
[77] Filippi C, Zaccheddu M, Buda F. J. Chem. Theory Comput., 2009, 5: 2074-2087
[78] Creemers T M H, Lock A J, Subramaniam V, Jovin T M, Völker S. Nat. Struct. Biol., 1999, 6: 557-560
[79] Lammich L, Åxman Pertersen M., Bröndsted Nielsen M, Andersen L H. Biophys. J., 2007, 92: 201-207
[80] Nielsen I B, Lammich L, Andersen L H. Phys. Rev. Lett., 2006, 96: art. no. 018304
[81] Andersen L H, Nielsen I B, Kristensen M B, El Ghazaly M O A, Haacke S, Nielsen M B, Petersen M Å. J. Am. Chem. Soc., 2005, 127: 12347-12350
[82] Sekharan S, Weingart O, Buss V. Biophys. J., 2006, 91: L07-L09
[83] Sun M T, Ding Y, Cui G L, Liu Y J. J. Phys. Chem. A, 2007, 111: 2946-2950
[84] Cembran A, González-Luque R, Altoè P, Merchán M, Bernardi F, Olivucci M, Garavelli M. J. Phys. Chem. A, 2005, 109: 6597-6605
[85] Bravaya K, Bochenkova A, Granovsky A, Nemukhin A. J. Am. Chem. Soc., 2007, 129: 13035-13042
[86] Send R, Sundholm D. Phys. Chem. Chem. Phys., 2007, 9: 2862-2867
[87] Ma Y, Rohlfing M, Molteni C. J. Chem. Theory Comput., 2010, 6: 257-265
[88] Baumeier B, Andrienko D, Ma Y, Rohlfing M. J. Chem. Theory Comput., 2012, 8: 997-1002
[89] Grüning M, Marini A, Gonze X. Nano Lett., 2009, 9: 2820-2824
[90] Runge E, Gross E K U. Phys. Rev. Lett., 1984, 52: 997-1000
[91] Hsu C P, Hirata S, Head-Gordon M. J. Phys. Chem. A, 2001, 105: 451-458
[92] Olovsson W, Tanaka I, Mizoguchi T, Puschnig P, Ambrosch-Draxl C. Phys. Rev. B, 2009, 79: art. no. 041102(R)
[93] Ankudinov A L, Nesvizhskii A I, Rehr J J. Phys. Rev. B, 2003, 67: art. no. 115120
[94] Rohlfing M, Pollmann J. Phys. Rev. B, 2001, 63: art. no. 125201
[95] Ma Y C, Rohlfing M, Gali A. Phys. Rev. B, 2010, 81: art. no. 041204(R)
[96] Hellmann H. Einfuhrung in die Quantumchemie. Leipzig: Franz Deuticke, 1937. p285
[97] Feynman R P. Phys. Rev., 1939, 56: 340-343
[98] Stanton J F, Gauss J, Ishikawa N, Head-Gordon M. J. Chem. Phys., 1995, 103: 4160-4174
[99] Liu J, Liang W Z. J. Chem. Phys., 2011, 134: art. no. 044114
[100] Liu J, Liang W Z. J. Chem. Phys., 2011, 135: art. no. 014113
[101] Liu J, Liang W Z. J. Chem. Phys., 2011, 135: art. no. 184111
[102] Ma Y C, Rohlfing M. Phys. Rev. B, 2008, 77: art. no. 115118
[103] Artacho E, Rohlfing M, Cté M, Haynes P D, Needs R J, Molteni C. Phys. Rev. Lett., 2004, 93: art. no. 116401
[104] Umari P, Stenuit G, Baroni S. Phys. Rev. B, 2010, 81: art. no. 115104
[105] Friedrich C, Blügel S, Schindlmayr A. Phys. Rev. B, 2010, 81: art. no. 125102
[106] Bruneval F, Gonze X. Phys. Rev. B, 2008, 78: art. no. 085125
[107] Reining L, Onida G, Godby R W. Phys. Rev. B, 1997, 56: R4301-R4304
[108] Rohlfing M. Phys. Rev. B, 2010, 82: art. no. 205127
[109] Shishkin M, Kresse G. Phys. Rev. B, 2007, 75: art. no. 235102
[110] Rostgaard C, Jacobsen K W, Thygesen K S. Phys. Rev. B, 2010, 81: art. no. 085103
[2] Kohn W, Sham L J. Phys. Rev., 1965, 140: A1133-A1138
[3] Perdew J P, Burke K, Ernzerhof M. Phys. Rev. Lett., 1996, 77: 3865-3868
[4] Van Schilfgaarde M, Kotani T, Faleev S. Phys. Rev. Lett., 2006, 96: art. no. 226402
