Advertisement

Molecular Fine Structure

  • Stephen R. Langhoff
  • C. William Kern
Part of the Modern Theoretical Chemistry book series (MTC, volume 4)

Abstract

Weak interactions in the complete many-body Hamiltonian are an important part of modern chemical theory and experiment.(1) As we have seen in this volume, these terms are usually neglected in the construction of electronic wave functions because they account for relatively small amounts of energy, typically on the order of cm-1. Nevertheless, they can be measured to very high accuracy, such as in EPR experiments that probe the coupling of the electron spins with themselves and with the angular momenta of their orbital motion.(2,3) These particular interactions, which are the subject of this chapter, are so small that they act as perturbations to split the nonrelativistic electronic states into a “fine structure” of levels. Since the spacings between these spin multiplets are often very sensitive to the details of the charge distribution, they provide a test of the zero-order wave functions that are used to calculate them.* We are, therefore, dealing with weak forces that have conspicious, measurable, and calculable effects.

Keywords

Triplet State Diatomic Molecule Phosphorescent Lifetime Electron Repulsion Integral Repulsive State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Karplus, Weak interactions in molecular quantum mechanics, Rev. Mod. Phys. 32, 455–460(1960).Google Scholar
  2. 2.
    A. B. Zahlan (ed.), The Triplet State, Proceedings of an International Symposium held at the American University of Beirut, Lebanon, Cambridge Univ. Press, Cambridge (1967).Google Scholar
  3. 3.
    A. Carrington, D. H. Levy, and T. A. Miller, in: Advances in Chemical Physics (I. Prigogine and S. A. Rice, eds.), Vol. 18, pp. 149–248, Wiley-Interscience, New York (1970).Google Scholar
  4. 4.
    C. A. Hutchison and B. W. Mangum, Paramagnetic resonance absorption in naphthalene in its phosphorescent state, J. Chem. Phys. 29, 952–953 (1958);Google Scholar
  5. 4a.
    C. A. Hutchison and B. W. Mangum, Paramagnetic resonance absorption in naphthalene in its phosphorescent state, J. Chem. Phys. 34, 908–922 (1961).Google Scholar
  6. 5.
    M. Gouterman and W. Moffitt, Origin of zero-field splittings in triplet states of aromatic hydrocarbons, J. Chem. Phys. 30, 1107–1108 (1959).Google Scholar
  7. 6.
    G. N. Lewis and M. Kasha, Phosphorescence and the triplet state, J. Am. Chem. Soc. 66, 2100–2116 (1944);Google Scholar
  8. 6a.
    G. N. Lewis and M. Kasha, Phosphorescence in fluid media and the reverse process of singlet-triplet absorption, J. Am. Chem. Soc. 67, 994–1003 (1945).Google Scholar
  9. 7.
    L. C. Allen and A. M. Karo, Basis functions for ab initio calculations, Rev. Mod. Phys. 32, 275–285 (1960).Google Scholar
  10. 8.
    H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two- Electron Atoms, Springer-Verlag, Berlin (1957).Google Scholar
  11. 9.
    T. E. H. Walker and W. G. Richards, Ab initio computation of spin-orbit coupling constants in diatomic molecules, Symp. Faraday Soc. 2, 64–68 (1968).Google Scholar
  12. 10.
    H. Ito and Y. J. I’haya, Evaluation of molecular spin-orbit integrals by a Gaussian expansion method, Mol. Phys. 24, 1103–1115 (1972).Google Scholar
  13. 11.
    R. H. Pritchard, M. L. Sink, and C. W. Kern, Theoretical study of the fine-structure coupling constants in the 2p 3 u state of H2, Mol. Phys. 30, 1273–1282 (1975).Google Scholar
  14. 12.
    S. Fraga and K. M. S. Saxena, Electronic Structure of Atoms, Division of Theoretical Chemistry, University of Alberta, Technical Report TC-AS-I-72, 1972;Google Scholar
  15. 12a.
    S. Fraga and J. Karwowski, Electronic Structure of Atoms, Division of Theoretical Chemistry, University of Alberta, Technical Report TC-AS-II-73, 1973.Google Scholar
  16. 13.
    G. Malli, Spin-other-orbit interaction in many-electron atoms, J. Chem. Phys. 48, 1088–1091 (1968).Google Scholar
  17. 14.
    G. Malli, Spin-spin interaction in many-electron atoms, J. Chem. Phys. 48, 1092–1094 (1968).Google Scholar
  18. 15.
    K. M. S. Saxena and G. Malli, Spin-orbit and spin-other-orbit interactions for f 4 electron configuration, Can. J. Phys. 47, 1829–1862 (1969).Google Scholar
  19. 16.
    S. Fraga, K. M. S. Saxena, and B. W. N. Lo, Hartree-Fock values of energies, interaction constants, and atomic properties for the ground states of the negative ions, neutral atoms, and first four positive ions from helium to krypton, At. Data 3, 323–361 (1971);Google Scholar
  20. 16.
    S. Fraga, K. M. S. Saxena, and B. W. N. Lo, Hartree-Fock values of energies, interaction constants, and atomic properties for excited states with p N configurations of the negative ions, neutral atoms, and first positive ions from boron to bromine, At. Data 4, 255–267 (1972);Google Scholar
  21. 16b. S. Fraga and K. M. S. Saxena, Hartree-Fock values of energies, interaction constants, and atomic properties for excited states with 3dN4s 0 and 3d N4s 2 configurations of the negative ions, neutral atoms, and first four positive ions of the transition elements, At. Data 4, 269–287 (1972).Google Scholar
  22. 17.
