Calculation of Bethe Logarithm by Drake-Goldman Method to Hydrogen Molecular Ions

Precision spectroscopy of hydrogen molecular ions can be used to determinate proton to electron mass ratio $m_p/m_e$. Several experiments are carrying out to measure rovibrational transition of HD$^+$ to a precision of $10^{-9}$, and two photon transition of H$_2^+$ to $10^{-10}$ level. Current theoretical calculations are as precision as $10^{-10}$, which are done by calculate relativistic and radiative corrections up to $R_{\infty}\alpha^5$ order. Uncertainty of $R_{\infty}\alpha^2$ order corrections have been reduced to sub kHz for H$_2^+$ states $(v,L): (0,0), (0,1),(1,0)$ and HD$^+$ rovibrational states $(v=0-4, L=0-4)$. Bethe Logarithm appeared in $R_{\infty}\alpha^3$ order corrections has been calculated to more than 8 digits for H$_2^+$ and HD$^+$ rovibrational states $(v=0-4, L=0-4)$. In this poster, we will report our recent work on Bethe Logarithm, which is calculated by Drake-Goldman method. The work was supported by the NSFC under Grants No. 11004221 and No. 10974224, by the National Basic Research Program of China (973 Program) under Grant No. 2010CB832803, and by the NSERC of Canada. The work was carried out at the computing facilities of SHARCnet of Canada and Wuhan University of China.