The density matrix renormalization group for ab initio quantum chemistry

Topical Review

Abstract

During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. Its underlying wavefunction ansatz, the matrix product state (MPS), is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS, the rank of the decomposition, controls the size of the corner of the many-body Hilbert space that can be reached with the ansatz. This parameter can be systematically increased until numerical convergence is reached. The MPS ansatz naturally captures exponentially decaying correlation functions. Therefore DMRG works extremely well for noncritical one-dimensional systems. The active orbital spaces in quantum chemistry are however often far from one-dimensional, and relatively large virtual dimensions are required to use DMRG for ab initio quantum chemistry (QC-DMRG). The QC-DMRG algorithm, its computational cost, and its properties are discussed. Two important aspects to reduce the computational cost are given special attention: the orbital choice and ordering, and the exploitation of the symmetry group of the Hamiltonian. With these considerations, the QC-DMRG algorithm allows to find numerically exact solutions in active spaces of up to 40 electrons in 40 orbitals.

Keywords

Molecular Physics and Chemical Physics 

References

  1. 1.
    D.R. Hartree, Math. Proc. Cambridge Philos. Soc. 24, 89 (1928)MATHADSGoogle Scholar
  2. 2.
    J.C. Slater, Phys. Rev. 32, 339 (1928)MATHADSGoogle Scholar
  3. 3.
    V. Fock, Z. Phys. 61, 126 (1926)ADSGoogle Scholar
  4. 4.
    T. Helgaker, P. Jørgensen, J. Olsen, Molecular electronic-structure theory, 1st edn. (Wiley, New-York, 2000)Google Scholar
  5. 5.
    P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964)MathSciNetADSGoogle Scholar
  6. 6.
    W. Kohn, L.J. Sham, Phys. Rev. 140, A1133 (1965)MathSciNetADSGoogle Scholar
  7. 7.
    R.M. Dickson, A.D. Becke, J. Chem. Phys. 123, 111101 (2005)ADSGoogle Scholar
  8. 8.
    C. Møller, M.S. Plesset, Phys. Rev. 46, 618 (1934)MATHADSGoogle Scholar
  9. 9.
    J.C. Slater, Phys. Rev. 34, 1293 (1929)MATHADSGoogle Scholar
  10. 10.
    E.U. Condon, Phys. Rev. 36, 1121 (1930)MATHADSGoogle Scholar
  11. 11.
    F. Coester, Nucl. Phys. 7, 421 (1958)Google Scholar
  12. 12.
    F. Coester, H. Kümmel, Nucl. Phys. 17, 477 (1960)MATHGoogle Scholar
  13. 13.
    J. Čížek, J. Chem. Phys. 45, 4256 (1966)Google Scholar
  14. 14.
    B.O. Roos, Int. J. Quantum Chem. 18, 175 (1980)Google Scholar
  15. 15.
    B.O. Roos, P.R. Taylor, P.E.M. Siegbahn, Chem. Phys. 48, 157 (1980)MathSciNetADSGoogle Scholar
  16. 16.
    P.E.M. Siegbahn, J. Almlöf, A. Heiberg, B.O. Roos, J. Chem. Phys. 74, 2384 (1981)ADSGoogle Scholar
  17. 17.
    P.-Å. Malmqvist, A. Rendell, B.O. Roos, J. Phys. Chem. 94, 5477 (1990)Google Scholar
  18. 18.
    K. Andersson, P.-Å. Malmqvist, B.O. Roos, J. Chem. Phys. 96, 1218 (1992)ADSGoogle Scholar
  19. 19.
    R.J. Buenker, S.D. Peyerimhoff, Theor. Chim. Acta 35, 33 (1974)Google Scholar
  20. 20.
    H.-J. Werner, E.-A. Reinsch, J. Chem. Phys. 76, 3144 (1982)ADSGoogle Scholar
  21. 21.
    P.E.M. Siegbahn, J. Chem. Phys. 70, 5391 (1979)ADSGoogle Scholar
  22. 22.
    P.E.M. Siegbahn, J. Chem. Phys. 72, 1647 (1980)MathSciNetADSGoogle Scholar
  23. 23.
