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Min–max and max–min approaches to the solution of wave equation in relativistic quantum chemistry

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Abstract

The variation problem associated with the solution of Dirac’s relativistic electron equation is reviewed here. Derivations of the min–max and max–min theorems are discussed: A new observation is that the spurious roots of negative energy satisfy a max–min theorem. Min–max principle (MMP) for solution of Dirac equation, extendable to the Dirac–Fock case, is concisely inspected and illustrated. MMP for two-electron Dirac–Coulomb equation is also considered. The min–max relation is physically interpreted for both Dirac and Dirac–Coulomb problems. Applications to chemical systems are investigated. Mathematical advances are reviewed. Limitations of MMP are spelt out, recent mathematical developments are collated, and an outline is given of the associated theoretical and computational developments. For a complementary purpose, the case of spin-zero is briefly examined. The variation principle for Dirac fermions in nanoscience is discussed. A brief outline of future prospects is given.

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Fig. 1

Reproduced and modified from Ref. [27]

Fig. 2

Reproduced and modified from Ref. [29]

Fig. 3

Reproduced and modified from Ref. [29]

Fig. 4

Reproduced and modified from Ref. [42]

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Fig. 7

Reproduced and modified from Ref. [40]

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References

  1. G E Brown and D G Ravenhall Proc. R. Soc. A208 552 (1951)

    ADS  Google Scholar 

  2. G Breit Phys. Rev. 34 553 (1929)

    Article  ADS  Google Scholar 

  3. J Sucher Phys. Rev. A 22 348 (1980); Erratum: Phys. Rev. A 23 388 (1981)

  4. J Sucher Phys. Scr. 36 271 (1987)

    Article  ADS  Google Scholar 

  5. I P Grant Adv. Phys. 19 747 (1970)

    Article  ADS  Google Scholar 

  6. J-P Desclaux Comput. Phys. Commun. 9 31 (1975)

    Article  ADS  Google Scholar 

  7. D R Hartree The Calculation of Atomic Structures (New York: Wiley) (1957)

    MATH  Google Scholar 

  8. S N Datta Chem. Phys. Lett. 74 568 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  9. J D Weeks, A Hazi and S A Rice Adv. Chem. Phys. 16 283 (1969)

    Google Scholar 

  10. H Wallmeier and W Kutzelnigg Chem. Phys. Lett. 78 341 (1981)

    Article  ADS  Google Scholar 

  11. Y Ishikawa and G Malli Chem. Phys. Lett. 80 111 (1981)

    Article  ADS  Google Scholar 

  12. G W F Drake and S P Goldman Phys. Rev. A23 2093 (1981)

    Article  ADS  Google Scholar 

  13. F Mark and W H E Schwarz Phys. Rev. Lett. 48 673 (1982)

    Article  ADS  Google Scholar 

  14. W H E Schwarz and H Wallmeier Mol. Phys. 46 1045 (1982)

    Article  ADS  Google Scholar 

  15. W H E Schwarz and E Wechsel-Trakowski Chem. Phys. Lett. 85 94 (1982)

    Article  ADS  Google Scholar 

  16. B Gazdy Chem. Phys. Lett. 99 41 (1983)

    Article  ADS  Google Scholar 

  17. B Gazdy and K Ladanyi J. Chem. Phys. 80 4333 (1984)

    Article  ADS  Google Scholar 

  18. F Rosicky and F Mark J. Phys. B At. Mol. Phys. 8 2581 (1975)

    Article  ADS  Google Scholar 

  19. S N Datta and S Jagannathan Pramana 23 467 (1984)

    Article  ADS  Google Scholar 

  20. R E Stanton and S Havriliak J. Chem. Phys. 81 1910 (1984)

    Article  ADS  Google Scholar 

  21. K G Dyall and K Faegri Chem. Phys. Lett. 174 25 (1990)

    Article  ADS  Google Scholar 

  22. K G Dyall, I P Grant and S Wilson J. Phys. B At. Mol. Opt. Phys. 17 L45 (1984)

    Article  ADS  Google Scholar 

  23. K G Dyall, I P Grant and S Wilson J. Phys. B At. Mol. Opt. Phys. 17 493 (1984)

    Article  ADS  Google Scholar 

  24. J Wood, I P Grant and S Wilson J. Phys. B At. Mol. Opt. Phys. 18 3027 (1985)

    Article  ADS  Google Scholar 

  25. I P Grant J. Phys. B At. Mol. Opt. Phys. 19 3187 (1986)

    Article  ADS  Google Scholar 

  26. J D Talman Phys. Rev. Lett. 57 1091 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  27. S N Datta and G Devaiah Pramana 30 387 (1988); Also see: S N Datta Adriatico Research Conference on Relativistic Many-Body Effects (Trieste) Abstracts (1986)

