If we define a new creation operator $$ \begin{equation} b^{\dagger}_{i}=a^{\dagger}_{i}+\sum_{a>F}C_{ai}a_{a}^{\dagger}, \tag{88} \end{equation} $$ we have $$ |c'\rangle=\prod_{i}b^{\dagger}_{i}|0\rangle=\prod_{i}\left(a^{\dagger}_{i}+\sum_{a>F}C_{ai}a_{a}^{\dagger}\right)|0\rangle, $$ meaning that the new representation of the Slater determinant in second quantization, \( |c'\rangle \), looks like our previous ones. However, this representation is not general enough since we have a restriction on the sum over single-particle states in Eq. (88). The single-particle states have all to be above the Fermi level. The question then is whether we can construct a general representation of a Slater determinant with a creation operator $$ \tilde{b}^{\dagger}_{i}=\sum_{p}f_{ip}a_{p}^{\dagger}, $$ where \( f_{ip} \) is a matrix element of a unitary matrix which transforms our creation and annihilation operators \( a^{\dagger} \) and \( a \) to \( \tilde{b}^{\dagger} \) and \( \tilde{b} \). These new operators define a new representation of a Slater determinant as $$ |\tilde{c}\rangle=\prod_{i}\tilde{b}^{\dagger}_{i}|0\rangle. $$