I have wired my window a/c but can not get it to opperate. I have 12-2 wire running to a split (two narrow) 20 amp breakers for 220v. The black and white are both hooked up to the breaker and on both gold screws for the 20 amp 220 volt outlet. There is no place for a common so I used 12- 2 instead of 12-3. However the system does not turn on. Any suggestions?

You CANNOT use a twin breaker (skinny, tandem, etc) for a 240v circuit. You MUST use a two-pole breaker that connects to BOTH legs of the panel.

What brand panel is it?

I have wired my window a/c but can not get it to opperate. I have 12-2 wire running to a split (two narrow) 20 amp breakers for 220v. The black and white are both hooked up to the breaker and on both gold screws for the 20 amp 220 volt outlet. There is no place for a common so I went ahead and used 12- 2 instead of 12-3. However the system does not turn on. Any suggestions?
I can ue Murry or Simmons Breakers. However, I can not find any Murry breakers in my area (Phoenix).

Murray will be hard to find IMO. Siemens is a VERY popular brand and a HUGE company.
You can also use a "classified" BR style breaker.
You can find any of these at a real electrical supply house. A big city like (or suburb of) Phoenix should have several.