    B.R. Brooks, H.F. Schaefer, J. Chem. Phys. 70, 5092 (1979)ADSGoogle Scholar
  24. 24.
    N. Oliphant, L. Adamowicz, J. Chem. Phys. 96, 3739 (1992)ADSGoogle Scholar
  25. 25.
    L.Z. Stolarczyk, Chem. Phys. Lett. 217, 1 (1994)ADSGoogle Scholar
  26. 26.
    T. Yanai, G.K.-L. Chan, J. Chem. Phys. 124, 194106 (2006)ADSGoogle Scholar
  27. 27.
    S.R. White, R.L. Martin, J. Chem. Phys. 110, 4127 (1999)ADSGoogle Scholar
  28. 28.
    J.F. Cornwell, in Group theory in physics, 1st edn. (Academic Press Inc., Ltd., London, 1984), Vols. 1 and 2Google Scholar
  29. 29.
    S.R. White, Phys. Rev. Lett. 69, 2863 (1992)ADSGoogle Scholar
  30. 30.
    S.R. White, Phys. Rev. B 48, 10345 (1993)ADSGoogle Scholar
  31. 31.
    S. Östlund, S. Rommer, Phys. Rev. Lett. 75, 3537 (1995)ADSGoogle Scholar
  32. 32.
    S. Rommer, S. Östlund, Phys. Rev. B 55, 2164 (1997)ADSGoogle Scholar
  33. 33.
    M.B. Hastings, J. Stat. Mech.: Theor. Exp. 2007, P08024 (2007)MathSciNetGoogle Scholar
  34. 34.
    T. Nishino, Origin of Matrix Product State in Statistical Mechanics, in International Workshop on Density Matrix Renormalization Group and Other Advances in Numerical Renormalization Group Methods, August 23 - September 3, 2010 Google Scholar
  35. 35.
    H.A. Kramers, G.H. Wannier, Phys. Rev. 60, 263 (1941)MathSciNetMATHADSGoogle Scholar
  36. 36.
    R.J. Baxter, J. Math. Phys. 9, 650 (1968)ADSGoogle Scholar
  37. 37.
    M.P. Nightingale, H.W.J. Blöte, Phys. Rev. B 33, 659 (1986)ADSGoogle Scholar
  38. 38.
    I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki, Phys. Rev. Lett. 59, 799 (1987)ADSGoogle Scholar
  39. 39.
    M. Fannes, B. Nachtergaele, R.F. Werner, Europhys. Lett. 10, 633 (1989)ADSGoogle Scholar
  40. 40.
    M. Fannes, B. Nachtergaele, R.F. Werner, Commun. Math. Phys. 144, 443 (1992)MathSciNetMATHADSGoogle Scholar
  41. 41.
    I. Oseledets, SIAM J. Sci. Comput. 33, 2295 (2011)MathSciNetMATHGoogle Scholar
  42. 42.
    D.V. Savostyanov, S.V. Dolgov, J.M. Werner, I. Kuprov, Phys. Rev. B 90, 085139 (2014)Google Scholar
  43. 43.
    S. Tomonaga, Prog. Theor. Phys. 1, 27 (1946)MathSciNetMATHADSGoogle Scholar
  44. 44.
    J. Schwinger, Phys. Rev. 73, 416 (1948)MathSciNetMATHADSGoogle Scholar
  45. 45.
    J. Schwinger, Phys. Rev. 74, 1439 (1948)MathSciNetMATHADSGoogle Scholar
  46. 46.
    R.P. Feynman, Phys. Rev. 76, 769 (1949)MathSciNetMATHADSGoogle Scholar
  47. 47.
    R.P. Feynman, Phys. Rev. 76, 749 (1949)MathSciNetMATHADSGoogle Scholar
  48. 48.
    K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975)ADSGoogle Scholar
  49. 49.
    S.R. White, R.M. Noack, Phys. Rev. Lett. 68, 3487 (1992)ADSGoogle Scholar
  50. 50.
    J. von Neumann, Mathematisch-Physikalische Klasse 1927, 273 (1927)Google Scholar
  51. 51.
