What's the point of doing high accuracy spectroscopy calculations?
I think there's a more fundamental answer.
But I'm going to argue from the experimental side. Of course implicit in my answer is the idea that these higher accuracy experiments will then require higher accuracy calculations, which will then demand higher accuracy experiments...
How does science progress?
- look at the world
- spot some thing interesting
- make a model
- test the model
- despair when it works perfectly, rejoice when it breaks down
- revise the model or throw it out and build a new one
- lather, rinse, repeat (i.e. back to step #4)
The more accurate the measurement is, usually the more it challenges a model. Most models these days include implicit or explicit approximations, and the deviations between models and accurate measurements challenge the models limits.
Talk about the very fundamental The Standard Model which has continued to fit new measurements in high energy particle physics. What has bothered at least some physicists is that predictions of this model has continued to agree with new experimental results better than it should. Some might say better than it "has a right to."
It makes some folks at least uncomfortable.
Suddenly there is news:
Contrary to popular headlines, there is much rejoicing!
Finally sufficiently accurate measurement has been made that disagrees with the best model available. This is gold for theorists, a chance to find either a refinement/adjustment of the current model or a new model (or perhaps new particle or new effect).
Another example is Eric Adelberger's (also here) and the The Eöt-Wash Group's insistence on super high accuracy of gravitation attraction between two objects in his lab.
Who else would stack lead bricks around their experiment to cancel the gravitational quadrupole moment of the hill behind the physics building?
These super-duper accurate torsional pendulum measurements have put serious constraints on theories of gravity and space.
Science often advances by measuring the next few digits of a quantity that models can predict. Nobody knows when and where the surprise will happen, so we simply pursue higher and higher accuracy wherever we can.