Atomic properties of selected biomolecules. Part 1. The interpretation of atomic integration errors

Reliable atomic properties can be obtained via the theory of "Atoms in Molecules" (AIM) via integration over a finite volume. These integrations are challenging because of the variety and complexity of the shape of the AIM atoms. In practice the integration of a large number of atoms (100-1000, sampled from many molecules) yields integration errors L(Omega) of varying magnitude. We prove that it is impossible to predict the size of an angular Gauss-Legendre grid (outside the beta sphere) that guarantees a pre-set error. Hence it is incorrect to assume that a large grid (similar to 23 000 angular grid points) will automatically yield a low L(Omega) value. The erratic relationship between the integration error and the grid size prompts a statistical interpretation of atomic integration, at a purely practical level. More importantly we have investigated the relationship between L(Omega) and seven atomic properties which include volume, energy, and the magnitudes of five electrostatic multipole moments. The electronic population (N(Omega)) and the volume (v(Omega)) of carbon is linearly correlated with L(Omega), enabling the interpolation or extrapolation of N(Omega) and v(Omega). Other properties of carbon and other atoms (N, O, and S) yield low correlation coefficients but occasionally trends can be observed. For example, we find that some properties are systematically underestimated if L(Omega) is negative. This work has led to an estimate of safe error bars of atomic properties for atoms occurring in biological molecules with reasonably sized integration grids. The most stable properties were found to be the energy and the population. Finally, we have observed that the influence of the grid orientation is less if L(Omega) is small, and that population and energy are the least affected.