Excited-state machine-learned potentials get a local Hamiltonian: two molecules 10 Å apart keep exactly the union of their levels

The readout of an excited-state machine-learned potential becomes a symmetric matrix that grows with the molecule, and its eigenvalues are the singlet excitation energies, so the spectrum of separated molecules is exactly the union of their spectra. Earlier multistate readouts sum, average or soft-minimize atomic contributions; summing makes excitation energies grow with system size, averaging blurs the fragments, and an extremal pool recovers only the lowest excitation, so none reproduces the union of the whole low-lying spectrum. For two molecules 10 Å apart, the levels LEH-MACE predicts for the pair agree with those it predicts for the molecules separately within eV, and for two identical molecules every level appears twice, as in TD-B97X-D, whereas sum, mean and soft-minimum readouts shift and spread the levels.

The construction's error grows more slowly when test molecules move away from the training distribution, and it keeps single-molecule accuracy for molecule pairs. For molecules similar to the training molecules, the direct softmin readout and X-MACE had slightly lower errors than LEH-MACE; their errors grew faster as test molecules moved away from the training distribution. For larger molecules LEH-MACE reached 0.17 eV (direct softmin 0.19 eV, X-MACE 0.21 eV), for the 3,875 sulfur-free dyes unseen in training 0.29 eV (0.35 eV and 0.96 eV), and for ladder molecules 0.36 eV (0.61 eV and 0.77 eV); for the 180 dimers its errors were 0.15 eV and 0.17 eV, no larger than the 0.18 eV for the same molecules alone, while direct softmin and X-MACE reached 0.55 eV and 0.64 eV on dimers.

A MACE encoder supplies atomic features that generate the local blocks: one diagonal element per atom and one coupling element per atom pair within the cutoff, and the ascending eigenvalues of the matrix are the excitation energies, with excited-state forces from the derivatives of the eigenvalues. Ground-state energies can be written as sums of atomic energies, but excitation energies are not additive, so the readout layer must be an object whose spectrum has the union property by construction rather than a pooling over atomic contributions. Training and testing used TD-B97X-D/6-31G* energies and forces for 303,061 geometries of 106,933 molecules, 203,040 of them with excited-state forces, spanning equilibrium, thermal and excited-state-sampled geometries; benchmark models trained on 168,587 geometries of 76,621 molecules, with test sets of 500 molecules of 11–20 heavy atoms, 5,018 conjugated dyes and 1,393 ladder molecules with up to 30 heavy atoms.

The demonstrated scope is singlet states of closed-shell C, H, N, O, F molecules, with energies and forces at the TD-B97X-D/6-31G* level, tested on larger neutral molecules, conjugated dyes, ladder molecules, molecule pairs and the public QCDGE benchmark; for dynamics of a single larger chromophore, the direct softmin readout is the more accurate model.

Size consistency holds only within the sector of local single excitations the model represents, and the short-range blocks do not represent charge-transfer states between fragments, double excitations or long-range excitonic couplings; the eigenvalues determine the Hamiltonian only up to orthogonal transformations, so couplings and eigenvectors carry no diabatic meaning and the model provides no nonadiabatic couplings. The learning curves had not saturated with the full training set, and fitted power laws reach 0.1 eV only at 0.6 to 4 million training molecules; for larger chromophores all seeds shared a too-high excitation energy at twisted geometries, so disagreement between seeds is not a sufficient error estimate far from the training data.