新一代相对论量子化学计算方法与应用
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摘要
本论文主要包括三部分,即相对论电子结构理论、相对论含时密度泛函理论和双值群对称性,在每一方向上都取得较好进展,简述如下:
(1)发展了新一代相对论量子化学方法
相对论量子化学方法包括四分量全相对论和二分量准相对论两大类,长期以来,人们关于这两类方法孰优孰劣一直存在很大分歧。四分量方法形式简单,计算精度高,所以有人说:“四分量好,二分量差!”;但四分量方法计算量很大,只能用于小分子计算,而二分量方法计算量较小,且对价电子有较高的计算精度,因而也有人说:“二分量好,四分量差!”。之所以有如此争议,是因为人们希望好的方法应该简单、精度好、效率高,而这三者往往不可兼得。在充分考察相对论效应局域性的基础上,我们提出了“用原子(或分子片)合成分子”的思想,来大大简化分子的相对论计算,使得四分量完全相对论和二分量准相对论方法在简洁性、计算精度和计算效率诸方面达到完全一致,从而可以说:“四分量、二分量同样好!”。尤其是,我们提出的新一代二分量准相对论XQR(exact matrix quasi-relativistic theory)方法不仅准确,而且比原有的近似方法还要简单,该方法可以对重、轻原子分别进行二分量相对论与标量相对论(非相对论)处理,从而成为联系相对论狄拉克方程与非相对论薛定谔方程的“无缝桥梁”。这是概念上的一大突破。我们完全有理由说,化学(和普通物理)中的相对论问题已得到解决!
(2)进一步发展了相对论含时密度泛函理论以研究重元素体系激发态的性质
相对论密度泛函理论是迄今唯一可用于含重元素复杂大分子体系计算的第一性原理方法,因此有必要将其推广到含时领域,以描述体系激发态的电子结构和动力学。实际上,刘文剑教授研究组在国际上首次实现了完全相对论含时密度泛函方法并用于重元素体系激发态计算。本论文对该理论进行了进一步发展,通过利用非共线(noncollinear)型交换相关核(kernel),提出了含时密度泛函理论的统一形式和高精度简化形式,即我们的理论公式同时适用于四分量、二分量、标量相对论以及非相对论含时密度泛函理论。
(3)发展了一个新的处理分子点群和时间反演对称性的方法和程序
在全面分析了分子点群玻色表示和费米表示特点的基础上,我们发展了一个新的能处理任意单值群、双值群和时间反演对称性的方法和程序,并用于密度泛函与含时密度泛函理论计算,大大提高了计算效率。
上述工作都在BDF(Beijing Density Functional)程序上完成。
The dissertation is composed of three major parts, viz. relativistic electronic structuretheory, time-dependent relativistic density functional theory, and group symmetries. Goodprogresses have been made along each of the directions.
(1) New generation relativistic quantum chemical methods
While the importance of relativistic e?ects for the chemistry and physics of heavy ele-ments has widely been recognized, there is little consensus on how to account for such e?ectsin the actual calculations. That is, which of the two sets of methods, four-component andtwo-component, is superior has long been controversial, and one often hears statements like“four-component good, two-component bad!”or“two-component good, four-component bad!”.In the present dissertation we will show that four- and two-component methods can actuallybe made fully equivalent in all aspects of simplicity, accuracy, and e?ciency, which then allowsus to speak of“four- and two-component equally good!”. Note that this has been achievedbased solely on physical arguments but not on mathematical tricks. One of the essential ideasis“from atoms to molecule”employing the spatial locality of relativistic e?ects and the knowl-edge about the atoms. The other is to formulate two-component theories at matrix level ratherthan at operator level, by one-step block-diagonalization of the matrix representation of theDirac operator in a restricted kinetically balanced basis. The resulting matrix two-componenttheories are far simpler than the operator counterparts and exact in that they can reproduce theelectronic (and positronic) eigenstates of the original Dirac matrix. The SESC (symmetrizedelimination of the small component) variant is particularly attractive because it requires onlynonrelativistic-type molecular ERI integrals and outperforms all previous quasirelativistic the-ories, whether finite- or infinite-order. In addition, SESC furnishes a seamless bridge betweenthe Dirac and Schro¨dinger equations allowing for hybrid treatments of heavy and light elements in the system. This is indeed a conceptual breakthrough. These findings allow us to claim thatrelativity in chemistry (and ordinary physics) has been solved!
(2)Further development of time-dependent relativistic density functional theory for excitedstates
Since relativistic density functional theory is so far the only first principles method fordescribing large and complex systems containing heavy elements, it is of great value to extendit to the time-dependent domain in order to describe excited states and dynamical propertiesof heavy elements. This is actually done for the first time by Prof. W. Liu’s group. Inthe present work we have further extended the theory by using a noncollinear form for theexchange-correlation kernel. We have thus obtained a general matrix formalism and a highlyaccurate simplified formulation, by which four-, two-, and one-component relativistic as wellas nonrelativistic time-dependent density functional methods can be expressed with the sameequations.
(3)A novel method and routines for generating double group and time-reversal symmetriesBy analyzing the characteristics of the Boson and Fermion irreducible representations ofmolecular point groups we have developed a novel method and corresponding routines forgenerating arbitrary single and double point group symmetries as well as time-reversal sym-metry. Full use of such symmetries greatly enhances the e?ciency of density functional andtime-dependent density functional calculations.
All the works have been accomplished on top of the BDF (Beijing Density Functional)package.
(1)发展了新一代相对论量子化学方法
相对论量子化学方法包括四分量全相对论和二分量准相对论两大类,长期以来,人们关于这两类方法孰优孰劣一直存在很大分歧。四分量方法形式简单,计算精度高,所以有人说:“四分量好,二分量差!”;但四分量方法计算量很大,只能用于小分子计算,而二分量方法计算量较小,且对价电子有较高的计算精度,因而也有人说:“二分量好,四分量差!”。之所以有如此争议,是因为人们希望好的方法应该简单、精度好、效率高,而这三者往往不可兼得。在充分考察相对论效应局域性的基础上,我们提出了“用原子(或分子片)合成分子”的思想,来大大简化分子的相对论计算,使得四分量完全相对论和二分量准相对论方法在简洁性、计算精度和计算效率诸方面达到完全一致,从而可以说:“四分量、二分量同样好!”。尤其是,我们提出的新一代二分量准相对论XQR(exact matrix quasi-relativistic theory)方法不仅准确,而且比原有的近似方法还要简单,该方法可以对重、轻原子分别进行二分量相对论与标量相对论(非相对论)处理,从而成为联系相对论狄拉克方程与非相对论薛定谔方程的“无缝桥梁”。这是概念上的一大突破。我们完全有理由说,化学(和普通物理)中的相对论问题已得到解决!
