Unable to create output/error file with current time as filename

I have tried multiple ways to save the output and error file to get current time as file names of respective files still couldnt match the expected outcome.

#PBS -l select=1:ncpus=2:mem=1GB,walltime=00:02:00
#PBS -N PBS_test
#PBS -o "path_to _dir/output_$(date +%Y%m%d_%H%M%S).log"
#PBS -e "path_to _dir/error_$(date +%Y%m%d_%H%M%S).log"
#PBS -q queue
#PBS -v CONTAINER_IMAGE=docker_img

#rest of my PBS commands.

current o/p: output_$(date +%Y%m%d_%H%M%S).log
expected o/p: output_<current date>_<current time>.log

same goes with the error file.

You can do variable subsitution but evaluation is not supported.
Our forum collegues would voice in otherwise