Salient signature of van der Waals interactions

Abstract

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...

Range separation of the Coulomb hole

“…c II evaluated at R, R being the distance between any two atoms in the molecule will decay like R −3 when R → ∞. 74 This dependency is connected with the well-known R −6 decay of the van der Waals (vdW) dispersion energy. Because dispersion interactions are weak, the region in h c II (s) is proportionally small.…”

“…The SD approximation includes neither short- 73 nor long-range 74 dynamic correlation, that is, it only considers nondynamic correlation. Even though it cannot be guaranteed that all nondynamic correlation in the system is accounted for, it is legitimate to claim that the SD pair density, eq 6, includes some extent of it.…”

“…98 The only peak in the Coulomb hole corresponds to dispersion interactions and, thus, it is captured by h c II (s). 74 h c II (s) peaks around the interatomic separation, R HH ′ , indicating that HF underestimates the number of electron pairs at R HH ′ , which are placed at shorter and longer distances. On the other hand, there is almost no h c I (s) contribution at equilibrium, and it is zero at the stretched geometry.…”

Singling Out Dynamic and Nondynamic Correlation

“…However, ρ SD 2 (ρ 1 , 1, 2) can be regarded as an approximation to the actual pair density; an approximation which does not account for dynamic correlation either at short 30 or at long range. 28 Figure 1 depicts the two paths of arriving at the exact ρ 2 (ρ 1 , 1, 2) from ρ SD 2 (ρ HF 1 , 1, 2), either straightforwardly or through the intermediate SD approximation. The latter path defines the decomposition of the correlation part of the pair density,…”

Electron-Pair Distribution in Chemical Bond Formation

“…Interestingly, the intracule density is related to an experimental observable, as it can be obtained from X-ray scattering techniques. [40][41][42] The intracule density has been previously used to analyze the electronic structure and electron correlation of some molecular systems 24,[43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59] but very few studies of the intracule density have been devoted to the investigation of bond formation. [60][61][62] The aim of this paper is to understand the changes occurring in the intracule density during the chemical bond formation.…”