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

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
Revised Date: 3 November 2017
Accepted Date: 10 November 2017
Available Online: 22 June 2017
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

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.
- Conceptual DFT,
- Response kernels,
- Analogy with thermodynamics,
- Stability-concavity/convexity
1. [1]
  Parr, R. G.; Yang, W. Ann. Rev. Phys. Chem. 1995, 46, 701.doi: 10.1146/annurev.pc.46.100195.003413 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 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 doi: 10.1021/cr990029p
4. [4]
  De Proft, F.; Geerlings, P. Chem. Rev. 2001, 101, 1451. doi: 10.1021/cr9903205 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 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 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 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.
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 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 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 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 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 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 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 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 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 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 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 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 doi: 10.1063/1.4913415
23. [23]
  Nalewajski, R. F.; Parr, R. G. J. Chem. Phys. 1982, 77, 399. doi: 10.1063/1.443620 doi: 10.1063/1.443620
24. [24]
  Berkowitz, M.; Parr, R. G. J. Chem. Phys. 1988, 88, 2554. doi: 10.1063/1.454034 doi: 10.1063/1.454034
25. [25]
  Senet, P. J. Chem. Phys. 1996, 105, 6471. doi: 10.1063/1.472498 doi: 10.1063/1.472498
26. [26]
  Ayers, P. W.; Parr, R. G. J. Am. Chem. Soc. 2001, 123, 2007. doi: 10.1021/ja002966g doi: 10.1021/ja002966g
27. [27]
  Ayers, P. W. Theor. Chem. Acc. 2001, 106, 271. doi: 10.1007/PL00012385 doi: 10.1007/PL00012385
28. [28]
  Liu, S.; Li, T.; Ayers, P. W. J. Chem. Phys. 2009, 131, 114106. doi: 10.1063/1.3231687 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 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 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 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 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 doi: 10.1016/j.cplett.2016.07.039
34. [34]
  Kohn, W. Phys. Rev. Lett. 1996, 76, 3168. doi: 10.1103/PhysRevLett.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 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 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 doi: 10.1021/ja00310a011
38. [38]
  Ghosh, S. K.; Berkowitz, M. J. Chem. Phys. 1985, 83, 2976. doi: 10.1063/1.449846 doi: 10.1063/1.449846
39. [39]
  Ghosh, S. K. Chem. Phys. Lett. 1990, 172, 77. doi: 10.1016/0009-2614(90)87220-L 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 doi: 10.1021/j100017a022
42. [42]
  Chamorro, E.; De Proft, F.; Geerlings, P. J. Chem. Phys. 2005, 123, 154104. doi: 10.1063/1.2072907 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 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 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 doi: 10.1039/c7cp00691h
46. [46]
  Liu, S.; Parr, R. G. J. Chem. Phys. 1997, 106, 5578. doi: 10.1063/1.473580 doi: 10.1063/1.473580
47. [47]
  Lieb, E. H. Int. J. Quantum Chem. 1983, 24, 243. doi: 10.1002/qua.560240302 doi: 10.1002/qua.560240302
48. [48]
  Eschrig, H. The Fundamentals of Density Functional Theory; Teubner: Stuttgart-Leipzig, Germany, 1996.
49. [49]
  Kvaal, S.; Ekstrom, U.; Teale, A. M.; Helgaker, T. J. Chem. Phys. 2014, 140, 18A518. doi: 10.1063/1.4867005 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 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 doi: 10.1073/pnas.81.24.8028
53. [53]
  Nagy, A.; Parr, R. G. Proc. Indian Acad. Sci. 1994, 106, 217.
54. [54]
  Nagy, A.; Parr, R. G. J. Mol. Struct. THEOCHEM 2000, 501–502, 101. doi: 10.1016/S0166-1280[99]00418-2 doi: 10.1016/S0166-1280[99]00418-2
55. [55]
  Nagy, A. Int. J. Quantum Chem. 2017, 117, e25396. doi: 10.1002/qua.25396 doi: 10.1002/qua.25396
56. [56]
  Callen, H. B. Thermodynamics and an Introduction to Thermostatistics; John Wiley: New York, NY, USA, 1985.
57. [57]
  Prigogine, I.; Defay, R. Chemical Thermodynamics; Longman: London, UK, 1954.
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 doi: 10.1063/1.3151599
60. [60]
  Ayers, P. W. Phys. Rev. A 2006, 73, 012513. doi: 10.1103/PhysRevA.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 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 doi: 10.1063/1.4932539
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