Thouless transformed operators
In order to calculate a matrix element between two different determinants \[\begin{equation} \require{physics} \bra{^x\Phi}(a^\mu)^\dagger (a^\nu)^\dagger \dots (a^\sigma)(a^\tau)\ket{^w\Phi} \end{equation}\] we need the extended Thouless theorem in order to transform one of the determinants into the other \[\begin{align} \require{physics} \bra{^x\Phi}(a^\mu)^\dagger (a^\nu)^\dagger \dots (a^\sigma)(a^\tau)\ket{^w\Phi} &= {^{xw}\tilde{S}} \sum^m_{n=0} (-1)^{m-n} \\ &\sum_{c_n} \bra{^x\Phi}(a^\mu)^\dagger (a^\nu)^\dagger \dots (a^\sigma)(a^\tau) \exp(\tilde{\mathcal{Z}}^{c_n})\ket{^x\Phi} \thinspace . \end{align}\] Essentially, in order to to work out this expression we want to perform a similarity transformation on the operators \((a^p)^\dagger / (a^q)\) using the operator \(\tilde{\mathcal{Z}}^{c_n}\).
\(\tilde{\mathcal{Z}}^{c_n}\) is skew-Hermitian, meaning that we can introduce the transformed operators (T. Helgaker, Jørgensen, and Olsen 2000) \[\begin{align} \require{physics} (c[c_n]^\mu)^\dagger &= \sum_\nu (a^\nu)^\dagger \tilde{\mathcal{Z}}^{c_n}_{\nu \mu} \\ &= \sum_\nu (a^\nu)^\dagger \exp(- \tilde{\mathcal{Z}}^{c_n}) \thinspace , \\ (c[c_n]^\mu) &= \sum_\nu (a^\nu) \tilde{\mathcal{Z}}^{c_n}_{\nu \mu} \\ &= \sum_\nu (a^\nu) \exp(- \tilde{\mathcal{Z}}^{c_n}) \thinspace . \end{align}\] If we now want to introduce the similarity transformed operators \[\begin{align} (d[c_n]^\mu)^\dagger &= \exp(- \tilde{\mathcal{Z}}^{c_n}) (a^\mu)^\dagger \exp(\tilde{\mathcal{Z}}^{c_n}) \thinspace , \\ (d[c_n]^\mu) &= \exp(- \tilde{\mathcal{Z}}^{c_n}) (a^\mu) \exp(\tilde{\mathcal{Z}}^{c_n}) \thinspace , \label{eq:similarity-transformed} \end{align}\] we have to show that \[\begin{align} (d[c_n]^\mu)^\dagger &= (c[c_n]^\mu)^\dagger \thinspace , \\ (d[c_n]^\mu) &= (c[c_n]^\mu) \thinspace . \end{align}\]
Using the Baker-Campbell-Hausdorff (BCH) expansion (T. Helgaker, Jørgensen, and Olsen 2000), we can write the operator \((d[c_n]^\mu)^\dagger\) as \[\begin{equation} (d[c_n]^\mu)^\dagger = (a^\mu)^\dagger + [(a^\mu)^\dagger, \tilde{\mathcal{Z}}^{c_n}] + \frac{1}{2!} [[(a^\mu)^\dagger, \tilde{\mathcal{Z}}^{c_n}], \tilde{\mathcal{Z}}^{c_n}] + \dots \thinspace , \end{equation}\] where the first commutator can be simplified to \(-\sum_\nu (a^\nu)^\dagger \tilde{\mathcal{Z}}^{c_n}_{\nu \mu}\) and the first nested commutator becomes \(\sum_\nu (a^\nu)^\dagger [\tilde{\mathcal{Z}}^{c_n}]^2_{\nu \mu}\). Any n-fold nested commutator can the be written as \[\begin{equation} [\dots [[(a^\mu)^\dagger, \tilde{\mathcal{Z}}^{c_n}], \tilde{\mathcal{Z}}^{c_n}] \dots] = (-1)^n \sum_\nu (a^\nu)^\dagger [\tilde{\mathcal{Z}}^{c_n}]^n_{\nu \mu} \thinspace . \end{equation}\] Using this relation gives us \[\begin{align} (d[c_n]^\mu)^\dagger &= \sum_\nu (a^\nu)^\dagger \{ \delta_{\nu \mu} - [\tilde{\mathcal{Z}}^{c_n}]_{\nu \mu} + \dots + \frac{(-1)^n}{n!} [\tilde{\mathcal{Z}}^{c_n}]^n_{\nu \mu} + \dots \} \\ &= \sum_\nu (a^\nu)^\dagger \sum_n \frac{(-1)^n}{n!} [\tilde{\mathcal{Z}}^{c_n}]^n_{\nu \mu} \\ &= \sum_\nu (a^\nu)^\dagger \exp(-[\tilde{\mathcal{Z}}^{c_n}]^n_{\nu \mu}) \\ &= (c[c_n]^\mu)^\dagger \thinspace . \end{align}\] The proof for the annihilation operators is completely analogous.
