Collect. Czech. Chem. Commun. 2005, 70, 1225-1271
https://doi.org/10.1135/cccc20051225

The Spherical Tensor Gradient Operator

Ernst Joachim Weniger

Institut für Physikalische und Theoretische Chemie, Universität Regensburg, D-93040 Regensburg, Germany

References

1. Kline M.: Mathematical Thought from Ancient to Modern Times. Oxford University Press, Oxford 1972.
2. Watson G. N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge 1966.
3. Weniger E. J., Steinborn E. O.: J. Chem. Phys. 1983, 78, 6121. <https://doi.org/10.1063/1.444574>
4. Bracken P., Bartlett R. J.: Int. J. Quantum Chem. 1997, 62, 557. <https://doi.org/10.1002/(SICI)1097-461X(1997)62:6<557::AID-QUA1>3.0.CO;2-V>
5. Hobson E. W.: Proc. Math. Soc. London 1892, 24, 54.
6. Weniger E. J.: Ph.D. Thesis. Universität Regensburg, Regensburg 1982. A short abstract of this thesis was published in Zentralbl. Math. 1984, 523, 444.
7. Bracher C., Kramer T., Kleber M.: Phys. Rev. A 2003, 67, 043601. <https://doi.org/10.1103/PhysRevA.67.043601>
8. Bott E., Methfessel M., Krabs W., Schmidt P. C.: J. Math. Phys. 1998, 39, 3393. <https://doi.org/10.1063/1.532437>
9. Methfessel M., van Schilfgaarde M., Casali R. A.: Lect. Notes Phys. 1999, 535, 114. <https://doi.org/10.1007/3-540-46437-9_3>
10. Grotendorst J., Steinborn E. O.: J. Comput. Phys. 1985, 61, 195. <https://doi.org/10.1016/0021-9991(85)90082-8>
11. Grotendorst J., Steinborn E. O.: Phys. Rev. A 1988, 38, 3857. <https://doi.org/10.1103/PhysRevA.38.3857>
12. Niukkanen A. W.: J. Math. Phys. 1983, 24, 1989. <https://doi.org/10.1063/1.525957>
13. Niukkanen A. W.: J. Math. Phys. 1984, 25, 698. <https://doi.org/10.1063/1.526178>
14. Niukkanen A. W.: J. Math. Phys. 1985, 26, 1540. <https://doi.org/10.1063/1.526914>
15. Novosadov B. K.: Int. J. Quantum Chem. 1983, 24, 1. <https://doi.org/10.1002/qua.560240102>
16. Novosadov B. K. in: Theory and Methods of Calculation of Molecular Spectra (I. A. Gribov and W. J. Orville-Thomas, Eds), pp. 596–625. Wiley, Chichester 1988.
