σ-moments: compressing a σ-profile into features that predict

A σ-profile is a rich curve, but many downstream models want a fixed-length feature vector. σ-moments are the standard way to compress the profile into a few physically-interpretable numbers without throwing away the parts that matter for thermodynamics.

What a σ-moment is

A σ-moment is an integral of the σ-profile weighted by a function of σ. Low-order moments capture coarse features (total area, overall polarity); higher-order and hydrogen-bonding moments capture the wings of the profile — the donor and acceptor regions that drive specific interactions.

The moments you'll actually use

• Zeroth moment — total molecular surface area; a size descriptor.
• Second moment — overall polarity / total screening; how charged the surface is on average.
• Third moment — asymmetry between positive and negative regions; a donor-vs-acceptor balance.
• Hydrogen-bond moments — area in the donor and acceptor wings beyond a σ threshold; these correlate with explicit H-bond capacity.

Why they predict

Because each moment is a real integral over the screening-charge surface, a moment-based model is not a black box fit to arbitrary descriptors — its inputs carry units and meaning. That makes the resulting structure–property relationships easier to interpret and to defend, and less prone to spurious correlation than generic fingerprint features.

Using them responsibly

• Keep the recipe fixed — moments computed from profiles made with different cavities or basis sets are not comparable.
• Prefer physically-grounded moments over an ever-growing pile of engineered features; more descriptors than data points is how you overfit.
• Validate on held-out scaffolds, not just random splits — chemistry generalization is the real test.
• Report the moment definitions you used; 'the third moment' is ambiguous without the convention.

From profile to model

A common, robust pattern: generate σ-profiles with a validated recipe, compute σ-moments, and fit a small, regularized model to your target property. Because the features are few and meaningful, such models tend to be stable and transferable. mfsig.com returns σ-moments alongside the full profile, computed from the same validated surface, so the features you model on are consistent with the profile you can inspect.