    S. Fraga and G. Malli, Many-Electron Systems: Properties and Interactions, W. B. Saunders Company, Philadelphia (1968).Google Scholar
  23. 18.
    P. A. M. Dirac, Quantum theory of the electron, Proc. R. Soc. London, Ser. A 117, 610–624 (1928).Google Scholar
  24. 19.
    H. A. Kramers, Structure of the multiplet 5-states in molecules with two atoms. Parts I and II, Z. Phys. 53, 422–438(1929).Google Scholar
  25. 20.
    J. H. van Vleck, The coupling of angular momentum vectors in molecules, Rev. Mod. Phys. 23, 213–227 (1951).Google Scholar
  26. 21.
    G. Breit, Effect of retardation on the interaction of two electrons, Phys. Rev. 34, 553–573 (1929).Google Scholar
  27. 22.
    W. Pauli, Quantum mechanics of the magnetic electron, Z. Phys. 43, 601–623 (1927).Google Scholar
  28. 23.
    J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, pp. 1044–1046, John Wiley and Sons, New York (1954).Google Scholar
  29. 24.
    M. Tinkham and M. W. P. Strandberg, Theory of the fine structure of the molecular oxygen ground state, Phys. Rev. 97, 937–951 (1955).Google Scholar
  30. 25.
    A. Carrington and A. D. McLachlan, Introduction to Magnetic Resonance, Harper and Row, New York (1967);Google Scholar
  31. 25a.
    A. D. McLachlan, Spin-spin coupling Hamiltonian in spin multiplets, Mol. Phys. 6, 441–444 (1963).Google Scholar
  32. 26.
    R. D. Sharma, Spin-spin interaction in methylene, J. Chem. Phys. 38, 2350–2352 (1963);Google Scholar
  33. 26a.
    R. D. Sharma, Spin-spin interaction in methylene, erratum J. Chem. Phys. 41, 3259 (1964).Google Scholar
  34. 27.
    A. L. Kwiram, in M.T.P. International Review of Science (C. A. McDowell, ed.), Vol. 4, pp. 271–315, Butterworths, London (1972).Google Scholar
  35. 28.
    D. De Santis, A. Lurio, T. A. Miller, and R. S. Freund, Radio-frequency spectrum of metastable N2(A 3+ u). II. Fine structure, magnetic hyperfine structure, and electric quad-rupole constants in the lowest 13 vibrational levels, J. Chem. Phys. 58, 4625–4665 (1973); see also references contained therein.Google Scholar
  36. 29.
    M. A. El-Sayed,M.T.P. International Review of Science (A. D. Buckingham and D. A. Ramsay, eds.), Vol. 3, pp. 119–153, Butterworths, London (1972).Google Scholar
  37. 30.
    R. S. Freund and T. A. Miller, Microwave optical magnetic resonance induced by electrons (MOMRIE) in H2 G(3d 1+ g), J. Chem. Phys. 56, 2211–2219 (1972).Google Scholar
  38. 31.
    H. F. Hameka, Advanced Quantum Chemistry, Addison-Wesley, Reading, Massachusetts (1965).Google Scholar
  39. 32.
    O. Zamani-Khamiri and H. F. Hameka, Spinorbit contribution to the zero-field splitting of the oxygen molecule, J. Chem. Phys. 55, 2191–2196 (1971);Google Scholar
  40. 32.
    R. H. Pritchard, C. W. Kern, O. Zamani-Khamiri, and H. F. Hameka, Comment on the spin-orbit contribution to the zero-field splitting of the oxygen molecule, J. Chem. Phys. 56, 5744–5745 (1972).Google Scholar
  41. 33.
    A. Messiah, Quantum Mechanics, Appendix C, North-Holland, Amsterdam (1962).Google Scholar
  42. 34.
    D. R. Beck, C. A. Nicolaides, and J. I. Musher, Calculation on the fine structure of the a 3+ u state of molecular helium, Phys. Rev. A 10, 1522–1527 (1974).Google Scholar
  43. 35.
    J. B. Lounsbury, Calculation of zero-field splitting in NH. II. One-center representation of triplet states, J. Chem. Phys. 46, 2193–2200 (1967);Google Scholar
  44. 35.
    J. B. Lounsbury, I. One-center minimal basis and atomic orbital representations of the ground state, J. Chem. Phys. 42, 1549–1554 (1965).Google Scholar
  45. 36.
    S. R. Langhoff, Spin-orbit contribution to the zero-field splitting in CH2, J. Chem. Phys. 61, 3881–3885(1974).Google Scholar
  46. 37.
    S. A. Boorstein and M. Gouterman, Zero-field splittings. IV. Gaussian approximation of integrals, J. Chem. Phys. 41, 2776–2781 (1964).Google Scholar
  47. 38.
    S. R. Langhoff, E. R. Davidson, M. Gouterman, W. R. Leenstra, and A. L. Kwiram, Zero-field splitting of the triplet state of porphyrins. II, J. Chem. Phys. 62, 169–176 (1975).Google Scholar
  48. 39.
    P. S. Han, T. P. Das, and M. F. Rettig, Calculation of the spin-spin and spin-orbit contribution to the zero-field splitting in hemin, J. Chem. Phys. 56, 3861–3873 (1972).Google Scholar
  49. 40.
    K. Kayama and J. C. Baird, Spin-orbit effects and the fine structure in the 3- g ground state of O 2, J. Chem. Phys. 46, 2604–2618 (1967).Google Scholar
  50. 41.
