An accelerated linear method for optimizing non-linear wavefunctions in variational Monte Carlo.

Although the linear method is one of the most robust algorithms for optimizing nonlinearly parametrized wavefunctions in variational Monte Carlo, it suffers from a memory bottleneck due to the fact that at each optimization step, a generalized eigenvalue problem is solved in which the Hamiltonian and overlap matrices are stored in memory. Here, we demonstrate that by applying the Jacobi-Davidson algorithm, one can solve the generalized eigenvalue problem iteratively without having to build and store the matrices in question. The resulting direct linear method greatly lowers the cost and improves the scaling of the algorithm with respect to the number of parameters. To further improve the efficiency of optimization for wavefunctions with a large number of parameters, we use the first order method AMSGrad far from the minimum as it is very inexpensive and only switch to the direct linear method near the end of the optimization where methods such as AMSGrad have long convergence tails. We apply this improved optimizer to wavefunctions with real and orbital space Jastrow factors applied to a symmetry-projected generalized Hartree-Fock reference. Systems addressed include atomic systems such as beryllium and neon, molecular systems such as the carbon dimer and iron(ii) porphyrin, and model systems such as the Hubbard model and hydrogen chains.

Multireference configuration interaction and perturbation theory without reduced density matrices.

The computationally expensive evaluation and storage of high-rank reduced density matrices (RDMs) has been the bottleneck in the calculation of dynamic correlation for multireference wave functions in large active spaces. We present a stochastic formulation of multireference configuration interaction and perturbation theory that avoids the need for these expensive RDMs. The algorithm presented here is flexible enough to incorporate a wide variety of active space reference wave functions, including selected configuration interaction, matrix product states, and symmetry-projected Jastrow mean field wave functions. It enjoys the usual attractive features of Monte Carlo methods, such as embarrassing parallelizability and low memory costs. We find that the stochastic algorithm is already competitive with the deterministic algorithm for small active spaces, containing as few as 14 orbitals. We illustrate the utility of our stochastic formulation using benchmark applications.

Improved Speed and Scaling in Orbital Space Variational Monte Carlo.

In this work, we introduce three algorithmic improvements to reduce the cost and improve the scaling of orbital space variational Monte Carlo (VMC). First, we show that, by appropriately screening the one- and two-electron integrals of the Hamiltonian, one can improve the efficiency of the algorithm by several orders of magnitude. This improved efficiency comes with the added benefit that the cost of obtaining a constant error per electron scales as the second power of the system size O( N2), down from the fourth power O( N4). Using numerical results, we demonstrate that the practical scaling obtained is, in fact, O( N1.5) for a chain of hydrogen atoms. Second, we show that, by using the adaptive stochastic gradient descent algorithm called AMSGrad, one can optimize the wave function energies robustly and efficiently. Remarkably, AMSGrad is almost as inexpensive as the simple stochastic gradient descent but delivers a convergence rate that is comparable to that of the Stochastic Reconfiguration algorithm, which is significantly more expensive and has a worse scaling with the system size. Third, we introduce the use of the rejection-free continuous time Monte Carlo (CTMC) to sample the determinants. Unlike the first two improvements, CTMC does come at an overhead that the local energy must be calculated at every Monte Carlo step. However, this overhead is mitigated to a large extent because of the reduced scaling algorithm, which ensures that the asymptotic cost of calculating the local energy is equal to that of updating the walker. The resulting algorithm allows us to calculate the ground state energy of a chain of 160 hydrogen atoms using a wave function containing ∼2 × 105 variational parameters with an accuracy of 1 mEh/particle at a cost of just 25 CPU h, which when split over 2 nodes of 24 processors each amounts to only about half hour of wall time. This low cost coupled with embarrassing parallelizability of the VMC algorithm and great freedom in the forms of usable wave functions, represents a highly effective method for calculating the electronic structure of model and ab initio systems.