Radial shells are defined by some kind of a mapping between the interval [0,1], or sometimes [-1,1], onto the semi-infinite interval [0,infinity). Precise mapping function that is used varies depending on the grid. See my SG-2/SG-3 paper cited above for one example and references to others, e.g., one of the original ones is Murray, Handy, & Laming, Ref. 26

The idea is that if there are Nr radial points (e.g., Nr = 50 for SG-1, Nr = 75 for SG-2, Nr = 99 for SG-3), one obtains them by dividing [0,1] up onto Nr equal segments and mapping those out onto the semi-infinite interval. Practical upshot is that the radial shells get farther apart as one moves away from the nucleus, which is appropriate because the density is slowly-varying when it is far away from all nuclei.

The mapping can be atomic-number dependent, e.g., the alpha exponents in Table 1 of the SG-2/SG-3, which are different for different nuclei and are used in the mapping function of Eq. (6).

**N.B.**, the radial mapping functions are not the same for different grids. In Q-Chem specifically, SG-1 uses and different radial mapping as compared to SG-2 and SG-3, and the unpruned Euler-Maclaurin-Lebedev (EML) grids use a different mapping altogether. This means, for example, that XC_GRID = 000075000302, which requests an unpruned (75,302) grid, uses different radial values as compared to SG-2, even though the latter is a pruned version of a (75,302) grid – but it’s a different radial quadrature scheme. This is done for a reason (discovery of new and better quadrature schemes over time) but is an occasional point of confusion.