Energetic ground state calculation and Electronic band structure at surfaces are an important field of research in surface science, since crystallographic surfaces have been grown under ultra high vacuum conditions and several surface sensitive techniques have been developed. Currently clean, almost 2D periodic surfaces can be obtained even after adsorbing complex organic molecules.Adsorption is a very relevant phenomenon and well known are its implications to corrosion, heterogeneous catalysis and more recently to organic electronics. But the understanding of such processes requires that of bonding of atoms and molecules at solid surfaces. Nowadays a joint experimental and theoretical efforts, based on first principles calculations, strongly propelled by the enormously increased computer power, is often successful in determining adsorption sites, geometry, possible surface reconstruction, orientation of molecules, potential energy barriers, wavefunctions, vibrations, and more recently in a few cases many-body properties such as plasmon, polariton on metals, magnons, etc.. The density of states of the clean surface and of that of the projected surface bands onto the molecular eigenstates enjoy now clearer pictures. We can distinguish surface projected bulk states, resonant surface ones but also localized surface electronic states. Refined two photon photoemission experiments or inverse photoemission ones have detected image potential states determined by the long range Coulomb tails of the surface potential. Quantum well states localized between an adsorbate overlayer and the surface are also important. Surface states can be used as a tool to monitor adsorption but may also occur as a consequence of such phenomena.The study of molecule-surface interaction boast a great variety of systems and of