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}\]