Why not free electrons in atom doesn't radiates em waves\photons? Why not free electrons in atom doesn't radiates em waves\photons, although they move with acceleration? Like 1s electron of Titan, it doesn't emits em waves, yes? Why?
 A: It's because the classical model of an atom as a little solar system simply doesn't work for atoms. That was the message of the quantum physicists starting with Bohr (1913) and Heisenberg et al (1925).
We don't know what an atom "looks like" inside. Hard little balls going in orbits? Um, we cannot watch them. Some quantum field sloshing about? Some strange state that we cannot observe in detail.
So the early quantum physicists, starting with Bohr, simply declared that electrons in atoms can only have various discrete states, and that no radiation is emitted while an electron remains in a state, only when it transitions between two states. Why is this? Well, we don't know. But it's what the experimental data tells us. Quantum physics tells us how to calculate things, like energies and chemical bonds, but it doesn't tell us why it is so.
A: The free electrons you refer to, are free electrons in a vacuum, not bount to atoms.
Now electrons in an atom are bound to the nucleus. These electrons exist as per QM at a certain energy level around the nucleus. Their wavefunction describes the probability distribution of their position for all of space. They do not classically orbit the nucleus, but as per QM they exist around it, and the wavefunction describes the probability of finding the electron at a certain position around the nucleus.
Now you are referring to accelerating charges, electrons, emitting photons. That is how radio antennas work, and in that case, the antenna (metal lattice) has electrons that are loosely bound to the atoms. These electrons are not free. They are loosely bound, so they can (because of an external EM field) move to the next atom. This is called drift velocity, and is pretty slow, but because the electrons are densely packed, the electricity moves in the metal with close to light speed. Now these electrons in the metal antenna are accelerating, and emitting photons.
In your case, the electrons around the nucleus are not accelerating. They exist around the nucleus as per QM at a certain stable energy level. These electrons do not accelerate. They have a certain kinetic energy, that keeps them away from the nucleus, and they have a certain EM charge, that pulls them to the nucleus, and the two forces equalize, and keep the electron at a stable energy level.
These bound electrons around the nucleus can emit photons, for a few reasons. There is spontaneous emission, when the electron emits a photon and moves to a certain lower energy level when a lower energy level is available.
A: In quantum mechanics there are no individual electrons which dart about like ping bong balls. But there is an electric charge density which can be derived from the total wave function of the system. In general this charge density is not stationary. To the extent that it is moving, oscillating or accelerating, it will emit or absorb radiation. And that emission or absorption can be calculated exactly by using Maxwell's Equations.
The absoption of light, which occurs as a result of the coupling of the conduction band electrons in a metal to the free electron  states outside the metal is, governed by Maxwell's Equations (together with Schroedinger's Equation). The linewidths of the absorption lines in the hydrogen spectrum are governed by Maxwell's Equations (together with Schroedinger's equation). The emission of light from a heated tungsten filament is governed by Maxwell's Equations, as applied to the oscillating charge distributions within that metal filament (which are given by Schroedinger's Equation.)
It is not necessary to talk about photons to explain any of these things, and it is not particularly helpful either.
