The clock has an RTC built in, so it doesn’t need the Raspberry Pi to operate. It will, of course, be capable of a lot more when the Pi is connected, as it relies on the Pi for the current time and anything internet related.
The Pi will probably consume quite a bit of power too. Perhaps the clock could control the Pi with a relay, and start it up half an hour prior to the alarm going off so it has time to boot and so it can fetch the latest weather forecast to show the User when he/she finally wakes up. This would remove the ability to have real-time notifications though.
It doesn’t need to be a Raspberry Pi, by the way. The communication program is written in Java, so any computer with a USB or Serial port, an internet connection and Java can be used, for example an old laptop.