In this paper we analyze again a transition from the classical to quantum description of bound charged particles, which involves a substantial modification of the structure of their electromagnetic (EM) fields related to the well-known fact that bound micro-particles do not radiate in sta-tionary energy states. We show that a simple exclusion of the radiative component of EM field produced by bound particles leads to a violation of the energy-momentum conservation law, if the non-radiative EM field is left unmodified. In order to restore the energy-momentum conservation, we make a closer look at the interaction of two hypothetical classical charges with the prohibited radiation component of their EM field and bring the appropriate modifications in the structure of their bound EM field and, accordingly, in the Hamilton function of this system. In comparison with the common Hamilton function for the one-body problem, the electric interaction energy is multiplied by the Lorentz factor of orbiting charged particle, and its rest mass m is replaced by an effective rest mass parameter, which includes the interaction EM energy. We introduce, as a novel postulate, these replacements into the Dirac equation for the bound electron and show that the solution of the modified Dirac Coulomb equation gives the same gross and fine structure of energy levels, as the one furnished by the conventional approach, for hydrogen-like atoms. The correction to spin-spin splitting of 1S state of hydrogen and heavier atoms is much smaller than nuclear structure contribution and can be ignored. However, as discussed in the part 2 of this paper, our approach does induce corrections to the energy levels at the scale of hyperfine interactions, which at once remove a number of long-standing discrepancies between theory and experiment in the atomic physics.
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