17. Novosadov B. K.: J. Struct. Chem. 2001, 42, 355. <https://doi.org/10.1023/A:1012444517593>
18. Novosadov B. K.: J. Struct. Chem. 2002, 43, 383. <https://doi.org/10.1023/A:1020372612789>
19. Novosadov B. K.: J. Struct. Chem. 2002, 43, 390. <https://doi.org/10.1023/A:1020324729627>
20. Novosadov B. K.: J. Mol. Struct. (THEOCHEM) 2003, 664–665, 55. <https://doi.org/10.1016/S0166-1280(03)00559-1>
21. Tai H.: Chin. J. Phys. 1994, 32, 847.
22. Steinborn E. O., Weniger E. J.: Theor. Chim. Acta 1992, 83, 105. <https://doi.org/10.1007/BF01113245>
23. Weniger E. J.: Int. J. Quantum Chem. 2000, 76, 280. <https://doi.org/10.1002/(SICI)1097-461X(2000)76:2<280::AID-QUA16>3.0.CO;2-C>
24. Weniger E. J.: Int. J. Quantum Chem. 2002, 90, 92. <https://doi.org/10.1002/qua.948>
25. Weniger E. J., Grotendorst J., Steinborn E. O.: Phys. Rev. A 1986, 33, 3688. <https://doi.org/10.1103/PhysRevA.33.3688>
26. Weniger E. J., Steinborn E. O.: J. Math. Phys. 1983, 24, 2553. <https://doi.org/10.1063/1.525649>
27. Weniger E. J., Steinborn E. O.: J. Math. Phys. 1985, 26, 664. <https://doi.org/10.1063/1.526604>
28. Weniger E. J., Steinborn E. O.: J. Math. Phys. 1989, 30, 774. <https://doi.org/10.1063/1.528396>
29. Dunlap B. I.: Phys. Rev. A 1990, 42, 1127. <https://doi.org/10.1103/PhysRevA.42.1127>
30. Dunlap B. I.: Int. J. Quantum Chem. 2001, 81, 373. <https://doi.org/10.1002/1097-461X(2001)81:6<373::AID-QUA1007>3.0.CO;2-3>
31. Dunlap B. I.: Phys. Rev. A 2002, 66, 032502. <https://doi.org/10.1103/PhysRevA.66.032502>
32. Dunlap B. I.: J. Chem. Phys. 2003, 118, 1036. <https://doi.org/10.1063/1.1528935>
33. Dunlap B. I.: Comput. Phys. Commun. 2005, 165, 18. <https://doi.org/10.1016/j.cpc.2004.09.002>
34. Biedenharn L. C., Louck J. D.: Angular Momentum in Quantum Physics. Addison–Wesley, Reading (MS) 1981.
35. Bayman B. F.: J. Math. Phys. 1978, 19, 2558. <https://doi.org/10.1063/1.523640>
36. Santos F. D.: Nucl. Phys. A 1973, 212, 341. <https://doi.org/10.1016/0375-9474(73)90568-X>
37. Stuart S. N.: J. Aust. Math. Soc. B 1981, 22, 368. <https://doi.org/10.1017/S0334270000002708>
38. Rashid M. A.: J. Math. Phys. 1986, 27, 549. <https://doi.org/10.1063/1.527205>
39. Hobson E. W.: The Theory of Spherical and Ellipsoidal Harmonics. Chelsea, New York 1965. Originally published by Cambridge University Press, Cambridge 1931.
40. Avery J.: Hyperspherical Harmonics – Applications in Quantum Theory. Kluwer, Dordrecht 1989.
41. Avery J.: Hyperspherical Harmonics and Generalized Sturmians. Kluwer, Dordrecht 2000.
42. Judd B. R.: Angular Momentum Theory for Diatomic Molecules. Academic Press, New York 1975.
43. Weniger E. J.: J. Math. Phys. 1985, 26, 276. <https://doi.org/10.1063/1.526970>
44. Arakane F., Matsuoka O.: Int. J. Quantum Chem. 1998, 66, 273. <https://doi.org/10.1002/(SICI)1097-461X(1998)66:4<273::AID-QUA2>3.0.CO;2-S>
