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arXiv:1704.00864 · The Journal of Chemical Physics

Density matrices in full configuration interaction quantum Monte Carlo: Excited states, transition dipole moments, and parallel distribution

Nick S. Blunt, George H. Booth, Ali Alavi

Published Jun 26, 2017• 54 citations
arXivHTMLPDF

Abstract

We present developments in the calculation of reduced density matrices (RDMs) in the full configuration interaction quantum Monte Carlo (FCIQMC) method. An efficient scheme is described to allow storage of RDMs across distributed memory, thereby allowing their calculation and storage in large basis sets. We demonstrate the calculation of RDMs for general states by using the recently introduced excited-state FCIQMC approach [N. S. Blunt et al., J. Chem. Phys. 143, 134117 (2015)] and further introduce calculation of transition density matrices in the method. These approaches are combined to calculate excited-state dipole and transition dipole moments for heteronuclear diatomic molecules, including LiH, BH, and MgO, and initiator error is investigated in these quantities.

DOI
10.1063/1.4986963

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We present developments in the calculation of reduced density matrices (RDMs) in the full configuration interaction quantum Monte Carlo (FCIQMC) method. An efficient scheme is described to allow storage of RDMs across distributed memory, thereby allowing their calculation and storage in large basis sets. We demonstrate the calculation of RDMs for general states by using the recently introduced excited-state FCIQMC approach [N. S. Blunt et al., J. Chem. Phys. 143, 134117 (2015)] and further introduce calculation of transition density matrices in the method. These approaches are combined to calculate excited-state dipole and transition dipole moments for heteronuclear diatomic molecules, including LiH, BH, and MgO, and initiator error is investigated in these quantities.
Full configuration interaction quantum Monte Carlo for accurate ab initio\textit{ab initio}ab initio nuclear structure calculations: algorithms and calculation details
Full configuration interaction quantum Monte Carlo (FCIQMC) is a stochastic many-body solver that has been widely applied to electronic, molecular, and condensed-matter systems. In this work we apply FCIQMC to ab initio\textit{ab initio}ab initio nuclear structure calculations using interactions derived from chiral effective field theory. We describe the algorithm in detail, including imaginary-time propagation, excitation generation, estimator choices, the initiator approximation with adaptive shift correction, and reduced-density-matrix (RDM) sampling. Benchmark calculations in small model spaces, where deterministic full configuration interaction (FCI) results are available, validate the stochastic calculation of energies, radii, and RDM-based pure estimators. For large model spaces, we analyze the residual finite-walker bias through systematic walker-number convergence and infinite-walker extrapolations. We also demonstrate that FCIQMC can be extended beyond ground-state calculations by computing the low-lying spectrum of 6^66Li.
We expand upon the recent semi-stochastic adaptation to full configuration interaction quantum Monte Carlo (FCIQMC). We present an alternate method for generating the deterministic space without a priori knowledge of the wave function and present stochastic efficiencies for a variety of both molecular and lattice systems. The algorithmic details of an efficient semi-stochastic implementation are presented, with particular consideration given to the effect that the adaptation has on parallel performance in FCIQMC. We further demonstrate the benefit for calculation of reduced density matrices in FCIQMC through replica sampling, where the semi-stochastic adaptation seems to have even larger efficiency gains. We then combine these ideas to produce explicitly correlated corrected FCIQMC energies for the beryllium dimer, for which stochastic errors on the order of wavenumber accuracy are achievable.
The initiator full configuration interaction quantum Monte Carlo (i-FCIQMC) method has recently been developed as a highly accurate stochastic electronic structure technique. It has been shown to calculate the exact basis-set ground state energy of small molecules, to within modest stochastic error bars, using tractable computational cost. Here, we use this technique to elucidate an often troublesome series of first-row diatomics consisting of Be2, C2, CN, CO, N2, NO, O2, and F2. Using i-FCIQMC, the dissociation energies of these molecules are obtained almost entirely to within chemical accuracy of experimental results. Furthermore, the i-FCIQMC calculations are performed in a relatively black-box manner, without any a priori knowledge or specification of the wave function. The size consistency of i-FCIQMC is also demonstrated with regards to these diatomics at their more multiconfigurational stretched geometries. The clear and simple i-FCIQMC wave functions obtained for these systems are then compared and investigated to demonstrate the dynamic identification of the dominant determinants contributing to significant static correlation. The appearance and nature of such determinants is shown to provide insight into both the i-FCIQMC algorithm and the diatomics themselves.
We here apply the recently developed initiator density matrix quantum Monte Carlo (i-DMQMC) to a variety of atoms and molecules in vacuum. i-DMQMC samples the exact density matrix of a Hamiltonian at finite temperature and combines the accuracy of full configuration interaction quantum Monte Carlo (FCIQMC)—full configuration interaction (FCI) or exact energies in a finite basis set—with finite temperature. In order to explore the applicability of i-DMQMC for molecular systems, we choose to study a recently developed test set by Rubenstein and co-workers: Be, H 2 O, and H 10 at near-equilibrium and stretched geometries. We find that, for Be and H 2 O, i-DMQMC delivers energies with submillihartree accuracy when compared with finite temperature FCI. For H 2 O and both geometries of H 10, we examine the difference between FT-AFQMC and i-DMQMC, which, in turn, estimates the difference in canonical versus grand canonical energies. We close with two discussions: one of simulation settings (initiator error, the interaction picture, and different basis sets), and another of energy difference calculations in the form of specific heat capacity and ionization potential calculations.
Two distinct types of Quantum Monte Carlo (QMC) calculations are applied to electronic structure problems such as calculating potential energy curves and producing benchmark values for reaction barriers. First, Variational and Diffusion Monte Carlo (VMC and DMC) methods using a trial wavefunction subject to the fixed node approximation were tested using the CASINO code.[1] Next, Full Configuration Interaction Quantum Monte Carlo (FCIQMC), along with its initiator extension (i-FCIQMC) were tested using the NECI code.[2] FCIQMC seeks the FCI energy for a specific basis set. At a reduced cost, the efficient i-FCIQMC method can be applied to systems in which the standard FCIQMC approach proves to be too costly. Since all of these methods are statistical approaches, uncertainties (error-bars) are introduced for each calculated energy. This study tests the performance of the methods relative to traditional quantum chemistry for some benchmark systems. References: [1] R. J. Needs et al., J. Phys.: Condensed Matter 22, 023201 (2010). [2] G. H. Booth et al., J. Chem. Phys. 131, 054106 (2009).
We here apply the recently developed initiator density matrix quantum Monte Carlo (i-DMQMC) to a wide range of chemical environments using atoms and molecules in vacuum. i-DMQMC samples the exact density matrix of a Hamiltonian at finite temperature and combines the accuracy of full configuration interaction quantum Monte Carlo (FCIQMC) - full configuration interaction (FCI) or exact energies in a finite basis set - with finite temperature. By way of exploring the applicability of i-DMQMC for molecular systems, we choose to study a recently developed test set by Rubenstein and coworkers: Be, H2O, and H10 at near-equilibrium and stretched geometries. We find that, for Be and H2O, i-DMQMC delivers energies which are sub-millihartree accuracy when compared with finite temperature FCI. For H2O and both geometries of H10 we examine the difference between FT-AFQMC and i-DMQMC which in turn is an estimate of the difference in canonical versus grand canonical energies. We close with a discussion of simulation parameters (initiator error and different basis sets) and by showing energy difference calculations in the form of specific heat capacity and ionization potential calculations.