Short-Range Cut-Off of the Summed-Up van der Waals Series: Rare-Gas Dimers

Abstract

van der Waals interactions are important in typical van der Waals-bound systems such as noble gas, hydrocarbon, and alkaline earth dimers. The summed-up van der Waals series of Perdew et al. 2012 works well and is asymptotically correct at large separation between two atoms. However, as with the Hamaker 1937 expression, it has a strong singularity at short non-zero separation, where the two atoms touch. In this work we remove that singularity (and most of the short-range contribution) by evaluating the summed-up series at an effective distance between the atom centers. Only one fitting parameter is introduced for this short-range cut-off. The parameter in our model is optimized for each system, and a system-averaged value is used to make the final binding energy curves. This method is applied to different noble gas dimers such as Ar–Ar, Kr–Kr, Ar–Kr, Ar–Xe, Kr–Xe, Xe–Xe, Ne–Ne, He–He, and also to the Be₂ dimer. When this correction is added to the binding energy curve from the semilocal density functional meta-GGA-MS2, we get a vdW-corrected binding energy curve. These curves are compared with the results of other vdW-corrected methods such as PBE-D2 and vdW-DF2, and found to be typically better. Binding energy curves are in reasonable agreement with those from experiment.

Appendices

Appendix 1 Summed-Up Series Expression

The Casimir–Polder [50, 51] formula for the van der Waals coefficients between two objects A and B to the second-order in electron–electron interaction is

$$ {C}_{2k}^{\mathrm{A}\mathrm{B}}=\frac{\left(2k-2\right)!}{2\pi }{\displaystyle \sum_{l_1=1}^{k-2}\frac{1}{\left(2{l}_1\right)!\left(2{l}_2\right)!}}{\displaystyle {\int}_0^{\infty }du{\alpha}_{l_1}^{\mathrm{A}}(iu){\alpha}_{l_2}^{\mathrm{B}}(iu)}, $$

(19)

where ${l}_2=k-{l}_1-1$ and ${\alpha}_{l_1}^{\mathrm{A}}\to {2}^{l_1}$ pole dynamic polarizability of A at imaginary frequency ω = iu. It should be noted that $l=1,\ l=2,\ l=3$ are for dipole, quadrupole, and octupole interactions, respectively.

The dynamic multipole polarizabilities for a classical conducting spherical shell can be found from the following expressions:

$$ {\alpha}_l(iu)={R}^{2l+1}\frac{\omega_l^2}{{\omega_l}^2+{u}^2}\frac{1-{\theta}_l}{1-{\beta}_l{\theta}_l}, $$

(20)

where

$$ {\beta}_l=\frac{{\omega_l}^2{\tilde{\omega}}_l^2}{\left({\omega_l}^2+{u}^2\right)\left({\tilde{\omega}}_l^2+{u}^2\right)} $$

(21)

and

$$ {\theta}_l={\left(\frac{R-t}{R}\right)}^{2l+1}={\left(1-t/R\right)}^{2l+1} $$

(22)

from the work of Lucas et al. [52]. Here ${\omega}_l={\omega}_{\mathrm{p}}\sqrt{l/\left(2l+1\right)}$ and ${\tilde{\omega}}_l={\omega}_{\mathrm{p}}\sqrt{\left(l+1\right)/\left(2l+1\right)}$. The plasma frequency of the system is ${\omega}_{\mathrm{p}}=\sqrt{4\pi \rho }$ with $\rho =\frac{N}{\left[\frac{4}{3}\pi \left\{{R}^3-{\left(R-t\right)}^3\right\}\right]}$ for the spherical shell (with radius R and thickness t) and $\rho =\frac{N}{\left(\frac{4}{3}\pi {R}^3\right)}$ for the sphere (with radius R). N is the total number of valence electrons, equal to 2 for He and 8 for other rare gas atoms.

