Yes. You can place the following script into the 'Run_AfterPost' action to notify the user after the run has completed:

Sub Run_AfterPost
On Error Resume Next
Dim strSubject, strDes, sendTo, sendFrom
strSubject = ''Test Change Notification'' & '' for project '' & TDConnection.ProjectName & '' in domain '' & TDConnection.DomainName
strDes = ''A user has run the following test: Test Instance ID = '' & Run_Fields(''RN_TEST_INSTANCE'').Value & ''The test run ended with the following status '' & Run_Fields(''RN_STATUS'').Value
sendTo = Run_Fields(''RN_TESTER_NAME'').Value
sendFrom = ''''
TDConnection.SendMail sendTo, sendFrom, strSubject, strDes, NULL,''HTML''
On Error GoTo 0
End Sub

You may need to edit the script some to suite your specific needs. Hope this helps!