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Sum MPO Formalism in Unrestricted Spatial Orbitals

Hamiltonian

The quantum chemistry Hamiltonian is written as follows

\[\hat{H} = \sum_{ij,\sigma} t_{ij,\sigma} \ a_{i\sigma}^\dagger a_{j\sigma} + \frac{1}{2} \sum_{ijkl, \sigma\sigma'} v_{ijkl, \sigma\sigma'}\ a_{i\sigma}^\dagger a_{k\sigma'}^\dagger a_{l\sigma'}a_{j\sigma}\]

where

\[\begin{split}t_{ij,\sigma} =&\ t_{(ij),\sigma} = \int \mathrm{d}\mathbf{x} \ \phi_{i\sigma}^*(\mathbf{x}) \left( -\frac{1}{2}\nabla^2 - \sum_a \frac{Z_a}{r_a} \right) \phi_{j\sigma}(\mathbf{x}) \\ v_{ijkl,\sigma\sigma'} =&\ v_{(ij)(kl),\sigma\sigma'} = v_{(kl)(ij),\sigma\sigma'} = \int \mathrm{d} \mathbf{x}_1 \mathrm{d} \mathbf{x}_2 \ \frac{\phi_{i\sigma}^*(\mathbf{x}_1)\phi_{k\sigma'}^*(\mathbf{x}_2) \phi_{l\sigma'}(\mathbf{x}_2)\phi_{j\sigma}(\mathbf{x}_1)}{r_{12}}\end{split}\]

Note that here the order of \(ijkl\) is the same as that in FCIDUMP (chemist’s notation \([ij|kl]\)).

Derivation

Sum of MPO

\[\hat{H} = \sum_{m\sigma} a_{m\sigma}^\dagger \hat{H}_{m\sigma} = \sum_{m\sigma} a_{m\sigma}^\dagger \left[ \sum_{j} t_{mj,\sigma} \ a_{j\sigma} + \frac{1}{2} \sum_{jkl, \sigma'} v_{mjkl, \sigma\sigma'}\ a_{k\sigma'}^\dagger a_{l\sigma'}a_{j\sigma} \right]\]

Now consider \(LR\) partition. There are 8 possibilities: \(LLL, LRR, RLR, RRL, LLR, LRL, RLL, RRR\).

\[\begin{split}\hat{H}_{m\sigma} =&\ \left[ \sum_{j \in L} t_{mj,\sigma} \ a_{j\sigma} + \frac{1}{2} \sum_{jkl\in L, \sigma'} v_{mjkl, \sigma\sigma'} \ a_{k\sigma'}^\dagger a_{l\sigma'}a_{j\sigma} \right] + \left[ \sum_{j \in R} t_{mj,\sigma} \ a_{j\sigma} + \frac{1}{2} \sum_{jkl\in R, \sigma'} v_{mjkl, \sigma\sigma'} \ a_{k\sigma'}^\dagger a_{l\sigma'}a_{j\sigma} \right] \\ +&\ \left[ \frac{1}{2} \sum_{j \in L} a_{j\sigma} \sum_{kl\in R, \sigma'} v_{mjkl, \sigma\sigma'}\ a_{k\sigma'}^\dagger a_{l\sigma'} + \frac{1}{2} \sum_{k \in L, \sigma'} a_{k\sigma'}^\dagger \sum_{jl \in R} v_{mjkl, \sigma\sigma'}\ a_{l\sigma'}a_{j\sigma} -\frac{1}{2} \sum_{l \in L, \sigma'} a_{l\sigma'} \sum_{jk\in R} v_{mjkl, \sigma\sigma'}\ a_{k\sigma'}^\dagger a_{j\sigma} \right]\\ +&\ \left[ \frac{1}{2} \sum_{j\in R} \left( \sum_{kl \in L, \sigma'} v_{mjkl, \sigma\sigma'}\ a_{k\sigma'}^\dagger a_{l\sigma'} \right) a_{j\sigma} + \frac{1}{2} \sum_{k\in R, \sigma'} \left( \sum_{jl \in L} v_{mjkl, \sigma\sigma'}\ a_{l\sigma'}a_{j\sigma} \right) a_{k\sigma'}^\dagger - \frac{1}{2} \sum_{l\in R, \sigma'} \left( \sum_{jk \in L} v_{mjkl, \sigma\sigma'}\ a_{k\sigma'}^\dagger a_{j\sigma} \right) a_{l\sigma'} \right]\end{split}\]

