As the trial function or the coefficients vary in the search for the ground state, so do the NSOs, as in any NOFT. The exponential complexity of determining the ground-state energy by variation of the $C_{{{\bf{k}}}}$ is composed of the separate exponential complexities of the signs and the magnitudes of the coefficients $C_{{{\bf{k}}}}$. We use distinct reductive approximations for their signs and magnitudes. The signs and consequently the sign-approximation scheme depend on the sign convention chosen for the SDs. We use the Leibniz form for the SDs, $$\Phi_{{{\bf{k}}}}(x_1,\cdots,x_N) =
\frac1{\sqrt{N!}} \sum_p \text{sgn}\{P_p\} P_p \psi_{k_1}(x_1) \cdots
\psi_{k_N}(x_N).
\label{SDs}$$ The sum is over the elements of the symmetric group of order $N$, the permutations $P_p$. The sign of $\Phi_{{{\bf{k}}}}$ is fixed by the ordering $k_1 < k_2 < \cdots < k_N$ in the product of the NSO $\psi_{k_i}$ in [[**\[SDs\]**]{}]{}. Once this sign convention is understood, the SDs and their coefficients can be specified simply by listing the NSOs occupied in the SDs, i.e. by the index ${{\bf{k}}}$.

The 1-DM, the orthogonality constraint, the PDC
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The 1-DM of $\Psi$, $$\begin{aligned}
\rho(x',x) = N \int dx_2\cdots dx_N \nonumber \\
\Psi(x',x_2,\cdots,x_N) \Psi^*(x,x_2,\cdots,x_N),
\label{rho}\end{aligned}$$ becomes $$\rho(x',x) = \left[ \sum_{{{\bf{k}}}; i,j\not\in{{\bf{k}}}} C_{i,{{\bf{k}}}} C_{j,{{\bf{k}}}}
  \right] \, 