DOI: https://doi.org/10.15587/2313-8416.2015.42643

Quantum-chemical studies of quasi-one-dimensional electron systems. 1. Polyenes

Yuriy Kruglyak

Abstract


This review is devoted to the basic problem in quantum theory of quasi-one-dimensional electron systems like polyenes (Part 1) and cumulenes (Part 2) – physical origin of the forbidden zone in these and analogous 1D electron systems due to two possible effects – Peierls instability (bond alternation) and Mott instability (electron correlation). Both possible contradiction and coexistence of the Mott and Peierls instabilities are summerized on the basis of the Kiev quantum chemistry team research projects.


Keywords


quasi-one-dimensional electron system; polyenes; 1D electron systems; local states; impurity states; generalized HF method

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References


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GOST Style Citations


1. Kventsel, G. F. Local electronic states in long polyene chains [Text] / G. F. Kventsel, Yu. A. Kruglyak // Theoretica chimica Acta. – 1968. ‑ Vol. 12, Issue 1. ‑ P. 1–17. doi: 10.1007/bf00527002

2. Kventsel, G. F. Local electronic states in bounded polyene chains [Text] / G. F. Kventsel // Teoreticheskaya i Eksperimental’naya Khimiya. – 1968. ‑ Vol. 4, Issue 3. ‑ P. 291–298.

3. Kventsel, G. F. Double substitution in long polyene chains [Text] / G. F. Kventsel // Teoreticheskaya i Eksperimental’naya Khimiya. – 1969. ‑ Vol. 5, Issue 1. ‑ P. 26–31.

4. Kventsel, G. F. Local electronic states in chains with two atoms in the unit cell [Text] / G. F. Kventsel // Teoreticheskaya i Eksperimental’naya Khimiya. – 1969. ‑ Vol. 5, Issue 4. ‑ P. 435–445.

5. Kruglyak, Yu. A. Generalized Hartree – Fock method and its versions: from atoms and molecules up to polymers [Text] / Yu. A. Kruglyak // ScienceRise. – 2014. ‑ Vol. 5, Issue 3(5). ‑ P. 6–21. doi: 10.15587/2313-8416.2014.30726

6. Kruglyak, Yu. A. Study of the electronic structure of alternant radicals by the DODS method [Text] / Yu. A. Kruglyak, I. I. Ukrainsky // Ukrainskii Fizicheskii Zhurnal. – 1970. ‑ Vol. 15, Issue 7. ‑ P. 1068–1081.

7. Ukrainsky, I. I. Electronic structure of long polyene chains with an impurity atom [Text] / I. I. Ukrainsky, G. F. Kventsel // Theoretica chimica Acta. – 1972. ‑ Vol. 25, Issue 4. ‑ P. 360–371. doi: 10.1007/bf00526568

8. Kruglyak, Yu. A. Torsion barriers of end-groups in cumulenes. I. General consideration [Text] / Yu. A. Kruglyak, G. G. Dyadyusha // Theoretica chimica Acta. – 1968. ‑ Vol. 10, Issue 1. ‑ P. 23–32. doi: 10.1007/bf00529040

9. Kruglyak, Yu. A. Rotating barrier of end groups in organic cumulenes [Text] / Yu. A. Kruglyak, G. G. Dyadyusha // Teoreticheskaya i Eksperimental’naya Khimiya. – 1968. ‑ Vol. 4, Issue 4. ‑ P. 431–437.

10. Ukrainsky, I. I. Electronic structure of long cumulene chains [Text] / I. I. Ukrainsky // International Journal of Quantum Chemistry. – 1972. ‑ Vol. 6, Issue 3. ‑ P. 473–489. doi: 10.1002/qua.560060309

11. Kventsel, G. F. Peierls- and Mott-type instabilities in one-dimensional chains – coexistence or contradiction [Text] / G. F. Kventsel // International Journal of Quantum Chemistry. – 1982. ‑ Vol. 22, Issue 4. ‑ P. 825‑835. doi: 10.1002/qua.560220412

12. Ukrainskii, I. I. Coexistence of Mott and Peierls instabilities in quasi-one-dimensional organic conductors [Text] / I. I. Ukrainskii, O. V. Shramko; A. A. Ovchinnikov, I. I. Ukrainskii (Eds.). ‑ Electron – electron correlation effects in low-dimensional conductors and superconductors, Springer Verlag, Berlin, 1991. ‑ P. 62–72. doi: 10.1007/978-3-642-76753-1_8

13. Ovchinnikov, A. A. Introduction [Text] / A. A. Ovchinnikov, I. I. Ukrainskii; A. A. Ovchinnikov, I. I. Ukrainskii (Eds.). ‑ Electron – electron correlation effects in low-dimensional conductors and superconductors, Springer Verlag, Berlin, 1991. ‑ P. 1–9.

