How ambiguous is the local kinetic energy?

Journal of Physical Chemistry A

Volumn 114, Issue 33, 2010, Pages 8884-8895

  Anderson, James S M ^a   Ayers, Paul W ^a   Hernandez, Juan I Rodriguez ^a,b  

^a MCMASTER UNIVERSITY   (Canada)

^b INSTITUTO POLITÉCNICO NACIONAL   (Mexico)

Author keywords

[No Author keywords available]

Indexed keywords

CHEMICAL ANALYSIS; CHEMICAL BONDS; DENSITY FUNCTIONAL THEORY; KINETIC ENERGY; MOLECULES; TENSORS;

ATOMS IN MOLECULES; CHEMICAL BONDINGS; ENERGY FUNCTIONS; QUASIPROBABILITY DISTRIBUTIONS; STRESS TENSORS; VIRIAL THEOREMS;

KINETICS;

EID: 77955879914     PISSN: 10895639     EISSN: 15205215     Source Type: Journal    
DOI: 10.1021/jp1029745     Document Type: Article

References (144)

1. Hellmann, H. Z. Phys. 1933, 35, 180
2. Ruedenberg, K. Rev. Mod. Phys. 1962, 34, 326
3. Goddard, W. A.; Wilson, C. W. Theor. Chim. Acta 1972, 26, 195
4. Goddard, W. A.; Wilson, C. W. Theor. Chim. Acta 1972, 26, 211
5. Kutzelnigg, W. Angew. Chem., Int. Ed. 1973, 12, 546
6. Bader, R. F. W.; Preston, H. J. T. Int. J. Quantum Chem. 1969, 3, 327
7. Feinberg, M. J.; Ruedenberg, K. J. Chem. Phys. 1971, 55, 5804
8. Feinberg, M. J.; Ruedenberg, K. J. Chem. Phys. 1971, 54, 1495
9. Feinberg, M. J.; Ruedenberg, K.; Mehler, E. L. Adv. Quantum Chem. 1970, 5, 27
10. Ghosh, S. K.; Berkowitz, M.; Parr, R. G. Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 8028
11. Lombard, R. J.; Mas, D.; Moszkowski, S. A. J. Phys. G: Nucl. Part. Phys. 1991, 17, 455
12. Lombard, R. J.; Moszkowski, S. A. Nuovo Cimento Della Societa Italiana Di Fisica B-General Physics Relativity Astronomy and Mathematical Physics and Methods 1994, 109, 1291
13. Equations 1 and 2 are related by partial integration, subject only to the vanishing of a surface term. It is intuitively clear that there are many other forms for the local kinetic energy that also reduce to eq 1 by partial integration, but which have more complicated forms for the vanishing surface term.
14. Ziff, R. M.; Uhlenbeck, G. E.; Kac, M. Physics Reports 1977, 32, 169
15. Tal, Y.; Bader, R. F. W. Int. J. Quantum Chem. 1978, 153
16. Bader, R. F. W.; Nguyendang, T. T. Adv. Quantum Chem. 1981, 14, 63
17. Yang, Z. Z.; Liu, S. B.; Wang, Y. A. Chem. Phys. Lett. 1996, 258, 30
18. Yang, Z. Z.; Wang, Y. A.; Liu, S. B. Sci. China, Ser. B: Chem. 1998, 41, 174
19. Ayers, P. W.; Parr, R. G.; Nagy, A. Int. J. Quantum Chem. 2002, 90, 309
20. Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Clarendon: Oxford, U.K., 1990.
21. Bader, R. F. W.; Beddall, P. M. J. Chem. Phys. 1972, 56, 3320
22. Cohen, L. J. Chem. Phys. 1979, 70, 788
23. Real, C. C.; Muga, J. G.; Brouard, S. Am. J. Phys. 1997, 65, 157
24. Muga, J. G.; Seidel, D.; Hegerfeldt, G. C. J. Chem. Phys. 2005, 122, 154106
25. Ruedenberg, K.; Schmidt, M. W. J. Phys. Chem. A 2009, 113, 1954
26. Tachibana, A. J. Chem. Phys. 2001, 115, 3497