In Perdew et al. [37] it is shown that, for a classical conducting spherical shell of radius R, thickness t, and uniform density ρ, the above integration at (18) can be performed to get all the higher order vdW coefficients. For two identical spheres, i.e., when A = B, one can get

$$ {C}_{2k}^{\mathrm{AA}}={\omega}_{\mathrm{p}}{(2R)}^{2k}\frac{1}{2^{2k}}\frac{\left(2k-2\right)!}{4}{\displaystyle \sum_{l=1}^{k-2}\frac{1}{(2l)!\left(2k-2l-1\right)!}\times \frac{1}{\sqrt{\left(2l+1\right)/l}+\sqrt{\left(2k-2l-1\right)/\left(k-l-1\right)}}}. $$

(23)

Then the van der Waals interaction of (2) can be written as

$$ E(d)=-\sqrt{4\pi \rho }{\displaystyle \sum_{k=3}^{\infty }{c}_k\left(t/R\right){z}^k}, $$

(24)

where c _k (t/R) is related to C _2k by (4) and $z={\left(\frac{2R}{d}\right)}^2$.

Now, by the introduction of the geometric series of ${\displaystyle {\sum}_{k=1}^{\infty }{z}^k}={\left(1-z\right)}^{-1}$ for 0 ≤ z < 1 and approximating ${c}_k\to {c}_{\infty }$ for k > 5, we find

$$ {E}^{\mathrm{geo}}(d)=-\sqrt{4\pi \rho}\left[{c}_3\left(\frac{t}{R}\right){z}^3+{c}_4\left(\frac{t}{R}\right){z}^4+{c}_5\left(\frac{t}{R}\right){z}^5+{c}_{\infty}\left\{{\left(1-z\right)}^{-1}-{\displaystyle \sum_{k=0}^5{z}^k}\right\}\right]. $$

(25)

This expression interpolates between the very large d and $d\to 2R$ limits. The above expression for E ^geo(d) has an unphysical divergence at z = 1 or d = 2R where the two spheres touch each other. This divergence appears because we sum up all the terms. However, in reality there is no divergence in (2) because it is an asymptotic expansion for large value of d.

This is true because at large d the exponential density overlap between the two real quantum-mechanical objects may be neglected. This divergence in the expression of E ^geo(d) can be removed by replacing z by z′ where z′ = (2R/d′)² with a proper choice of d′. The expression for E ^geo(d) is true for the interaction between identical spheres but it can be generalized to non-identical spheres $2R\to {R}_{\mathrm{A}}+{R}_{\mathrm{B}}$, which leads to an equation such as (14) for the expression of E _vdW(d′).

In the pair interaction picture, Hamaker’s [41] expression of the van der Waals interaction between two solid spheres of uniform density ρ is

$$ E(d)=-\beta {\displaystyle {\int}_{\mathrm{A}}{d}^3r}{\displaystyle {\int}_{\mathrm{B}}{d}^3{r}^{\prime}\frac{1}{\left|r-{r}^{\prime}\right|{}^6}}, $$

(26)

where $\beta ={c}_3(1)\sqrt{4\pi \rho }{\left(\frac{3}{4\pi}\right)}^2{2}^6=0.006766\sqrt{4\pi \rho }{\left(\frac{3}{4\pi}\right)}^2{2}^6$ can be evaluated using the value of c ₃(1) from Table 1.

Appendix 2 Binding Energy Curves from Geometric Series

The summed-up van der Waals series expression of (4) can also be used to obtain the binding energy curves for the rare-gas dimers if we use our short-range cut-off idea. Reduced van der Waals coefficients c ₃(t/R), c ₄(t/R), c ₅(t/R) and c _∞(t/R) are taken from Table 1 for t/R = 1, e.g., for solid spheres. For identical solid-spheres the electron density is $\rho =N/\left(4\pi {R}^3/3\right)$ for a sphere with radius R and number of total valence electrons N (N is 2 for He and 8 for the other rare-gas atoms). The electron density for non-identical spheres can be evaluated using $2\sqrt{\rho_{\mathrm{A}}}\sqrt{\rho_{\mathrm{B}}}/\left(\sqrt{\rho_{\mathrm{A}}}+\sqrt{\rho_{\mathrm{B}}}\right)$. We could not optimize the fitting parameter for every dimer. We have used a = 4.09, the average of the optimum values for Ar–Ar and Kr–Kr. Figure 12 shows the results.

Fig. 12

Binding energy curves for different dimers using the geometric-series expression of (25) a = 4.09