14. Salem, L. The molecular orbital theory of conjugated systems [Text] / L. Salem, W. A. Benjamin. ‑ New York, 1966.

15. Peierls, R. E. Quantum Theory of Solids [Text] / R. E. Peierls. ‑ Clarendon Press, Oxford, 1955.

16. Misurkin, I. A. Electronic structure of long molecules with conjugated bonds [Text] / I. A. Misurkin, A. A. Ovchinnikov // Teoreticheskaya i Eksperimental’naya Khimiya. – 1967. ‑ Vol. 3, Issue 4. ‑ P. 431–436. doi: 10.1007/bf01112374

17. Misurkin, I. A. The electronic structures of large π-electron systems (graphite, polyacenes, cumulenes) [Text] / I. A. Misurkin, A. A. Ovchinnikov // Teoreticheskaya i Eksperimental’naya Khimiya. – 1968. ‑ Vol. 4, Issue 1. ‑ P. 3–11.

18. Ruedenberg, K. Quantum Mechanics of Mobile Electrons in Conjugated Bond Systems. I. General Analysis of the Tight-Binding Approximation [Text] / K. Ruedenberg // Journal of Chemical Physics. – 1961. ‑ Vol. 34, Issue 6. ‑ P. 1861–1878. doi: 10.1063/1.1731785

19. Murell, J. N. Energies of excited electronic states as calculated with ZDO approximation [Text] / J. N. Murell, L. Salem // Journal of Chemical Physics. – 1961. ‑ Vol. 34, Issue 6. ‑ P. 1914. doi: 10.1063/1.1731791

20. Lifshits, I. M. About degenerate regular perturbations. 1. Discrete spectrum [Text] / I. M. Lifshits // Zhurnal eksperimentalnoi i teoreticheskoi fiziki. – 1947. ‑ Vol. 17. ‑ P. 1017.

21. Lifshits, I. M. About degenerate regular perturbations. 2. Quasicontinuous spectrum [Text] / I. M. Lifshits // Zhurnal eksperimentalnoi i teoreticheskoi fiziki. – 1947. ‑ Vol. 17. ‑ P. 1076.

22. Lifshits, I. M. On a problem of the theory of perturbations connected with quantum statistics [Text] / I. M. Lifshits // Uspekhi matematicheskikh nauk. – 1952. ‑ Vol. 7, Issue 1 (47). ‑ P. 171–180.

23. Lifshits, I. M. Some problems of the dynamic theory of non-ideal crystal lattices [Text] / I. M. Lifshits // Il Nuovo Cimento. – 1956. ‑ Vol. 3, Issue S4. ‑ P. 716–734. doi: 10.1007/bf02746071

24. Koster, G. F. Wave functions for impurity levels [Text] / G. F. Koster, J. C. Slater // Physical Review. – 1954. ‑ Vol. 95, Issue 5. ‑ P. 1167‑1176. doi: 10.1103/physrev.95.1167

25. Koutecky, J. Contribution to the theory of the surface electronic states in the one-electron approximation [Text] / J. Koutecky // Physical Review. – 1957. ‑ Vol. 108, Issue 1. ‑ P. 13‑18. doi: 10.1103/physrev.108.13

26. Koutecky, J. A contribution to the molecular-orbital theory of chemisorption [Text] / J. Koutecky // Transaction Faraday Society. – 1958. ‑ Vol. 54. ‑ P. 1038‑1052. doi: 10.1039/tf9585401038

27. Montroll, E. W. Effects of defects on lattice vibrations [Text] / E. W. Montroll, R. B. Potts // Physical Review. – 1955. ‑ Vol. 100, Issue 2. ‑ P. 525‑543. doi: 10.1103/physrev.100.525

28. Kruglyak, Yu. A. Methods of computations in quantum chemistry. Calculation of π-electronic molecular structure by simple molecular orbital methods [Text] / Yu. A. Kruglyak et. al. – Kiev: Naukova Dumka, 1967.

