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.
Similar content being viewed by others
References
D.R. Hartree, Math. Proc. Cambridge Philos. Soc. 24, 89 (1928)
J.C. Slater, Phys. Rev. 32, 339 (1928)
V. Fock, Z. Phys. 61, 126 (1926)
T. Helgaker, P. Jørgensen, J. Olsen, Molecular electronic-structure theory, 1st edn. (Wiley, New-York, 2000)
P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964)
W. Kohn, L.J. Sham, Phys. Rev. 140, A1133 (1965)
R.M. Dickson, A.D. Becke, J. Chem. Phys. 123, 111101 (2005)
C. Møller, M.S. Plesset, Phys. Rev. 46, 618 (1934)
J.C. Slater, Phys. Rev. 34, 1293 (1929)
E.U. Condon, Phys. Rev. 36, 1121 (1930)
F. Coester, Nucl. Phys. 7, 421 (1958)
F. Coester, H. Kümmel, Nucl. Phys. 17, 477 (1960)
J. Čížek, J. Chem. Phys. 45, 4256 (1966)
B.O. Roos, Int. J. Quantum Chem. 18, 175 (1980)
B.O. Roos, P.R. Taylor, P.E.M. Siegbahn, Chem. Phys. 48, 157 (1980)
P.E.M. Siegbahn, J. Almlöf, A. Heiberg, B.O. Roos, J. Chem. Phys. 74, 2384 (1981)
P.-Å. Malmqvist, A. Rendell, B.O. Roos, J. Phys. Chem. 94, 5477 (1990)
K. Andersson, P.-Å. Malmqvist, B.O. Roos, J. Chem. Phys. 96, 1218 (1992)
R.J. Buenker, S.D. Peyerimhoff, Theor. Chim. Acta 35, 33 (1974)
H.-J. Werner, E.-A. Reinsch, J. Chem. Phys. 76, 3144 (1982)
P.E.M. Siegbahn, J. Chem. Phys. 70, 5391 (1979)
P.E.M. Siegbahn, J. Chem. Phys. 72, 1647 (1980)
B.R. Brooks, H.F. Schaefer, J. Chem. Phys. 70, 5092 (1979)
N. Oliphant, L. Adamowicz, J. Chem. Phys. 96, 3739 (1992)
L.Z. Stolarczyk, Chem. Phys. Lett. 217, 1 (1994)
T. Yanai, G.K.-L. Chan, J. Chem. Phys. 124, 194106 (2006)
S.R. White, R.L. Martin, J. Chem. Phys. 110, 4127 (1999)
J.F. Cornwell, in Group theory in physics, 1st edn. (Academic Press Inc., Ltd., London, 1984), Vols. 1 and 2
S.R. White, Phys. Rev. Lett. 69, 2863 (1992)
S.R. White, Phys. Rev. B 48, 10345 (1993)
S. Östlund, S. Rommer, Phys. Rev. Lett. 75, 3537 (1995)
S. Rommer, S. Östlund, Phys. Rev. B 55, 2164 (1997)
M.B. Hastings, J. Stat. Mech.: Theor. Exp. 2007, P08024 (2007)
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
H.A. Kramers, G.H. Wannier, Phys. Rev. 60, 263 (1941)
R.J. Baxter, J. Math. Phys. 9, 650 (1968)
M.P. Nightingale, H.W.J. Blöte, Phys. Rev. B 33, 659 (1986)
I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki, Phys. Rev. Lett. 59, 799 (1987)
M. Fannes, B. Nachtergaele, R.F. Werner, Europhys. Lett. 10, 633 (1989)
M. Fannes, B. Nachtergaele, R.F. Werner, Commun. Math. Phys. 144, 443 (1992)
I. Oseledets, SIAM J. Sci. Comput. 33, 2295 (2011)
D.V. Savostyanov, S.V. Dolgov, J.M. Werner, I. Kuprov, Phys. Rev. B 90, 085139 (2014)
S. Tomonaga, Prog. Theor. Phys. 1, 27 (1946)
J. Schwinger, Phys. Rev. 73, 416 (1948)
J. Schwinger, Phys. Rev. 74, 1439 (1948)
R.P. Feynman, Phys. Rev. 76, 769 (1949)
R.P. Feynman, Phys. Rev. 76, 749 (1949)
K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975)
S.R. White, R.M. Noack, Phys. Rev. Lett. 68, 3487 (1992)
J. von Neumann, Mathematisch-Physikalische Klasse 1927, 273 (1927)
