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

  • Atsushi Ishikawa
  • Hiroyuki Nakashima
  • Hiroshi Nakatsuji
Conference paper
Part of the Progress in Theoretical Chemistry and Physics book series (PTCP, volume 26)


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.


Wave Function Initial Function Strong Magnetic Field Complement Function Parallel Magnetic Field 
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  1. 1.
    Bhaduri RK, Nogami Y, Warke CS (1977) Astrophys J 217:324CrossRefGoogle Scholar
  2. 2.
    Bignami GF, Caraveo PA, De Luca A, Mereghetti S (2003) Nature 423:725CrossRefGoogle Scholar
  3. 3.
    Vink J, de Vries CP, Mendez M, Verbunt F (2004) Astrophys J 609:L75CrossRefGoogle Scholar
  4. 4.
    van Kerkwijk MH, Kaplan DL, Durant M, Kulkarni SR, Paerels F (2004) Astrophys J 608:432CrossRefGoogle Scholar
  5. 5.
    Mori K, Chonko JC, Hailey CJ (2005) Astrophys J 631:1082CrossRefGoogle Scholar
  6. 6.
    Mori K, Hailey CJ (2006) Astrophys J 648:1139CrossRefGoogle Scholar
  7. 7.
    Helgaker T, Jorgensen P (1991) J Chem Phys 95:2595CrossRefGoogle Scholar
  8. 8.
    Ruud K, Helgaker T, Bak KL, Jorgensen P, Jensen HJA (1993) J Chem Phys 99:3847CrossRefGoogle Scholar
  9. 9.
    Barszczewicz A, Helgaker T, Jaszunski M, Jorgensen P, Ruud K (1994) J Chem Phys 101:6822CrossRefGoogle Scholar
  10. 10.
    Jonsson D, Norman P, Ruud K, Agren H, Helgaker T (1998) J Chem Phys 109:572CrossRefGoogle Scholar
  11. 11.
    Helgaker T, Jaszuński M, Ruud K (1998) Chem Rev 99:293CrossRefGoogle Scholar
  12. 12.
    Tellgren EI, Soncini A, Helgaker T (2008) J Chem Phys 129:154114CrossRefGoogle Scholar
  13. 13.
    Hylleraas EA (1931) Z Physik 71:739CrossRefGoogle Scholar
  14. 14.
    Jaffe G (1934) Z Physik 87:535CrossRefGoogle Scholar
  15. 15.
    Bates DR, Ledsham K, Stewart AL (1953) Philos Trans R Soc Lond Ser A 246:215CrossRefGoogle Scholar
  16. 16.
    Wind H (1965) J Chem Phys 42:2371CrossRefGoogle Scholar
  17. 17.
    Peek JM (1965) J Chem Phys 43:3004CrossRefGoogle Scholar
  18. 18.
    Demelo CP, Ferreira R, Brandi HS, Miranda LCM (1976) Phys Rev Lett 37:676CrossRefGoogle Scholar
  19. 19.
    Peek JM, Katriel J (1980) Phys Rev A 21:413CrossRefGoogle Scholar
  20. 20.
    Larsen DM (1982) Phys Rev A 25:1295CrossRefGoogle Scholar
  21. 21.
    Turbiner AV (1983) JETP Lett 38:618Google Scholar
  22. 22.
    Turbiner AV, Lopez Vieyra JC (2003) Phys Rev A 68:012504CrossRefGoogle Scholar
  23. 23.
    Turbiner AV, Lopez Vieyra JC (2004) Phys Rev A 69:053413CrossRefGoogle Scholar
  24. 24.
    Turbiner AV, Lopez Vieyra JC (2005) Mod Phys Lett A 20:2845CrossRefGoogle Scholar
  25. 25.
    Turbiner AV, Lopez Vieyra JC (2006) Phys Rep 424:309CrossRefGoogle Scholar
  26. 26.
    Turbiner AV, Olivares-Pilon H (2011) J Phys B Atom Mol Opt Phys 44:101002CrossRefGoogle Scholar
  27. 27.
    Khersonskij VK (1984) Astrophys Space Sci 98:255CrossRefGoogle Scholar
  28. 28.
    Khersonskij VK (1984) Astrophys Space Sci 103:357CrossRefGoogle Scholar
  29. 29.
    Khersonskij VK (1985) Astrophys Space Sci 117:47CrossRefGoogle Scholar
  30. 30.
    Wille U (1988) Phys Rev A 38:3210CrossRefGoogle Scholar
  31. 31.
    Kappes U, Schmelcher P, Pacher T (1994) Phys Rev A 50:3775CrossRefGoogle Scholar
  32. 32.
    Kappes U, Schmelcher P (1995) Phys Rev A 51:4542CrossRefGoogle Scholar
