Relativistic quantum theory for atomic and molecular response properties
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Ignacio Agustín Aucar
Abstract
This perspective revisits key developments and challenges in the theoretical description of atomic and molecular electronic properties within the framework of relativistic quantum chemistry. Fundamental conceptual issues arising from the extension of non-relativistic theories to the relativistic domain are discussed, with particular emphasis on four-component formalisms, the treatment and interpretation of negative-energy states, including virtual excitations involving these states, the reinterpretation of operators and observables, and the introduction of weak interactions into this field of research. Special attention is given to the construction of model Hamiltonians as accurately as possible and the analysis of the electronic contributions due to spin-dependent and spin-independent effective Hamiltonians to molecular properties. In this context, the generalization of the spin-symmetry to time-reversal-restricted symmetry is emphasized. Recent methodological advances, including the application of the relativistic polarization propagator theory, are also highlighted.
Funding source: Fondo para la Investigación Científica y Tecnológica
Award Identifier / Grant number: PICT-2021 I-A-00933
Funding source: Consejo Nacional de Investigaciones Científicas y Técnicas
Award Identifier / Grant number: PIP 112202101 00483
Acknowledgments
We are grateful to the great efforts of leading scientists that made it possible for relativistic quantum physics to be applied to the atomic and molecular issues. GAA gives special thanks to the small, encouraging and ever visionary group of young, though not all scientists, that have met for the first time in Odense, DK, in the early 1990’s to start developing some of the, at the time very promising and exciting, areas of the relativistic quantum chemistry. We also would like to thank Professors Manuel Yañez and Russell Boyd for the invitation to write this article.
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Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Use of Large Language Models, AI and Machine Learning Tools: None declared.
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Conflict of interest: All other authors state no conflict of interest.
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Research funding: None declared.
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Data availability: Not applicable.
References
1. Schrödinger, E. Ann. Phys. (Berl.) 1926, 384, 361–376. https://doi.org/10.1002/andp.19263840404.Search in Google Scholar
2. Schrödinger, E. Ann. Phys. (Berl.) 1926, 385, 437–490. https://doi.org/10.1002/andp.19263851302.Search in Google Scholar
3. Dirac, P. A. M. Proc. R. Soc. Lond. A 1928, 117, 610–624. https://doi.org/10.1098/rspa.1928.0023.Search in Google Scholar
4. Swirles, B. Proc. R. Soc. Lond. A 1935, 152, 625–649. https://doi.org/10.1098/rspa.1935.0211.Search in Google Scholar
5. Dirac, P. A. M. Proc. R. Soc. Lond. A 1929, 123, 714–733. https://doi.org/10.1098/rspa.1929.0094.Search in Google Scholar
6. Norman, P.; Ruud, K.; Saue, T. Principles and Practices of Molecular Properties: Theory, Modeling, and Simulations; Wiley: Oxford, UK, 2018.10.1002/9781118794821Search in Google Scholar
7. Dyall, K. G.; Faegri, K.Jr. Introduction to Relativistic Quantum Chemistry; Oxford University Press: Oxford, UK, 2007.10.1093/oso/9780195140866.001.0001Search in Google Scholar
8. Grandy, W. T. Relativistic Quantum Mechanics of Leptons and Fields, Vol. 41; Kluwer Academic Publishers: Dordrecht, Boston, London, 1991.10.1007/978-94-011-3302-9Search in Google Scholar
9. Gell-Mann, M. Nuovo Cimento 1956, 4, 848–866. https://doi.org/10.1007/bf02748000.Search in Google Scholar
10. Goldstein, H.; Poole, C. P.; Safko, J. L. Classical Mechanics, 3rd ed.; Addison Wesley: San Francisco, 2002.10.1119/1.1484149Search in Google Scholar
11. Breit, G. Phys. Rev. 1929, 34, 553–573. https://doi.org/10.1103/physrev.34.553.Search in Google Scholar
12. Bethe, H. A.; Salpeter, E. E. Quantum Mechanics of One- and Two-Electron Atoms; Springer: Berlin, Heidelberg, 1957.10.1007/978-3-662-12869-5Search in Google Scholar
