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Convergence problems of Coulomb and multipole sums in crystals


A.V. Nikolaev Institute of Inorganic Chemistry, Siberian Branch of the Russian Academy of Sciences, prosp. akad. Lavrenteva 3, Novosibirsk, 630090, Russian Federation

Different ways of calculating Coulomb and dipole sums over crystal lattices are analyzed comparatively. It is shown that the currently alleged disagreement between various approaches originates in ignoring the requirement for the self-consistency of surface conditions, which are of fundamental importance due to the long-range nature of the bulk interactions that these sums describe. This is especially true of surfaces arising when direct sums for infinite translation-invariant structures are truncated. The charge conditions for actual surfaces being self-consistently adjusted to the bulk state are formally the same as those on the truncation surface, consistent with the concept of the thermodynamic limit for the bulk-state absolute equilibrium and with the fact that the surface energy contribution in this case is, naturally, statistically small compared to the bulk contribution. Two-point multipole expansions are briefly discussed, and the problems associated with the boundary of their convergence circle are pointed out.

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Fulltext is also available at DOI: 10.1070/PU2004v047n10ABEH001720
PACS: 41.20.Cv, 61.50.Lt, 75.10.−b, 81.40.Rs (all)
DOI: 10.1070/PU2004v047n10ABEH001720
URL: https://ufn.ru/en/articles/2004/10/a/
000226730200001
2004PhyU...47..965K
Citation: Kholopov E V "Convergence problems of Coulomb and multipole sums in crystals" Phys. Usp. 47 965–990 (2004)
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Оригинал: Холопов Е В «Проблемы сходимости кулоновских и мультипольных сумм в кристаллах» УФН 174 1033–1060 (2004); DOI: 10.3367/UFNr.0174.200410a.1033

References (354) ↓ Cited by (15) Similar articles (20)

  1. Sherman J Chem. Rev. 11 93 (1932)
  2. Waddington T C, in Advances in Inorganic Chemistry and Radiochemistry Vol. 1 (Eds H J Emeléus, A G Sharpe, New York: Academic Press, 1959) p. 157
  3. Ladd M F C, Lee W H Prog. Solid State Chem. 1 37 (1964)
  4. Tosi M P, in Solid State Physics Vol. 16 (Eds F Seitz, D Turnbull, New York: Academic Press, 1964) p. 1
  5. Baur W H Acta Crystallogr. 19 909 (1965)
  6. Trefry M G, Maslen E N, Spackman M A J. Phys. C: Solid State Phys. 20 19 (1987)
  7. Van Genechten K A, Mortier W J, Geerlings P J. Chem. Phys. 86 5063 (1987)
  8. Sproul G J. Phys. Chem. 98 6699 (1994)
  9. Van Gool W, Piken A G J. Mater. Sci. 4 95, 105 (1969)
  10. Tsukada M J. Phys. Soc. Jpn. 49 1183 (1980)
  11. Wang J C J. Chem. Phys. 73 5786 (1980)
  12. Satoh M, Taki T Phys. Rev. B 23 6711, 6732 (1981)
  13. Spackman M A, Stewart R F, in Chemical Applications of Atomic and Molecular Electrostatic Potentials (Eds P Politzer, D G Truhlar, New York: Plenum Press, 1981) p. 407
  14. Kondo J, Asai Y, Nagai S J. Phys. Soc. Jpn. 57 4334 (1988)
  15. Cole R J, Weightman P J. Phys.: Condens. Matter 9 5609 (1997)
  16. Fuchs K Proc. R. Soc. London Ser. A 151 585 (1935)
