You are correct that according to classical electromagnetism, an accelerating charged particle emits electromagnetic (EM) radiation. This is known as radiation reaction. According to classical physics, an orbiting electron should emit EM radiation because it is constantly changing direction and accelerating due to its circular motion.
However, classical electromagnetism alone is not sufficient to fully explain the behavior of electrons in atoms. The classical model of an electron orbiting the nucleus, known as the Rutherford-Bohr model, is an oversimplified representation. It suggests that electrons move in fixed circular orbits, which would result in radiation emission and eventual collapse into the nucleus.
In reality, electron behavior is better described by quantum mechanics. According to quantum mechanics, electrons exist in discrete energy levels within an atom. These energy levels are quantized, meaning they have specific values and are distinct from one another. The electrons can transition between these energy levels by absorbing or emitting discrete packets of energy called photons.
When an electron is in a stable energy state, such as in a bound atomic orbital, it does not undergo continuous acceleration or change in direction. Therefore, it does not emit EM radiation in the form of photons. The electron's energy remains constant, and it stays in its orbit without radiating energy away.
However, when an electron undergoes a transition between energy levels, such as when it jumps to a lower energy level, it can emit or absorb a photon of corresponding energy. This emission or absorption of photons occurs at specific frequencies or wavelengths, giving rise to spectral lines observed in atomic spectra.
In summary, the behavior of electrons in atoms is better described by quantum mechanics, which accounts for the discrete energy levels and quantized transitions. Electrons do not emit continuous EM radiation as long as they remain in stable energy states and do not undergo acceleration or change in direction.