  28. G W F Drake and S P Goldman Adv. At. Mol. Phys. 25 393 (1989)

    Article  ADS  Google Scholar 

  29. S N Datta Pramana 1992, 38 51 (1992)

  30. A Kolakowska, J D Talman and K A Ashamar Phys. Rev. A53 R168 (1996)

    Article  ADS  Google Scholar 

  31. M Griesemer and H Siedentop J. Lond. Math. Soc. 60 490 (1999)

    Article  Google Scholar 

  32. M Griesemer, R T Lewis and H Siedentop Doc. Math. 4 275 (1999)

    MathSciNet  Google Scholar 

  33. J Dolbeault, M J Esteban and É Séré J. Funct. Anal. 174 208 (2000)

    Article  MathSciNet  Google Scholar 

  34. S N Datta Pramana J. Phys. 55 383 (2000)

    Article  ADS  Google Scholar 

  35. W Kutzelnigg Chem. Phys. 225 203 (1997)

    Article  Google Scholar 

  36. F Mark Schriftenreihe des Max-Planck-Instituts für Strahlenchemiei, No. 24 (Mülheim: MPI) (1985)

  37. S N Datta Pramana 41 363 (1993)

    Article  ADS  Google Scholar 

  38. S N Datta Int. J. Quantum Chem. 56 91 (1995)

    Article  Google Scholar 

  39. H Partridge NASA Technical Memorandum No. 101044 (Moffett Field: NASA) (1989)

  40. A G Darwin Proc. Roy. Soc. Lond. A 118 654 (1928)

    Article  ADS  Google Scholar 

  41. F J Belifante Comput. Phys. 1 319 (1991)

    Article  ADS  Google Scholar 

  42. S Datta and S N Datta Pramana 38 521 (1992)

    Article  ADS  Google Scholar 

  43. S N Datta Pramana 23 L275 (1984)

    Article  ADS  Google Scholar 

  44. S N Datta Pramana 28 633 (1987)

    Article  ADS  Google Scholar 

  45. L LaJohn and J D Talman Chem. Phys. Lett. 189 383 (1992)

    Article  ADS  Google Scholar 

  46. Y Ishikawa, R C Binning Jr. and K M Sando, Chem. Phys. Lett. 101 111 (1983)

    Article  ADS  Google Scholar 

  47. S N Datta and C S Ewig Chem. Phys. Lett. 85 443 (1982)

    Article  ADS  Google Scholar 

  48. P I Pavlik and S M Blinder J. Chem. Phys. 46 2749 (1967)

    Article  ADS  Google Scholar 

  49. S K Luke, G Hunter, R P McEachran and M Cohen J. Chem. Phys. 50 1644 (1969)

    Article  ADS  Google Scholar 

  50. J Oreg and G Malli J. Chem. Phys. 61 4349 (1974)

    Article  ADS  Google Scholar 

  51. G Malli and J Oreg J. Chem. Phys. 63 830 (1975)

    Article  ADS  Google Scholar 

  52. J Oreg and G Malli J. Chem. Phys. 65 1746 (1976)

    Article  ADS  Google Scholar 

  53. J Oreg and G Malli J. Chem. Phys. 65 1755 (1976)

    Article  ADS  Google Scholar 

  54. G Malli Chem. Phys. Lett. 73 510 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  55. P Pyykkö and J-P Desclaux Chem. Phys. Lett. 42 545 (1976)

    Article  ADS  Google Scholar 

  56. Y S Lee and A D McLean J. Chem. Phys. 76 735 (1982)

    Article  ADS  Google Scholar 

  57. S N Datta Proc. Ind. Acad. Sci. Chem. Sci., Special Issue on Theoretical Models for Molecular Structure, Properties and Dynamics 106 445 (1994)