  17. Callaway J, Glasser M L Phys. Rev. 112 73 (1958)
  18. Kohn W, Sham L J Phys. Rev. 140 A1133 (1965)
  19. Rudge W E Phys. Rev. 181 1020, 1024 (1969)
  20. Harris F E, in Theoretical Chemistry: Advances and Perspectives Vol. 1 (Eds H Eyring, D Henderson, New York: Academic Press, 1975) p. 147
  21. Dovesi R et al. Phys. Rev. B 28 5781 (1983)
  22. Ángyán J G, Silvi B J. Chem. Phys. 86 6957 (1987)
  23. Ihm J Rep. Prog. Phys. 51 105 (1988)
  24. Alatalo M, Weinert M, Watson R E Phys. Rev. B 60 7680 (1999)
  25. Herzfeld C M, Meijer P H E, in Solid State Physics Vol. 12 (Eds F Seitz, D Turnbull, New York: Academic Press, 1961) p. 1
  26. Hutchings M T, in Solid State Physics Vol. 16 (Eds F Seitz, D Turnbull, New York: Academic Press, 1964) p. 227
  27. Faulkner J S, Wang Y, Stocks G M Phys. Rev. B 55 7492 (1997)
  28. Levine J D, Mark P Phys. Rev. 144 751 (1966)
  29. Nosker R W, Mark P, Levine J D Surf. Sci. 19 291 (1970)
  30. Lang N D, Kohn W Phys. Rev. B 3 1215 (1971)
  31. Tasker P W J. Phys. C: Solid State Phys. 12 4977 (1979)
  32. Lee W W, Choi S-i J. Chem. Phys. 72 6164 (1980)
  33. Heyes D M, van Swol F J. Chem. Phys. 75 5051 (1981)
  34. Watson R E et al. Solid State Commun. 41 151 (1982)
  35. Becker P, Coppens P Acta Crystallogr. A 46 254 (1990)
  36. Wolf D Phys. Rev. Lett. 68 3315 (1992)
  37. Kokko K et al. Phys. Rev. B 52 1536 (1995)
  38. Plihal M, Langreth D C Phys. Rev. B 58 2191 (1998)
  39. Glasser M L, Zucker I J, in Theoretical Chemistry: Advances and Perspectives Vol. 5 (Eds H Eyring, D Henderson, New York: Academic Press, 1980) p. 67
  40. Cohen M H, Keffer F Phys. Rev. 99 1128 (1955); Phys. Rev. B 10 787 (1974)
  41. Appell P Acta Math. IV 313 (1884); VIII 265 (1886)
  42. Jackson J D Classical Electrodynamics (New York: Wiley, 1962) [Translated into Russian (Moscow: Mir, 1965)]
  43. Oh Y et al. Phys. Rev. B 46 4495 (1992)
  44. Fisher M E, Ruelle D J. Math. Phys. 7 260 (1966)
  45. Born M, Bormann E Ann. Phys. (Leipzig) 62 218 (1920)
  46. Wasastjerna J A Soc. Sci. Fennica, Comment. Phys.-Math. 8 (21) 1 (1936)
  47. Onsager L J. Phys. Chem. 43 189 (1939)
  48. Bradburn M Proc. Camb. Philos. Soc. 39 113 (1943)
  49. Devonshire A F Philos. Mag. 40 1040 (1949)
  50. Dick B G (Jr), Overhauser A W Phys. Rev. 112 90 (1958)
  51. Tripathy S K, Hopfinger A J, Taylor P L J. Phys. Chem. 85 1371 (1981)
  52. Frech R Phys. Rev. B 32 6832 (1985)
  53. Epstein P Math. Ann. 56 615 (1903)
  54. Whittaker E T, Watson G N A Course of Modern Analysis (Cambridge: Cambridge Univ. Press, 1952) [Translated into Russian (Moscow: Fizmatgiz, 1962, 1963)]
  55. Madelung E Phys. Z. 19 524 (1918)
  56. Emersleben O Z. Phys. Chem. 204 43 (1955)
  57. Jahnke E, Emde F, Lösch F Tables of Higher Functions 6th ed. (New York: McGraw-Hill, 1960) [Translated into Russian (Moscow: Nauka, 1964)]
  58. Benson G C Can. J. Phys. 34 888 (1956)
  59. Sholl C A Proc. Phys. Soc. London 92 434 (1967)
  60. Madelung E Die mathematischen Hilfsmittel des Physikers 5th ed. (Berlin: Springer-Verlag, 1954) [Translated into Russian (Moscow: Fizmatgiz, 1961)]
  61. Stein E M, Weiss G Introduction to Fourier Analysis on Euclidean Spaces (Princeton, NJ: Princeton Univ. Press, 1971)