M.B. Plenio, J. Eisert, J. Dreißig, M. Cramer, Phys. Rev. Lett. 94, 060503 (2005)
J. Eisert, M. Cramer, M.B. Plenio, Rev. Mod. Phys. 82, 277 (2010)
K. Van Acoleyen, M. Mariën, F. Verstraete, Phys. Rev. Lett. 111, 170501 (2013)
G. Vidal, J.I. Latorre, E. Rico, A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003)
G. Evenbly, G. Vidal, J. Stat. Phys. 145, 891 (2011)
E.M. Stoudenmire, S.R. White, Ann. Rev. Condens. Matter Phys. 3, 111 (2012)
F. Verstraete, J.I. Cirac, Phys. Rev. Lett. 104, 190405 (2010)
F. Verstraete, J.I. Cirac, arXiv:cond-mat/0407066 (2004)
G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)
F. Verstraete, D. Porras, J.I. Cirac, Phys. Rev. Lett. 93, 227205 (2004)
Y.-Y. Shi, L.-M. Duan, G. Vidal, Phys. Rev. A 74, 022320 (2006)
A.J. Ferris, Phys. Rev. B 87, 125139 (2013)
V. Murg, F. Verstraete, Ö. Legeza, R.M. Noack, Phys. Rev. B 82, 205105 (2010)
V. Murg, F. Verstraete, R. Schneider, P.R. Nagy, Ö. Legeza, arXiv:1403.0981 (2014)
T. Xiang, Phys. Rev. B 53, R10445 (1996)
S. Daul, I. Ciofini, C. Daul, S.R. White, Int. J. Quantum Chem. 79, 331 (2000)
A.O. Mitrushchenkov, G. Fano, F. Ortolani, R. Linguerri, P. Palmieri, J. Chem. Phys. 115, 6815 (2001)
G.K.-L. Chan, M. Head-Gordon, J. Chem. Phys. 116, 4462 (2002)
Ö. Legeza, J. Röder, B.A. Hess, Phys. Rev. B 67, 125114 (2003)
G.K.-L. Chan, M. Head-Gordon, J. Chem. Phys. 118, 8551 (2003)
Ö. Legeza, J. Röder, B.A. Hess, Mol. Phys. 101, 2019 (2003)
A.O. Mitrushchenkov, R. Linguerri, P. Palmieri, G. Fano, J. Chem. Phys. 119, 4148 (2003)
Ö. Legeza, J. Sólyom, Phys. Rev. B 68, 195116 (2003)
G.K.-L. Chan, J. Chem. Phys. 120, 3172 (2004)
G.K.-L. Chan, M. Kállay, J. Gauss, J. Chem. Phys. 121, 6110 (2004)
Ö. Legeza, J. Sólyom, Phys. Rev. B 70, 205118 (2004)
G. Moritz, B.A. Hess, M. Reiher, J. Chem. Phys. 122, 024107 (2005)
G.K.-L. Chan, T. Van Voorhis, J. Chem. Phys. 122, 204101 (2005)
G. Moritz, A. Wolf, M. Reiher, J. Chem. Phys. 123, 184105 (2005)
G. Moritz, M. Reiher, J. Chem. Phys. 124, 034103 (2006)
J. Hachmann, W. Cardoen, G.K.-L. Chan, J. Chem. Phys. 125, 144101 (2006)
J. Rissler, R.M. Noack, S.R. White, Chem. Phys. 323, 519 (2006)
G. Moritz, M. Reiher, J. Chem. Phys. 126, 244109 (2007)
J.J. Dorando, J. Hachmann, G.K.-L. Chan, J. Chem. Phys. 127, 084109 (2007)
J. Hachmann, J.J. Dorando, M. Avilés, G.K.-L. Chan, J. Chem. Phys. 127, 134309 (2007)
K.H. Marti, I.M. Ondík, G. Moritz, M. Reiher, J. Chem. Phys. 128, 014104 (2008)
D. Zgid, M. Nooijen, J. Chem. Phys. 128, 014107 (2008)
D. Zgid, M. Nooijen, J. Chem. Phys. 128, 144115 (2008)
D. Zgid, M. Nooijen, J. Chem. Phys. 128, 144116 (2008)
D. Ghosh, J. Hachmann, T. Yanai, G.K.-L. Chan, J. Chem. Phys. 128, 144117 (2008)
G.K.-L. Chan, Phys. Chem. Chem. Phys. 10, 3454 (2008)
T. Yanai, Y. Kurashige, D. Ghosh, G.K.-L. Chan, Int. J. Quantum Chem. 109, 2178 (2009)
J.J. Dorando, J. Hachmann, G.K.-L. Chan, J. Chem. Phys. 130, 184111 (2009)
Y. Kurashige, T. Yanai, J. Chem. Phys. 130, 234114 (2009)
T. Yanai, Y. Kurashige, E. Neuscamman, G.K.-L. Chan, J. Chem. Phys. 132, 024105 (2010)