13. Kutzelnigg, W. Chem. Phys. 2012, 395, 16–34. https://doi.org/10.1016/j.chemphys.2011.06.001.Search in Google Scholar
14. Pyykko, P. Chem. Rev. 1988, 88, 563–594. https://doi.org/10.1021/cr00085a006.Search in Google Scholar
15. Quiney, H. M.; Skaane, H.; Grant, I. P. Adv. Quantum Chem. 1998, 32, 1–49. https://doi.org/10.1016/S0065-3276(08)60405-0.Search in Google Scholar
16. Hess, B. A.; Marian, C. M. Relativistic Electronic Structure Theory. In Relativistic Electronic Structure Theory. Part 1. Fundamentals; Schwerdtfeger, P., Ed.; Elsevier: London, 2002; pp 78–133.Search in Google Scholar
17. Pyykkö, P. Chem. Rev. 2012, 112, 371–384. https://doi.org/10.1021/cr200042e.Search in Google Scholar PubMed
18. Liu, W. Phys. Rep. 2014, 537, 59–89. https://doi.org/10.1016/j.physrep.2013.11.006.Search in Google Scholar
19. Reiher, M.; Wolf, A. Relativistic Quantum Chemistry. The Fundamental Theory of Molecular Science; Wiley-VCH: Weinheim, 2015; pp. 78–133.10.1002/9783527667550Search in Google Scholar
20. Dolg, M.; Cao, X. Chem. Rev. 2012, 112, 403–480. https://doi.org/10.1021/cr2001383.Search in Google Scholar PubMed
21. Zhou, H.; Kincaid, B.; Wang, G.; Annaberdiyev, A.; Ganesh, P.; Mitas, L. J. Chem. Phys. 2024, 160, 084302–084318. https://doi.org/10.1063/5.0180057.Search in Google Scholar PubMed
22. Kim, Y.-K. Phys. Rev. 1967, 154, 17–39. https://doi.org/10.1103/physrev.154.17.Search in Google Scholar
23. Stanton, R. E.; Havriliak, S. J. Chem. Phys. 1984, 81, 1910–1918. https://doi.org/10.1063/1.447865.Search in Google Scholar
24. Grant, I. P.; Quiney, H. M. Adv. At. Mol. Phys. 1988, 23, 37–86. https://doi.org/10.1016/S0065-2199(08)60105-0.Search in Google Scholar
25. Talman, J. D. Phys. Rev. Lett. 1986, 57, 1091–1094. https://doi.org/10.1103/physrevlett.57.1091.Search in Google Scholar
26. Lohr, L. L.Jr.; Pyykkö, P. Chem. Phys. Lett. 1979, 62, 333–338. https://doi.org/10.1016/0009-2614(79)80191-8.Search in Google Scholar
27. Aerts, P.; Nieuwpoort, W. Int. J. Quantum Chem. 1985, 28, 267–277. https://doi.org/10.1002/qua.560280826.Search in Google Scholar
28. DIRAC, a relativistic ab initio electronic structure program, Release DIRAC25 (2025), written by T. Saue, L. Visscher, H. J. A. Jensen, R. Bast and A. S. P. Gomes, with contributions from I. A. Aucar, V. Bakken, J. Brandejs, C. Chibueze, J. Creutzberg, K. G. Dyall, et al. https://doi.org/10.5281/zenodo.14833106, see also https://www.diracprogram.org.Search in Google Scholar
29. Saue, T.; Bast, R.; Gomes, A. S. P.; Jensen, H. J. A.; Visscher, L.; Aucar, I. A.; Di Remigio, R.; Dyall, K. G.; Eliav, E.; Fasshauer, E.; Fleig, T.; Halbert, L.; Hedegård, E. D.; Helmich-Paris, B.; Iliaš, M.; Jacob, C. R.; Knecht, S.; Laerdahl, J. K.; Vidal, M. L.; Nayak, M. K.; Olejniczak, M.; Olsen, J. M. H.; Pernpointner, M.; Senjean, B.; Shee, A.; Sunaga, A.; van Stralen, J. N. P. J. Chem. Phys. 2020, 152, 204104. https://doi.org/10.1063/5.0004844.Search in Google Scholar PubMed
30. Saue, T.; Faegri, K.; Gropen, O. Chem. Phys. Lett. 1996, 263, 360–366. https://doi.org/10.1016/s0009-2614(96)01250-x.Search in Google Scholar
31. Saue, T.; Faegri, K.; Helgaker, T.; Gropen, O. Mol. Phys. 1997, 91, 937–950. https://doi.org/10.1080/002689797171058.Search in Google Scholar
32. Anderson, C. D. Phys. Rev. 1933, 43, 491–494. https://doi.org/10.1103/physrev.43.491.Search in Google Scholar
33. Aucar, I. A.; Gómez, S. S.; de Azúa, M. C. R.; Giribet, C. G. J. Chem. Phys. 2012, 136, 204119. https://doi.org/10.1063/1.4721627.Search in Google Scholar PubMed
34. Aucar, G. A.; Aucar, I. A. Annu. Rep. NMR Spectrosc.; Webb, G. A., Ed.; Academic Press: London, 2019, ch. 3; pp. 78–133.Search in Google Scholar
35. Flygare, W. H. J. Chem. Phys. 1964, 41, 793–800. https://doi.org/10.1063/1.1725962.Search in Google Scholar
36. Flygare, W. H. Chem. Rev. 1974, 74, 653–687. https://doi.org/10.1021/cr60292a003.Search in Google Scholar
37. Sternheim, M. M. Phys. Rev. 1962, 128, 676–677. https://doi.org/10.1103/physrev.128.676.Search in Google Scholar
38. Pyykkö, P. Chem. Phys. 1983, 74, 1–7. https://doi.org/10.1016/0301-0104(83)80001-9.Search in Google Scholar
