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:

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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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