Abstract
Complex symbolic algebra, such as the manipulation of second-quantized operators, Slater determinants, Feynman diagrams, is inevitable in quantum chemistry. Increasingly, these operations are performed by the computerized systems that can handle higher mathematical constructs than just numbers and simple arithmetic. This article reviews these new algorithms that automate the algebraic transformation and computer implementation of many-body quantum-mechanical methods for electron correlation. They enable a whole new class of highly complex but vastly accurate methods, the manual development of which is no longer practical
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References
Szabo A, Ostlund NS (1989) Modern quantum chemistry: Introduction to advanced electronic structure theory. McGraw-Hill, New York
McWeeny R (1992) Methods of molecular quantum mechanics. Academic Press, San Diego
Roos BO (1992) Lecture notes in quantum chemistry. Springer, Berlin Heidelberg New York
Roos BO (1994) Lecture notes in quantum chemistry II. Springer, Berlin Heidelberg New York
Schleyer PvR, Allinger NL, Clark T, Gasteiger J, Kollman PA, Schaefer III HF, Schreiner PR (1998) Encyclopedia of computational chemistry. Wiley, Chichester, New York
Helgaker T, Jørgensen P, Olsen J (2000) Molecular electronic-structure theory. Wiley, Chichester New York
Shavitt I, Bartlett RJ (2005) Many-body methods in quantum chemistry. Cambridge University Press, Cambridge
Ruscic B, Feller D, Dixon DA, Peterson KA, Harding LB, Asher RL, Wagner AF (2001). J Phys Chem A 105:1
Ruscic B, Wagner AF, Harding LB, Asher RL, Feller D, Dixon DA, Peterson KA, Song Y, Qian X, Ng CY, Liu J, Chen W, Schwenke DW (2002). J Phys Chem A 106:2727
Császár AG, Leininger ML, Szalay V (2003). J Chem Phys 118:10631
Hirata S, Yanai T, de Jong WA, Nakajima T, Hirao K (2004). J Chem Phys 120:3297
Tajti A, Szalay PG, Császár AG, Kállay M, Gauss J, Valeev EF, Flowers BA, Vázquez J, Stanton JF (2004). J Chem Phys 121:11599
Schaefer III HF (1986). Science 231:1100
Bartlett RJ (1991). Theor Chim Acta 80:71
Paldus J (1992) In: Wilson S, Diercksen GHF (ed) Methods in computational molecular physics (NATO ASI Series B: Physics), vol 293 Plenum, New York, p 99
Paldus J (1994) In: Malli GL (ed) Relativistic and correlation effects in molecules and solids (NATO ASI Series B: Physics), vol 318. Plenum, New York, p 207
Bartlett RJ, Stanton J F (1994) In: Lipkowitz KB, Boyd DB (ed) Reviews in computational chemistry, vol 5. VCH, New York, p 65
Bartlett RJ (1995) In: Yarkony DR (ed) Modern electronic structure theory, Part II. World Scientific, Singapore, p 1047
Lee TJ, Scuseria GE (1995) In: Langhoff SR (ed) Quantum mechanical electronic structure calculations with chemical accuracy. Kluwer, Dordrecht, p 47
