Relationship between atmospheric density and atmospheric pressure?
There are two fundamental gas laws that need to be understood:

Boyle's Law which states     ${P_1.V_1 = P_2.V_2}$ 

From this formula if volume increases pressure decreases and vice versa

The second law is Charles's Law which states ${V_1/V_2 = T_1/T_2}$   or    ${V_1/T_1 = V_2/T_2}$

This law states that if the temperature increases there must be a corresponding increase in volume of the gas and vice versa. Another way of looking at it is if the volume drops so must the temperature and vice versa.

The ideal gas law that you mention in your question can be also be written as:

${PV = mRT}$

Where:


P is pressure,
V is volume
m is mass
R is the universal gas constant
T is temperature


Now, density, ${ρ = m/V}$

Which leads to the equation you used:

${P = ρRT}$

One of the primary ways the atmosphere changes is via changes in temperature. The atmosphere on the side of the Earth facing the Sun warms up, while the atmosphere of the other side of the Earth cools.

According to Charles's Law, for the side facing the Sun, the increase in temperature will cause the volume to increase. An increase in volume will reduce the density, because mass doesn't change. With an increase in temperature and a decrease in density the pressure will remain unchanged.