  This paper studies the gamma-ray spectra of positron annihilation processes
in a series of molecules. The results show that the average valence electron
energy of the molecules has a linear correlation with the full width at half
maximum (FWHM) of the gamma-ray spectra. In addition, we defined a new physical
quantity Average Doppler Shift (ADS), which can be used as the eigenvalue to
describe the characteristics of the gamma-ray spectra. Since ADS contains all
the information about the gamma-ray spectra, it can more accurately represent
the characteristics of the gamma-ray spectra. For a series of molecules, this
paper compares the ADS and FWHM of their gamma-ray spectra and the average
valence electron energy. The results show that ADS has a linear correlation
with the average valence electron energy and the FWHM. Further, this proves
that the annihilation mainly occurs on valence electrons, and it also
illustrates that the ADS has certain applicability. It is expected that this
will provide us with a deeper understanding of the positron annihilation
process.