  33. 33.
    Kappes U, Schmelcher P (1996) Phys Rev A 53:3869CrossRefGoogle Scholar
  34. 34.
    Kappes U, Schmelcher P (1996) Phys Lett A 210:409CrossRefGoogle Scholar
  35. 35.
    Kravchenko YP, Liberman MA (1997) Phys Rev A 55:2701CrossRefGoogle Scholar
  36. 36.
    Kaschiev MS, Vinitsky SI, Vukajlovic FR (1980) Phys Rev A 22:557CrossRefGoogle Scholar
  37. 37.
    Ozaki J, Tomishima Y (1980) J Phys Soc Jpn 49:1497CrossRefGoogle Scholar
  38. 38.
    Ozaki J, Tomishima Y (1983) J Phys Soc Jpn 52:1142CrossRefGoogle Scholar
  39. 39.
    Vincke M, Baye D (2006) J Phys B Atom Mol Opt Phys 39:2605CrossRefGoogle Scholar
  40. 40.
    Baye D, de ter Beerst AJ, Sparenberg JM (2009) J Phys B Atom Mol Opt Phys 42:225102CrossRefGoogle Scholar
  41. 41.
    Nakatsuji H (2000) J Chem Phys 113:2949CrossRefGoogle Scholar
  42. 42.
    Nakatsuji H, Davidson ER (2001) J Chem Phys 115:2000CrossRefGoogle Scholar
  43. 43.
    Nakatsuji H (2002) Phys Rev A 65:052122CrossRefGoogle Scholar
  44. 44.
    Nakatsuji H, Ehara M (2002) J Chem Phys 117:9CrossRefGoogle Scholar
  45. 45.
    Nakatsuji H (2004) Phys Rev Lett 93:030403CrossRefGoogle Scholar
  46. 46.
    Nakatsuji H (2005) Phys Rev A 72:062110CrossRefGoogle Scholar
  47. 47.
    Nakatsuji H, Nakashima H (2005) Phys Rev Lett 95:050407CrossRefGoogle Scholar
  48. 48.
    Kurokawa Y, Nakashima H, Nakatsuji H (2005) Phys Rev A 72:062502CrossRefGoogle Scholar
  49. 49.
    Nakatsuji H, Nakashima H, Kurokawa Y, Ishikawa A (2007) Phys Rev Lett 99:240402CrossRefGoogle Scholar
  50. 50.
    Nakashima H, Nakatsuji H (2008) J Chem Phys 128:154107CrossRefGoogle Scholar
  51. 51.
    Nakashima H, Nakatsuji H (2008) Phys Rev Lett 101:240406CrossRefGoogle Scholar
  52. 52.
    Ishikawa A, Nakashima H, Nakatsuji H (2008) J Chem Phys 128:124103CrossRefGoogle Scholar
  53. 53.
    Hijikata Y, Nakashima H, Nakatsuji H (2009) J Chem Phys 130:024102CrossRefGoogle Scholar
  54. 54.
    Nakatsuji H, Nakashima H (2009) Int J Quant Chem 109:2248CrossRefGoogle Scholar
  55. 55.
    Bande A, Nakashima H, Nakatsuji H (2010) Chem PhysLett 496:347Google Scholar
  56. 56.
    Nakashima H, Nakatsuji H (2010) Astrophys J 725:528CrossRefGoogle Scholar
  57. 57.
    Nakashima H, Nakatsuji H (2011) Theor Chem Acc 129:567CrossRefGoogle Scholar
  58. 58.
    Nakatsuji H (2011) Phys Rev A 84:062507CrossRefGoogle Scholar
  59. 59.
    GMP, the GNU multiple precision arithmetic library.Google Scholar
  60. 60.
    Maple, Waterloo Maple Inc., Ontario, Canada.Google Scholar
  61. 61.
    This technique was first applied to H2+ 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. 62.
    Light JC, Carrington T (2000) Adv Chem Phys 114:263CrossRefGoogle Scholar
  63. 63.
    Laaksonen L, Pyykko P, Sundholm D (1983) Int J Quant Chem 23:309CrossRefGoogle Scholar
  64. 64.
    Ditchfie R (1974) Mol Phys 27:789CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Atsushi Ishikawa
    • 1
  • Hiroyuki Nakashima
    • 1
  • Hiroshi Nakatsuji
    • 2
    • 3
  1. 1.Quantum Chemistry Research Institute & JST CRESTNishikyo-kuJapan
  2. 2.Institute of Multidisciplinary Research for Advanced Materials (IMRAM)Tohoku UniversityAoba-kuJapan
  3. 3.Quantum Chemical Research Institute (QCRI)KyotoJapan

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