Let

\[\begin{split}\hat{H}^{L/R}_{m\sigma} =&\ \sum_{j \in L/R} t_{mj,\sigma} \ a_{j\sigma} + \frac{1}{2} \sum_{jkl\in L/R, \sigma'} v_{mjkl, \sigma\sigma'} \ a_{k\sigma'}^\dagger a_{l\sigma'}a_{j\sigma} \\ \hat{P}_{ik,\sigma\sigma'}^{L/R} =&\ \sum_{jl\in L/R} v_{ijkl,\sigma\sigma'} a_{l\sigma'} a_{j\sigma}, \\ \hat{Q}_{ij,\sigma}^{L/R} =&\ \sum_{kl\in L/R,\sigma'} v_{ijkl,\sigma\sigma'} a_{k\sigma'}^\dagger a_{l\sigma'}, \\ {\hat{Q}}_{il,\sigma\sigma'}^{\prime L/R} =&\ \sum_{jk\in L/R} v_{ijkl,\sigma\sigma'} a_{k\sigma'}^\dagger a_{j\sigma} \\ {\hat{Q}}_{ij,\sigma\sigma'}^{\prime\prime R} =&\ \delta_{\sigma\sigma'} \hat{Q}^{R}_{ij\sigma} - \hat{Q}^{\prime R}_{ij\sigma\sigma'}\end{split}\]

we have

\[\begin{split}\hat{H}_{m\sigma} =&\ \hat{H}^{L}_{m\sigma} \otimes \hat{1}^R + \hat{1}^L \otimes \hat{H}^{R}_{m\sigma} + \frac{1}{2} \sum_{j \in L} a_{j\sigma} \hat{Q}_{mj,\sigma}^{R} + \frac{1}{2} \sum_{k \in L, \sigma'} a_{k\sigma'}^\dagger \hat{P}_{mk,\sigma\sigma'}^{R} - \frac{1}{2} \sum_{l \in L, \sigma'} a_{l\sigma'} {\hat{Q}}_{ml,\sigma\sigma'}^{\prime R} + \frac{1}{2} \sum_{j \in R} \hat{Q}_{mj,\sigma}^{L} a_{j\sigma} + \frac{1}{2} \sum_{k \in R, \sigma'} \hat{P}_{mk,\sigma\sigma'}^{L} a_{k\sigma'}^\dagger - \frac{1}{2} \sum_{l \in R, \sigma'} {\hat{Q}}_{ml,\sigma\sigma'}^{\prime L} a_{l\sigma'} \\ =&\ \hat{H}^{L}_{m\sigma} \otimes \hat{1}^R + \hat{1}^L \otimes \hat{H}^{R}_{m\sigma} + \frac{1}{2} \sum_{k \in L, \sigma'} a_{k\sigma'}^\dagger \hat{P}_{mk,\sigma\sigma'}^{R} + \frac{1}{2} \sum_{j \in L, \sigma'} a_{j\sigma'} \left( \delta_{\sigma\sigma'} \hat{Q}_{mj,\sigma}^{R} - {\hat{Q}}_{mj,\sigma\sigma'}^{\prime R} \right) + \frac{1}{2} \sum_{k \in R, \sigma'} \hat{P}_{mk,\sigma\sigma'}^{L} a_{k\sigma'}^\dagger + \frac{1}{2} \sum_{j \in R, \sigma'} \left( \delta_{\sigma\sigma'} \hat{Q}_{mj,\sigma}^{L} - {\hat{Q}}_{mj,\sigma\sigma'}^{\prime L} \right) a_{j\sigma'} \\ =&\ \hat{H}^{L}_{m\sigma} \otimes \hat{1}^R + \hat{1}^L \otimes \hat{H}^{R}_{m\sigma} + \frac{1}{2} \sum_{k \in L, \sigma'} a_{k\sigma'}^\dagger \hat{P}_{mk,\sigma\sigma'}^{R} + \frac{1}{2} \sum_{j \in L, \sigma'} a_{j\sigma'} {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime R} + \frac{1}{2} \sum_{k \in R, \sigma'} \hat{P}_{mk,\sigma\sigma'}^{L} a_{k\sigma'}^\dagger + \frac{1}{2} \sum_{j \in R, \sigma'} {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime L} a_{j\sigma'}\end{split}\]

