In this paper, the unconditionally energy-stable and orthonormality-preserving iterative scheme proposed in [X. Wang et al. (2024), J. Comput. Phys., 498:112670] is extended both theoretically and numerically, including (i) the exchange-correlation energy is introduced into the model for a more comprehensive description of the quantum system, utilizing the local density approximation used by the National Institution of Science and Technology Standard Reference Database; (ii) both the unconditional energy-stability and orthonormality-preservation are attained in the newly derived scheme; (iii) a $C^0$ tetrahedral spectral element method is adopted for the quality spatial discretization, of which a quality initial condition can be designed using low order one for effectively accelerating the simulation. A series of numerical experiments validate the effectiveness of our method, encompassing various atoms and molecules. All the computations successfully reveal the anticipated spectral accuracy and the exponential error dependence to the cubic root of the degree of freedom number. Moreover, the efficiency of the extended framework is discussed in detail on updating schemes.