The semiclassical Ehrenfest theory as a special case

Let us take an average of the force operator defined in Eq. (6.18) over the running total electron wavepacket of Eq. (6.13) such that 

Upon comparing this force with that in Eq. (4.43), gives rise to the Ehrenfest (mean-field) force. Therefore, the mean-field force does not arise from a physical origin but from our mathematical or artificial operation of averaging. However, this averaging is not necessarily bad in case where the nonadiabatic interaction is so strong that the electronic states may largely fluctuate from time to time. 

Note that the above derivation of SET is not intuitive one in contrast to that starting from Eq. (4.38). Also the equations of motion for electron dynamics, Eq. (6.14) are different from Eq. (4.41) in that the former includes the second order terms like Y j + Uj/. 

Thus the theoretical basis of the semiclassical Ehrenfest theory is found on a purely theoretical ground. 

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