With all respect to the very good advice given by members above, my approach would be slightly different.
Where the employee is in "Good Bye Mood" and is not doing any work, then he/she is no longer an asset to the company. Therefore, you can ask that all pending work that cannot be transferred to another employee, must to completed, under strict supervision, during the notice period, upon such completion of the work, the employee can be placed on "Garden Leave".
Garden Leave means that the employee does not need to come into the office and can stay at home, however, the employee must make themselves available either by phone or personal attendance. During this time the employee is not allowed to work elsewhere, if they do they will lose their notice period pay. The Garden Leave can be from day one or partial.
This is an approach that ensures commercial viability, without incurring avoidable costs and leave HR to continue their daily work.
In any case, the employee leaving should never be allowed to interview their replacement!!
I appreciate that this approach is not suitable for all employees in all sectors, but I hope the above gives some food for thought.
Regards
Harsh