[5] Onida G, Reining L, Rubio A. Rev. Mod. Phys., 2002, 74: 601-659
[6] Koopmans T. Physica, 1934, 1: 104-113
[7] Hedin L, Lundqvist S. Solid State Physics: Advances in Research and Application. NY: Academic Press, 1969. 23: 1-181
[8] Hedin L. Phys. Rev., 1965, 139: A796-A823
[9] Hybertsen M S, Louie S G. Phys. Rev. B, 1986, 34: 5390-5413
[10] Hybertsen M S, Louie S G. Phys. Rev. B, 1987, 35: 5585-5601
[11] Hybertsen M S, Louie S G. Phys. Rev. B, 1987, 35: 5602-5610
[12] Aulbur W G, Jönsson L, Wilkins J W. Solid State Physics: Advances in Research and Application. NY: Academic Press, 2000. 54: 1-218
[13] Aryasetiawan F, Gunnarsson O. Rep. Prog. Phys., 1998, 61: 237-312
[14] Rohlfing M, Krüger P, Pollmann J. Phys. Rev. B, 1993, 48: 17791-17805
[15] Rohlfing M, Krüger P, Pollmann J. Phys. Rev. B, 1995, 52: 1905-1917
[16] Spataru C D, Cazalilla M A, Rubio A, Benedict L X, Echenique P M, Louie S G. Phys. Rev. Lett., 2001, 87: art. no. 246405
[17] Godby R W, Schlüter M, Sham L J. Phys. Rev. B, 1988, 37: 10159-10175
[18] Shishkin M, Kresse G. Phys. Rev. B, 2006, 74: art. no. 035101
[19] Strinati G. Riv. Nuovo Cimento, 1988, 11: 1-86
[20] Strinati G. Phys. Rev. Lett., 1982, 49: 1519-1522
[21] Spataru C D, Ismail-Beigi S, Capaz R B, Louie S G. Phys. Rev. Lett., 2005, 95: art. no. 247402
[22] Strinati G. Phys. Rev. B, 1984, 29: 5718-5726
[23] Puschnig P, Ambrosch-Draxl C. Phys. Rev. B, 2002, 66: art. no. 165105
[24] Rohlfing M, Louie S G. Phys. Rev. B, 2000, 62: 4927-4944
[25] Tiago M L, Chelikowsky J R. Phys. Rev. B, 2006, 73: art. no. 205334
[26] Rödl C, Fuchs F, Fruthmüller J, Bechstedt F. Phys. Rev. B, 2008, 77: art. no. 184408
[27] Marini A, Hogan C, Grüning M, Varsano D. Comput. Phys. Commun., 2009, 180: 1392-1403
[28] Albrecht S, Reining L, Del Sole R, Onida G. Phys. Rev. Lett., 1998, 80: 4510-4513
[29] Shirley E L. Phys. Rev. Lett., 1998, 80: 794-797
[30] Benedict L X, Shirley E L, Bohn R B. Phys. Rev. B, 1998, 80: 4514-4517
[31] Rohlfing M, Louie S G. Phys. Rev. Lett., 1999, 82: 1959-1962
[32] Rohlfing M, Louie S G. Phys. Rev. Lett., 1998, 81: 2312-2315
[33] Rohlfing M, Louie S G. Phys. Rev. Lett., 1998, 80: 3320-3323
[34] van der Horst J W, Bobbert P A, Michels M A J, Brocks G, Kelly P J. Phys. Rev. Lett., 1999, 83: 4413-4416
[35] Kaczmarski M S, Ma Y C, Rohlfing M. Phys. Rev. B, 2010, 81: art. no. 115433
[36] Ismail-Beigi S, Louie S G. Phys. Rev. Lett., 2003, 90: art. no. 076401
[37] Ismail-Beigi S, Louie S G. Phys. Rev. Lett., 2005, 95: art. no. 156401
[38] Linderberg J, Öhrn Y. Propagator in Quantum Chemistry, Second Edition. Hoboken-New Jersey: John Wiley &Sons, Inc., 2004
[39] Von Niessen W, Schirmer J, Cederbaum L S. Comp. Phys. Rep., 1984, 1: 57-125
[40] Ortiz J V. J. Chem. Phys., 1988, 89: 6348-6352
[41] Zakrzewski V G, Ortiz J V. Int. J. Quant. Chem., 1995, 53: 583-590
[42] 徐光宪(Xu G X),黎乐民(Li L M),王德民(Wang D M),陈敏伯(Chen M B). 量子化学-基本原理和从头计算法(Quantum Chemistry-Basic Principles and Ib Inition Calcuation)(下册)(2nd ed.). 北京: 科学出版社(Science Press), 2008