(2)进一步发展了相对论含时密度泛函理论以研究重元素体系激发态的性质
相对论密度泛函理论是迄今唯一可用于含重元素复杂大分子体系计算的第一性原理方法,因此有必要将其推广到含时领域,以描述体系激发态的电子结构和动力学。实际上,刘文剑教授研究组在国际上首次实现了完全相对论含时密度泛函方法并用于重元素体系激发态计算。本论文对该理论进行了进一步发展,通过利用非共线(noncollinear)型交换相关核(kernel),提出了含时密度泛函理论的统一形式和高精度简化形式,即我们的理论公式同时适用于四分量、二分量、标量相对论以及非相对论含时密度泛函理论。
(3)发展了一个新的处理分子点群和时间反演对称性的方法和程序
在全面分析了分子点群玻色表示和费米表示特点的基础上,我们发展了一个新的能处理任意单值群、双值群和时间反演对称性的方法和程序,并用于密度泛函与含时密度泛函理论计算,大大提高了计算效率。
上述工作都在BDF(Beijing Density Functional)程序上完成。
The dissertation is composed of three major parts, viz. relativistic electronic structuretheory, time-dependent relativistic density functional theory, and group symmetries. Goodprogresses have been made along each of the directions.
(1) New generation relativistic quantum chemical methods
While the importance of relativistic e?ects for the chemistry and physics of heavy ele-ments has widely been recognized, there is little consensus on how to account for such e?ectsin the actual calculations. That is, which of the two sets of methods, four-component andtwo-component, is superior has long been controversial, and one often hears statements like“four-component good, two-component bad!”or“two-component good, four-component bad!”.In the present dissertation we will show that four- and two-component methods can actuallybe made fully equivalent in all aspects of simplicity, accuracy, and e?ciency, which then allowsus to speak of“four- and two-component equally good!”. Note that this has been achievedbased solely on physical arguments but not on mathematical tricks. One of the essential ideasis“from atoms to molecule”employing the spatial locality of relativistic e?ects and the knowl-edge about the atoms. The other is to formulate two-component theories at matrix level ratherthan at operator level, by one-step block-diagonalization of the matrix representation of theDirac operator in a restricted kinetically balanced basis. The resulting matrix two-componenttheories are far simpler than the operator counterparts and exact in that they can reproduce theelectronic (and positronic) eigenstates of the original Dirac matrix. The SESC (symmetrizedelimination of the small component) variant is particularly attractive because it requires onlynonrelativistic-type molecular ERI integrals and outperforms all previous quasirelativistic the-ories, whether finite- or infinite-order. In addition, SESC furnishes a seamless bridge betweenthe Dirac and Schro¨dinger equations allowing for hybrid treatments of heavy and light elements in the system. This is indeed a conceptual breakthrough. These findings allow us to claim thatrelativity in chemistry (and ordinary physics) has been solved!
(2)Further development of time-dependent relativistic density functional theory for excitedstates
Since relativistic density functional theory is so far the only first principles method fordescribing large and complex systems containing heavy elements, it is of great value to extendit to the time-dependent domain in order to describe excited states and dynamical propertiesof heavy elements. This is actually done for the first time by Prof. W. Liu’s group. Inthe present work we have further extended the theory by using a noncollinear form for theexchange-correlation kernel. We have thus obtained a general matrix formalism and a highlyaccurate simplified formulation, by which four-, two-, and one-component relativistic as wellas nonrelativistic time-dependent density functional methods can be expressed with the sameequations.
(3)A novel method and routines for generating double group and time-reversal symmetriesBy analyzing the characteristics of the Boson and Fermion irreducible representations ofmolecular point groups we have developed a novel method and corresponding routines forgenerating arbitrary single and double point group symmetries as well as time-reversal sym-metry. Full use of such symmetries greatly enhances the e?ciency of density functional andtime-dependent density functional calculations.
All the works have been accomplished on top of the BDF (Beijing Density Functional)package.
引文
[1] P. A. M. Dirac. Proc. Roy. Soc. (London), A117:610, 1928.
[2] P. A. M. Dirac. Proc. Roy. Soc. (London), A118:351, 1928.
[3] P. Pyykk¨o. Chem. Rev., 88:563, 1988.
[4] K. Stark, H.-J. Werner. J. Chem. Phys., 104:6515, 1996.
[5] W. Liu, G. Hong, D. Dai, L. Li, M. Dolg. Theor. Chem. Acc., 96:75, 1997.
[6] Wenjian Liu, Fan Wang, Lemin Li. J. Thero. and Comput. Chem., 2:257–272, 2003.
[7] W. Liu, F. Wang, L. Li. Encyclopedia of Computational Chemistry, electronic edition. Wiley,Chichester, UK, 2004.
[8] Attila Szabo, Neil S. Ostlund. Morden Quantum Chemistry, Introduction to Advanced Elec-tronic Structure Theory. Dover Publications, Inc. Mineola, New York, 1996.
[9] 徐光宪, 黎乐民. 量子化学(上册). 科学出版社, 北京, 2001.
[10] 徐光宪, 黎乐民, 王德民. 量子化学(中册). 科学出版社, 北京, 2001.
[11] R. G. Parr, W.T. Yang. Density-Functional Theory of Atoms and Molecules. OxfordUniversity Press, New York, 1989.
[12] C. C. J. Roothaan. Rev. Mod. Phys., 32(6):179, 1960.
[13] P. Hohenberg, W. Kohn. Phys. Rev. B, 136:864, 1964.
[14] L. H. Thomas. Proc. Camb. Phil. Soc., 23:542, 1927.
[15] E. Fermi. Rend. Accad. Lincei., 26:376, 1927.
