van der Waals interactions govern the physics of a plethora of molecular structures. It is well known that the leading term in the distance-based London expansion of the van der Waals energy for atomic and molecular dimers decays as 1/R 6 , where R is the dimer distance. Using perturbation theory, we find the leading term in the distance-based expansion of the intracule pair density at the interatomic distance. Our results unveil a universal 1/R 3 decay, which is less prone to numerical errors than the 1/R 6 dependency, and it is confirmed numerically in H2 and He2 molecules. This signature of van der Waals interactions can be directly used in the construction of approximate pair density and energy functionals including vdW corrections.Dispersion or van der Waals (vdW) interactions are ubiquitous in nature, governing the stability of molecules and materials, 1 and having an essential role in molecular recognition, 2 the double-helical structure of DNA, 3 molecular adsorption on surfaces, 4,5 and the adhesion of micromachined surfaces.6 They are so important in physics, chemistry and biology, that even the most simple electronic structure methods consider corrections for vdW interactions. Due to their long-range dynamiccorrelation nature, they are not well modelled by standard functionals in density functional theory (DFT), 1 which by construction are essentially local or semi-local in nature.7 Hence, except for a few functionals, 8,9 most DFT functionals include ad hoc energy corrections to account for vdW interactions.
10-12vdW forces arise from the electrostatic interaction between fluctuations in the electron density, and the pairwise effect in the energy shows a leading 1/R 6 dependency, where R is the interaction distance between fragments. This fact is often exploited in the construction of effective pairwise potentials that enter the expressions of various methods. On the other hand, the effect of vdW interactions in the wavefunction or related quantities has been less discussed in the literature. [13][14][15][16] This knowledge could shed some light in the design of computational approaches including vdW interactions and provide further tests to calibrate electronic structure theory methods.In this letter we use perturbation theory to find the leading term in the expansion of the intracule pair density in terms of R, the interatomic distance. Our results reveal a universal 1/R 3 dependency that is corroborated by numerical calculations in H 2 and He 2 molecules. Upon integration of the vdW contribution to the intracule we recover the vdW energy that follows the stablished 1/R 6 dependency.
FIG. 1. Two atoms,A and B, separated by a distance R and two electrons with coordinates rA1 and rB2 defined with respect to the position of atoms A and B.
I. THEORYWe start from the unperturbed wavefunction for a system of two hydrogen atoms, A and B, separated a distance R, given by the product of two hydrogenoid 1s functions,For convenience we have chosen to represent the hydrogenoid functions by one Gaussian functio...