    M.B. Plenio, J. Eisert, J. Dreißig, M. Cramer, Phys. Rev. Lett. 94, 060503 (2005)MathSciNetADSGoogle Scholar
  52. 52.
    J. Eisert, M. Cramer, M.B. Plenio, Rev. Mod. Phys. 82, 277 (2010)MathSciNetMATHADSGoogle Scholar
  53. 53.
    K. Van Acoleyen, M. Mariën, F. Verstraete, Phys. Rev. Lett. 111, 170501 (2013)Google Scholar
  54. 54.
    G. Vidal, J.I. Latorre, E. Rico, A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003)ADSGoogle Scholar
  55. 55.
    G. Evenbly, G. Vidal, J. Stat. Phys. 145, 891 (2011)MathSciNetMATHADSGoogle Scholar
  56. 56.
    E.M. Stoudenmire, S.R. White, Ann. Rev. Condens. Matter Phys. 3, 111 (2012)Google Scholar
  57. 57.
    F. Verstraete, J.I. Cirac, Phys. Rev. Lett. 104, 190405 (2010)MathSciNetADSGoogle Scholar
  58. 58.
    F. Verstraete, J.I. Cirac, arXiv:cond-mat/0407066 (2004)
  59. 59.
    G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)ADSGoogle Scholar
  60. 60.
    F. Verstraete, D. Porras, J.I. Cirac, Phys. Rev. Lett. 93, 227205 (2004)ADSGoogle Scholar
  61. 61.
    Y.-Y. Shi, L.-M. Duan, G. Vidal, Phys. Rev. A 74, 022320 (2006)ADSGoogle Scholar
  62. 62.
    A.J. Ferris, Phys. Rev. B 87, 125139 (2013)ADSGoogle Scholar
  63. 63.
    V. Murg, F. Verstraete, Ö. Legeza, R.M. Noack, Phys. Rev. B 82, 205105 (2010)ADSGoogle Scholar
  64. 64.
    V. Murg, F. Verstraete, R. Schneider, P.R. Nagy, Ö. Legeza, arXiv:1403.0981 (2014)
  65. 65.
    T. Xiang, Phys. Rev. B 53, R10445 (1996)ADSGoogle Scholar
  66. 66.
    S. Daul, I. Ciofini, C. Daul, S.R. White, Int. J. Quantum Chem. 79, 331 (2000)Google Scholar
  67. 67.
    A.O. Mitrushchenkov, G. Fano, F. Ortolani, R. Linguerri, P. Palmieri, J. Chem. Phys. 115, 6815 (2001)ADSGoogle Scholar
  68. 68.
    G.K.-L. Chan, M. Head-Gordon, J. Chem. Phys. 116, 4462 (2002)ADSGoogle Scholar
  69. 69.
    Ö. Legeza, J. Röder, B.A. Hess, Phys. Rev. B 67, 125114 (2003)ADSGoogle Scholar
  70. 70.
    G.K.-L. Chan, M. Head-Gordon, J. Chem. Phys. 118, 8551 (2003)ADSGoogle Scholar
  71. 71.
    Ö. Legeza, J. Röder, B.A. Hess, Mol. Phys. 101, 2019 (2003)ADSGoogle Scholar
  72. 72.
    A.O. Mitrushchenkov, R. Linguerri, P. Palmieri, G. Fano, J. Chem. Phys. 119, 4148 (2003)ADSGoogle Scholar
  73. 73.
    Ö. Legeza, J. Sólyom, Phys. Rev. B 68, 195116 (2003)ADSGoogle Scholar
  74. 74.
    G.K.-L. Chan, J. Chem. Phys. 120, 3172 (2004)ADSGoogle Scholar
  75. 75.
    G.K.-L. Chan, M. Kállay, J. Gauss, J. Chem. Phys. 121, 6110 (2004)ADSGoogle Scholar
  76. 76.
    Ö. Legeza, J. Sólyom, Phys. Rev. B 70, 205118 (2004)ADSGoogle Scholar
  77. 77.
    G. Moritz, B.A. Hess, M. Reiher, J. Chem. Phys. 122, 024107 (2005)ADSGoogle Scholar
  78. 78.