[16] P. A. M. Dirac. Proc. Camb. Phil. Soc., 26:376, 1930.
[17] W. Kohn, L.J. Sham. Phys. Rev. A, 140:1133, 1965.
[18] S.H. Vosko, L. Wilk, M. Nusair. Can. J. Phys., 58:1200, 1980.
[19] J. P. Perdew, Y. Wang. Phys. Rev. B, 45:13244, 1992.
[20] A. D. Becke. Phys. Rev. A, 38:3098, 1988.
[21] J. P. Perdew, Y. Wang. Phys. Rev. B, 33:8822, 1986.
[22] C. Lee, W. Yang, R. G. Parr. Phys. Rev. B, 37:785, 1988.
[23] A. D. Becke, M. R. Roussel. Phys. Rev. A, 39:3761, 1989.
[24] A. D. Becke. J. Chem. Phys., 98:5648, 1993.
[25] A. D. Becke. J. Chem. Phys., 104:1040, 1996.
[26] P. J. Stevens, J. F. Devlin, C. F. Chabalowski, M. J. Frisch. J. Phys. Chem., 98:11623,1994.
[27] Fan Wang, Wenjian Liu. J. Chin. Chem. Soc., 50:597–606, 2003.
[28] Jun Gao, Wenjian Liu, Bo Song, Chengbu Liu. J. Chem. Phys., 121:6658, 2004.
[29] Jun Gao, Wenli Zou, Wenjian Liu, Yunlong Xiao, Daoling Peng, Bo Song, Chengbu Liu.J. Chem. Phys., 123:054102, 2005.
[30] Daoling Peng, Wenli Zou, Wenjian Liu. J. Chem. Phys., 123:144101, 2005.
[31] Yunlong Xiao, Daoling Peng, Wenjian Liu. J. Chem. Phys., 126:081101, 2007.
[32] A. K. Rajagopal. J. Phys. C, 11:L943, 1978.
[33] A. H. MacDonald, S. H. Vosko. J. Phys. C, 12:2977, 1979.
[34] 曾谨言. 量子力学(卷II). 科学出版社, 北京, 2001.
[35] Erik van Lenthe. The ZORA Equation. Ph.D. thesis, Vrije University, 1996.
[36] 王繁. 含重元素分子的相对论密度泛函理论计算方法研究. Ph.D. thesis, 北京大学,2001.
[37] Y. Ishikawa, R.C. Binning, K.M. Sando. Chem. Phys. Lett., 101:111, 1985.
[38] R.E. Stanton, S. Havriliak. J. Chem. Phys., 81:1910, 1984.
[39] P. Pulay. Chem. Phys. Lett., 73:193, 1980.
[40] P. Pulay. J. Comput. Chem., 3:556, 1982.
[41] T.P. Hamilton, P. Pulay. J. Chem. Phys., 84:5728, 1986.
[42] G. Pongor. Chem. Phys. Lett., 24:603, 1973.
[43] A.D. Becke. J. Chem. Phys., 88:2547, 1988.
[44] C.W. Murry, N.C. Handy, G.J. Laming. Mol. Phys., 78:997, 1993.
[45] V.I. Lebdev. Zh. Vychisl. Mat. Mat. Fiz., 16:293, 1976.
[46] P.H.T. Philipsen, E. van Lenthe, J.G. Snijders, E.J. Baerends. Phys. Rev. B, 56:13556,1997.
[47] B. Delly. J. Chem. Phys., 92:508, 1990.
[48] G. Hong, L. Li. Chin. J. Chem., 14:289, 1996.
[49] A. Rosen, D.E. Ellis. J. Chem. Phys., 62:3039, 1975.
[50] D.E.Ellis, G.L. Goodman. Int. J. Quantum Chem., 25:185, 1984.
[51] D. Guenzburger, D.E. Ellis, J.A. Gomez. Phys. Rev. B, 115:8267, 2001.
[52] H. Eschrig, V. D. P. Servedio. J. Comput. Chem., 20:23, 1999.
[53] T. Ziegler, A. Rauk. Theor. Chim. Acta., 46:1, 1977.
[54] M. Douglas, N. M. Kroll. Acta Physica, 82:89, 1974.
[55] B. A. Hess. Phys. Rev. A, 33:3742, 1986.
[56] G. Jansen, B. A. Hess. Phys. Rev. A, 39:6016, 1989.
[57] A. Matveev, N. R¨osch. J. Chem. Phys., 118:3997, 2002.
[58] T. Nakajima, K. Hirao. J. Chem. Phys., 119:4105, 2003.
[59] T. Nakajima, K. Hirao. J. Chem. Phys., 113:7786, 2000.
[60] T. Nakajima, K. Hirao. Chem. Phys. Lett., 329:511, 2000.
[61] A. Wolf, M. Reiher, B. A. Hess. J. Chem. Phys., 117:9215, 2002.
[62] C. van Wu¨llen. J. Chem. Phys., 120:7307, 2004.
[63] Ch. Chang, M. P′elissier, Ph. Durand. Phys. Scr., 34:394, 1986.
[64] J.-L. Heully, I. Lindgren, E. Lindroth, S. Lundqvist, A.-M. Martensson-Pendrill. J. Phys.B, 19:2799, 1986.
[65] E. van Lenthe, E. J. Baerends, J. G. Snijders. J. Chem. Phys., 99:4597, 1993.
[66] E. van Lenthe, E. J. Baerends, J. G. Snijders. J. Chem. Phys., 101:9783, 1994.
[67] E. van Lenthe, J. G. Snijders, E. J. Baerends. J. Chem. Phys., 105:6505, 1996.
[68] C. van Wu¨llen. J. Chem. Phys., 109:392, 1998.
[69] F. Wang, G. Hong, L. Li. Chem. Phys. Lett., 316:318, 2000.
[70] G. Breit. Phys. Rev., 34:553, 1929.
[71] J. A. Gaunt. Proc. Roy. Soc. (London), A122:513, 1929.
[72] O. Visser, L. Visscher, P. J. C. Aerts, W. C. Niewpoort. Theor. Chim. Acta, 81:405, 1992.
[73] T. Saue L. Vissche and, W. C. Nieuwpoort, K. F?grijr., O. Gropen. J. Chem. Phys.,99:6704, 1993.
