How to send an email to an author on a specific date using date field specified in a node?
You might use the Rules Scheduler module (included with Rules) for this. The following tutorial explains a case similar to yours: http://www.webwash.net/tutorials/how-send-follow-emails-using-rules-scheduler-drupal-7

In short, you would need to:


create a "send email" component that sends an email with a user as
parameter
create a rule triggered when a node of your content type is
modified
set the action for that rule to an "scheduled component
evaluation" that uses the "send email" component you created.
in the "scheduled evaluation date" of the action you need to press the "switch to data selection" button, and then to select the date field of your node from the dropdown menu.


Note that the scheduled component evaluation is triggered by cron. You need to make sure that cron is configured correctly in your site.

Hope that helps,