Thermodynamical transcription of the density functional theory with constant temperature

Abstract

The thermodynamical interpretation of the density functional theory for an electronic ground state is revisited. Ghosh et al. invented the thermodynamical transcription of the ground-state density functional theory into a local thermodynamics. They introduced the idea of the local temperature that varies from point to point. The local temperature is defined via the kinetic energy density. The kinetic energy density is not uniquely defined, usually the everywhere positive gradient form is applied. Now we prove that it is possible selecting the kinetic energy density so that the local temperature be a constant for the whole system under consideration. The kinetic energy density is proportional to the electron density and the temperature is proportional to the kinetic energy. Furthermore, the kinetic energy density corresponding to the constant temperature, maximizes the information entropy.density functional theory, information entropy, thermodynamical transcription | I N T R O D U C T I O NMore than three decades ago Ghosh et al. wrote a paper [1] entitled "Transcription of ground-state density functional theory into a local thermodynamics." They obtained a phase-space distribution function fðr; pÞ by maximizing a phase-space Shannon information entropy subject to the conditions that f yields the density nðrÞ and the local kinetic energy density of the system. A local Maxwell-Boltzmann distribution function was resulted.They introduced the concept of local temperature. This formalism has been used in several applications. [2,3] For example, approximate expressions for the exchange energy were derived. [4,5] It was possible to define local entropy, free energy, and to establish an analogy with the classical thermodynamics of fluids. [2] There have been several extension of the formalism. [6][7][8][9][10][11][12][13][14][15][16] There are some new developments in applying and understanding the Ghosh-Berkowitz-Parr entropy. [17][18][19][20][21][22][23] The local temperature of Ghosh et al. was defined via the kinetic energy density and varies from point to point. However, the kinetic energy density is not uniquely defined. Adding a term that integrates to zero to the kinetic energy density results another kinetic energy density with the same kinetic energy but different local temperature. Usually, the gradient form of the kinetic energy density is applied, because it is everywhere positive. Now we prove that it is possible selecting the kinetic energy density so that the local temperature be a constant for the whole system under consideration. The kinetic energy density is proportional to the electron density and the temperature is proportional to the kinetic energy. Moreover, the kinetic energy density corresponding to the constant temperature, maximizes the information entropy. This very simple interpretation gives a new insight into the density functional theory. Analytical examples for the linear harmonic oscillator and the H-atom are presented.In the following section, the Ghosh-Berkowitz-Parr theory is ...

Kinetic-Energy Density-Functional Theory on a Lattice

“…The concept of local temperature was also introduced in the local thermodynamic ansatz of DFT. 27 − 29 Furthermore, it is important to note that the kinetic-energy density is already used extensively in DFT, for instance, as an integral part of the so-called meta-GGAs. When treated within the generalized KS framework, 30 meta-GGAs lead to a local potential coupling to the kinetic-energy density, which can be interpreted as a position-dependent mass.…”

Conceptual density functional theory based electronic structure principles

“…[87][88][89][90][91] Even though a condensed-to-atom variant of Q(r) has been developed, such a condensation suffers from a mathematical drawback. [92][93][94][95][96][97][98] As Q(r) is non-linearly dependent on r(r), its condensation using the associated electron population becomes ad hoc. Attempts to dene a local variant of hardness also suffer from this mathematical inexact formulation.…”

Spatially resolved characterization of electron localization and delocalization in molecules: Extending the Kohn‐Resta approach

“…It is interesting to reveal interrelation between our approach and other electron localization descriptors. This is, for example, Ghosh-Berkowitz-Parr "local temperature" of electron gas [45,55,[120][121][122] that is used as a measure of electron localization and metallicity of chemical bond [118] :…”