  62. Fumi F G, Tosi M P Phys. Rev. 117 1466 (1960)
  63. Massidda V, Hernando J A Physica B+C 101 159 (1980)
  64. McNeil M B J. Phys. C: Solid State Phys. 3 2020 (1970)
  65. Brown R D, Lo B W N J. Phys. C: Solid State Phys. 4 263, 277 (1971)
  66. Mackenzie J K J. Chem. Phys. 26 1769 (1957)
  67. Massidda V Physica B+C 114 327 (1982)
  68. Waddington T C Trans. Faraday Soc. 56 305 (1960)
  69. Benson G C, Van Zeggeren F J. Chem. Phys. 26 1083 (1957)
  70. Shuttleworth R Proc. Phys. Soc. London Ser. A 62 167 (1949)
  71. Van Zeggeren F, Benson G C J. Chem. Phys. 26 1077 (1957)
  72. Bertaut F C.R. Acad. Sci. 246 3447 (1958)
  73. Emersleben O Phys. Z. 24 73, 97 (1923)
  74. Jones J E, Ingham A E Proc. R. Soc. London Ser. A 107 636 (1925)
  75. Crandall R E, Buhler J P J. Phys. A: Math. Gen. 20 5497 (1987)
  76. Ewald P P Ann. Phys. (Leipzig) 64 253 (1921)
  77. Krazer A, Prym F Neue Grundlagen einer Theorie der allgemeinen Thetafunctionen (Leipzig: Teubner, 1892)
  78. Dasgupta S, Sen D, Sengupta S Pramvna — J. Phys. 25 597 (1985)
  79. Van der Hoff B M E, Benson G C Can. J. Phys. 31 1087 (1953)
  80. Pis’mennyi A A Kristallografiya 25 383 (1980) [Sov. Phys. Crystallogr. 25 220 (1980)]
  81. Lekner J Physica A 176 485 (1991)
  82. Mestechkin M M J. Phys.: Condens. Matter 9 157 (1997)
  83. Marshall S L J. Phys.: Condens. Matter 12 4575 (2000); 14 3175 (2002)
  84. Rozenbaum V M Phys. Rev. B 53 6240 (1996)
  85. Hautot A J. Math. Phys. 15 1722 (1974)
  86. Miller A R J. Phys. A: Math. Gen. 28 735 (1995)
  87. Poisson S D J. l’École Polytech. 12 420 (1823)
  88. Misra R D Proc. Camb. Philos. Soc. 36 173 (1940)
  89. Erdélyi A Higher Transcendental Functions (Bateman Project) Vol. 3 (New York: McGraw-Hill, 1955) [Translated into Russian (Moscow: Nauka, 1967)]
  90. Medeiros e Silva J, Mokross B J Phys. Rev. B 21 2972 (1980)
  91. Vohra Y K, Duclos S J, Ruoff A L J. Phys. Chem. Solids 46 515 (1985)
  92. Dienes G J J. Chem. Phys. 16 620 (1948)
  93. Bethe H A Ann. Phys. (Leipzig) 87 55 (1928)
  94. Bertaut F J. Phys. Radium 13 499 (1952)
  95. Boys S F Proc. R. Soc. London Ser. A 200 542 (1950)
  96. McWeeny R Acta Crystallogr. 6 631 (1953)
  97. Birman J Phys. Rev. 97 897 (1955)
  98. Jones R E, Templeton D H J. Chem. Phys. 25 1062 (1956)
  99. Chatterjee S, Ghosh S, Basu A N Phys. Status Solidi B 106 693 (1981)
  100. Fuchizaki K J. Phys. Soc. Jpn. 63 4051 (1994)
  101. Andrés J et al. J. Mol. Struct.: THEOCHEM 254 329 (1992)
  102. Darden T, York D, Pedersen L J. Chem. Phys. 98 10089 (1993)
  103. Petersen H G J. Chem. Phys. 103 3668 (1995)
  104. Templeton D H J. Chem. Phys. 23 1629 (1955)
  105. Weenk J W, Harwig H A J. Phys. Chem. Solids 36 783 (1975)
  106. Bertaut E F J. Phys. Chem. Solids 39 97 (1978)
  107. Herzig P Chem. Phys. Lett. 68 207 (1979)
  108. Heyes D M J. Chem. Phys. 74 1924 (1981)
  109. Crandall R E, Delord J F J. Phys. A: Math. Gen. 20 2279 (1987)
  110. Johnson Q C, Templeton D H J. Chem. Phys. 34 2004 (1961)
  111. Lombardi E, Jansen L Phys. Rev. 151 694 (1966)
  112. Su Z, Coppens P Acta Crystallogr. A 51 27 (1995)
  113. Jenkins H D B, Waddington T C J. Chem. Phys. 56 5323 (1972)
