Did you completely remove the other trial antivirus you had?
I suggest doing that, then a clean install of Avast from scratch:
-Remove the other AV (if not done already),
-Download the latest Avast home (if you don’t already have the installer file on your computer),
-Download the Avast Uninstall Utility,
-Disconnect from the net,
-Turn off the Avast self-protection module,
-Uninstall Avast via the control panel, reboot.
-Run the Avast uninstall utility. Reboot if prompted.
-Install Avast. (Personally, I’d let it run a boot scan at this point, as part of the install process. You have had malware on the computer in the past. You may feel confident skipping that.)
Reconnect to the net after it is installed. Update.
avast! freezes or crashes, it’s probably avast! problem.
The computer freezes or restarts, it could be e.g. a conflict with some other program, or a hardware problem.
The computer powers down - it’s probably a hardware problem (e.g. overheating). The scan generates a lot of hard disk activity (and is also CPU intensive), which may increase the temperature of your hardware. I suggest to check the coolers in the case.
If you get a blue screen, it could have a bug in avast! drivers (but then, there is not a power down).