[74] H. M. Quiney, H. Skaane, I. P. Grant. Chem. Phys. Lett., 290:473, 1998.
[75] J. Sucher. Phys. Rev. A, 22:348, 1980.
[76] J. Sucher. Int. J. Quantum Chem. Quantum Chem. Symp., 25:3, 1984.
[77] J-L. Heully, I. Lindgren, E. Lindroth, A-M. Martensson-Pendrill. Phys. Rev. A, 33:4426,1985.
[78] G. Hardekopf, J. Sucher. Phys. Rev. A, 30:703, 1984.
[79] W. H. E. Schwarz, E. M. van Wezenbeek, E. J. Baerends, J. G. Snijders. J. Phys. B,22:1515, 1989.
[80] L. L. Foldy, S. A. Wouthuysen. Phys. Rev., 78:29, 1950.
[81] A. J. Sadlej, J. G. Snijders. Chem. Phys. Lett., 229:435, 1994.
[82] W. Kutzelnigg. Int. J. Quantum Chem., 25:107, 1984.
[83] K. G. Dyall. J. Chem. Phys., 100:2118, 1994.
[84] K. G. Dyall. J. Chem. Phys., 106:9618, 1997.
[85] L. Visscher, E. van Lenthe. Chem. Phys. Lett., 306:357, 1999.
[86] W. Kutzelnigg, W. Liu. J. Chem. Phys., 123:241102, 2005.
[87] Wenjian Liu, Daoling Peng. J. Chem. Phys., 125:044102, 2006.
[88] W. Kutzelnigg. Chem. Phys., 225:203, 1997.
[89] C. van Wu¨llen, C. Michauk. J. Chem. Phys., 123:204113, 2005.
[90] A. Rosen, D. E. Ellis. J. Chem. Phys., 62:3039, 1975.
[91] G. L. Malli, N. C. Pyper. Proc. R. Soc. London A, 407:377, 1986.
[92] K. G. Dyall, T. Enevoldsen. J. Chem. Phys., 111:10000, 1999.
[93] W. Liu, C. van Wu¨llen, Y.-K. Han, Y.-J. Choi, Y.-S. Lee. Adv. Quantum Chem., 39:325,2001.
[94] W. Liu, C. van Wu¨llen, F. Wang, L. Li. J. Chem. Phys., 116:3626, 2002.
[95] E. van Lenthe, E. J. Baerends. J. Comput. Chem., 24:1142, 2003.
[96] B. A. Hess. Phys. Rev. A, 32:756, 1985.
[97] J. del Bene, R. Ditchfield, J. A. Pople. J. Chem. Phys., 55:2236, 1971.
[98] A. D. McLachlan, M. A. Ball. Rev. Mod. Phys., 36:844, 1964.
[99] M. Head-Gordon, R. J. Rico, M. Oumi, T. J. Lee. Chem. Phys. Lett., 219:21, 1994.
[100] R. J. Buenker, S. D. Peyerimho?, W. Butscher. Mol. Phys., 35:771, 1978.
[101] B. O. Roos. Adv. Chem. Phys., 69:399, 1987.
[102] J. J. McDonall, K. Peasley, M. R. Robb. Chem. Phys. Lett., 148:183, 1988.
[103] K. Andersoson, B. O. Roos. Modern Electronic Structure Theory, volume 1. World Sci-entific: New York, 1995.
[104] Geertsen, M. Rittby, R. J. Bartlett. Chem. Phys. Lett., 164:57, 1989.
[105] O. Christiansen, J. Gauss, B. Schimmelpfennig. Phys. Chem. Chem. Phys., 2:965, 2000.
[106] H. Nakatsuji, K. Hirao. J. Chem. Phys, 68:2053, 1978.
[107] C. Haettig, F. Weigend. J. Chem. Phys., 113:5154, 2000.
[108] E. Runge, E. K. U. Gross. Phys. Rev. Lett., 52:997, 1984.
[109] E. K. U. Gross, W. Kohn. Adv. Quantum Chem., 21:255, 1990.
[110] Mark E. Casida. Recent Advances in Density Functional Methods, Part I. World Scien-tific, Singapore, 1995.
[111] A. Dreum, M. Head-Gordon. Chem. Rev., 105:4009, 2005.
[112] R. van Leeuwen. Int. J. Mod. Phys., 15:1969, 2001.
[113] V. Peuckert. J. Phys. C, 11:4945, 1978.
[114] Zangwill, P. Soven. Phys. Rev. A, 21:1561, 1980.
[115] K. L. Liu, S. H. Vosko. Can. J. Phys., 67:1015, 1989.
[116] S. K. Gosh, A. K. Dhara. Phys. Rev. A, 38:1149, 1988.
[117] R. van Leeuwen. Phys. Rev. Lett., 82:3863, 1999.
[118] M. H. Cohen, A. Wasserman. Phys. Rev. A, 71:032515, 2005.
[119] M. Levy. Phys. Rev. A, 26:1200, 1982.
[120] E. H. Lieb. Int. J. Quantum Chem., 24:243, 1983.
[121] M. Petersilk, U. J. Gossmann, E. K. U. Gross. Phys. Rev. Lett., 76:1212, 1996.
[122] C. Jamorski, M. E. Casida, D. R. Salahub. J. Chem. Phys., 104:5134, 1996.
[123] R. Bauernschmitt, R. Ahlrichs. Chem. Phys. Lett., 256:454, 1996.
[124] S. Hirata, M. Head-Gordon. Chem. Phys. Lett., 302:375, 1999.
[125] C. Van Caillie, R. D. Amos. Chem. Phys. Lett., 317:159, 2000.
[126] F. Furche, R. Ahlrichs. J. Chem. Phys., 117:7433, 2002.
[127] S.J.A. van Gisbergen, J.G. Snijders, E.J. baerends. Phys. Rev. Lett., 78:3097, 1997.
[128] S.J.A. van Gisbergen, J.G. Snijders, E.J. baerends. J. Chem. Phys., 109:10644, 1998.
[129] S.J.A. van Gisbergen, J.G. Snijders, E.J. baerends. J. Chem. Phys., 109:10657, 1998.. 124 .