Now consider \(m \in L\) or \(m \in R\). For \(m \in L\):

\[\begin{split}\sum_{m\in L, \sigma} a_{m\sigma}^\dagger \hat{H}_{m\sigma} =&\ \left( \sum_{m\in L, \sigma} a_{m\sigma}^\dagger \hat{H}^L_{m\sigma} \right) \otimes \hat{1}^R + \sum_{m\in L, \sigma} a_{m\sigma}^\dagger \otimes \hat{H}^{R}_{m\sigma} \\ +&\ \frac{1}{2} \sum_{mk \in L, \sigma\sigma'} a_{m\sigma}^\dagger a_{k\sigma'}^\dagger \hat{P}_{mk,\sigma\sigma'}^{R} + \frac{1}{2} \sum_{mj \in L, \sigma\sigma'} a_{m\sigma}^\dagger a_{j\sigma'} {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime R} + \frac{1}{2} \sum_{k \in R, \sigma'} \left( \sum_{m\in L,\sigma} a_{m\sigma}^\dagger \hat{P}_{mk,\sigma\sigma'}^{L} \right) a_{k\sigma'}^\dagger + \frac{1}{2} \sum_{j \in R, \sigma'} \left( \sum_{m\in L,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime L} \right) a_{j\sigma'} \\ =&\ \hat{H}^{ML} \otimes \hat{1}^R + \sum_{m\in L, \sigma} a_{m\sigma}^\dagger \otimes \hat{H}^{R}_{m\sigma} + \frac{1}{2} \sum_{mk \in L, \sigma\sigma'} \hat{A}_{mk,\sigma\sigma'} \hat{P}_{mk,\sigma\sigma'}^{R} + \frac{1}{2} \sum_{mj \in L, \sigma\sigma'} \hat{B}_{mj,\sigma\sigma'} {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime R} + \frac{1}{2} \sum_{k \in R, \sigma'} \hat{P}_{k\sigma'}^{ML} a_{k\sigma'}^\dagger + \frac{1}{2} \sum_{j \in R, \sigma'} \hat{Q}_{j\sigma'}^{ML} a_{j\sigma'}\end{split}\]

where

\[\begin{split}\hat{A}_{ik,\sigma\sigma'} =&\ a_{i\sigma}^\dagger a_{k\sigma'}^\dagger, \\ \hat{B}_{il,\sigma\sigma'} =&\ a_{i\sigma}^\dagger a_{l\sigma'}, \\ \hat{H}^{ML/R} =&\ \sum_{m\in L/R, \sigma} a_{m\sigma}^\dagger \hat{H}^{L/R}_{m\sigma} \\ \hat{P}_{k\sigma'}^{ML/R} =&\ \sum_{m\in L/R,\sigma} a_{m\sigma}^\dagger \hat{P}_{mk,\sigma\sigma'}^{L/R} \\ \hat{Q}_{j\sigma'}^{ML/R} =&\ \sum_{m\in L/R,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime L/R}\end{split}\]

For \(m \in R\):

\[\begin{split}\sum_{m\in R, \sigma} a_{m\sigma}^\dagger \hat{H}_{m\sigma} =&\ -\sum_{m \in R,\sigma} \hat{H}^{L}_{m\sigma} \otimes a_{m\sigma}^\dagger + \hat{1}^L \otimes \left( \sum_{m \in R,\sigma} a_{m\sigma}^\dagger \hat{H}^{R}_{m\sigma} \right) \\ -&\ \frac{1}{2} \sum_{k \in L, \sigma'} a_{k\sigma'}^\dagger \left( \sum_{m \in R,\sigma} a_{m\sigma}^\dagger \hat{P}_{mk,\sigma\sigma'}^{R} \right) - \frac{1}{2} \sum_{j \in L, \sigma'} a_{j\sigma'} \left( \sum_{m \in R,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime R} \right) + \frac{1}{2} \sum_{mk \in R, \sigma\sigma'} \hat{P}_{mk,\sigma\sigma'}^{L} a_{m\sigma}^\dagger a_{k\sigma'}^\dagger + \frac{1}{2} \sum_{mj \in R, \sigma\sigma'} {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime L} a_{m\sigma}^\dagger a_{j\sigma'} \\ =&\ -\sum_{m \in R,\sigma} \hat{H}^{L}_{m\sigma} \otimes a_{m\sigma}^\dagger + \hat{1}^L \otimes \hat{H}^{MR} - \frac{1}{2} \sum_{k \in L, \sigma'} a_{k\sigma'}^\dagger \hat{P}_{k,\sigma'}^{MR} - \frac{1}{2} \sum_{j \in L, \sigma'} a_{j\sigma'} {\hat{Q}}_{j,\sigma'}^{MR} + \frac{1}{2} \sum_{mk \in R, \sigma\sigma'} \hat{P}_{mk,\sigma\sigma'}^{L} \hat{A}_{mk,\sigma\sigma'} + \frac{1}{2} \sum_{mj \in R, \sigma\sigma'} {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime L} \hat{B}_{mj,\sigma\sigma'}\end{split}\]

