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I am computing the expectation value of the kinetic term of a tight-binding model, respect to the Gutzwiller wavefunction, in the limit of infinite lattice-coordination, i.e using these constraints (the notation i am using is here)

\begin{align} \left\langle\varPsi_0\right| \hat{P}^{\dagger}_{i}\hat{P}_{i}\left|\varPsi_0\right\rangle &=1 \label{eq:GAnorm}\\ \left\langle\varPsi_0\right| \hat{P}^{\dagger}_{i}\hat{P}_{i}\hat{c}^{\dagger}_{i,a}\hat{c}_{i,b} \left|\varPsi_0\right\rangle &=\left\langle\varPsi_0\right| \hat{c}^{\dagger}_{i,a}\hat{c}_{i,b}\left|\varPsi_0\right\rangle \label{eq:GAvasp} \end{align}

I can apply the Wick's theorem

\begin{align} \sum_{i,j,a,b} t_{i,j}^{a,b} \left\langle\varPsi_0\right| \hat{P}^{\dagger} \hat{c}^{\dagger}_{i;a} \hat{c}_{j;b} \hat{P} \left|\varPsi_0\right\rangle &= \sum_{i,j,a,b} t_{i,j}^{a,b} \left\langle\varPsi_0\right| \left( \prod_{h \neq i,j} \hat{P}^{\dagger}_{h} \hat{P}_{h} \right) \hat{P}^{\dagger}_{i} \hat{c}^{\dagger}_{i;a} \hat{P}_{i} \hat{P}^{\dagger}_{j} \hat{c}_{j;b} \hat{P}_{j} \left|\varPsi_0\right\rangle \nonumber \\ &= \sum_{i,j,a,b} t_{i,j}^{a,b} \left[ \prod_{h \neq i,j} \overbrace{ \left\langle\varPsi_0\right| \hat{P}^{\dagger}_{h} \hat{P}_{h} \left|\varPsi_0\right\rangle}^{1} %%%%% \left\langle\varPsi_0\right| \hat{P}^{\dagger}_{i} \hat{c}^{\dagger}_{i;a} \hat{P}_{i} \hat{P}^{\dagger}_{j} \hat{c}_{j;b} \hat{P}_{j} \left|\varPsi_0\right\rangle \right.\nonumber \\ &\left.+\ \left\langle\varPsi_0\right| \left( \prod_{h \neq i,j} \hat{P}^{\dagger}_{h} \hat{P}_{h} \right) \hat{P}^{\dagger}_{i} \hat{c}^{\dagger}_{i;a} \hat{P}_{i} \hat{P}^{\dagger}_{j} \hat{c}_{j;b} \hat{P}_{j} \left|\varPsi_0\right\rangle_{\text{conn.}} \right] \nonumber \\ &= \sum_{i,j,a,b} t_{i,j}^{a,b} \left\langle\varPsi_0\right| \hat{P}^{\dagger}_{i} \hat{c}^{\dagger}_{i;a} \hat{P}_{i} \hat{P}^{\dagger}_{j} \hat{c}_{j;b} \hat{P}_{j} \left|\varPsi_0\right\rangle \label{eq:Hopping_GA_1}\tag{1} \end{align}

Here, \eqref{eq:Hopping_GA_1} is rewritten in terms of the renormalization Gutzwiller factors $R_{i;a,b}$ \begin{equation} \left\langle\varPsi_0\right| \hat{P}^{\dagger}_{i}\, \hat{c}^{\dagger}_{i;a} \hat{P}_{i}\, \hat{c}_{j;b} \left|\varPsi_0\right\rangle = \sum_{c} R_{i;a,c}^{*} \left\langle\varPsi_0\right| \hat{c}^{\dagger}_{i;c} \hat{c}_{j;b} \left|\varPsi_0\right\rangle \label{eq:Renormalization_GW_factors} \tag{2} \end{equation}

So the expectation value in \eqref{eq:Hopping_GA_1} becomes

\begin{equation} \left\langle\varPsi_0\right| \hat{P}^{\dagger}_{i} \hat{c}^{\dagger}_{i;a} \hat{P}_{i} \hat{P}^{\dagger}_{j} \hat{c}_{j;b} \hat{P}_{j} \left|\varPsi_0\right\rangle = \sum_{c,d} R^{*}_{i,a,c} R_{j,b,d} \left\langle\varPsi_0\right| \hat{c}^{\dagger}_{i;a} \hat{c}_{j;b} \left|\varPsi_0\right\rangle \label{eq:prova}\tag{3} \end{equation}

I want to prove \eqref{eq:prova}

To do this, I am computing the expectation value in \eqref{eq:Hopping_GA_1} trying to isolate

$\left\langle\varPsi_0\right| \hat{c}^{\dagger}_{i;a} \hat{c}_{j;b} \left|\varPsi_0\right\rangle $ but maybe this never get anywhere

Can you give me some hint?

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