45. Chow Chiu L.-Y., Moharerrzadeh M.: J. Chem. Phys. 1998, 108, 5230. <https://doi.org/10.1063/1.475960>
46. Chow Chiu L.-Y., Moharerrzadeh M.: Int. J. Quantum Chem. 1999, 73, 265. <https://doi.org/10.1002/(SICI)1097-461X(1999)73:3<265::AID-QUA1>3.0.CO;2-7>
47. Chow Chiu L.-Y., Moharerrzadeh M.: J. Mol. Struct. (THEOCHEM) 2001, 536, 263. <https://doi.org/10.1016/S0166-1280(00)00704-1>
48. Fieck G.: J. Phys. B 1979, 12, 1063. <https://doi.org/10.1088/0022-3700/12/7/010>
49. Fieck G.: Theor. Chim. Acta 1980, 54, 323. <https://doi.org/10.1007/BF00552466>
50. Fortunelli A., Carrravetta V.: Phys. Rev. A 1992, 45, 4438. <https://doi.org/10.1103/PhysRevA.45.4438>
51. Fortunelli A., Salvetti O.: Int. J. Quantum Chem. 1993, 48, 257. <https://doi.org/10.1002/qua.560480407>
52. Fujimura N., Matsuoka O.: Int. J. Quantum Chem. 1992, 42, 751. <https://doi.org/10.1002/qua.560420415>
53. Hu A., Staufer M., Birkenheuer U., Igoshine V., Rösch N.: J. Quantum Chem. 2000, 79, 209. <https://doi.org/10.1002/1097-461X(2000)79:4<209::AID-QUA2>3.0.CO;2-J>
54. Ishida K.: J. Chem. Phys. 1998, 109, 881. <https://doi.org/10.1063/1.476628>
55. Ishida K.: J. Chem. Phys. 1999, 111, 4913. <https://doi.org/10.1063/1.479785>
56. Ishida K.: J. Chem. Phys. 2000, 113, 7818. <https://doi.org/10.1063/1.1316013>
57. Ishida K.: J. Comput. Chem. 2002, 23, 378. <https://doi.org/10.1002/jcc.10016>
58. Ishida K.: J. Comput. Chem. 2003, 24, 1874. <https://doi.org/10.1002/jcc..10348>
59. Kuang J., Lin C. D.: J. Phys. B 1997, 30, 2529. <https://doi.org/10.1088/0953-4075/30/11/007>
60. Kuang J., Lin C. D.: J. Phys. B 1997, 30, 2549. <https://doi.org/10.1088/0953-4075/30/11/008>
61. Maretis D. K.: J. Chem. Phys. 1979, 71, 917. <https://doi.org/10.1063/1.438381>
62. Matsuoka O.: J. Chem. Phys. 1992, 92, 4364. <https://doi.org/10.1063/1.457744>
63. Matsuoka O.: Can. J. Chem. 1992, 70, 388. <https://doi.org/10.1139/v92-055>
64. Matsuoka O.: J. Chem. Phys. 1998, 108, 1063. <https://doi.org/10.1063/1.475468>
65. Matsuoka O.: J. Mol. Struct. (THEOCHEM) 1998, 451, 35. <https://doi.org/10.1016/S0166-1280(98)00157-2>
66. Matsuoka O.: Mol. Phys. 2003, 101, 33. <https://doi.org/10.1080/00268970210158704>
67. Saunders V. R. in: Methods in Computational Molecular Physics (G. Diercksen and S. Wilson, Eds), pp. 1–36. Reidel, Dordrecht 1983.
68. Prosser F. P., Blanchard C. H.: J. Chem. Phys. 1962, 36, 1112. <https://doi.org/10.1063/1.1732673>
69. Geller M.: J. Chem. Phys. 1963, 39, 853. <https://doi.org/10.1063/1.1734348>
70. Reed M., Simon B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness. Academic Press, New York 1975.
71. Steinborn O.: Chem. Phys. Lett. 1969, 3, 671. <https://doi.org/10.1016/0009-2614(69)87006-5>
72. Shavitt I. in: Methods in Computational Physics (B. Alder, S. Fernbach and M. Rotenberg, Eds), Vol 2, pp. 1–45. Academic Press, New York 1963.
