## 3 Dilute Alkali Atoms in a Solid Inert-Gas Matrix: Theoretical Analysis
### The Theoretical Model
We immediately find  
\[A=\frac{6C_{6}R_{rep}}{R_{eq}^{7}}e^{R_{eq}/R_{rep}}, \tag{5}\]yielding  
\[V_{\text{BH}}(r)=\frac{6C_{6}R_{rep}}{R_{eq}^{2}}e^{(R_{eq}-r)/R_{rep}}-\frac{C_{6 }}{r^{6}}. \tag{6}\]  
By taking into account that the quantities experimentally measured are the van der Waals constant \(C_{6}\), the equilibrium distance \(R_{eq}\), and the dissociation energy \(V(R_{eq})\) (that is the potential energy at the equilibrium distance), from (6) we can obtain the following expression for \(R_{rep}\)  
\[R_{rep}=\frac{R_{eq}^{7}}{6C_{6}}\Bigg{(}V_{\text{BH}}(R_{eq})+\frac{C_{6}}{R _{eq}^{6}}\Bigg{)}. \tag{7}\]  
The complete expression of the potential is then  
\[V_{\text{BH}}(r)=-\frac{C_{6}}{r^{6}}+\Bigg{(}V_{\text{BH}}(R_{eq})+\frac{C_{6 }}{R_{eq}^{6}}\Bigg{)}\exp\Bigg{[}\frac{6C_{6}}{R_{eq}^{7}(V_{\text{BH}}(R_{ eq})+\frac{C_{6}}{R_{eq}^{6}})}(R_{eq}-r)\Bigg{]} \tag{8}\]  
Thus, using Equation (8), the Buckingham-Hill potential is entirely expressed in terms of hopefully known and tabulated parameters (for example, the various parameters for several alkali and RG atoms can be found in [47; 51; 52]). The Buckingham-Hill potential is a good approximation for distances around \(R_{eq}\), yielding a good approximation to the equilibrium distance, and for larger distances, where it correctly reduces to the van der Waals potential, but not for shorter distances: in fact, for distances smaller than \(R_{eq}\) the Buckingham-Hill potential decreases to negative values, where it should grow and give the short-distance repulsion. In other words, the equilibrium position of the potential (6) is an unstable point. We thus need an appropriate approximation to the potential energy valid for distances smaller than the equilibrium distance. As will emerge in the next subsection, the short-distance case is what we are mainly interested to. In the short-distance region, that is, for distances smaller than the dimer's equilibrium distance, we can use the Morse potential, given by [36; 37]  
\[V_{\text{M}}(r)=D\Big{(}1-e^{-\alpha(r-R_{eq})}\Big{)}^{2}-D, \tag{9}\]  
\(D=|V_{\text{M}}(R_{eq})|\) being the dissociation energy of the dimer, that is the energy at the equilibrium distance \(R_{eq}\), \(\alpha=\sqrt{\kappa/(2D)}\), with \(\kappa=\partial^{2}V(r)/\partial r^{2}|_{r=R_{eq}}\) the curvature of the potential at the equilibrium distance; \(\kappa\) is a second parameter characterizing the interatomic potential energy. Although the Morse potential (9) is a good approximation to the short-range exchange/Coulomb potential, it fails at large distances. The Buckingham-Hill and the Morse potential are both expected to give a reasonably good approximation around the equilibrium distance. The Morse potential (9) contains three free parameters, but its mathematical expression does not allow to obtain a relation between them exploiting that its derivative must vanish at the equilibrium distance \(R_{eq}\). For this reason, since the constant \(\kappa\) (that is the curvature of the potential at the equilibrium distance) is not always available from tabulated data in the literature, we suggest that in such cases we can resort to the Buckingham-Hill potential for its estimation. This could be done by calculating the modulus of the second derivative of the Buckingham-Hill potential at the equilibrium distance, and finally obtain the parameter \(\alpha\) in the Morse potential (9). We stress that some care should be used in this procedure, because, at the equilibrium distance, the BH potential has a maximum while the Morse potential has a minimum. Thus, using in the Morse potential the value of \(\kappa\) obtainedfrom the BH potential assumes that, around the equilibrium distance, the modulus of the curvature of both potentials are close to each other.