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Fukui function and response function for nonlocal and fractional systems

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10.1063/1.4803101

### Abstract

We present extensions to our previous work on Fukui functions and linear-response functions[W. Yang, A. J. Cohen, F. D. Proft, and P. Geerlings, J. Chem. Phys.136, 144110 (Year: 2012)10.1063/1.3701562]. Viewed as energy derivatives with respect to the number of electrons and the external potential, all second-order derivatives (the linear-response function, the Fukui function, and the chemical hardness) are extended to fractional systems, and all third-order derivatives (the second-order response function, the Fukui response function, the dual descriptor, and the hyperhardness) for integer systems are also obtained. These analytical derivatives are verified by finite difference numerical derivatives. In the context of the exact linearity condition and the constancy condition, these analytical derivatives enrich greatly the information of the exact conditions on the energy functional through establishing real-space dependency. The introduction of an external nonlocal potential defines the nonlocal Fukui function and the nonlocal linear-response function. The nonlocal linear-response function so defined also provides the precise meaning for the time-dependent linear-response density-functional theory calculations with generalized Kohn-Sham functionals. These extensions will be useful to conceptual density-functional theory and density functional development.

© 2013 AIP Publishing LLC

Received 29 January 2013
Accepted 15 April 2013
Published online 13 May 2013

Acknowledgments: Support from the Office of Naval Research (Grant No. N00014-09-0576)(W.Y.) and National Science Foundation (NSF) (Grant No. CHE-09-11119)(W.Y.) is greatly appreciated. D.P. has also been supported by William Krigbaum and Marcus Hobbs Fellowship from Duke University.

Article outline:

I. INTRODUCTION

II. *E*[*N*, *v*] AND ITS DERIVATIVES

III. THE LINEARITY CONDITION AND ITS EXTENSIONS

IV. ANALYTICAL EXPRESSIONS FOR DERIVATIVES IN KOHN-SHAM AND GENERALIZED KOHN-SHAM FRAMEWORK

A. *p* + *q* = 2 derivatives of a system with a fractional number of electrons

B. δ^{3} *E*/δ*v* ^{3} and δ^{3} *E*/δ*N*δ*v* ^{2} for a system with an integer number of electrons

C. δ^{3} *E*/δ*N* ^{2}δ*v* and δ^{3} *E*/δ*N* ^{3} for a system with an integer number of electrons

D. Numerical verification

V. EXTENSIONS TO NONLOCAL FUKUI FUNCTIONS AND LINEAR-RESPONSE FUNCTIONS

VI. THE CONSTANCY CONDITION AND ITS EXTENSIONS

VII. CONCLUSIONS

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2013-05-13

2014-04-16

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