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arXiv:1909.12319 · Physical Review Research

Accurate many-body electronic structure near the basis set limit: Application to the chromium dimer

Junhao Li, Yuan Yao, Adam A. Holmes, Matthew Otten, Qiming Sun, Sandeep Sharma, Cyrus J. Umrigar

Published Jan 15, 2020• 49 citations
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Abstract

The authors use the recently developed semistochastic heat bath configuration interaction method to calculate the potential energy curve of the very challenging Cr${}_{2}$ dimer. Despite the fact that the largest Hilbert space has dimension ${10}^{42}$, the paper obtains energies with estimated errors of a few mHa or less.

DOI
10.1103/physrevresearch.2.012015

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The authors use the recently developed semistochastic heat bath configuration interaction method to calculate the potential energy curve of the very challenging Cr2{}_{2}2​ dimer. Despite the fact that the largest Hilbert space has dimension 1042{10}^{42}1042, the paper obtains energies with estimated errors of a few mHa or less.
This paper presents in detail our fast semistochastic heat-bath configuration interaction (SHCI) method for solving the many-body Schrödinger equation. We identify and eliminate computational bottlenecks in both the variational and perturbative steps of the SHCI algorithm. We also describe the parallelization and the key data structures in our implementation, such as the distributed hash table. The improved SHCI algorithm enables us to include in our variational wavefunction two orders of magnitude more determinants than has been reported previously with other selected configuration interaction methods. We use our algorithm to calculate an accurate benchmark energy for the chromium dimer with the X2C relativistic Hamiltonian in the cc-pVDZ-DK basis, correlating 28 electrons in 76 spatial orbitals. Our largest calculation uses two billion Slater determinants in the variational space and semistochastically includes perturbative contributions from at least trillions of additional determinants with better than 10−5 Ha statistical uncertainty.
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 2 with an active space of (14e, 108o), Mn-Salen cluster with an active space of (28e, 22o), and Cr 2 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.
The recently developed semistochastic heat-bath configuration interaction (SHCI) method is a systematically improvable selected configuration interaction plus perturbation theory method capable of giving essentially exact energies for larger systems than is possible with other such methods. We compute SHCI atomization energies for 55 molecules that have been used as a test set in prior studies because their atomization energies are known from experiment. Basis sets from cc-pVDZ to cc-pV5Z are used, totaling up to 500 orbitals and a Hilbert space of 1032 Slater determinants for the largest molecules. For each basis, an extrapolated energy well within chemical accuracy (1 kcal/mol or 1.6 mHa/mol) of the exact energy for that basis is computed using only a tiny fraction of the entire Hilbert space. We also use our almost exact energies to benchmark energies from the coupled cluster method with single, double, and perturbative triple excitations. The energies are extrapolated to the complete basis set limit and compared to the experimental atomization energies. The extrapolations are done both without and with a basis-set correction based on density-functional theory. The mean absolute deviations from experiment for these extrapolations are 0.46 kcal/mol and 0.51 kcal/mol, respectively. Orbital optimization methods used to obtain improved convergence of the SHCI energies are also discussed.
The chromium dimer (Cr2) presents an outstanding challenge for many-body electronic structure methods. Its complicated nature of binding, with a formal sextuple bond and an unusual potential energy curve (PEC), is emblematic of the competing tendencies and delicate balance found in many strongly correlated materials. We present an accurate calculation of the PEC and ground state properties of Cr2, using the auxiliary-field quantum Monte Carlo (AFQMC) method. Unconstrained, exact AFQMC calculations are first carried out for a medium-sized but realistic basis set. Elimination of the remaining finite-basis errors and extrapolation to the complete basis set limit are then achieved with a combination of phaseless and exact AFQMC calculations. Final results for the PEC and spectroscopic constants are in excellent agreement with experiment.