The equilibrium isotope fractionations among C-O-H gases in a variety of geological settings are commonly used as isotope geothermometers to evaluate the temperatures of geothermal fluids at depth, subsurface fluid rock interactions, volcanic-hydrothermal systems and natural gas pools. However, due to limited experimental data and sophisticated theoretical calculations, applications of these geothermometers have been restricted. This study uses the carbon isotope exchange reaction between CH4 and CO2 as a case study to develop theoretical methods that can improve accuracies in calculating harmonic vibrational frequencies for CH4 and CO2, the equilibrium constants and temperatures for the carbon isotope exchange reaction between CH4 and CO2. Results suggest that the Bigeleisen-Mayer equation is sufficient to calculate the equilibrium constants and temperatures associated with the isotope exchange reaction between CH4 and CO2 with an accurate estimation of molecular harmonic vibrational frequencies. Calculations of the harmonic frequencies of CH4 and CO2 are achieved using the B3LYP density functional method with the 6-311 + G(d) basis set, and the calculated harmonic frequencies are highly consistent with experimental values. The frequency correction factor is taken as 1.022 which puts the calculated fractionation factors in good agreement with experimental values. The calculated equilibrium constants are comparable to experimental data and a theoretical data set. They are highly consistent. In order to improve the accuracy and efficiency of solving for equilibrium temperatures using the Bigeleisen-Mayer equation, symbol operation and iterative algorithm in the MatLab software have been applied to compute the temperatures instead of using limited theoretical data sets or empirical fit equations. Our calculated results suggest that this algorithm can rapidly and conveniently yield relatively precise equilibrium temperatures. This algorithm can thus provide an important tool to evaluate whether the carbon isotope exchange reaction for CH4 and CO2 has attained equilibrium and estimate the formation temperature of CH4 and CO2 in high temperature geothermal systems.