    G.K.-L. Chan, T. Van Voorhis, J. Chem. Phys. 122, 204101 (2005)ADSGoogle Scholar
  79. 79.
    G. Moritz, A. Wolf, M. Reiher, J. Chem. Phys. 123, 184105 (2005)ADSGoogle Scholar
  80. 80.
    G. Moritz, M. Reiher, J. Chem. Phys. 124, 034103 (2006)ADSGoogle Scholar
  81. 81.
    J. Hachmann, W. Cardoen, G.K.-L. Chan, J. Chem. Phys. 125, 144101 (2006)ADSGoogle Scholar
  82. 82.
    J. Rissler, R.M. Noack, S.R. White, Chem. Phys. 323, 519 (2006)ADSGoogle Scholar
  83. 83.
    G. Moritz, M. Reiher, J. Chem. Phys. 126, 244109 (2007)ADSGoogle Scholar
  84. 84.
    J.J. Dorando, J. Hachmann, G.K.-L. Chan, J. Chem. Phys. 127, 084109 (2007)ADSGoogle Scholar
  85. 85.
    J. Hachmann, J.J. Dorando, M. Avilés, G.K.-L. Chan, J. Chem. Phys. 127, 134309 (2007)ADSGoogle Scholar
  86. 86.
    K.H. Marti, I.M. Ondík, G. Moritz, M. Reiher, J. Chem. Phys. 128, 014104 (2008)ADSGoogle Scholar
  87. 87.
    D. Zgid, M. Nooijen, J. Chem. Phys. 128, 014107 (2008)ADSGoogle Scholar
  88. 88.
    D. Zgid, M. Nooijen, J. Chem. Phys. 128, 144115 (2008)ADSGoogle Scholar
  89. 89.
    D. Zgid, M. Nooijen, J. Chem. Phys. 128, 144116 (2008)ADSGoogle Scholar
  90. 90.
    D. Ghosh, J. Hachmann, T. Yanai, G.K.-L. Chan, J. Chem. Phys. 128, 144117 (2008)ADSGoogle Scholar
  91. 91.
    G.K.-L. Chan, Phys. Chem. Chem. Phys. 10, 3454 (2008)Google Scholar
  92. 92.
    T. Yanai, Y. Kurashige, D. Ghosh, G.K.-L. Chan, Int. J. Quantum Chem. 109, 2178 (2009)ADSGoogle Scholar
  93. 93.
    J.J. Dorando, J. Hachmann, G.K.-L. Chan, J. Chem. Phys. 130, 184111 (2009)ADSGoogle Scholar
  94. 94.
    Y. Kurashige, T. Yanai, J. Chem. Phys. 130, 234114 (2009)ADSGoogle Scholar
  95. 95.
    T. Yanai, Y. Kurashige, E. Neuscamman, G.K.-L. Chan, J. Chem. Phys. 132, 024105 (2010)ADSGoogle Scholar
  96. 96.
    E. Neuscamman, T. Yanai, G.K.-L. Chan, J. Chem. Phys. 132, 024106 (2010)ADSGoogle Scholar
  97. 97.
    K.H. Marti, M. Reiher, Mol. Phys. 108, 501 (2010)ADSGoogle Scholar
  98. 98.
    H.-G. Luo, M.-P. Qin, T. Xiang, Phys. Rev. B 81, 235129 (2010)ADSGoogle Scholar
  99. 99.
    W. Mizukami, Y. Kurashige, T. Yanai, J. Chem. Phys. 133, 091101 (2010)ADSGoogle Scholar
  100. 100.
    K.H. Marti, B. Bauer, M. Reiher, M. Troyer, F. Verstraete, New J. Phys. 12, 103008 (2010)ADSGoogle Scholar
  101. 101.
    K.H. Marti, M. Reiher, Phys. Chem. Chem. Phys. 13, 6750 (2011)Google Scholar
  102. 102.
    G. Barcza, Ö. Legeza, K.H. Marti, M. Reiher, Phys. Rev. A 83, 012508 (2011)ADSGoogle Scholar
  103. 103.
    K. Boguslawski, K.H. Marti, M. Reiher, J. Chem. Phys. 134, 224101 (2011)ADSGoogle Scholar
  104. 104.
