arXiv:1407.1959v1 [cond-mat.mtrl-sci] 8 Jul 2014 Excited-state density-functional theory revisited: study based on Hohenberg-Kohn, Gunnarsson-Lundqvist and constrained-search formalism Prasanjit Samal, 1, ∗ Subrata Jana, 1 and Sourabh S. Chauhan 1 1 School of Physical Sciences, National Institute of Science Education & Research, Bhubaneswar 751005, INDIA. (Dated: July 9, 2014) The Hohenberg-Kohn theorem and constrained search formalism of density functional theory are in principle sufficient to address the question of mapping from an excited-state density of an interacting many-electron system to the external potential. Which is also equally applicable to the model noninteracting system. Be it the lowest (non)degenerate excited state of a given symmetry or any other many-particle quantum states with differing symmetry and thus corresponding to arbitrary order of excitations, the analogue of Hohenberg-Kohn theorem and constrained search formalism can uniquely establish the one-to-one density-to-potential mapping. We have rigorously investigated and also shown by examples that the existence of multiple effective/external potentials for excited- states and the seemingly contradictory results in connection to the applicability of Gunnarsson- Lundqvist theorem even for the lowest excited-state is not truly a failure or violation of Hohenberg- Kohn/Gunnarsson-Lundqvist theorem. Rather these are nothing but our limited understanding of the subtle differences between the ground and excited-state density-functional theory. So these are in fact no issues in the context of the foundational aspects of excited-state density-functional formalism. Our critical analysis outlines that the multiplicity of potentials for a given density (i.e. the symmetries of the quantum states involve) never guarantee the violation of one-to-one correspondence between the two most valuable physical quantities of interest in modern density- functional theory. By furthering the existing theories and basic principles, we have provided a firm footing to the density -to-potential mapping for excited-states in general. We have shown that our proposed criterions based on generalized constrained search formalism keep the excited-state density-to-potential mapping intact. INTRODUCTION Since its advent, density-functional theory (DFT) is the most widely used, popular and succssful many body quantum mechanical (QM) approach for describing mat- ter and still continuing to be the same. DFT is now rou- tinely applied for calculating the e.g. electronic, mag- netic, spectroscopic and thermodynamic properties of atoms, molecules and materials for both ground and ex- cited states [1–3]. Its because the most straightforward approach to characterizing electronic system i.e. solv- ing the many particle Schr¨ odinger equation is impracti- cal except for very small systems, as the wavefunction’s complexity grows rapidly with increasing size. Thus the earlier attempts were to simplify the many-body problem using particle density as a basic variable started with the Thomas-Fermi approximation [1–3] and later Hohenberg, Kohn and Sham (HKS) [4, 5] formulation of modern DFT for non-degenerate ground state. Hohenberg and Kohn (HK) [4] proved that ground state of a many-electron sytem is uniquely determined by it’s particle density. Analogus to the QM variational approach, HK developed the variational DFT principle which states, the energy can be expressed as a functional of the density which assume its minimum value for the correct ground-state density. The HK theorems proved via the variational principle of energy also gives the one-to-one mapping be- tween the ground state density ρ( r) and the external po- tential ˆ v( r). Hence, the knowledge of density ρ( r), which in turn predicts the external potential v( r) to within an additive constant and under the constraint of particle number N conservations is sufficient to address and de- scribe the many-body problem. The significance of HK theorem is that the ground state density ρ( r) of a physical system uniquely determines the Hamiltonian ˆ H [ˆ v( r),N ] and other physical properties. In the past couple of decades, it became apparent to ask whether the HK and KS ground-state DFT can be extended for successfully studying the excited-states [1–3, 6–16, 20–34] and perform self- consistent Kohn- Sham calculations for energy and other desired proper- ties. As several excitonic phenomenas attributing to var- ious effects are now an area of active research, so to do such calculations time dependent density functional the- ory (TDDFT) gained overwhelming response [2, 28, 35– 40]. But the theory has its limitations [41] and several foundational as well as technical issues to handle dif- ferent types of excitation effects. Then one of a most natural approach to excited-state DFT is to use time- independent density functionals [21, 27, 42], where the