Bartlett RJ (1997) Recent advances in coupled-cluster methods. World Scientific, Singapore
Crawford TD, Schaefer III HF (2000). Rev Comp Chem 14:33
Paldus J (2003) In: Wilson S (ed) Handbook of molecular physics and quantum chemistry, vol 2. Wiley, Chichester, New York, p 272
Purvis III GD, Bartlett RJ (1982). J Chem Phys 76:1910
Scuseria GE, Scheiner AC, Lee TJ, Rice JE, Schaefer III HF (1987). J Chem Phys 86:2881
Noga J, Bartlett RJ (1987). J Chem Phys 86:7041
Scuseria GE, Schaefer III HF (1988). Chem Phys Lett 152:382
Oliphant N, Adamowicz L (1991). J Chem Phys 95:6645
Kucharski SA, Bartlett RJ (1992). J Chem Phys 97:4282
Wick GC (1950). Phys Rev 80:268
Paldus J, Čížek J (1975). Adv Quantum Chem 9:105
Surján PR (1989) Second quantized approach to quantum chemistry: an elementary introduction with 11 figures. Springer, Berlin Heidelberg New York
Harris FE, Monkhorst HJ, Freeman DL (1992) Algebraic and diagrammatic methods in many-fermion theory. Oxford University Press, New York
Goldstone J (1957). Proc R Soc Lon Ser-A 239:267
Hugenholtz NM (1957). Physica 23:481
Čížek J (1966). J Chem Phys 45:4256
Kucharski SA, Bartlett RJ (1986). Adv Quantum Chem 18:281
Kucharski SA, Bartlett RJ (1991). Theor Chim Acta 80:387
Stanton JF, Gauss J, Watts JD, Bartlett RJ (1991). J Chem Phys 94:4334
Kállay M, Surján PR (2001). J Chem Phys 115:2945
Barnett MP, Capitani JF, von zur Gathen J, Gerhard J (2004). Int J Quantum Chem 100:80
Brueckner KA (1955). Phys Rev 100:36
Paldus J, Wong HC (1973). Comput Phys Commun 6:1
Wong HC, Paldus J (1973). Comput Phys Commun 6:9
Csépes Z, Pipek J (1988). J Comput Phys 77:1
Herbert JM, Ermler WC (1998). Comput Chem 22:169
Coester F (1958). Nucl Phys 7:421
Coester F, Kümmel H (1960). Nucl Phys 17:477
Čížek J (1969). Adv Chem Phys 14:35
Paldus J, Čížek J, Shavitt I (1972). Phys Rev A 5:50
Čížek J, Paldus J (1980). Phys Scripta 21:251
Paldus J (2005) In: Dykstra CF, Franking G, Kim KS, Scuseria GE (ed) Theory and applications of computational chemistry: the first 40,years, Elsevier, Amsterdam
Bartlett RJ (2005) In: Dykstra CF, Franking G, Kim KS, Scuseria GE (ed) Theory and applications of computational chemistry: the first 40,years, Elsevier Science, Amsterdam
Pople JA, Krishnan R, Schlegel HB, Binkley JS (1978). Int J Quantum Chem 14:545
Bartlett RJ, Purvis GD (1978). Int J Quantum Chem 14:561
Harris FE (1999). Int J Quantum Chem 75:593
Berente I, Szalay PG, Gauss J (2002). J Chem Phys 117:7872
Bochevarov AD, Sherrill CD (2004). J Chem Phys 121:3374
Knowles PJ, Handy NC (1988). J Chem Phys 88:6991
Čížek J, Vinette F, Paldus J (1990). Int J Quantum Chem 38:831
Piecuch P, Toboła R, Paldus J (1996). Phys Rev A 54:1210
Janssen CL, Schaefer III HF (1991). Theor Chim Acta 79:1
Cole SJ, Purvis GD (1986). Int J Quantum Chem Symp 20:665
