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We study several approaches to orbital optimization in selected configuration interaction plus perturbation theory (SCI+PT) methods, and test them on the ground and excited states of three molecules using the semistochastic heatbath configuration interaction (SHCI) method. We discuss the ways in which the orbital optimization problem in SCI resembles and differs from that in complete active space self-consistent field (CASSCF). Starting from natural orbitals, these approaches divide into three classes of optimization methods according to how they treat coupling between configuration interaction (CI) coefficients and orbital parameters, namely uncoupled, fully coupled, and quasi-fully coupled methods. We demonstrate that taking the coupling into account is crucial for fast convergence and recommend two quasi-fully coupled methods for such applications: accelerated diagonal Newton and Broyden-Fletcher-Goldfarb-Shanno (BFGS).
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We extend the recently proposed heat-bath configuration interaction (HCI) method [Holmes, Tubman, Umrigar, J. Chem. Theory Comput. 12, 3674 (2016)], 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 wavefunction, 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 F2 with an active space of (14e, 108o), Mn-Salen cluster with an active space of (28e, 22o), and Cr2 dimer with up to a quadruple-zeta 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 seconds, 37 seconds, and 56 minutes on 1, 1, and 4 nodes, respectively.
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 tradeoff between speed and accuracy, one which controls the selection of determinants to add to a variational wavefunction, and one which controls the 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 3x10^14 and 2x10^22 determinants respectively.
Even when starting with a very poor initial guess, the iterative configuration interaction (iCI) approach can converge from above to full CI very quickly by constructing and diagonalizing a small Hamiltonian matrix at each macro/micro-iteration. However, iCI scales exponentially with respect to the numbers of electrons and orbitals. The problem can be mitigated by observing that a vast number of configurations have little weights in the wave function and do not contribute to the correlation energy. The real questions are then (a) how to identify important configurations in the early stage of the calculation and (b) how to account for the residual contributions of those unimportant configurations. It is generally true that if a high-quality yet compact variational space can be determined for describing the static correlation, a low-order treatment of the residual dynamic correlation would be sufficient. While this is common to all selected CI schemes, the `iCI with selection scheme presented here has the following distinctive features: (1) Full spin symmetry is maintained. (2) Although the selection is performed on individual CSFs, it is orbital configurations (oCFG) that are used as the organizing units. (3) Given a coefficient pruning-threshold, the selection of important oCFGs/CSFs is performed iteratively until convergence. (4) At each iteration for the growth of wave function, the first-order interacting space is decomposed into disjoint subspaces to reduce memory requirement and facilitate parallelization. (5) Upper bounds for the interactions between oCFG pairs are used to screen each first-order interacting subspace before the first-order coefficients of individual CSFs are evaluated. (6) Upon termination of the selection, dynamic correlation is estimated by using the ENPT2 theory. The efficacy of the iCIPT2 scheme is demonstrated numerically by taking several examples.
We introduce vibrational heat-bath configuration interaction (VHCI) as an accurate and efficient method for calculating vibrational eigenstates of anharmonic systems. Inspired by its origin in electronic structure theory, VHCI is a selected CI approach that uses a simple criterion to identify important basis states with a pre-sorted list of anharmonic force constants. Screened second-order perturbation theory and simple extrapolation techniques provide significant improvements to variational energy estimates. We benchmark VHCI on four molecules with 12 to 48 degrees of freedom and use anharmonic potential energy surfaces truncated at fourth and sixth order. For all molecules studied, VHCI produces vibrational spectra of tens or hundreds of states with sub-wavenumber accuracy at low computational cost.
Full configuration interaction (FCI) solvers are limited to small basis sets due to their expensive computational costs. An optimal orbital selection for FCI (OptOrbFCI) is proposed to boost the power of existing FCI solvers to pursue the basis set limit under a computational budget. The optimization problem coincides with that of the complete active space SCF method (CASSCF), while OptOrbFCI is algorithmically quite different. OptOrbFCI effectively finds an optimal rotation matrix via solving a constrained optimization problem directly to compress the orbitals of large basis sets to one with a manageable size, conducts FCI calculations only on rotated orbital sets, and produces a variational ground-state energy and its wave function. Coupled with coordinate descent full configuration interaction (CDFCI), we demonstrate the efficiency and accuracy of the method on the carbon dimer and nitrogen dimer under basis sets up to cc-pV5Z. We also benchmark the binding curve of the nitrogen dimer under the cc-pVQZ basis set with 28 selected orbitals, which provide consistently lower ground-state energies than the FCI results under the cc-pVDZ basis set. The dissociation energy in this case is found to be of higher accuracy.
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