In summary

\[\begin{split}\hat{H} =&\ \hat{H}^{ML} \otimes \hat{1}^R + \sum_{m\in L, \sigma} a_{m\sigma}^\dagger \otimes \hat{H}^{R}_{m\sigma} + \frac{1}{2} \sum_{mj \in L, \sigma\sigma'} \hat{A}_{mj,\sigma\sigma'} \hat{P}_{mj,\sigma\sigma'}^{R} + \frac{1}{2} \sum_{mj \in L, \sigma\sigma'} \hat{B}_{mj,\sigma\sigma'} {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime R} + \frac{1}{2} \sum_{k \in R, \sigma'} \hat{P}_{k\sigma'}^{ML} a_{k\sigma'}^\dagger + \frac{1}{2} \sum_{k \in R, \sigma'} \hat{Q}_{k\sigma'}^{ML} a_{k\sigma'} \\ -&\ \sum_{n \in R,\sigma} \hat{H}^{L}_{n\sigma} \otimes a_{n\sigma}^\dagger + \hat{1}^L \otimes \hat{H}^{MR} - \frac{1}{2} \sum_{j \in L, \sigma'} a_{j\sigma'}^\dagger \hat{P}_{j,\sigma'}^{MR} - \frac{1}{2} \sum_{j \in L, \sigma'} a_{j\sigma'} {\hat{Q}}_{j,\sigma'}^{MR} + \frac{1}{2} \sum_{nk \in R, \sigma\sigma'} \hat{P}_{nk,\sigma\sigma'}^{L} \hat{A}_{nk,\sigma\sigma'} + \frac{1}{2} \sum_{nk \in R, \sigma\sigma'} {\hat{Q}}_{nk,\sigma\sigma'}^{\prime\prime L} \hat{B}_{nk,\sigma\sigma'}\end{split}\]

The operators required in left block are

\[\big\{ \hat{H}^{ML}, a_{m\sigma}^\dagger, \hat{A}_{mj,\sigma\sigma'}, \hat{B}_{mj,\sigma\sigma'}, \hat{P}_{k\sigma'}^{ML}, \hat{Q}_{k\sigma'}^{ML}, \hat{H}^{L}_{n\sigma} ,\hat{1}^L, a_{j\sigma'}^\dagger, a_{j\sigma'}, \hat{P}_{nk,\sigma\sigma'}^{L}, {\hat{Q}}_{nk,\sigma\sigma'}^{\prime\prime L} \big\} \quad (m,j\in L, \ n,k \in R)\]

The total number of operators is

\[\begin{split}N =&\ 1 + 2 K_{ML} + 4 K_{ML} K_{L} + 4 K_{ML} K_{L} + 2 K_{R} + 2 K_{R} + 2 K_{MR} + 1 + 2 K_{L} + 2 K_{L} + 4 K_{MR} K_{R} + 4 K_{MR} K_{R} \\ =&\ 2 + 2 K_M + 4 K + 8 K_{ML} K_{L} + 8 K_{MR} K_{R}\end{split}\]

Reordered left and right block operators

\[\begin{split}L =&\ \big\{ \hat{H}^{ML}, \hat{1}^L, a_{m\sigma}^\dagger, \hat{H}^{L}_{n\sigma} ,a_{j\sigma'}^\dagger, a_{j\sigma'}, \hat{P}_{k\sigma'}^{ML}, \hat{Q}_{k\sigma'}^{ML}, \hat{A}_{mj,\sigma\sigma'}, \hat{B}_{mj,\sigma\sigma'}, \hat{P}_{nk,\sigma\sigma'}^{L}, {\hat{Q}}_{nk,\sigma\sigma'}^{\prime \prime L} \big\} \quad (m,j\in L, \ n,k \in R) \\ R =&\ \big\{ \hat{1}^R, \hat{H}^{MR}, \hat{H}^{R}_{m\sigma}, a_{n\sigma}^\dagger, \hat{P}_{j,\sigma'}^{MR}, {\hat{Q}}_{j,\sigma'}^{MR}, a_{k\sigma'}^\dagger, a_{k\sigma'}, \hat{P}_{mj,\sigma\sigma'}^{R}, {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime R}, \hat{A}_{nk,\sigma\sigma'}, \hat{B}_{nk,\sigma\sigma'} \big \}\end{split}\]

