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

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

GEERLINGS Paul*(),DE PROFT Frank,FIAS Stijn   

  • 收稿日期:2017-09-11 发布日期:2018-03-20
  • 通讯作者: GEERLINGS Paul E-mail:pgeerlin@vub.ac.be
  • 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

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

Paul GEERLINGS*(),Frank DE PROFT,Stijn FIAS   

  • Received:2017-09-11 Published:2018-03-20
  • Contact: Paul GEERLINGS E-mail:pgeerlin@vub.ac.be
  • Supported by:
    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

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 DFT, Response kernels, Analogy with thermodynamics, Stability-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