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Exchange symmetry in a system of nonrelativistic spin-1/2 fermions in the Feynman quantum statistics representation

  • Atoms, Spectra, Radiation
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Abstract

A compact representation is obtained for the quantum statistical sum of indistinguishable nonrelativistic spin-1/2 fermions in the form of Feynman path integrals which can be used as the basis to develop a fundamentally exact method of computer modeling for systems of strongly interacting electrons at nonzero temperature. A basis of symmetrized wave functions is constructed using Young symmetry operators. An exact permutation symmetrization procedure leads to an avalanche-like multiplication in the number of diagrams of linked Feynman integrals of the order of N!. The partition function can be simplified without introducing any approximations and this is performed numerically by computer by direct sorting of diagrams. The control tables obtained, containing combinatorial weights of diagrams, direct the Markov random walk process in virtual trajectory space which is achieved numerically by computer. The equilibrium characteristics of the quantum system are calculated by averaging. This approach is an expansion of the Monte Carlo-Metropolis method to systems of quantum indistinguishable particles with spin. Demonstration numerical calculations using this method were made for the simplest exchange systems, for a hydrogen molecule, a Be+ ion, and a Li atom. The ground state of the hydrogen molecule is reproduced with a statistical error of 0.2%. Exchange-correlation effects lead to nontrivial structural changes in the thermally excited electron shells of ions in a state of strong plasma compression.

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References

  1. Physics of Simple Liquids, Ed. by H. N. V. Temperley, J. S. Rowlinson, and G. S. Rushbrooke (North-Holland, Amsterdam, 1968; Mir, Moscow, 1973).

    Google Scholar 

  2. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, and H. A. Teller, J. Chem. Phys. 21, 1087 (1953).

