Density-functional exchange correlation through coordinate scaling in adiabatic connection and correlation hole

Abstract

The exact exchange-correlation functional Exc[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 Exc[n]=F01(Vee±[n]-U[n])d±, where Vee±[n] is the electron-electron repulsion energy of nmin,±, which is that wave function that yields the density n and minimizes T+±V. Here ± is the coupling constant. Consequently, knowledge of the behavior of Vee±[n] as a function of ± ensures knowledge of Exc[n]. With this in mind and for the purpose of approximating Exc, it was previously established that (Vee±/±)0. The present paper reveals that Vee±[n]= ±Vee1[n1/±], where n(x,y,z)=3n(x,y,z), and where is a coordinate scale factor. In other words, knowledge of Vee1[n] implies knowledge of Vee±[n] for all ±. Alternatively, knowledge of Vee±[n] for some small ± implies knowledge of all of the Vee±[n]. In any case, any viable approximation to Vee±[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, xc([n,±];r1,r2)=±3xc([n1/±,1]; ±r1,±r2), where xc([n,±]; r1,r2) is the exact exchange-correlation hole of nmin,±. (A corresponding expression holds for the correlation hole alone.) Further, when n belongs to a noninteracting ground state that is nondegenerate, then lim0 Vee±[n]=A[n]+fn(±)B[n]+..., where fn(±) must vanish at least as rapidly as ±, and lim Ec[n]>-, where Ec[n]=Exc[n]- lim' -1Exc[n], and where Ec is a familiar exact density-functional correlation energy. In contrast, in the local-density approximation and in certain nonlocal approximations, fn(±) is replaced by a function that goes as ±[ln(±-1)], 0, and Ec is replaced by a functional that is unbounded as . Further, lim Exc[nx]>- and lim Ex[nx]>-, which are also not generally satisfied by common approximations. Here nx(x,y,z)=n(x,y,z) and Ex is a familiar exact density-functional exchange energy. Finally, comparison is made between Ec and the traditional quantum-mechanical correlation energy, which is expressed exactly as a functional of the Hartree-Fock density. © 1991 The American Physical Society.