Optimizing Effective Potentials via Gradient-Enhanced Surrogates for Quantum Embedding

Accurate reaction energetics in complex environments require correlated electronic-structure accuracy in the reactive region while retaining the influence of the surrounding system. Density functional embedding theory (DFET) enables this balance by introducing a local embedding potential that makes the cluster and environment densities reproduce the full-system Kohn-Sham density. However, obtaining this potential through optimized effective potential (OEP) or Wu-Yang optimization can be expensive, especially for metallic systems where dense frontier orbital manifolds make direct second-order optimization difficult. Here, we introduce a gradient-enhanced surrogate strategy for DFET embedding-potential optimization. The method builds a kernel-based surrogate of the Wu-Yang objective from previously evaluated functional values and gradients, optimizes this surrogate within a trust region and adaptive Newton subspace, and evaluates the true Wu-Yang functional only for selected trial potentials. The resulting surrogate-optimized embedding potentials are used in embedded correlated wave function (ECW) calculations and benchmarked against trust-exact OEP potentials and all-atom NEVPT2 reaction energies. Across representative metallic and molecular reaction benchmarks, the surrogate approach produces embedding potentials comparable to trust-exact baselines while reducing the number of expensive true OEP evaluations for challenging metallic systems. ECW energies driven by surrogate-optimized embedding potentials closely reproduce all-atom NEVPT2 reaction profiles and improve over parent KS-DFT. These results show that surrogate OEP provides a practical and accurate route to DFET embedding potentials for large and electronically complex systems.