Optimizing the Energy with Quantum Monte Carlo: A Lower Numerical Scaling for Jastrow-Slater Expansions

We present an improved formalism for quantum Monte Carlo calculations of energy derivatives and properties (e.g., the interatomic forces), with a multideterminant Jastrow-Slater function. As a function of the number N-e of Slater determinants, the numerical scaling of O(N-e) per derivative we have recently reported is here lowered to O(N-e) for the entire set of derivatives. As a function of the number of electrons N, the scaling to optimize the wave function and the geometry of a molecular system is lowered to O(N-3) + O(NNe), the same as computing the energy alone in the sampling process. The scaling is demonstrated on linear polyenes up to C60H62 and the efficiency of the method is illustrated with the structural optimization of butadiene and octatetraene with Jastrow-Slater wave functions comprising as many as 200 000 determinants and 60 000 parameters.