[43] Schindlmayr A, Godby R W. Phys. Rev. Lett., 1998, 80: 1702-1705
[44] Marini A, Onida G, Del Sole R. Phys. Rev. Lett., 2001, 88: art. no. 016403
[45] Marini A, Del Sole R, Onida G. Phys. Rev. B, 2002, 66: art. no. 115101
[46] Marini A, Del Sole R, Rubio A, Onida G. Phys. Rev. B, 2002, 66: art. no. 161104(R)
[47] Yi Z J, Ma Y C, Rohlfing M, Silkin V M, Chulkov E V. Phys. Rev. B, 2010, 81: art. no. 125125
[48] Johnson D L. Phys. Rev. B, 1974, 9: 4475-4484
[49] von der Linden W, Horsch P. Phys. Rev. B, 1988, 37: 8351-8362
[50] Hott R. Phys. Rev. B, 1991, 44: 1057-1065
[51] Hybertsen M S, Louie S G. Phys. Rev. B, 1988, 37: 2733-2736
[52] Bechstedt F, Del Sole R, Cappellini G, Reining L. Solid State Commun., 1992, 84: 765-770
[53] Ma Y C, Rohlfing M. Phys. Rev. B, 2007, 75: art. no. 205114
[54] Rohlfing M, Krüger P, Pollmann J. Phys. Rev. Lett., 1995, 75: 3489-3492
[55] Rohlfing M, Krüger P, Pollmann J. Phys. Rev. B, 1998, 57: 6485-6492
[56] Hamann D R, Schlüter M, Chiang C. Phys. Rev. Lett., 1979, 43: 1494-1497
[57] Louie S G, Froyen S, Cohen M L. Phys. Rev. B, 1982, 26: 1738-1742
[58] Grossman J C, Rohlfing M, Mitas L, Louie S G, Cohen M L. Phys. Rev. Lett., 2001, 86: 472-475
[59] Faber C, Attaccalite C, Olevano V, Runge E, Blase X. Phys. Rev. B, 2011, 83: art. no. 115123
[60] Blase X, Attaccalite C, Olevano V. Phys. Rev. B, 2011, 83: art. no. 115103
[61] Ma Y C, Rohlfing M, Molteni C. Phys. Rev. B, 2009, 80: art. no. 241405(R)
[62] Papakonstantinou P. EPL, 2007, 78: art. no. 12001
[63] De Groot M, Buma W J, Gromov E V, Burghardt I, Köppel H, Cederbaum L S. J. Chem. Phys., 2006, 125: art. no. 204303
[64] González E M, Guidoni L, Molteni C. Phys. Chem. Chem. Phys., 2009, 11: 4556-4563
[65] De Groot M, Gromov E V, Köppel H, Buma W J. J. Phys. Chem. B, 2008, 112: 4427-4434
[66] Putschügl M, Zirak P, Penzkofer A. Chem. Phys., 2008, 343: 107-120
[67] Sergi A, Grüning M, Ferrario M, Buda F. J. Phys. Chem. B, 2001, 105: 4386-4391
[68] Ko C, Levine B, Toniolo A, Manohar L, Olsen S, Werner H J, Martínez T J. J. Am. Chem. Soc., 2003, 125: 12710-12711
[69] Nielsen I B, Boyé-Péronne S, El Ghazaly M O A, Kristensen M B, Nielsen S B, Andersen L H. Biophy. J., 2005, 89: 2597-2604
[70] Rocha-Rinza T, Christiansen O, Rajput J, Gopalan A, Rahbek D B, Andersen L H, Bochenkova A V, Granovsky A A, Bravaya K B, Nemukhin A V, Christiansen K L, Nielsen M B. J. Phys. Chem. A, 2009, 113: 9442-9449
[71] Nemukhin A V, Bochenkova A V, Bravaya K B, Granovsky A A. Proc. SPIE, 2007, 6449: art. no. 64490N
[72] Andersen L H, Bochenkova A V. Eur. Phys. J. D, 2009, 51: 5-14
[73] Molina V, Merchán M. Proc. Natl. Acad. Sci. USA, 2001, 98: 4299-4304
[74] Gromov E V, Burghardt I, Köppel H, Cederbaum L S. J. Am. Chem. Soc., 2007, 129: 6798-6806
[75] Nielsen S B, Lapierre A, Andersen J U, Pedersen U V, Tomita S, Andersen L H. Phys. Rev. Lett., 2001, 87: art. no. 228102
[76] Toniolo A, Olsen S, Manohar L, Martínez T J. Faraday Discuss., 2004, 127: 149-163