Now let

\[\begin{split}\hat{R}_{k\sigma}^{ML/R} =&\ -2 \delta(k\in M) \hat{H}^{L/R}_{k\sigma} + \hat{P}_{k\sigma'}^{ML/R} \\ \hat{S}_{k\sigma}^{ML/R} =&\ \hat{Q}_{k\sigma'}^{ML/R}\end{split}\]

we have

\[\begin{split}L =&\ \big\{ \hat{H}^{ML}, \hat{1}^L, a_{j\sigma'}^\dagger, a_{j\sigma'}, \hat{R}_{k\sigma'}^{ML}, \hat{S}_{k\sigma'}^{ML}, \hat{A}_{mj,\sigma\sigma'}, \hat{B}_{mj,\sigma\sigma'}, \hat{P}_{nk,\sigma\sigma'}^{L}, {\hat{Q}}_{nk,\sigma\sigma'}^{\prime \prime L} \big\} \quad (m,j\in L, \ n,k \in R) \\ R =&\ \big\{ \hat{1}^R, \hat{H}^{MR}, \hat{R}_{j,\sigma'}^{MR}, \hat{S}_{j,\sigma'}^{MR}, a_{k\sigma'}^\dagger, a_{k\sigma'}, \hat{P}_{mj,\sigma\sigma'}^{R}, {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime R}, \hat{A}_{nk,\sigma\sigma'}, \hat{B}_{nk,\sigma\sigma'} \big \}\end{split}\]

The total number of operators is

\[N = 2 + 4 K + 8 K_{ML} K_{L} + 8 K_{MR} K_{R}\]

Blocking

\[\begin{split}\hat{P}_{k\sigma'}^{ML*} =&\ \sum_{m\in L*,\sigma} a_{m\sigma}^\dagger \hat{P}_{mk,\sigma\sigma'}^{L*} = \sum_{m\in L*,\sigma} a_{m\sigma}^\dagger \sum_{jl\in L*} v_{mjkl,\sigma\sigma'} a_{l\sigma'} a_{j\sigma} \\ =&\ \hat{P}_{k\sigma'}^{ML} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{P}_{k\sigma'}^{M*} + \sum_{m\in *,\sigma} a_{m\sigma}^\dagger \hat{P}_{mk,\sigma\sigma'}^{L} + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger \sum_{j\in *, l\in L} v_{mjkl,\sigma\sigma'} a_{l\sigma'} a_{j\sigma} + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger \sum_{j\in L, l\in *} v_{mjkl,\sigma\sigma'} a_{l\sigma'} a_{j\sigma} \\ &\ + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger \hat{P}_{mk,\sigma\sigma'}^{*} + \sum_{m\in *,\sigma} a_{m\sigma}^\dagger \sum_{j\in *, l\in L} v_{mjkl,\sigma\sigma'} a_{l\sigma'} a_{j\sigma} + \sum_{m\in *,\sigma} a_{m\sigma}^\dagger \sum_{j\in L, l\in *} v_{mjkl,\sigma\sigma'} a_{l\sigma'} a_{j\sigma} \\ =&\ \hat{P}_{k\sigma'}^{ML} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{P}_{k\sigma'}^{M*} + \sum_{m\in *,\sigma} \hat{P}_{mk,\sigma\sigma'}^{L} a_{m\sigma}^\dagger + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger \hat{P}_{mk,\sigma\sigma'}^{*} + \sum_{ml\in L,j\in *,\sigma} v_{mjkl,\sigma\sigma'} a_{m\sigma}^\dagger a_{l\sigma'} a_{j\sigma} - \sum_{mj\in L,l\in *,\sigma} v_{mjkl,\sigma\sigma'} a_{m\sigma}^\dagger a_{j\sigma} a_{l\sigma'} \\ &\ - \sum_{mj\in *,l\in L,\sigma} v_{mjkl,\sigma\sigma'} a_{l\sigma'} a_{m\sigma}^\dagger a_{j\sigma} + \sum_{ml\in *,j\in L,\sigma} v_{mjkl,\sigma\sigma'} a_{j\sigma} a_{m\sigma}^\dagger a_{l\sigma'} \\ =&\ \hat{P}_{k\sigma'}^{ML} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{P}_{k\sigma'}^{M*} + \sum_{m\in *,\sigma} \hat{P}_{mk,\sigma\sigma'}^{L} a_{m\sigma}^\dagger + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger \hat{P}_{mk,\sigma\sigma'}^{*} + \sum_{ml\in L,j\in *,\sigma} v_{mjkl,\sigma\sigma'} a_{m\sigma}^\dagger a_{l\sigma'} a_{j\sigma} - \sum_{ml\in L,j\in *,\sigma} v_{mlkj,\sigma\sigma'} a_{m\sigma}^\dagger a_{l\sigma} a_{j\sigma'} \\ &\ - \sum_{mj\in *,l\in L,\sigma} v_{mjkl,\sigma\sigma'} a_{l\sigma'} a_{m\sigma}^\dagger a_{j\sigma} + \sum_{mj\in *,l\in L,\sigma} v_{mlkj,\sigma\sigma'} a_{l\sigma} a_{m\sigma}^\dagger a_{j\sigma'} \\ =&\ \hat{P}_{k\sigma'}^{ML} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{P}_{k\sigma'}^{M*} + \sum_{m\in *,\sigma} \hat{P}_{mk,\sigma\sigma'}^{L} a_{m\sigma}^\dagger + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger \hat{P}_{mk,\sigma\sigma'}^{*} \\ &\ + \sum_{ml\in L,j\in *,\sigma} v_{mjkl,\sigma\sigma'} \hat{B}_{ml,\sigma\sigma'} a_{j\sigma} - \sum_{ml\in L,j\in *,\sigma} v_{mlkj,\sigma\sigma'} \hat{B}_{ml,\sigma\sigma} a_{j\sigma'} + \sum_{mj\in *,l\in L,\sigma} v_{mlkj,\sigma\sigma'} a_{l\sigma} \hat{B}_{mj,\sigma\sigma'} - \sum_{mj\in *,l\in L,\sigma} v_{mjkl,\sigma\sigma'} a_{l\sigma'} \hat{B}_{mj,\sigma\sigma}\end{split}\]