The similarity transformed expressions for \((d[c_n]^\mu)^\dagger\) and \((d[c_n]^\mu)\) can now be used to derive their explicit forms. First, consider the expansion of \((d[c_n]^\mu)^\dagger\) \[\begin{align} (d[c_n]^\mu)^\dagger &= \exp(- \tilde{\mathcal{Z}}^{c_n}) (a^\mu)^\dagger \exp(\tilde{\mathcal{Z}}^{c_n}) \\ &= (a^\mu)^\dagger - [\tilde{\mathcal{Z}}^{c_n}, (a^\mu)^\dagger] \thinspace . \end{align}\] Now transform \((a^\mu)^\dagger\) to its covariant counterpart using \((a^\mu)^\dagger = \sum_\nu (a_\nu)^\dagger g^{\mu \nu}\) \[\begin{equation} (d[c_n]^\mu)^\dagger = (\sum_\nu (a_\nu)^\dagger g^{\mu \nu} - [\tilde{\mathcal{Z}}^{c_n}, \sum_\nu (a_\nu)^\dagger g^{\mu \nu}] \thinspace . \end{equation}\] The covariant operator can be expanded in the bi-orthogonal molecular orbital basis \[\begin{align} (d[c_n]^\mu)^\dagger &= \sum_\nu \sum_{p \mu} {^x\tilde{b}^\dagger_p} (^x\tilde{C}^*)^{• \mu}_{p •} g_{\mu \nu} g^{\mu \nu} - [\tilde{\mathcal{Z}}^{c_n}, \sum_{p \mu} {^x\tilde{b}^\dagger_p} (^x\tilde{C}^*)^{• \mu}_{p •}] \\ &= \sum_{p} {^x\tilde{b}^\dagger_p} (^x\tilde{C}^*)^{• \mu}_{p •} - [\tilde{\mathcal{Z}}^{c_n}, \sum_{p \mu} {^x\tilde{b}^\dagger_p} (^x\tilde{C}^*)^{• \mu}_{p •}] \thinspace . \end{align}\] Substituting \(\tilde{\mathcal{Z}}^{c_n}\) by its definition (see here) leads to \[\begin{align} (d[c_n]^\mu)^\dagger &= \sum_{p} {^x\tilde{b}^\dagger_p} (^x\tilde{C}^*)^{• \mu}_{p •} \\ &- \sum_{ap} \big( \sum_{i | ^{xw}\tilde{S}_i \neq 0} {^{xw}\tilde{\mathcal{Z}}_{ai}} [{^x\tilde{b}^\dagger_a}{^x\tilde{b}_i}, {^x\tilde{b}^\dagger_p}] \\ &+ \sum_{k \in c_n} {^{xw}\tilde{S}_{ak}} [{^x\tilde{b}^\dagger_a}{^x\tilde{b}_k}, {^x\tilde{b}^\dagger_p}] \big) (^x\tilde{C}^*)^{• \mu}_{p •} \thinspace . \end{align}\] Exploiting the commutation relation \([{^x\tilde{b}^\dagger_a}{^x\tilde{b}_i}, {^x\tilde{b}^\dagger_p}] = {^x\tilde{b}^\dagger_a} \delta_{ip}\) then yields \[\begin{align} (d[c_n]^\mu)^\dagger &= \sum_{p} {^x\tilde{b}^\dagger_p} (^x\tilde{C}^*)^{• \mu}_{p •} \\ &- \sum_{a} {^x\tilde{b}^\dagger_a} \big( \sum_{i | ^{xw}\tilde{S}_i \neq 0} {^{xw}\tilde{\mathcal{Z}}_{ai}} (^x\tilde{C}^*)^{• \mu}_{i •} \\ &+ \sum_{k \in c_n} {^{xw}\tilde{S}_{ak}} (^x\tilde{C}^*)^{• \mu}_{k •} \big) \thinspace . \end{align}\] The