39. Aucar, G.; Saue, T.; Visscher, L.; Jensen, H. J. A. J. Chem. Phys. 1999, 110, 6208–6218. https://doi.org/10.1063/1.479181.Search in Google Scholar
40. Aucar, G. A.; Oddershede, J. Int. J. Quantum Chem. 1993, 47, 425–435. https://doi.org/10.1002/qua.560470603.Search in Google Scholar
41. Aucar, G. A.; Romero, R. H.; Maldonado, A. F. Int. Rev. Phys. Chem. 2010, 29, 1–64. https://doi.org/10.1080/01442350903432865.Search in Google Scholar
42. Kutzelnigg, W. Phys. Rev. A 2003, 67, 032109–032121. https://doi.org/10.1103/physreva.67.032109.Search in Google Scholar
43. Xiao, Y.; Sun, Q.; Liu, W. Theor. Chem. Acc. 2012, 131, 1080–1097. https://doi.org/10.1007/s00214-011-1080-z.Search in Google Scholar
44. Gomez, S. S.; Maldonado, A.; Aucar, G. A. J. Chem. Phys. 2005, 123, 214108–214115. https://doi.org/10.1063/1.2133729.Search in Google Scholar PubMed
45. Bauke, H.; Ahrens, S.; Keitel, C. H.; Grobe, R. New J. Phys. 2014, 16, 04312–04321. https://doi.org/10.1088/1367-2630/16/4/043012.Search in Google Scholar
46. Bauke, H.; Ahrens, S.; Keitel, C. H.; Grobe, R. Phys. Rev. A 2014, 89, 052101–052116. https://doi.org/10.1103/physreva.89.052101.Search in Google Scholar
47. Aucar, G. A.; Jensen, H. J. A.; Oddershede, J. Chem. Phys. Lett. 1995, 232, 47–53. https://doi.org/10.1016/0009-2614(94)01332-p.Search in Google Scholar
48. Aucar, G. A. NMR Spectroscopic Parameters: Theories and Models, Computational Codes and Calculations, 1st ed.; Aucar, G. A., Ed.; Royal Society of Chemistry: London, 2025; pp. 231–261.10.1039/9781837678020-00231Search in Google Scholar
49. Jensen, H. J. J.; Dyall, K. G.; Saue, T.; Fægri, K.Jr. J. Chem. Phys. 1996, 104, 4083–4097. https://doi.org/10.1063/1.471644.Search in Google Scholar
50. Saue, T.; Jensen, H. J. A. J. Chem. Phys. 1999, 111, 6211–6222. https://doi.org/10.1063/1.479958.Search in Google Scholar
51. Fleig, T. Phys. Rev. A 2008, 77, 062503–062512. https://doi.org/10.1103/physreva.77.062503.Search in Google Scholar
52. Kutzelnigg, W. Relativistic Electronic Structure Theory. Part 1. Fundamentals; Schwerdtfeger, P., Ed.; Elsevier: London, 2002, ch. 12; pp. 78–133.Search in Google Scholar
53. Kutzelnigg, W. Int. J. Quantum Chem. 1984, 25, 107–129. https://doi.org/10.1002/qua.560250112.Search in Google Scholar
54. Dyall, K. G. J. Chem. Phys. 1994, 100, 2118–2127. https://doi.org/10.1063/1.466508.Search in Google Scholar
55. Malkin, V. G.; Malkina, O. L.; Salahub, D. R. Chem. Phys. Lett. 1996, 261, 335–345. https://doi.org/10.1016/0009-2614(96)00988-8.Search in Google Scholar
56. Kaupp, M.; Malkin, O. L.; Malkin, V. G. Chem. Phys. Lett. 1997, 265, 55–59. https://doi.org/10.1016/s0009-2614(96)01425-x.Search in Google Scholar
57. Vaara, J.; Ruud, K.; Vahtras, O.; Ågren, H.; Jokisaari, J. J. Chem. Phys. 1998, 109, 1212–1222. https://doi.org/10.1063/1.476672.Search in Google Scholar
58. Pyykkö, P. Annu. Rev. Phys. Chem. 2012, 63, 45–64. https://doi.org/10.1146/annurev-physchem-032511-143755.Search in Google Scholar PubMed
59. Liu, W. WIREs Comput. Mol. Sci. 2023, 13, e1652. https://doi.org/10.1002/wcms.1652.Search in Google Scholar
60. Aucar, G. A.; Melo, J. I.; Aucar, I. A.; Maldonado, A. F. Int. J. Quantum Chem. 2018, 118, e25487. https://doi.org/10.1002/qua.25487.Search in Google Scholar
61. Aucar, I. A.; Gomez, S. S.; Giribet, C. G.; Aucar, G. A. Phys. Chem. Chem. Phys. 2016, 18, 23572–23586. https://doi.org/10.1039/c6cp03355e.Search in Google Scholar PubMed
62. Aucar, I. A.; Gomez, S. S.; Giribet, C. G.; Aucar, G. A. J. Phys. Chem. Lett. 2016, 7, 5188–5192. https://doi.org/10.1021/acs.jpclett.6b02361.Search in Google Scholar PubMed
63. Aucar, I. A.; Gomez, S. S.; Giribet, C. G.; Ruiz de Azúa, M. C. J. Chem. Phys. 2014, 141, 194103. https://doi.org/10.1063/1.4901422.Search in Google Scholar PubMed
64. Helgaker, T.; Coriani, S.; Jørgensen, P.; Kristensen, K.; Olsen, J.; Ruud, K. Chem. Rev. 2012, 112, 543–631. https://doi.org/10.1021/cr2002239.Search in Google Scholar PubMed
65. Repisky, M.; Komorovsky, S.; Kadek, M.; Konecny, L.; Ekström, U.; Malkin, E.; Kaupp, M.; Ruud, K.; Malkina, O. L.; Malkin, V. G. J. Chem. Phys. 2020, 152, 184101. https://doi.org/10.1063/5.0005094.Search in Google Scholar PubMed
