An Improved Penalty-Based\nExcited-State Variational\nMonte Carlo Approach with Deep-Learning Ansatzes

We introduce several\nimprovements to the penalty-based variational\nquantum Monte Carlo (VMC) algorithm for computing electronic excited\nstates of Entwistle et al. [Nat. Commun. <b>14</b>, 274 (2023)]\nand demonstrate that the accuracy of the updated method is competitive\nwith other available excited-state VMC approaches. A theoretical comparison\nof the computational aspects of these algorithms is presented, where\nseveral benefits of the penalty-based method are identified. Our main\ncontributions include an automatic mechanism for tuning the scale\nof the penalty terms, an updated form of the overlap penalty with\nproven convergence properties, and a new term that penalizes the spin\nof the wave function, enabling the selective computation of states\nwith a given spin. With these improvements, along with the use of\nthe latest self-attention-based ansatz, the penalty-based method achieves\na mean absolute error below 1 kcal/mol for the vertical excitation\nenergies of a set of 26 atoms and molecules, without relying on variance\nmatching schemes. Considering excited states along the dissociation\nof the carbon dimer, the accuracy of the penalty-based method is on\npar with that of natural-excited-state (NES) VMC, while also providing\nresults for additional sections of the potential energy surface, which\nwere inaccessible with the NES method. Additionally, the accuracy\nof the penalty-based method is improved for a conical intersection\nof ethylene, with the predicted angle of the intersection agreeing\nwell with both NES-VMC and multireference configuration interaction.