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 energies 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 spatially local approximation for the kernel. The analysis indicates a complex structure of the function f(xc)(r,r('),omega), which is revealed in a different behavior of the various matrix elements K-1c,1c(xc) (between the highest 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 proposed, while for other matrix elements the standard ALDA expressions are retained. This approach provides substantial improvement over the standard ALDA. In particular, the ALDA-AC curve for the lowest singlet excitation qualitatively reproduces the shape of the exact curve. It displays a minimum and approaches a relatively large positive energy at large R(H-H). ALDA-AC also produces a substantial improvement for the calculated lowest triplet excitation, which is known to suffer from the triplet instability problem of the restricted KS ground state. Failure of the ALDA for the excitation energies is related to the failure of the local density as well as generalized gradient approximations to reproduce correctly the polarizability of dissociating 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 lacking in the local density and generalized gradient approximations and which is required to obtain a correct polarizability.