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Solving the Schrödinger Equation for the Hydrogen Molecular Ion in a Magnetic Field Using the Free-Complement Method

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Quantum Systems in Chemistry and Physics

Part of the book series: Progress in Theoretical Chemistry and Physics ((PTCP,volume 26))

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Abstract

The hydrogen molecular ion (H +2 ) in a magnetic field is investigated theoretically using the free-complement (FC) method for solving the Schrödinger equation. H +2 was placed in magnetic fields of moderate strengths. Our results were shown to be highly accurate. Total energies, dissociation energies, quadrupole moments, and electron densities were calculated for parallel and perpendicular fields. The gauge-origin dependence of the wave function was examined in detail. It was shown that the results of the FC method are always gauge independent when the gauge-including function is employed as the initial function. Even when we start from the gauge-nonincluding functions, the FC method gives the gauge-independent result in some order, because the FC wave function becomes exact as the order of the FC calculations increases. We observed that properties such as total energy, potential energy curve, vibrational level, and electron density distribution became gauge-origin independent as the order of the FC wave function increased.

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References

  1. Bhaduri RK, Nogami Y, Warke CS (1977) Astrophys J 217:324

    Article  CAS  Google Scholar 

  2. Bignami GF, Caraveo PA, De Luca A, Mereghetti S (2003) Nature 423:725

    Article  CAS  Google Scholar 

  3. Vink J, de Vries CP, Mendez M, Verbunt F (2004) Astrophys J 609:L75

    Article  Google Scholar 

  4. van Kerkwijk MH, Kaplan DL, Durant M, Kulkarni SR, Paerels F (2004) Astrophys J 608:432

