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The Relaxed Scan as a Constrained Optimization

Let \(q\) denote the single internal coordinate being scanned (a bond length, angle, or dihedral), and let \(q'\) denote the vector of all remaining internal coordinates. At scan point \(n\), Gaussian fixes \(q = q_n\) and solves

\[ \min_{q'} \; E(q_n, q') \]

This is a constrained optimization: minimize the energy over \(q'\) while \(q\) is held at a fixed value. Introducing a Lagrange multiplier \(\lambda\) for the constraint \(q = q_n\), the stationary conditions of the Lagrangian

\[ \mathcal{L}(q', \lambda) = E(q_n, q') + \lambda\,(q - q_n) \]

require that the gradient with respect to every unconstrained coordinate vanishes,

\[ \frac{\partial E}{\partial q'_i} = 0 \quad \text{for all } i, \]

while the gradient along the constrained coordinate is, in general, non-zero — it is balanced by the multiplier:

\[ \lambda = -\frac{\partial E}{\partial q}\bigg|_{q=q_n} \]

This is exactly what Gaussian prints as the -DE/DX value next to a constrained variable in the optimized parameters table: the residual force the constraint is holding back. A large -DE/DX at a scan point signals that the surrounding geometry is straining noticeably against the fixed coordinate; a value near zero — as in this tutorial's example — means the scan point sits close to where the coordinate would relax to on its own.