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 ${n}_{o}$(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.

Physical Review AJun 1, 198744 citations

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.