27. Szarek, P.; Sueda, Y.; Tachibana, A. J. Chem. Phys. 2008, 129 094102
28. Bitter, T.; Ruedenberg, K.; Schwarz, W. H. E. J. Comput. Chem. 2007, 28, 411
29. Ruedenberg, K.; Schmidt, M. W. J. Comput. Chem. 2007, 28, 391
30. Wang, S. G.; Qiu, Y. X.; Schwarz, W. H. E. Chem.-Eur. J. 2009, 15, 6032
31. Wang, Shu-Guang; Qiu, Yi-Xiang; Eugen Schwarz, W. H. Manuscript submitted for publication.
32. Sears, S. B.; Parr, R. G.; Dinur, U. Isr. J. Chem. 1980, 19, 165
33. Nalewajski, R. F. J. Math. Chem. 2010, 47, 667
34. Becke, A. D.; Edgecombe, K. E. J. Chem. Phys. 1990, 92, 5397
35. Savin, A.; Becke, A. D.; Flad, J.; Nesper, R.; Preuss, H.; Vonschnering, H. G. Angew. Chem. 1991, 30, 409
36. Savin, A.; Nesper, R.; Wengert, S.; Fassler, T. F. Angew. Chem. 1997, 36, 1809
37. Silvi, B.; Savin, A. Nature 1994, 371, 683
38. Ayers, P. W. J. Chem. Sci. 2005, 117, 441
39. Schmider, H. L.; Becke, A. D. J. Mol. Struct. (THEOCHEM) 2000, 527, 51
40. Jacobsen, H. Canadian Journal of Chemistry-Revue Canadienne De Chimie 2008, 86, 695
41. Jacobsen, H. J. Comput. Chem. 2009, 30, 1093
42. Jacobsen, H. Dalton Trans. 2009, 4252
43. Jacobsen, H. Canadian Journal of Chemistry-Revue Canadienne De Chimie 2009, 87, 965
44. Chattaraj, P. K.; Chamorro, E.; Fuentealba, P. Chem. Phys. Lett. 1999, 314, 114
45. Wigner, E. Phys. Rev. 1932, 40, 749
46. Wigner, E. P.; Yourgrau, W.; van der Merwe, A. Perspectives in Quantum Theory; MIT: Cambridge, MA, 1971; p 25.
47. Rozendaal, A.; Baerends, E. J. Chem. Phys. 1985, 95, 57
48. Schmider, H. L.; Becke, A. D. J. Chem. Phys. 2002, 116, 3184
49. Bohorquez, H. J.; Boyd, R. J. J. Chem. Phys. 2008, 129 024110
50. Pauli, W. Handbuch der physik; Springer: Berlin, 1958.
51. Schrödinger, E. Ann. Phys. 1927, 387, 265
52. Epstein, S. T. J. Chem. Phys. 1975, 63, 3573
53. Holas, A.; March, N. H. Phys. Rev. A 1995, 51, 2040
54. Bader, R. F. W. J. Chem. Phys. 1980, 73, 2871
55. Tao, J. M.; Vignale, G.; Tokatly, I. V. Phys. Rev. Lett. 2008, 100, 206405
56. Grafenstein, J.; Ziesche, P. Phys. Rev. B 1996, 53, 7143
57. Morante, S.; Rossi, G. C.; Testa, M. J. Chem. Phys. 2006, 125 034101
58. Bader, R. F. W.; Essen, H. J. Chem. Phys. 1984, 80, 1943
59. Hernandez-Trujillo, J.; Cortes-Guzman, F.; Fang, D. C.; Bader, R. F. W. Faraday Discuss. 2007, 135, 79
60. Bader, R. F. W.; Austen, M. A. J. Chem. Phys. 1997, 107, 4271
61. Bader, R. F. W.; Fang, D. C. J. Chem. Theory Comput. 2005, 1, 403
62. Deb, B. M.; Bamzai, A. S. Mol. Phys. 1978, 35, 1349
63. Deb, B. M.; Bamzai, A. S. Mol. Phys. 1979, 38, 2069
64. Deb, B. M.; Ghosh, S. K. J. Phys. B: At. Mol. Opt. Phys. 1979, 12, 3857
65. Bartolotti, L. J.; Parr, R. G. J. Chem. Phys. 1980, 72, 1593
66. Tachibana, A. Int. J. Quantum Chem. 2004, 100, 981
67. Tachibana, A. J. Mol. Model. 2005, 11, 301
68. Szarek, P.; Tachibana, A. J. Mol. Model. 2007, 13, 651
69. Ichikawa, K.; Tachibana, A. Phys. Rev. A 2009, 80 062507
70. Jenkins, S.; Heggie, M. I. J. Phys.: Condens. Matter 2000, 12, 10325