[77] Filippi C, Zaccheddu M, Buda F. J. Chem. Theory Comput., 2009, 5: 2074-2087
[78] Creemers T M H, Lock A J, Subramaniam V, Jovin T M, Völker S. Nat. Struct. Biol., 1999, 6: 557-560
[79] Lammich L, Åxman Pertersen M., Bröndsted Nielsen M, Andersen L H. Biophys. J., 2007, 92: 201-207
[80] Nielsen I B, Lammich L, Andersen L H. Phys. Rev. Lett., 2006, 96: art. no. 018304
[81] Andersen L H, Nielsen I B, Kristensen M B, El Ghazaly M O A, Haacke S, Nielsen M B, Petersen M Å. J. Am. Chem. Soc., 2005, 127: 12347-12350
[82] Sekharan S, Weingart O, Buss V. Biophys. J., 2006, 91: L07-L09
[83] Sun M T, Ding Y, Cui G L, Liu Y J. J. Phys. Chem. A, 2007, 111: 2946-2950
[84] Cembran A, González-Luque R, Altoè P, Merchán M, Bernardi F, Olivucci M, Garavelli M. J. Phys. Chem. A, 2005, 109: 6597-6605
[85] Bravaya K, Bochenkova A, Granovsky A, Nemukhin A. J. Am. Chem. Soc., 2007, 129: 13035-13042
[86] Send R, Sundholm D. Phys. Chem. Chem. Phys., 2007, 9: 2862-2867
[87] Ma Y, Rohlfing M, Molteni C. J. Chem. Theory Comput., 2010, 6: 257-265
[88] Baumeier B, Andrienko D, Ma Y, Rohlfing M. J. Chem. Theory Comput., 2012, 8: 997-1002
[89] Grüning M, Marini A, Gonze X. Nano Lett., 2009, 9: 2820-2824
[90] Runge E, Gross E K U. Phys. Rev. Lett., 1984, 52: 997-1000
[91] Hsu C P, Hirata S, Head-Gordon M. J. Phys. Chem. A, 2001, 105: 451-458
[92] Olovsson W, Tanaka I, Mizoguchi T, Puschnig P, Ambrosch-Draxl C. Phys. Rev. B, 2009, 79: art. no. 041102(R)
[93] Ankudinov A L, Nesvizhskii A I, Rehr J J. Phys. Rev. B, 2003, 67: art. no. 115120
[94] Rohlfing M, Pollmann J. Phys. Rev. B, 2001, 63: art. no. 125201
[95] Ma Y C, Rohlfing M, Gali A. Phys. Rev. B, 2010, 81: art. no. 041204(R)
[96] Hellmann H. Einfuhrung in die Quantumchemie. Leipzig: Franz Deuticke, 1937. p285
[97] Feynman R P. Phys. Rev., 1939, 56: 340-343
[98] Stanton J F, Gauss J, Ishikawa N, Head-Gordon M. J. Chem. Phys., 1995, 103: 4160-4174
[99] Liu J, Liang W Z. J. Chem. Phys., 2011, 134: art. no. 044114
[100] Liu J, Liang W Z. J. Chem. Phys., 2011, 135: art. no. 014113
[101] Liu J, Liang W Z. J. Chem. Phys., 2011, 135: art. no. 184111
[102] Ma Y C, Rohlfing M. Phys. Rev. B, 2008, 77: art. no. 115118
[103] Artacho E, Rohlfing M, Cté M, Haynes P D, Needs R J, Molteni C. Phys. Rev. Lett., 2004, 93: art. no. 116401
[104] Umari P, Stenuit G, Baroni S. Phys. Rev. B, 2010, 81: art. no. 115104
[105] Friedrich C, Blügel S, Schindlmayr A. Phys. Rev. B, 2010, 81: art. no. 125102
[106] Bruneval F, Gonze X. Phys. Rev. B, 2008, 78: art. no. 085125
[107] Reining L, Onida G, Godby R W. Phys. Rev. B, 1997, 56: R4301-R4304
[108] Rohlfing M. Phys. Rev. B, 2010, 82: art. no. 205127
[109] Shishkin M, Kresse G. Phys. Rev. B, 2007, 75: art. no. 235102
[110] Rostgaard C, Jacobsen K W, Thygesen K S. Phys. Rev. B, 2010, 81: art. no. 085103
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脚注
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基金
国家自然科学基金项目(No.21173130)资助
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