    Article  Google Scholar 

  5. Mori K, Chonko JC, Hailey CJ (2005) Astrophys J 631:1082

    Article  CAS  Google Scholar 

  6. Mori K, Hailey CJ (2006) Astrophys J 648:1139

    Article  CAS  Google Scholar 

  7. Helgaker T, Jorgensen P (1991) J Chem Phys 95:2595

    Article  CAS  Google Scholar 

  8. Ruud K, Helgaker T, Bak KL, Jorgensen P, Jensen HJA (1993) J Chem Phys 99:3847

    Article  CAS  Google Scholar 

  9. Barszczewicz A, Helgaker T, Jaszunski M, Jorgensen P, Ruud K (1994) J Chem Phys 101:6822

    Article  Google Scholar 

  10. Jonsson D, Norman P, Ruud K, Agren H, Helgaker T (1998) J Chem Phys 109:572

    Article  CAS  Google Scholar 

  11. Helgaker T, Jaszuński M, Ruud K (1998) Chem Rev 99:293

    Article  Google Scholar 

  12. Tellgren EI, Soncini A, Helgaker T (2008) J Chem Phys 129:154114

    Article  Google Scholar 

  13. Hylleraas EA (1931) Z Physik 71:739

    Article  CAS  Google Scholar 

  14. Jaffe G (1934) Z Physik 87:535

    Article  CAS  Google Scholar 

  15. Bates DR, Ledsham K, Stewart AL (1953) Philos Trans R Soc Lond Ser A 246:215

    Article  Google Scholar 

  16. Wind H (1965) J Chem Phys 42:2371

    Article  CAS  Google Scholar 

  17. Peek JM (1965) J Chem Phys 43:3004

    Article  CAS  Google Scholar 

  18. Demelo CP, Ferreira R, Brandi HS, Miranda LCM (1976) Phys Rev Lett 37:676

    Article  CAS  Google Scholar 

  19. Peek JM, Katriel J (1980) Phys Rev A 21:413

    Article  CAS  Google Scholar 

  20. Larsen DM (1982) Phys Rev A 25:1295

    Article  CAS  Google Scholar 

  21. Turbiner AV (1983) JETP Lett 38:618

    Google Scholar 

  22. Turbiner AV, Lopez Vieyra JC (2003) Phys Rev A 68:012504

    Article  Google Scholar 

  23. Turbiner AV, Lopez Vieyra JC (2004) Phys Rev A 69:053413

    Article  Google Scholar 

  24. Turbiner AV, Lopez Vieyra JC (2005) Mod Phys Lett A 20:2845

    Article  CAS  Google Scholar 

  25. Turbiner AV, Lopez Vieyra JC (2006) Phys Rep 424:309

    Article  CAS  Google Scholar 

  26. Turbiner AV, Olivares-Pilon H (2011) J Phys B Atom Mol Opt Phys 44:101002

    Article  Google Scholar 

  27. Khersonskij VK (1984) Astrophys Space Sci 98:255

    Article  Google Scholar 

  28. Khersonskij VK (1984) Astrophys Space Sci 103:357

    Article  Google Scholar 

  29. Khersonskij VK (1985) Astrophys Space Sci 117:47

    Article  Google Scholar 

  30. Wille U (1988) Phys Rev A 38:3210

    Article  CAS  Google Scholar 

  31. Kappes U, Schmelcher P, Pacher T (1994) Phys Rev A 50:3775

    Article  CAS  Google Scholar 

  32. Kappes U, Schmelcher P (1995) Phys Rev A 51:4542

    Article  CAS  Google Scholar 

  33. Kappes U, Schmelcher P (1996) Phys Rev A 53:3869

    Article  CAS  Google Scholar 

  34. Kappes U, Schmelcher P (1996) Phys Lett A 210:409

    Article  CAS  Google Scholar 

  35. Kravchenko YP, Liberman MA (1997) Phys Rev A 55:2701

    Article  CAS  Google Scholar 

  36. Kaschiev MS, Vinitsky SI, Vukajlovic FR (1980) Phys Rev A 22:557

    Article  CAS  Google Scholar 

  37. Ozaki J, Tomishima Y (1980) J Phys Soc Jpn 49:1497

    Article  CAS  Google Scholar 

  38. Ozaki J, Tomishima Y (1983) J Phys Soc Jpn 52:1142

    Article  CAS  Google Scholar 

  39. Vincke M, Baye D (2006) J Phys B Atom Mol Opt Phys 39:2605

    Article  CAS  Google Scholar 

  40. Baye D, de ter Beerst AJ, Sparenberg JM (2009) J Phys B Atom Mol Opt Phys 42:225102

    Article  Google Scholar 

  41. Nakatsuji H (2000) J Chem Phys 113:2949

    Article  CAS  Google Scholar 

  42. Nakatsuji H, Davidson ER (2001) J Chem Phys 115:2000

    Article  CAS  Google Scholar 

  43. Nakatsuji H (2002) Phys Rev A 65:052122

    Article  Google Scholar 

  44. Nakatsuji H, Ehara M (2002) J Chem Phys 117:9

    Article  CAS  Google Scholar 

  45. Nakatsuji H (2004) Phys Rev Lett 93:030403

    Article  Google Scholar 

  46. Nakatsuji H (2005) Phys Rev A 72:062110

    Article  Google Scholar 

  47. Nakatsuji H, Nakashima H (2005) Phys Rev Lett 95:050407

    Article  Google Scholar 

  48. Kurokawa Y, Nakashima H, Nakatsuji H (2005) Phys Rev A 72:062502

    Article  Google Scholar 

  49. Nakatsuji H, Nakashima H, Kurokawa Y, Ishikawa A (2007) Phys Rev Lett 99:240402

    Article  CAS  Google Scholar 

  50. Nakashima H, Nakatsuji H (2008) J Chem Phys 128:154107

    Article  Google Scholar 

  51. Nakashima H, Nakatsuji H (2008) Phys Rev Lett 101:240406

    Article  Google Scholar 

  52. Ishikawa A, Nakashima H, Nakatsuji H (2008) J Chem Phys 128:124103

    Article  Google Scholar 

  53. Hijikata Y, Nakashima H, Nakatsuji H (2009) J Chem Phys 130:024102

    Article  Google Scholar 

  54. Nakatsuji H, Nakashima H (2009) Int J Quant Chem 109:2248

    Article  CAS  Google Scholar 

  55. Bande A, Nakashima H, Nakatsuji H (2010) Chem PhysLett 496:347

    CAS  Google Scholar 

  56. Nakashima H, Nakatsuji H (2010) Astrophys J 725:528

    Article  CAS  Google Scholar 

  57. Nakashima H, Nakatsuji H (2011) Theor Chem Acc 129:567

    Article  CAS  Google Scholar 

  58. Nakatsuji H (2011) Phys Rev A 84:062507

    Article  Google Scholar 

  59. GMP, the GNU multiple precision arithmetic library.

    Google Scholar 

  60. Maple, Waterloo Maple Inc., Ontario, Canada.

    Google Scholar 

  61. This technique was first applied to H +2 in a magnetic field by Wille [30], and is similar to the gauge-including (or independent) atomic orbital (GIAO) or London orbital often used in standard ab initio calculations. See references 7–12 and 64.

    Google Scholar 

  62. Light JC, Carrington T (2000) Adv Chem Phys 114:263

    Article  Google Scholar 

  63. Laaksonen L, Pyykko P, Sundholm D (1983) Int J Quant Chem 23:309

    Article  CAS  Google Scholar 

  64. Ditchfie R (1974) Mol Phys 27:789

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Quantum Chemistry Research Institute & JST CREST, Kyodai Katsura Venture Plaza 107, Goryo Oohara 1-36, Nishikyo-ku, Kyoto, 615-8245, Japan

    Atsushi Ishikawa & Hiroyuki Nakashima

  2. Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University, Aoba-ku, Sendai, Japan

    Hiroshi Nakatsuji

  3. Quantum Chemical Research Institute (QCRI), Kyoto, Japan

    Hiroshi Nakatsuji

Authors
  1. Atsushi Ishikawa
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  2. Hiroyuki Nakashima
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  3. Hiroshi Nakatsuji
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Corresponding author

Correspondence to Hiroshi Nakatsuji .

Editor information

Editors and Affiliations

  1. , Division of Mathem. and Physical Science, Kanazawa University, Kanazawa, 920-1192, Japan

    Kiyoshi Nishikawa

  2. Laboratoire de Chimie Physique, rue Pierre et Marie Curie 11, Paris, 75005, France

    Jean Maruani

  3. Dept. Quantum Chemistry, Uppsala University, Uppsala, Sweden

    Erkki J. Brändas

  4. Inst. Matemáticas y Física, Fundamental (IMAFF), CSIC Madrid, C/ Serrano 123, Madrid, 28006, Spain

    Gerardo Delgado-Barrio

  5. Dept. Chemistry, Michigan State University, Chemistry Building, East Lansing, 48824, Michigan, USA

    Piotr Piecuch

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Ishikawa, A., Nakashima, H., Nakatsuji, H. (2012). Solving the Schrödinger Equation for the Hydrogen Molecular Ion in a Magnetic Field Using the Free-Complement Method. In: Nishikawa, K., Maruani, J., Brändas, E., Delgado-Barrio, G., Piecuch, P. (eds) Quantum Systems in Chemistry and Physics. Progress in Theoretical Chemistry and Physics, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5297-9_13

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