    J. S. Griffith, The Theory of Transition-Metal Ions, Cambridge Univ. Press, Cambridge (1961).Google Scholar
  51. 42.
    H. Hayashi and S. Nagakura, Correlation of the zero-field splittings with the nπ* and ππ* triplet levels of benzaldehydes, Chem. Phys. Lett. 18, 63–66 (1973);Google Scholar
  52. 42.
    H. Hayashi and S. Nagakura, The lowest nπ* and ππ* triplet levels of benzaldehydes and their correlation with the zero-field splittings, Mol. Phys. 27, 969–979 (1974).Google Scholar
  53. 43.
    D. S. McClure, Spin-orbit interaction in aromatic molecules, J. Chem. Phys. 20, 682–686 (1952).Google Scholar
  54. 44.
    S. A. Boorstein and M. Gouterman, Theory for zero-field splittings in aromatic hydrocarbons, J. Chem. Phys. 39, 2443–2452 (1963).Google Scholar
  55. 45.
    M. S. de Groot, I. A. M. Hesselmann, and J. H. van der Waals, Paramagnetic resonance in phosphorescent aromatic hydrocarbons, Mol. Phys. 16, 45–60 (1969).Google Scholar
  56. 46.
    R. F. Curl, The relationship between electron spin-rotation coupling constants and gtensor components, Mol. Phys. 9, 585–597 (1965).Google Scholar
  57. 47.
    A. N. Jette, Fine-structure of the metastable, c 3u (1s, 2p), state of molecular hydrogen, Chem. Phys. Lett. 25, 590–592 (1974);Google Scholar
  58. 47.
    A. N. Jette and T. A. Miller, Fine structure in Rydberg states of the H2 molecule, Chem. Phys. Lett. 29, 547–550 (1974).Google Scholar
  59. 48.
    W. Lichten, M. V. McCusker, and T. L. Vierima, Fine structure of the metastable a 3+ u state of the helium molecule, J. Chem. Phys. 61, 2200–2212 (1974).Google Scholar
  60. 49.
    R. N. Dixon, Spin-rotation interaction constants for bent AH2 molecules in doublet electronic states, Mol. Phys. 10, 1–6 (1966).Google Scholar
  61. 50.
    S. K. Luke, The radio-frequency spectrum of H+ 2, Astrophys. J. 156, 761–769 (1969).Google Scholar
  62. 50a.
    See also P. M. Kalaghan and A. Dalgarno, Hyperfine structure of the molecular ion H+ 2, Phys. Lett. 38A, 485–486 (1972).Google Scholar
  63. 51.
    F. E. Harris, Matrix elements of spin-interaction operators, J. Chem. Phys. 47, 1047–1061 (1967).Google Scholar
  64. 52.
    I. L. Cooper and J. I. Musher, Evaluation of matrix elements of spin-dependent operators for N-electron systems. I. One-body operators, J. Chem. Phys. 57, 1333–1342 (1972);Google Scholar
  65. 52.
    I. L. Cooper and J. I. Musher, II. Two-body operators, J. Chem. Phys. 59, 929–938 (1973).Google Scholar
  66. 53.
    C. Böttcher and J. C. Browne, Matrix elements of spin-dependent operators over total molecular wavefunctions, J. Chem. Phys. 52, 3197–3201 (1970);Google Scholar
  67. 53a.
    C. Bottcher, Calculations on the Small Terms in the Hamiltonian of a Diatomic Molecule (Part I) and Variational Principles for the Study of Resonances (Part II), Ph.D. thesis, The Queen’s University of Belfast, 1968.Google Scholar
  68. 54.
    Y-N. Chiu, On singlet-triplet transitions induced by exchange with paramagnetic molecules and the intermolecular coupling of spin angular momenta, J. Chem. Phys. 56, 4882–4898 (1972).Google Scholar
  69. 55.
    L. C. Chiu, Electron magnetic perturbation in diatomic molecules of Hund’s case (b), J. Chem. Phys. 40, 2276–2285 (1964).Google Scholar
  70. 56.
    R. McWeeny, On the origin of spin-Hamiltonian parameters, J. Chem. Phys. 42, 1717–1725 (1965).Google Scholar
  71. 57.
    K. F. Freed, Theory of the hyperfine structure of molecules: Application to 3∏ states of diatomic molecules intermediate between Hund’s cases (a) and (b), J. Chem. Phys. 45, 4214–4241 (1966).Google Scholar
  72. 58.
    S. R. Langhoff, Spin Dipole-Dipole Contribution to the Zero-Field Splitting in Methylene and Formaldehyde, Ph.D. thesis, University of Washington (1973).Google Scholar
  73. 59.
    H. M. McConnell, A theorem on zero-field splittings, Proc. Natl. Acad. Sci. USA 45, 172–174 (1959).Google Scholar
  74. 60.
    A. C. Wahl, P. E. Cade, and C. C. J. Roothaan, Study of two-center integrals useful in calculations on molecular structure. V. General methods for diatomic integrals applicable to digital computers, J. Chem. Phys. 41, 2578–2599 (1964).Google Scholar
  75. 61.
    R. H. Pritchard, A Theoretical Study of Fine Structure Interactions in Diatomic Molecules, Ph.D. thesis, The Ohio State University, 1974.Google Scholar
  76. 62.
    R. H. Pritchard, M. L. Sink, J. D. Allen, and C. W. Kern, Theoretical studies of fine structure in the ground state of O2, Chem. Phys. Lett. 17, 157–159 (1972).Google Scholar
  77. 63.
