How to evaluate fermion operator in product state?

I have a question concerning the implementation of DMRG for fermion chains. Suppose I have a simple chain $$H=t\sum_{<i,j>}\sum_{\sigma}c^{\dagger}_{i\sigma}c_{j\sigma},$$ in which $i$ and $j$ are nearest neighbor site indices. Suppose using infinite DMRG at some stage, I want to calculate configuration $B\cdot\cdot B$, for block $B$ I have a bunch of state ${\left|\alpha\right\rangle}$. We define matrix product state is that $$\left|\alpha_1\alpha_2\right\rangle = c^{\dagger}_{2}\cdots c^{\dagger}_{1}\cdots\left|0\right\rangle.$$ Therefore I want to find ground state of superblock in terms of $\left|\alpha_Ls_L\alpha_Rs_R\right\rangle$, in which $s$ is the site wavefunction, and in our case, could be $\left|0\right\rangle$, $\left|\uparrow\right\rangle$, $\left|\downarrow\right\rangle$, and $\left|2\right\rangle$.

Then question is , when I want to evaluate the matrix element in the basis of $\left|\alpha_Ls_L\alpha_Rs_R\right\rangle$, I found that the hopping between $s_L$ and $s_R$ is inevitably depending on the electron number in $\alpha$, which should be fractional (because subsystem $B$ is part of the entire system, and usually the subsystem contains fractional number of electrons).

To be specific, see the hopping operator between $s_L$ and $s_R$: $$V=\sum_{\sigma}(d^{\dagger}_{L\sigma}d_{R\sigma}+d^{\dagger}_{R\sigma}d_{L\sigma}).$$ The matrix element for $d^{\dagger}_{L\sigma}d_{R\sigma}$ is $$\left\langle \alpha_Ls_L\alpha_Rs_R\right|d^{\dagger}_{L\sigma}d_{R\sigma}\left|\alpha'_Ls'_L\alpha'_Rs'_R\right\rangle = (-1)^{n_{s_R}}\delta_{\alpha_L\alpha'_L}[d_{R\sigma}]_{s_Rs'_R}\left\langle s_L\alpha_R\right|d^{\dagger}_{L\sigma}\left|s'_L\alpha'_R\right\rangle.$$ Here $n_{s_R}$ is the number of electrons in $s_R$. We can see that the matrix element of $\left\langle s_L\alpha_R\right|d^{\dagger}_{L\sigma}\left|s'_L\alpha'_R\right\rangle$ is depending on the number of electrons in $\alpha_R$.

So could anyone tell me how to proceed if $\alpha$ contains fractional number of electrons, and what is the correct procedure for treating operators like this in fermionic DMRG procedure?