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My understanding is that geometry relaxation (i.e. letting the nuclei follow the force gradients of the PES until they reach local minima) should usually be performed in order to get the correct molecular or material energies from electronic structure calculations.

Are there times when doing so would actually give an incorrect/inaccurate answer?

One example that I can think of (which may be incorrect) would be when trying to study molecular ionization by an applied electric field. That is, the electronic dynamics may be so fast that the nuclei would not have the time to respond to the forces before the electron actually leaves the molecule. Thus, to get an accurate understanding of the electron dynamics, you should not let the nuclei relax. To get the final, post-ionization molecular energy, though, you would want to let the nuclei relax again.

Are there other, less clear, situations?

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Geometry optimization corresponds to a system in equilibrium. It is the "average" position of a molecule vibrating in a well.

However, there are many cases where the system is "non-equilibrium" which are important to model.

For example, photochemistry can involve important non-equilibrium processes. In photochemistry, relaxation to the ground-state can involve passage through the crossing seam between potential energy surface. At each point in this crossing is a so called "Conical intersection"

enter image description here

The dimensions of the seam space is $3N-2$ (or $3N-8$ when excluding translations and rotations) which means that there is a lot of places where the molecule can decay to the ground-state.

If there is no local excited-state minimum, then the reactant will simply roll down hill and hit a "tree in the forest"

enter image description here

This however, has not stopped computational chemists from employing equilibrium principles to photochemistry by optimizing the "Minimum Energy Conical Intersections". However, the value in MECI is more qualitative than quantitative. Sometimes they work sometimes they don't!

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