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物理化学学报  2018, Vol. 34 Issue (6): 699-707    DOI: 10.3866/PKU.WHXB201711221
所属专题: 密度泛函理论中的化学概念特刊
论文     
Analogies between Density Functional Theory Response Kernels and Derivatives of Thermodynamic State Functions
GEERLINGS Paul*(),DE PROFT Frank,FIAS Stijn
Analogies between Density Functional Theory Response Kernels and Derivatives of Thermodynamic State Functions
Paul GEERLINGS*(),Frank DE PROFT,Stijn FIAS
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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, ?, 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 ?, F is eliminated as it has two dependent (extensive) variables, N and ρ. For ?, 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: Conceptual DFTResponse kernelsAnalogy with thermodynamicsStability-concavity/convexity    
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, ?, 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 ?, F is eliminated as it has two dependent (extensive) variables, N and ρ. For ?, 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: Conceptual DFT    Response kernels    Analogy with thermodynamics    Stability-concavity/convexity
收稿日期: 2017-09-11 出版日期: 2017-11-22
基金资助: 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
通讯作者: GEERLINGS Paul     E-mail: pgeerlin@vub.ac.be
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GEERLINGS Paul,DE PROFT Frank,FIAS Stijn. Analogies between Density Functional Theory Response Kernels and Derivatives of Thermodynamic State Functions[J]. 物理化学学报, 2018, 34(6): 699-707, 10.3866/PKU.WHXB201711221

Paul GEERLINGS,Frank DE PROFT,Stijn FIAS. Analogies between Density Functional Theory Response Kernels and Derivatives of Thermodynamic State Functions. Acta Phys. -Chim. Sin., 2018, 34(6): 699-707, 10.3866/PKU.WHXB201711221.

链接本文:

http://www.whxb.pku.edu.cn/CN/10.3866/PKU.WHXB201711221        http://www.whxb.pku.edu.cn/CN/Y2018/V34/I6/699

S V T P N ρ μ v
U + + F / +
F + - R + 0
H + - E + - -
G - - ? - -
extensive intensive   extensive intensive
Table 1  Analogy between the sign of the second order derivatives of the thermodynamic functions and those of the DFT functionals (positive or negative semidefiniteness in the case of functional derivatives).
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