Excitation energies of dissociating H-2: A problematic case for the adiabatic approximation of time-dependent density functional theory

Time-dependent density functional theory (TDDFT) is applied for calculation of the excitation energies of the dissociating H-2 molecule. The standard TDDFT method of adiabatic local density approximation (ALDA) totally fails to reproduce the potential curve for the lowest excited singlet (1)Sigma ()(u) state of H-2. Analysis of the eigenvalue problem for the excitation en ergies as well as direct derivation of the exchange-correlation (xc) kernel f(xc)(r,r('),omega) shows that ALDA fails due to breakdown of its simple s patially local approximation for the kernel. The analysis indicates a compl ex structure of the function f(xc)(r,r('),omega), which is revealed in a di fferent behavior of the various matrix elements K-1c,1c(xc) (between the hi ghest occupied Kohn-Sham molecular orbital psi (1) and virtual MOs psi (c)) as a function of the bond distance R(H-H). The effect of nonlocality of f( xc)(r,r(')) is modeled by using different expressions for the corresponding matrix elements of different orbitals. Asymptotically corrected ALDA (ALDA -AC) expressions for the matrix elements K-12,12(xc(sigma tau)) are propose d, while for other matrix elements the standard ALDA expressions are retain ed. This approach provides substantial improvement over the standard ALDA. In particular, the ALDA-AC curve for the lowest singlet excitation qualitat ively reproduces the shape of the exact curve. It displays a minimum and ap proaches a relatively large positive energy at large R(H-H). ALDA-AC also p roduces a substantial improvement for the calculated lowest triplet excitat ion, which is known to suffer from the triplet instability problem of the r estricted KS ground state. Failure of the ALDA for the excitation energies is related to the failure of the local density as well as generalized gradi ent approximations to reproduce correctly the polarizability of dissociatin g H-2. The expression for the response function chi is derived to show the origin of the field-counteracting term in the xc potential, which is lackin g in the local density and generalized gradient approximations and which is required to obtain a correct polarizability. (C) 2000 American Institute o f Physics. [S0021-9606(00)31143-6].