The model that i am studying is the s-f model.
I wrote some post about it, then, in order to understund better my notation go to this question
Now, I am computing some parameters that emerge out of the equations of motion for the Green's function.
For this purpose i am using the spectral theorem $$ \left\langle \hat{B}\,\hat{A} \right\rangle = \frac{1}{\hbar} \int_{-\infty}^{+\infty} \, \text{d}E \, \frac{S_{\hat{A}\,\hat{B}}\left(E\right)}{e^{\beta\,E} + 1 } $$ Since i know the Green function i can compute the spectral density through $$ \text{Im}\,\text{G}_{\hat{A},\hat{B}}^{r} \left(E \right) = -\,\pi \,\text{S}_{\hat{A},\hat{B}}\left( E \right) $$
The parameter that i have to found is $$ \Delta^{-\sigma}_{i} = \frac{1}{\hbar} \left\langle \hat{S}^{-\sigma}_{i} \,\hat{a}^{\dagger}_{i,\sigma}\,\hat{a}_{i,-\sigma} \right\rangle - \frac{1}{\hbar}\, z_{\sigma} \left\langle \hat{S}^{z}_{i}\, \hat{n}_{i,-\sigma} \right\rangle \tag{1} \label{eq:1} $$ where $\hat{a}_{i,\sigma},\,\hat{a}^{\dagger}_{i,\sigma}$ are the fermionic operators referred to the conduction electrons, $i$ is the index of the $i-th$ lattice site while $\sigma=\uparrow,\downarrow$ the spin number.
$\hat{S}^{z}_{i}$ and $\hat{S}^{-\sigma}_{i}$ are the spin operators referred to the isolated electrons in f-shells.
They commute to each other because they belong to different degrees of freedom of the system
I computed a Green function that could be used for this purpose $$ \ll \hat{d}_{i,\sigma}; \, \hat{a}^{\dagger}_{i,\sigma}\, \gg \equiv G_{d,\sigma}^{r}\left(E\right) $$ where $$ \hat{d}_{i,\sigma} = z_{\sigma}\, \hat{S}^{z}_{i} \,\hat{a}_{i,\sigma} + \hat{S}^{-\sigma}_{i} \,\hat{a}_{i,-\sigma} $$ so the spectral theorem could compute the following correlation function \begin{align} \left\langle \hat{a}^{\dagger}_{i,\sigma}\, \hat{d}_{i,\sigma} \right\rangle &= \left\langle \hat{a}^{\dagger}_{i,\sigma}\, \left( z_{\sigma}\, \hat{S}^{z}_{i} \,\hat{a}_{i,\sigma} + \hat{S}^{-\sigma}_{i} \,\hat{a}_{i,-\sigma} \right) \right\rangle \\ &= z_{\sigma}\, \left\langle \hat{S}^{z}_{i}\, \hat{a}^{\dagger}_{i,\sigma} \,\hat{a}_{i,\sigma} \right\rangle + \left\langle \hat{S}^{-\sigma}_{i} \hat{a}^{\dagger}_{i,\sigma} \,\hat{a}_{i,-\sigma} \right\rangle \\ &= z_{\sigma}\, \left\langle \hat{S}^{z}_{i}\, \hat{n}_{i,\sigma} \right\rangle + \left\langle \hat{S}^{-\sigma}_{i} \hat{a}^{\dagger}_{i,\sigma} \,\hat{a}_{i,-\sigma} \right\rangle \tag{2} \label{eq:2} \end{align}
I noticed that changing the spin index of \eqref{eq:2} from $\sigma$ to $-\sigma$
\begin{equation} \left\langle \hat{a}^{\dagger}_{i,-\sigma}\, \hat{d}_{i,-\sigma} \right\rangle = - z_{\sigma}\, \left\langle \hat{S}^{z}_{i}\, \hat{n}_{i,-\sigma} \right\rangle + \left\langle \hat{S}^{\sigma}_{i} \hat{a}^{\dagger}_{i,-\sigma} \,\hat{a}_{i,\sigma} \right\rangle \tag{3} \label{eq:3} \end{equation} and summing \eqref{eq:2} and \eqref{eq:3}, i have the following relation \begin{equation} \left\langle \hat{a}^{\dagger}_{i,\sigma}\, \hat{d}_{i,\sigma} \right\rangle + \left\langle \hat{a}^{\dagger}_{i,-\sigma}\, \hat{d}_{i,-\sigma} \right\rangle = \hbar\, \left( \Delta_{i,\sigma} + \Delta_{i,-\sigma} \right) \tag{4} \label{eq:4} \end{equation} in \eqref{eq:3} i used the propieties of $z_{\sigma}$. Since it could be 1 or -1 , it satisfies \begin{align*} z_{\sigma}\cdot z_{\sigma} &= 1 \,\forall\,\sigma \\ z_{-\sigma} &= -z_{\sigma} \,\forall\,\sigma \end{align*} The rhs of \eqref{eq:4} is computed through the spectral theorem. knowing that $$ S_{d,\sigma} = \hbar^{2}\,\sum_{i=1}^{4} \, \beta_{i,\sigma}\,\delta\left(E - E_{i} + \mu\right) $$ where $\beta_{i,\sigma}$ is the i-th spectral weight referred to the i-th excitation energy $E_{i}$, while $\mu$ is the chemical potential
Then \eqref{eq:4} becomes \begin{equation} \sum_{i=1}^{4}\,\left(\beta_{i,\sigma} + \beta_{i,-\sigma}\right) f_{i}\left(T\right) = \Delta_{i}^{\sigma} + \Delta_{i}^{-\sigma} \tag{5} \label{eq:5} \end{equation} where $$ f_{i}\left(T\right) = \frac{1}{1+e^{\beta\left(E_{i} - \mu\right)}} $$ Unfortunatly i have not a relation only for the parameter \eqref{eq:1}
The result of the book is $$ \Delta_{i}^{-\sigma} = \sum_{i=1}^{4}\,\beta_{i,-\sigma} f_{i}\left(T\right) $$ but i can't find other relations that allow me to find this one.
I can't use directly the spectral theorem for the spectral density related to $S_{d,-\sigma}$, beacuse it does not reproduce the correlation function \eqref{eq:1}
