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Rigid vs. Relaxed Scans, Formally

For a scanned coordinate \(q\) and remaining coordinates \(q'\), define the two scan energies at each grid point \(q_n\):

\[ E_{\text{rigid}}(q_n) = E(q_n, q'_0) \]
\[ E_{\text{relaxed}}(q_n) = \min_{q'} \, E(q_n, q') \]

where \(q'_0\) is the initial (starting-geometry) value of every other coordinate, frozen throughout a rigid scan.

Because the relaxed energy is defined as a minimum over \(q'\) while the rigid energy uses one specific, generally sub-optimal choice of \(q'\), it follows directly that

\[ E_{\text{relaxed}}(q_n) \le E_{\text{rigid}}(q_n) \quad \text{for every } n \]

with equality only at points where the starting geometry happens to already be fully relaxed for that value of \(q\). In practice this means a rigid scan systematically overestimates the energy away from the starting structure, and can distort the shape of the profile — including the position and height of any apparent barrier — whenever the neglected coordinates would have relaxed significantly. This is why a relaxed scan, at the cost of one geometry optimization per point, is the standard choice whenever the energy profile itself is the object of interest.