66. Belpassi, L.; De Santis, M.; Quiney, H. M.; Tarantelli, F.; Storchi, L. J. Chem. Phys. 2020, 152, 164118. https://doi.org/10.1063/5.0002831.Search in Google Scholar PubMed
67. Zhang, Y.; Suo, B.; Wang, Z.; Zhang, N.; Li, Z.; Lei, Y.; Zou, W.; Gao, J.; Peng, D.; Pu, Z.; Xiao, Y.; Sun, Q.; Wang, F.; Ma, Y.; Wang, X.; Guo, Y.; Liu, W. J. Chem. Phys. 2020, 152, 064113. https://doi.org/10.1063/1.5143173.Search in Google Scholar PubMed
68. Yanai, T.; Nakano, H.; Nakajima, T.; Tsuneda, T.; Hirata, S.; Kawashima, Y.; Nakao, Y.; Kamiya, M.; Sekino, H.; Hirao, K. Computational Science – ICCS 2003: Berlin, Heidelberg, 2003; pp. 84–95.10.1007/3-540-44864-0_9Search in Google Scholar
69. Sun, Q.; Zhang, X.; Banerjee, S.; Bao, P.; Barbry, M.; Blunt, N. S.; Bogdanov, N. A.; Booth, G. H.; Chen, J.; Cui, Z.-H.; Eriksen, J. J.; Gao, Y.; Guo, S.; Hermann, J.; Hermes, M. R.; Koh, K.; Koval, P.; Lehtola, S.; Li, Z.; Liu, J.; Mardirossian, N.; McClain, J. D.; Motta, M.; Mussard, B.; Pham, H. Q.; Pulkin, A.; Purwanto, W.; Robinson, P. J.; Ronca, E.; Sayfutyarova, E. R.; Scheurer, M.; Schurkus, H. F.; Smith, J. E. T.; Sun, C.; Sun, S.-N.; Upadhyay, S.; Wagner, L. K.; Wang, X.; White, A.; Whitfield, J. D.; Williamson, M. J.; Wouters, S.; Yang, J.; Yu, J. M.; Zhu, T.; Berkelbach, T. C.; Sharma, S.; Sokolov, A. Y.; Chan, G. K.-L J. Chem. Phys. 2020, 153, 024109. https://doi.org/10.1063/5.0006074.Search in Google Scholar PubMed
70. Williams-Young, D. B.; Petrone, A.; Sun, S.; Stetina, T. F.; Lestrange, P.; Hoyer, C. E.; Nascimento, D. R.; Koulias, L.; Wildman, A.; Kasper, J.; Goings, J. J.; Ding, F.; DePrince, A. E.III; Valeev, E. F.; Li, X. WIREs Comput. Mol. Sci. 2020, 10, e1436. https://doi.org/10.1002/wcms.1436.Search in Google Scholar
71. Dyall, K. G. J. Chem. Phys. 1997, 106, 9618–9626. https://doi.org/10.1063/1.473860.Search in Google Scholar
72. Kutzelnigg, W.; Liu, W. Mol. Phys. 2006, 104, 2225–2240. https://doi.org/10.1080/00268970600662481.Search in Google Scholar
73. Liu, W.; Kutzelnigg, W. J. Chem. Phys. 2007, 126, 114107. https://doi.org/10.1063/1.2710258.Search in Google Scholar PubMed
74. Liu, W.; Peng, D. J. Chem. Phys. 2009, 131, 031104. https://doi.org/10.1063/1.3159445.Search in Google Scholar
75. Reiher, M.; Wolf, A. J. Chem. Phys. 2004, 121, 10945–10956. https://doi.org/10.1063/1.1818681.Search in Google Scholar
76. Douglas, M.; Kroll, N. M. Ann. Phys. 1974, 82, 89–155. https://doi.org/10.1016/0003-4916(74)90333-9.Search in Google Scholar
77. Hess, B. A. Phys. Rev. A 1985, 32, 756. https://doi.org/10.1103/physreva.32.756.Search in Google Scholar
78. Hess, B. A. Phys. Rev. A 1986, 33, 3742. https://doi.org/10.1103/physreva.33.3742.Search in Google Scholar
79. Barysz, M.; Sadlej, A. J.; Snijders, J. G. Int. J. Quantum Chem. 1997, 65, 225–239. https://doi.org/10.1002/(sici)1097-461x(1997)65:3<225::aid-qua4>3.0.co;2-y.10.1002/(SICI)1097-461X(1997)65:3<225::AID-QUA4>3.0.CO;2-YSearch in Google Scholar
80. Foldy, L. L.; Wouthuysen, S. A. Phys. Rev. 1950, 78, 29. https://doi.org/10.1103/physrev.78.29.Search in Google Scholar
81. Iliaš, M.; Jensen, H. J. A.; Kellö, V.; Roos, B. O.; Urban, M. Chem. Phys. Lett. 2005, 408, 210–215. https://doi.org/10.1016/j.cplett.2005.04.027.Search in Google Scholar
82. Chang, C.; Pelissier, M.; Durand, P. Phys. Scr. 1986, 34, 394. https://doi.org/10.1088/0031-8949/34/5/007.Search in Google Scholar
83. van Lenthe, E.; Baerends, E. J.; Snijders, J. G. J. Chem. Phys. 1994, 101, 9783–9792. https://doi.org/10.1063/1.467943.Search in Google Scholar
84. Filatov, M.; Cremer, D. J. Chem. Phys. 2005, 122, 064104. https://doi.org/10.1063/1.1844298.Search in Google Scholar PubMed
85. Zou, W.; Filatov, M.; Cremer, D. Theor. Chem. Acc. 2011, 130, 633–644. https://doi.org/10.1007/s00214-011-1007-8.Search in Google Scholar
86. Filatov, M.; Zou, W.; Cremer, D. J. Chem. Phys. 2013, 139, 014106. https://doi.org/10.1063/1.4811776.Search in Google Scholar PubMed
87. Cremer, D.; Zou, W.; Filatov, M. WIREs Comput. Mol. Sci. 2014, 4, 436–467. https://doi.org/10.1002/wcms.1181.Search in Google Scholar
88. Cheng, L.; Gauss, J. J. Chem. Phys. 2011, 135, 084114. https://doi.org/10.1063/1.3624397.Search in Google Scholar PubMed