29. Pople, J. A. Bond alternation defects in long polyene molecules [Text] / J. A. Pople, S. H. Walmsley // Molecular Physics. – 1962. ‑ Vol. 5, Issue 1. ‑ P. 15‑20. doi: 10.1080/00268976200100021

30. Berlin, A. A. Polymers with conjugated bonds in the macromolecular chains. II. Magnetic and some other properties of polyarylvinylenes [Text] / A. A. Berlin, L. A. Blumenfeld, M. I. Cherkashin, A. E. Kalmanson, O. G. Selskaia // Vysokomolekulyarnye Soedineniya. – 1959. ‑ Vol. 1, Issue 9. ‑ P. 1361–1363.

31. Blumenfeld, L. A. Application of electron paramagnetic resonance in chemistry [Text] / L. A. Blumenfeld, V. V. Voevodsky, A. G. Semenov. ‑ Academy of Sciences of the USSR, Novosibirsk, 1962.

32. Pen’kovsky, V. V. Compounds with conjugated double bonds [Text] / V. V. Pen’kovsky // Uspekhi khimii (USSR). – 1964. ‑ Vol. 33, Issue 10. ‑ P. 1232–1263.

33. Kruglyak, Yu. A. The electronic properties of polyenes and polyphenylacetylenes. I. Ionization potentials, electron affinity, and energy of transition to the lowest triplet state [Text] / Yu. A. Kruglyak // Zhurnal Strukturnoi Khimii. – 1969. ‑ Vol. 10, Issue 1. ‑ P. 26–31.

34. Kruglyak, Yu. A. The electronic properties of polyenes and polyphenylacetylenes. II. Delocalization of the upaired electron and spin density [Text] / Yu. A. Kruglyak, V. V. Pen’kovskii // Zhurnal Strukturnoi Khimii. – 1969. ‑ Vol. 10, Issue 2. ‑ P. 223–229.

35. Pen’kovskii, V. V. The electronic properties of polyenes and polyphenylacetylenes. III. Intermolecular charge transfer in polymers with conjugated bonds and their ESR spectra [Text] / V. V. Pen’kovskii, Yu. A. Kruglyak // Zhurnal Strukturnoi Khimii. – 1969. ‑ Vol. 10, Issue 3. ‑ P. 459–464.

36. Blyumenfel’d, L. A. States with charge transfer in organic systems [Text] / L. A. Blyumenfel’d, V. A. Benderskii // Zhurnal Strukturnoi Khimii. – 1964. ‑ Vol. 4, Issue 3. ‑ P. 405–414.

37. Blyumenfel’d, L. A. Charge transfer states in organic systems [Text] / L. A. Blyumenfel’d, V. A. Benderskii, P. A. Stunzhas // Zhurnal Strukturnoi Khimii. – 1966. ‑ Vol. 7, Issue 5. ‑ P. 686–693.

38. Lutoshkin, V. I. Quantitative estimation of the bond alternation in polyenes [Text] / V. I. Lutoshkin, G. G. Dyadyusha, Yu. A. Kruglyak; A. I. Brodsky (Ed.). ‑ Structure of molecules and quantum chemistry, Naukova Dumka, Kiev, Ukraine, 1970. ‑ P. 132–138.

39. Lutoshkin, V. I. Calculation of electronic structure and force field of alternant radicals with allowance for deformation of the σ-core [Text] / V. I. Lutoshkin, Yu. A. Kruglyak, G. G. Dyadyusha // Teoreticheskaya i Eksperimental’naya Khimiya. – 1971. ‑ Vol. 7, Issue 5. ‑ P. 579–584.

40. Hartree, D. R. The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods [Text] / D. R. Hartree // Proceedings Cambridge Philosophical Society. – 1928. ‑ Vol. 24, Issue 1. ‑ P. 89–110. doi: 10.1017/s0305004100011919

41. Hartree, D. R. The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part II. Some Results and Discussion [Text] / D. R. Hartree // Proceedings Cambridge Philosophical Society. – 1928. ‑ Vol. 24, Issue 1. ‑ P. 111–132.

42. Fock, V. A. An approximate method for solving the quantum many-body problem [Text] / V. A. Fock // Zeitschrift fur Physik. – 1930. ‑ Vol. 61, Issue 1-2. ‑ P. 126‑148.

