Variational Density-Functional Theory for an Individual Excited State

Abstract

It is shown that there exists a variational Kohn-Sham density-functional theory, with a minimum principle, for the self-consistent determination of an individual excited-state energy and density. Exact properties of the required functional are ascertained, including a Koopmans theorem. This knowledge allows the employment of an effective potential that gives encouraging numerical results, and also helps to explain the success of a recent perturbation theory and its time-dependent counterpart. 31.15.Ew, 31.50. + w Density-functional theory (DFT) is now in widespread use as an effective approach for ground-state electronic structure calculations. The development of accurate functionals within the popular Kohn-Sham formulation of DFT has enabled us to perform efficient ground-state variational calculations with remarkable accuracy. In Kohn-Sham theory, the simplicity of the three-dimensional electron density is coupled with the use of a relatively small number of orbitals to ensure Fermi statistics, through the use of an auxiliary noninteracting system (see, for instance,There has also been noteworthy progress in excited-state DFT (see, for example, Refs. ). These studies have stimulated us into asking if there exists a variational KohnSham theory for an individual excited state, which is analogous to the ground-state theory, because an affirmative answer implies the possibility that accurate excited-state calculations might be performed routinely, in a manner comparable to today's ground-state calculations. Accordingly, it is our purpose to show that there does indeed exist such a variational Kohn-Sham theory, with a minimum principle, for an individual excited state. In our proof, the necessary universal functional is identified and several of its properties are ascertained for the purpose of approximation. This enables us to actually carry out illustrative self-consistent calculations, and encouraging results are obtained for the systems studied.Consider the Hamiltonian of interestĤ y :whereT is the kinetic energy operator,V ee is the electron-electron repulsion operator, and y͑r͒ is the localmultiplicative attractive potential of interest. Assume we want the energy and density of the kth state ofĤ y . (In this Letter, all interacting and noninteracting states shall be assumed nondegenerate to facilitate the presentation.) For this purpose we start by generalizing earlier excited-state functionals [12,13] and define the universal bywhere both r and r 0 are arbitrary electron densities. In Eq. (2), it is understood that each C is orthogonal to the first k 2 1 states of that Hamiltonian,Ĥ y 0 T 1V ee 1 P N i 1 y 0 ͑r i ͒, for which r 0 is the ground-state density. It follows from the definition of F͓r, r 0 ͔ that E k , the energy of the kth state ofĤ y , is given bywhere r 0 is the ground-state density ofĤ y and r k is the density of its kth state. Analogous with the constrainedsearch proof of the ground-state Hohenberg-Kohn variational theorem, Eq. (3) is true becausewhere the C's are understood to be restric...

Excitation Energies from Spin-Restricted Ensemble-Referenced Kohn−Sham Method: A State-Average Approach

“…[18][19][20][21] However, if such a state is the excited state of a molecule, the direct optimization of its energy with respect to the excited-state density is questionable within the framework of Kohn-Sham DFT. [28][29][30][31][32][33] This problem can be bypassed with the use of the KS orbitals optimized for the ground state or for an ensemble of the ground and the excited states. 5 In the present work, we have formulated and tested a state-average version of the REKS method, in which the KS orbitals are optimized with respect to a weighted sum of the energies of the ground state and the singly excited state.…”

“…29,30 In developing a time-independent density functional theory for excited states, one has to include the ground-state density into consideration. [31][32][33] In the present work, we suggest a time-independent method to the calculation of excited states in DFT which is based on the ensemble approach. Within the ensemble approach to DFT, 34,35 the variational principle can be applied strictly to the † Part of the "Sason S. Shaik Festschrift".…”

Analysis of Floquet formulation of time-dependent density-functional theory

“…Finally, one may raise a question if Floquet theory of TDDFT is applicable only to "ground-states". The answer is that even for steady-state "excited-states", a theory similar to the stationary-state excited-state DFT [4,5,6,7,8,9,10,11] can be developed but that is not our main concern here. In this appendix, we show that if one confines the quasienergies to a range such that they go to their unperturbed counterpart when the time-periodic potential is turned off, then there is a well defined ground-state quasienergy satisfying the minimum variational principle.…”

Excited-state pair-density-functional theory

“…(4). We mention in passing that the Kohn-Sham theory was also extended to excited states [12,13]. Note that instead of the ground-state electron density 0 , we could use the external potential v or any ground-state Kohn-Sham orbital, etc.…”

“…Local self-interactionfree approximate exchange-correlation potentials have been proposed for this purpose [20]. Orbital-dependent functionals (optimized potential method (OPM) [21] and the KriegerLi-Iafrate (KLI) method [22]) were also generalized and tested [10,12,13,23,24]. Glushkov and Levy [25] presented an OPM algorithm that takes the necessary orthogonality constraints to lower states into account.…”