Analogies between Density Functional Theory Response Kernels and Derivatives of Thermodynamic State Functions

Citation: GEERLINGS Paul, DE PROFT Frank, FIAS Stijn. Analogies between Density Functional Theory Response Kernels and Derivatives of Thermodynamic State Functions[J]. Acta Physico-Chimica Sinica, 2018, 34(6): 699-707. doi: 10.3866/PKU.WHXB201711221 shu

Analogies between Density Functional Theory Response Kernels and Derivatives of Thermodynamic State Functions

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General Chemistry (ALGC), Free University Brussels (VUB), Pleinlaan 2, 1050 Brussel, Belgium

通讯作者: GEERLINGSPaul, pgeerlin@vub.ac.be

摘要: In view of its use as reactivity theory, Conceptual Density Functional Theory (DFT), introduced by Parr et al., has mainly concentrated up to now on the E = E[N, v] functional. However, different ensemble representations can be used involving other variables also, such as ρ and µ. In this study, these different ensemble representations (E, Ω, F, and R) are briefly reviewed. Particular attention is then given to the corresponding second-order (functional) derivatives, and their analogies with the second-order derivatives of thermodynamic state functions U, F, H, and G, which are related to each other via Legendre transformations, just as the DFT functionals (Nalewajski and Parr, 1982). Starting from an analysis of the convexity/concavity of the DFT functionals, for which explicit proofs are discussed for some cases, the positive/negative definiteness of the associated kernels is derived and a detailed comparison is made with the thermodynamic derivatives.The stability conditions in thermodynamics are similar in structure to the convexity/concavity conditions for the DFT functionals. Thus, the DFT functionals are scrutinized based on the convexity/concavity of their two variables, to yield the possibility of establishing a relationship between the three second-order reactivity descriptors derived from the considered functional. Considering two ensemble representations, F and Ω, F is eliminated as it has two dependent (extensive) variables, N and ρ. For Ω, on the other hand, which is concave for both of its intensive variables (µ and υ), an inequality is derived from its three second-order (functional) derivatives: the global softness, the local softness, and the softness kernel. Combined with the negative value of the diagonal element of the linear response function, this inequality is shown to be compatible with the Berkowitz-Parr relationship, which relates the functional derivatives of ρ with υ, at constant N and µ. This was recently at stake upon quantifying Kohn’s Nearsightedness of Electronic Matter. The analogy of the resulting inequality and the thermodynamic inequality for the G derivatives is highlighted. Potential research paths for this study are briefly addressed; the analogies between finite-temperature DFT response functions and their thermodynamic counterparts and the quest for analogous relationships, as derived in this paper, for DFT functionals that are analogues of entropy-dimensioned thermodynamic functions such as the Massieu function.

关键词:

English

Analogies between Density Functional Theory Response Kernels and Derivatives of Thermodynamic State Functions

General Chemistry (ALGC), Free University Brussels (VUB), Pleinlaan 2, 1050 Brussel, Belgium

Corresponding author: GEERLINGS Paul, pgeerlin@vub.ac.be

Received Date: 11 September 2017
Accepted Date: 10 November 2017
Revised Date: 03 November 2017
Available Online: 15 June 2018
Fund Project:
S.F. wishes to thank the Research Foundation Flanders (FWO) and the European Union's Horizon 2020 Marie Sklodowska-Curie grant (No. 706415) for financially supporting his post-doctoral research at the ALGC group. F.D.P. and P.G. acknowledge the Research Foundation-Flanders (FWO) and the Vrije Universiteit Brussel (VUB) for continuous support to the ALGC research group, in particular the VUB for a Strategic Research Program awarded to ALGC, started up at January 1, 2013. F.D.P. also acknowledges the Francqui foundation for a position as Francqui Research Professor