43. Rutherford, D. E. Substitutional Analysis [Text] / D. E. Rutherford. ‑ Edinburgh University Press, London, 1948.

44. Hammermesh, M. Group theory and its application to physical problems [Text] / M. Hammermesh. ‑ Addison-Wesley, Reading, 1962.

45. Kaplan, I. G. Symmetry of many-electron systems [Text] / I. G. Kaplan. ‑ Nauka, Moscow, 1969.

46. Goddard III, W. A. Improved quantum theory of many-electron systems: I. Construction of eigenfunctions of  which satisfy Pauli's principle [Text] / W. A. Goddard III // Physical Review. – 1967. ‑ Vol. 157, Issue 1. ‑ P. 73–80. doi: 10.1103/physrev.157.73

47. Goddard III, W. A. Improved quantum theory of many-electron systems: II. The basic method [Text] / W. A. Goddard III // Physical Review. – 1967. ‑ Vol. 157, Issue 1. ‑ P. 81–93. doi: 10.1103/physrev.157.81

48. Goddard III, W. A. Improved quantum theory of many-electron systems: III. The GF method [Text] / W. A. Goddard III // Journal Chemical Physics. – 1968. ‑ Vol. 48, Issue 1. ‑ P. 450–461. doi: 10.1063/1.1667943

49. Goddard III, W. A. Wavefunctions and correlation energies for two-, three-, and four-electron atoms [Text] / W. A. Goddard III // Journal Chemical Physics. – 1968. ‑ Vol. 48, Issue 3. ‑ P. 1008–1017. doi: 10.1063/1.1668754

50. Goddard III, W. A. Improved quantum theory of many-electron systems: IV. Properties of GF wavefunctions [Text] / W. A. Goddard III // Journal Chemical Physics. – 1968. ‑ Vol. 48, Issue 12. ‑ P. 5337–5347. doi: 10.1063/1.1668225

51. Ladner, R. C. Improved quantum theory of many-electron systems: V. The spin-coupling optimized GI method [Text] / R. C. Ladner, W. A. Goddard III // Journal Chemical Physics. – 1969. ‑ Vol. 51, Issue 3. ‑ P. 1073–1087. doi: 10.1063/1.1672106

52. Goddard III, W. A. The symmetric group and the spin generalized SCF method [Text] / W. A. Goddard III // International Journal of Quantum Chemistry. – 1970. ‑ Vol. IIIs. ‑ P. 593–600. doi: 10.1002/qua.560040720

53. J. C. Slater, “The theory of complex spectra”, Physical Review, vol. 34, pp. 1293 – 1323, 1929.

54. Slater, J. C. Quantum theory of molecules and solids [Text] / J. C. Slater // Physical Review. – 1930. ‑ Vol. 35, Issue 2. ‑ P. 210–211.

55. Roothaan, C. C. J. New developments in molecular orbital theory [Text] / C. C. J. Roothaan // Review of Modern Physics. – 1951. ‑ Vol. 23, Issue 2. ‑ P. 69‑89. doi: 10.1103/revmodphys.23.69

56. Amos, A. T. Single determinant wave functions [Text] / A. T. Amos, G. G. Hall // Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. – 1961. ‑ Vol. 263, Issue 1315. ‑ P. 483–493. doi: 10.1098/rspa.1961.0175

57. Ukrainskii, I. I. Projection of the wave function of the unlimited Hartree – Fock method on the doublet state in the case of benzyl radical [Text] / . I. Ukrainskii, Yu. A. Kruglyak, H. Preuss, R. Yanoshek // Teoreticheskaya i Eksperimental’naya Khimiya. – 1971. ‑ Vol. 8, Issue 3. ‑ P. 299–308.

58. Amos, A. T. Some properties of π-ions and triplets [Text] / A. T. Amos; O. Sinanoğlu (Ed.). ‑ Modern quantum chemistry , Academic Press, New York, 1965. ‑ P. 157–170.