89. Cheng, L.; Gauss, J. J. Chem. Phys. 2011, 135, 244104. https://doi.org/10.1063/1.3667202.Search in Google Scholar PubMed
90. Cheng, L.; Gauss, J.; Stanton, J. F. J. Chem. Phys. 2013, 139, 054105. https://doi.org/10.1063/1.4816130.Search in Google Scholar PubMed
91. Knecht, S.; Repisky, M.; Jensen, H. J. A.; Saue, T. J. Chem. Phys. 2022, 157, 114106. https://doi.org/10.1063/5.0095112.Search in Google Scholar PubMed
92. Autschbach, J.; Peng, D.; Reiher, M. J. Chem. Theory Comput. 2012, 8, 4239–4248. https://doi.org/10.1021/ct300623j.Search in Google Scholar PubMed
93. Melo, J. I.; Ruiz de Azua, M. C.; Giribet, C. G.; Aucar, G. A.; Romero, R. H. J. Chem. Phys. 2003, 118, 471–486. https://doi.org/10.1063/1.1525808.Search in Google Scholar
94. Aucar, I. A.; Colombo Jofré, M. T.; Aucar, G. A. J. Chem. Phys. 2023, 158, 094306; https://doi.org/10.1063/5.0141176.Search in Google Scholar PubMed
95. Melo, J. I.; Maldonado, A. F. Int. J. Quantum Chem. 2019, 119, e25935. https://doi.org/10.1002/qua.25935.Search in Google Scholar
96. Bajac, D. F. E.; Aucar, I. A.; Aucar, G. A. Phys. Rev. A 2021, 104, 012805. https://doi.org/10.1103/physreva.104.012805.Search in Google Scholar
97. Aucar, J. J.; Maldonado, A. F.; Melo, J. I. J. Chem. Phys. 2025, 162, 194103. https://doi.org/10.1063/5.0264596.Search in Google Scholar PubMed
98. Ruiz De Azua, M. C.; Giribet, C. G.; Melo, J. I. J. Chem. Phys. 2011, 134, 034123. https://doi.org/10.1063/1.3528717.Search in Google Scholar PubMed
99. Fukui, H.; Baba, T.; Inomata, H. J. Chem. Phys. 1996, 105, 3175–3186. https://doi.org/10.1063/1.472165.Search in Google Scholar
100. Manninen, P.; Lantto, P.; Vaara, J.; Ruud, K. J. Chem. Phys. 2003, 119, 2623–2637. https://doi.org/10.1063/1.1586912.Search in Google Scholar
101. Lantto, P.; Vaara, J. J. Chem. Phys. 2007, 127, 084312. https://doi.org/10.1063/1.2759205.Search in Google Scholar PubMed
102. Aucar, I. A. Ph.D. Thesis; National Northeastearn University, 2015. https://repositorio.unne.edu.ar/handle/123456789/452.Search in Google Scholar
103. Lantto, P.; Standara, S.; Riedel, S.; Vaara, J.; Straka, M. Phys. Chem. Chem. Phys. 2012, 14, 10944. https://doi.org/10.1039/c2cp41240c.Search in Google Scholar PubMed
104. Aucar, J. J.; Maldonado, A. F.; Melo, J. I. J. Chem. Phys. 2022, 157, 244105. https://doi.org/10.1063/5.0124701.Search in Google Scholar PubMed
105. Aucar, J. J.; Melo, J. I.; Maldonado, A. F. J. Phys. Chem. A 2024, 128, 5089–5099. https://doi.org/10.1021/acs.jpca.4c00426.Search in Google Scholar PubMed
106. Pyykkö, P.; Seth, M. Theor. Chem. Acc. 1997, 96, 92–104. https://doi.org/10.1007/s002140050209.Search in Google Scholar
107. Stopkowicz, S.; Gauss, J. J. Chem. Phys. 2011, 134, 204106. https://doi.org/10.1063/1.3587633.Search in Google Scholar PubMed
108. Sakurai, J. J. Modern Quantum Mechanics; Addison-Wesley: New York, USA, 1994.Search in Google Scholar
109. Feynman, R. P.; Hibbs, A. R. Quantum Mechanics and Path Integrals; McGraw-Hill: New York, 1965.Search in Google Scholar
110. Huang, K. Quantum Field Theory: From Operators to Path Integrals; WILEY-VCH: New York, 2010.Search in Google Scholar
111. Zubarev, D. N. Usp. Fiz. Nauk 1960, 71, 71 [Sov. Phys. Usp. 3, 320 (1960), English translation by D. ter Haar]. https://doi.org/10.3367/ufnr.0071.196005c.0071.Search in Google Scholar
112. Oddershede, J.; Jørgensen, P.; Yeager, D. L. Comput. Phys. Rep. 1984, 2, 33–92. https://doi.org/10.1016/0167-7977(84)90003-0.Search in Google Scholar
113. Aucar, G. A.; Maldonado, A. F. Comprehensive Computational Chemistry, 1st ed.; Boyd, R. J.; Yáñez, M., Eds.; Elsevier: Oxford, 2024, ch. 7; pp. 175–199.10.1016/B978-0-12-821978-2.00122-7Search in Google Scholar
114. Aucar, G. A. Phys. Chem. Chem. Phys. 2014, 16, 4420–4438. https://doi.org/10.1039/c3cp52685b.Search in Google Scholar PubMed
115. Safronova, M. S.; Budker, D.; DeMille, D.; Kimball, D. F. J.; Derevianko, A.; Clark, C. W. Rev. Mod. Phys. 2018, 90, 3–40. https://doi.org/10.1103/RevModPhys.90.025008.Search in Google Scholar
116. Lee, T. D.; Yang, C. N. Phys. Rev. 1956, 104, 254–258. https://doi.org/10.1103/physrev.104.254.Search in Google Scholar