    Y. Kurashige and T. Yanai, J. Chem. Phys. 135, 094104 (2011)ADSGoogle Scholar
  105. 105.
    A.O. Mitrushchenkov, G. Fano, R. Linguerri, P. Palmieri, Int. J. Quantum Chem. 112, 1606 (2012)Google Scholar
  106. 106.
    S. Sharma, G.K.-L. Chan, J. Chem. Phys. 136, 124121 (2012)ADSGoogle Scholar
  107. 107.
    S. Wouters, P.A. Limacher, D. Van Neck, P.W. Ayers, J. Chem. Phys. 136, 134110 (2012)ADSGoogle Scholar
  108. 108.
    K. Boguslawski, K.H. Marti, Ö. Legeza, M. Reiher, J. Chem. Theor. Comput. 8, 1970 (2012)Google Scholar
  109. 109.
    T. Yanai, Y. Kurashige, E. Neuscamman, G.K.-L. Chan, Phys. Chem. Chem. Phys. 14, 7809 (2012)Google Scholar
  110. 110.
    K. Boguslawski, P. Tecmer, Ö. Legeza, M. Reiher, J. Phys. Chem. Lett. 3, 3129 (2012)Google Scholar
  111. 111.
    W. Mizukami, Y. Kurashige, T. Yanai, J. Chem. Theor. Comput. 9, 401 (2013)Google Scholar
  112. 112.
    N. Nakatani, G.K.-L. Chan, J. Chem. Phys. 138, 134113 (2013)ADSGoogle Scholar
  113. 113.
    K. Boguslawski, P. Tecmer, G. Barcza, Ö. Legeza, M. Reiher, J. Chem. Theor. Comput. 9, 2959 (2013)Google Scholar
  114. 114.
    Y. Kurashige, G.K.-L. Chan, T. Yanai, Nat. Chem. 5, 660 (2013)Google Scholar
  115. 115.
    Y. Ma, H. Ma, J. Chem. Phys. 138, 224105 (2013)ADSGoogle Scholar
  116. 116.
    M. Saitow, Y. Kurashige, T. Yanai, J. Chem. Phys. 139, 044118 (2013)ADSGoogle Scholar
  117. 117.
    F. Liu, Y. Kurashige, T. Yanai, K. Morokuma, J. Chem. Theor. Comput. 9, 4462 (2013)Google Scholar
  118. 118.
    P. Tecmer, K. Boguslawski, Ö. Legeza, M. Reiher, Phys. Chem. Chem. Phys. 16, 719 (2014)Google Scholar
  119. 119.
    N. Nakatani, S. Wouters, D. Van Neck, G.K.-L. Chan, J. Chem. Phys. 140, 024108 (2014)ADSGoogle Scholar
  120. 120.
    S. Knecht, Ö. Legeza, M. Reiher, J. Chem. Phys. 140, 041101 (2014)ADSGoogle Scholar
  121. 121.
    S. Wouters, W. Poelmans, P.W. Ayers, D. Van Neck, Comput. Phys. Commun. 185, 1501 (2014)MathSciNetADSGoogle Scholar
  122. 122.
    T.V. Harris, Y. Kurashige, T. Yanai, K. Morokuma, J. Chem. Phys. 140, 054303 (2014)ADSGoogle Scholar
  123. 123.
    M. Mottet, P. Tecmer, K. Boguslawski, Ö. Legeza, M. Reiher, Phys. Chem. Chem. Phys. 16, 8872 (2014)Google Scholar
  124. 124.
    T.N. Lan, Y. Kurashige, T. Yanai, J. Chem. Theor. Comput. 10, 1953 (2014)Google Scholar
  125. 125.
    S. Sharma, T. Yanai, G.H. Booth, C.J. Umrigar, G.K.-L. Chan, J. Chem. Phys. 140, 104112 (2014)ADSGoogle Scholar
  126. 126.
    Y. Kurashige, M. Saitow, J. Chalupsky, T. Yanai, Phys. Chem. Chem. Phys. 16, 11988 (2014)Google Scholar
  127. 127.
