Hamiltonian with periodic potential in second quantization
I'm working with the following Hamiltonian
$\hat{H}=\int\mathrm{d}\mathbf{x}\sum_{\sigma\in\left\lbrace\uparrow,\downarrow\right\rbrace}\hat{\psi}\sigma^\dagger(\mathbf{x})\left[-\frac{\hbar^2\nabla^2}{2m}+V(\mathbf{x})\right]\hat{\psi}\sigma(\mathbf{x})$
(plus a term concerning an interaction potential, which I can do myself when I understand this part.)
The $\hat\psi^\dagger(\mathbf{x})$ and $\hat\psi(\mathbf{x})$ are the fermionic field creation and annihilation operators. $V(\mathbf{x})=V(\mathbf{x}+a\mathbf{\hat{e}}i)$ is a periodic potential with cubic symmetry, where $a$ is the lattice constant. The minima of the potential are denoted by $\mathbf{x}\mathbf{j}$, with $\mathbf{j}=j_x \mathbf{\hat{e}}_x+j_y \mathbf{\hat{e}}_y+j_z \mathbf{\hat{e}}_z$ the vector that labels the minima. 281
The coordinates are chosen in such a way that $\mathbf{x}=0$ coincides with a minimum.
The single particle <PERSON> eqn. is solved according to <PERSON>'s theorem by eigenfunctions $\phi_{\mathbf{k},m}(\mathbf{x})$ with energies $\epsilon_{\mathbf{k},m}$. The Wannier functions $W_m(\mathbf{x}-\mathbf{x_j})$ that describe a localized particle at site $\mathbf{x_j}$ are determined by
$\phi_{\mathbf{k},m}(\mathbf{x})=\frac{1}{\sqrt{N}}\sum_\mathbf{j}\mathrm{e}^{i\mathbf{k}\cdot\mathbf{x_j}}W_m(\mathbf{x}-\mathbf{x_j})$
The aim is to write the Hamiltonian in terms of Wannier functions and the creation/annihilation operators, $\hat{c}{\mathbf{j},m,\sigma}^\dagger$ and $\hat{c}{\mathbf{j},m,\sigma}$, that create/annihilate an electron with spin state $\left|{\sigma}\right\rangle$ at lattice site $\mathbf{x_j}$ in the band with index $m$.
To do so, I started by expanding the field operators in terms of the <PERSON> functions and the operators $\hat{c}{\mathbf{j},m,\sigma}^\dagger$ and $\hat{c}{\mathbf{j},m,\sigma}$, i.e. 526