73. Filter E., Steinborn E. O.: Phys. Rev. A 1978, 18, 1. <https://doi.org/10.1103/PhysRevA.18.1>
74. Homeier H. H. H.: Ph.D. Thesis, Universität Regensburg 1990. Published by S. Roderer Verlag, Regensburg 1990.
75. Magnus W., Oberhettinger F., Soni R. P.: Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, New York 1966.
76. Steinborn E. O., Filter E.: Theor. Chim. Acta 1975, 38, 273. <https://doi.org/10.1007/BF00963467>
77. Weniger E. J., Steinborn E. O.: Phys. Rev. A 1983, 28, 2026. <https://doi.org/10.1103/PhysRevA.28.2026>
78. Grosswald E.: Bessel Polynomials. Springer, Berlin 1978.
79. Baker G. A., Jr., Graves-Morris P.: Padé Approximants, 2nd ed. Cambridge University Press, Cambridge 1996.
80. Padé H.: Ann. Sci. Ec. Norm. Sup. 1892, 9, 1.
81. Niukkanen A. W.: Int. J. Quantum Chem. 1984, 25, 941. <https://doi.org/10.1002/qua.560250603>
82. Filter E., Steinborn E. O.: J. Math. Phys. 1978, 19, 79. <https://doi.org/10.1063/1.523517>
83. Steinborn E. O., Homeier H. H. H., Fernández Rico J., Ema I., López R., Ramírez G.: J. Mol. Struct. (THEOCHEM) 1999, 490, 201. <https://doi.org/10.1016/S0166-1280(99)00099-8>
84. Steinborn E. O., Homeier H. H. H., Ema I., López R., Ramírez G.: Int. J. Quantum Chem. 2000, 76, 244. <https://doi.org/10.1002/(SICI)1097-461X(2000)76:2<244::AID-QUA13>3.0.CO;2-T>
85. Sidi A.: J. Inst. Math. Appl. 1980, 26, 9. <https://doi.org/10.1093/imamat/26.1.1>
86. Levin D., Sidi A.: Appl. Math. Comput. 1981, 9, 175. <https://doi.org/10.1016/0096-3003(81)90028-X>
87. Berlu L., Safouhi H.: J. Phys. A 2003, 36, 11267. <https://doi.org/10.1088/0305-4470/36/44/007>
88. Berlu L., Safouhi H.: J. Phys. A 2003, 36, 11791. <https://doi.org/10.1088/0305-4470/36/47/007>
89. Berlu L., Safouhi H.: J. Phys. A 2004, 37, 3393. <https://doi.org/10.1088/0305-4470/37/10/006>
90. Berlu L., Safouhi H., Hoggan P.: Int. J. Quantum Chem. 2004, 99, 221. <https://doi.org/10.1002/qua.10853>
91. Safouhi H.: J. Comput. Phys. 2000, 165, 473. <https://doi.org/10.1006/jcph.2000.6621>
92. Safouhi H.: J. Phys. A 2001, 34, 881. <https://doi.org/10.1088/0305-4470/34/4/314>
93. Safouhi H.: J. Phys. A 2001, 34, 2801. <https://doi.org/10.1088/0305-4470/34/13/311>
94. Safouhi H.: J. Math. Chem. 2001, 29, 213. <https://doi.org/10.1023/A:1010994517423>
95. Safouhi H.: J. Comput. Phys. 2002, 176, 1. <https://doi.org/10.1006/jcph.2001.6925>
96. Safouhi H.: J. Phys. A 2002, 35, 9685. <https://doi.org/10.1088/0305-4470/35/45/314>
97. Safouhi H., Hoggan P. E.: Int. J. Quantum Chem. 2001, 84, 580. <https://doi.org/10.1002/qua.1412>
98. Safouhi H., Hoggan P. E.: Int. J. Quantum Chem. 2002, 90, 119. <https://doi.org/10.1002/qua.962>
99. Safouhi H., Hoggan P. E.: Mol. Phys. 2003, 101, 19. <https://doi.org/10.1080/0026897021000026809>
100. Dirac P. A. M.: The Principles of Quantum Mechanics, 4th ed. Clarendon Press, Oxford 1958.
101. Schwartz L.: Théorie des Distributions, 2nd ed. Hermann, Paris 1966.
102. Jackson J. D.: Classical Electrodynamics, 2nd ed. Wiley, New York 1975.
103. Rowe E. G. P.: J. Math. Phys. 1978, 19, 1962. <https://doi.org/10.1063/1.523927>
104. Gel’fand I. M., Shilov G. E.: Generalized Functions. I. Properties and Operations. Academic Press, New York 1964.
105. Yukawa H.: Proc. Phys.-Math. Soc. Jpn. 1935, 17, 48.
106. Arfken G. B.: Mathematical Methods for Physicists, 3rd ed. Academic Press, Orlando 1985.
107. Jones M. N.: Spherical Harmonics and Tensors for Classical Field Theory. Wiley, New York 1985.
108. Stone A. J.: The Theory of Intermolecular Forces. Clarendon Press, Oxford 1996.
109. Weniger E. J.: M.S. Thesis. Universität Regensburg, Regensburg 1977.
110. Steinborn E. O., Weniger E. J.: Int. J. Quantum Chem. Symp. 1977, 11, 509.
111. Homeier H. H. H., Weniger E. J., Steinborn E. O.: Int. J. Quantum Chem. 1992, 44, 405. <https://doi.org/10.1002/qua.560440308>
112. Filter E., Steinborn E. O.: J. Math. Phys. 1980, 21, 2725. <https://doi.org/10.1063/1.524390>
113. Weniger E. J., Steinborn E. O.: Phys. Rev. A 1984, 29, 2268. <https://doi.org/10.1103/PhysRevA.29.2268>
114. Varshalovich D. A., Moskalev A. N., Khersonskii V. K.: Quantum Theory of Angular Momentum. World Scientific, Singapore 1988.