    R. L. Matcha, C. W. Kern, and D. M. Schrader, Fine-structure studies of diatomic molecules: Two-electron spin-spin and spin-orbit integrals, J. Chem. Phys. 51, 2152–2170 (1969);Google Scholar
  78. 63a.
    R. L. Matcha, C. W. Kern, and D. M. Schrader, Fine-structure studies of diatomic molecules: Two-electron spin-spin and spin-orbit integrals, erratum, J. Chem. Phys. 57, 2598 (1972).Google Scholar
  79. 64.
    R. M. Pitzer and H. F. Hameka, Evaluation of the spin-spin interaction in benzene, J. Chem. Phys. 37, 2725 (1962);Google Scholar
  80. 64a.
    H. F. Hameka, Theory of the electron spin resonance of benzene in the triplet state, J. Chem. Phys. 31, 315–321 (1959).Google Scholar
  81. 65.
    M. Blume and R. E. Watson, Theory of spin-orbit coupling in atoms. I. Derivation of the spin-orbit coupling constant, Proc. R. Soc. London, Ser. A 270, 127–143 (1962);Google Scholar
  82. 65a.
    M. Blume and R. E. Watson, Theory of spin-orbit coupling in atoms. II. Comparison of theory with experiment, Proc. R. Soc. London, Ser. A 271, 565–578 (1963).Google Scholar
  83. 66.
    Z. Kopal, Numerical Analysis, John Wiley and Sons, New York (1961).Google Scholar
  84. 67.
    R. E. Christoffersen and K. Ruedenberg, Hybrid integrals over Slater-type atomic orbitals, J. Chem. Phys. 49, 4285–4292 (1968).Google Scholar
  85. 68.
    R. L. Matcha, G. Malli, and M. B. Milleur, Two-center two-electron spin-spin and spinorbit hybrid integrals, J. Chem. Phys. 56, 5982–5989 (1972);Google Scholar
  86. 68.
    G. Malli, M. B. Milleur, and R. L. Matcha, Two-center hybrid integrals, J. Chem. Phys. 54, 4964–4965 (1971).Google Scholar
  87. 69.
    R. L. Matcha, R. H. Pritchard, and C. W. Kern, Prolate-spheroidal expansions of the spinorbit, spin-spin, and orbit-orbit operators, J. Math. Phys. 12, 1155–1159 (1971) and references therein.Google Scholar
  88. 70.
    D. M. Schrader, Calculation of spin-spin interaction integrals, J. Chem. Phys. 41, 3266–3267 (1964).Google Scholar
  89. 71.
    G.G. Hall and A. Hardisson, The anistropy of the g-factor for polycyclic hydrocarbons, Proc. R. Soc. London, Ser. A 278, 129–136 (1964).Google Scholar
  90. 72.
    K. Rüdenberg, A study of two-center integrals useful in calculations on molecular structure. II. The two-center exchange integrals, J. Chem. Phys. 19, 1459–1477 (1951).Google Scholar
  91. 73.
    R. L. Matcha, D. J. Kouri, and C. W. Kern, Relativistic effects in diatomic molecules: Evaluation of one-electron integrals, J. Chem. Phys. 53, 1052–1059 (1970).Google Scholar
  92. 74.
    R. L. Matcha and C. W. Kern, Evaluation of three- and four-center integrals for operators appearing in the Breit-Pauli Hamiltonian, J. Chem. Phys. 55, 469 (1971).Google Scholar
  93. 75.
    A. D. McLean, LCAO-MO-SCF ground state calculations on C2H2 and CO2, J. Chem. Phys. 32, 1595–1597 (1960).Google Scholar
  94. 76.
    E. A. Magnusson and C. Zauli, Evaluation of molecular integrals by a numerical method, Proc. Phys. Soc., London 78, 53–64 (1961).Google Scholar
  95. 77.
    A. C. Wahl and R. H. Land, The evaluation of multicenter integrals by polished brute force techniques I. Analysis, numerical methods, and computational design of the potential-charge distribution scheme, Int. J. Quantum Chem. IS, 375–401 (1967).Google Scholar
  96. 78.
    I. Shavitt and M. Karplus, Gaussian-transform method for molecular integrals. I. Formulation for energy integrals, J. Chem. Phys. 43, 398–414 (1965).Google Scholar
  97. 79.
    C. W. Kern and M. Karplus, Gaussian-transform method for molecular integrals. II. Evaluation of molecular properties, J. Chem. Phys. 43, 415–429 (1965).Google Scholar
  98. 80.
    S. F. Boys, Electronic wavefunctions. I. A general method of calculation for the stationary states of any molecular system, Proc. R. Soc. London, Ser. A 200, 542–554 (1950).Google Scholar
  99. 81.
    I. Shavitt, in:Methods in Computational Physics (B. Alder, S. Fernbach, and M. Rotenberg, eds.) vol. 2, pp. 1–45, Academic Press, New York (1963).Google Scholar
  100. 82.
    R. L. Matcha and C. W. Kern, Relationships between spin-spin and electron repulsion integrals, J. Phys. B 4, 1102–1108 (1971).Google Scholar
  101. 83.
    M. Geller and R. W. Griffith, Zero-field splitting, one- and two-center Coulomb-type integrals, J. Chem. Phys. 40, 2309–2325 (1964);Google Scholar
  102. 83a.
    M. Geller and R. W. Griffith, Zero-field splitting, one- and two-center Coulomb-type integrals, erratum, J. Chem. Phys. 40, 2309–2310 (1964).Google Scholar
  103. 84.
    F. E. Harris and H. H. Michels, in: Advances in Chemical Physics (I. Prigogine, ed.), Vol. 13, pp. 205–266, Interscience, New York (1967).Google Scholar
  104. 85.