Li X, Paldus J (1994). J Chem Phys 101:8812
Jankowski P, Jeziorski B (1999). J Chem Phys 111:1857
Crawford TD, Lee TJ, Schaefer III HF (1997). J Chem Phys 107:7943
Lee TJ, Jayatilaka D (1993). Chem Phys Lett 201:1
Nooijen M, Lotrich V (2001). J Mol Struc-Theochem 547:253
Nooijen M, Lotrich V (2000). J Chem Phys 113:494
Nooijen M, Lotrich V (2000). J Chem Phys 113:4549
Nooijen M (2002). Int J Mol Sci 3:656
Nooijen M, Wladyslawski M, Hazra A (2003). J Chem Phys 118:4832
Hirata S (2003). J Phys Chem A 107:9887
Hirata S (2004). J Chem Phys 121:51
Kállay M, Gauss J (2004). J Chem Phys 121:9257
Hirata S, Fan PD, Auer AA, Nooijen M, Piecuch P (2004). J Chem Phys 121:12197
Head-Gordon M, Rico RJ, Oumi M, Lee TJ (1994). Chem Phys Lett 219:21
Hirata S (2005). J Chem Phys 122:094105
Post DE, Votta LG (2005) Physics Today January:35
Cociorva D, Baumgartner G, Lam C, Sadayappan P, Ramanujam J, Nooijen M, Bernholdt D, Harrison R (2002). Acm Sigplan Notices 37:177
Krishnan S, Krishnamoorthy S, Baumgartner G, Cociorva D, Lam C, Sadayappan P, Ramanujam J, Bernholdt D, Choppella V (2003). High Perform Comput - HIPC 2913:406
Bibireata A, Krishnan S, Baumgartner G, Cociorva D, Lam C, Sadayappan P, Ramanujam J, Bernholdt D, Choppella V (2004). Lect Notes Comput Sci 2958:93
Baumgartner G, Auer AA, Bernholdt DE, Bibireata A, Choppella V, Cociorva D, Gao X, Harrison RJ, Hirata S, Krishnamoorthy S, Krishnan S, Lam CC, Lu Q, Nooijen M, Pitzer RM, Ramanujam J, Sadayappan P, Sibiryakov A (2005). Proc IEEE 93:276
Musiał M, Kucharski SA, Bartlett RJ (2000). Chem Phys Lett 320:542
Musiał M, Kucharski SA, Bartlett RJ (2002). J Chem Phys 116:4382
Kowalski K, Piecuch P (2000). J Chem Phys 113:8490
Kowalski K, Piecuch P (2001). J Chem Phys 115:643
Kowalski K, Piecuch P (2001). Chem Phys Lett 347:237
Kucharski SA, Włoch M, Musiał M, Bartlett RJ (2001). J Chem Phys 115:8263
Head-Gordon M, Maslen PE, White CA (1998). J Chem Phys 108:616
Kállay M, Gauss J, Szalay PG (2003). J Chem Phys 119:2991
Duch W (1985). J Phys A-Math Gen 18:3283
Knowles PJ, Handy NC (1984). Chem Phys Lett 111:315
Olsen J, Roos BO, Jørgensen P, Jensen HJA (1988). J Chem Phys 89:2185
Knowles PJ, Handy NC (1989). Comput Phys Commun 54:75
Harrison RJ, Zarrabian S (1989). Chem Phys Lett 158:393
Zarrabian S, Sarma CR, Paldus J (1989). Chem Phys Lett 155:183
Olsen J, Jørgensen P, Simons J (1990). Chem Phys Lett 169:463
Sherrill CD, Schaefer III HF (1999). Adv Quantum Chem 34:143
Knowles PJ, Somasundram K, Handy NC, Hirao K (1985). Chem Phys Lett 113:8
Laidig WD, Fitzgerald G, Bartlett RJ (1985). Chem Phys Lett 113:151
Christiansen O (2003). J Chem Phys 119:5773
Olsen J, Christiansen O, Koch H, Jørgensen P (1996). J Chem Phys 105:5082
Halkier A, Larsen H, Olsen J, Jørgensen P (1999). J Chem Phys 110:7127
Larsen H, Halkier A, Olsen J, Jørgensen P (2000). J Chem Phys 112:1107