  114. Jenkins H D B Mol. Phys. 30 1843 (1975)
  115. Grzybowski A, Gwózdz E, Bródka A Phys. Rev. B 61 6706 (2000)
  116. Argyriou D N, Howard C J Aust. J. Phys. 45 239 (1992)
  117. Kholopov E V Zh. Strukt. Khimii 43 600 (2002) [J. Struct. Chem. 43 552 (2002)]; cond-mat/0208248
  118. Nijboer B R A, De Wette F W Physica 23 309 (1957)
  119. Leung C H, Song K S Physica B+C 114 323 (1982)
  120. Sugiyama A J. Phys. Soc. Jpn. 53 1624 (1984)
  121. Kozarzewski B Phys. Status Solidi B 129 459 (1985)
  122. Wheeler D R, Newman J Chem. Phys. Lett. 366 537 (2002)
  123. O’Keeffe M, Spence J C H Acta Crystallogr. A 50 33 (1994)
  124. Placzek G, Nijboer B R A, Van Hove L Phys. Rev. 82 392 (1951)
  125. Ewald P P, Juretschke H, in Structure and Properties of Solid Surfaces (Eds R Gomer, C S Smith, Chicago: Univ. of Chicago Press, 1953, 1955) p. 82 (and accompanying discussion)
  126. De Leeuw S W, Perram J W, Smith E R Proc. R. Soc. London Ser. A 373 27 (1980)
  127. Campbell E S J. Phys. Chem. Solids 24 197 (1963)
  128. Morgan J van W J. Phys. C: Solid State Phys. 10 1181 (1977)
  129. Born M, Huang K Dynamical Theory of Crystal Lattices (Oxford: Clarendon Press, 1954) [Translated into Russian (Moscow: IL, 1958)]
  130. Ihm J, Cohen M L Phys. Rev. B 21 3754 (1980)
  131. Kleinman L Phys. Rev. B 24 7412 (1981)
  132. Restori R Chem. Phys. 143 57 (1990)
  133. Bagno P, Doná dalle Rose L F, Toigo F Adv. Phys. 40 685 (1991)
  134. Makov G, Payne M C Phys. Rev. B 51 4014 (1995)
  135. Hall G L Phys. Rev. B 19 3921 (1979)
  136. von Laue M Z. Kristallogr. 103 54 (1940)
  137. Evjen H M Phys. Rev. 39 675 (1932)
  138. Coogan C K Aust. J. Chem. 20 2551 (1967)
  139. Calara J V, Miller J D J. Chem. Phys. 65 843 (1976)
  140. Birman J L J. Phys. Chem. Solids 6 65 (1958)
  141. Frenkel J Z. Phys. 59 649 (1930)
  142. Euwema R N, Surratt G T J. Phys. Chem. Solids 36 67 (1975)
  143. Coldwell-Horsfall R A, Maradudin A A J. Math. Phys. 1 395 (1960)
  144. Hall G L Phys. Rev. B 24 7415 (1981)
  145. Kleinman L Phys. Rev. B 3 3083 (1971)
  146. Picard E C.R. Acad. Sci. 90 601 (1880)
  147. Redlack A, Grindlay J Can. J. Phys. 50 2815 (1972)
  148. Smith P E, Pettitt B M Comput. Phys. Commun. 91 339 (1995)
  149. de Wette F W Phys. Rev. B 21 3751 (1980)
  150. Deem M W, Newsam J M, Sinha S K J. Phys. Chem. 94 8356 (1990)
  151. Rivière J C, in Solid State Surface Science Vol. 1 (Ed. M Green, New York: M. Dekker, 1969) p. 179
  152. Tull V F G Proc. R. Soc. London Ser. A 206 232 (1951)
  153. Wigner E Trans. Faraday Soc. 34 678 (1938)
  154. Care C M, March N H Adv. Phys. 24 101 (1975)
  155. Aguilera-Navarro V C, Baker G A (Jr), de Llano M Phys. Rev. B 32 4502 (1985)
  156. Russier V, Badiali J P Phys. Rev. B 39 13193 (1989)
  157. Hartford J, Hansen L B, Lundqvist B I J. Phys.: Condens. Matter 8 7379 (1996)
  158. Shore H B, Rose J H Phys. Rev. B 59 10485 (1999)
  159. Wigner E Phys. Rev. 46 1002 (1934)
  160. Wyckoff R W G Crystal Structures Vol. 1, 2nd ed. (New York: Interscience Publ., 1963)
  161. Carr W J (Jr) Phys. Rev. 122 1437 (1961)
  162. Foldy L L Phys. Rev. B 17 4889 (1978)
  163. de Wette F W Phys. Rev. 135 A287 (1964)
