User:Szggr/sandbox/Tkatchenko-Scheffler method

The Tkatchenko-Scheffler (TS) method is a computational chemistry method to model van der Waals parameters based on free atomic data. The model gives a good approximation for the $$C_6$$ pairwise interaction term.

Formulation
A good approximation for van der Waals interactions is a pairwise approximation in the form:

The sum is over all atoms in the system $$A, B$$, the function $$f_{damp}(R_{AB},R_A^0,R_B^0)$$ is a damping function that, in general, can depend on the free atom van der Waals radii and the distance between atoms $$A$$ and $$B$$. The damping function used in the original method is:

with using the sum of atomic radii $$R_{AB}^0=R_A^0+R_B^0$$ and having $$s_R$$ and $$d$$ as free parameters to fit.

According to the formula of Equation ($$), a full description of the van der Waals forces in this approximation consists of determining the $$C_{6AB}$$ coefficients for the set of atomic radii used. The Tkatchenko-Scheffler method expressed the heteronuclear parameter $$C_{6AB}$$ as a function of homonuclear $$C_{6AA}$$ parameters with the following combination rule:

The original work is based on free atom reference data of Chu and Dalgarno.