In density-functional theory (DFT) of the ground state, a density distribution ${n}_{o}$(r) is called v-representable (VR) if it is the ground-state density in some external potential. [It is known that not all ``reasonable'' ${n}_{0}$(r) are VR.] In DFT of time-dependent linear response of a nondegenerate ground state, a similar question arises: Is a response density, ${n}_{1}$(r,\ensuremath{\omega})${e}^{\mathrm{\ensuremath{-}}i\ensuremath{\omega}t}$, VR, i.e., is it the response to some perturbing potential ${v}_{1}$(r,\ensuremath{\omega})${e}^{\mathrm{\ensuremath{-}}i\ensuremath{\omega}t}$ [E.K.U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850 (1985)]? In the present paper we show that (1) if the frequency \ensuremath{\omega}${\ensuremath{\omega}}_{\mathrm{min}}$ (the lowest excitation frequency), the answer is affirmative; and (2) if \ensuremath{\omega}>${\ensuremath{\omega}}_{\mathrm{min}}$, the answer is not necessarily affirmative, as demonstrated by counterexamples. [We exhibit ``reasonable'' functions ${n}_{1}$(r,\ensuremath{\omega})${e}^{\mathrm{\ensuremath{-}}i\ensuremath{\omega}t}$ which, at isolated frequencies, are not VR.] Implications for time-dependent DFT of linear response are discussed.
In density-functional theory (DFT) of the ground state, a density distribution no(r) is called v-representable (VR) if it is the ground-state density in some external potential. [It is known that not all reasonable'' ${n}_{0}$(r) are VR.] In DFT of time-dependent linear response of a nondegenerate ground state, a similar question arises: Is a response density, ${n}_{1}$(r,\ensuremath{\omega})${e}^{\mathrm{\ensuremath{-}}i\ensuremath{\omega}t}$, VR, i.e., is it the response to some perturbing potential ${v}_{1}$(r,\ensuremath{\omega})${e}^{\mathrm{\ensuremath{-}}i\ensuremath{\omega}t}$ [E.K.U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850 (1985)]? In the present paper we show that (1) if the frequency \ensuremath{\omega}${\ensuremath{\omega}}_{\mathrm{min}}$ (the lowest excitation frequency), the answer is affirmative; and (2) if \ensuremath{\omega}>${\ensuremath{\omega}}_{\mathrm{min}}$, the answer is not necessarily affirmative, as demonstrated by counterexamples. [We exhibit reasonable'' functions n1(r,\ensuremath{\omega})e\ensuremath−i\ensuremathωt which, at isolated frequencies, are not VR.] Implications for time-dependent DFT of linear response are discussed.
It is shown that if n(r) is the discrete density on a lattice (enclosed in a finite box) associated with a nondegenerate ground state in an external potential v(r) (i.e., is "v-representable"), then the density $n(r)+\ensuremath{\mu}m(r)$, with m(r) arbitrary (apart from trivial constraints) and $\ensuremath{\mu}$ small enough, is also associated with a nondegenerate ground state in an external potential ${v}^{\ensuremath{'}}(r)$ near v(r); i.e., $n(r)+\ensuremath{\mu}m(r)$ is also v-representable. Implications for the Hohenberg-Kohn variational principle and the Kohn-Sham equations are discussed.
A particle-density distribution is said to be v-representable if it is the ground-state density for some potential v(r). By developing the idea of proper order, proposed earlier [J. Chen and M. J. Stott, Phys. Rev. A 44, 2816 (1991)], to any number of energy levels which are involved in possible ground-state configurations, we show that any reasonable density with an arbitrary number of spinless fermions is either pure-state or ensemble v-representable. This result supports the general applicability of the Kohn-Sham method [Phys. Rev. 140, A1133 (1965)] in density-functional theory.
On the non-interacting v-representability conjecture within density functional theory
Density functional theory together with the Kohn-Sham (KS) scheme represent an efficient framework to recover the ground state density and energy of a many-body quantum system from an auxiliary system (one-body with a local potential that is a density functional). However, to fully achieve establishing the KS scheme, a general proof of the validity of the so-called v-representability conjecture is still required. In this article, we propose such a proof. We reduce the demonstration to proving (1) that the KS potential is differentiable for all non-interacting v-representable densities and (2) that its derivative is bounded. Then, we demonstrate points (1) and (2) applying static linear response to the non-interacting system.
Recent development in particle-particle random phase approximation (pp-RPA) broadens the perspective on ground state correlation energies [H. van Aggelen, Y. Yang, and W. Yang, Phys. Rev. A 88, 030501 (2013), Y. Yang, H. van Aggelen, S. N. Steinmann, D. Peng, and W. Yang, J. Chem. Phys. 139, 174110 (2013); D. Peng, S. N. Steinmann, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 104112 (2013)] and N ± 2 excitation energies [Y. Yang, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 224105 (2013)]. So far Hartree-Fock and approximated density-functional orbitals have been utilized to evaluate the pp-RPA equation. In this paper, to further explore the fundamentals and the potential use of pairing matrix dependent functionals, we present the linear-response time-dependent density-functional theory with pairing fields with both adiabatic and frequency-dependent kernels. This theory is related to the density-functional theory and time-dependent density-functional theory for superconductors, but is applied to normal non-superconducting systems for our purpose. Due to the lack of the proof of the one-to-one mapping between the pairing matrix and the pairing field for time-dependent systems, the linear-response theory is established based on the representability assumption of the pairing matrix. The linear response theory justifies the use of approximated density-functionals in the pp-RPA equation. This work sets the fundamentals for future density-functional development to enhance the description of ground state correlation energies and N ± 2 excitation energies.
The possibility of formulating a density-functional theory for time-dependent systems is studied in the linear-response regime. Two results are obtained: (i) The work of Runge and Gross is generalized to a finite time interval with an arbitrary form of the external potential; and (ii) there exists no zero in the mapping $n(\mathrm{r}, \ensuremath{\omega})=\ensuremath{\int}{d}^{3}{r}^{\ensuremath{'}}\ensuremath{\chi}(\mathrm{r}, {\mathrm{r}}^{\ensuremath{'}};\ensuremath{\omega})V({\mathrm{r}}^{\ensuremath{'}}, \ensuremath{\omega})$ for all $\ensuremath{\omega}$ under a very general thermodynamic condition.
The mapping of time-dependent densities on potentials in systems of identical quantum mechanical particles is examined. This mapping is of significance ever since Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] established its uniqueness, forming the theoretical basis for time-dependent density functional theory. Beyond uniqueness there are two important issues: existence, often called v-representability, and stability. We show that v-representability for localized densities in turn-on situations is not assured and we give a simple example of nonexistence. As for stability, we discuss an inversion procedure and by computing its Lyapunov exponents we demonstrate that the mapping is unstable with respect to fluctuations in the initial state. We argue that such instabilities will plague any inversion procedure.