The exact exchange-correlation functional ${\mathit{E}}_{\mathrm{xc}}$[n] must be approximated in density-functional theory for the computation of electronic properties. By the coupling-constant integration (adiabatic-connection) formula we know that ${\mathit{E}}_{\mathrm{xc}}$[n]=${\mathcal{F}}_0^1$(${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n]-U[n])dα, where ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n] is the electron-electron repulsion energy of ${\mathrm{\Psi }}_{\mathit{n}}^{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{\alpha }}$, which is that wave function that yields the density n and minimizes 〈T^+αV${\mathrm{^}}_{\mathit{e}\mathit{e}}$〉. Here α is the coupling constant. Consequently, knowledge of the behavior of ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n] as a function of α ensures knowledge of ${\mathit{E}}_{\mathrm{xc}}$[n]. With this in mind and for the purpose of approximating ${\mathit{E}}_{\mathrm{xc}}$, it was previously established that (∂${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$/∂α)≤0. The present paper reveals that ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n]=α${\mathit{V}}_{\mathit{e}\mathit{e}}^1$[${\mathit{n}}_{1/\mathrm{\alpha }}$], where ${\mathit{n}}_{\mathrm{\beta }}$(x,y,z)=${\mathrm{\beta }}^3$n(βx,βy,βz), and where β is a coordinate scale factor.

In other words, knowledge of ${\mathit{V}}_{\mathit{e}\mathit{e}}^1$[n] implies knowledge of ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n] for all α. Alternatively, knowledge of ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n] for some small α implies knowledge of all of the ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n]. In any case, any viable approximation to ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n] should be made to satisfy the above displayed equality. Analogous conclusions hold for the second-order density matrix, the pair-correlation function, the exchange-correlation hole, and the correlation component of the exchange-correlation hole, etc. For example, ${\mathrm{\rho }}_{\mathrm{xc}}$([n,α];${\mathbf{r}}_1$,${\mathbf{r}}_2$)=${\mathrm{\alpha }}^3$${\mathrm{\rho }}_{\mathrm{xc}}$([${\mathit{n}}_{1/\mathrm{\alpha }}$,1]; α${\mathbf{r}}_1$,α${\mathbf{r}}_2$), where ${\mathrm{\rho }}_{\mathrm{xc}}$([n,α]; ${\mathbf{r}}_1$,${\mathbf{r}}_2$) is the exact exchange-correlation hole of ${\mathrm{\Psi }}_{\mathit{n}}^{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{\alpha }}$.

(A corresponding expression holds for the correlation hole alone.) Further, when n belongs to a noninteracting ground state that is nondegenerate, then ${\lim }_{\mathrm{\alpha }\to 0}$ ${\mathrm{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n]=A[n]+${\mathit{f}}_{\mathit{n}}$(α)B[n]+..., where ${\mathit{f}}_{\mathit{n}}$(α) must vanish at least as rapidly as α, and ${\lim }_{\lambda \to \mathrm{\infty }}$ ${\mathrm{E}}_{\mathit{c}}$[${\mathit{n}}_{\lambda }$]>-∞, where ${\mathit{E}}_{\mathit{c}}$[n]=${\mathit{E}}_{\mathrm{xc}}$[n]- ${\lim }_{\gamma \to \mathrm{\infty }}$ ${\gamma }^{\mathrm{-}1}$${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\gamma }$], and where ${\mathit{E}}_{\mathit{c}}$ is a familiar exact density-functional ‘‘correlation energy.’’ In contrast, in the local-density approximation and in certain nonlocal approximations, ${\mathit{f}}_{\mathit{n}}$(α) is replaced by a function that goes as α[ln(${\mathrm{\alpha }}^{\mathrm{-}1}$)], α→0, and ${\mathit{E}}_{\mathit{c}}$ is replaced by a functional that is unbounded as λ→∞. Further, ${\lim }_{\lambda \to \mathrm{\infty }}$ ${\mathrm{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\lambda }^{\mathit{x}}$]>-∞ and ${\lim }_{\lambda \to \mathrm{\infty }}$ ${\mathit{E}}_{\mathit{x}}$[${\mathit{n}}_{\lambda }^{\mathit{x}}$]>-∞, which are also not generally satisfied by common approximations. Here ${\mathit{n}}_{\lambda }^{\mathit{x}}$(x,y,z)=λn(λx,y,z) and ${\mathit{E}}_{\mathit{x}}$ is a familiar exact density-functional ‘‘exchange energy.’’ Finally, comparison is made between ${\mathit{E}}_{\mathit{c}}$ and the traditional quantum-mechanical correlation energy, which is expressed exactly as a functional of the Hartree-Fock density.

• Received 19 October 1990

DOI:https://doi.org/10.1103/PhysRevA.43.4637