    Article  Google Scholar 

  3. R. D. Etters and J. Kaelberer, Phys. Rev. A 11, 1068 (1975).

    Article  ADS  Google Scholar 

  4. Ch. Margheritis and C. Sinistri, Z. Naturforsch. A 30, 83 (1975).

    Google Scholar 

  5. P. N. Vorontsov-Vel’yaminov and V. P. Chasovskikh, Teplofiz. Vys. Temp. 13, 1153 (1975).

    Google Scholar 

  6. S. V. Shevkunov, Zh. Éksp. Teor. Fiz. 104, 3032 (1993) [JETP 77, 413 (1993)].

    Google Scholar 

  7. S. V. Shevkunov, Zh. Éksp. Teor. Fiz. 105, 1258 (1994) [JETP 78, 677 (1994)].

    Google Scholar 

  8. S. V. Shevkunov, Zh. Éksp. Teor. Fiz. 108, 1373 (1995) [JETP 81, 753 (1995)].

    Google Scholar 

  9. A. A. Martsinovski, S. V. Shevkunov, and P. N. Vorontsov-Velyaminov, Mol. Simul. 6, 143 (1991).

    Google Scholar 

  10. S. V. Shevkunov, P. N. Vorontsov-Velyaminov, and A. A. Martsinovski, Mol. Simul. 5, 119 (1990).

    Google Scholar 

  11. S. V. Shevkunov and A. Vegiri, J. Chem. Phys. 111, 9303 (1999).

    Article  ADS  Google Scholar 

  12. K. Kiyohara, K. E. Gubbins, and A. Z. Panagiotopoulos, Mol. Phys. 94, 803 (1998).

    Google Scholar 

  13. K. J. Oh and X. C. Zeng, J. Chem. Phys. 110, 4471 (1999).

    Article  ADS  Google Scholar 

  14. D. Colognesi, A. de Santis, and D. Rocca, Mol. Phys. 88, 465 (1996).

    Article  Google Scholar 

  15. P. Jedlovszky, I. Bako, G. Palinkas, et al., J. Chem. Phys. 105, 245 (1996).

    ADS  Google Scholar 

  16. P. Jedlovszky, J. P. Brodholt, F. Bruni, et al., J. Chem. Phys. 108, 8528 (1998).

    Article  ADS  Google Scholar 

  17. P. Jedlovszky and J. Richardi, J. Chem. Phys. 110, 8019 (1999).

    ADS  Google Scholar 

  18. P. Jedlovszky, J. Chem. Phys. 111, 5975 (1999).

    ADS  Google Scholar 

  19. E. Allahyarov, I. D’Amico, and H. L. Lowen, Phys. Rev. Lett. 81, 1334 (1998).

    Article  ADS  Google Scholar 

  20. J. Wu, D. Bratko, and J. M. Prausnitz, Proc. Natl. Acad. Sci. USA 95, 15 169 (1998).

    Google Scholar 

  21. J. Z. Wu, D. Bratko, H. W. Blanch, and J. M. Praushitz, J. Chem. Phys. 111, 7084 (1999).

    ADS  Google Scholar 

  22. N. Gronbech-Jensen, K. M. Beardmore, and P. Pincus, Physica A (Amsterdam) 261, 74 (1998).

    Google Scholar 

  23. L. Degreve and F. L. B. da Silva, J. Chem. Phys. 110, 3070 (1999).

    ADS  Google Scholar 

  24. A. D. Mackie, J. Hernandez-Cobos, and L. F. Vega, J. Chem. Phys. 111, 2103 (1999).

    Article  ADS  Google Scholar 

  25. E. V. Akhmatskaya, M. D. Cooper, N. A. Burton, et al., Chem. Phys. Lett. 267, 105 (1997).

    Article  Google Scholar 

  26. S. J. Vaughn, E. V. Akhmatskaya, M. A. Vincent, et al., J. Chem. Phys. 110, 4338 (1999).

    Article  ADS  Google Scholar 

  27. F. Calvo, Phys. Rev. B 60, 15 601 (1999).

    Google Scholar 

  28. U. Hansen, P. Vogl, and V. Fiorentini, Phys. Rev. B 60, 5055 (1999).

    ADS  Google Scholar 

  29. N. Matubayasi, Ch. Wakai, and M. Nakahara, J. Chem. Phys. 110, 8000 (1999).

    ADS  Google Scholar 

  30. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965; Mir, Moscow, 1968).

