Here's a method of estimating the theoretical temperature increase of the atmosphere caused by carbon dioxide, which is 240 trillionths of a degree (240 x 10^-12 °C):

If the sun heats the Earth's surface 20°C every day, while cooling occurs that much each night, and radiation is 2% of the energy leaving the surface of the Earth, then cooling of the Earth's surface is 2% of 20°C, which is 0.4°C as radiation, while the other 19.6°C is due to conduction, convection and evaporation.

This type of radiation is called black body radiation. It is the wide bandwidth of radiation emitted by all matter based on temperature. But CO2 in the air only absorbs 8% of the black body radiation as fingerprint radiation. So 8% of 0.4°C is 0.032°C as the temperature decrease at the surface of the Earth attributable to CO2 absorption.

The 8% number was determined during the early 1950s before global warming became a social issue. Eight percent is approximately the proportion of the bandwidth attributable to CO2 absorption. This number does not evaluate total atmospheric dynamics but only the percent of the bandwidth available to CO2.

 
Temperature can be related between ground and CO2 in the air on a wave by wave basis. The temperature decrease on the surface of the Earth attributed to these wavelengths will approximately be the temperature increase of the CO2 molecules in the air which absorb the radiation. When the wavelength emitted and cooling is the wavelength absorbed and heating, heat capacities and temperature changes will be similar on a molecule for molecule basis. Wavelength has a correlation with molecular mass and energy transfer.

It means the 0.032°C cooling of the surface of the Earth will approximately equal 0.032°C heating of CO2 in the atmosphere.

Each CO2 molecule in the air would have to be 2,500°C to average 1°C, since each CO2 is surrounded by 2,500 air molecules. But it reduces to 1875°C to produce an average temperature increase of 1°C for the entire atmosphere, after CO2 transfers one fourth of its energy by conduction to nearby molecules.

A small amount of energy would be spread to nearby molcules before being radiated away. A CO2 molecule would lose about half of its gained energy in one half cycle as it bumped a nearby molecule (usually nitrogen). It would impart half of its excess energy into the molecule it bumps. One half of one half equals one fourth of the increased energy being imparted into the bumped molecule. The bumped molecule would do the same thing and impart one fourth of its added energy into the molecule which it bumps.

This means that three fourth of the energy picked up by CO2 is radiated away, while one fourth if added to nearby molecules. The energy cannot spread significantly before being radiated away by a small number of molecules.

Dividing 1875°C by 0.032°C equals 58,594 times too little heating of the CO2 to increase the air temperature by 1°C average.

Conservatives are saying they agree that some heating is due to a greenhouse effect, but it might be only half as much as claimed. Maybe it's 1/58,594 times as much as claimed. That's 0.000017 times as much as claimed. That's 17 millionths of a degree instead of 1°C heating upon doubling the amount of CO2 in the atmosphere.

These temperatures are the amount of heating that would occur for each wave absorbed by CO2. With CO2 emitting radiation easier than it is absorbing, the heat is also lost with each wave being emitted. With one wave emitted while one wave is absorbed, the temperature increase would be reduced by a factor of about 0.14 times as much, which would be 2.4 millionths of a degree (0.0000024°C) instead of 17 millionths of a degree.

Here's were the 0.14 comes from: The main bandwidth for CO2 fingerprint radiation is 15 microns, while the average emission for black body radiation is 25 microns. So waves are being absorbed 1.7 times faster than emitted (25 ÷ 17 = 1.7). This effect would increase heating of CO2 by a factor of 1.7. But the radiation absorbed by CO2 is only 8% of the black body bandwidth. So 8% of 1.7 equals 0.14.

This result would occur if there were no such thing as saturation. Saturation means all radiation available is absorbed by a tiny amount of CO2 in the air, so more CO2 cannot absorb more radiation. Heinz Hug did a laboratory measurement and said all radiation is absorbed by CO2 as it travels 10 meters near the Earth's surface. That means nothing more happens when CO2 is doubled in the atmosphere besides reducing the 10 meters to 5 meters, which is irrelevant.
 
Climatologists admit that saturation occurs, but they claim only the center of the absorption peak saturates, while the shoulders do not saturate.

This means the shoulders of the absorption curve represent CO2 molecules in a different energy state due to bonds vibrating. There are very few of these CO2 molecules, so radiation must travel farther before being completely absorbed at shoulder wavelengths. But to not saturate in the troposphere would mean there are extremely few. Most shoulder molecules do saturate in the troposphere.

One thousandths of the CO2 molecules would absorb in 10 kilometers instead of 10 meters, which is still in the troposphere. So divide the result by about one ten thousandths for actual absorption increase. Then, 2.4 millionths of a degree becomes 240 trillionths of a degree.

The atmosphere heats 240 x 10^-12 °C upon doubling the amount of CO2 in the atmosphere.

This number is the hypothetical potential of CO2 to heat the atmosphere. In actuality, water vapor absorbs most of the same radiation due to overlap of absorption spectra, so there is even less heating due to CO2 under atmospheric conditions.