Extended Thouless theorem
In a biorthogonal basis, it is possible for orbital pairs to have a zero overlap \(\bra{^w\Phi}\ket{^x\Phi} = ^{xw}\tilde{S}_i = 0\). If such zero overlaps are present, Thouless theorem no longer holds, since it requires the transformation matrix between the determinants to be invertible. For such cases we modify the Thouless theorem in the so-called extended Thouless theorem.
If \(m\) zero-overlaps are present, we construct the reduced determinants by removing electrons in the zero-overlap orbitals \[\begin{align} \require{physics} \ket{^x\Phi_{k_1 \dots k_m}} &= ^x\tilde{b}_{k_m} \dots ^x\tilde{b}_{k_1}\ket{^x\Phi} \\ \ket{^w\Phi_{k_1 \dots k_m}} &= ^w\tilde{b}_{k_m} \dots ^w\tilde{b}_{k_1}\ket{^w\Phi} \thinspace . \end{align}\] These determinants are strictly non-orthogonal \[\begin{equation} ^{xw}\tilde{S} = \bra{^w\Phi_{k_1 \dots k_m}}\ket{^x\Phi_{k_1 \dots k_m}} = \prod_{i | ^xw\tilde{S}_i \neq 0} {^{xw}\tilde{S}_i} \thinspace . \end{equation}\]
In order to relate these determinants, we can use traditional Thouless theorem \[\begin{align} \ket{^w\Phi_{k_1 \dots k_m}} &= \exp(\tilde{\mathcal{Z}})\ket{^x\Phi_{k_1 \dots k_m}} \bra{^w\Phi_{k_1 \dots k_m}}\ket{^x\Phi_{k_1 \dots k_m}} \\ &= \exp(\tilde{\mathcal{Z}})\ket{^x\Phi_{k_1 \dots k_m}} ^{xw}\tilde{S} \thinspace , \end{align}\] with \[\begin{equation} \require{physics} \tilde{\mathcal{Z}} = \sum_{i | ^{xw}\tilde{S}_i \neq 0} {\sum_a} {^{xw}Z}_{ai} {^x\tilde{b}_{a}^\dagger} {^x\tilde{b}_{i}} \thinspace . \end{equation}\]
The two full determinants can now be related as follows \[\begin{align} \require{physics} {^w\tilde{b}_{k_m}} \dots {^w\tilde{b}_{k_1}} \ket{^w\Phi} &= \exp(\tilde{\mathcal{Z}}) {^x\tilde{b}_{k_m}} \dots {^x\tilde{b}_{k_1}} \ket{^x\Phi} ^{xw}\tilde{S} \\ \ket{^w\Phi} &= {^w\tilde{b}_{k_1}^\dagger} \dots {^w\tilde{b}_{k_m}^\dagger} \exp(\tilde{\mathcal{Z}}) {^x\tilde{b}_{k_m}} \dots {^x\tilde{b}_{k_1}} \ket{^x\Phi} {^{xw}\tilde{S}} \\ &= {^w\tilde{b}_{k_1}^\dagger} \dots {^w\tilde{b}_{k_m}^\dagger} {^x\tilde{b}_{k_m}} \dots {^x\tilde{b}_{k_1}} \exp(\tilde{\mathcal{Z}}) \ket{^x\Phi} {^{xw}\tilde{S}} \thinspace , \end{align}\] where we exploited the commutation relation \([\tilde{\mathcal{Z}}, ^x\tilde{b}_k] = 0\).
In order to simplyfy this relation, we will write \({^w\tilde{b}^\dagger_k}\) in terms of \({^x\tilde{b}^\dagger_k}\) \[\begin{equation} ^w\tilde{b}^\dagger_k = \sum_p \sum_{\mu \nu} (^x\tilde{C}^*)^{• \mu}_{p •} g_{\mu \nu} (^w\tilde{C})^{\nu •}_{• k} {^x\tilde{b}^\dagger_k} \thinspace , \end{equation}\] where we used the tensorial notation of Martin Head-Gordon to represent co- and contravariant indices (Head-Gordon, Maslen, and White 1998).
The product of the operators \({^w\tilde{b}^\dagger_k}{^x\tilde{b}_k}\) then becomes \[\begin{align} {^w\tilde{b}^\dagger_k}{^x\tilde{b}_k} &= \sum_p (\sum_{\mu \nu}(^x\tilde{C}^*)^{• \mu}_{p •} g_{\mu \nu} (^w\tilde{C})^{\nu •}_{• k} {^x\tilde{b}^\dagger_p} {^x\tilde{b}_k} \\ &= \sum_a {^{xw}\tilde{S}_{ak}} {^x\tilde{b}^\dagger_a} {^x\tilde{b}_k} \\ &= \hat{z}_k \thinspace , \end{align}\] where the terms corresponding to occupied indices disappear as their overlap \(^{xw}\tilde{S}_{ik}\) is zero and \({^{xw}\tilde{S}_{ak}} = \sum_a\sum_{\mu \nu}(^x\tilde{C}^*)^{• \mu}_{a •} g_{\mu \nu} (^w\tilde{C})^{\nu •}_{• k}\).
If we fill in this newly derived \(\hat{z}_k\) operator in our previous expression we get \[\begin{equation} \require{physics} \ket{^w\Phi} = \prod_{k | ^{xw}\tilde{S}_k = 0} \hat{z}_k \exp(\tilde{\mathcal{Z}}) \ket{^x\Phi} {^{xw}\tilde{S}} \thinspace . \end{equation}\] By introducing the relation \(\hat{z}_k = \exp(\hat{z}_k) - 1\), we get \[\begin{equation} \require{physics} \ket{^w\Phi} = \prod_{k | ^{xw}\tilde{S}_k = 0} (\exp(\hat{z}_k) - 1) \exp(\tilde{\mathcal{Z}}) \ket{^x\Phi} {^{xw}\tilde{S}} \thinspace . \end{equation}\] In this expression, expanding the product \(\prod_{k | ^{xw}\tilde{S}_k = 0} (\exp(\hat{z}_k) - 1)\) corresponds with letting the \(\hat{z}_k\) operator act on the expression and introducing a phase factor. This yields \[\begin{equation} \require{physics} \ket{^w\Phi} = \sum^m_{n=0}(-1)^{m-n} \sum_{c_n} \exp(\tilde{\mathcal{Z}}^{c_n}) \ket{^x\Phi} {^{xw}\tilde{S}} \thinspace , \end{equation}\] where \[\begin{equation} \require{physics} \tilde{\mathcal{Z}}^{c_n} = \mathcal{\tilde{Z}} + \sum_{k \in {c_n}} \hat{z}_k \thinspace , \end{equation}\] in which \(c_n\) represents a compound index that denotes a particular combination of \(n\) zero-overlap orbitals out of \(m\). This expression is the extended Thouless theorem.