    J. B. Lounsbury and G. W. Barry, General solution for one-center zero-field splitting integrals, J. Chem. Phys. 44, 4367–4372 (1966).Google Scholar
  105. 86.
    J. W. McIver, Jr. and H. F. Hameka, Effect of spin-orbit interactions on the zero-field splitting of the NH radical, J. Chem. Phys. 45, 767–773 (1966).Google Scholar
  106. 87.
    F. P. Prosser and C. H. Blanchard, On the evaluation of two-center integrals, J. Chem. Phys. 36, 1112(1962).Google Scholar
  107. 88.
    R. L. Matcha and C. W. Kern, Identities relating spin-spin and orbit-orbit to spin-orbit interactions, Phys. Rev. Lett. 25, 981–982 (1970).Google Scholar
  108. 89.
    R. M. Pitzer, C. W. Kern, and W. N. Lipscomb, Evaluation of molecular integrals by solid spherical harmonic expansions, J. Chem. Phys. 37, 267–274 (1962).Google Scholar
  109. 90.
    R. H. Pritchard and C. W. Kern, Spin-spin and spin-other-orbit integrals for diatomic molecules, J. Chem. Phys. 57, 2590–2591 (1972);Google Scholar
  110. 90a.
    R. H. Pritchard and C. W. Kern, Spin-spin and spin-other-orbit integrals for diatomic molecules, erratum, J. Chem. Phys. 61, 754 (1974).Google Scholar
  111. 91.
    S. R. Langhoff, Ab initio evaluation of the fine structure of the oxygen molecule, J. Chem. Phys. 61, 1708–1716(1974).Google Scholar
  112. 92.
    P. W. Abegg and T-K. Ha, Ab initio calculation of the spin-orbit coupling constant from gaussian lobe SCF molecular wavefunctions, Mol. Phys. 27, 763–767 (1974).Google Scholar
  113. 93.
    R. K. Hinkley, T. E. H. Walker, and W. G. Richards, Spin-orbit coupling constants from Gaussian wavefunctions, J. Chem. Phys. 52, 5975–5976 (1970).Google Scholar
  114. 94.
    G. Herzberg, Spectra of Diatomic Molecules, 2nd ed., D. Van Nostrand, Princeton (1950).Google Scholar
  115. 95.
    K. P. Huber, Constants of diatomic molecules, in: American Institute of Physics Handbook, McGraw-Hill, New York (1972).Google Scholar
  116. 96.
    S. P. McGlynn, T. Azumi, and M. Kinoshita, Molecular Spectroscopy of the Triplet State, Prentice-Hall, Englewood Cliffs, New Jersey (1969).Google Scholar
  117. 97.
    P. R. Fontana, Spin-orbit and spin-spin interactions in diatomic molecules. I. Fine structure of H2, Phys. Rev. 125, 220–228 (1962).Google Scholar
  118. 98.
    L. C. Chiu, Fine structure constants of metastable H2 in the c 3 u state, Phys. Rev. 137, A384-A387 (1965);Google Scholar
  119. 98.
    L. C. Chiu, Fine-structure constants of the metastable c 3 u-state hydrogen molecule, J. Chem. Phys. 41, 2197–2198 (1964).Google Scholar
  120. 99.
    W. T. Zemke and P. G. Lykos, Double configuration self-consistent field study of the 1 u, 3 u and 1 g states of H2, J. Chem. Phys. 51, 5635–5650 (1969).Google Scholar
  121. 100.
    M. Lombardi, Fine and hyperfine structure of the 2p and 3p 3 u states of H2, J. Chem. Phys. 58, 797–802 (1973).Google Scholar
  122. 101.
    S. Rothenberg and E. R. Davidson, Natural orbitals for hydrogen-molecule excited states, J. Chem. Phys. 45, 2560–2576 (1966).Google Scholar
  123. 102.
    H. H. Nielsen, The vibration-rotation energies of molecules, Rev. Mod. Phys. 23, 90–136 (1951).Google Scholar
  124. 103.
    C. W. Kern and R. L. Matcha, Nuclear corrections to electronic expectation values: Zero-point vibrational effects in the water molecule, J. Chem. Phys. 49, 2081–2091 (1968).Google Scholar
  125. 104.
    W. C. Ermler and C. W. Kern, Zero-point vibrational corrections to one-electron properties of the water molecule in the near Hartree-Fock limit, J. Chem. Phys. 55, 4851–4860 (1971).Google Scholar
  126. 105.
    B.J. Krohn, W. C. Ermler, and C. W. Kern, Nuclear corrections to molecular properties. IV. Theory for low-lying vibrational states of polyatomic molecules with application to the water molecule near the Hartree-Fock limit, J. Chem. Phys. 60, 22–33 (1974).Google Scholar
  127. 106.
    L. L. Sprandel and C. W. Kern, A test of perturbation theory for determining anharmonic vibrational corrections to properties of diatomic molecules, Mol. Phys. 24, 1383–1389 (1972).Google Scholar
  128. 107.
    G. D. Carney and R. N. Porter, Abstracts of the 1972 Molecular Structure and Spectroscopy Symposium held at The Ohio State University;Google Scholar
  129. 107a.
    G. D. Carney, Ab-initio Calculation of Vibration-Rotation Properties for the Ground Electronic State of the H+ 3 Molecular Ion, Ph.D. thesis, University of Arkansas, 1973.Google Scholar
  130. 108.
