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arXiv:2103.03176 · Physical Review Research

Towards efficient and accurate ab initio solutions to periodic systems via transcorrelation and coupled cluster theory

Ke Liao, Thomas Schraivogel, Hongjun Luo, Daniel Kats, Ali Alavi

Published Jul 20, 2021• 41 citations
arXivHTMLPDF

Abstract

We propose a streamlined combination of the transcorrelation (TC) and coupled cluster (CC) theory, which not only increases the convergence rate with respect to the basis set but also extends the applicability of the lowest-order CC approximations to strongly correlated systems. Using the three-dimensional uniform electron gas as a model for real periodic solids and highly accurate ground-state energies from the state-of-the-art quantum Monte Carlo methods as benchmark, we show that, with the correct physical insights incorporated, our TC-CC methods gain drastic improvements in accuracy compared with canonical CC methods, and the errors of our methods remain $\ifmmode\pm\else\textpm\fi{}0.001$ Hartree per electron (Ha/electron) across a wide range of densities. With the greatly improved efficiencies and the balanced and accurate performances at both low and high densities, our methods present great promise for applications on real solids with weak and strong correlations.

DOI
10.1103/physrevresearch.3.033072

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We propose a streamlined combination of the transcorrelation (TC) and coupled cluster (CC) theory, which not only increases the convergence rate with respect to the basis set but also extends the applicability of the lowest-order CC approximations to strongly correlated systems. Using the three-dimensional uniform electron gas as a model for real periodic solids and highly accurate ground-state energies from the state-of-the-art quantum Monte Carlo methods as benchmark, we show that, with the correct physical insights incorporated, our TC-CC methods gain drastic improvements in accuracy compared with canonical CC methods, and the errors of our methods remain \ifmmode±\else\textpm\fi0.001\ifmmode\pm\else\textpm\fi{}0.001\ifmmode±\else\textpm\fi0.001 Hartree per electron (Ha/electron) across a wide range of densities. With the greatly improved efficiencies and the balanced and accurate performances at both low and high densities, our methods present great promise for applications on real solids with weak and strong correlations.
Accurate wave-function descriptions of pristine and defected solids remain challenging because of the simultaneous presence of finite-size, basis-set, and correlation errors. While embedding techniques alleviate finite-size effects and correlated wave-function approaches systematically improve correlation, basis-set incompleteness continues to limit practical accuracy. Here, we present a study of transcorrelated (TC) many-body wave-function methods on properties of solid state systems. We augment the existing xTC theory to periodic systems, and establish an unified transcorrelated embedding framework that integrates periodic TC theory with fragment-based correlated solvers. Using silicon as a test case, we validate the method against coupled-cluster, FCIQMC, and diffusion Monte Carlo benchmarks for bulk. Then, we apply transcorrelated embedding to the calculation of formation energies of two silicon self-interstitials. The TC Hamiltonian yields rapid basis convergence and quantitatively reliable defect formation energies at the triple- ζ level, substantially reducing the basis-set bottleneck for wave-function treatments of crystalline defects.
Coupled-cluster theories can be used to compute ab initio electronic correlation energies of real materials with systematically improvable accuracy. However, the widely used coupled cluster singles and doubles plus perturbative triples [CCSD(T)] method is only applicable to insulating materials. For zero-gap materials the truncation of the underlying many-body perturbation expansion leads to an infrared catastrophe. Here, we present a novel perturbative triples formalism denoted as (cT) that yields convergent correlation energies in metallic systems. Furthermore, the computed correlation energies for the three-dimensional uniform electron gas at metallic densities are in good agreement with quantum Monte Carlo results. At the same time the newly proposed method retains all desirable properties of CCSD(T) such as its accuracy for insulating systems as well as its low computational cost compared to a full inclusion of the triples. This paves the way for ab initio calculations of real metals with chemical accuracy.
The workhorse method of computational materials science is undeniably density functional theory in the Kohn-Sham framework of approximate exchange and correlation energy functionals. However, the need for highly accurate predictions of ground and excited state properties in materials science motivates the further development and exploration of alternative as well as complementary techniques. Among these alternative approaches, quantum chemical wavefunction based theories and in particular coupled cluster theory hold the promise to fill a gap in the toolbox of computational materials scientists. Coupled cluster (CC) theory provides a compelling framework of approximate infinite-order perturbation theory in the form of an exponential of cluster operators describing the true quantum many-body effects of the electronic wave function at a computational cost that, despite being significantly more expensive than DFT, scales polynomially with system size. The hierarchy of size-extensive approximate methods established in the framework of CC theory achieves systematic improvability for many materials properties. This is in contrast to currently available density functionals that often suffer from uncontrolled approximations that limit the accuracy in the