E. Neuscamman, T. Yanai, G.K.-L. Chan, J. Chem. Phys. 132, 024106 (2010)
K.H. Marti, M. Reiher, Mol. Phys. 108, 501 (2010)
H.-G. Luo, M.-P. Qin, T. Xiang, Phys. Rev. B 81, 235129 (2010)
W. Mizukami, Y. Kurashige, T. Yanai, J. Chem. Phys. 133, 091101 (2010)
K.H. Marti, B. Bauer, M. Reiher, M. Troyer, F. Verstraete, New J. Phys. 12, 103008 (2010)
K.H. Marti, M. Reiher, Phys. Chem. Chem. Phys. 13, 6750 (2011)
G. Barcza, Ö. Legeza, K.H. Marti, M. Reiher, Phys. Rev. A 83, 012508 (2011)
K. Boguslawski, K.H. Marti, M. Reiher, J. Chem. Phys. 134, 224101 (2011)
Y. Kurashige and T. Yanai, J. Chem. Phys. 135, 094104 (2011)
A.O. Mitrushchenkov, G. Fano, R. Linguerri, P. Palmieri, Int. J. Quantum Chem. 112, 1606 (2012)
S. Sharma, G.K.-L. Chan, J. Chem. Phys. 136, 124121 (2012)
S. Wouters, P.A. Limacher, D. Van Neck, P.W. Ayers, J. Chem. Phys. 136, 134110 (2012)
K. Boguslawski, K.H. Marti, Ö. Legeza, M. Reiher, J. Chem. Theor. Comput. 8, 1970 (2012)
T. Yanai, Y. Kurashige, E. Neuscamman, G.K.-L. Chan, Phys. Chem. Chem. Phys. 14, 7809 (2012)
K. Boguslawski, P. Tecmer, Ö. Legeza, M. Reiher, J. Phys. Chem. Lett. 3, 3129 (2012)
W. Mizukami, Y. Kurashige, T. Yanai, J. Chem. Theor. Comput. 9, 401 (2013)
N. Nakatani, G.K.-L. Chan, J. Chem. Phys. 138, 134113 (2013)
K. Boguslawski, P. Tecmer, G. Barcza, Ö. Legeza, M. Reiher, J. Chem. Theor. Comput. 9, 2959 (2013)
Y. Kurashige, G.K.-L. Chan, T. Yanai, Nat. Chem. 5, 660 (2013)
Y. Ma, H. Ma, J. Chem. Phys. 138, 224105 (2013)
M. Saitow, Y. Kurashige, T. Yanai, J. Chem. Phys. 139, 044118 (2013)
F. Liu, Y. Kurashige, T. Yanai, K. Morokuma, J. Chem. Theor. Comput. 9, 4462 (2013)
P. Tecmer, K. Boguslawski, Ö. Legeza, M. Reiher, Phys. Chem. Chem. Phys. 16, 719 (2014)
N. Nakatani, S. Wouters, D. Van Neck, G.K.-L. Chan, J. Chem. Phys. 140, 024108 (2014)
S. Knecht, Ö. Legeza, M. Reiher, J. Chem. Phys. 140, 041101 (2014)
S. Wouters, W. Poelmans, P.W. Ayers, D. Van Neck, Comput. Phys. Commun. 185, 1501 (2014)
T.V. Harris, Y. Kurashige, T. Yanai, K. Morokuma, J. Chem. Phys. 140, 054303 (2014)
M. Mottet, P. Tecmer, K. Boguslawski, Ö. Legeza, M. Reiher, Phys. Chem. Chem. Phys. 16, 8872 (2014)
T.N. Lan, Y. Kurashige, T. Yanai, J. Chem. Theor. Comput. 10, 1953 (2014)
S. Sharma, T. Yanai, G.H. Booth, C.J. Umrigar, G.K.-L. Chan, J. Chem. Phys. 140, 104112 (2014)
Y. Kurashige, M. Saitow, J. Chalupsky, T. Yanai, Phys. Chem. Chem. Phys. 16, 11988 (2014)
S. Wouters, T. Bogaerts, P. Van Der Voort, V. Van Speybroeck, D. Van Neck, J. Chem. Phys. 140, 241103 (2014)
E. Fertitta, B. Paulus, G. Barcza, Ö. Legeza, arXiv:1406.7038 (2014)
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. 49–65
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. 149–162
K.H. Marti, M. Reiher, Z. Phys. Chem. 224, 583 (2010)
G.K.-L. Chan, S. Sharma, Ann. Rev. Phys. Chem. 62, 465 (2011)
G.K.-L. Chan, WIREs Comput. Mol. Sci. 2, 907 (2012)
Y. Kurashige, Mol. Phys. 112, 1485 (2014)
S.F. Keller, M. Reiher, Chimia 68, 200 (2014)