  58. C Tix Bull. Lond. Math. Soc. 30 283 (1998)

    Article  MathSciNet  Google Scholar 

  59. J Karwowski, G Pestica, M Stancke and F E Harris Int. J. Quantum Chem. 106 3129 (2006)

    Article  ADS  Google Scholar 

  60. M Stancke, J Karwowski and H Tatewaki Mol. Phys. 104 2085 (2006)

    Article  ADS  Google Scholar 

  61. W D Evans Proc. Lond. Math. Soc. 20 537 (1970)

    Article  Google Scholar 

  62. U W Schmincke Math. Z. 129 335 (1972)

    Article  MathSciNet  Google Scholar 

  63. R Wüst Math. Z. 141 93 (1975)

    Article  MathSciNet  Google Scholar 

  64. G Nenciu Commun. Math. Phys. 48 235 (1976)

    Article  ADS  Google Scholar 

  65. J J Landgren and P A Rejto J. Math. Phys. 20 2204 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  66. J J Landgren, P A Rejto and M Klaus J. Math. Phys. 21 1210 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  67. J Dolbeault, M J Esteban and É Séré Calc. Var. Partial Differ. Equ. 10 321 (2000)

    Article  Google Scholar 

  68. J Dolbeault, M J Esteban, É Séré and M Vanbreugel Phys. Rev. Lett. 85 4020 (2000)

    Article  ADS  Google Scholar 

  69. J Dolbeault, M J Esteban and É Séré Int. J. Quantum Chem. 93 149 (2003)

    Article  Google Scholar 

  70. J Dolbeault, M J Esteban and É Séré J. Eur. Math. Soc. 8 243 (2006)

    Article  Google Scholar 

  71. M J Esteban, M Lewin and É Séré Bull. Am. Math. Soc. 45 535 (2008)

    Article  Google Scholar 

  72. J Dolbeault, M J Esteban, M Loss and L Vega J. Funct. Anal. 216 1 (2004)

    Article  MathSciNet  Google Scholar 

  73. J Dolbeault, M J Esteban, J Duoandikoetxea and L Vega Ann. Sci. Ecole Norm. Sup. 40 885 (2007)

    Article  Google Scholar 

  74. M J Esteban and M Loss J. Math. Phys. 48 112107 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  75. M J Esteban and M Loss Mathematical Results in Quantum Mechanics (eds.) I Beltita, G Nenciu and R Purice (Hackensack: World Sci.) p 41 (2008)

  76. M J Esteban, M Lewin and É Séré https://arxiv.org/pdf/1702.04976 (2017)

  77. H Hogreve J. Phys. A Math. Gen. 46 025301 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  78. V C Zelati and M J Nolasco Elliptic Parabol. Equ. 1 89 (2015)

    Article  MathSciNet  Google Scholar 

  79. S Morozov and D Müller Math. Z. 280 733 (2015)

    Article  MathSciNet  Google Scholar 

  80. D Müller arXiv:1603.01557v1 [math-ph] (2016)

  81. P Pyykkö Relativistic Theory of Atoms and Molecules: A Bibliography 1916-1985, Lecture Notes in Chemistry Vol. 41 (Berlin: Springer) (1986)

    Google Scholar 

  82. A K Mohanty, F A Parpia and E Clementi Modern Techniques in Computational Chemistry: MOTECC-91 (ed.) E Clementi (Leiden: ESCOM) Chapter 4 (1991)

  83. F Parpia and A K Mohanty Modern Techniques in Computational Chemistry: MOTECC-91 (ed.) E Clementi (Leiden: ESCOM) Chapter 5 (1991)

  84. A K Mohanty, S Panigrahy and E Clementi Modern Techniques in Computational Chemistry: MOTECC-91 (ed.) E Clementi (Leiden: ESCOM) Chapter 15 (1991)

  85. I Lindgren and J Morrison Atomic Many-Body Theory, 2nd Edn. (Berlin: Springer) Chapter 3, 4, 14 (1985)

  86. B Thaller The Dirac Equation. Texts and Monographs in Physics (Berlin: Springer) (1992)

    Google Scholar 

  87. T Kato Perturbation Theory for Linear Operators. Classics in Mathematics (Berlin: Springer) (1995)

    Book  Google Scholar 

  88. A A Balinsky and W D Evans Spectral Analysis of Relativistic Operators (London: Imperial College Press) (2011)

    MATH  Google Scholar 

  89. R L Hall Z. Phys. A291 255 (1979)

    Article  ADS  Google Scholar 

  90. R L Hall J. Phys. A Math. Gen. 22 179 (1989)

    Article  ADS  Google Scholar 

  91. R L Hall, W Lucha and F F Schöberl J. Math. Phys. 43 1237 (2002); Erratum, J. Math. Phys. 44 2724 (2003)

  92. R L Hall, W Lucha and F F Schöberl J. Math. Phys. 45 3086 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  93. R L Hall and W Lucha J. Phys. A Math. Gen. 41 355202 (2008)

