This is not currently possible. Applications are prevented from sending private messages to their users (or any users for that matter) because it would be a serious vulnerability for spam.
One Alternative
would be to request the email
permission from your applications users and then using
you'll be able to request the email of the user and send him an email using some code on the server.
You can read more about requesting permissions for facebook applications at these links :
https://developers.facebook.com/docs/reference/api/permissions/
https://developers.facebook.com/docs/authentication/
Another Alternative
would be to utilize facebook dialogs and initiate a 'request' or 'apprequest' dialog. For this you'll need to implement the Facebook JavaScript SDK and the Request Dialog.
The Request Dialog
When you initiate this dialog you can specify a message to be sent along with the request. When a user responds to one of these requests they are redirected to your application.

(source: fbcdn.net)