    S. Wouters, T. Bogaerts, P. Van Der Voort, V. Van Speybroeck, D. Van Neck, J. Chem. Phys. 140, 241103 (2014)ADSGoogle Scholar
  128. 128.
    E. Fertitta, B. Paulus, G. Barcza, Ö. Legeza, arXiv:1406.7038 (2014)
  129. 129.
    G.K.-L. Chan, J.J. Dorando, D. Ghosh, J. Hachmann, E. Neuscamman, H. Wang, T. Yanai, Frontiers in Quantum Systems in Chemistry and Physics, in Progress in Theoretical Chemistry and Physics, edited by S. Wilson, P.J. Grout, J. Maruani, G. Delgado-Barrio, P. Piecuch (Springer, 2008), Vol. 18, pp. 49–65Google Scholar
  130. 130.
    G.K.-L. Chan, D. Zgid, The Density Matrix Renormalization Group in Quantum Chemistry, in Annual Reports in Computational Chemistry (Elsevier, 2009), Vol. 5, Chap. 7, pp. 149–162Google Scholar
  131. 131.
    K.H. Marti, M. Reiher, Z. Phys. Chem. 224, 583 (2010)Google Scholar
  132. 132.
    G.K.-L. Chan, S. Sharma, Ann. Rev. Phys. Chem. 62, 465 (2011)ADSGoogle Scholar
  133. 133.
    G.K.-L. Chan, WIREs Comput. Mol. Sci. 2, 907 (2012)Google Scholar
  134. 134.
    Y. Kurashige, Mol. Phys. 112, 1485 (2014)ADSGoogle Scholar
  135. 135.
    S.F. Keller, M. Reiher, Chimia 68, 200 (2014)Google Scholar
  136. 136.
    C. Lanczos, J. Res. Nat. Bureau Stand. 45, 255 (1950)MathSciNetGoogle Scholar
  137. 137.
    E.R. Davidson, J. Comput. Phys. 17, 87 (1975)MATHADSGoogle Scholar
  138. 138.
    P. Jordan, E. Wigner, Z. Phys. 47, 631 (1928)MATHADSGoogle Scholar
  139. 139.
    S. Wouters, Ph.D. thesis, Ghent University, 2014Google Scholar
  140. 140.
    C. Edmiston, K. Ruedenberg, Rev. Mod. Phys. 35, 457 (1963)MATHADSGoogle Scholar
  141. 141.
    Ö. Legeza, G. Fáth, Phys. Rev. B 53, 14349 (1996)ADSGoogle Scholar
  142. 142.
    G.K.-L. Chan, P.W. Ayers, E.S. Croot III, J. Stat. Phys. 109, 289 (2002)MathSciNetMATHGoogle Scholar
  143. 143.
    S.R. White, Phys. Rev. B 72, 180403 (2005)ADSGoogle Scholar
  144. 144.
    S.R. White, Phys. Rev. Lett. 77, 3633 (1996)ADSGoogle Scholar
  145. 145.
    B.C. Carlson, J.M. Keller, Phys. Rev. 105, 102 (1957)MathSciNetMATHADSGoogle Scholar
  146. 146.
    J. Pipek, P.G. Mezey, J. Chem. Phys. 90, 4916 (1989)ADSGoogle Scholar
  147. 147.
    A.O. Mitrushchenkov, G. Fano, R. Linguerri, P. Palmieri, arXiv:cond-mat/0306058 (2003)
  148. 148.
    J. Hubbard, Proc. Roy. Soc. Lond. Ser. A 276, 238 (1963)ADSGoogle Scholar
  149. 149.
    K. Hallberg, Density Matrix Renormalization, in Theoretical Methods for Strongly Correlated Electrons, edited by D. Sénéchal, A.-M. Tremblay, C. Bourbonnais, CRM Series in Mathematical Physics (Springer, New York, 2004), Chap. 1, pp. 3–37Google Scholar
  150. 150.
    B. Pirvu, J. Haegeman, F. Verstraete, Phys. Rev. B 85, 035130 (2012)ADSGoogle Scholar
  151. 151.
