We had previously shown that deep learning could be used for gold-standard calculations of the *lowest* energy states of real molecules. This is already quite challenging, but ignores all of the exotic physics that happens when molecules are stimulated.
Quantum Monte Carlo methods using neutral-network ansatzes can provide virtually exact solutions to the electronic Schrödinger equations for small systems and are comparable to conventional quantum chemistry methods when investigating systems with dozens of electrons.
We present a variational Monte Carlo algorithm for estimating the lowest excited states of a quantum system which is a natural generalization of the estimation of ground states. The method has no free parameters and requires no explicit orthogonalization of the different states, instead transforming the problem of finding excited states of a given system into that of finding the ground state of an expanded system. Expected values of arbitrary observables can be calculated, including off-diagonal expectations between different states such as the transition dipole moment. Although the method is entirely general, it works particularly well in conjunction with recent work on using neural networks as variational Ansatze for many-electron systems, and we show that by combining this method with the FermiNet and Psiformer Ansatze we can accurately recover vertical excitation energies and oscillator strengths on molecules as large as benzene. Beyond the examples on molecules presented here, we expect this technique will be of great interest for applications of variational quantum Monte Carlo to atomic, nuclear and condensed matter physics.