It seems almost all the common implementations of Generalized Born (GB) implicit solvent models (in Amber, OpenMM etc) use the "pairwise descreening" approximation, which is described in Hawkins, Cramer & Truhlar 1995[1]:

... $H_k$ is eclipsed surface area of sphere of radius $r$ centered at atom $k$. Therefore $$\alpha_k^{-1}=\rho_k^{-1}-\int_{\rho_k}^{\infty}\frac{dr}{r^2}H_k(r,\{R_{kk'},\rho_{k'}\}_\text{all k'})\tag{9}$$ Using the pairwise descreening approximation, we can replace Eq. (9) with $$\alpha_k^{-1}=\rho_k^{-1}-\sum_{k'}\int_{\rho_k}^{\infty}\frac{dr}{r^2}H_{kk'}(R_{kk'},\rho_{k'})\tag{10}$$ where $H_{kk'}$ is the fraction of the area of a sphere with radius $r$ centered at atom $k$ that is shielded by a sphere of radius $\rho_{k'}$ at a distance $R_{kk'}$. Eq. (10) is exact provided there is no overlap between any atomic spheres $k',k''\neq k$.

$\alpha_k$ is the Generalized Born radius of atom $k$. Equation 9 is exact, where the integral is over the area eclipsed by all nearby atoms; equation 10 is the approximation, which is a sum of integrals over the area eclipsed by one atom at a time.

I understand what the approximation does, but I don't understand how it is not a really inaccurate approximation, especially with bonded atoms. Consider a sphere of radius r (dashed line) around atom A, which intersects two atoms B and C bonded to each other (where $\text{bond length} \ll \text{sum of atom radiuses}$).

Overlapping spheres B and C within a radius r of sphere A

Under the pairwise approximation, the area where the two atoms overlap would count as being eclipsed twice (instead of the correct method which is to count it once). In a typical biomolecular simulation (e.g. a protein), each atom is bonded to a few others and the Born radius (~$\pu{0.27nm}$ for carbon) is much more than the bond length (~$\pu{0.15nm}$ for C-C bond). Most bonded atoms would have a lot of overlap - back of the envelope, two carbons bonded to each other would have about 65% area overlap in a plane that passes through their nuclei, and similar percentage overlap in the integral in equation 10. In a crowded environment where a lot of the space is occupied (relatively little solvent volume) and most atoms each have a few bonds (and hence overlap the atoms they're bonded to), this would seem to result in a very large relative error in calculated GB radius.

How is this approximation okay? Are there any systematic studies of the degree of error this introduces?

P.S. I think pairwise descreening idea originally comes from Schaeffer and Froemmel 1995[2]; the Hawkins paper above may be the first to derive the analytical expression for the GB radius.


  1. Hawkins, G. D.; Cramer, C. J.; Truhlar, D. G. Pairwise solute descreening of solute charges from a dielectric medium. Chem. Phys. Lett. 1995, 246 (1-2), 122–129. DOI: 10.1016/0009-2614(95)01082-K.
  2. Schaefer, M.; Froemmel, C. A precise analytical method for calculating the electrostatic energy of macromolecules in aqueous solution. J. Mol. Biol. 1990, 216 (4), 1045–1066. DOI: 10.1016/S0022-2836(99)80019-9.
  • $\begingroup$ @NikeDattani Unfortunately I have not figured this out at all; it still seems GB with the pairwise approximation is just wrong, and I don't understand why people thought it would work in the original papers, or why this version is so widely used. I would be very interested in an answer though. $\endgroup$
    – Alex I
    Sep 8, 2022 at 5:38
  • $\begingroup$ @NikeDattani I have however done a little bit of looking for examples of simple molecules where GB disagrees with other methods. For example: try calculating the (pairwise approximation)GB energy of neutral triethylamine in water with any reasonable geometry and charges, and compare with the PB results. $\endgroup$
    – Alex I
    Sep 8, 2022 at 5:43

1 Answer 1


This would seem to result in a very large relative error in calculated GB radius. How is this approximation okay?

The systematic error is corrected in an average way by the scale parameters on page 125 of the Hawkins et al. paper.

Are there any systematic studies of the degree of error this introduces?

The mean-unsigned error in the polarization free energies between the pairwise method and the method that does not make ths simplification is 0.38 kcal/mol. See Table 2 of the Hawkins et al. paper.


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