Simon B (1982). Int J Quantum Chem 21:3
Silverstone HJ, Harris JG, Čížek J, Paldus J (1985). Phys Rev A 32:1965
Kendall RA, Dunning TH Jr, Harrison RJ (1992). J Chem Phys 96:6796
Christiansen O, Olsen J, Jørgensen P, Koch H, Malmqvist PÅ (1996). Chem Phys Lett 261:369
Sergeev AV, Goodson DZ, Wheeler SE, Allen WD (2005). J Chem Phys 123:064105
Leininger ML, Allen WD, Schaefer III HF, Sherrill CD (2000). J Chem Phys 112:9213
Hirata S, Bartlett RJ (2000). Chem Phys Lett 321:216
Laidig W, Fitzgerald G, Bartlett R (1985). Chem Phys Lett 113:151
Schmidt C, Warken M, Handy NC (1993). Chem Phys Lett 211:272
Kállay M, Surján PR (2000). J Chem Phys 113:1359
Olsen J (2000). J Chem Phys 113:7140
Christiansen O (2004). J Chem Phys 120:2140
Christiansen O (2004). J Chem Phys 120:2149
Van Voorhis T, Head-Gordon M (2000). J Chem Phys 113:8873
Szalay PG, Nooijen M, Bartlett RJ (1995). J Chem Phys 103:281
Van Voorhis T, Head-Gordon M (2001). J Chem Phys 115:5033
Piecuch P, Kowalski K, Fan PD, Jedziniak K (2003). Phys Rev Lett 90:113001
Meissner L, Hirata S, Bartlett RJ (2005) Theor Chem Acc (to appear)
Hirata S, Nooijen M, Bartlett RJ (2000). Chem Phys Lett 326:255
Hirata S, Nooijen M, Bartlett RJ (2000). Chem Phys Lett 328:459
Emrich K (1981). Nucl Phys A351:379
Emrich K (1981). Nucl Phys A351:397
Sekino H, Bartlett RJ (1984). Int J Quantum Chem Symp 18:255
Geertsen J, Rittby M, Bartlett RJ (1989). Chem Phys Lett 164:57
Comeau DC, Bartlett RJ (1993). Chem Phys Lett 207:414
Stanton JF, Bartlett RJ (1993). J Chem Phys 98:7029
Monkhorst HJ (1977). Int J Quantum Chem Symp 11:421
Mukherjee D, Mukherjee PK (1979). Chem Phys 39:325
Ghosh S, Mukherjee D, Bhattacharyya S (1981). Mol Phys 43: 173
Dalgaard E, Monkhorst HJ (1983). Phys Rev A 28:1217
Takahashi M, Paldus J (1986). J Chem Phys 85:1486
Koch H, Jørgensen P (1990). J Chem Phys 93:3333
Koch H, Jensen HJA, Jørgensen P, Helgaker T (1990). J Chem Phys 93:3345
Rico RJ, Head-Gordon M (1993). Chem Phys Lett 213:224
Nakatsuji H, Hirao K (1977). Chem Phys Lett 47:569
Nakatsuji H, Hirao K (1978). J Chem Phys 68:2053
Nakatsuji H (1978). Chem Phys Lett 59:362
Nakatsuji H (1979). Chem Phys Lett 67:329
Nakatsuji H (1979). Chem Phys Lett 67:334
Hirao K (1983). J Chem Phys 79:5000
Ehara M, Ishida M, Toyota K, Nakatsuji H (2003) In: Sen KD (ed) Reviews in Modern Quantum Chemistry (A Tribute to Professor Robert G. Parr), World Scientific, Singapore, p 293
Nakatsuji H (1991). J Chem Phys 94:6716
Nakatsuji H (1983). Chem Phys 75:425
Toyota K, Hasegawa J, Nakatsuji H (1996). Chem Phys Lett 250:437
Tokita Y, Hasegawa J, Nakatsuji H (1998). J Phys Chem A 102: 1843
Nooijen M, Bartlett RJ (1997). J Chem Phys 106:6449
Gwaltney SR, Bartlett RJ (1998). J Chem Phys 108:6790
Hasegawa J, Ozeki Y, Ohkawa K, Hada M, Nakatsuji H (1998). J Phys Chem B 102:1320
