A second variation of the energy functional: III. Quantum force for surface hardness from chemical hardness

In the previous two papers, we have found that energy is not conserved for molecular processes with charge transfers if the chemical hardness is introduced, while the hardness must be zero for a gas-phase free-particle model, such as in the density-functional theories, the Thomas-Fermi model, and the free electron model of metals. Non-zero chemical hardness provides a non-gaseous collection of surface-atoms having surface hardness. We apply our results to an adatom on a surface in analogue with the molecular reaction processes except that the total surface energy is kept constant. We find the energy increments due to quantization of the integer number of electrons in the charge transfer processes. Two kinds of increments are obtained from Koopmans' theorem and from the affinity respectively. We attribute these energy increments to the hardness acting in a very similar way to the macroscopic static friction before moving. The sum of the original energy and the increment energy provides the surface hardness in mechanics far surface plus surface adatoms. This energy increment can, in turn, be attributed to a kind of quantum force, an increment of the elastic force. This increment provides a stiffer or harder surface. We demonstrate our theory on H/Al(111) case. Conceptually we find that static frictional forces have the same origin as this surface hardness. Any changes of energy-levels-matching between surface and the adatom will cause energy non-conservation just like the particle collision processes. This is also one possible way for particle-atomic adsorbates to form liquid-laminar- or solid-plane-surfaces by a dimensional change of the system, from lower-dimension to two-dimension.