Origin of the mysterious factor 2.6752 in COSMO

 Background

In 2007, we reported on a sort of non-empirical approach to obtain solvent parameters in dielectric continuum models, in particular for use with COSMO in ADF (see also this post for how the atomic radii were obtained, for instance the value of 1.858 for Fe):

M. Swart, E. Rösler and F.M. Bickelhaupt
"Proton Affinities in Water of Maingroup-Element Hydrides. Effects of Hydration and Methyl Substitution"
Eur. J. Inorg. Chem. 2007, 3646-3654
DOI 10.1002/ejic.200700228

with the specific formula R³ = 2.6752*M/rho (here M is the molecular mass of the solvent [amu], and rho the macroscopic density [kg/l]).

Explanation

This week I received a message from Eva Blokker from Amsterdam, asking me about the origin of this factor, and after helping her out, she kindly sent me a Excel sheet with all the conversion factors.

Conversion factors needed

As usual, it is best to do all conversion in atomic units (m_e = E_h = a₀ = 4pi·eps₀ = 1).

• From atomic mass unit (amu) to electron-mass (me): 1822.888486209 {F_m}
• Electron-mass (me): 9.1093837015E-31 kg
• Bohr radius (a0): 0.529177210903E-10 m
• Volume in atomic units [1L=1dm^3 => a0^3]: 6.74833449460E+27
• Density in atomic units [kg/l => me/a0]: 162.6725982 {F_r}
• (4/3)*pi: 4.188790205 {F_p}

By combining all factors and formulas, one obtains easily 2.6752 = (1/F_p)*F_m/F_r

Acknowledgments

Thank you Eva for the question and for the Excel sheet!