71. Jenkins, S.; Morrison, I. J. Phys.: Condens. Matter 2001, 13, 9207
72. Jenkins, S.; Kirk, S. R.; Cote, A. S.; Ross, D. K.; Morrison, I. Can. J. Phys. 2003, 81, 225
73. Ayers, P. W.; Jenkins, S. J. Chem. Phys. 2009, 130, 154104
74. Ziesche, P.; Lehmann, D. Phys. Status Solidi B 1987, 139, 467
75. Tokatly, I. V. Phys. Rev. B 2005, 71, 165104
76. Tao, J. M.; Vignale, G.; Tokatly, I. V. Phys. Rev. B 2007, 76, 195126
77. Godfrey, M. J. Phys. Rev. B 1988, 37, 10176
78. Grafenstein, J.; Ziesche, P. Phys. Rev. B 1990, 41, 3245
79. Godfrey, M. J. J. Phys. B: At. Mol. Opt. Phys. 1990, 23, 2427
80. Ramprasad, R. J. Phys.: Condens. Matter 2002, 14, 5497
81. Rogers, C. L.; Rappe, A. M. Phys. Rev. B 2002, 65, 224117
82. Maranganti, R.; Sharma, P.; Wheeler, L. Journal of Aerospace Engineering 2007, 20, 22
83. Popelier, P. L. A. Atoms in Molecules: An Introduction; Pearson: Harlow, U.K., 2000.
84. Matta, C. F.; Bader, R. F. W. J. Phys. Chem. A 2006, 110, 6365
85. Bader, R. F. W. J. Phys. Chem. A 2007, 111, 7966
86. Nasertayoob, P.; Shahbazian, S. Int. J. Quantum Chem. 2009, 109, 726
87. Bader, R. F. W.; Srebrenik, S.; Nguyendang, T. T. J. Chem. Phys. 1978, 68, 3680
88. Bader, R. F. W. Phys. Rev. B 1994, 49, 13348
89. Bader, R. F. W. In Quantum theory of atoms in molecules: recent progress in theory and application; Matta, C. F.; Boyd, R. J., Eds.; Wiley-VCH: Weinheim, 2007, p 37.
90. Nasertayoob, P.; Shahbazian, S. Int. J. Quantum Chem. 2010, 110, 1188
91. Cioslowski, J.; Karwowski, J. In Fundamentals of molecular similarity; Kluwer: Dordrecht, The Netherlands, 2001, p 101.
92. Cohen, L. J. Math. Phys. (Cambridge, Mass.) 1966, 7, 781
93. Cohen, L. Philosophy of Science 1966, 33, 317
94. Weyl, H. Z. Phys. 1927, 46, 1
95. Margenau, H.; Hill, R. N. Prog. Theor. Phys. 1961, 26, 722
96. Groblicki, R. J. Math. Phys. (Melville, NY, U. S.) 1983, 24, 841
97. Cohen, L.; Zaparovanny, Y. I. J. Math. Phys 1980, 21, 794
98. Cohen, L. J. Math. Phys. (Melville, NY, U. S.) 1984, 25, 2402
99. Cohen, L. J. Chem. Phys. 1984, 80, 4277
100. Cohen, L. Phys. Lett. A 1996, 212, 315
101. More generally, the result is true whenever the second derivative of h, at zero, vanishes.
102. I.e., one now allows coupling between the different Cartesian directions in the local kinetic energy formula.
103. Levy, M. Phys. Rev. A 1982, 26, 1200
104. Levy, M.; Perdew, J. P. Phys. Rev. A 1985, 32, 2010
105. Ghosh, S. K.; Parr, R. G. J. Chem. Phys. 1985, 82, 3307
106. Kohn, W.; Sham, L. J. Phys. Rev. 1965, 140, A1133
107. Ou-Yang, H.; Levy, M. Phys. Rev. A 1990, 42, 155
108. Nagy, A. Phys. Rev. A 1993, 47, 2715
109. Parr, R. G.; Liu, S. B.; Kugler, A. A.; Nagy, A. Phys. Rev. A 1995, 52, 969
110. Nagy, A.; Liu, S. B.; Parr, R. G. Phys. Rev. A 1999, 59, 3349
111. Gal, T.; Nagy, A. J. Mol. Struct. (THEOCHEM) 2000, 501, 167
112. Ayers, P. W.; Lucks, J. B.; Parr, R. G. Acta University Debreceniensis de Ludovio Kossuth Nominatae, Series Physica et Chemica 2002, 34-35, 223
