Pseudoscalars

This page enumerates the pseudoscalar paths available from a set of e3nn irreps, and converts equivariant features into pseudoscalar (chirality-sensitive) invariants.

Why pseudoscalars

An ordinary invariant (0e) is unchanged by any rotation or reflection, so it gives a molecule and its mirror image the same value and cannot see chirality. A pseudoscalar (0o) is rotation-invariant but flips sign under reflection, so it distinguishes enantiomers.

Pseudoscalars can't be read off directly: from ordinary (polar) tensors, 0o is reachable only via two tensor products. You need an axial (l ≥ 1) intermediate that contracts with a third input to 0o (equivalently, an odd l₁+l₂+l₃ path). The classic example is the scalar triple product of three vectors, (a × b) · c.

The universal function

find_pseudoscalar_paths(irreps) enumerates every valid 2-tensor-product path to 0o for a given irrep set, using the same selection rules (triangle + parity) e3nn and MACE use. It's pure and cheap; use it to check what's reachable.

from remedi.utils.pseudoscalar_paths import find_pseudoscalar_paths

# e.g. MACE-style node irreps
paths = find_pseudoscalar_paths("8x0e + 8x1o + 8x2e")
for p in paths:
    print(p)   # ir_in1 ⊗ ir_in2 → ir_mid ; ir_mid ⊗ ir_in3 → 0o

Each result is a PseudoscalarPath (ir_in1, ir_in2, ir_mid, ir_in3, ir_out=0o). An empty list means no pseudoscalar is reachable from those irreps.

Producing pseudoscalars from features

Rem3DiPseudoScalarTP is an nn.Module that realizes all discovered paths as two fused tensor products and maps equivariant features [..., irreps.dim] to K pseudoscalars [..., K]:

import torch
from e3nn.o3 import Irreps
from remedi.model.preprocessing.pseudoscalar_tp import Rem3DiPseudoScalarTP

irreps = Irreps("8x0e + 8x1o + 8x2e")
ps_tp = Rem3DiPseudoScalarTP(
    input_irreps=irreps,
    pseudoscalar_dimension=4,      # K
    invariant_conditioned=False,   # standalone: no invariant conditioning
)

x = torch.randn(16, irreps.dim)    # a batch of equivariant features
ps = ps_tp(x)                      # -> (16, 4) pseudoscalars

By construction the output is K x 0o, so reflecting the input negates it; that sign flip is the chirality signal (the notebook checks it explicitly).

invariant_conditioned=True (the default in the model) makes the tensor-product weights a function of invariant features: pass invariant_irreps=... at construction and invariants=... to forward. Use False for a standalone block.

Outputs

• find_pseudoscalar_paths returns a list of PseudoscalarPath (possibly empty).
• Rem3DiPseudoScalarTP(...)(x) returns a [..., K] tensor of pseudoscalars.

Next steps

• Notebook: examples/05_pseudoscalars.ipynb: enumerate paths, build the block, and verify the reflection sign-flip (runs on CPU, no model needed).
• Concepts: descriptor shape.