Abstract
We review recently developed methods to efficiently utilize the Cholesky decomposition technique in electronic structure calculations. The review starts with a brief introduction to the basics of the Cholesky decomposition technique. Subsequently, examples of applications of the technique to ab inito procedures are presented. The technique is demonstrated to be a special type of a resolution-of-identity or density-fitting scheme. This is followed by explicit examples of the Cholesky techniques used in orbital localization, computation of the exchange contribution to the Fock matrix, in MP2, gradient calculations, and so-called method specific Cholesky decomposition. Subsequently, examples of calibration of the method with respect to computed total energies, excitation energies, and auxiliary basis set pruning are presented. In particular, it is demonstrated that the Cholesky method is an unbiased method to derive auxiliary basis sets. Furthermore, details of the implementational considerations are put forward and examples from a parallel Cholesky decomposition scheme is presented. Finally, an outlook and perspectives are presented, followed by a summary and conclusions section. We are of the opinion that the Cholesky decomposition method is a technique that has been overlooked for too long. We have just recently started to understand how to efficiently incorporate the method in existing ab initio programs. The full potential of the Cholesky technique has not yet been fully explored.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Whitten JL (1973) J Chem Phys 58:4496
Baerends EJ, Ellis DE, Ros P (1973) Chem Phys 2:41
Jafri JA, Whitten JL (1974) J Chem Phys 61:2116
Sambe H, Felton RH (1975) J Chem Phys 62:1122
Dunlap BI, Connolly JWD, Sabin JR (1979) J Chem Phys 71:3396
Dunlap BI, Connolly JWD, Sabin JR (1979) J Chem Phys 71, 4993
Feyereisen M, Fitzgerald G, Komornicki A (1993) Chem Phys Lett 208:359
Vahtras O, Almlöf, Feyereisen M (1993) Chem Phys Lett 213:514
Gallant RT, St-Amant A (1996) Chem Phys Lett 256:569
Fonseca Guerra C, Snijders JG, te Velde G, Baerends EJ (1998) Theor Chem Acc 99:391
Dunlap BI (2000) J Mol Struct Theochem 501–502:221
Sierka M, Hogekamp A, Ahlrichs R (2003) J Chem Phys 118:9136
Sodt A, Subotnik JE, Head-Gordon M (2006) J Chem Phys 125:194109
Sałek P, Høst S, Thøgersen L, Jørgensen P, Manninen P, Olsen J, Jansík B, Reine S, Pawłowski F, Tellgren E, Helgaker T, Coriani S (2007) J Chem Phys 126:Art.Nr. 114110
Reine S, Tellgren E, Krapp A, Kjærgaard T, Helgaker T, Jansík B, Høst S, Sałek P (2008) J Chem Phys 129:Art.Nr. 104101
Köster AM del Campo JM, Janetzko F, Zuniga-Gutierrez B (2009) J Chem Phys 130:Art.Nr. 114106
Domínguez-Soria, G Geudtner, JL Morales, P Calaminici, Köster, AM (2009) J Chem Phys 131:Art.Nr. 124102
Burow AM, Sierka M, Mohamed F (2009) J Chem Phys 131:Art.Nr. 214101
Weigend F (2002) Phys Chem Chem Phys 4:4285
Polly R, Werner HJ, Manby FR, Knowles P (2004) Mol Phys 102:2311
Aquilante F, Pedersen TB, Lindh R (2007) J Chem Phys 126:Art.Nr. 194106
Sodt AM, Head-Gordon (2008) J Chem Phys 128:Art.Nr. 104106
Benoît Cdt (1924) Bull Géodésique 2:67
Beebe NHF, Linderberg J (1977) Int J Quantum Chem 12:683
