  In this paper, we study a priori error estimates for the finite element
approximation of the nonlinear Schr\"{o}dinger-Poisson model. The electron
density is defined by an infinite series over all eigenvalues of the
Hamiltonian operator. To establish the error estimate, we present a unified
theory of error estimates for a class of nonlinear problems. The theory is
based on three conditions: 1) the original problem has a solution $u$ which is
the fixed point of a compact operator $\Ca$, 2) $\Ca$ is
Fr\'{e}chet-differentiable at $u$ and $\Ci-\Ca'[u]$ has a bounded inverse in a
neighborhood of $u$, and 3) there exists an operator $\Ca_h$ which converges to
$\Ca$ in the neighborhood of $u$. The theory states that $\Ca_h$ has a fixed
point $u_h$ which solves the approximate problem. It also gives the error
estimate between $u$ and $u_h$, without assumptions on the well-posedness of
the approximate problem. We apply the unified theory to the finite element
approximation of the Schr\"{o}dinger-Poisson model and obtain optimal error
estimate between the numerical solution and the exact solution. Numerical
experiments are presented to verify the convergence rates of numerical
solutions.