    Google Scholar 

  31. L. D. Fosdick and H. F. Jordan, Phys. Rev. 143, 58 (1966).

    Article  ADS  Google Scholar 

  32. H. F. Jordan and L. D. Fosdick, Phys. Rev. 171, 128 (1968).

    Article  ADS  Google Scholar 

  33. V. R. Panharipande, S. C. Pieper, and R. B. Wiringa, Phys. Rev. B 34, 4571 (1986).

    ADS  Google Scholar 

  34. D. S. Lewart, V. R. Pandharipande, and S. C. Pieper, Phys. Rev. B 37, 4950 (1988).

    Article  ADS  Google Scholar 

  35. P. A. Whitlock, G. V. Chester, and M. H. Kalos, Phys. Rev. B 38, 2418 (1988).

    Article  ADS  Google Scholar 

  36. B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 (1989).

    Article  ADS  Google Scholar 

  37. H. Yokoyama and H. Shiba, J. Phys. Soc. Jpn. 56, 3570 (1987); 59, 3669 (1990).

    Google Scholar 

  38. Y. Liu, J. Dong, Chang-de Gong, and T. Chen, Phys. Rev. B 48, 1308 (1993).

    ADS  Google Scholar 

  39. D. Bressanini, G. Fabbri, M. Mella, and G. Morosi, J. Chem. Phys. 111, 6230 (1999).

    ADS  Google Scholar 

  40. J. B. Anderson, J. Chem. Phys. 63, 1499 (1975); 65, 4121 (1976); 73, 3897 (1980).

    ADS  Google Scholar 

  41. R. D. Amos, Adv. Chem. Phys. 67, 99 (1987).

    Google Scholar 

  42. P. Sandler, V. Buch, and D. C. Clary, J. Chem. Phys. 101, 6353 (1994).

    ADS  Google Scholar 

  43. M. H. Muser and J. Ankerhold, Europhys. Lett. 44, 216 (1998).

    ADS  Google Scholar 

  44. R. N. Barnett, P. J. Reynolds, and W. A. Lester, J. Chem. Phys. 84, 4992 (1986).

    Article  ADS  Google Scholar 

  45. M. Caffarel and P. Claverie, J. Chem. Phys. 88, 1088 (1988).

    ADS  MathSciNet  Google Scholar 

  46. D. F. Coker and R. O. Watts, J. Chem. Phys. 86, 5703 (1987).

    ADS  Google Scholar 

  47. D. M. Ceperley and H. Partridge, J. Chem. Phys. 84, 820 (1986).

    Article  ADS  Google Scholar 

  48. R. M. Grimes, B. L. Hammond, P. J. Reynolds, and W. A. Lester, J. Chem. Phys. 85, 4779 (1986).

    Article  ADS  Google Scholar 

  49. M. Caffarel and P. Claverie, J. Chem. Phys. 88, 1100 (1988).

    ADS  MathSciNet  Google Scholar 

  50. G. J. Martyna and B. J. Berne, J. Chem. Phys. 88, 4516 (1988); 90, 3744 (1989).

    Article  ADS  Google Scholar 

  51. P. A. Christensen, J. Chem. Phys. 88, 4867 (1988).

    ADS  Google Scholar 

  52. T. Yoshida and K. Iguchi, J. Chem. Phys. 88, 1032 (1988).

    ADS  Google Scholar 

  53. M. Caffarel, P. Claverie, C. Mijoule, et al., J. Chem. Phys. 90, 990 (1989).

    Article  ADS  Google Scholar 

  54. B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 (1989).

    Article  ADS  Google Scholar 

  55. J. Carlson, J. W. Moskowitz, and K. E. Schmidt, J. Chem. Phys. 90, 1003 (1989).

    Article  ADS  Google Scholar 

  56. J. E. Hirsch, D. J. Scalapino, and R. L. Sugar, Phys. Rev. Lett. 47, 1628 (1981).

    Article  ADS  Google Scholar 

  57. J. E. Hirsch, R. L. Sugar, D. J. Scalapino, and R. Blankenbecler, Phys. Rev. B 26, 5033 (1982).

    Article  ADS  Google Scholar 

  58. D. J. Scalapino and R. L. Sugar, Phys. Rev. B 24, 4295 (1981).

    Article  ADS  Google Scholar 

  59. J. E. Hirsch and D. J. Scalapino, Phys. Rev. B 27, 7169 (1983).

    ADS  Google Scholar 

  60. J. E. Hirsh, Phys. Rev. B 31, 4403 (1985).

    ADS  Google Scholar 

  61. Ad. Lagendijk and H. DeRaedt, Phys. Rev. Lett. 49, 602 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  62. S. R. White, D. J. Scalapino, R. L. Sugar, and N. E. Bickers, Phys. Rev. Lett. 63, 1523 (1989).

    ADS  Google Scholar 

  63. M. Imada, J. Phys. Soc. Jpn. 57, 3128 (1988).

    Google Scholar 

  64. M. Imada and Y. Hatsugai, J. Phys. Soc. Jpn. 58, 3752 (1989).

    Google Scholar 

  65. H. Otsika, J. Phys. Soc. Jpn. 59, 2916 (1990).

    ADS  Google Scholar 

  66. W. von der Linden, I. Morgenstern, and H. DeRaedt, Phys. Rev. B 41, 4669 (1990).

    ADS  Google Scholar 

  67. H. Aoki and K. Kuroki, Phys. Rev. B 42, 2125 (1990).

    ADS  Google Scholar 

  68. Y. Takada and T. Kita, J. Phys. Soc. Jpn. 60, 25 (1991).

    Google Scholar 

  69. R. T. Scalettar, D. J. Scalapino, R. L. Sugar, and S. R. White, Phys. Rev. B 44, 770 (1991).

    ADS  Google Scholar 

  70. L. Tan, Q. Li, and J. Callaway, Phys. Rev. B 44, 341 (1991).

    ADS  Google Scholar 

  71. A. Moreo, D. Scalapino, and E. Dagatto, Phys. Rev. B 43, 11 442 (1991).

  72. G. G. Batrouni and R. T. Scalettar, Phys. Rev. B 46, 9051 (1992).

    Article  ADS  Google Scholar 

  73. R. Valenti and C. Gros, Phys. Rev. Lett. 68, 2402 (1992).

    ADS  Google Scholar 

  74. A. F. Elesin and V. A. Kashurnikov, Zh. Éksp. Teor. Fiz. 106, 1773 (1994) [JETP 79, 961 (1994)].

    Google Scholar 

  75. N. V. Prokof’ev, B. V. Svistunov, and I. S. Tupitsyn, Pis’ma Zh. Éksp. Teor. Fiz. 64, 853 (1996) [JETP Lett. 64, 911 (1996)].