59. Pople, J. A. Self-consistent orbitals for radicals [Text] / J. A. Pople, R. K. Nesbet // Journal Chemical Physics. – 1954. ‑ Vol. 22, Issue 3. ‑ P. 571–572. doi: 10.1063/1.1740120

60. Lowdin, P.-O. Quantum theory of many-particle systems. III. Extension of the Hartree-Fock scheme to include degenerate systems and correlation effects [Text] / P.-O. Lowdin // Physical Review. – 1955. ‑ Vol. 97, Issue 6. ‑ P. 1509–1520. doi: 10.1103/physrev.97.1509

61. P.-O. Lowdin, Correlation problem in many-electron quantum mechanics. I. Review of different approaches and discussion of some current ideas [Text] / P.-O. Lowdin; I. Prigogine (Ed.) // Advances in Chemical Physics. – 1959. ‑ Vol. 2. ‑ P. 207–322.

62. Lowdin, P.-O. Angular momentum wave functions constructed by projection operators [Text] / P.-O. Lowdin // Review of Modern Physics. – 1964. ‑ Vol. 36, Issue 4. ‑ P. 966–976. doi: 10.1103/revmodphys.36.966

63. Sasaki, F. Spin-component analysis of single-determinant wavefunctions [Text] / F. Sasaki, K. Ohno // Journal Mathematical Physics. – 1963. ‑ Vol. 4, Issue 9. ‑ P. 1140–1147. doi: 10.1063/1.1704044

64. Smith, V. H. Projection of exact spin eigenfunctions [Text] / V. H. Smith // The Journal of Chemical Physics. – 1964. ‑ Vol. 41, Issue 1. ‑ P. 277. doi: 10.1063/1.1725634

65. Sando, K. M. Spin-projected and extended SCF calculations [Text] / K. M. Sando, J. E. Harriman // The Journal of Chemical Physics. – 1967. ‑ Vol. 47, Issue 1. ‑ P. 180. doi: 10.1063/1.1711843

66. Harris, F. On the calculation of spin densities [Text] / F. Harris // Molecular Physics. – 1966. ‑ Vol. 11, Issue 3. ‑ P. 243–256. doi: 10.1080/00268976600101081

67. Pauncz, R. Alternant Molecular Orbital Method [Text] / R. Pauncz. ‑ W. B. Saunders, London, 1967.

68. Lowdin, P.-O. Band theory, valence band and tight-binding calculations [Text] / P.-O. Lowdin // Journal of Applied Physics Supplement. – 1962. ‑ Vol. 33, ‑ Issue 1. ‑ P. 251–280. doi: 10.1063/1.1777106

69. Pauncz, R. Studies of the alternant molecular orbital method. I. General energy expression for an alternant system with closed-shell structure [Text] / R. Pauncz, J. de Heer, P.-O. Lowdin // Journal Chemical Physics. – 1962. ‑ Vol. 36, Issue 9. ‑ P. 2247–2256. doi: 10.1063/1.1732872

70. Pauncz, R.Studies of the alternant molecular orbital method. II. Application to Cyclic Systems [Text] / R. Pauncz, J. de Heer, P.-O. Lowdin // Journal Chemical Physics. – 1962. ‑ Vol. 36. ‑ P. 2257–2265. doi: 10.1063/1.1732873

71. Hückel, E. Zur Quantentheorie der Doppelbindung [Text] / E. Hückel // Zeitschrift für Physik. – 1930. ‑ Vol. 60, Issue 7-8. ‑ P. 423–456. doi: 10.1007/bf01341254

72. Hückel, E. Quantentheoretische Beiträge zum Benzolproblem [Text] / E. Hückel // Zeitschrift für Physik. – 1931. ‑ Vol. 70, Issue 3-4. ‑ P. 204–286. doi: 10.1007/bf01339530

73. Coulson, C. A. The electronic structure of conjugated systems. I. General theory [Text] / C. A. Coulson, H. C. Longuet-Higgins // Proceedings Royal Society – 1947. ‑ Vol. 191, Issue 1024. ‑ P. 39–60. doi: 10.1098/rspa.1947.0102

74. Brickstock, A. Resonance energies and charge distributions of unsaturated hydrocarbon radicals and ions [Text] / A. Brickstock, J. A. Pople // Transaction Faraday Society. – 1954. ‑ Vol. 50. ‑ P. 901–911. doi: 10.1039/tf9545000901

75. Koopmans, T. A. Über die zuordnung von wellenfunktionen und eigenwerten zu den einzelnen elektronen eines atoms [Text] / T. A. Koopmans // Physica. – 1933. ‑ Vol. 1, Issue 1-6. ‑ P. 104–113. doi: 10.1016/s0031-8914(34)90011-2

76. Brillouin, L. La méthode du champ self-consistent, (Actualites Scientifiques et Industrielles, Vol. 71) [Text] / L. Brillouin. ‑ Hermann, Paris, 1933.

