(a) Activation energies are calculated as the energy difference between two geometries; the corresponding alchemical derivative at atom is defined in Eq. (9) . (b) In a dissociation reaction where atom is only present in the reactant and product space and , respectively, space does not contribute to the alchemical derivative .
(a) The minimum energy path for the umbrella inversion of passes through a planar transition state. This path is shown for several values of , where is alchemically transformed to . (b) for the inversion process as a function of . The structures of the transition states are shown for select values of .
The bond distance between the central atom and the hydrogen being annihilated increases along for both the relaxed initial structure and the relaxed transition state for the umbrella flipping process.
Isosurfaces of the alchemical potential from Eq. (12) are shown for the protonated methane series using geometries fixed to .
Left to right is the protonation energy as is alchemically transformed into Ne. The structures shown as insets are the protonated end points whose pseudopotentials have been interpolated when following alchemical paths. For each alchemical path the number of protons on the central nucleus is increased by one and a hydrogen nucleus is concurrently annihilated. The solid symbols represent configurations fixed to the optimal geometry and the empty symbols are relaxed. A consequence of using fixed structures is that all hydrogens are symmetrically equivalent for the geometry; two separate paths emerge by selecting which hydrogen is to be annihilated.
Correlation of the analytical [Eq. (9) ] and numerical [Eq. (13) ] alchemical derivatives and , for all paths and all values of . The inset shows a similar correlation for the sum of these atomic components, which give derivatives of the binding and activation energies along the alchemical paths in Table I .
The alchemical potential , calculated from Eqs. (8) and (17) , for the binding of O to a 79 atom Pd nanoparticle. Colors represent the value of the alchemical potential on each atom. To weaken oxygen binding atoms colored in red should be changed to the nobler Ag and atoms colored in blue should be changed to Rh. The opposite transformations should be made to strengthen the oxygen bond.
Correlation of alchemically predicted changes, for , with actual changes to the binding energy, , of molecular oxygen to a 79 atom Pd cluster (initial) due to integer variations in atomic numbers (final).
Investigated properties, reaction enthalpies and energy barriers , and chemical species connected by alchemical paths via the order parameter .
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