    Google Scholar 

  76. D. Chandler and P. G. Wolynes, J. Chem. Phys. 74, 4078 (1981).

    Article  ADS  Google Scholar 

  77. V. S. Filinov, Phys. Lett. A 54, 259 (1975).

    Article  ADS  Google Scholar 

  78. V. S. Filinov and G. E. Norman, Phys. Lett. A 55, 219 (1975).

    Article  ADS  Google Scholar 

  79. R. M. Fye, Phys. Rev. B 33, 6271 (1986).

    Article  ADS  Google Scholar 

  80. H. DeRaedt and B. DeRaedt, Phys. Rev. A 28, 3575 (1983).

    ADS  MathSciNet  Google Scholar 

  81. V. S. Filinov, Nucl. Phys. B 271, 717 (1986).

    ADS  Google Scholar 

  82. N. Makri and W. H. Miller, Chem. Phys. Lett. 139, 10 (1987).

    Article  ADS  Google Scholar 

  83. J. D. Doll, D. L. Freeman, and M. J. Gillan, Chem. Phys. Lett. 143, 277 (1988).

    Article  ADS  Google Scholar 

  84. S. V. Shevkunov, O. M. Roschinenko, and P. N. Vorontsov-Velyaminov, Mol. Simul. 7, 205 (1991).

    Google Scholar 

  85. M. Imada, J. Phys. Soc. Jpn. 53, 2861 (1984).

    Google Scholar 

  86. D. M. Ceperley and E. L. Pollock, Phys. Rev. Lett. 56, 351 (1986).

    Article  ADS  Google Scholar 

  87. E. L. Pollock and D. M. Ceperley, Phys. Rev. B 36, 8343 (1987).

    Article  ADS  Google Scholar 

  88. D. M. Ceperley and E. L. Pollock, Phys. Rev. B 39, 2084 (1989).

    Article  ADS  Google Scholar 

  89. S. R. White and J. W. Wilkins, Phys. Rev. B 37, 5024 (1988).

    ADS  Google Scholar 

  90. H. Q. Lin and J. E. Hirsch, Phys. Rev. B 34, 1964 (1986).

    ADS  Google Scholar 

  91. M. Takahashi and M. Imada, J. Phys. Soc. Jpn. 53, 963 (1983).

    Google Scholar 

  92. C. L. Cleveland, U. Landman, and R. N. Barnett, Phys. Rev. B 39, 117 (1989).

    Article  ADS  Google Scholar 

  93. E. L. Pollock and D. M. Ceperley, Phys. Rev. B 30, 2555 (1984).

    Article  ADS  Google Scholar 

  94. B. A. Mason and K. Hess, Phys. Rev. B 39, 5051 (1989).

    ADS  Google Scholar 

  95. B. J. Berne and D. Thirumalai, Annu. Rev. Phys. Chem. 37, 401 (1986).

    Article  Google Scholar 

  96. K. Mackeown, Am. J. Phys. 53, 880 (1985).

    Article  ADS  Google Scholar 

  97. D. Marx and M. Parrinello, Z. Phys. B 95, 143 (1994); J. Chem. Phys. 104, 4077 (1996).

    Article  Google Scholar 

  98. M. E. Tuckerman, D. Marx, M. E. Klein, and M. Parrinello, J. Chem. Phys. 104, 5579 (1996).

    Article  ADS  Google Scholar 

  99. Ph. Sindzingre, D. M. Ceperley, and M. L. Klein, Phys. Rev. Lett. 67, 1871 (1991).

    Article  ADS  Google Scholar 

  100. D. Scharf, G. J. Martyna, and M. L. Klein, J. Chem. Phys. 99, 8997 (1993).

    ADS  Google Scholar 

  101. D. Marx, S. Sengupta, and P. Nielaba, J. Chem. Phys. 99, 6031 (1993).

    Article  ADS  Google Scholar 

  102. Ch. Chakravarty, J. Chem. Phys. 99, 8038 (1993).

    Article  ADS  Google Scholar 

  103. Ch. Chakravarty, J. Chem. Phys. 102, 956 (1995).

    ADS  Google Scholar 

  104. L. M. Sese and R. Ledesma, J. Chem. Phys. 102, 3776 (1995).

    ADS  Google Scholar 

  105. D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995).

    Article  ADS  Google Scholar 

  106. M. Skorobogatiy and J. D. Joannopoulos, Phys. Rev. B 60, 1433 (1999).

    Article  ADS  Google Scholar 

  107. M. C. Gordillo, Phys. Rev. B 60, 6790 (1999).

    Article  ADS  Google Scholar 

  108. B. Hetenyi, E. Rabani, and B. J. Berne, J. Chem. Phys. 110, 6143 (1999).

    ADS  Google Scholar 