    Article  Google Scholar 

  94. R L Hall and M D Aliyu Phys. Rev. A 78 052115 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  95. R L Hall Phys. Rev. A 81 052101 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  96. S N Datta, A Ghosh and R Chakraborty Ind. J. Phys. 89 181 (2015)

    Article  Google Scholar 

  97. L Lamata, M A Martin-Delgado and E Solano Phys. Rev. Lett. 97 250502 (2006)

    Article  ADS  Google Scholar 

  98. A Bermudez, M A Martin-Delgado and E Solano Phys. Rev. A 76 041801(R) (2007)

    Article  ADS  Google Scholar 

  99. E T Jaynes and F W Cummings Proc. IEEE 51 89 (1963)

    Article  Google Scholar 

  100. A Bermudez, M A Martin-Delgado and E Solano Phys. Rev. Lett. 99 123602 (2007)

    Article  ADS  Google Scholar 

  101. A Bermudez, M A Martin-Delgado and A Luis Phys. Rev. A 77 063815 (2008)

    Article  ADS  Google Scholar 

  102. S N Datta Chem. Phys. Lett. 692 313 (2018)

    Article  ADS  Google Scholar 

  103. P R Wallace Phys. Rev. 71 622 (1947)

    Article  ADS  Google Scholar 

  104. A H Castro Neto, F Guinea, N M R Peres, K S Novoselov and A K Geim Rev. Mod. Phys. 2009, 81 109 (2009)

  105. G W Semenoff Phys. Rev. Lett. 53 2449 (1984)

    Article  ADS  Google Scholar 

  106. R Mondal, M Berritta, A K Nandy and P M Oppeneer Phys. Rev. B 96 024425 (2017)

    Article  ADS  Google Scholar 

  107. T Tudorovskiy and M I Katsnelson arXiv:1206.2242v2 [cond-mat.mes-hall] (2012)

  108. A K Rajagopal and J Callaway Phys. Rev. B 7 1912 (1973)

    Article  ADS  Google Scholar 

  109. D Bagayoko J. Mod. Phys. 7 911 (2016)

    Article  Google Scholar 

  110. E Engel and R M Dreizler Top. Curr. Chem. 181 1 (1996)

    Article  Google Scholar 

  111. R M Dreizler Lecture Notes in Physics Vol. 620 (eds.) C Fiolhais, F Nogueira and M Marques (Berlin: Springer) p 123 (2003)

  112. E Eliav, U Kaldor and Y Ishikawa Int. J. Quantum Chem. 52 (Issue Supplement S28) 205 (1994)

  113. U Kaldor Recent Advances in Coupled-cluster Methods (ed.) R J Bartlett (Singapore: World Sci.) p 125 (1997)

  114. R F Bishop Lecture Notes in Physics Vol. 510 (eds.) J Navarro and A Polls (Berlin: Springer) p 1 (1998)

  115. H G Kummel Recent Progress in Many-Body Theories, Proceedings of the 11th International Conference (eds.) R F Bishop, R N W Bishop, T Brandes, K A Gernoth, N R Walet and Y Xiam (Singapore: World Sci.) p 334 (2002)

  116. B K Sahoo, R Chaudhuri, B P Das and D Mukherjee Phys. Rev. Lett. 96 163003 (2006)

    Article  ADS  Google Scholar 

  117. A Shee, L Visscher and T Saue J. Chem. Phys. 145 184107 (2016)

    Article  ADS  Google Scholar 

  118. B K Sahoo and B P Das arXiv:1704.04340v2 [atom-ph] (2017)

  119. S Hirata, T Yanai and W A De Jong J. Chem. Phys. 120 3297 (2004)

    Article  ADS  Google Scholar 

  120. A Ghosh, R K Chaudhuri and S Chattopadhyay J. Chem. Phys. 145 124303 (2016)

    Article  ADS  Google Scholar 

  121. S N Datta arXiv:1709.07061 [quant-ph] (2017)

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Acknowledgements

I am grateful to A. K. Pal for help in preparing the figures.

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  1. Department of Chemistry, Indian Institute of Technology – Bombay, Powai, Mumbai, 400 076, India

    Sambhu Nath Datta

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Datta, S.N. Min–max and max–min approaches to the solution of wave equation in relativistic quantum chemistry. Indian J Phys 93, 285–299 (2019). https://doi.org/10.1007/s12648-018-1271-8

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