and

\[\begin{split}\hat{Q}_{j\sigma'}^{ML*} =&\ \sum_{m\in L*,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime L*} = \sum_{m\in L*,\sigma} a_{m\sigma}^\dagger \sum_{kl\in L*} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{k\sigma''}^\dagger a_{l\sigma''} - v_{mlkj,\sigma\sigma'} a_{k\sigma'}^\dagger a_{l\sigma} \right) \\ =&\ \hat{Q}_{j\sigma'}^{ML} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{Q}_{j\sigma'}^{M*} + \sum_{m\in *,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime L} + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime *} \\ &\ + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger \sum_{k\in *, l\in L} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{k\sigma''}^\dagger a_{l\sigma''} - v_{mlkj,\sigma\sigma'} a_{k\sigma'}^\dagger a_{l\sigma} \right) + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger \sum_{k\in L, l\in *} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{k\sigma''}^\dagger a_{l\sigma''} - v_{mlkj,\sigma\sigma'} a_{k\sigma'}^\dagger a_{l\sigma} \right) \\ &\ + \sum_{m\in *,\sigma} a_{m\sigma}^\dagger \sum_{k\in *, l\in L} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{k\sigma''}^\dagger a_{l\sigma''} - v_{mlkj,\sigma\sigma'} a_{k\sigma'}^\dagger a_{l\sigma} \right) + \sum_{m\in *,\sigma} a_{m\sigma}^\dagger \sum_{k\in L, l\in *} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{k\sigma''}^\dagger a_{l\sigma''} - v_{mlkj,\sigma\sigma'} a_{k\sigma'}^\dagger a_{l\sigma} \right) \\ =&\ \hat{Q}_{j\sigma'}^{ML} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{Q}_{j\sigma'}^{M*} + \sum_{m\in *,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime L} + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime *} \\ &\ + \sum_{ml\in L,k\in *, \sigma} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{m\sigma}^\dagger a_{k\sigma''}^\dagger a_{l\sigma''} - v_{mlkj,\sigma\sigma'} a_{m\sigma}^\dagger a_{k\sigma'}^\dagger a_{l\sigma} \right) + \sum_{mk\in L, l\in *,\sigma} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{m\sigma}^\dagger a_{k\sigma''}^\dagger a_{l\sigma''} - v_{mlkj,\sigma\sigma'} a_{m\sigma}^\dagger a_{k\sigma'}^\dagger a_{l\sigma} \right) \\ &\ + \sum_{mk\in *, l\in L,\sigma} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{m\sigma}^\dagger a_{k\sigma''}^\dagger a_{l\sigma''} - v_{mlkj,\sigma\sigma'} a_{m\sigma}^\dagger a_{k\sigma'}^\dagger a_{l\sigma} \right) + \sum_{ml\in *,k\in L,\sigma} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{m\sigma}^\dagger a_{k\sigma''}^\dagger a_{l\sigma''} - v_{mlkj,\sigma\sigma'} a_{m\sigma}^\dagger a_{k\sigma'}^\dagger a_{l\sigma} \right) \\ =&\ \hat{Q}_{j\sigma'}^{ML} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{Q}_{j\sigma'}^{M*} + \sum_{m\in *,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime L} + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime *} \\ &\ - \sum_{ml\in L,k\in *, \sigma} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{m\sigma}^\dagger a_{l\sigma''} a_{k\sigma''}^\dagger - v_{mlkj,\sigma\sigma'} a_{m\sigma}^\dagger a_{l\sigma} a_{k\sigma'}^\dagger \right) + \sum_{mk\in L, l\in *,\sigma} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{m\sigma}^\dagger a_{k\sigma''}^\dagger a_{l\sigma''} - v_{mlkj,\sigma\sigma'} a_{m\sigma}^\dagger a_{k\sigma'}^\dagger a_{l\sigma} \right) \\ &\ + \sum_{mk\in *, l\in L,\sigma} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{l\sigma''} a_{m\sigma}^\dagger a_{k\sigma''}^\dagger - v_{mlkj,\sigma\sigma'} a_{l\sigma} a_{m\sigma}^\dagger a_{k\sigma'}^\dagger \right) - \sum_{ml\in *,k\in L,\sigma} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{k\sigma''}^\dagger a_{m\sigma}^\dagger a_{l\sigma''} - v_{mlkj,\sigma\sigma'} a_{k\sigma'}^\dagger a_{m\sigma}^\dagger a_{l\sigma} \right)\end{split}\]

and

\[\begin{split}=&\ \hat{Q}_{j\sigma'}^{ML} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{Q}_{j\sigma'}^{M*} + \sum_{m\in *,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime L} + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime *} \\ &\ - \sum_{ml\in L,k\in *, \sigma} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} \hat{B}_{ml\sigma\sigma''} a_{k\sigma''}^\dagger - v_{mlkj,\sigma\sigma'} \hat{B}_{ml\sigma\sigma} a_{k\sigma'}^\dagger \right) + \sum_{mk\in L, l\in *,\sigma} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} \hat{A}_{mk\sigma\sigma''} a_{l\sigma''} - v_{mlkj,\sigma\sigma'} \hat{A}_{mk\sigma\sigma'} a_{l\sigma} \right) \\ &\ + \sum_{mk\in *, l\in L,\sigma} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{l\sigma''} \hat{A}_{mk\sigma\sigma''} - v_{mlkj,\sigma\sigma'} a_{l\sigma} \hat{A}_{mk\sigma\sigma'} \right) - \sum_{ml\in *,k\in L,\sigma} \left( \delta_{\sigma\sigma'} \sum_{\sigma''} v_{mjkl,\sigma\sigma''} a_{k\sigma''}^\dagger \hat{B}_{ml\sigma\sigma''} - v_{mlkj,\sigma\sigma'} a_{k\sigma'}^\dagger \hat{B}_{ml\sigma\sigma} \right)\end{split}\]