    L. L. Sprandel, Quantum Mechanical Studies of Molecular Vibrations, Part II—A Study of the Use of the Self-Consistent Field and Configuration Interaction Methods for Solving the Vibrational Schrodinger Equation of a Bent AB2 Molecule With Application to Water, Ph.D. thesis, The Ohio State University 1974.Google Scholar
  131. 109.
    M. G. Bucknell, N. C. Handy, and S. F. Boys, Vibration-rotation wave-functions and energies for any molecule obtained by a variational method, Mol. Phys. 28, 759–776 (1974).Google Scholar
  132. 110.
    M. G. Bucknell and N. C. Handy, Vibration-rotation wavefunctions and energies for the ground electronic state of the water molecule by a vibrational method, Mol. Phys. 28, 777–792 (1974).Google Scholar
  133. 111.
    S. A. Gribov and G. V. Khovrin, Determination of the potential surface and analysis of the anharmonic vibrations of the water molecule, Opt. Spectrosc. 36, 274–279 (1974).Google Scholar
  134. 112.
    E. K. Lai, M. S. Thesis, Department of Chemistry, Indiana University, 1975.Google Scholar
  135. 113.
    G. D. Carney and C. W. Kern, Vibration-rotation analysis of some nonlinear molecules by a variational method, Int. J. Quantum Chem. Symp. 9, 317–323 (1975).Google Scholar
  136. 114.
    J. K. G. Watson, Simplification of the molecular vibration-rotation Hamiltonian, Mol. Phys. 15, 479–490 (1968).Google Scholar
  137. 115.
    H. Lefebvre-Brion and C. M. Moser, Calculation of valence states of NO and NO+, J. Chem. Phys. 44. 2951–2954 (1966).Google Scholar
  138. 116.
    T. E. H. Walker and W. G. Richards, The nature of the first excited electronic state in BeF, Proc. Phys. Soc., London 92, 285–290 (1967).Google Scholar
  139. 117.
    T. E. H. Walker and W. G. Richards, The nature of the first excited electronic state in MgF, Proc. Phys. Soc., London 1 (Ser. 2), 1061–1065 (1968).Google Scholar
  140. 118.
    A. L. Roche and H. Lefebvre-Brion, Valence-shell states of PO: An example of the variation of the spin-orbit coupling constants with internuclear distance, J. Chem. Phys. 59, 1914–1921 (1973).Google Scholar
  141. 119.
    W. H. Moores and R. McWeeny, The calculation of spin-orbit splitting and g tensors for small molecules and radicals, Proc. R. Soc. London, Ser. A 332, 365–384 (1973).Google Scholar
  142. 120.
    E. Ishiguro and M. Kobori, Spin-orbit coupling constants in simple diatomic molecules, J. Phys. Soc. Japan 22, 263–270 (1967).Google Scholar
  143. 121.
    C. E. Moore, Atomic Energy Levels, I., II., and III. National Bureau of Science, Circular No. 467, U.S. Government Printing Office, Washington, D.C. (1949).Google Scholar
  144. 122.
    L. Veseth, Spin-orbit and spin-other-orbit interaction in diatomic molecules, Theor. Chim. Acta, 18, 368–384 (1970).Google Scholar
  145. 123.
    T. E. H. Walker and W. G. Richards, Calculation of spin-orbit coupling constants in diatomic molecules from Hartree-Fock wavefunctions, Phys. Rev. 177, 100–101 (1969).Google Scholar
  146. 124.
    T. E. H. Walker and W. G. Richards, Molecular spin-orbit coupling constants. The role of core polarization, J. Chem. Phys. 52, 1311–1314 (1970).Google Scholar
  147. 125.
    H. Lefebvre-Brion and C. M. Moser, On the calculation of spin-orbit interaction in diatomic molecules, J. Chem. Phys. 46, 819–820 (1967).Google Scholar
  148. 126.
    H. Lefebvre-Brion and N. Bessis, Spin-orbit splitting in 2Δ states of diatomic molecules, Can. J. Phys. 47, 2727–2730 (1969).Google Scholar
  149. 127.
    J. A. Hall, J. Schamps, J. M. Robbe, and H. Lefebvre-Brion, A theoretical study of the perturbation parameters in the a 3∏ and A 1 states of CO. J. Chem. Phys. 62, 1802–1805 (1975).Google Scholar
  150. 128.
    E. A. Ballik and D. A. Ramsay, The A3- g-X3u band system of the C2 molecule, Astrophys. J. 137, 61–83 (1963);Google Scholar
  151. 128.
    E. A. Ballik and D. A. Ramsay, An extension of the Phillips system of C2 and a survey of C2 states, Astrophys. J. 137, 84–101 (1963);Google Scholar
  152. 128a.
    G. Herzberg, A. Lagerqvist, and C. Malmberg, New electronic transitions of the C2 molecule in absorption in the vacuum ultravoilet region, Can. J. Phys. 47, 2735–2743 (1969).Google Scholar
  153. 129.
    S. R. Langhoff, M. L. Sink, R. H. Pritchard, C. W. Kern, S. J. Strickler, and M. J. Boyd, Ab initio study of perturbations between the X 1+ g and b 3- g states of the C2 molecule, J. Chem. Phys. (submitted for publication).Google Scholar
  154. 130.
    K. Kayama, Spin dipole-dipole interaction in O2, J. Chem. Phys. 42, 622–630 (1965).Google Scholar
  155. 131.
    R. H. Pritchard, C. F. Bender, and C. W. Kern, Fine-structure interactions in the ground state of O2, Chem. Phys. Lett. 5, 529–532 (1970).Google Scholar
  156. 132.