Røeggen I, Wisløff-Nilssen E (1986) Chem Phys Lett 132:154
Røeggen I, Veseth L (2005) Int J Quantum Chem 101:201
Røeggen I (2006) J Chem Phys 124:Art.Nr.184502
Røeggen I (2006) Theor Chem Acc 116:683
Røeggen I (2007) J Chem Phys 126:Art.Nr. 204303
Røeggen I, Johansen T (2008) J Chem Phys 128:Art.Nr. 194107
O’Neal DW, Simons J (1989) Int J Quantum Chem 36:673
Koch, H Sánchez de Merás A, Pedersen TB (2003) J Chem Phys 118:9481
Pedersen TB, Sánchez de Merás A, Koch H (2004) J Chem Phys 120:8887
Christiansen O, Koch H, Jörgensen P (1995) Chem Phys Lett 243:409
García Cuesta I, Pedersen TB, Koch H, Sánchez de Merás AMJ (2004) Chem Phys Lett 390:170
Pedersen TB, Koch H, Boman L, Sánchez de Merás AMJ (2004) Chem Phys Lett 393:319
García Cuesta I, Pedersen TB, Koch H, Sánchez de Merás A (2006) Chem Phys Chem 7:2503
Fernández B, Pedersen TB, Sánchez de Merás A, Koch H (2007) Chem Phys Lett 441:332
García Cuesta I, Sánchez Marín J, Pedersen TB, Koch H, Sánchez de Merás A (2008) Phys Chem Chem Phys 10:361
Boman L, Koch H, Sánchez de Merás A (2008) J Chem Phys 129:Art.Nr. 134107
Pedersen TB, Kongsted J, Crawford TD, Ruud K (2009) J Chem Phys 130:Art.Nr. 034310
Pedersen TB, Kongsted J, Crawford TD (2009) Chirality 21:S68
Sánchez de Merás A, Koch H, García Cuesta I, Boman L (2010) J Chem Phys 132: Art.Nr. 204105
Aquilante F, De Vico L, Ferré N, Ghigo G, Malmqvist PA, Neogrády P,Pedersen TB, Pitoňák M, Reiher M, Roos BO, Serrano-Andrés L, Urban, M Veryazov V, Lindh R (2010) J Comput Chem 31:224
Aquilante F, Pedersen TB, Roos BO, Sánchez de Merás A, Koch H (2008) J Chem Phys 129:Art.Nr. 24113
Aquilante F, Malmqvist PÅ, Pedersen TB, Ghosh A, Roos BO (2006) J Chem Theory Comput 4:694
Aquilante F, Pedersen TB (2007) Chem Phys Lett 449:354
Aquilante F, Pedersen TB, Sánchez de Merás A, Koch H (2006) J Chem Phys 125:Art.Nr. 174101
Aquilante F, Lindh R, Pedersen TB (2007) J Chem Phys 127:Art.Nr. 114107
Öhrn A, Aquilante F (2007) Phys Chem Chem Phys 9:470
Roos BO, Veryazov V, Conradie J, Taylor PR, Ghosh A (2008) J Phys Chem B 112:14099
Aquilante F, Lindh R, Pedersen TB (2008) J Chem Phys 129:Art.Nr. 034106
Pierloot K, Vancoillie S (2008) J Chem Phys 128:Art.Nr. 034104
Radoń M, Pierloot K (2008) J Phys Chem A 112:11824
Bozoglian F, Romain S, Ertem MZ, Todorova TK, Sens C, Mola J, Montserrat R, Romero I, Benet-Buchholz J, Fontrodona X, Cramer CJ, Gagliardi L, Llobet A (2009) J Am Chem Soc 131:15176
Söderhjelm P, Aquilante F, Ryde U (2009) J Phys Chem B 113:11085
Srnec M, Aquilante F, Ryde U, Rulíšek L (2009) J Phys Chem B 113:6074
Pedersen TB, Aquilante F, Lindh R (2009) Theor Chem Acc 124:1
Huber SM, Shahi ARM, Aquilante F, Cramer CJ, Gagliardi L (2009) J Chem Theory Comput 5:2967
Huber SM, Ertem MZ, Aquilante F, Gagliardi L, Tolman WB, Cramer CJ (2009) Chem Eur J 15:4886
Boström J, Aquilante F, Pedersen TB, Lindh R (2009) J Chem Theory Comput 5:1545
Aquilante F, Gagliardi L, Pedersen TB, Lindh R (2009) J Chem Phys 130:Art.Nr. 154107
Riley KE, Pitonňák M, Černý J, Hobza P (2010) J Chem Theory Comput 6:66
Pitoňák M, Heßelmann A (2010) J Chem Theory Comput 6:168
Pitoňák M, Neogŕady P, Hobza P (2010) Phys Chem Chem Phys 12:1369
