A detailed review of a novel density-functional theory (DFT) [1] is described in this chapter, where the electron density \(n^e(\epsilon )\) on the energy coordinate \(\epsilon \) serves as the fundamental variable for the DFT. The definition of the energy electron density \(n^e(\epsilon )\) is provided in the first Section. The exchange functional \(E_x^{e,\text {LDA}}[n^e]\) based on the local density approximation (LDA) for the variable \(n^e(\epsilon )\) is also introduced. In the Sects. 3.2 and 3.3, the Hohenberg-Kohn (HK) functional \(F_\text {HK}^e[n^e]\) is formulated for the variable \(n^e(\epsilon )\) in parallel to the original HK functional. It is shown in Sect. 3.4 that the domain of definition of the functional can be extended to the set of N-rep. energy electron densities by means of Levy’s constraint search method. In Sect. 3.5, it is shown that the exchange-correlation functional \(E_{xc}^e[n^e]\) of the density \(n^e(\epsilon )\) can be incorporated into the equation for the Kohn-Sham DFT. In the last section, the correction based on the generalized gradient approximation (GGA) is applied to the LDA functional \(E_{xc}^{e,\text {LDA}}[n^e]\) by introducing an inhomogeneity parameter defined on the energy coordinate. The functional is employed to calculate the potential energies of the chemical bonds. In this Chapter, the atomic units (a.u.) are used unless otherwise stated.