    T. J. Cook, B. R. Zegarski, W. H. Breckenridge, and T. A. Miller, Gas phase EPR of vibrationally excited O2, J. Chem. Phys. 58, 1548–1552 (1972) and references therein.Google Scholar
  157. 133.
    T. Amano and E. Hirota, Microwave spectrum of the molecular oxygen in the excited vibrational state, J. Mol. Spectrosc. 53, 346–363 (1974).Google Scholar
  158. 134.
    E. Wasserman, R. S. Hutton, V. J. Kuck, and W. A. Yager, Zero-field parameters of “free” CH2: Spin-orbit contributions in xenon, J. Chem. Phys. 55, 2593–2594 (1971);Google Scholar
  159. 134.
    E. Wasserman, R. S. Hutton, V. J. Kuck, and W. A. Yager, Electron paramagnetic resonance of CD2 and CHD isotope effects, motion and geometry of methylene, J. Am. Chem. Soc. 92, 7491–7493 (1970);Google Scholar
  160. 134a.
    E. Wasserman, V J. Kuck, R. S. Hutton, E. D. Anderson, and W. A. Yager, 13C hyperfine interactions and geometry of methylene, J. Chem. Phys. 54, 4120–4121 (1971).Google Scholar
  161. 135.
    R. A. Bernheim, H. W. Bernard, P. S. Wang, L. S. Wood, and P. S. Skell, Electron paramagnetic resonance of triplet CH2, J. Chem. Phys. 53, 1280–1281 (1970);Google Scholar
  162. 135.
    R. A. Bernheim, H. W. Bernard, P. S. Wang, L. S. Wood, and P. S. Skell, 13C hyperfine interaction in CD2, J. Chem. Phys. 53, 1280–1281 (1970).Google Scholar
  163. 136.
    J. Higuchi, On the effect of the bond angle in the electron spin-spin interaction. Methylene derivatives, J. Chem. Phys. 39, 1339–1341 (1963);Google Scholar
  164. 136a.
    J. Higuchi, Zero-field splittings in molecular multiplets. Spin-spin interaction of methylene derivatives, J. Chem. Phys. 38, 1237–1245 (1963).Google Scholar
  165. 137.
    J. F. Harrison, An ab initio study of the zero field splitting parameters of 3 B 1 methylene, J. Chem. Phys. 54, 5413–5417 (1971).Google Scholar
  166. 138.
    S. R. Langhoff and E. R. Davidson, An ab initio calculation of the spin dipole-dipole parameters for methylene, Int. J. Quantum Chem. 7, 759–777 (1973).Google Scholar
  167. 139.
    S. R. Langhoff, S. T. Elbert, and E. R. Davidson, A configuration interaction study of the spin dipole-dipole parameters for formaldehyde and methylene, Int. J. Quantum Chem. 7, 999–1019 (1973).Google Scholar
  168. 140.
    S. H. Glarum, Spin-orbit interactions in molecular radicals, J. Chem. Phys. 39, 3141–3144 (1963).Google Scholar
  169. 141.
    S. J. Fogel and H. F. Hameka, Spin-orbit interactions and their effect on the zero-field splitting of the methylene radical, J. Chem. Phys. 42, 132–136 (1965).Google Scholar
  170. 142.
    W. R. Hall and H. F. Hameka, Second-order effect of spin-orbit coupling on the angular dependence of the zero-field splitting in CH2, J. Chem. Phys. 58, 226–231 (1973).Google Scholar
  171. 142a.
    W. R. Hall and H. F. Hameka, See also erratum, J. Chem. Phys. 60, 4104 (1974).Google Scholar
  172. 143.
    S. R. Langhoff and G. D. Carney, to be published.Google Scholar
  173. 144.
    W. T. Raynes, Rotational analysis of some bands of the triplet←singlet transition in formaldehyde, J. Chem. Phys. 44, 2755–2777 (1966);Google Scholar
  174. 144.
    W. T. Raynes, Spin splittings and rotational structure of nonlinear molecules in doublet and triplet electronic states, J. Chem. Phys. 41, 3020–3032 (1964).Google Scholar
  175. 145.
    S. R. Langhoff and E. R. Davidson, Ab initio evaluation of the fine structure and radiative lifetime of the 3 A 2(nπ*) state of formaldehyde, J. Chem. Phys. 64, 4699–4710 (1976).Google Scholar
  176. 146.
    S. D. Peyerimhoff, R. J. Buenker, W. E. Kammer, and H. Hsu, Calculation of the electronic spectrum of formaldehyde, Chem. Phys. Lett. 8, 129–135 (1971).Google Scholar
  177. 147.
    A. M. Ponte Goncalves and C. A. Hutchinson, Jr., Electron nuclear double resonance in photoexcited triplet-state benzene-h 6 molecules in benzene-d 6 single crystals, J. Chem. Phys. 49, 4235–4236 (1968).Google Scholar
  178. 148.
    Y-N. Chiu, Zero-field splittings in some triplet-state aromatic molecules, J. Chem. Phys. 39, 2736–2748 (1963).Google Scholar
  179. 149.
    M. Geller, Two-electron, one- and two-center integrals, J. Chem. Phys. 39, 853–854 (1963).Google Scholar
  180. 150.
    J. H. van der Waals and G. ter Maten, Zero-field splitting of the lowest triplet state of some aromatic hydrocarbons: Calculation and comparison with experiment, Mol. Phys. 8, 301–318 (1964).Google Scholar
  181. 151.
    M. Godfrey, C. W. Kern, and M. Karplus, Studies of zero-field splittings in aromatic molecules, J. Chem. Phys. 44, 4459–4469 (1966).Google Scholar
  182. 152.
