Conceptual DFT
In DFT, the ground-state electronic energy of a molecule is a functional of the electron density \(\require{physics} \rho( \vb{r} )\) (Parr et al. 1978; Parr and Yang 1989): \[\begin{align} E [ \rho( \vb{r} )] % &= T[\rho] % + V_{ee}[\rho( \vb{r} )] % + V_{ne}[\rho( \vb{r} )] % \\ &= T[\rho( \vb{r} )] % + V_{ee}[\rho( \vb{r} )] % + \int \rho( \vb{r} ) % v( \vb{r} ) % \dd{\vb{r}} \end{align}\] where \(\require{physics} v( \vb{r} )\) is the external electron-nuclei potential. Minimizing the total energy w.r.t. \(\rho( \vb{r} )\) and with the condition that the total number of electrons is fixed (Parr et al. 1978) \[\begin{equation} \require{physics} N = % \int \dd{ \vb{r} } % \rho( \vb{r} ) \end{equation}\] leads to the stationary principle \[\begin{equation} \delta \{ E [ N, v( \vb{r} ) ] % - \mu N[\rho( \vb{r} )] \} = 0 \end{equation}\] with \(\mu\) the Lagrange multiplier. From this equation follows that \[\begin{equation} \require{physics} \mu % = \qty( \fdv{E}{ \rho( \vb{r} ) } )_{ v( \vb{r} ) } % \thinspace . \end{equation}\] Since \(N\) is a functional of \(\rho( \vb{r} )\), one can use the chain rule to obtain \[\begin{equation} \mu % = \qty( \pdv{E}{N} )_{ v( \vb{r} ) } % \thinspace . \end{equation}\] The corresponding derivative w.r.t. the external potential \(v( \vb{r} )\) is \[\begin{equation} \rho( \vb{r} ) % = \qty( \fdv{E}{ v( \vb{r} ) } )_N \thinspace , \end{equation}\] making the total differential of \(E [ N, v(\vb{r}) ]\) equal to (Parr et al. 1978; Nalewajski and Parr 1982) \[\begin{align} \dd{E} [ N, v( \vb{r} ) ] % &= \mu \dd{N} % + \int \rho( \vb{r} ) % \var{ v( \vb{r} ) } % \dd{ \vb{r} } % \\ &= \qty( \pdv{E}{N} )_{ v( \vb{r} ) } % \dd{N} % + \int \qty( \fdv{E}{ v( \vb{r} ) } )_N % \var{ v( \vb{r} ) } % \dd{ \vb{r} } % \thinspace , \end{align}\] where the chemical potential’s physical interpretation (Ayers and Levy 2000; Parr and Yang 1989) in DFT is as follows: it represents the escaping tendency of electrons from a system, since electrons migrate from a point in space with high \(\mu\) to a point in space with low \(\mu\). In conceptual DFT, the chemical potential is linked to the electronegativity as \[\begin{equation} \mu % = - \chi % \thinspace , \end{equation}\] which by definition is a measure of an element’s power to attract electrons (Parr et al. 1978).