Vysotskiy VP, Cederbaum LS (2010) J Chem Phys 132:Art.Nr. 044110
Chwee TS, Carter EA (2010) J Chem Phys 132:074104
Tabookht Z, López X, de Graaf C (2010) J Phys Chem A 114:2028
Boström J, Delcey MG, Aquilante F, Serrano-Andrés L, Pedersen TB, Lindh R (2010) J Chem Theory Comput 6:P747
Löwdin PO (1965) J Chem Phys 43:S175
Löwdin PO (1970) Int J Quantum Chem S4:231
Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. Johns Hopkins University Press, Baltimore, MD
Koch H, Sánchez de Merás A (2000) J Chem Phys 113:508
Eichkorn K, Treutler O, Öhm H, Häser M, Ahlrichs R (1995) Chem Phys Lett 240:283
Weigend F, Köhn A, Hättig C (2002) J Chem Phys 116:3175
Weigend F (2006) Phys Chem Chem Phys 8:1057
Jung Y, Sodt A, Gill PMW, Head-Gordon M (2005) Proc Natl Acad Sci USA 102:6692
Werner HJ, Manby FR (2006) J Chem Phys 124:Art.Nr. 054114
Boys SF (1960) Rev Mod Phys 32:296
Edmiston C, Ruedenberg K (1963) Rev Mod Phys 35:457
Pipek J, Mezey PG (1989) J Chem Phys 90:4916
Schweizer S, Kussmann J, Doser B, Ochsenfeld C (2008) J Comput Chem 29:1004
Ziołkowski M, Jansík B, Jørgensen P, Olsen J (2009) J Chem Phys 131:Art.Nr. 124112
Dunning TH Jr (1989) J Chem Phys 90:1007
Fernández B, Koch H, Makarewicz J (1999) J Chem Phys 111:5922
Scuseria GE, Ayala PY (1999) J Chem Phys 111:8330
Weigend F, Häser M (1997) Theor Chem Acc 97:331
Hättig C (2003) J Chem Phys 118:7751
Köhn A, Hättig C J Chem Phys 119:5021
Deglmann P, May K, Furche F, Ahlrichs R (2004) Chem Phys Lett 384:103
Rappoport D, Furche F (2005) J Chem Phys 122:Art.Nr. 064105
Baker J, Chan F (1996) J Comput Chem 17:888
Curtiss LA, Raghavachari K, Redfern PC, Pople JA (1997) J Chem Phys 106:1063
Schultz NE, Zhao Y, Truhlar DG (2005) J Phys Chem A 109:11127
Schreiber M, Silva-Junior M, Sauer SPA, Thiel W (2008) J Chem Phys 128:Art.Nr. 134110
Acknowledgments
ASM acknowledges financial suport from Spanish FEDER+MEC project CTQ2007-67143-C02-01/BQU. RL and JB would like to thank the Swedish Research Council directly and through the Linnaeus Center of Excellence on Organizing Molecular Matter at Lund University, Sweden, for financial support. TBP acknowledges financial support from the CoE Centre for Theoretical and Computational Chemistry (179568/V30). FA would like to acknowledge funding from the Swiss National Science Foundation (SNF), Grant No. 200020-120007. HK and LB would like to thank the Norwegian Research Council for support (Grant No. 154011/420). The authors thank Eirik Hjertenæs at the Norwegian University of Science and Technology for proof-reading the manuscript.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Aquilante, F. et al. (2011). Cholesky Decomposition Techniques in Electronic Structure Theory. In: Zalesny, R., Papadopoulos, M., Mezey, P., Leszczynski, J. (eds) Linear-Scaling Techniques in Computational Chemistry and Physics. Challenges and Advances in Computational Chemistry and Physics, vol 13. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2853-2_13
Download citation
DOI: https://doi.org/10.1007/978-90-481-2853-2_13
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-2852-5
Online ISBN: 978-90-481-2853-2
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)