77. Brillouin, L. Les champs "self-consistents" de Hartree et de Fock, (Actualites Scientifiques et Industrielles, Vol. 159) [Text] / L. Brillouin. ‑ Hermann, Paris, 1934.

78. Mozdor, E. V. Matrix elements of the physical value operators on single-configurational functions for radicals [Text] / E. V. Mozdor, Yu. A. Kruglyak, ad V. A. Kuprievich // Teoreticheskaya i Eksperimentalnaya Khimiya. – 1969. ‑ Vol. 5, Issue 6. ‑ P. 723–730.

79. Kruglyak, Yu. A. Study of the electronic structure of radicals by the CI method. I. Matrix elements of the physical value operators [Text] / Yu. A. Kruglyak, E. V. Mozdor, V. A. Kuprievich // Croatica Chemica Acta. – 1971. ‑ Vol. 43. ‑ P. 15–22.

80. Popov, N. A. Symmetry properties of one-electron orbitals in the method of different orbitals for different spins [Text] / N. A. Popov // Zhurnal Strukturnoi Khimii. – 1970. ‑ Vol. 11, Issue 4. ‑ P. 727–733.

81. Hylleraas, E. A. Neue berechnung der energie des Heliums im grundzustande, sowie des tiefsten terms von ortho-helium [Text] / E. A. Hylleraas // Zeitschrift für Physik. – 1929. ‑ Vol. 54, Issue 5-6. ‑ P. 347–366. doi: 10.1007/bf01375457

82. Ekkart, C. The theory and calculation of screening constants [Text] / C. Ekkart // Physical Review. – 1930. ‑ Vol. 36, Issue 5. ‑ P. 878–892. doi: 10.1103/physrev.36.878

83. Shull, H. Superposition of configurations and natural spin orbitals. Applications to the He problem [Text] / H. Shull, P.-O. Lowdin // Journal Chemical Physics. – 1959. ‑ Vol. 30, Issue 3. ‑ P. 617–626. doi: 10.1063/1.1730019

84. Chong, D. P. Different orbitals for different spins. Singlet ground state of Helium [Text] / D. P. Chong // The Journal of Chemical Physics. – 1966. ‑ Vol. 45, Issue 9. ‑ P. 3317–3318. doi: 10.1063/1.1728108

85. Green, L. Correlation energies and angular components of the wave functions of the ground states of H, He, and Li+ [Text] / L. Green, M. Lewis, M. Mulder, C. W. Wyeth, J. W. Woll // Physical Review. – 1954. ‑ Vol. 93, Issue 2. ‑ P. 273‑279. doi: 10.1103/physrev.93.273

86. Bonham, R. A. correlated wavefunctions for the ground state of Heliumlike atoms [Text] / R. A. Bonham, D. A. Kohl // Journal Chemical Physics. – 1966. ‑ Vol. 45, Issue 7. ‑ P. 2471. doi: 10.1063/1.1727963

87. Dolgushin, M. D. Splitted orbitals and correlation energies for ground state of two-electron atoms [Text] / M. D. Dolgushin; A. Jucys (Ed.). ‑ Theory of electronic shells in atoms and molecules, Mintis, Vilnius, 1971. ‑ P. 108–111.

88. Fraga, S. Studies in molecular structure. VI. Potential curve for the interaction of two hydrogen atoms in the LCAO MO SCF approximation [Text] / S. Fraga, B. J. Ransil // The Journal of Chemical Physics. – 1967. ‑ Vol. 35, Issue 6. ‑ P. 2471. doi: 10.1063/1.1732194

89. Kolos, W. Accurate adiabatic treatment of the ground state of the hydrogen molecule [Text] / W. Kolos, L. Wolniewicz // The Journal of Chemical Physics. – 1964. ‑ Vol. 41, Issue 12. ‑ P. 3663. doi: 10.1063/1.1725796

90. Swalen, J. D. Many‐parameter alternant molecular orbital calculations for large cyclic systems with closed‐shell structure [Text] / J. D. Swalen, J. de Heer // The Journal of Chemical Physics. – 1964. ‑ Vol. 40, Issue 2. ‑ P. 378‑384. doi: 10.1063/1.1725122