  164. Hall G L, Rice T R Phys. Rev. B 21 3757 (1980)
  165. Hall G L Phys. Rev. B 24 2881 (1981)
  166. Redlack A, Grindlay J J. Phys. Chem. Solids 36 73 (1975)
  167. Harris F E, Monkhorst H J Chem. Phys. Lett. 4 181 (1969); Phys. Rev. B 2 4400 (1970)
  168. Reining L, Del Sole R Phys. Status Solidi B 162 K37 (1990)
  169. Hund F Z. Phys. 94 11 (1935)
  170. Kihara T, Koba S J. Phys. Soc. Jpn. 7 348 (1952)
  171. Mohazzabi P, Behroozi F Phys. Status Solidi B 144 459 (1987)
  172. Kukhtin V V, Shramko O V Phys. Lett. A 156 257 (1991)
  173. Born M Z. Phys. 7 124 (1921)
  174. Chelikowsky J R, Burdett J K Phys. Rev. Lett. 56 961 (1986)
  175. Hunt A, Pollak M Philos. Mag. B 53 353 (1986)
  176. Natoli V, Martin R M, Ceperley D M Phys. Rev. Lett. 70 1952 (1993)
  177. Wolverton C, Zunger A Phys. Rev. B 51 6876 (1995)
  178. Leslie M, Gillan N J J. Phys. C: Solid State Phys. 18 973 (1985)
  179. Schultz P A Phys. Rev. B 60 1551 (1999)
  180. Ziman J M Electrons and Phonons: the Theory of Transport Phenomena in Solids (Oxford: Clarendon Press, 1960) [Translated into Russian (Moscow: IL, 1962)]
  181. Born M, von Kármán Th Phys. Z. 13 297 (1912)
  182. Naor P Z. Kristallogr. 110 112 (1958)
  183. Jain A K, Verma M P Phys. Status Solidi B 102 705 (1980)
  184. Streitwolf H-W Gruppentheorie in der Festkörperphysik (Leipzig: Akad. Verlagsgesellschaft, 1967) [Translated into English: Group Theory in Solid-State Physics (London: Macdonald and Co., 1971); translated into Russian (Moscow: Mir, 1971)]
  185. Brush S G, Sahlin H L, Teller E J. Chem. Phys. 45 2102 (1966)
  186. Jansoone V M Chem. Phys. 3 78 (1974)
  187. Barandiarán Z, Seijo L, Pueyo L J. Solid State Chem. 55 236 (1984)
  188. He Xiaguang, Huang Meichun, Lin Hong J. Phys.: Condens. Matter 4 5977 (1992)
  189. Stefanovich E V, Truong T N J. Phys. Chem. B 102 3018 (1998)
  190. Kummer J T, Yao Y F Y Can. J. Chem. 45 421 (1967)
  191. Mark P J. Phys. Chem. Solids 29 689 (1968)
  192. Nafari N, Awin A M Phys. Status Solidi A 104 K21 (1987)
  193. Sakamoto Y, Takahasi U J. Chem. Phys. 30 337 (1959)
  194. Gupta K C, Agrawal R M, Rathore R P S Phys. Status Solidi B 118 399 (1983)
  195. Sakamoto Y J. Chem. Phys. 28 164 (1958)
  196. Metropolis N et al. J. Chem. Phys. 21 1087 (1953)
  197. Barker A A Aust. J. Phys. 18 119 (1965)
  198. Sangster M J L, Dixon M Adv. Phys. 25 247 (1976)
  199. Slattery W L, Doolen G D, DeWitt H E Phys. Rev. A 21 2087 (1980)
  200. Danese G et al. Comput. Phys. Commun. 108 211 (1998)
  201. Woodcock L V, Singer K Trans. Faraday Soc. 67 12 (1971)
  202. Maass P et al. Phys. Rev. Lett. 66 52 (1991)
  203. Clarke J H R, Smith W, Woodcock L V J. Chem. Phys. 84 2290 (1986)
  204. Chadi D J, Cohen M L Phys. Rev. B 8 5747 (1973); Chadi D J Phys. Rev. B 16 1746 (1977)
  205. Monkhorst H J, Pack J D Phys. Rev. B 13 5188 (1976); Pack J D, Monkhorst H J Phys. Rev. B 16 1748 (1977)
  206. Cooley J W, Tukey J W Math. Comput. 19 297 (1965)
  207. Alder B J, Wainwright T W J. Chem. Phys. 31 459 (1959)
  208. Essmann U et al. J. Chem. Phys. 103 8577 (1995)
  209. Kawata M, Mikami M Chem. Phys. Lett. 313 261 (1999); 317 515 (2000)
  210. Perram J W, Petersen H G, De Leeuw S W Mol. Phys. 65 875 (1988)