115. Knopp K.: Theorie und Anwendung der unendlichen Reihen. Springer, Berlin 1964.
116. Grotendorst J., Weniger E. J., Steinborn E. O.: Phys. Rev. A 1986, 33, 3706. <https://doi.org/10.1103/PhysRevA.33.3706>
117. Steinborn E. O., Weniger E. J.: J. Mol. Struct. (THEOCHEM) 1990, 210, 71. <https://doi.org/10.1016/0166-1280(90)80026-K>
118. Weniger E. J., Grotendorst J., Steinborn E. O.: Int. J. Quantum Chem., Quantum Chem. Symp. 1986, 19, 181.
119. Weniger E. J., Steinborn E. O.: J. Chem. Phys. 1987, 87, 3709. <https://doi.org/10.1063/1.452975>
120. Weniger E. J., Steinborn E. O.: Theor. Chim. Acta 1988, 73, 323. <https://doi.org/10.1007/BF00527739>
121. Weniger E. J.: Comput. Phys. Rep. 1989, 10, 189. Los Alamos Preprint math-ph/0306302, http://arXiv.org. <https://doi.org/10.1016/0167-7977(89)90011-7>
122. Bender C. M., Weniger E. J.: J. Math. Phys. 2001, 42, 2167. <https://doi.org/10.1063/1.1362287>
123. Weniger E. J.: J. Comput. Appl. Math. 2000, 122, 329. Reprinted in: Numerical Analysis 2000 (C. Brezinski, Ed.), Vol. 2, pp. 329–356. Elsevier, Amsterdam 2000. <https://doi.org/10.1016/S0377-0427(00)00363-0>
124. Weniger E. J.: Numer. Algor. 2003, 33, 499. <https://doi.org/10.1023/A:1025517617217>
125. Weniger E. J.: J. Math. Phys. 2004, 45, 1209. <https://doi.org/10.1063/1.1643787>
126. Weniger E. J., Kirtman B.: Comput. Math. Appl. 2003, 45, 189. <https://doi.org/10.1016/S0898-1221(03)80014-7>
127. Condon E. U., Shortley G. H.: The Theory of Atomic Spectra. Cambridge University Press, Cambridge 1970.
128. Condon E. U., Odabasi H.: Atomic Structure. Cambridge University Press, Cambridge 1980.
129. Steinborn E. O., Ruedenberg K.: Adv. Quantum Chem. 1973, 7, 1. <https://doi.org/10.1016/S0065-3276(08)60558-4>
130. Normand J.-M.: A Lie Group: Rotations in Quantum Mechanics. North-Holland, Amsterdam 1980.
131. Gaunt J. A.: Philos. Trans. R. Soc. London, Ser. A 1929, 228, 151. <https://doi.org/10.1098/rsta.1929.0004>
132. Weniger E. J., Steinborn E. O.: Comput. Phys. Commun. 1982, 25, 149. <https://doi.org/10.1016/0010-4655(82)90031-5>
133. Schulten K., Gordon R. G.: J. Math. Phys. 1975, 16, 1961. <https://doi.org/10.1063/1.522426>
134. Schulten K., Gordon R. G.: Comput. Phys. Commun. 1976, 11, 269. <https://doi.org/10.1016/0010-4655(76)90058-8>
135. Homeier H. H. H., Steinborn E. O.: J. Mol. Struct. (THEOCHEM) 1996, 368, 31. <https://doi.org/10.1016/S0166-1280(96)90531-X>
136. Xu Y.-L.: Math. Comput. 1996, 65, 1601. <https://doi.org/10.1090/S0025-5718-96-00774-0>
137. Xu Y.-L.: J. Comput. Appl. Math. 1997, 85, 53. <https://doi.org/10.1016/S0377-0427(97)00128-3>
138. Xu Y.-L.: J. Comput. Phys. 1998, 139, 137. <https://doi.org/10.1006/jcph.1997.5867>
139. Sébilleau D.: J. Phys. A 1998, 31, 7157. <https://doi.org/10.1088/0305-4470/31/34/017>
140. Mavromatis H. A., Alassar R. S.: Appl. Math. Lett. 1999, 12, 101. <https://doi.org/10.1016/S0893-9659(98)00180-3>