Table 2.

Parameter Sets for the Different DFTB Variants

variant	seta	ζ b	$U_C^{\mathrm{d}}$ c	$U_H^{\mathrm{d}}$ c	$U_N^{\mathrm{d}}$ c	$U_O^{\mathrm{d}}$ c	$U_P^{\mathrm{d}}$ c	$U_S^{\mathrm{d}}$ c
DFTB2		-	-	-	-	-	-	-
DFTB2-γh	calc	3.70d	-	-	-	-	-	-
DFTB3-γ	calc	-	−0.1492	−0.1857	−0.1535	−0.1575	−0.0702	−0.0695
DFTB3-diag	calc	4.53	−0.1492	−0.1857	−0.1535	−0.1575	−0.0702	−0.0695
DFTB3-diag	fit	5.0	−0.04	−0.14	−0.11	−0.17	−0.07	-
DFTB3	calc	4.05	−0.1492	−0.1857	−0.1535	−0.1575	−0.0702	−0.0695
DFTB3	fit	4.2	−0.23	−0.16	−0.13	−0.19	−0.14	-

^a“calc” stands for calculated Hubbard derivatives U^d and/or ζ fitted to the water dimer; “fit” for a set of parameters fitted to a large set of binding energies and proton affinities. For details see text.

^bζ is the unitless parameter as defined in eq 21

^c$U_X^{\mathrm{d}}={\phantom{\mid }\frac{\partial U_X}{\partial n_X}\mid }_{n_X^0}$ is the Hubbard derivative with respect to the occupation number of the highest occupied atomic orbital n_X of atom type X in atomic units. In our third order formalism we define ${\phantom{\mid }\frac{\partial U_X}{\partial q_X}\mid }_{q_X^0}={\phantom{\mid }\frac{\partial U_X}{\partial n_X}\mid }_{n_X^0}$

^dNote that ζ is fitted to yield a binding energy for the water dimer of −4.9 kcal/mol in contrast to ζ = 4.5 as reported in ref³⁹ for DFTB2-γ^h where ζ was fitted to minimize the error of 22 selected binding energies.