  211. Hummer G Chem. Phys. Lett. 235 297 (1995)
  212. Hockney R W, Goel S P, Eastwood J W Chem. Phys. Lett. 21 589 (1973)
  213. Luty B A, Tironi I G, van Gunsteren W F J. Chem. Phys. 103 3014 (1995)
  214. Pollock E L, Glosli J Comput. Phys. Commun. 95 93 (1996)
  215. Toukmaji A Y, Board J A (Jr) Comput. Phys. Commun. 95 73 (1996)
  216. Payne M C et al. Rev. Mod. Phys. 64 1045 (1992)
  217. Roy S K Can. J. Phys. 32 509 (1954)
  218. Højendahl K Kgl. Danske Videnskab. Selskab. Math.-Fys. Medd. 16 (2) 133 (1938)
  219. Gurney I D C Phys. Rev. 90 317 (1953)
  220. Chaba A N, Pathria R K J. Phys. A: Math. Gen. 9 1411, 1801 (1976)
  221. Weinert M J. Math. Phys. 22 2433 (1981)
  222. Taylor K F J. Comput. Chem. 8 291 (1987)
  223. Grønbech-Jensen N, Hummer G, Beardmore K M Mol. Phys. 92 941 (1997)
  224. Thompson C J, Matsubara T Prog. Theor. Phys. 86 1191 (1991)
  225. Zeiner P, Dirl R, Davies B L Phys. Rev. B 58 7681 (1998)
  226. Blount E I, in Solid State Physics Vol. 13 (Eds F Seitz, D Turnbull, New York: Academic Press, 1962) p. 305
  227. King-Smith R D, Vanderbilt D Phys. Rev. B 47 1651 (1993)
  228. Olszewski S, Dubejko M J. Chem. Phys. 48 5576 (1968)
  229. Gordon R G, Kim Y S J. Chem. Phys. 56 3122 (1972)
  230. Su Z, Coppens P Acta Crystallogr. A 48 188 (1992)
  231. Pendás A M, Costales A, Luaña V J. Phys. Chem. B 102 6937 (1998)
  232. Dahl J P J. Phys. Chem. Solids 26 33 (1965)
  233. Hajj F Y J. Chem. Phys. 56 891 (1972)
  234. Fisenko I G, Kholopov E V Phys. Status Solidi B 173 515 (1992)
  235. Bell D G Rev. Mod. Phys. 26 311 (1954)
  236. Martin R M, Ortiz G Phys. Rev. B 56 1124 (1997)
  237. Frank F C Philos. Mag. Ser. VII 41 1287 (1950)
  238. Wigner E P, Seitz F, in Solid State Physics Vol. 1 (Eds F Seitz, D Turnbull, New York: Academic Press, 1955) p. 97
  239. Hajj F Y J. Chem. Phys. 70 4369 (1979)
  240. Coker H J. Phys. Chem. 87 2512 (1983)
  241. Boeyens J C A, Gafner G Acta Crystallogr. A 25 411 (1969)
  242. Borwein D, Borwein J M, Taylor K F J. Math. Phys. 26 2999 (1985)
  243. Kholopov E V, Preprint No. 2001-02 (Novosibirsk: A V Nikolaev Institute of Inorganic Chemistry, Siberian Branch of the Russ. Acad. of Sciences, 2001)
  244. Bhowmick S, Roy D, Bhattacharya R Chem. Phys. Lett. 148 317 (1988)
  245. Sarkar B, Bhattacharyya K Chem. Phys. Lett. 150 419 (1988)
  246. Baker G A (Jr) Essentials of Padé Approximants (New York: Academic Press, 1975)
  247. Sarkar B, Bhattacharyya K Phys. Rev. B 48 6913 (1993)
  248. Chaba A N, Pathria R K J. Math. Phys. 16 1457 (1975)
  249. Kholopov E V, Preprint No. 2001-01 (Novosibirsk: A V Nikolaev Institute of Inorganic Chemistry, Siberian Branch of the Russ. Acad. of Sciences, 2001); cond-mat/0203530; cond-mat/0207396
  250. Steiger D, Ahlbrandt C, Glaser R J. Phys. Chem. B 102 4257 (1998)
  251. Gonis A, Sowa E C, Sterne P A Phys. Rev. Lett. 66 2207 (1991)
  252. Schadler G H Phys. Rev. B 45 11314 (1992)
  253. Slater J C Phys. Rev. 51 846 (1937)
  254. Herzig P Theor. Chim. Acta 67 323 (1985)
  255. Korringa J Physica 13 392 (1947)
  256. Kohn W, Rostoker N Phys. Rev. 94 1111 (1954)
  257. Hama J J. Phys. Soc. Jpn. 51 116 (1982)
