Excited-state surfaces without point-by-point solves: transferable deep quantum Monte Carlo cuts cost by up to two orders of magnitude

An excited-state potential energy surface can now come from a single neural-network wavefunction trained jointly across many molecular geometries and several electronic states, instead of a separate solve at each geometry. Producing a whole surface previously forced a cheaper single-reference method such as time-dependent density functional theory, which breaks down at conical intersections and in strongly multi-reference systems, while multi-reference methods such as CASSCF need expert active-space selection and scale steeply with system size. Along ethylene's one-dimensional torsion and pyramidalization coordinates, the transferable simulation reproduces the conical intersection at 98.5° and reaches relative-energy mean absolute errors of 29.4(1) meV and 4.7(7) meV against single-point NES-VMC references, using roughly five times less compute than single-point penalty-based deep-learning quantum Monte Carlo and requiring no supervised pretraining.

The method shares wavefunction parameters across electronic states and reorders the orthogonalization direction dynamically by per-geometry energy, keeping states continuous through level crossings. Penalty-based excited-state optimization originally orthogonalized each excited state against lower states in a fixed order; order is irrelevant in single-point runs, but under transferable training a change of energy ordering at a conical intersection makes the wavefunction jump. In the ethylene pyramidalization scan, dynamic ordering keeps the excited-state wavefunction's overlap with itself near 1 and with the ground state near 0 at 90°; with fixed ordering the excited-state overlap drops to 0 at 100° while the ground-state overlap rises to 1, and spurious non-adiabatic couplings appear.

Cost grows sublinearly with the number of geometries, so denser two-dimensional surfaces cost little extra. Single-point calculations optimize each geometry independently, so cost accumulates linearly and refining the grid directly raises the total. On the methylenimmonium cation's 10×10 grid in torsion angle and C–N bond length, transferable training cuts cost by about one order of magnitude relative to single-point calculations for the ten τ = 0° geometries and by a further order of magnitude for the full 100-point grid, while lowering the relative-energy mean absolute error on the τ = 0° geometries from 134(2) meV for single point to 96(2) meV and 97(5) meV.

The demonstration covers four systems: ethylene's one-dimensional torsion and pyramidalization coordinates (9 and 14 geometries), the eight lowest states of the carbon dimer across 10 separations from 1.0 Å to 1.9 Å, the methylenimmonium cation's 10×10 two-dimensional grid, and a one-dimensional planar-to-tetrahedral path of the simplified 96-electron rubredoxin active-site model Fe(SH)4²−. The method targets fixed datasets of molecular geometries and currently scales to hundreds or thousands of them.

Dynamic state ordering handles reordering only among the n lowest states, and states entering from outside that subspace can still introduce discontinuities; increasing the number of simulated states mitigates this, but a general solution remains open. The method currently relies on exponential moving averages to estimate per-geometry energies, and reaching high-dimensional surfaces will require Monte Carlo sampling that combines electron and nuclear moves plus a replacement for that estimate. Parameter sharing is neither necessary for high accuracy nor guaranteed to improve convergence, and the optimal sharing strategy may depend on the system's electronic structure. The rubredoxin active site uses the simplified Fe(SH)4²− model, and its crossing geometry is sensitive to the functional, with PBE systematically overstabilizing the low-spin state.