Unifying Microscopic and Continuum Treatments of van der Waals and Casimir Interactions

We present an approach for computing long-range van der Waals (vdW) interactions between complex molecular systems and arbitrarily shaped macroscopic bodies, melding atomistic treatments of electronic fluctuations based on density functional theory in the former with continuum descriptions of strongly shape-dependent electromagnetic fields in the latter, thus capturing many-body and multiple scattering effects to all orders. Such a theory is especially important when considering vdW interactions at mesoscopic scales, i.e., between molecules and structured surfaces with features on the scale of molecular sizes, in which case the finite sizes, complex shapes, and resulting nonlocal electronic excitations of molecules are strongly influenced by electromagnetic retardation and wave effects that depend crucially on the shapes of surrounding macroscopic bodies. We show that these effects together can modify vdW interaction energies and forces, as well as molecular shapes deformed by vdW interactions, by orders of magnitude compared to previous treatments based on Casimir-Polder, nonretarded, or pairwise approximations, which are valid only at macroscopically large or atomic-scale separations or in dilute insulating media, respectively.

Figure 1

Schematic of molecular bodies described by electric susceptibilities ${\mathbb{V}}_n$ in the vicinity of and interacting with macroscopic bodies described by a collective susceptibility ${\mathbb{V}}_{\text{env}}$, where the interactions are mediated by vacuum electromagnetic fields ${\mathbb{G}}_0$.

Figure 2

(a) Energy ratio $\mathcal{E}/{\mathcal{E}}_{\mathrm{PWS}}$ versus $z$ for a fullerene (solid blue), protein (solid green), or wire in the parallel (solid red) or perpendicular (solid black) orientations, above the gold plate; ${\mathcal{E}}_{\mathrm{PWS}}$ is the energy obtained by a pairwise approximation defined in (10). Also shown are the predictions of both $CP$ (dotted red) and nonretarded (dashed red) approximations for the case of a parallel wire. Inset: power law $\partial \ln (\mathcal{E})/\partial \ln (z)$ (solid blue) of the fullerene-plate system with respect to $z$, compared to both $CP$ (dotted blue) and nonretarded (dashed blue) approximations. (b) $CP$ ${\mathcal{E}}_{CP}$ (dotted black) and nonretarded ${\mathcal{E}}_0$ (dashed black) energies of a perpendicular carbyne wire separated from a gold plate by a vertical distance $z$, normalized to the RMB energy $\mathcal{E}$ of Eq. (6), as a function of $z$.

Figure 3

Energy ratio $\mathcal{E}/{\mathcal{E}}_{\mathrm{PWS}}$ versus vertical distance $z$ for two fullerenes at fixed horizontal separation $d=3\mathrm{nm}$ (solid blue) or two wires at $d=10\mathrm{nm}$, in either the parallel (solid red) or perpendicular (solid black) orientations, above a gold plate. Top inset: horizontal-force ratio $F_y/F_{y,\mathrm{PWS}}$ versus $z$ for the parallel wires at $d=10\mathrm{nm}$. Bottom inset: $\mathcal{E}/{\mathcal{E}}_{\mathrm{PWS}}$ versus $d$ for the fullerenes and the parallel wires at several values of $z$; also shown are the corresponding ratios obtained via $CP$ (dot-dashed red) and nonretarded (dashed red) approximations, specifically for $z=10\mathrm{nm}$.

Figure 4

(a) Energy ${\mathcal{E}}_{\text{cone}}$ of either a fullerene (solid blue) or carbyne wire (solid red/black) above a gold cone, normalized to the energy ${\mathcal{E}}_{\text{plate}}$ of the same molecule but separated from a gold plate by the same surface-surface vertical distance $z$. (b) Energy variations $1-\mathcal{E}(\beta )/\mathcal{E}(0)$, with (solid black) or without (dashed black) retardation, for a clamped long vertical carbyne wire as a function of dimensionless curvature $\beta$. Inset schematically shows the wire shape for $\beta =0.5$.