We introduce a new selected configuration interaction plus perturbation theory algorithm that is based on a deterministic analog of our recent efficient heat-bath sampling algorithm. This Heat-bath Configuration Interaction (HCI) algorithm makes use of two parameters that control the trade-off between speed and accuracy, one which controls the selection of determinants to add to a variational wave function and one which controls the selection of determinants used to compute the perturbative correction to the variational energy. We show that HCI provides an accurate treatment of both static and dynamic correlation by computing the potential energy curve of the multireference carbon dimer in the cc-pVDZ basis. We then demonstrate the speed and accuracy of HCI by recovering the full configuration interaction energy of both the carbon dimer in the cc-pVTZ basis and the strongly correlated chromium dimer in the Ahlrichs VDZ basis, correlating all electrons, to an accuracy of better than 1 mHa, in just a few minutes on a single core. These systems have full variational spaces of 3 × 10(14) and 2 × 10(22) determinants, respectively.

We extend the recently proposed heat-bath configuration interaction (HCI) method [Holmes, Tubman, Umrigar, J. Chem. Theory Comput. 2016, 12, 3674], by introducing a semistochastic algorithm for performing multireference Epstein-Nesbet perturbation theory, in order to completely eliminate the severe memory bottleneck of the original method. The proposed algorithm has several attractive features. First, there is no sign problem that plagues several quantum Monte Carlo methods. Second, instead of using Metropolis-Hastings sampling, we use the Alias method to directly sample determinants from the reference wave function, thus avoiding correlations between consecutive samples. Third, in addition to removing the memory bottleneck, semistochastic HCI (SHCI) is faster than the deterministic variant for many systems if a stochastic error of 0.1 mHa is acceptable. Fourth, within the SHCI algorithm one can trade memory for a modest increase in computer time. Fifth, the perturbative calculation is embarrassingly parallel. The SHCI algorithm extends the range of applicability of the original algorithm, allowing us to calculate the correlation energy of very large active spaces. We demonstrate this by performing calculations on several first row dimers including F with an active space of (14e, 108o), Mn-Salen cluster with an active space of (28e, 22o), and Cr dimer with up to a quadruple-ζ basis set with an active space of (12e, 190o). For these systems we were able to obtain better than 1 mHa accuracy with a wall time of merely 55 s, 37 s, and 56 min on 1, 1, and 4 nodes, respectively.

We use the recently developed Heat-bath Configuration Interaction (HCI) algorithm as an efficient active space solver to perform multiconfiguration self-consistent field calculations (HCISCF) with large active spaces. We give a detailed derivation of the theory and show that difficulties associated with non-variationality of the HCI procedure can be overcome by making use of the Lagrangian formulation to calculate the HCI relaxed two-body reduced density matrix. HCISCF is then used to study the electronic structure of butadiene, pentacene, and Fe-porphyrin. One of the most striking results of our work is that the converged active space orbitals obtained from HCISCF are relatively insensitive to the accuracy of the HCI calculation. This allows us to obtain nearly converged CASSCF energies with an estimated error of less than 1 mHa using the orbitals obtained from the HCISCF procedure in which the integral transformation is the dominant cost. For example, an HCISCF calculation on the Fe-porphyrin model complex with an active space of (44e, 44o) took only 412 s per iteration on a single node containing 28 cores, out of which 185 s was spent in the HCI calculation and the remaining 227 s was used mainly for integral transformation. Finally, we also show that active space orbitals can be optimized using HCISCF to substantially speed up the convergence of the HCI energy to the Full CI limit because HCI is not invariant to unitary transformations within the active space.

We introduce a semistochastic implementation of the power method to compute, for very large matrices, the dominant eigenvalue and expectation values involving the corresponding eigenvector. The method is semistochastic in that the matrix multiplication is partially implemented numerically exactly and partially stochastically with respect to expectation values only. Compared to a fully stochastic method, the semistochastic approach significantly reduces the computational time required to obtain the eigenvalue to a specified statistical uncertainty. This is demonstrated by the application of the semistochastic quantum Monte Carlo method to systems with a sign problem: the fermion Hubbard model and the carbon dimer. Introduction.-Consider the computation of the dominant eigenvalue of an N Â N matrix, with N so large that the matrix cannot be stored. Transformation methods cannot be used in this case, but one can still proceed with the power method, also known as the projection method, as long as one can compute and store the result of multiplication of an arbitrary vector by the matrix. When, for sufficiently large N, this is no longer feasible, Monte Carlo methods can be used to represent stochastically both the vector and multiplication by the matrix. This suffices to implement the power method to compute the dominant eigenvalue and averages involving its corresponding eigenvector.In this Letter, we propose a hybrid method consisting of numerically exact representation and multiplication in a small deterministic subspace, complemented by stochastic treatment of the rest of the space. This semistochastic projection method combines the advantages of both approaches: it greatly reduces the statistical uncertainty of averages relative to purely stochastic projection while allowing N to be large. These advantages are realized if one succeeds in choosing a deterministic subspace that carries a substantial fraction of the total spectral weight of the dominant eigenstate.Semistochastic projection has numerous potential applications: transfer matrix [1] and quantum Monte Carlo (QMC) [2][3][4] calculations, respectively for classical statistical mechanical and quantum mechanical systems, and the calculation of subdominant eigenvalues [5].In this Letter we apply the semistochastic method to compute the ground state energy of quantum mechanical Hamiltonians represented in a discrete basis. In this context, deterministic projection is known as full configuration interaction (FCI) to chemists and as exact diagonalization to physicists, whereas stochastic projection is the essence of various projector QMC methods [2,3]. Hence, semistochastic projection shall be referred to as the SQMC method. The benefit of the SQMC method over the