# Does this averaging function have a name? (“sigma profiles” in chemistry)

I am looking for a mathematical (!!) answer, NOT a chemical one, hence the post to Mathematics.

Quick description: Solvation modelling with for example COSMO-SAC uses something called a sigma profile calculated over a cavity (volume). We have surface segments with a known area, charge and position in 3D space. (Basically a long table of data.)

It is described as the "average surface charge density" - but no explicit argument for the use of the formula is given. Does it have a name or specific components?

$$\sigma_m = \frac{ \sum_n{\sigma_n\frac{r^2_{n} r^2_{eff}}{r_n^2 + r^2_{eff}}\exp\left(\frac{-d^2_{mn}}{r^2_n +r^2_{eff}}\right)} }{ \sum_n{\frac{r_n^2 r^2_{eff}}{r_n^2 + r^2_{eff}}\exp\left(\frac{-d^2_{mn}}{r^2_n +r^2_{eff}}\right)} }$$

$r_n$ is the segment area over $\pi$

$r_{eff}$ is a constant (dependent on a parameter used in the calculation)

$d_{mn}$ is the distance (via 3D Pythagoras for example) between area segment $m$ and $n$ (with co-ordinates known from the software that created the cavity). $\sigma_n$ is the charge of element $n$.

Searching for "sigma profile" or related, hasn't given me any clearer indication as to what the mathematical foundation of this expression is.

Does anybody know what branch of mathematics this came from? Where I could find a proper argument for the expression used?

(N.B. there is a correction to the original paper with a parameter c, which accounts for a unit conversion issue, but is effectively arbitrary...)

Original paper where the function was introduced https://pubs.acs.org/doi/abs/10.1021/ie001047w (At least for me.)

• An "average surface charge density" would be something like $\dfrac{\sum_n\sigma_nA_n}{\sum_nA_n}$, where $\sigma_n$ and $A_n$ are the charge density and area of each element. If the term in the summation in the denominator is some kind of area estimate, the formula would make sense. – Rahul Apr 18 at 13:15
• In electrical engineering they frequently call it a circuit. An averaging circuit. Even if its in software. Its a circuit because its an in-out path for data. – CogitoErgoCogitoSum Apr 18 at 15:17
• @Rahul in the next steps of the calculation, there is a multiplication with the area to get a "sigma profile" (in this case a fancy name for the histogram) of the charge distribution - scaled to a specific range. – DetlevCM Apr 18 at 17:27
• @CogitoErgoCogitoSum Do you have any good (general) references to recommend or is "just googling it" the best I can do? – DetlevCM Apr 18 at 17:31

This is generically called a weighted average (of the $\sigma_n$'s). The weighting coefficients have an expression very specific to the domain and have no reason to get a particular mathematical name.