C. Lanczos, J. Res. Nat. Bureau Stand. 45, 255 (1950)
E.R. Davidson, J. Comput. Phys. 17, 87 (1975)
P. Jordan, E. Wigner, Z. Phys. 47, 631 (1928)
S. Wouters, Ph.D. thesis, Ghent University, 2014
C. Edmiston, K. Ruedenberg, Rev. Mod. Phys. 35, 457 (1963)
Ö. Legeza, G. Fáth, Phys. Rev. B 53, 14349 (1996)
G.K.-L. Chan, P.W. Ayers, E.S. Croot III, J. Stat. Phys. 109, 289 (2002)
S.R. White, Phys. Rev. B 72, 180403 (2005)
S.R. White, Phys. Rev. Lett. 77, 3633 (1996)
B.C. Carlson, J.M. Keller, Phys. Rev. 105, 102 (1957)
J. Pipek, P.G. Mezey, J. Chem. Phys. 90, 4916 (1989)
A.O. Mitrushchenkov, G. Fano, R. Linguerri, P. Palmieri, arXiv:cond-mat/0306058 (2003)
J. Hubbard, Proc. Roy. Soc. Lond. Ser. A 276, 238 (1963)
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. 3–37
B. Pirvu, J. Haegeman, F. Verstraete, Phys. Rev. B 85, 035130 (2012)
J. Haegeman, B. Pirvu, D.J. Weir, J.I. Cirac, T.J. Osborne, H. Verschelde, F. Verstraete, Phys. Rev. B 85, 100408 (2012)
S. Wouters, N. Nakatani, D. Van Neck, G.K.-L. Chan, Phys. Rev. B 88, 075122 (2013)
J. Haegeman, T.J. Osborne, F. Verstraete, Phys. Rev. B 88, 075133 (2013)
J. Haegeman, J.I. Cirac, T.J. Osborne, I. Pižorn, H. Verschelde, F. Verstraete, Phys. Rev. Lett. 107, 070601 (2011)
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. 179–192
F. Mezzacapo, N. Schuch, M. Boninsegni, J.I. Cirac, New J. Phys. 11, 083026 (2009)
H. Weyl, Gruppentheorie und Quantenmechanik (Hirzel, Leipzig, 1928)
E. Wigner, Ann. Math. 40, 149 (1939)
G. Sierra, T. Nishino, Nucl. Phys. B 495, 505 (1997)
I.P. McCulloch, M. Gulácsi, Austr. J. Phys. 53, 597 (2000)
I.P. McCulloch, M. Gulácsi, Philos. Mag. Lett. 81, 447 (2001)
I.P. McCulloch, M. Gulácsi, Europhys. Lett. 57, 852 (2002)
I.P. McCulloch, J. Stat. Mech.: Theory Exp. 2007, P10014 (2007)
S. Singh, H.-Q. Zhou, G. Vidal, New J. Phys. 12, 033029 (2010)
S. Singh, R.N.C. Pfeifer, G. Vidal, Phys. Rev. A 82, 050301 (2010)
S. Singh, G. Vidal, Phys. Rev. B 86, 195114 (2012)
S. Pittel, N. Sandulescu, Phys. Rev. C 73, 014301 (2006)
J. Rotureau, N. Michel, W. Nazarewicz, M. Płoszajczak, J. Dukelsky, Phys. Rev. Lett. 97, 110603 (2006)
B. Thakur, S. Pittel, N. Sandulescu, Phys. Rev. C 78, 041303 (2008)
A. Weichselbaum, Ann. Phys. 327, 2972 (2012)
Ö. Legeza, J. Sólyom, Phys. Rev. B 56, 14449 (1997)
W.H. Dickhoff, D. Van Neck, Many-body theory exposed!, 2nd edn. (World Scientific, 2008)
S. Sharma, G.K.-L. Chan, Block code for DMRG (2012), http://www.princeton.edu/chemistry/chan/software/dmrg/
S. Wouters, CheMPS2: a spin-adapted implementation of DMRG for ab initio quantum chemistry (2014), https://github.com/SebWouters/CheMPS2
E.M. Stoudenmire, S.R. White, Phys. Rev. B 87, 155137 (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wouters, S., Van Neck, D. The density matrix renormalization group for ab initio quantum chemistry. Eur. Phys. J. D 68, 272 (2014). https://doi.org/10.1140/epjd/e2014-50500-1
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjd/e2014-50500-1