## 3 Dilute Alkali Atoms in a Solid Inert-Gas Matrix: Theoretical Analysis
### Numerical Estimates of the Atoms-Matrix Interaction Energies  
We can now use the potentials of Section 3.1 to obtain approximated estimates of the value of the interaction energy between an alkali atom (Lithium, Sodium, Potassium, Rubidium) and an atom or molecule of the solid matrix (in this section we will consider Rubidium in Argon and Neon, and then Lithium in parahydrogen), assuming that the alkali atom replaces one atom or molecule of the solid crystal. From that, the interaction energy between a guest alkali atom and the solid RG or parahydrogen matrix can be easily obtained, assuming only pairwise interactions between the nearest neighbours. As mentioned before, this should be a fairly good approximation due to the large reticular distance of the solid matrix. Possible trapping sites of alkali atoms in RG crystals, and their stability, have been recently studied in the literature [53; 54; 55]. We first consider the Rb-Ar case and then the Rb-Ne case, comparing our theoretical estimates with the measured spectra reported in the previous section. At the end, we will also analyze the Li-(p-H\({}_{2}\)) case. Throughout this subsection, the units used are eV for the energy and A for the distance. Solid Argon and solid Neon have both a face-centered cubic (fcc) crystal structure; the reticular distance is \(a_{Ar}=5.26\) A for Ar and \(a_{Ne}=4.43\) A for Ne [11; 48; 49; 50]. The guest alkali atom considered is Rubidium (Rb), as in the measurements reported in the previous section. We assume that the Rubidium atom substitutes one of the atoms of the matrix. Taking into account the fcc reticular structure of Argon and Neon, each Rubidium atom interacts with 12 atoms of the crystal at a distance \(a_{Ar(Ne)}/\sqrt{2}\), \(a\) being the reticular distance. In order to evaluate the potential energy through the Morse or the Buckingham-Hill potential, we need the numerical values of the equilibrium distance \(R_{eq}\), the dissociation energy \(D=\big{|}V(R_{eq})\big{|}\), the curvature \(\kappa\) of the potential energy at the equilibrium distance or the van der Waals constant \(C_{6}\). Theoretical estimates of the equilibrium distance \(R_{eq}\) and the dissociation energy \(D\) can be found in several works [46, 56, 57, 58]. The curvature of the potential energy at the equilibrium distance can be found in [56, 57]. The \(C_{6}\) coefficients for several alkali atoms and RG atoms, both in the ground state and in excited levels, can be found in [47, 59, 59, 60]. Although there is a good agreement in the literature on the values of \(R_{eq}\) and \(C_{6}\), the tabulated values of \(D\) and \(\kappa\) for Rb-Ne can significantly differ in different papers, and even in the same paper when different methods of calculation are used; the available data for Rb-Ar are instead much more uniform. For this reason we will concentrate mainly on the Rb-Ar case. We have chosen to use the data reported in [57], where the three necessary parameters \(R_{eq}\), \(D\), \(\kappa\) are calculated. We will also compare these data with those reported in [58]. However, the small variability of the tabulated data for Rb-Ar is not much relevant for our purposes, because we only need an order-of-magnitude estimate of the interaction energy, and not its exact value. _Rb-Ar_  
We first consider the ground-state configuration \(X^{2}\Sigma_{1/2}^{+}\) of the Rb-Ar dimer; it corresponds to the electronic configuration of both the Rubidium and Argon atoms in the ground state. The relevant data are [57]  
\[R_{eq}^{Rb-Ar} = 5.498\ \text{\AA},\] \[D^{Rb-Ar} = 4.45\cdot 10^{-3}\ \text{eV},\] \[\kappa^{Rb-Ar} = 5.96\cdot 10^{-3}\ \text{eV\AA}^{-2}. \tag{13}\]  
For the same configuration, the data in [58] for the equilibrium distance and the dissociation energy (the curvature of the potential is not explicitly given) are not very dissimilar: \(R_{eq}^{Rb-Ar}=5.482\ \text{\AA}\), \(D^{Rb-Ar}=6.07\cdot 10^{-3}\ \text{eV}\).