  258. Krasovskii E E, Starrost F, Schattke W Phys. Rev. B 59 10504 (1999)
  259. Knox R S, Gold A Symmetry in the Solid State (New York: W.A. Benjamin, 1964) [Translated into Russian (Moscow: Nauka, 1970)]
  260. Buehler R J, Hirschfelder J O Phys. Rev. 83 628 (1951); 85 149 (1952)
  261. Chiu Y N J. Math. Phys. 5 283 (1964)
  262. Gaunt J A Philos. Trans. R. Soc. London Ser. A 228 151 (1929)
  263. Carlson B C, Rushbrooke G S Proc. Camb. Philos. Soc. 46 626 (1950)
  264. Sack R A J. Math. Phys. 5 245, 252, 260 (1964)
  265. Yasuda H, Yamamoto T Prog. Theor. Phys. 45 1458 (1971)
  266. Zhang X-G et al. Phys. Rev. B 49 13383 (1994)
  267. Gonis A et al. J. Phys.: Condens. Matter 8 7869 (1996)
  268. Vitos L, Kollár J Phys. Rev. B 51 4074 (1995)
  269. Kanamori J et al. J. Phys. Soc. Jpn. 10 93 (1955)
  270. Bersohn R J. Chem. Phys. 29 326 (1958)
  271. Simmons W W, Slichter C P Phys. Rev. 121 1580 (1961)
  272. De Wette F W Phys. Rev. 123 103 (1961)
  273. Cohen M H, Reif F, in Solid State Physics Vol. 5 (Eds F Seitz, D Turnbull, New York: Academic Press, 1957) p. 321
  274. Kaufmann E N, Vianden R J Rev. Mod. Phys. 51 161 (1979)
  275. De Wette F W, Schacher G E Phys. Rev. 137 A78, A92 (1965)
  276. Greengard L, Rokhlin V J. Comput. Phys. 73 325 (1987)
  277. Board J A (Jr) et al. Chem. Phys. Lett. 198 89 (1992)
  278. Williams D E Acta Crystallorg. A 27 452 (1971)
  279. Olives J Phys. Status Solidi B 138 457 (1986)
  280. Smith E R Proc. R. Soc. London Ser. A 375 475 (1981)
  281. Massidda V Physica B 145 124 (1987)
  282. Born M, Misra R D Proc. Camb. Philos. Soc. 36 466 (1940)
  283. Born M, Bradburn M Proc. Camb. Philos. Soc. 39 104 (1943)
  284. Griffiths R B Phys. Rev. 176 655 (1968)
  285. Lucas A A J. Chem. Phys. 48 3156 (1968)
  286. Colpa J H P Physica 56 185, 205 (1971)
  287. Aharony A, Fisher M E Phys. Rev. B 8 3323 (1973)
  288. Purvis C K, Taylor P L Phys. Rev. B 26 4547 (1982)
  289. Fujiki N M, De’Bell K, Geldart D J W Phys. Rev. B 36 8512 (1987)
  290. Lorentz H A Ann. Phys. Chem. (Leipzig) 9 641 (1880)
  291. Möller H Phys. Rev. 50 547 (1936)
  292. Brüesch P, Lietz M J. Phys. Chem. Solids 31 1137 (1970)
  293. Hartman P Z. Kristallogr. 161 259 (1982)
  294. Wind P, Guihéry N, Malrieu J-P Phys. Rev. B 59 2556 (1999)
  295. Adams D J, McDonald I R Mol. Phys. 32 931 (1976)
  296. Van Vleck J H J. Chem. Phys. 5 320 (1937)
  297. Sauer J A Phys. Rev. 57 142 (1940)
  298. Luttinger J M, Tisza L Phys. Rev. 70 954 (1946); 72 257 (1947)
  299. Lorentz H A The Theory of Electrons and Its Applications to the Phenomena of Light and Heat Radiation (Leipzig: B.G. Teubner, 1915) [Translated into Russian (Leningrad – Moscow: ONTI, 1934)
  300. Kittel C Rev. Mod. Phys. 21 541 (1949); Kittel C Usp. Fiz. Nauk 41 452 (1950)
  301. Kittel Ch Introduction to Solid-State Physics (New York: Wiley, 1956) [Translated into Russian (Moscow: Fizmatgiz, 1963)]
  302. Nijboer B R A, De Wette F W Physica 24 422 (1958)
  303. Shaked H Phys. Lett. A 50 385 (1974)
  304. Kornfeld H Z. Phys. 22 27 (1924)
  305. Toupin R A, Lax M J. Chem. Phys. 27 458 (1957)
  306. Holmes L M, Als-Nielsen J, Guggenheim H J Phys. Rev. B 12 180 (1975)
