 Contracted nonlinear susceptibility notation

For identical input frequencies, intrinsic permutation symmetry of nonlinear susceptibility makes the last two Cartesian indices symmetric. Store their six independent pairs as one index, so the quadratic response uses a three-by-six array $d_{iJ}$. In the convention $d_{ijk}={\chi }_{ijk}^{(2)}/2$, the contracted input vector is $(A_x^2,A_y^2,A_z^2,2A_y{A}_z,2A_z{A}_x,2A_x{A}_y{)}^T$. The factors of two preserve the sum over both off-diagonal orderings. This bookkeeping does not assert the stronger Kleinman symmetry and does not change the tensor's physical order.