The property of the anomalous inverted region in the energy gap law which was found in the recent non-Condon theory of the electron transfer (ET) in protein media is investigated in more detail in relation to the inelastic electron tunneling. The physical aspect of the inelastic electron tunneling is theoretically discussed and schematically explained. Since it was previously shown that the inelastic electron tunneling mechanism worked significantly due to an exponential-like decay of the autocorrelation function of the electron tunneling matrix element with a small correlation time at 300 K, we investigated whether the similar exponential-like decay of the autocorrelation function is obtained or not and how the correlation time is changed at low temperatures. For this purpose, numerical calculations for the electron tunneling matrix element in the ET at 77 K from bacteriopheophytin (Bph) anion to the primary quinone in the reaction center of photosynthetic bacteria Rhodobacter sphaeroides are made. The results are that almost exponential-like decay of the autocorrelation function was obtained even at 77 K and the half decay time was of the similar magnitude to that at 300 K. Physical meaning of this temperature dependence is discussed.