Accuracy of approximation in the Poisson theorem in terms of the $\chi^2$-distance

We study the limit behavior of the $\chi^2$-distance between the distributions of the $n$th partial sum of independent not necessarily identically distributed Bernoulli random variables and the accompanying Poisson law. As a consequence in the i.i.d. case we make the multiplicative constant preciser in the available upper bound for the rate of convergence in the Poisson limit theorem.