Time-dependent density functional theory (TDDFT) calculations of charge-transfer excitation energies omega(CT) are significantly in error when the adiabatic local density approximation (ALDA) is employed for the exchange-correlation kernel f(xc). We relate the error to the physical meaning of the orbital energy of the Kohn-Sham lowest unoccupied molecular orbital (LUMO). The LUMO orbital energy in Kohn-Sham DFT-in contrast to the Hartree-Fock model-approximates an excited electron, which is correct for excitations in compact molecules. In CT transitions the energy of the LUMO of the acceptor molecule should instead describe an added electron, i.e., approximate the electron affinity. To obtain a contribution that compensates for the difference, a specific divergence of f(xc) is required in rigorous TDDFT, and a suitable asymptotically correct form of the kernel f(xc)(asymp) is proposed. The importance of the asymptotic correction of f(xc) is demonstrated with the calculation of omega(CT)(R) for the prototype diatomic system HeBe at various separations R(He-Be). The TDDFT-ALDA curve omega(CT)(R) roughly resembles the benchmark ab initio curve omega(CT)(CISD)(R) of a configuration interaction calculation with single and double excitations in the region R=1-1.5 A, where a sizable He-Be interaction exists, but exhibits the wrong behavior omega(CT)(R)much less thanomega(CT)(CISD)(R) at large R. The TDDFT curve obtained with f(xc)(asymp) however approaches omega(CT)(CISD)(R) closely in the region R=3-10 Angstrom. Then, the adequate rigorous TDDFT approach should interpolate between the LDA/GGA ALDA xc kernel for excitations in compact systems and f(xc)(asymp) for weakly interacting fragments and suitable interpolation expressions are considered. (C) 2004 American Institute of Physics.