[130] E. R. Davidson. J. Comput. Phys., 17:87, 1975.
[131] A. Stathopoulos, C. F. Fischer. Comput. Phys. Commun, 79:268, 1994.
[132] M. Petersilka, E. K. U. Gross. Int. J. Quantum Chem. Symp., 30:181, 1996.
[133] K. L. Liu. Can. J. Phys., 69:573, 1991.
[134] Y. Shao, M. Head-Gordon. J. Chem. Phys., 118:4807, 2003.
[135] S. Varga, B. Fricke, H. Nakamatsu, J. Anton T. Mukoyama, D. Geschke, A. Heitmann, E.Engel, T. Bastug. J. Chem. Phys., 112:3499, 2000.
[136] T. Yanai, H. Iikura, T. Nakajima, Y. Ishikawa, K. Hirao. J. Chem. Phys., 115:8267, 2001.
[137] T. Saue, T. Helgaker. J. Comput. Chem., 23:814, 2001.
[138] H. M. Quiney, P. Belanzoni. J. Chem. Phys., 117:5550, 2002.
[139] E. Engel, R. M. Dreizler. Density Functional Theory II, Topics in Current Chemistry,volume 181. Springer, Berlin, 1996.
[140] A. K. Rajagopal, J. Callaway. Phys. Rev. B, 7:1912, 1973.
[141] A. K. Rajagopal. Phys. Rev. A, 50:3759, 1994.
[142] E. Engel, S. Keller, R. M. Dreizler. Phys. Rev. A, 53:1367, 1996.
[143] E. Engel, A. Facco Bonetti, S. Keller, I. Andrejkovics, R. M. Dreizle. Phys. Rev. A, 58:964,1998.
[144] A. H. Macdonald, S. H. Vosko. J. Phys. C, 12:2977, 1979.
[145] M. V. Ramana, A.K. Rajagopal. J. Phys. C, 14:4291, 1981.
[146] 高军. 含时相对论密度泛函理论及应用. Ph.D. thesis, 山东大学, 2005.
[147] F. Wang, T. Ziegler, E. van Lenthe, S. van Gisbergen, E. J. Baerends. J. Chem. Phys.,122:204103, 2005.
[148] W. Liu, C. van Wu¨llen. J. Chem. Phys., 110:3730, 1999.
[149] P .R. T. Schipper, O. V. Gritsenko, S. J. A. van Gisbergen, E. J. Baerends. J. Chem.Phys., 112:1344, 2000.
[150] P. Celani, H.-J. Werner. J. Chem. Phys., 112:5546, 2000.
[151] A. Berning, M. Schweizer, H.-J. Werner, P. J. Knowles, P. Palmieri. Mol. Phys., 98:1823,2000.
[152] D. Figgen, G. Rauhut, M. Dolg, H. Stoll. Chem. Phys., 311:227, 2005.
[153] T. Heimer. Naturwiss, 24:78, 1936.
[154] E. Hulthe`n, R. V. Zumstein. Phys. Rev., 28:13, 1926.
[155] S. Imanishi. Sci. Pap. Inst. Phys. Chem. Res. (Jpn.), 31:247, 1937.
[156] U. Ringstro¨m. Ark. Fys., 27:227, 1964.
[157] U. Ringstro¨m. Nature (London), 198:981, 1963.
[158] C. E. Fellows, M. Rosberg, A. P. C. Campos, R. F. Gutteres, C. Amiot. J. Mol. Spectrosc.,185:420, 1997.
[159] J. Y. Seto, Z. Morbi, F. Charron, S. K. Lee, P. F. Bernath, R. J. Le Roy. J. Chem. Phys.,110:11756, 1999.
[160] G. Jansen, B. A. Hess. Z. Phys. D: At., Mol. Clusters, 13:363, 1989.
[161] H. A. Witek, T. Nakijima, K. Hirao. J. Chem. Phys., 113:8015, 2000.
[162] 高松, 陈志达, 黎乐民. 分子对称性群. 北京大学出版社, 1996.
[163] Simon L. Altmann, Peter Herzig. Point-Group Theory Tables. Clarendon Press Oxford,1994.
[164] 徐婉棠, 喀兴林. 群论及其在固体物理中的应用. 高等教育出版社, 1999.
[165] 杨泽森. 高等量子力学. 北京大学出版社, 2002.
[166] 曾谨言. 量子力学(卷I). 科学出版社, 北京, 2001.
[167] E. P. Wigner. Group Theory and Its Application to the Quantum Mechanics of AtomicSpactra. New York : Academic Press Inc., 1959.
[168] 马中骥. 物理学中的群论. 科学出版社, 2001.
[169] Simon L. Altmann. Rotations, Quaternions, and Double Groups. Clarendon Press Oxford,1982.
[170] T.D. Bouman, G. L. Goodman. J. Chem. Phys., 56:2478, 1972.
[171] A. L. H. Chung, G. L. Goodman. J. Chem. Phys., 56:4125, 1972.
[172] Fan Wang, Lemin Li. J. Mol. Struct. (THEOCHEM), 586:193, 2002.
[173] 王竹溪, 郭敦仁. 特殊函数概论. 北京大学出版社, 2000.
[174] J. Meyer, W.-D. Sepp, B. Fricke, A. Rosen. Comput. Phys. Commun., 96:263–287, 1996.
[175] E. Wigner. G¨ottinger Nachrichten, 31:546, 1932.
[2] P. A. M. Dirac. Proc. Roy. Soc. (London), A118:351, 1928.
[3] P. Pyykk¨o. Chem. Rev., 88:563, 1988.
[4] K. Stark, H.-J. Werner. J. Chem. Phys., 104:6515, 1996.
[5] W. Liu, G. Hong, D. Dai, L. Li, M. Dolg. Theor. Chem. Acc., 96:75, 1997.
[6] Wenjian Liu, Fan Wang, Lemin Li. J. Thero. and Comput. Chem., 2:257–272, 2003.
[7] W. Liu, F. Wang, L. Li. Encyclopedia of Computational Chemistry, electronic edition. Wiley,Chichester, UK, 2004.
[8] Attila Szabo, Neil S. Ostlund. Morden Quantum Chemistry, Introduction to Advanced Elec-tronic Structure Theory. Dover Publications, Inc. Mineola, New York, 1996.