prediction of materials properties. In this tutorial-style review we will introduce basic concepts of coupled cluster theory and recent developments that increase its computational efficiency for calculations of molecules, solids and materials in general. We will touch upon the connection between coupled cluster theory and the random-phase approximation to bridge the gap between traditional quantum chemistry and many-body Green’s function theories that are widely-used in the field of solid state physics. We will discuss various approaches to improve the computational performance without compromising accuracy. These approaches include large-scale parallel design as well as techniques that reduce the prefactor of the computational complexity. A central part of this article discusses the convergence of calculated properties to the thermodynamic limit which is of significant importance for reliable predictions of materials properties and constitutes an additional challenge compared to calculations of large molecules. Furthermore we mention technical aspects of computer code implementations of periodic coupled cluster theories in different numerical frameworks of the one-electron orbital basis; the projector-augmented-wave formalism using a plane wave basis set and the numeric atom-centered-orbital with resolution-of-identity.
The transcorrelated (TC) method dramatically accelerates the convergence of correlated calculations toward the complete-basis-set (CBS) limit by folding a Jastrow correlator into the Hamiltonian via a similarity transformation, incorporating the electron--electron cusp into the effective interaction. We make the TC framework practical for large systems and flexible, multi-center correlators by compressing the grid-evaluated TC integrals with the interpolative separable density-fitting (ISDF) approximation, combined with the effective two-body (xTC) treatment of the three-body operator. This low-rank representation reduces storage and integration costs by orders of magnitude, and a multi-GPU implementation with automatic differentiation of the correlator makes the construction routine for large basis sets. We demonstrate the resulting ISDF-xTC-CCSD method on the linear hydrogen chain, reaching the joint thermodynamic and CBS limits with basis sets up to cc-pV5Z in agreement with state-of-the-art many-body references to within about 1~mHa/atom, and on the benzene ground-state energy with up to 1200 orbitals (cc-pCV5Z), where the method attains state-of-the-art accuracy at the coupled cluster singles and doubles level and its CBS extrapolation is markedly more robust than that of conventional coupled-cluster methods.
Coupled-cluster theory with single, double, and perturbative triple excitations (CCSD(T))-often considered the "gold standard" of main-group quantum chemistry-is inapplicable to three-dimensional metals due to an infrared divergence, preventing its application to many important problems in materials science. We study the full, nonperturbative inclusion of triple excitations (CCSDT) and propose a new, iterative method, which we call ring-CCSDT, that resums the essential triple excitations with the same N^{7} run-time scaling as CCSD(T). CCSDT and ring-CCSDT are used to calculate the correlation energy of the uniform electron gas at metallic densities and the structural properties of solid lithium. Inclusion of connected triple excitations is shown to be essential to achieving high accuracy. We also investigate semiempirical CC methods based on spin-component scaling and the distinguishable cluster approximation and find that they enhance the accuracy of their parent ab initio methods.
Simulation of electronic structure is one of the most promising applications on noisy intermediate-scale quantum (NISQ) era devices. However, NISQ devices suffer from a number of challenges like limited qubit connectivity, short coherence times, and sizable gate error rates. Thus, desired quantum algorithms should require shallow circuit depths and low qubit counts to take advantage of these devices. Here, we attempt to reduce quantum resource requirements for molecular simulations on a quantum computer while maintaining the desired accuracy with the help of classical quantum chemical theories of canonical transformation and explicit correlation. In this work, compact ab initio Hamiltonians are generated classically, in the second quantized form, through an approximate similarity transformation of the Hamiltonian with (a) an explicitly correlated two-body unitary operator with generalized pair excitations that remove the Coulombic electron-electron singularities from the Hamiltonian and (b) a unitary one-body operator to efficiently capture the orbital relaxation effects required for accurate description of the excited states. The resulting transcorrelated Hamiltonians are able to describe both the ground and the excited states of molecular systems in a balanced manner. Using the variational quantum eigensolver (VQE) method based on the unitary coupled cluster with singles and doubles (UCCSD) ansatz and only a minimal basis set (ANO-RCC-MB), we demonstrate that the transcorrelated Hamiltonians can produce ground state energies comparable to the reference CCSD energies with the much larger cc-pVTZ basis set. This leads to a reduction in the number of required CNOT gates by more than 3 orders of magnitude for the chemical species studied in this work. Furthermore, using the quantum equation of motion (qEOM) formalism in conjunction with the transcorrelated Hamiltonian, we are able to reduce the deviations in the excitation energies from the reference EOM-CCSD/cc-pVTZ values by an order of magnitude. The transcorrelated Hamiltonians developed here are Hermitian and contain only one- and two-body interaction terms and thus can be easily combined with any quantum algorithm for accurate electronic structure simulations.