    S. R. Langhoff, E. R. Davidson, and C. W. Kern, Ab initio study of the zero-field splitting parameters of 3 B 1u benzene, J. Chem. Phys. 63, 4800–4807 (1975).Google Scholar
  183. 153.
    M. S. de Groot and J. H. van der Waals, Paramagnetic resonance in phosphorescent aromatic hydrocarbons. III. Conformational isomerism in benzene and triptycene, Mol. Phys. 6, 545–562 (1963).Google Scholar
  184. 154.
    M. Kasha, Characterization of electronic transitions in complex molecules, Discuss. Faraday Soc. 9, 14–19 (1950).Google Scholar
  185. 155.
    L. Goodman and B. J. Laurenzi, in: Advances in Quantum Chemistry (P-O. Lowdin, ed.), Vol. 4, pp. 153–169, Academic Press, New York (1968).Google Scholar
  186. 156.
    R. L. Ellis, R. Squire, and H. H. Jaffe, Use of the CNDO method in spectroscopy. V. Spin-orbit coupling, J. Chem. Phys. 55, 3499–3505 (1971).Google Scholar
  187. 157.
    G. Herzberg, Electronic Spectra of Polyatomic Molecules, pp. 417–419 Van Nostrand, Princeton, New Jersey (1966), and references therein.Google Scholar
  188. 158.
    J. W. Sidman, Spin-orbit coupling in the 3 A 2– 1 A 1 transition of formaldehyde, J. Chem. Phys. 29, 644–652 (1958).Google Scholar
  189. 159.
    T. Yonezawa, H. Kato, and H. Kato, Oscillator strength of singlet-triplet transition in formaldehyde, J. Mol Spectrosc. 24, 500–503 (1967).Google Scholar
  190. 160.
    L. L. Lohr, Spin-forbidden electric-dipole transition moments, J. Chem. Phys. 45, 1362–1363 (1966).Google Scholar
  191. 161.
    S. R. Langhoff, S. T. Elbert, C. F. Jackeis, and E. R. Davidson, Chem. Phys. Lett. 29, 247–249 (1974).Google Scholar
  192. 162.
    J. E. Mentall, E. P. Gentieu, M. Krauss, and D. Neumann, Photoionization and absorption spectrum of formaldehyde in the vacuum ultraviolet, J. Chem. Phys. 55, 5471–5479 (1971).Google Scholar
  193. 163.
    D. L. Yeager and V. McKoy, Equations of motion method: Excitation energies and intensities in formaldehyde, J. Chem. Phys. 60, 2714–2716 (1974).Google Scholar
  194. 164.
    G. L. Bendazzoli and P. Palmieri, Spin-orbit interaction in polyatomic molecules: Ab initio computations with gaussian orbitals, Int. J. Quantum Chem. 8, 941–950 (1974).Google Scholar
  195. 165.
    H. F. Schaefer III and W. H. Miller, Curve crossing of the B 3- u and 3u states of O2 and its relation to predissociation in the Schumann-Runge bands, J. Chem. Phys. 55, 4107–4115 (1971).Google Scholar
  196. 166.
    J. A. Hall and W. G. Richards, A theoretical study of the spectroscopic states of the CF molecule, Mol. Phys. 23, 331–343 (1972).Google Scholar
  197. 167.
    P. S. Julienne, M. Krauss, and B. Donn, Formation of OH through inverse predissociation, Astrophys. J. 170, 65–70 (1971).Google Scholar
  198. 168.
    M. Ackerman and F. Biaume, Structure of the Schumann-Runge bands from the 0–0 to the 13–0 band, J. Mol. Spectrosc. 35, 73–82 (1970).Google Scholar
  199. 169.
    I. Riess and Y. Ben-Aryeh, Application of the quantum Franck-Condon principle to predissociation in oxygen, J. Quant. Spectrosc. Radiat. Transfer 9, 1463–1468 (1969).Google Scholar
  200. 170.
    J. N. Murrell and J. M. Taylor, Predissociation in diatomic spectra with special reference to the Schumann-Runge bands of O2, Mol. Phys. 16, 609–621 (1969).Google Scholar
  201. 171.
    R. D. Hudson and S. H. Mahle, Photodissociation rates of molecular oxygen in the mesophere and lower thermosphere, J. Geophys. Res. 77, 2902–2914 (1972).Google Scholar
  202. 172.
    P. S. Julienne and M. Krauss, Predissociation of the Schumann-Runge Bands of O2, NRL Memorandum Report 2900, Naval Research Laboratory, Washington, D.C. (1974).Google Scholar
  203. 173.
    P. S. Julienne and M. Krauss, Predissociation of the Schumann-Runge bands of O2, J. Mol. Spectrosc. 56, 270–308 (1975).Google Scholar
  204. 174.
    G. Herzberg, Predissociation and similar phenomenon, Ergeh. Exakten. Naturwiss. 10, 207–284 (1931).Google Scholar
  205. 175.
    R. S. Mulliken, Some neglected subcases of predissociation in diatomic molecules, J. Chem. Phys. 33, 247–252 (1960).Google Scholar
  206. 176.
    P. S. Julienne, private communication.Google Scholar

Copyright information

© Plenum Press, New York 1977

Authors and Affiliations

  • Stephen R. Langhoff
    • 1
  • C. William Kern
    • 2
  1. 1.Battelle Columbus LaboratoriesColumbusUSA
  2. 2.Department of ChemistryBattelle Columbus Laboratories and The Ohio State UniversityColumbusUSA

Personalised recommendations