[9] 徐光宪, 黎乐民. 量子化学(上册). 科学出版社, 北京, 2001.
[10] 徐光宪, 黎乐民, 王德民. 量子化学(中册). 科学出版社, 北京, 2001.
[11] R. G. Parr, W.T. Yang. Density-Functional Theory of Atoms and Molecules. OxfordUniversity Press, New York, 1989.
[12] C. C. J. Roothaan. Rev. Mod. Phys., 32(6):179, 1960.
[13] P. Hohenberg, W. Kohn. Phys. Rev. B, 136:864, 1964.
[14] L. H. Thomas. Proc. Camb. Phil. Soc., 23:542, 1927.
[15] E. Fermi. Rend. Accad. Lincei., 26:376, 1927.
[16] P. A. M. Dirac. Proc. Camb. Phil. Soc., 26:376, 1930.
[17] W. Kohn, L.J. Sham. Phys. Rev. A, 140:1133, 1965.
[18] S.H. Vosko, L. Wilk, M. Nusair. Can. J. Phys., 58:1200, 1980.
[19] J. P. Perdew, Y. Wang. Phys. Rev. B, 45:13244, 1992.
[20] A. D. Becke. Phys. Rev. A, 38:3098, 1988.
[21] J. P. Perdew, Y. Wang. Phys. Rev. B, 33:8822, 1986.
[22] C. Lee, W. Yang, R. G. Parr. Phys. Rev. B, 37:785, 1988.
[23] A. D. Becke, M. R. Roussel. Phys. Rev. A, 39:3761, 1989.
[24] A. D. Becke. J. Chem. Phys., 98:5648, 1993.
[25] A. D. Becke. J. Chem. Phys., 104:1040, 1996.
[26] P. J. Stevens, J. F. Devlin, C. F. Chabalowski, M. J. Frisch. J. Phys. Chem., 98:11623,1994.
[27] Fan Wang, Wenjian Liu. J. Chin. Chem. Soc., 50:597–606, 2003.
[28] Jun Gao, Wenjian Liu, Bo Song, Chengbu Liu. J. Chem. Phys., 121:6658, 2004.
[29] Jun Gao, Wenli Zou, Wenjian Liu, Yunlong Xiao, Daoling Peng, Bo Song, Chengbu Liu.J. Chem. Phys., 123:054102, 2005.
[30] Daoling Peng, Wenli Zou, Wenjian Liu. J. Chem. Phys., 123:144101, 2005.
[31] Yunlong Xiao, Daoling Peng, Wenjian Liu. J. Chem. Phys., 126:081101, 2007.
[32] A. K. Rajagopal. J. Phys. C, 11:L943, 1978.
[33] A. H. MacDonald, S. H. Vosko. J. Phys. C, 12:2977, 1979.
[34] 曾谨言. 量子力学(卷II). 科学出版社, 北京, 2001.
[35] Erik van Lenthe. The ZORA Equation. Ph.D. thesis, Vrije University, 1996.
[36] 王繁. 含重元素分子的相对论密度泛函理论计算方法研究. Ph.D. thesis, 北京大学,2001.
[37] Y. Ishikawa, R.C. Binning, K.M. Sando. Chem. Phys. Lett., 101:111, 1985.
[38] R.E. Stanton, S. Havriliak. J. Chem. Phys., 81:1910, 1984.
[39] P. Pulay. Chem. Phys. Lett., 73:193, 1980.
[40] P. Pulay. J. Comput. Chem., 3:556, 1982.
[41] T.P. Hamilton, P. Pulay. J. Chem. Phys., 84:5728, 1986.
[42] G. Pongor. Chem. Phys. Lett., 24:603, 1973.
[43] A.D. Becke. J. Chem. Phys., 88:2547, 1988.
[44] C.W. Murry, N.C. Handy, G.J. Laming. Mol. Phys., 78:997, 1993.
[45] V.I. Lebdev. Zh. Vychisl. Mat. Mat. Fiz., 16:293, 1976.
[46] P.H.T. Philipsen, E. van Lenthe, J.G. Snijders, E.J. Baerends. Phys. Rev. B, 56:13556,1997.
[47] B. Delly. J. Chem. Phys., 92:508, 1990.
[48] G. Hong, L. Li. Chin. J. Chem., 14:289, 1996.
[49] A. Rosen, D.E. Ellis. J. Chem. Phys., 62:3039, 1975.
[50] D.E.Ellis, G.L. Goodman. Int. J. Quantum Chem., 25:185, 1984.
[51] D. Guenzburger, D.E. Ellis, J.A. Gomez. Phys. Rev. B, 115:8267, 2001.
[52] H. Eschrig, V. D. P. Servedio. J. Comput. Chem., 20:23, 1999.
[53] T. Ziegler, A. Rauk. Theor. Chim. Acta., 46:1, 1977.
[54] M. Douglas, N. M. Kroll. Acta Physica, 82:89, 1974.
[55] B. A. Hess. Phys. Rev. A, 33:3742, 1986.
[56] G. Jansen, B. A. Hess. Phys. Rev. A, 39:6016, 1989.
[57] A. Matveev, N. R¨osch. J. Chem. Phys., 118:3997, 2002.
[58] T. Nakajima, K. Hirao. J. Chem. Phys., 119:4105, 2003.
[59] T. Nakajima, K. Hirao. J. Chem. Phys., 113:7786, 2000.
[60] T. Nakajima, K. Hirao. Chem. Phys. Lett., 329:511, 2000.
[61] A. Wolf, M. Reiher, B. A. Hess. J. Chem. Phys., 117:9215, 2002.
[62] C. van Wu¨llen. J. Chem. Phys., 120:7307, 2004.
[63] Ch. Chang, M. P′elissier, Ph. Durand. Phys. Scr., 34:394, 1986.
[64] J.-L. Heully, I. Lindgren, E. Lindroth, S. Lundqvist, A.-M. Martensson-Pendrill. J. Phys.B, 19:2799, 1986.
[65] E. van Lenthe, E. J. Baerends, J. G. Snijders. J. Chem. Phys., 99:4597, 1993.
