Why does the local density approximation (LDA) overestimate atomization energy?
What is the cause?


The LDA exchange hole is spherically symmetric and centered on a reference electron, which is a fairly good approximation for the case of molecules, but not for the case of atoms. For instance, in an atom the exact exchange hole is displaced toward the nucleus, while the LDA exchange hole is always centered on its reference electron. This inability of LDA exchange hole to appropriately describe the situation in atoms leads to significant deviations in energies upon bond formation or breaking.


  Also, how can inclusion of gradient can help solve this problem?


In general, the electron density in atoms is more inhomogeneous than in molecules, and GGA is ntroduced exactly in order to account for the non-homogeneity. In particular, the gradients in the separated atoms are usually bigger than the gradients in the corresponding molecule, and therefore gradient corrections lower the energy of the atoms more than the energy of the molecule, which leads to a reduction of the overbinding.

See, e.g. Ernzerhof, M., Perdew, J. P. and Burke, K. (1997), Coupling-constant dependence of atomization energies. Int. J. Quantum Chem., 64: 285–295. DOI: 10.1002/(SICI)1097-461X(1997)64:3&lt;285::AID-QUA2>3.0.CO;2-S