91. Hall, G. G. Molecular orbital theory of the spin properties of conjugated molecules [Text] / G. G. Hall, A. T. Amos; D. R. Bates, I. Estermann (Eds.) // Advances in Atomic and Molecular Physics. – 1965. ‑ Vol. 1. ‑ P. 1–59. doi: 10.1016/s0065-2199(08)60279-1

92. Pople, J. A. Molecular orbital theory of the electronic structure of organic compounds. II. Spin densities in paramagnetic species [Text] / J. A. Pople, D. L. Beveridge, P. A. Dobosh // Journal of the American Chemical Society. – 1968. ‑ Vol. 90, Issue 16. ‑ P. 4201‑4209. doi: 10.1021/ja01018a003

93. Kruglyak, Yu. A. Non-empirical computation of the electronic structure of benzyl radical [Text] / Yu. A. Kruglyak, H. Preuss, R. Yanoshek // Ukrainskii Fizichnyi Zhurnal. – 1970. ‑ Vol. 15, Issue 6. ‑ P. 977–985.

94. Kruglyak, Yu. A. Calculation of the electron shells of the benzyl radical by the unrestricted Hartree-Fock method on a Gaussian basis [Text] / Yu. A. Kruglyak, H. Preuss, R. Yanoshek // Zhurnal Strukturnoi Khimii. – 1971. ‑ Vol. 12, Issue 4. ‑ P. 689–696.

95. Kruglyak, Yu. A. An orbital analysis of the ab initio electron and spin populations of the atoms in the benzyl radical [Text] / Yu. A. Kruglyak, I. I. Ukrainskii, H. Preuss, R. Yanoshek // Teoreticheskaya i Eksperimental’naya Khimiya. – 1970. ‑ Vol. 7, Issue 6. ‑ P. 815–819.

96. Carrington, A. The electron spin resonance spectrum and spin density distribution of the benzyl radical [Text] / A. Carrington, I. C. P. Smith // Molecular Physics. – 1965. ‑ Vol. 9, Issue 2. ‑ P. 137–147. doi: 10.1080/00268976500100151

97. Benson, H. G. On the spin density distribution in the benzyl radical [Text] / H. G. Benson, A. Hudson // Molecular Physics. – 1971. ‑ Vol. 20, Issue 1. ‑ P. 185–187. doi: 10.1080/00268977100100181

98. Lloyd, R. V. Free radicals in adamantane matrix. EPR and Indo study of the benzyl, aniline, and phenoxy radicals and their fluorinated derivatives [Text] / R. V. Lloyd, D. E. Wood // Journal American Chemical Society. – 1974. ‑ Vol. 96, Issue 3. ‑ P. 659–665. doi: 10.1021/ja00810a004

99. McConnell, H. M. Electron densities in semiquinones by paramagnetic resonance [Text] / H. M. McConnell // H. M. McConnell // Journal Chemical Physics. – 1956. ‑ Vol. 24, Issue 3. ‑ P. 632. doi: 10.1063/1.1742580

100. McConnell, H. M. Indirect hyperfine interactions in the paramagnetic resonance spectra of aromatic free radicals [Text] / H. M. McConnell // The Journal of Chemical Physics. – 1956. ‑ Vol. 24, Issue 4. ‑ P. 764. doi: 10.1063/1.1742605

101. Fessenden, R. W. Electron spin resonance studies of transient alkyl radicals [Text] / R. W. Fessenden, R. H. Schuler // The Journal of Chemical Physics. – 1963. ‑ Vol. 39, Issue 9. ‑ P. 2147. doi: 10.1063/1.1701415

102. Kruglyak, Yu. A. Full configuration interaction of the benzyl radical [Text] / Yu. A. Kruglyak, E. V. Mozdor, V. A. Kuprievich // Ukrainsky Fizichnyi Zhurnal. – 1970. ‑ Vol. 15, Issue 1. ‑ P. 47–57.

103. Kruglyak, Yu. A. Electronic structure of the ground state of the benzyl radical in equilibrium geometry [Text] / Yu. A. Kruglyak, G. Hibaum, N. E. Radomyselskaya // Revue Roumaine de Chimie. – 1972. ‑ Vol. 17, Issue 5. ‑ P. 781–799.

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