Functional Derivatives and Differentiability
in Density-Functional Theory
Ping Xiang and Yan Alexander Wang
Abstract Based on Lindgren and Salomonson’s analysis on Fréchet differentiability
[Phys Rev A 67:056501 (2003)], we showed a specific variational path along which
the Fréchet derivative of the Levy-Lieb functional does not exist in the unnormalized density domain. This conclusion still holds even when the density is restricted
within a normalized space. Furthermore, we extended our analysis to the Lieb functional and demonstrated that the Lieb functional is not Fréchet differentiable. Along
our proposed variational path, the Gâteaux derivative of the Levy-Lieb functional or
the Lieb functional takes a different form from the corresponding one along other
more conventional variational paths. This fact prompted us to define a new class of
unconventional density variations and inspired us to present a modified density variation domain to eliminate the problems associated with such unconventional density
variations.
Keywords Density functional ⋅ Density variation ⋅ Functional differentiability
Functional derivative
1 Introduction
Functional differentiability plays crucial roles in density-functional theory (DFT)
[1]. Within DFT, the total electronic energy is expressed as a functional of the electron density í µí¼(í µí°«) [1, 2]. To find the ground-state (GS) density and its corresponding
energy, we need to perform variational calculations based on the functional derivative [1–4]. Naturally, researchers have been interested in studying the differentiability
of density functionals since the beginning of DFT.
More than twenty years ago, based on Lieb’s early work [5], Englisch and Englisch
proved the Gâteaux differentiability of a large class of density functionals and settled
P. Xiang ⋅ Y. A. Wang ( ✉ )
Department of Chemistry, University of British Columbia, 2036 Main Mall,
Vancouver, BC V6T 1Z1, Canada
e-mail: yawang@chem.ubc.ca
© Springer International Publishing AG, part of Springer Nature 2018
Y. A. Wang et al. (eds.), Concepts, Methods and Applications of Quantum Systems
in Chemistry and Physics, Progress in Theoretical Chemistry and Physics 31,
https://doi.org/10.1007/978-3-319-74582-4_18
331
in Density-Functional Theory
Ping Xiang and Yan Alexander Wang
Abstract Based on Lindgren and Salomonson’s analysis on Fréchet differentiability
[Phys Rev A 67:056501 (2003)], we showed a specific variational path along which
the Fréchet derivative of the Levy-Lieb functional does not exist in the unnormalized density domain. This conclusion still holds even when the density is restricted
within a normalized space. Furthermore, we extended our analysis to the Lieb functional and demonstrated that the Lieb functional is not Fréchet differentiable. Along
our proposed variational path, the Gâteaux derivative of the Levy-Lieb functional or
the Lieb functional takes a different form from the corresponding one along other
more conventional variational paths. This fact prompted us to define a new class of
unconventional density variations and inspired us to present a modified density variation domain to eliminate the problems associated with such unconventional density
variations.
Keywords Density functional ⋅ Density variation ⋅ Functional differentiability
Functional derivative
1 Introduction
Functional differentiability plays crucial roles in density-functional theory (DFT)
[1]. Within DFT, the total electronic energy is expressed as a functional of the electron density í µí¼(í µí°«) [1, 2]. To find the ground-state (GS) density and its corresponding
energy, we need to perform variational calculations based on the functional derivative [1–4]. Naturally, researchers have been interested in studying the differentiability
of density functionals since the beginning of DFT.
More than twenty years ago, based on Lieb’s early work [5], Englisch and Englisch
proved the Gâteaux differentiability of a large class of density functionals and settled
P. Xiang ⋅ Y. A. Wang ( ✉ )
Department of Chemistry, University of British Columbia, 2036 Main Mall,
Vancouver, BC V6T 1Z1, Canada
e-mail: yawang@chem.ubc.ca
© Springer International Publishing AG, part of Springer Nature 2018
Y. A. Wang et al. (eds.), Concepts, Methods and Applications of Quantum Systems
in Chemistry and Physics, Progress in Theoretical Chemistry and Physics 31,
https://doi.org/10.1007/978-3-319-74582-4_18
331