    J. Haegeman, B. Pirvu, D.J. Weir, J.I. Cirac, T.J. Osborne, H. Verschelde, F. Verstraete, Phys. Rev. B 85, 100408 (2012)ADSGoogle Scholar
  152. 152.
    S. Wouters, N. Nakatani, D. Van Neck, G.K.-L. Chan, Phys. Rev. B 88, 075122 (2013)ADSGoogle Scholar
  153. 153.
    J. Haegeman, T.J. Osborne, F. Verstraete, Phys. Rev. B 88, 075133 (2013)ADSGoogle Scholar
  154. 154.
    J. Haegeman, J.I. Cirac, T.J. Osborne, I. Pižorn, H. Verschelde, F. Verstraete, Phys. Rev. Lett. 107, 070601 (2011)ADSGoogle Scholar
  155. 155.
    J.M. Kinder, C.C. Ralph, G.K.-L. Chan, Quantum Information and Computation for Chemistry, in Advances in Chemical Physics, edited by S. Kais (John Wiley & Sons, 2014), Vol. 154, Chap. 7, pp. 179–192Google Scholar
  156. 156.
    F. Mezzacapo, N. Schuch, M. Boninsegni, J.I. Cirac, New J. Phys. 11, 083026 (2009)ADSGoogle Scholar
  157. 157.
    H. Weyl, Gruppentheorie und Quantenmechanik (Hirzel, Leipzig, 1928)Google Scholar
  158. 158.
    E. Wigner, Ann. Math. 40, 149 (1939)MathSciNetGoogle Scholar
  159. 159.
    G. Sierra, T. Nishino, Nucl. Phys. B 495, 505 (1997)MathSciNetMATHADSGoogle Scholar
  160. 160.
    I.P. McCulloch, M. Gulácsi, Austr. J. Phys. 53, 597 (2000)MATHGoogle Scholar
  161. 161.
    I.P. McCulloch, M. Gulácsi, Philos. Mag. Lett. 81, 447 (2001)Google Scholar
  162. 162.
    I.P. McCulloch, M. Gulácsi, Europhys. Lett. 57, 852 (2002)ADSGoogle Scholar
  163. 163.
    I.P. McCulloch, J. Stat. Mech.: Theory Exp. 2007, P10014 (2007)Google Scholar
  164. 164.
    S. Singh, H.-Q. Zhou, G. Vidal, New J. Phys. 12, 033029 (2010)ADSGoogle Scholar
  165. 165.
    S. Singh, R.N.C. Pfeifer, G. Vidal, Phys. Rev. A 82, 050301 (2010)MathSciNetADSGoogle Scholar
  166. 166.
    S. Singh, G. Vidal, Phys. Rev. B 86, 195114 (2012)ADSGoogle Scholar
  167. 167.
    S. Pittel, N. Sandulescu, Phys. Rev. C 73, 014301 (2006)ADSGoogle Scholar
  168. 168.
    J. Rotureau, N. Michel, W. Nazarewicz, M. Płoszajczak, J. Dukelsky, Phys. Rev. Lett. 97, 110603 (2006)ADSGoogle Scholar
  169. 169.
    B. Thakur, S. Pittel, N. Sandulescu, Phys. Rev. C 78, 041303 (2008)ADSGoogle Scholar
  170. 170.
    A. Weichselbaum, Ann. Phys. 327, 2972 (2012)MathSciNetMATHADSGoogle Scholar
  171. 171.
    Ö. Legeza, J. Sólyom, Phys. Rev. B 56, 14449 (1997)ADSGoogle Scholar
  172. 172.
    W.H. Dickhoff, D. Van Neck, Many-body theory exposed!, 2nd edn. (World Scientific, 2008)Google Scholar
  173. 173.
    S. Sharma, G.K.-L. Chan, Block code for DMRG (2012), http://www.princeton.edu/chemistry/chan/software/dmrg/
  174. 174.
    S. Wouters, CheMPS2: a spin-adapted implementation of DMRG for ab initio quantum chemistry (2014), https://github.com/SebWouters/CheMPS2
  175. 175.
    E.M. Stoudenmire, S.R. White, Phys. Rev. B 87, 155137 (2013)ADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Center for Molecular ModelingGhent UniversityZwijnaardeBelgium

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