  307. Cohen M H, Keffer F Phys. Rev. 99 1135 (1955)
  308. Benson H, Mills D L Phys. Rev. 178 839 (1969)
  309. Hummel M, Schwabl F, Pich C Phys. Rev. B 63 094425 (2001)
  310. Lax M Phys. Rev. 97 629 (1955)
  311. Vaks V G, Larkin A I, Pikin S A Zh. Eksp. Teor. Fiz. 51 361 (1966) [Sov. Phys. JETP 24 240 (1967)]
  312. Lo C K, Wan J T K, Yu K W J. Phys.: Condens. Matter 13 1315 (2001)
  313. Ledermann W Proc. R. Soc. London 182 362 (1944)
  314. Smith E R, Perram J W Phys. Lett. A 50 294 (1974)
  315. Pirenne J Helv. Phys. Acta 22 479 (1949)
  316. Fernández J F, Alonso J J Phys. Rev. B 62 53 (2000)
  317. Onsager L J. Am. Chem. Soc. 58 1486 (1936)
  318. Hiley B J, Joyce G S Proc. Phys. Soc. London 85 493 (1965)
  319. Landau L D, Lifshitz E M Elektrodinamika Sploshnykh Sred (Electrodynamics of Continuous Media, Moscow: Nauka, 1982) [Translated into English (Oxford: Pergamon Press, 1984)]
  320. Stoner E C Philos. Mag. 36 803 (1945)
  321. Osborn J A Phys. Rev. 67 351 (1945)
  322. Rosenberg R, Lax M J. Chem. Phys. 21 424 (1953)
  323. Levy P M Phys. Rev. 170 595 (1968)
  324. Horner H Phys. Rev. 172 535 (1968)
  325. Smith E R Proc. R. Soc. London Ser. A 381 241 (1982)
  326. Aizu K Rev. Mod. Phys. 34 550 (1962)
  327. Larmor J Proc. R. Soc. London Ser. A 99 1 (1921)
  328. Cady W G Piezoelectricity (New York: McGraw-Hill, 1946)
  329. Martin R M Phys. Rev. B 6 4874 (1972)
  330. Ginzburg V L Zh. Eksp. Teor. Fiz. 15 739 (1945); Ginzburg V L J. Phys. USSR 10 107 (1946)
  331. Kittel C Phys. Rev. 82 965 (1951)
  332. Lines M E, Glass M Principles and Applications of Ferroelectrics and Related Materials (Oxford: Clarendon Press, 1977) [Translated into Russian (Moscow: Mir, 1981)]
  333. Lundager-Madsen H E J. Cryst. Growth 112 458 (1991)
  334. Kallin C, Halperin B I Phys. Rev. B 29 2175 (1984)
  335. Tagantsev A K Usp. Fiz. Nauk 152 423 (1987) [Sov. Phys. Usp. 30 588 (1987)]
  336. White et al. Phys. Rev. Lett. 71 3553 (1993)
  337. Kholopov E V, cond-mat/0405458
  338. Niemeijer Th, Blõte H W J Physica 67 125 (1973)
  339. Weiss P J. Phys. (Paris) 6 661 (1907)
  340. Bouchaud J P, Zérah P G Phys. Rev. B 47 9095 (1993)
  341. Niemeijer Th Physica 57 281 (1972)
  342. Palmer S E, Chalker J T Phys. Rev. B 62 488 (2000)
  343. Smith J R Phys. Rev. 181 522 (1969)
  344. Alves J L A, Hebenstreit J, Scheffler M Phys. Rev. B 44 6188 (1991)
  345. Parry D E Surf. Sci. 49 433 (1975); 54 195 (1976)
  346. Phillips J C Rev. Mod. Phys. 42 317 (1970)
  347. Fall C J, Binggeli N, Baldereschi A Phys. Rev. B 61 8489 (2000)
  348. Fiorentini V, Methfessel M J. Phys.: Condens. Matter 8 6525 (1996); 10 895 (1998)
  349. Yokogawa K, Nomura Y, Iwata G J. Phys. C: Solid State Phys. 18 2033 (1985)
  350. Boettger J C Phys. Rev. B 49 16798 (1994)
  351. Boettger J C et al. J. Phys.: Condens. Matter 10 893 (1998)
  352. Kholopov E V, cond-mat/0307298 (v2); Comput. Mater. Sci. (to be published)
  353. Vasil’ev B V, Kaganov M I, Lyuboshits V L Usp. Fiz. Nauk 164 375 (1994) [Phys. Usp. 37 353 (1994)]
  354. Kholopov E V, in Progress in Mathematical Physics Research (Ed. F Columbus, New York: Nova Science Publ., to be published)
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