Background: Even-even isotopes of Mo (Z = 42) and Ru (Z = 44) are nuclei close to the subshell closure at Z = 40, where shape coexistence plays a significant role. As a result, their spectroscopic properties are expected to resemble those of Sr (Z = 38) and Zr (Z = 40). Exploring the evolution of these properties as they move away from the subshell closure is of great interest. Purpose: The purpose of this study is to reproduce the spectroscopic properties of even-even [96][97][98][99][100][101][102][103][104][105][106][107][108][109][110] 42 Mo and 98-11444 Ru isotopes and to determine the influence of shape coexistence. Method: We employed the interacting boson model with configuration mixing as the framework to calculate all the observables for Mo and Ru isotopes. We considered two types of configurations: 0-particle-0-hole and 2-particle-2-hole excitations. The model parameters were determined using a least-squares fitting to match the excitation energies and the B(E 2) transition rates. Results: We obtained the excitation energies, B(E 2) values, two-neutron separation energies, nuclear radii, and isotope shifts for the entire chain of isotopes. Our theoretical results show good agreement with experimental data. Furthermore, we conducted a detailed analysis of the wave functions and obtained the mean-field energy surfaces and the nuclear deformation parameter, β, for all considered isotopes. Conclusions: Our findings reveal that shape coexistence plays a significant role in Mo isotopes, with the crossing of intruder and regular configurations occurring at neutron number 60 (A = 102), which induces a quantum phase transition. In contrast, in Ru isotopes, the intruder states have minimal influence, remaining at higher energies. However, at neutron number 60, also a quantum phase transition occurs in Ru isotopes.