[66] E. van Lenthe, E. J. Baerends, J. G. Snijders. J. Chem. Phys., 101:9783, 1994.
[67] E. van Lenthe, J. G. Snijders, E. J. Baerends. J. Chem. Phys., 105:6505, 1996.
[68] C. van Wu¨llen. J. Chem. Phys., 109:392, 1998.
[69] F. Wang, G. Hong, L. Li. Chem. Phys. Lett., 316:318, 2000.
[70] G. Breit. Phys. Rev., 34:553, 1929.
[71] J. A. Gaunt. Proc. Roy. Soc. (London), A122:513, 1929.
[72] O. Visser, L. Visscher, P. J. C. Aerts, W. C. Niewpoort. Theor. Chim. Acta, 81:405, 1992.
[73] T. Saue L. Vissche and, W. C. Nieuwpoort, K. F?grijr., O. Gropen. J. Chem. Phys.,99:6704, 1993.
[74] H. M. Quiney, H. Skaane, I. P. Grant. Chem. Phys. Lett., 290:473, 1998.
[75] J. Sucher. Phys. Rev. A, 22:348, 1980.
[76] J. Sucher. Int. J. Quantum Chem. Quantum Chem. Symp., 25:3, 1984.
[77] J-L. Heully, I. Lindgren, E. Lindroth, A-M. Martensson-Pendrill. Phys. Rev. A, 33:4426,1985.
[78] G. Hardekopf, J. Sucher. Phys. Rev. A, 30:703, 1984.
[79] W. H. E. Schwarz, E. M. van Wezenbeek, E. J. Baerends, J. G. Snijders. J. Phys. B,22:1515, 1989.
[80] L. L. Foldy, S. A. Wouthuysen. Phys. Rev., 78:29, 1950.
[81] A. J. Sadlej, J. G. Snijders. Chem. Phys. Lett., 229:435, 1994.
[82] W. Kutzelnigg. Int. J. Quantum Chem., 25:107, 1984.
[83] K. G. Dyall. J. Chem. Phys., 100:2118, 1994.
[84] K. G. Dyall. J. Chem. Phys., 106:9618, 1997.
[85] L. Visscher, E. van Lenthe. Chem. Phys. Lett., 306:357, 1999.
[86] W. Kutzelnigg, W. Liu. J. Chem. Phys., 123:241102, 2005.
[87] Wenjian Liu, Daoling Peng. J. Chem. Phys., 125:044102, 2006.
[88] W. Kutzelnigg. Chem. Phys., 225:203, 1997.
[89] C. van Wu¨llen, C. Michauk. J. Chem. Phys., 123:204113, 2005.
[90] A. Rosen, D. E. Ellis. J. Chem. Phys., 62:3039, 1975.
[91] G. L. Malli, N. C. Pyper. Proc. R. Soc. London A, 407:377, 1986.
[92] K. G. Dyall, T. Enevoldsen. J. Chem. Phys., 111:10000, 1999.
[93] W. Liu, C. van Wu¨llen, Y.-K. Han, Y.-J. Choi, Y.-S. Lee. Adv. Quantum Chem., 39:325,2001.
[94] W. Liu, C. van Wu¨llen, F. Wang, L. Li. J. Chem. Phys., 116:3626, 2002.
[95] E. van Lenthe, E. J. Baerends. J. Comput. Chem., 24:1142, 2003.
[96] B. A. Hess. Phys. Rev. A, 32:756, 1985.
[97] J. del Bene, R. Ditchfield, J. A. Pople. J. Chem. Phys., 55:2236, 1971.
[98] A. D. McLachlan, M. A. Ball. Rev. Mod. Phys., 36:844, 1964.
[99] M. Head-Gordon, R. J. Rico, M. Oumi, T. J. Lee. Chem. Phys. Lett., 219:21, 1994.
[100] R. J. Buenker, S. D. Peyerimho?, W. Butscher. Mol. Phys., 35:771, 1978.
[101] B. O. Roos. Adv. Chem. Phys., 69:399, 1987.
[102] J. J. McDonall, K. Peasley, M. R. Robb. Chem. Phys. Lett., 148:183, 1988.
[103] K. Andersoson, B. O. Roos. Modern Electronic Structure Theory, volume 1. World Sci-entific: New York, 1995.
[104] Geertsen, M. Rittby, R. J. Bartlett. Chem. Phys. Lett., 164:57, 1989.
[105] O. Christiansen, J. Gauss, B. Schimmelpfennig. Phys. Chem. Chem. Phys., 2:965, 2000.
[106] H. Nakatsuji, K. Hirao. J. Chem. Phys, 68:2053, 1978.
[107] C. Haettig, F. Weigend. J. Chem. Phys., 113:5154, 2000.
[108] E. Runge, E. K. U. Gross. Phys. Rev. Lett., 52:997, 1984.
[109] E. K. U. Gross, W. Kohn. Adv. Quantum Chem., 21:255, 1990.
[110] Mark E. Casida. Recent Advances in Density Functional Methods, Part I. World Scien-tific, Singapore, 1995.
[111] A. Dreum, M. Head-Gordon. Chem. Rev., 105:4009, 2005.
[112] R. van Leeuwen. Int. J. Mod. Phys., 15:1969, 2001.
[113] V. Peuckert. J. Phys. C, 11:4945, 1978.
[114] Zangwill, P. Soven. Phys. Rev. A, 21:1561, 1980.
[115] K. L. Liu, S. H. Vosko. Can. J. Phys., 67:1015, 1989.
[116] S. K. Gosh, A. K. Dhara. Phys. Rev. A, 38:1149, 1988.
[117] R. van Leeuwen. Phys. Rev. Lett., 82:3863, 1999.
[118] M. H. Cohen, A. Wasserman. Phys. Rev. A, 71:032515, 2005.
[119] M. Levy. Phys. Rev. A, 26:1200, 1982.
[120] E. H. Lieb. Int. J. Quantum Chem., 24:243, 1983.
[121] M. Petersilk, U. J. Gossmann, E. K. U. Gross. Phys. Rev. Lett., 76:1212, 1996.
[122] C. Jamorski, M. E. Casida, D. R. Salahub. J. Chem. Phys., 104:5134, 1996.
