I don't think most thermostats are capable of driving a motor
contactor, directly. You need a small relay between them. 24 volt AC
coil and something like 5 or 10 amp contacts. Something with a less
than 1 watt coil. This might work:
KUP-11A15-24 at the bottom of http://dkc3.digikey.com/PDF/T052/1290.pdf
This can be socketed, or connected with separate wires with push on
Then get a contactor with a generous contact rating, compared to the
current your compressor takes. It can have a 120 or 240 VAC coil.
A 47 ohm in series with .47 uF across the thermostat contact will
reduce the turn off flash.