Miyahara T, Tokita Y, Nakatsuji H (2001). J Phys Chem B 105:7341
Miyahara T, Nakatsuji H, Hasegawa J, Osuka A, Aratani N, Tsuda A (2002). J Chem Phys 117:11196
Nakatsuji H (1985). J Chem Phys 83:713
Nakatsuji H (1985). J Chem Phys 83:5743
Nakatsuji H (1991). Chem Phys Lett 177:331
Hirata S, Nooijen M, Grabowski I, Bartlett RJ (2001). J Chem Phys 114:3919
Hirata S, Nooijen M, Grabowski I, Bartlett RJ (2001). J Chem Phys 115:3967
Stanton JF (1997). Chem Phys Lett 281:130
Urban M, Noga J, Cole SJ, Bartlett RJ (1985). J Chem Phys 83:4041
Raghavachari K, Trucks GW, Pople JA, Head-Gordon M (1989). Chem Phys Lett 157:479
Kucharski SA, Bartlett RJ (1998). J Chem Phys 108:5243
Kucharski SA, Bartlett RJ (1998). J Chem Phys 108:9221
Gwaltney SR, Head-Gordon M (2000). Chem Phys Lett 323:21
Gwaltney SR, Sherrill CD, Head-Gordon M, Krylov AI (2000). J Chem Phys 113:3548
Gwaltney SR, Head-Gordon M (2001). J Chem Phys 115:2014
Gwaltney SR, Byrd EFC, Van Voorhis T, Head-Gordon M (2002). Chem Phys Lett 353:359
Kowalski K, Piecuch P (2000). J Chem Phys 113:5644
Piecuch P, Kowalski K, Pimienta ISO, McGuire MJ (2002). Int Rev Phys Chem 21:527
Piecuch P, Kowalski K, Pimienta ISO, Fan PD, Lodriguito M, McGuire MJ, Kucharski SA, Kus T, Musiał M (2004). Theor Chem Acc 112:349
Zarrabian S, Bartlett RJ (1988). Chem Phys Lett 153:133
Zarrabian S, Laidig WD, Bartlett RJ (1990). Phys Rev A 41:4711
Zarrabian S, Paldus J (1990). Int J Quantum Chem 38:761
Lindgren I (1978). Int J Quantum Chem Symp 12:33
Kállay M, Szalay PG, Surján PR (2002). J Chem Phys 117:980
Oliphant N, Adamowicz L (1991). J Chem Phys 94:1229
Oliphant N, Adamowicz L (1992). J Chem Phys 96:3739
Piecuch P, Oliphant N, Adamowicz L (1993). J Chem Phys 99: 1875
Piecuch P, Kucharski SA, Bartlett RJ (1999). J Chem Phys 110:6103
Lyakh DI, Ivanov VV, Adamowicz L (2005). J Chem Phys 122: 024108
Kállay M, Gauss J (2004). J Chem Phys 120:6841
Boese AD, Oren M, Atasoylu O, Martin JML, Kállay M, Gauss J (2004). J Chem Phys 120:4129
Chan GKL, Kállay M, Gauss J (2004). J Chem Phys 121:6110
Rizzo A, Kállay M, Gauss J, Pawlowski F, Jørgensen P, Hattig C (2004). J Chem Phys 121:9461
Pople JA, Head-Gordon M, Fox DJ, Raghavachari K, Curtiss LA (1989). J Chem Phys 90:5622
Curtiss LA, Raghavachari K, Trucks GW, Pople JA (1991). J Chem Phys 94:7221
Curtiss LA, Raghavachari K, Redfern PC, Rassolov V, Pople JA (1998). J Chem Phys 109:7764
Bartlett RJ (2000) In: Keinan E, Schechter I (ed) Chemistry for the 21st century, Wiley-VCH, Weinheim
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Hirata, S. Symbolic Algebra in Quantum Chemistry. Theor Chem Acc 116, 2–17 (2006). https://doi.org/10.1007/s00214-005-0029-5
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DOI: https://doi.org/10.1007/s00214-005-0029-5