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arXiv:1703.05347 · The Journal of Chemical Physics

Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory

Yann Garniron, Anthony Scemama, Pierre‐François Loos, Michel Caffarel

Published Jul 17, 2017• 146 citations
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Abstract

A hybrid stochastic-deterministic approach for computing the second-order perturbative contribution E(2) within multireference perturbation theory (MRPT) is presented. The idea at the heart of our hybrid scheme—based on a reformulation of E(2) as a sum of elementary contributions associated with each determinant of the MR wave function—is to split E(2) into a stochastic and a deterministic part. During the simulation, the stochastic part is gradually reduced by dynamically increasing the deterministic part until one reaches the desired accuracy. In sharp contrast with a purely stochastic Monte Carlo scheme where the error decreases indefinitely as t−1/2 (where t is the computational time), the statistical error in our hybrid algorithm displays a polynomial decay ∼t−n with n = 3–4 in the examples considered here. If desired, the calculation can be carried on until the stochastic part entirely vanishes. In that case, the exact result is obtained with no error bar and no noticeable computational overhead compared to the fully deterministic calculation. The method is illustrated on the F2 and Cr2 molecules. Even for the largest case corresponding to the Cr2 molecule treated with the cc-pVQZ basis set, very accurate results are obtained for E(2) for an active space of (28e, 176o) and a MR wave function including up to 2×107 determinants.

DOI
10.1063/1.4992127

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A hybrid stochastic-deterministic approach for computing the second-order perturbative contribution E(2) within multireference perturbation theory (MRPT) is presented. The idea at the heart of our hybrid scheme—based on a reformulation of E(2) as a sum of elementary contributions associated with each determinant of the MR wave function—is to split E(2) into a stochastic and a deterministic part. During the simulation, the stochastic part is gradually reduced by dynamically increasing the deterministic part until one reaches the desired accuracy. In sharp contrast with a purely stochastic Monte Carlo scheme where the error decreases indefinitely as t−1/2 (where t is the computational time), the statistical error in our hybrid algorithm displays a polynomial decay ∼t−n with n = 3–4 in the examples considered here. If desired, the calculation can be carried on until the stochastic part entirely vanishes. In that case, the exact result is obtained with no error bar and no noticeable computational overhead compared to the fully deterministic calculation. The method is illustrated on the F2 and Cr2 molecules. Even for the largest case corresponding to the Cr2 molecule treated with the cc-pVQZ basis set, very accurate results are obtained for E(2) for an active space of (28e, 176o) and a MR wave function including up to 2×107 determinants.
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.
Full Configuration Interaction Quantum Monte Carlo (FCIQMC) has been effectively applied to very large configuration interaction (CI) problems and was recently adapted for use as an active space solver and combined with orbital optimization. In this work, we detail an approach within FCIQMC to allow for efficient sampling of fully internally contracted multireference perturbation theories within the same stochastic framework. Schemes are described to allow for the close control over the resolution of stochastic sampling of the effective higher-body intermediates within the active space. It is found that while complete active space second-order perturbation theory seems less amenable to a stochastic reformulation, strongly contracted N-Electron Valence second-order Perturbation Theory (NEVPT2) is far more stable, requiring a similar number of walkers to converge the sc-NEVPT2 expectation values as to converge the underlying CI problem. We demonstrate the application of the stochastic approach to the computation of sc-NEVPT2 within a (24, 24) active space in a biologically relevant system and show that small numbers of walkers are sufficient for a faithful sampling of the sc-NEVPT2 energy to chemical accuracy, despite the active space already exceeding the limits of practicality for traditional approaches. This raises prospects of an efficient stochastic solver for multireference chemical problems requiring large active spaces, with an accurate treatment of external orbitals.
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Development of exponentially scaling methods has seen great progress in tackling larger systems than previously thought possible. One such technique, full configuration interaction quantum Monte Carlo, is a useful algorithm that allows exact diagonalization through stochastically sampling determinants. The method derives its utility from the information in the matrix elements of the Hamiltonian, along with a stochastic projected wave function, to find the important parts of Hilbert space. However, the stochastic representation of the wave function is not required to search Hilbert space efficiently, and here we describe a highly efficient deterministic method that can achieve chemical accuracy for a wide range of systems, including the difficult Cr2 molecule. We demonstrate for systems like Cr2 that such calculations can be performed in just a few cpu hours which makes it one of the most efficient and accurate methods that can attain chemical accuracy for strongly correlated systems. In addition our method also allows efficient calculation of excited state energies, which we illustrate with benchmark results for the excited states of C2.
In this article we report a stochastic evaluation of the recently proposed multireference linearized coupled cluster theory [S. Sharma and A. Alavi, J. Chem. Phys. 143, 102815 (2015)]. In this method, both the zeroth-order and first-order wavefunctions are sampled stochastically by propagating simultaneously two populations of signed walkers. The sampling of the zeroth-order wavefunction follows a set of stochastic processes identical to the one used in the full configuration interaction quantum Monte Carlo (FCIQMC) method. To sample the first-order wavefunction, the usual FCIQMC algorithm is augmented with a source term that spawns walkers in the sampled first-order wavefunction from the zeroth-order wavefunction. The second-order energy is also computed stochastically but requires no additional overhead outside of the added cost of sampling the first-order wavefunction. This fully stochastic method opens up the possibility of simultaneously treating large active spaces to account for static correlation and recovering the dynamical correlation using perturbation theory. The method is used to study a few benchmark systems including the carbon dimer and aromatic molecules. We have computed the singlet-triplet gaps of benzene and m-xylylene. For m-xylylene, which has proved difficult for standard complete active space self consistent field theory with perturbative correction, we find the singlet-triplet gap to be in good agreement with the experimental values.
The interplay between advances in stochastic and deterministic algorithms has recently led to development of interesting new selected configuration interaction (SCI) methods for solving the many body Schrödinger equation. The performance of these SCI methods can be greatly improved with a second order perturbation theory (PT2) correction, which is often evaluated in a stochastic or hybrid-stochastic manner. In this work, we present a highly efficient, fully deterministic PT2 algorithm for SCI methods and demonstrate that our approach is orders of magnitude faster than recent proposals for stochastic SCI+PT2. We also show that it is important to have a compact reference SCI wave function, in order to obtain optimal SCI+PT2 energies. This indicates that it advantageous to use accurate search algorithms such as 'ASCI search' rather than more approximate approaches. Our deterministic PT2 algorithm is based on sorting techniques that have been developed for modern computing architectures and is inherently straightforward to use on parallel computing architectures. Related architectures such as GPU implementations can be also used to further increase the efficiency. Overall, we demonstrate that the algorithms presented in this work allow for efficient evaluation of trillions of PT2 contributions with modest computing resources.