after simplification

\[\begin{split}=&\ \hat{Q}_{j\sigma'}^{ML} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{Q}_{j\sigma'}^{M*} + \sum_{m\in *,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime L} + \sum_{m\in L,\sigma} a_{m\sigma}^\dagger {\hat{Q}}_{mj,\sigma\sigma'}^{\prime\prime *} \\ &\ - \sum_{ml\in L,k\in *, \sigma} \left( v_{mjkl,\sigma'\sigma} \hat{B}_{ml\sigma'\sigma} a_{k\sigma}^\dagger - v_{mlkj,\sigma\sigma'} \hat{B}_{ml\sigma\sigma} a_{k\sigma'}^\dagger \right) + \sum_{ml\in L, k\in *,\sigma} \left( v_{mjlk,\sigma'\sigma} \hat{A}_{ml\sigma'\sigma} a_{k\sigma} - v_{mklj,\sigma\sigma'} \hat{A}_{ml\sigma\sigma'} a_{k\sigma} \right) \\ &\ + \sum_{mk\in *, l\in L,\sigma} \left( v_{mjkl,\sigma'\sigma} a_{l\sigma} \hat{A}_{mk\sigma'\sigma} - v_{mlkj,\sigma\sigma'} a_{l\sigma} \hat{A}_{mk\sigma\sigma'} \right) - \sum_{mk\in *,l\in L,\sigma} \left( v_{mjlk,\sigma'\sigma} a_{l\sigma}^\dagger \hat{B}_{mk\sigma'\sigma} - v_{mklj,\sigma\sigma'} a_{l\sigma'}^\dagger \hat{B}_{mk\sigma\sigma} \right)\end{split}\]

For \(P, Q\), we have

\[\begin{split}\hat{P}_{ik,\sigma\sigma'}^{L*} =&\ \sum_{jl\in L*} v_{ijkl,\sigma\sigma'} a_{l\sigma'} a_{j\sigma} = \hat{P}_{ik,\sigma\sigma'}^{L} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{P}_{ik,\sigma\sigma'}^{*} + \sum_{j\in L, l \in *} v_{ijkl,\sigma\sigma'} a_{l\sigma'} a_{j\sigma} + \sum_{j\in *, l \in L} v_{ijkl,\sigma\sigma'} a_{l\sigma'} a_{j\sigma} \\ =&\ \hat{P}_{ik,\sigma\sigma'}^{L} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{P}_{ik,\sigma\sigma'}^{*} - \sum_{j\in L, l \in *} v_{ijkl,\sigma\sigma'} a_{j\sigma} a_{l\sigma'} + \sum_{j\in L, l \in *} v_{ilkj,\sigma\sigma'} a_{j\sigma'} a_{l\sigma} \\ \hat{Q}_{ij,\sigma\sigma'}^{\prime\prime L*} =&\ \delta_{\sigma\sigma'} \hat{Q}^{L*}_{ij\sigma} - \hat{Q}^{\prime L*}_{ij\sigma\sigma'} = \delta_{\sigma\sigma'} \sum_{kl\in L*,\sigma''} v_{ijkl,\sigma\sigma''} a_{k\sigma''}^\dagger a_{l\sigma''} - \sum_{kl\in L*} v_{ilkj,\sigma\sigma'} a_{k\sigma'}^\dagger a_{l\sigma} \\ =&\ \hat{Q}_{ij,\sigma\sigma'}^{\prime\prime L} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{Q}_{ij,\sigma\sigma'}^{\prime\prime *} + \delta_{\sigma\sigma'} \sum_{k\in L, l\in *,\sigma''} v_{ijkl,\sigma\sigma''} a_{k\sigma''}^\dagger a_{l\sigma''} - \sum_{k\in L, l\in *} v_{ilkj,\sigma\sigma'} a_{k\sigma'}^\dagger a_{l\sigma} + \delta_{\sigma\sigma'} \sum_{k\in *, l\in L,\sigma''} v_{ijkl,\sigma\sigma''} a_{k\sigma''}^\dagger a_{l\sigma''} - \sum_{k\in *, l\in L} v_{ilkj,\sigma\sigma'} a_{k\sigma'}^\dagger a_{l\sigma} \\ =&\ \hat{Q}_{ij,\sigma\sigma'}^{\prime\prime L} \otimes \hat{1}^* + \hat{1}^L \otimes \hat{Q}_{ij,\sigma\sigma'}^{\prime\prime *} + \delta_{\sigma\sigma'} \sum_{k\in L, l\in *,\sigma''} v_{ijkl,\sigma\sigma''} a_{k\sigma''}^\dagger a_{l\sigma''} - \sum_{k\in L, l\in *} v_{ilkj,\sigma\sigma'} a_{k\sigma'}^\dagger a_{l\sigma} - \delta_{\sigma\sigma'} \sum_{k\in L, l\in *,\sigma''} v_{ijlk,\sigma\sigma''} a_{k\sigma''} a_{l\sigma''}^\dagger + \sum_{k\in L, l\in *} v_{iklj,\sigma\sigma'} a_{k\sigma} a_{l\sigma'}^\dagger\end{split}\]