## 1. Introduction

Recently, a new spectroscopic technique has been developed that allows for a detection of electron spin resonance (ESR) transitions in doped helium nanodroplets.1, 2 It has the potential to monitor interactions between dopants sitting on the surface and inside a helium droplet of a given size.3 In ref. 4 our group presented hyperfine-resolved spectra of ESR transitions of rubidium atoms attached to superfluid helium nanodroplets. The high sensitivity of this experimental technique led to the observation of hyperfine shifts in the ESR spectra caused by the chemically inert but physically disturbing helium environment: shifts of the well-known hyperfine structure constants of ^{85}Rb and ^{87}Rb were detected as a function of the droplet size. In this theoretical follow-up article we try to explain and reproduce these observed shifts within a framework of relativistic ab initio calculations and a simulation of the helium droplet environment based on density functional theory.

In helium nanodroplet isolation spectroscopy5, 6 the alkali-metal atoms are particularly interesting as they stay on the surface of the superfluid He droplets, whereas most other atoms or molecules move to the center. For the alkali-metal atoms and their diffuse s electron the compensation of Pauli repulsion and van der Waals interaction forces leads to the formation of a “dimple” on the droplet surface in which the alkali-metal atom resides. Common methods employed to estimate the size and shape of these pockets, i.e. the resulting helium density distribution of such a weakly bound system, are density functional theory (DFT) or quantum Monte Carlo (QMC) approaches. A detailed study of dimple structures for alkali-metal-doped ^{3}He and ^{4}He droplets based on the latter is given in ref. 7. We will compare the results obtained with our method of choice, a time-independent DFT approach using the Orsay–Trento functional8 to map the He density onto the total energy, to the published QMC solvation energies. Previous studies mainly focused on the weak interaction between the dopants and the helium droplet and on those properties that can be derived from the variation of the total energy as a function of the geometry, for example, as a function of the distance between the dopant and the center of mass of the helium droplet. In this article, however, geometric effects on the total electronic wavefunction of the dopant are particularly interesting to us. Unfortunately, the problem of finding a computationally feasible bidirectional description of the coupling between the helium density distribution and the electronic wavefunction of the dopant has not yet been solved. Methods that in principle can account for dynamic effects such as the coupling between vibronic modes of the dopant and excitations of the helium droplet are diffusion or path integral techniques9, 10 or hybrid strategies, where the latter are combined with standard quantum chemistry approaches based on molecular orbital theory.11–14 Recently, the combination of time-dependent DFT with Bohmian dynamics has also been suggested.15 Despite this ongoing work there is still a huge technical gap between the two core aspects of simulating helium-droplet-induced ESR shifts: 1) A good description of the He density is needed, which includes all intrinsic quantum mechanical properties of the system but still scales reasonably with the system size. The first condition impedes the usage of molecular mechanics models, which are more likely to be applied to crystal-like clusters of larger rare gases such as Ar or Xe, but not to a superfluid quantum liquid such as a ^{4}He droplet. The second condition, a reasonable scaling with the size of the system, prevents us from using highly sophisticated standard methods of MO-based quantum chemistry. 2) Accurate predictions of ESR properties can hardly be achieved without the application of such MO-based quantum chemistry techniques.

Hence, mediating between these frontiers, we come up with a simple but effective combination of a time-independent, well-established DFT approach for the description of the helium density with a powerful ab initio post-Hartree–Fock method for the simulation of the electronic wavefunction of the dopant. Our article is structured as follows: In Section 2 we give a brief introduction to the origin of the hyperfine splitting in alkali-metal atoms, discuss the set of simplifications applied to obtain estimates for the ESR shifts, and elaborate on both parts of the strategy mentioned above, namely the He DFT and the ab initio part, in two separate subsections. A third subsection is dedicated to the connection of both aspects, linking the obtained dimple profiles to changes in the total electron wavefunctions of the dopants. Results are then presented in Section 3 and compared to experimental data that are available for Rb-doped He droplets at least. We summarize our work in Section 4.