  In this paper we continue to develop our approach to the chaoticity
properties of the quantum Hamiltonian systems. Our earlier suggested chaoticity
criterion characterizes the initial symmetry breaking and the destruction of
the corresponding integrals of motion in a perturbed system, which causes the
system's chaotisation. Transition from regularity to chaos in quantum systems
occurs at a certain critical value of the perturbation parameter. In our
previous papers we had to diagonalize the perturbed system's Hamiltonian matrix
in order to estimate this parameter. In the present paper we demonstrate that
the critical perturbation parameter for the transition from regularity to chaos
can be estimated in the framework of the first order perturbation theory. The
values of thus obtained critical parameter are in good agreement with the
results of our previous precise calculations for Hennon-Heiles Hamiltonian and
diamagnetic Kepler problem.