Abstract: In view of its use as reactivity theory, Conceptual Density Functional Theory (DFT), introduced by Parr et al., has mainly concentrated up to now on the E = E[N, v] functional. However, different ensemble representations can be used involving other variables also, such as ρ and µ. In this study, these different ensemble representations (E, Ω, F, and R) are briefly reviewed. Particular attention is then given to the corresponding second-order (functional) derivatives, and their analogies with the second-order derivatives of thermodynamic state functions U, F, H, and G, which are related to each other via Legendre transformations, just as the DFT functionals (Nalewajski and Parr, 1982). Starting from an analysis of the convexity/concavity of the DFT functionals, for which explicit proofs are discussed for some cases, the positive/negative definiteness of the associated kernels is derived and a detailed comparison is made with the thermodynamic derivatives.The stability conditions in thermodynamics are similar in structure to the convexity/concavity conditions for the DFT functionals. Thus, the DFT functionals are scrutinized based on the convexity/concavity of their two variables, to yield the possibility of establishing a relationship between the three second-order reactivity descriptors derived from the considered functional. Considering two ensemble representations, F and Ω, F is eliminated as it has two dependent (extensive) variables, N and ρ. For Ω, on the other hand, which is concave for both of its intensive variables (µ and υ), an inequality is derived from its three second-order (functional) derivatives: the global softness, the local softness, and the softness kernel. Combined with the negative value of the diagonal element of the linear response function, this inequality is shown to be compatible with the Berkowitz-Parr relationship, which relates the functional derivatives of ρ with υ, at constant N and µ. This was recently at stake upon quantifying Kohn's Nearsightedness of Electronic Matter. The analogy of the resulting inequality and the thermodynamic inequality for the G derivatives is highlighted. Potential research paths for this study are briefly addressed; the analogies between finite-temperature DFT response functions and their thermodynamic counterparts and the quest for analogous relationships, as derived in this paper, for DFT functionals that are analogues of entropy-dimensioned thermodynamic functions such as the Massieu function.

Key words:

1. [1]
  Parr, R. G.; Yang, W. Ann. Rev. Phys. Chem. 1995, 46, 701.doi: 10.1146/annurev.pc.46.100195.003413
2. [2]
  Chermette, H. J. Comput. Chem. 1999, 20, 129.doi: 10.1002/(SICI)1096-987X(19990115)20:1 < 129::AIDJCC13 > 3.0.CO; 2-A
3. [3]
  Geerlings, P.; De Proft, F.; Langenaeker, W. Chem. Rev. 2003, 103, 1793. doi: 10.1021/cr990029p
4. [4]
  De Proft, F.; Geerlings, P. Chem. Rev. 2001, 101, 1451. doi: 10.1021/cr9903205
5. [5]
  Ayers, P. W.; Anderson, J. S. M.; Bartolotti, L. J. Int. J. Quantum Chem. 2005, 101, 520. doi: 10.1002/qua.20307
6. [6]
  Gazquez, J. L. J. Mex. Chem. Soc. 2008, 52, 3.
7. [7]
  Liu, S. B. Acta Phys. -Chim. Sin. 2009, 25, 590. doi: 10.3866/PKU.WHXB20090332
8. [8]
  Gross, E. K. U.; Kohn, W. Phys. Rev. Lett. 1985, 55, 2850. doi: 10.1103/PhysRevLett.55.2850
9. [9]
  Casida, M. E. Recent Advances in Density Functional Methods; Chong, D. P. Ed.; World Scientific Pub. Co. Inc.: Singapore, 1995; p. 155. http://www.whxb.pku.edu.cn/CN/abstract/abstract30109.shtml
10. [10]
  Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford University Press: New York, NY, USA, 1989.
11. [11]
  Ayers, P.W.; De Proft, F.; Borgoo, A.; Geerlings, P. J. Chem. Phys. 2007, 126, 224107. doi: 10.1063/1.2736697
12. [12]
  Sablon, N.; De Proft, F.; Geerlings, P. J. Phys. Chem. Lett. 2010, 1, 1228. doi: 10.1021/jz1002132
13. [13]
  Sablon, N.; De Proft, F.; Ayers, P. W.; Geerlings, P. J. Chem. Theory Comput. 2010, 6, 3671. doi: 10.1021/ct1004577
14. [14]
  Fias, S.; Boisdenghien, Z.; Stuyver, T.; Audiffred, M.; Merino, G.; Geerlings, P.; De Proft, F. J. Phys. Chem. A 2013, 117, 3556. doi: 10.1021/jp401760j
15. [15]
  Fias, S.; Geerlings, P.; Ayers, P.; De Proft, F. Phys. Chem. Chem. Phys. 2013, 15, 2882. doi: 10.1039/c2cp43612d
16. [16]
  Boisdenghien, Z.; Van Alsenoy, C.; De Proft, F.; Geerlings, P. J. Chem. Theory Comp. 2013, 9, 1007. doi: 10.1021/ct300861r
17. [17]
  Yang, W.; Cohen, A. J.; De Proft, F.; Geerlings, P. J. Chem. Phys. 2012, 136, 144110. doi: 10.1063/1.3701562
18. [18]
  Boisenghien, Z.; Fias, S.; Van Alsenoy, C.; De Proft, F.; Geerlings, P. Phys. Chem. Chem. Phys. 2014, 16, 14614. doi: 10.1039/c4cp01331j
19. [19]
  Fias, S.; Boisdenghien, Z.; De Proft, F.; Geerlings, P. J. Chem. Phys. 2014, 141, 184107. doi: 10.1063/1.4900513
20. [20]
  Geerlings, P.; Fias, S.; Boisdenghien, Z.; De Proft, F. Chem. Soc. Rev. 2014, 43, 4989. doi: 10.1039/c3cs60456j
21. [21]
  Geerlings, P.; Boisdenghien, Z.; De Proft, F.; Fias, S. Theor. Chem. Acc. 2016, 135, 213. doi: 10.1007/s00214-016-1967-9
22. [22]
  Stuyver, T.; Fias, S.; De Proft, F.; Fowler, P.; Geerlings, P. J. Chem. Phys. 2015, 142, 094103. doi: 10.1063/1.4913415
23. [23]
  Nalewajski, R. F.; Parr, R. G. J. Chem. Phys. 1982, 77, 399. doi: 10.1063/1.443620
24. [24]
  Berkowitz, M.; Parr, R. G. J. Chem. Phys. 1988, 88, 2554. doi: 10.1063/1.454034
25. [25]
  Senet, P. J. Chem. Phys. 1996, 105, 6471. doi: 10.1063/1.472498
26. [26]
  Ayers, P. W.; Parr, R. G. J. Am. Chem. Soc. 2001, 123, 2007. doi: 10.1021/ja002966g
27. [27]
  Ayers, P. W. Theor. Chem. Acc. 2001, 106, 271. doi: 10.1007/PL00012385
28. [28]
  Liu, S.; Li, T.; Ayers, P. W. J. Chem. Phys. 2009, 131, 114106. doi: 10.1063/1.3231687
29. [29]
  Yang, W.; Parr, R. Proc. Natl. Acad. Sci. USA 1985, 82, 6723. doi: 10.1073/pnas.82.20.6723
30. [30]
  Mendez, F.; Gazquez, J. L. J. Am. Chem. Soc. 1994, 116, 9298. doi: 10.1021/ja00099a055
31. [31]
  Damoun, S.; Van de Woude, G.; Mendez, F.; Geerlings, P. J. Phys. Chem. 1997, 101, 886. doi: 10.1021/jp9611840
32. [32]
  Geerlings, P.; De Proft, F. Int. J. Quantum Chem. 2000, 80, 227. doi: 10.1002/1097-461X(2000)80:2 < 227::AID-QUA17 > 3.3.CO; 2-E
33. [33]
  Heidar-Zadeh, F.; Richer, M.; Fias, S.; Miranda-Quintana, R. A.; Chan, M.; Franco-Perez, M.; Gonzalez-Espinoza, C. E.; Kim, T. D.; Lanssens, C.; Patel, A. H. G.; et al. Chem. Phys. Lett. 2016, 660, 307. doi: 10.1016/j.cplett.2016.07.039
34. [34]
  Kohn, W. Phys. Rev. Lett. 1996, 76, 3168. doi: 10.1103/PhysRevLett.76.3168
35. [35]
  Prodan, E.; Kohn, W. Proc. Natl. Acad. Sci. USA 2005, 102, 11635. doi: 10.1073/pnas.0505436102
36. [36]
  Fias, S.; Heidar-Zadeh, F.; Geerlings, P.; Ayers, P. W. Proc. Natl. Acad. Sci. USA 2017, 114, 11633. doi: 10.1073/pnas.1615053114
37. [37]
  Berkowitz, M.; Ghosh, S. K.; Parr, R. J. Am. Chem. Soc. 1985, 107, 6811. doi: 10.1021/ja00310a011
38. [38]
  Ghosh, S. K.; Berkowitz, M. J. Chem. Phys. 1985, 83, 2976. doi: 10.1063/1.449846
39. [39]
  Ghosh, S. K. Chem. Phys. Lett. 1990, 172, 77. doi: 10.1016/0009-2614(90)87220-L
40. [40]
  Harbola, M. K.; Chattaraj, P. K.; Parr, R. G. Isr. J. Chem. 1991, 31, 395. doi: 10.1002/ijch.v31.4
41. [41]
  Langenaeker, W.; De Proft, F.; Geerlings, P. J. Phys. Chem. 1995, 99, 6424. doi: 10.1021/j100017a022
42. [42]
  Chamorro, E.; De Proft, F.; Geerlings, P. J. Chem. Phys. 2005, 123, 154104. doi: 10.1063/1.2072907
43. [43]
  Torrent-Sucarrat, M.; Salvador, P.; Sola, M.; Geerlings, P. J. Comp. Chem. 2007, 28, 574. doi: 10.1002/jcc.20535
44. [44]
  Chattaraj, P.; Roy, D. R.; Geerlings, P.; Torrent-Sucarrat, M. Theor. Chem. Acc. 2007, 118, 923. doi: 10.1007/s00214-007-0373-8
45. [45]
  Polanco-Ramirez, C. A.; Franco-Perez, M.; Carmona-Espindola, J.; Gazquez, J. L.; Ayers, P. W. Phys. Chem. Chem. Phys. 2017, 19, 12355. doi: 10.1039/c7cp00691h
46. [46]
  Liu, S.; Parr, R. G. J. Chem. Phys. 1997, 106, 5578. doi: 10.1063/1.473580
47. [47]
  Lieb, E. H. Int. J. Quantum Chem. 1983, 24, 243. doi: 10.1002/qua.560240302
48. [48]
  Eschrig, H. The Fundamentals of Density Functional Theory; Teubner: Stuttgart-Leipzig, Germany, 1996. https://link.springer.com/content/pdf/bfm:978-3-322-97620-8/1.pdf
49. [49]
  Kvaal, S.; Ekstrom, U.; Teale, A. M.; Helgaker, T. J. Chem. Phys. 2014, 140, 18A518. doi: 10.1063/1.4867005
50. [50]
  Perdew, J.; Parr, R.; Levy, M.; Balduz, J. L. J. Phys. Rev. Lett. 1982, 49, 1691. doi: 10.1103/PhysRevLett.49.1691
51. [51]
  Fias, S. ; Geerlings, P. ; De Proft, F. ; Ayers, P. W. in preparation.
52. [52]
  Ghosh, S. K.; Berkowitz, M.; Parr, R. G. Proc. Natl. Acad. Sci. USA 1984, 81, 8028. doi: 10.1073/pnas.81.24.8028
53. [53]
  Nagy, A.; Parr, R. G. Proc. Indian Acad. Sci. 1994, 106, 217. doi: 10.1007/BF02840745
54. [54]
  Nagy, A.; Parr, R. G. J. Mol. Struct. THEOCHEM 2000, 501–502, 101. doi: 10.1016/S0166-1280[99]00418-2
55. [55]
  Nagy, A. Int. J. Quantum Chem. 2017, 117, e25396. doi: 10.1002/qua.25396
56. [56]
  Callen, H. B. Thermodynamics and an Introduction to Thermostatistics; John Wiley: New York, NY, USA, 1985. http://www.oalib.com/references/13135886
57. [57]
  Prigogine, I.; Defay, R. Chemical Thermodynamics; Longman: London, UK, 1954. https://www.researchgate.net/publication/223991489_On_the_theoretical_determination_of_the_Prigogine-Defay_ratio_in_glass_transition
58. [58]
  Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry; Wiley: New York, NY, USA, 1980.
59. [59]
  Cardenas, C.; Echegaray, E.; Chakraborty, D.; Anderson, J. S. M.; Ayers, P. W. J. Chem. Phys. 2009, 130, 244105. doi: 10.1063/1.3151599
60. [60]
  Ayers, P. W. Phys. Rev. A 2006, 73, 012513. doi: 10.1103/PhysRevA.73.012513
61. [61]
  Franco-Perez, M.; Ayers, P. W.; Gazquez, J. L.; Vela, A. J. Chem. Phys. 2015, 143, 244117. doi: 10.1063/1.4938422
62. [62]
  Franco-Perez, M.; Gazquez, J. L.; Ayers, P. W.; Vela, A. J. Chem. Phys. 2015, 143, 154103. doi: 10.1063/1.4932539

