  This research will examine the computational methods to calculate the
nonlinear optical process of second harmonic generation (SHG) that will be
hypothesized to be present during lithium ion insertion into silicon nanowires.
First it will be determined whether the medium in which SHG is conveyed is
non-centrosymmetric or whether the medium is inversion symmetric where SHG as a
part of the second-order nonlinear optical phenomenon does not exist. It will
be demonstrated that the main interaction that determines SHG is multiphoton
absorption on lithium ions. The quantum harmonic oscillator (QHO) is used as
the background that generates coherent states for electrons and photons that
transverse the length of the silicon nanowire. The matrix elements of the
Hamiltonian which represents the energy of the system will be used to calculate
the probability density of second-order nonlinear optical interactions which
includes collectively SHG, sum-frequency generation (SFG) and
difference-frequency generation (DFG). As a result it will be seen that at
varies concentrations of lithium ions (Li+) within the crystallized silicon
(c-Si) matrix the second-order nonlinear optical process has probabilities
substantial enough to create second harmonic generation that could possibly be
used for such applications as second harmonic imaging microscopy.
