Strong Correlation of Electron Saddle Point Singularities to the Anomalous Isotope Effect in Zr, Nb$_{3}$Sn, and YBa$_{2}$Cu$_{3}$O$_{7}$*

Anomalously small isotope effect in some high and low T$_{\mathrm{c}}$ superconductors such as Zr, Nb$_{3}$Sn, and YBa$_{2}$Cu$_{3}$O$_{7}$ (YBCO) created a great challenge for understanding. It has been shown by experiments and first-principles calculations that there exist extended saddle point singularities in the electronic structures of these materials. In this work, a new methodology is further implemented by integrating first-principles calculations of electronic structures of the materials into the theory of many-body physics for superconductivity. The aim is to seek a unified methodology to calculate the electronic and superconducting properties of these materials. It is demonstrated from first-principles that the extended saddle point singularities in Zr, Nb$_{3}$Sn, and YBCO strongly correlate to the anomalous isotope effect in these superconductors. However, there still exist some differences between the calculated and experimental results that require further research work.