117. Wu, C. S.; Ambler, E.; Hayward, R. W.; Hoppes, D. D.; Hudson, R. P. Phys. Rev. 1957, 105, 1413–1415. https://doi.org/10.1103/physrev.105.1413.Search in Google Scholar
118. Khriplovich, I. B. Parity Nonconservation in Atomic Phenomena; Gordon and Breach Science Publishers: Philadelphia, PA, United States, 1991.Search in Google Scholar
119. Ginges, J. S. M.; Flambaum, V. V. Phys. Rep. 2004, 397, 63–154. https://doi.org/10.1016/j.physrep.2004.03.005.Search in Google Scholar
120. Bradley, L. C.; Wall, N. S. Nuovo Cim. 1962, 25, 48–54. https://doi.org/10.1007/bf02733315.Search in Google Scholar
121. Carhart, R. A. Phys. Rev. 1963, 132, 2337–2345. https://doi.org/10.1103/physrev.132.2337.Search in Google Scholar
122. Labzovsky, L. N. Zh. Eksp. Teor. Fiz. 1977, 73, 1623 [Sov. Phys. JETP 46, 853 (1977), English translation by E. Brunner].Search in Google Scholar
123. Bouchiat, M. A.; Bouchiat, C. J. Phys. France 1974, 35, 899–927. https://doi.org/10.1051/jphys:019740035012089900.10.1051/jphys:019740035012089900Search in Google Scholar
124. Laerdahl, J. K.; Schwerdtfeger, P. Phys. Rev. A 1999, 60, 4439–4453. https://doi.org/10.1103/physreva.60.4439.Search in Google Scholar
125. Kompanets, O.; Kukudzhanov, A.; Letokhov, V.; Gervits, L. Opt. Commun. 1976, 19, 414–416. https://doi.org/10.1016/0030-4018(76)90111-5.Search in Google Scholar
126. Arimondo, E.; Glorieux, P.; Oka, T. Opt. Commun. 1977, 23, 369–372. https://doi.org/10.1016/0030-4018(77)90384-4.Search in Google Scholar
127. Daussy, C.; Marrel, T.; Amy-Klein, A.; Nguyen, C. T.; Bordé, C. J.; Chardonnet, C. Phys. Rev. Lett. 1999, 83, 1554–1557. https://doi.org/10.1103/physrevlett.83.1554.Search in Google Scholar
128. Ziskind, M.; Daussy, C.; Marrel, T.; Chardonnet, C. Eur. Phys. J. D 2002, 20, 219–225. https://doi.org/10.1140/epjd/e2002-00133-0.Search in Google Scholar
129. DeMille, D.; Cahn, S. B.; Murphree, D.; Rahmlow, D. A.; Kozlov, M. G. Phys. Rev. Lett. 2008, 100, 023003. https://doi.org/10.1103/physrevlett.100.023003.Search in Google Scholar
130. Altuntaş, E.; Ammon, J.; Cahn, S. B.; DeMille, D. Phys. Rev. Lett. 2018, 120, 142501. https://doi.org/10.1103/physrevlett.120.142501.Search in Google Scholar
131. Fiechter, M. R.; Haase, P. A.; Saleh, N.; Soulard, P.; Tremblay, B.; Havenith, R. W.; Timmermans, R. G.; Schwerdtfeger, P.; Crassous, J.; Darquié, B.; Pašteka, L. F.; Borschevsky, A. J. Phys. Chem. Lett. 2022, 13, 10011–10017. https://doi.org/10.1021/acs.jpclett.2c02434.Search in Google Scholar PubMed PubMed Central
132. Berger, R.; Stohner, J. WIREs Comput. Mol. Sci. 2019, 9, e1396. https://doi.org/10.1002/wcms.1396.Search in Google Scholar
133. Titov, A.; Mosyagin, N.; Petrov, A.; Isaev, T.; DeMille, D. Recent Advances in the Theory of Chemical and Physical Systems; Springer Netherlands: Dordrecht, The Netherlands, 2006; pp 253–283.10.1007/1-4020-4528-X_12Search in Google Scholar
134. Isaev, T. A.; Hoekstra, S.; Berger, R. Phys. Rev. A 2010, 82, 052521. https://doi.org/10.1103/physreva.82.052521.Search in Google Scholar
135. Barron, L. D. Isr. J. Chem. 2021, 61, 517–529. https://doi.org/10.1002/ijch.202100044.Search in Google Scholar
136. Quack, M.; Seyfang, G.; Wichmann, G. Chem. Sci. 2022, 13, 10598–10643. https://doi.org/10.1039/d2sc01323a.Search in Google Scholar PubMed PubMed Central
137. Zabrodsky, H.; Avnir, D. J. Am. Chem. Soc. 1995, 117, 462–473. https://doi.org/10.1021/ja00106a053.Search in Google Scholar
138. Mislow, K. Molecular Chirality; John Wiley & Sons, Ltd: Chichester, UK, Vol. 73, 2007; pp 1–82.Search in Google Scholar
139. Wang, H.; Avnir, D.; Tuvi-Arad, I. Biochem. 2018, 57, 6395–6403. https://doi.org/10.1021/acs.biochem.8b00974.Search in Google Scholar PubMed
140. Grimme, S. Chem. Phys. Lett. 1998, 297, 15–22. https://doi.org/10.1016/s0009-2614(98)01101-4.Search in Google Scholar
141. Bellarosa, L.; Zerbetto, F. J. Am. Chem. Soc. 2003, 125, 1975–1979. https://doi.org/10.1021/ja028646+.10.1021/ja028646+Search in Google Scholar PubMed
142. Aucar, J. J.; Stroppa, A.; Aucar, G. A. J. Phys. Chem. Lett. 2023, 15, 234–240. https://doi.org/10.1021/acs.jpclett.3c03038.Search in Google Scholar PubMed
143. Hegstrom, R. A. Theochem (J. Mol. Struct.) 1991, 232, 17–21. https://doi.org/10.1016/0166-1280(91)85241-x.Search in Google Scholar
