An analytical treatment of disorder and resolution of atomic electron-density functions for carbon

Unresolved disorder can lead to structural distortions because of averaged atomic positions. The influence of the isotropic probability density function (p.d.f.), the distance between the disordered positions and the site occupation factors of a disordered C atom on the apparent position and anisotropic p.d.f. of the adjusted atom is studied with a simple model for the centrosymmetrical case. The electron density is derived from the STO-3G wave functions and convoluted analytically with the corresponding p.d.f. The optimal positional and displacement parameters are obtained by minimization of the integral of the square of the difference electron density. Several electron- and difference-density plots are shown in order to demonstrate the goodness of the adjustment and several correlations between the parameters of the disordered and adjusted atoms are discussed. The results are applied to some examples where unresolved disorder may be possible.