DENSITY-FUNCTIONAL EXCHANGE CORRELATION THROUGH COORDINATE SCALING IN ADIABATIC CONNECTION AND CORRELATION HOLE

The exact exchange-correlation functional E(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 E(xc)[n] = integral-0/1(V(ee)-alpha[n] - U[n])d-alpha, where V(ee)-alpha[n] is the electron-electron repulsion energy of PSI-n(min,alpha), which is that wave function that yields the density n and minimizes <T + alpha-V(ee)>. Here alpha is the coupling constant. Consequently, knowledge of the behavior of V(ee)-alpha[n] as a function of alpha ensures knowledge of E(xc)[n]. With this in mind and for the purpose of approximating E(xc), it was previously established that (partial V(ee)-alpha/partial-alpha) less-than-or-equal-to 0. The present paper reveals that V(ee)-alpha[n] = alpha-V(ee)1[n1/alpha], where n-beta(x,y,z) = beta-3n(beta-x,beta-y,beta-z), and where beta is a coordinate scale factor. In other words, knowledge of V(ee)1[n] implies knowledge of V(ee)-alpha[n] for all alpha. Alternatively, knowledge of V(ee)-alpha[n] for some small alpha implies knowledge of all of the V(ee)-alpha[n]. In any case, any viable approximation to V(ee)-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, rho-xc([n,alpha];r1,r2) = alpha-3-rho-xc([n1/alpha,1]; alpha-r1,alpha-r2), where rho-xc([n,alpha]; r1,r2) is the exact exchange-correlation hole of PSI-n(min,alpha). (A corresponding expression holds for the correlation hole alone.) Further, when n belongs to a noninteracting ground state that is nondegenerate, then lim-alpha --> 0 V(ee)-alpha[n] = A [n] + f(n)(alpha) B [n] + ..., where f(n)(alpha) must vanish at least as rapidly as alpha, and lim-lambda --> infinity E(c)[n-lambda] > - infinity, where E(c)[n] = E(xc)[n] - lim-gamma --> infinity gamma-1 E(xc)[n-gamma], and where E(c) is a familiar exact density-functional "correlation energy." In contrast, in the local-density approximation and in certain nonlocal approximations, f(n)(alpha) is replaced by a function that goes as alpha-[ln(alpha-1)], alpha --> 0, and E(c) is replaced by a functional that is unbounded as lambda --> infinity. Further, lim-lambda --> infinity E(xc)[n-lambda-x] > - infinity and lim-lambda --> infinity E(x)[n-lambda-x] > - infinity, which are also not generally satisfied by common approximations. Here n-lambda-x(x,y,z) = lambda-n (lambda-x,y,z) and E(x) is a familiar exact density-functional "exchange energy." Finally, comparison is made between E(c) and the traditional quantum-mechanical correlation energy, which is expressed exactly as a functional of the Hartree-Fock density.