113. Levy, M.; Ayers, P. W. Phys. Rev. A 2009, 79 064504
114. Ayers, P. W.; Rodriguez, J. I. Canadian Journal of Chemistry-Revue Canadienne De Chimie 2009, 87, 1540
115. Burke, K.; Cruz, F. G.; Lam, K. C. J. Chem. Phys. 1998, 109, 8161
116. Sim, E.; Larkin, J.; Burke, K.; Bock, C. W. J. Chem. Phys. 2003, 118, 8140
117. Hui, O. Y.; Levy, M. Phys. Rev. Lett. 1990, 65, 1036
118. Rodriguez, J. I.; Ayers, P. W.; Gotz, A. W.; Castillo-Alvarado, F. L. J. Chem. Phys. 2009, 131 021101
119. Gaiduk, A. P.; Chulkov, S. K.; Staroverov, V. N. J. Chem. Theory Comput. 2009, 5, 699
120. van Leeuwen, R.; Baerends, E. J. Phys. Rev. A 1995, 51, 170
121. It seems exceedingly unlikely that the local total kinetic energy would have the Laplacian form for the exact exchange-correlation energy functional. However, as this functional is unknown in any useful explicit form, it seems impossible to prove this.
122. This postulate is based on the incredible diversity of local kinetic energy forms that are attainable if f is allowed to be a functional of the wavefunction. This leads the authors to believe that virtually any well-behaved local kinetic energy function can be reverse-engineered to generate a quasiprobability distribution function.
123. Bader, R. F. W.; Heard, G. L. J. Chem. Phys. 1999, 111, 8789
124. Malcolm, N. O. J.; Popelier, P. L. A. Faraday Discuss. 2003, 124, 353
125. Gillespie, R. J.; Popelier, P. L. A. Chemical Bonding and Molecular Geometry; Oxford: New York, 2001.
126. Sagar, R. P.; Ku, A. C. T.; Smith, V. H.; Simas, A. M. J. Chem. Phys. 1988, 88, 4367
127. Bader, R. F. W.; Gillespie, R. J.; Macdougall, P. J. J. Am. Chem. Soc. 1988, 110, 7329
128. Gillespie, R. J.; Bytheway, I.; Dewitte, R. S.; Bader, R. F. W. Inorg. Chem. 1994, 33, 2115
129. Bader, R. F. W.; Macdougall, P. J.; Lau, C. D. H. J. Am. Chem. Soc. 1984, 106, 1594
130. Shi, Z.; Boyd, R. J. J. Chem. Phys. 1988, 88, 4375
131. Kiewisch, K.; Eickerling, G.; Reiher, M.; Neugebauer, J. J. Chem. Phys. 2008, 128 044114
132. Eickerling, G.; Reiher, M. J. Chem. Theory Comput. 2008, 4, 286
133. Cassam-Chenai, P.; Jayatilaka, D. Theor. Chem. Acc. 2001, 105, 213
134. Cassam-Chenai, P.; Jayatilaka, D. Theor. Chem. Acc. 2002, 107, 383
135. Cassam-Chenai, P. J. Math. Chem. 2002, 31, 145
136. Bader, R. F. W. Theor. Chem. Acc. 2002, 107, 381
137. Delle Site, L. Theor. Chem. Acc. 2002, 107, 378
138. Kryachko, E. S. Theor. Chem. Acc. 2002, 107, 375
139. Mohallem, J. R. Theor. Chem. Acc. 2002, 107, 372
140. Nasertayoob, P.; Shahbazian, S. Int. J. Quantum Chem. 2008, 108, 1477
141. Luana, V.; Costales, A.; Pendas, A. M. Phys. Rev. B 1997, 55, 4285
142. Pendas, A. M.; Costales, A.; Luana, V. Phys. Rev. B 1997, 55, 4275
143. Heidarzadeh, F.; Shahbazian, S. Int. J. Quantum Chem., DOI: 10.1002/qua.22629.
144. Heidarzadeh, F.; Shahbazian, S. Submitted for publication.