  109. S. Jang and G. A. Voth, J. Chem. Phys. 111, 2357 (1999); 111, 2371 (1999).

    ADS  Google Scholar 

  110. P.-N. Roy and G. A. Voth, J. Chem. Phys. 110, 3647 (1999).

    ADS  Google Scholar 

  111. R. Rousseau and D. Marx, J. Chem. Phys. 111, 5091 (1999).

    Article  ADS  Google Scholar 

  112. P.-N. Roy, S. Jang, and G. A. Voth, J. Chem. Phys. 111, 5303 (1999).

    Article  ADS  Google Scholar 

  113. J. P. Serre, Représentations linéaires des groupes finis (Herman, Paris, 1967; Mir, Moscow, 1970).

    Google Scholar 

  114. A. A. Kirillov, Elements of the Theory of Representations (Nauka, Moscow, 1978; Springer-Verlag, Berlin, 1976).

    Google Scholar 

  115. V. D. Lyakhovskii and A. A. Bolokhov, Symmetry Groups and Elementary Particles (Leningrad. Gos. Univ., Leningrad, 1983).

    Google Scholar 

  116. H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1950; Mir, Moscow, 1983).

    Google Scholar 

  117. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory (Nauka, Moscow, 1974; Pergamon, New York, 1987).

    Google Scholar 

  118. A. S. Davydov, Quantum Mechanics (Fizmatgiz, Moscow, 1963).

    Google Scholar 

  119. F. A. Berezin, Method of Second Quantization (Nauka, Moscow, 1986; Academic, New York, 1986).

    Google Scholar 

  120. V. M. Zamalin, G. É. Norman, and V. S. Filinov, Monte Carlo Method in Statistical Thermodynamics (Mir, Moscow, 1977).

    Google Scholar 

  121. S. M. Ermakov, Monte Carlo Method and Related Problems (Nauka, Moscow, 1975).

    Google Scholar 

  122. S. V. Shevkunov, Teplofiz. Vys. Temp. 28, 1 (1990).

    ADS  Google Scholar 

  123. N. Marki and W. H. Miller, J. Chem. Phys. 90, 904 (1989).

    ADS  Google Scholar 

  124. R. P. Feynman, Statistical Mechanics: A Set of Lectures (Benjamin, Reading, Mass., 1972; Mir, Moscow, 1978).

    Google Scholar 

  125. M. F. Herman, E. Bruskin, and B. J. Berne, J. Chem. Phys. 76, 5150 (1982).

    ADS  Google Scholar 

  126. M. Parrinello and A. Rahman, J. Chem. Phys. 80, 860 (1984).

    Article  ADS  Google Scholar 

  127. R. N. Barnett, U. Landman, C. L. Cleveland, and J. Jortner, J. Chem. Phys. 88, 4429 (1988).

    ADS  Google Scholar 

  128. T. L. Hill, Statistical Mechanics: Principles and Selected Applications (McGraw-Hill, New York, 1956; Inostrannaya Literatura, Moscow, 1960).

    Google Scholar 

  129. K. S. Krasnov, V. S. Timoshinin, T. G. Danilova, and S. V. Khandozhko, Molecular Constants of Inorganic Compounds (Khimiya, Leningrad, 1968).

    Google Scholar 

  130. A. A. Radtsig and B. M. Smirnov, Reference Data on Atoms, Molecules, and Ions (Atomizdat, Moscow, 1980; Springer-Verlag, Berlin, 1985).

    Google Scholar 

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    S. V. Shevkunov

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Translated from Zhurnal Éksperimental’no\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}\) i Teoretichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}\) Fiziki, Vol. 118, No. 1, 2000, pp. 36–55.

Original Russian Text Copyright © 2000 by Shevkunov.

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Shevkunov, S.V. Exchange symmetry in a system of nonrelativistic spin-1/2 fermions in the Feynman quantum statistics representation. J. Exp. Theor. Phys. 91, 31–47 (2000). https://doi.org/10.1134/1.1307232

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