The electric and magnetic fields in the waves that make up the emitted radiation are
proportional to various rates of change of the potentials, in space and time.
They get a factor of frequency from the fact that the potentials are proportional to frequency;
but they also get another factor of the frequency from the fact
that the shorter a wave is, the faster it varies in space; and the
higher its frequency is, the faster it varies in time. So the
fields are proportional to the square of the
frequency.
But we are not done yet! The important thing is how much
power is transmitted by the wave, and that is proportional
to the product of the electric field and the magnetic field. So the
power density in the wave goes up as the fourth power of
the frequency.
Therefore, the spectrum of the radiated light, and the scattered
light from an induced dipole, will be very strongly peaked at high
frequencies, or short wavelengths. However, there is some light of
lower frequencies as well, which is why we see
the color as a light turquoise rather than deep violet. The
light is scattered all across the spectrum, but high frequencies
are scattered much more than low frequencies.
This sort of scattering is called Rayleigh scattering, after
Lord Rayleigh, who first worked it out for a very small classical
dipole.
There are things I have neglected here, such as the fact that
sometimes, there are resonant frequencies at which a molecular dipole oscillates particularly
strongly when driven by an oscillating field. These resonances are
determined by the quantum mechanics of the molecule. However, in
this particular case, resonance is not a major contributor at
visible wavelengths.
A full analysis would also take into account the fact that the
electromagnetic field is quantized; the energy comes in photons.
But that turns out not to affect the fourth-power dependence of the
spectrum on frequency.