144. Senami, M.; Shimizu, T. Phys. Lett. A 2020, 384, 126796. https://doi.org/10.1016/j.physleta.2020.126796.Search in Google Scholar
145. Novikov, V.; Khriplovich, I. Pis’ma Zh. Eksp. Teor. Fiz. 1975, 22, 162 [JETP Lett. 22, 74 (1975), English translation by W. H. Furry].Search in Google Scholar
146. Flambaum, V.; Khriplovich, I. Zh. Eksp. Teor. Fiz. 1980, 79, 1656 [Sov. Phys. JETP 52, 835 (1980), English translation by J. G. Adashko].Search in Google Scholar
147. Blundell, S.; Sapirstein, J.; Johnson, W. Phys. Rev. D 1992, 45, 1602–1623. https://doi.org/10.1103/physrevd.45.1602.Search in Google Scholar PubMed
148. Hendrickson, K. R. G. Ph.D. Thesis, University of Washington, 1999. http://hdl.handle.net/1773/9668.Search in Google Scholar
149. Khriplovich, I. B. Parity Nonconservation in Atomic Phenomena; CRC Press: Boca Raton, FL, USA,1991.Search in Google Scholar
150. Novikov, V.; Sushkov, O.; Flambaum, V.; Khriplovich, I. Zh. Eksp. Teor. Fiz. 1977, 73, 802 [Sov. Phys. JETP 46, 420 (1977), English translation by W. H. Furry].Search in Google Scholar
151. Flambaum, V.; Khriplovich, I. Zh. Eksp. Teor. Fiz. 1985, 89, 1505 [Sov. Phys. JETP 62, 872 (1985), English translation by A. Brown].Search in Google Scholar
152. Zel’dovich, Y. B. Zh. Eksp. Teor. Fiz. 1957, 33, 1531 [Sov. Phys. JETP 6, 1184 (1957), English translation by G. E. Brown].Search in Google Scholar
153. Flambaum, V. V.; Khriplovich, I. B.; Sushkov, O. P. Phys. Lett. B 1984, 146, 367–369. https://doi.org/10.1016/0370-2693(84)90140-0.Search in Google Scholar
154. Flambaum, V. V.; Khriplovich, I. Phys. Lett. A 1985, 110, 121–125. https://doi.org/10.1016/0375-9601(85)90756-x.Search in Google Scholar
155. Tiesinga, E.; Mohr, P. J.; Newell, D. B.; Taylor, B. N. Rev. Mod. Phys. 2021, 93, 025010. https://doi.org/10.1103/revmodphys.93.025010.Search in Google Scholar
156. Gaul, K.; Marquardt, S.; Isaev, T.; Berger, R. Phys. Rev. A 2019, 99, 032509. https://doi.org/10.1103/physreva.99.032509.Search in Google Scholar
157. Zel’dovich, Y. B.; Saakyan, D.; Sobel’man, I. Pis’ma Zh. Eksp. Teor. Fiz. 1977, 25, 106 [JETP Lett. 25, 94 (1977)].Search in Google Scholar
158. Hegstrom, R.; Rein, D.; Sandars, P. J. Chem. Phys. 1980, 73, 2329–2341. https://doi.org/10.1063/1.440383.Search in Google Scholar
159. Bast, R.; Koers, A.; Gomes, A. S. P.; Iliaš, M.; Visscher, L.; Schwerdtfeger, P.; Saue, T. Phys. Chem. Chem. Phys. 2011, 13, 864–876. https://doi.org/10.1039/c0cp01483d.Search in Google Scholar
160. Forgeron, M. A. M.; Wasylishen, R. E.; Penner, G. H. J. Phys. Chem. A 2004, 108, 4751–4758. https://doi.org/10.1021/jp031279j.Search in Google Scholar
161. Gómez, S. S.; Aucar, G. A. J. Chem. Phys. 2011, 134, 204314. https://doi.org/10.1063/1.3587051.Search in Google Scholar
162. Boeckh, R. v.; Gräff, G.; Ley, R. Z. Phys. 1964, 179, 285–313. https://doi.org/10.1007/BF01381648.Search in Google Scholar
163. Brown, J. M.; Carrington, A. Rotational Spectroscopy of Diatomic Molecules; Cambridge University Press: Cambridge, UK, 2003.10.1017/CBO9780511814808Search in Google Scholar
164. Visscher, L.; Enevoldsen, T.; Saue, T.; Jensen, H. J. A.; Oddershede, J. J. Comput. Chem. 1999, 20, 1262–1273. https://doi.org/10.1002/(sici)1096-987x(199909)20:12<1262::aid-jcc6>3.3.co;2-8.10.1002/(SICI)1096-987X(199909)20:12<1262::AID-JCC6>3.3.CO;2-8Search in Google Scholar
165. Jackson, J. D. Classical Electrodynamics, 3rd ed.; Wiley: New York, 1998.10.1119/1.19136Search in Google Scholar
166. Jørgensen, P.; Simons, J. Second-Quantization-Based Methods in Quantum Chemistry; Academic Press: New York, 1981.10.1016/B978-0-12-390220-7.50005-2Search in Google Scholar
167. Saue, T.; Jensen, H. J. A. J. Chem. Phys. 2003, 118, 522–536. https://doi.org/10.1063/1.1522407.Search in Google Scholar
168. Iliaš, M.; Jensen, H. J. A.; Bast, R.; Saue, T. Mol. Phys. 2013, 111, 1373–1381. https://doi.org/10.1080/00268976.2013.798436.Search in Google Scholar
169. Quiney, H.; Skaane, H.; Grant, I. Chem. Phys. Lett. 1998, 290, 473–480. https://doi.org/10.1016/s0009-2614(98)00568-5.Search in Google Scholar
170. Aucar, I. A.; Gómez, S. S.; Melo, J. I.; Giribet, C. C.; Ruiz de Azúa, M. C. J. Chem. Phys. 2013, 138, 134107. https://doi.org/10.1063/1.4796461.Search in Google Scholar PubMed
