Derivation of nonlinear optical susceptibility based on quantum mechanical perturbation theory of the atomic wave function in this section.
But, relaxation processes ,which are important for the case of near-resonance excitation cannot be adquately described by this formalism, so after all we finally describe it with density matrix formalism. when using this method ,the calculation of the nonlinear susceptibility is much more comlicated.
Perturbation Solution to Schrodinger's equation
We now determine the form of amplitudes up to third-order.
To determine the first order amplitudes
,we set
equal to 
,for is zero without the groundstate.replacing
by 
,where 
