Solving Hamiltonian with chain of charge centers?

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I was thinking of how to solve the single particle Hamiltonian
[tex]H=...+\sum_i \frac{1}{\vec{r}-\vec{r}_i}[/tex]
where [itex]\vec{r}_i=i\cdot\vec{a}[/itex]
Transforming it into second quantization k-space I had terms like
[tex]H=...+\sum_G...c^\dag_{k+G}c_k[/tex]
But it seems to me that for the method of trial wavefunctions any wavefunction [itex]\psi[/itex] gives zero matrix elements?!
[tex]<\psi|c^\dag_{k+G}c_k|\psi>=<c_{k+G}\psi|c_k\psi>=0[/tex]
Is there anything wrong? How would I solve a potential from equally spaced chain of static point charges with a single electron moving?
 
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