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沈阳化工大学材料科学与工程学院沈阳 110142

Analogies between Density Functional Theory Response Kernels and Derivatives of Thermodynamic State Functions

通讯作者: GEERLINGSPaul, pgeerlin@vub.ac.be

English

Analogies between Density Functional Theory Response Kernels and Derivatives of Thermodynamic State Functions

Corresponding author: GEERLINGS Paul, pgeerlin@vub.ac.be

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通讯作者: 陈斌, bchen63@163.com

中国化学会秘书处

Analogies between Density Functional Theory Response Kernels and Derivatives of Thermodynamic State Functions

通讯作者: GEERLINGSPaul, pgeerlin@vub.ac.be

English

Analogies between Density Functional Theory Response Kernels and Derivatives of Thermodynamic State Functions

Corresponding author: GEERLINGS Paul, pgeerlin@vub.ac.be

CCS Chemistry：嵌段共聚物自组装成 “三明治”二维有机材料，实现纳米级压力传感功能

CCS Chemistry：颜徐州、鲍哲南： 你的皮肤可能是一片SPMs

CCS Chemistry：工作高效不疲劳，只须让催化剂动起来

CCS Chemistry：静电组装导向聚合- 聚电解质纳米凝胶量化制备新策略

CCS Chemistry：金黄色葡萄球菌醛基脱氢酶是如何识别特异性底物的？

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