171. Aucar, I. A.; Gómez, S. S.; Giribet, C. G.; Ruiz de Azúa, M. C. J. Chem. Phys. 2013, 139, 094112. https://doi.org/10.1063/1.4819958.Search in Google Scholar PubMed
172. Xiao, Y.; Liu, W. J. Chem. Phys. 2013, 138, 134104. https://doi.org/10.1063/1.4797496.Search in Google Scholar PubMed
173. Malkin, E.; Komorovsky, S.; Repisky, M.; Demissie, T. B.; Ruud, K. J. Phys. Chem. Lett. 2013, 4, 459–463. https://doi.org/10.1021/jz302146m.Search in Google Scholar PubMed
174. Laubender, G.; Berger, R. ChemPhysChem 2003, 4, 395–399. https://doi.org/10.1002/cphc.200390070.Search in Google Scholar PubMed
175. Bast, R.; Schwerdtfeger, P.; Saue, T. J. Chem. Phys. 2006, 125, 064504. https://doi.org/10.1063/1.2218333.Search in Google Scholar PubMed
176. Aucar, I. A.; Chamorro, Y.; Borschevsky, A. Phys. Rev. A 2022, 106, 062802. https://doi.org/10.1103/physreva.106.062802.Search in Google Scholar
177. Letokhov, V. S. Phys. Lett. A 1975, 53, 275–276. https://doi.org/10.1016/0375-9601(75)90064-x.Search in Google Scholar
178. Gorshkov, V. G.; Kozlov, M. G.; Labzovskii, L. N. Zh. Eksp. Teor. Fiz. 1982, 82, 1807 [Sov. Phys. JETP 55, 1042 (1982), English translation by J. G. Adashko].Search in Google Scholar
179. Barra, A. L.; Robert, J. B.; Wiesenfeld, L. Phys. Lett. A 1986, 115, 443–447. https://doi.org/10.1016/0375-9601(86)90072-1.Search in Google Scholar
180. Barra, A. L.; Robert, J. B.; Wiesenfeld, L. Europhys. Lett. 1988, 5, 217–222. https://doi.org/10.1209/0295-5075/5/3/006.Search in Google Scholar
181. Barra, A. L.; Robert, J. B. Mol. Phys. 1996, 88, 875–886. https://doi.org/10.1080/00268979609484479.Search in Google Scholar
182. Soncini, A.; Faglioni, F.; Lazzeretti, P. Phys. Rev. A 2003, 68, 033402. https://doi.org/10.1103/physreva.68.033402.Search in Google Scholar
183. Schwerdtfeger, P.; Bast, R. J. Am. Chem. Soc. 2004, 126, 1652–1653. https://doi.org/10.1021/ja038383z.Search in Google Scholar PubMed
184. Berger, R. Relativistic Electronic Structure Theory of Theoretical and Computational Chemistry; Schwerdtfeger, P., Ed.; Elsevier: Amsterdam, The Netherlands, Vol. 14, 2004; pp 188–288.10.1016/S1380-7323(04)80031-1Search in Google Scholar
185. Weijo, V.; Manninen, P.; Vaara, J. J. Chem. Phys. 2005, 123, 054501. https://doi.org/10.1063/1.1961321.Search in Google Scholar PubMed
186. Laubender, G.; Berger, R. Phys. Rev. A 2006, 74, 032105. https://doi.org/10.1103/physreva.74.032105.Search in Google Scholar
187. Weijo, V.; Bast, R.; Manninen, P.; Saue, T.; Vaara, J. J. Chem. Phys. 2007, 126, 074107. https://doi.org/10.1063/1.2436886.Search in Google Scholar PubMed
188. Nahrwold, S.; Berger, R. J. Chem. Phys. 2009, 130, 214101. https://doi.org/10.1063/1.3103643.Search in Google Scholar PubMed
189. Darquié, B.; Stoeffler, C.; Shelkovnikov, A.; Daussy, C.; Amy-Klein, A.; Chardonnet, C.; Zrig, S.; Guy, L.; Crassous, J.; Soulard, P.; Asselin, P.; Huet, T. R.; Schwerdtfeger, P.; Bast, R.; Saue, T. Chirality 2010, 22, 870–884. https://doi.org/10.1002/chir.20911.Search in Google Scholar PubMed
190. Hobi, F.; Berger, R.; Stohner, J. Mol. Phys. 2013, 111, 2345–2362. https://doi.org/10.1080/00268976.2013.816444.Search in Google Scholar
191. Eills, J.; Blanchard, J. W.; Bougas, L.; Kozlov, M. G.; Pines, A.; Budker, D. Phys. Rev. A 2017, 96, 042119. https://doi.org/10.1103/physreva.96.042119.Search in Google Scholar
192. Cournol, A.; Manceau, M.; Pierens, M.; Lecordier, L.; Tran, D. B. A.; Santagata, R.; Argence, B.; Goncharov, A.; Lopez, O.; Abgrall, M.; Le Coq, Y.; Le Targat, R.; Alvarez Martinez, H.; Lee, W. K.; Xu, D.; Pottie, P. E.; Hendricks, R. J.; Wall, T. E.; Bieniewska, J. M.; Sauer, B. E.; Tarbutt, M. R.; Amy-Klein, A.; Tokunaga, S. K.; Darquié, B. Quantum Electron. 2019, 49, 288–292. https://doi.org/10.1070/qel16880.Search in Google Scholar
193. Aucar, I. A.; Borschevsky, A. J. Chem. Phys. 2021, 155, 134307. https://doi.org/10.1063/5.0065487.Search in Google Scholar PubMed
194. Blanchard, J. W.; Budker, D.; DeMille, D.; Kozlov, M. G.; Skripnikov, L. V. Phys. Rev. Res. 2023, 5, 013191. https://doi.org/10.1103/physrevresearch.5.013191.Search in Google Scholar
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