[123] R. Bauernschmitt, R. Ahlrichs. Chem. Phys. Lett., 256:454, 1996.
[124] S. Hirata, M. Head-Gordon. Chem. Phys. Lett., 302:375, 1999.
[125] C. Van Caillie, R. D. Amos. Chem. Phys. Lett., 317:159, 2000.
[126] F. Furche, R. Ahlrichs. J. Chem. Phys., 117:7433, 2002.
[127] S.J.A. van Gisbergen, J.G. Snijders, E.J. baerends. Phys. Rev. Lett., 78:3097, 1997.
[128] S.J.A. van Gisbergen, J.G. Snijders, E.J. baerends. J. Chem. Phys., 109:10644, 1998.
[129] S.J.A. van Gisbergen, J.G. Snijders, E.J. baerends. J. Chem. Phys., 109:10657, 1998.. 124 .
[130] E. R. Davidson. J. Comput. Phys., 17:87, 1975.
[131] A. Stathopoulos, C. F. Fischer. Comput. Phys. Commun, 79:268, 1994.
[132] M. Petersilka, E. K. U. Gross. Int. J. Quantum Chem. Symp., 30:181, 1996.
[133] K. L. Liu. Can. J. Phys., 69:573, 1991.
[134] Y. Shao, M. Head-Gordon. J. Chem. Phys., 118:4807, 2003.
[135] S. Varga, B. Fricke, H. Nakamatsu, J. Anton T. Mukoyama, D. Geschke, A. Heitmann, E.Engel, T. Bastug. J. Chem. Phys., 112:3499, 2000.
[136] T. Yanai, H. Iikura, T. Nakajima, Y. Ishikawa, K. Hirao. J. Chem. Phys., 115:8267, 2001.
[137] T. Saue, T. Helgaker. J. Comput. Chem., 23:814, 2001.
[138] H. M. Quiney, P. Belanzoni. J. Chem. Phys., 117:5550, 2002.
[139] E. Engel, R. M. Dreizler. Density Functional Theory II, Topics in Current Chemistry,volume 181. Springer, Berlin, 1996.
[140] A. K. Rajagopal, J. Callaway. Phys. Rev. B, 7:1912, 1973.
[141] A. K. Rajagopal. Phys. Rev. A, 50:3759, 1994.
[142] E. Engel, S. Keller, R. M. Dreizler. Phys. Rev. A, 53:1367, 1996.
[143] E. Engel, A. Facco Bonetti, S. Keller, I. Andrejkovics, R. M. Dreizle. Phys. Rev. A, 58:964,1998.
[144] A. H. Macdonald, S. H. Vosko. J. Phys. C, 12:2977, 1979.
[145] M. V. Ramana, A.K. Rajagopal. J. Phys. C, 14:4291, 1981.
[146] 高军. 含时相对论密度泛函理论及应用. Ph.D. thesis, 山东大学, 2005.
[147] F. Wang, T. Ziegler, E. van Lenthe, S. van Gisbergen, E. J. Baerends. J. Chem. Phys.,122:204103, 2005.
[148] W. Liu, C. van Wu¨llen. J. Chem. Phys., 110:3730, 1999.
[149] P .R. T. Schipper, O. V. Gritsenko, S. J. A. van Gisbergen, E. J. Baerends. J. Chem.Phys., 112:1344, 2000.
[150] P. Celani, H.-J. Werner. J. Chem. Phys., 112:5546, 2000.
[151] A. Berning, M. Schweizer, H.-J. Werner, P. J. Knowles, P. Palmieri. Mol. Phys., 98:1823,2000.
[152] D. Figgen, G. Rauhut, M. Dolg, H. Stoll. Chem. Phys., 311:227, 2005.
[153] T. Heimer. Naturwiss, 24:78, 1936.
[154] E. Hulthe`n, R. V. Zumstein. Phys. Rev., 28:13, 1926.
[155] S. Imanishi. Sci. Pap. Inst. Phys. Chem. Res. (Jpn.), 31:247, 1937.
[156] U. Ringstro¨m. Ark. Fys., 27:227, 1964.
[157] U. Ringstro¨m. Nature (London), 198:981, 1963.
[158] C. E. Fellows, M. Rosberg, A. P. C. Campos, R. F. Gutteres, C. Amiot. J. Mol. Spectrosc.,185:420, 1997.
[159] J. Y. Seto, Z. Morbi, F. Charron, S. K. Lee, P. F. Bernath, R. J. Le Roy. J. Chem. Phys.,110:11756, 1999.
[160] G. Jansen, B. A. Hess. Z. Phys. D: At., Mol. Clusters, 13:363, 1989.
[161] H. A. Witek, T. Nakijima, K. Hirao. J. Chem. Phys., 113:8015, 2000.
[162] 高松, 陈志达, 黎乐民. 分子对称性群. 北京大学出版社, 1996.
[163] Simon L. Altmann, Peter Herzig. Point-Group Theory Tables. Clarendon Press Oxford,1994.
[164] 徐婉棠, 喀兴林. 群论及其在固体物理中的应用. 高等教育出版社, 1999.
[165] 杨泽森. 高等量子力学. 北京大学出版社, 2002.
[166] 曾谨言. 量子力学(卷I). 科学出版社, 北京, 2001.
[167] E. P. Wigner. Group Theory and Its Application to the Quantum Mechanics of AtomicSpactra. New York : Academic Press Inc., 1959.
[168] 马中骥. 物理学中的群论. 科学出版社, 2001.
[169] Simon L. Altmann. Rotations, Quaternions, and Double Groups. Clarendon Press Oxford,1982.
[170] T.D. Bouman, G. L. Goodman. J. Chem. Phys., 56:2478, 1972.
[171] A. L. H. Chung, G. L. Goodman. J. Chem. Phys., 56:4125, 1972.
[172] Fan Wang, Lemin Li. J. Mol. Struct. (THEOCHEM), 586:193, 2002.
[173] 王竹溪, 郭敦仁. 特殊函数概论. 北京大学出版社, 2000.
[174] J. Meyer, W.-D. Sepp, B. Fricke, A. Rosen. Comput. Phys. Commun., 96:263–287, 1996.
[175] E. Wigner. G¨ottinger Nachrichten, 31:546, 1932.