Transition State procedure
This phrase stands for an analysis method described in ref. [3] and has no relation to transition states in chemical reactions. An extensive discussion of bond energy analysis by ADF is given in [4, 5]
The energy associated with a change in charge density, say the relaxation to self-consistency from the sum-of-orthogonal-fragments, can be computed by subtracting final and initial energies, each obtained from the corresponding charge density. For purposes of analysis the change in energy de can be reformulated as
| dE = ∫ dr { [ρfinal(r)-ρinitial( r)] ∗ | ρfinal ∫ dρ ρinitial |
F[ρ(r)] } (1.2.8) |
F(ρ) is the Fock operator belonging to the charge density ρ
By writing the density difference ρfinal - ρinitial a summation over contributions from the different irreducible representations Γ of the molecular symmetry group, an expression is obtained that lends itself for a decomposition of the bond energy into terms from the different symmetry representations:
| dE = | ∑ Γ |
∫ dr { ΔρΓ(r) ∗ | ρfinal ∫ dρ ρinitial |
F[ρ(r)] } (1.2.9) |
The integral of the Fock operator over the charge density is now approximated by a weighted summation (in fact, a Simpson integration):
| ρfinal ∫ dρ F[ρ] ≈ ρinitial |
1/6 F(ρinitial) + 2/3 F(ρaverage) + 1/6 F(ρfinal) (1.2.10) |
ρaverage = 1/2 ρinitial + 1/2 ρfinal (1.2.11)
The term with the Fock operator due to the average charge density has given rise to the phrase transition state. To avoid confusion we will often refer to it as to the transition field.
The approximate integral (1.2.10) involves two errors. The first, rather obvious, is the approximation of the exact integral in (1.2.9) by the weighted sum in (1.2.10). Except in pathological cases this approximation is highly accurate.
The second error comes from the fact that the Coulomb and xc potentials in the Fock operator are computed from the fit density. This is only an approximation to the true density, while in the original bond-energy expression (energy due to the final density minus energy due to the initial density) no potentials occur and the exact charge density can be used. As mentioned before, these fit-related errors are usually small.
All such errors in the total bonding energy are easily corrected by comparing the summation over the Γs with the correct value for the total bonding interaction term. The difference is simply added to the total bond energy, so no true error remains. We only have a (correction) term that can't be split in contributions from the distinct symmetry representations. In the printed bond energy analysis such small corrections are 'distributed' over the other terms by scaling the other terms such that their sum is the correct total value.