summation over \(p\) contains the summation over all occupied, as well as all virtual indices. The summation is thus split in a summation over \(i\) and \(a\) respectively \[\begin{align} (d[c_n]^\mu)^\dagger &= \sum_{i} {^x\tilde{b}^\dagger_i} (^x\tilde{C}^*)^{• \mu}_{i •} + \sum_{a} {^x\tilde{b}^\dagger_a} (^x\tilde{C}^*)^{• \mu}_{a •} \\ &- \sum_{a} {^x\tilde{b}^\dagger_a} \big( \sum_{i | ^{xw}\tilde{S}_i \neq 0} {^{xw}\tilde{\mathcal{Z}}_{ai}} (^x\tilde{C}^*)^{• \mu}_{i •} \\ &+ \sum_{k \in c_n} {^{xw}\tilde{S}_{ak}} (^x\tilde{C}^*)^{• \mu}_{k •} \big) \thinspace , \end{align}\] which can be simplified by bringing all summations over \(a\) inside the brackets \[\begin{align} (d[c_n]^\mu)^\dagger &= \sum_{i} {^x\tilde{b}^\dagger_i} (^x\tilde{C}^*)^{• \mu}_{i •} \\ &+ \sum_{a} {^x\tilde{b}^\dagger_a} \big( (^x\tilde{C}^*)^{• \mu}_{a •} - \sum_{i | ^{xw}\tilde{S}_i \neq 0} {^{xw}\tilde{\mathcal{Z}}_{ai}} (^x\tilde{C}^*)^{• \mu}_{i •} \\ &+ \sum_{k \in c_n} {^{xw}\tilde{S}_{ak}} (^x\tilde{C}^*)^{• \mu}_{k •} \big) \thinspace . \end{align}\] Analogously we can derive an expression for the annihilation operator \((d[c_n]^\mu)\) \[\begin{align} (d[c_n]^\mu)^\dagger &= \sum_{i} (^x\tilde{C})^{\mu •}_{• i} {^x\tilde{b}_i} \\ &+ \sum_{a} \big( (^x\tilde{C})^{\mu •}_{• a} {^x\tilde{b}_a} + \sum_{i | ^{xw}\tilde{S}_i \neq 0} {^{xw}\tilde{\mathcal{Z}}_{ai}} {^x\tilde{b}_i} (^x\tilde{C})^{\mu •}_{• a} \\ &+ \sum_{k \in c_n} {^{xw}\tilde{S}_{ak}} {^x\tilde{b}_k} (^x\tilde{C})^{\mu •}_{• a} \big) \thinspace . \end{align}\]
Using the explicit forms of the transformed operators along with the relations \[\begin{align} \require{physics} \bra{^x\Phi} \exp(- \tilde{\mathcal{Z}}^{c_n}) &= \ket{^x\Phi} \\ \exp(\tilde{\mathcal{Z}}^{c_n}) \exp(- \tilde{\mathcal{Z}}^{c_n}) &= \vb{I} \thinspace , \end{align}\] the Thouless transformed expression \(\bra{^x\Phi}(a^\mu)^\dagger (a^\nu)^\dagger \dots (a^\sigma)(a^\tau) \exp(\tilde{\mathcal{Z}}^{c_n})\ket{^x\Phi}\) simply becomes \[\begin{align} \require{physics} \bra{^x\Phi}(a^\mu)^\dagger (a^\nu)^\dagger &\dots (a^\sigma)(a^\tau) \exp(\tilde{\mathcal{Z}}^{c_n})\ket{^x\Phi} = \\ &\bra{^x\Phi}(d[c_n]^\mu)^\dagger (d[c_n]^\nu)^\dagger \dots (d[c_n]^\sigma)(d[c_n]^\tau))\ket{^x\Phi} \thinspace . \end{align}\]