Non-orthogonal CI
In general, the Slater determinant built from the \(M\) non-orthogonal orbitals \(\require{physics} \set{\chi_P}\) would then look like \[\begin{equation} \ket*{\vb{k}'} = % \prod_P^M \qty( \hat{b}^\dagger_P )^{k'_P} % \ket{\text{vac}} \end{equation}\] and if we were to write out the product explicitly and choose occupied spin orbitals to be \(1 \cdots N\), we would get \[\begin{equation} \ket*{\vb{k}'} = % \hat{b}^\dagger_1 \cdots \hat{b}^\dagger_N % \ket{\text{vac}} % \thinspace . \end{equation}\] Inserting equation \(\eqref{eq:non-orthogonal:b^dagger_a}\), we arrive at: \[\begin{equation} \ket*{\vb{k}'} = % \sum_{Q_1}^M \cdots \sum_{Q_N}^N % \qty( % C^{-1}_{Q_1 1} \cdots C^{-1}_{Q_N N} % ) % \hat{a}^\dagger_{Q_1} \cdots \hat{a}^\dagger_{Q_N} % \ket{\text{vac}} % \thinspace , \end{equation}\] in which we have to note that the indices \(Q_1 \cdots Q_N\) for the creation operators should be all different (due to the Pauli principle). Furthermore, many of these creation operator strings are equal, so we will establish some ordering leading to a determinant of the coefficients that are in front of them, leading to \[\begin{equation} \ket*{\vb{k}'} = % \sum_{\vb{k}} \det( \vb{C}^{-1}(\vb{k}') ) % \ket{\vb{k}} % \thinspace , \end{equation}\] in which the notation \(\vb{C}^{-1}(\vb{k}')\) should get some clarification. We already know that \(\vb{C}^{-1}\) is an \((M \times M)\)-matrix, from equation \(\eqref{eq:non_orthogonal_overlap_matrix}\). I will denote \(\vb{C}^{-1}(\vb{k}')\) by the \((N \times N)\)-matrix that arises when we select the columns and rows that have indices \(P\) such that \(k'_P = 1\). In other words, we select the columns and rows that are occupied in \(\vb{k}'\). The normalization factor for this determinant then becomes (Péter. R. Surján 1989) \[\begin{equation} \braket{\vb{k}'} = \det\qty(\vb{S}(\vb{k}')) % \thinspace . \end{equation}\] Since one Slater determinant built from non-orthogonal orbitals corresponds to an expansion of Slater determinants in an orthonormal basis, in a sense, using non-orthogonal spinors already leads to some form of CI.