We develop a local correlation variant of auxiliary-field quantum Monte Carlo (AFQMC) based on local natural orbitals (LNO-AFQMC). In LNO-AFQMC, independent AFQMC calculations are performed for each localized occupied orbital using a truncated set of tailored orbitals. Because the size of this space does not grow with the system size for a target accuracy, the method has linear scaling. Applying LNO-AFQMC to molecular problems containing a few hundred to a thousand orbitals, we demonstrate convergence of total energies with significantly reduced costs. The savings are more significant for larger systems and larger basis sets. However, even for our smallest system studied, we find that LNO-AFQMC is cheaper than canonical AFQMC, in contrast with many other reduced-scaling methods. Perhaps most significantly, we show that energy differences converge much more quickly than total energies, making the method ideal for applications in chemistry and material science. Our work paves the way for linear scaling AFQMC calculations of strongly correlated systems, which would have a transformative effect on ab initio quantum chemistry.
We develop a local correlation variant of auxiliary-field quantum Monte Carlo (AFQMC) based on local natural orbitals (LNO-AFQMC). In LNO-AFQMC, independent AFQMC calculations are performed for each localized occupied orbital using a truncated set of tailored orbitals. Because the size of this space does not grow with the system size for a target accuracy, the method has linear scaling. Applying LNO-AFQMC to molecular problems containing a few hundred to a thousand orbitals, we demonstrate convergence of total energies with significantly reduced costs. The savings are more significant for larger systems and larger basis sets. However, even for our smallest system studied, we find that LNO-AFQMC is cheaper than canonical AFQMC, in contrast with many other reduced-scaling methods. Perhaps most significantly, we show that energy differences converge much more quickly than total energies, making the method ideal for applications in chemistry and material science. Our work paves the way for linear scaling AFQMC calculations of strongly correlated systems, which would have a transformative effect on ab initio quantum chemistry.
A local embedding and effective downfolding scheme has been developed and implemented in the auxiliary-field quantum Monte Carlo (AFQMC) method. A local cluster in which electrons are fully correlated is defined, and the frozen orbital method is used on the remainder of the system to construct an effective Hamiltonian, which operates within the local cluster. Local embedding, which involves only the occupied sector, has previously been employed in the context of Co/graphene. Here, the methodology is extended in order to allow for effective downfolding of the virtual sector, thus allowing for significant reduction in the computational effort required for AFQMC calculations. The system size, which can be feasibly treated with AFQMC, is therefore greatly extended as only a single local cluster is explicitly correlated at the AFQMC level of theory. The approximation is controlled by the separate choice of the spatial size of the active occupied region ( R o ) and of the active virtual region ( R v ). The systematic dependence of the AFQMC energy on R o and R v is investigated, and it is found that relative AFQMC energies of physical and chemical interest converge rapidly to the full AFQMC treatment (i.e., using no embedding or downfolding).
to develop quantum Monte Carlo (QMC) approaches for excited states and for more systematic and accurate treatment of ground states of strongly correlated sytems and heavier elements. We use the auxiliary-field quantum Monte Carlo (AFQMC) framework, which is a many-body total energy method that looks like many coupled mean-field calculations. I will discuss this method’s relations and dierences with traditional density-functional and quantum chemistry methods, and with other forms of QMC. The AFQMC method uses importance-sampled random walks in the space of nonorthogonal Slater determinants to project out the many-body ground state, or excited state. The random walks are constrained by a trial wave function (TWF) to remove the so-called sign problem, giving approximate (but non-perturbative) results. With singledeterminant TWFs from Hartree-Fock or density-functional theory (DFT), this approach can be thought of as a post-DFT method with no parameter. In this form, the accuracy is typically comparable to the quantum chemistry coupled-cluster CCSD(T) method near equilibrium geometries, and better than CCSD(T) when bonds are stretched. The computational cost of AFQMC scales as N 3 with characteristic size of the system. Its accuracy can be further improved at the cost of increased computing (releasing the constraint) or better TWF. With AFQMC, we can systematically target selected orbitals to correlate. This allows many-body calculations in solids with a frozen core, with down-folded Hamiltonians, or possibly embeded in a DFT calculations to extend scales. Results from a few on-going applications will be presented, including band structures in simple solids, magnetic phases in MnO, and Cobalt adsoption on graphene.
Phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) has recently emerged as a promising method for the production of benchmark-level simulations of medium- to large-sized molecules because of its accuracy and favorable polynomial scaling with system size. Unfortunately, the memory footprints of standard energy evaluation algorithms are nontrivial, which can significantly impact timings on graphical processing units (GPUs) where memory is limited. Previous attempts to reduce scaling by taking advantage of the low-rank structure of the Coulombic integrals have been successful but exhibit high prefactors, making their utility limited to very large systems. Here we present a complementary cubic-scaling route to reduce memory and computational scaling based on the low rank of the Coulombic interactions between localized orbitals, focusing on the application to ph-AFQMC. We show that the error due to this approximation, which we term localized-orbital AFQMC (LO-AFQMC), is systematic and controllable via a single variable and that the method is computationally favorable even for small systems. We present results demonstrating robust retention of accuracy versus both experiment and full ph-AFQMC for a variety of test cases chosen for their potential difficulty for localized-orbital-based methods, including the singlet–triplet gaps of the polyacenes benzene through pentacene, the heats of formation for a set of Platonic hydrocarbon cages, and the total energy of ferrocene, Fe(Cp) 2 . Finally, we reproduce our previous result for the gas-phase ionization energy of Ni(Cp) 2, agreeing with full ph-AFQMC to within statistical error while using less than 1/15th of the computer time.
auxiliary-field quantum Monte Carlo (AFQMC) is a systematically improvable many-body method, but its application to extended solids has been severely limited by unfavorable computational scaling and memory requirements that obstruct rigorous access to the thermodynamic and complete-basis-set limits. By combining tensor hypercontraction with k -point symmetry, we reduce the computational and memory scaling of AFQMC for solids to O ( N 3 ) and O ( N 2 ) , respectively, with an arbitrary basis, comparable to diffusion Monte Carlo. This enables simultaneous, controlled extrapolation of full-system AFQMC calculations to the thermodynamic and complete-basis-set limits across insulating, metallic, and strongly correlated solids, without embedding, local approximations, empirical finite-size corrections, or composite schemes. Our results establish AFQMC as a general-purpose, systematically improvable alternative to diffusion Monte Carlo and coupled-cluster methods for predictive simulations of solids, enabling accurate energies and magnetic observables within a unified framework.
The auxiliary‐field quantum Monte Carlo (AFQMC) method provides a computational framework for solving the time‐independent Schrödinger equation in atoms, molecules, solids, and a variety of model systems. AFQMC has recently witnessed remarkable growth, especially as a tool for electronic structure computations in real materials. The method has demonstrated excellent accuracy across a variety of correlated electron systems. Taking the form of stochastic evolution in a manifold of nonorthogonal Slater determinants, the method resembles an ensemble of density‐functional theory (DFT) calculations in the presence of fluctuating external potentials. Its computational cost scales as a low‐power of system size, similar to the corresponding independent‐electron calculations. Highly efficient and intrinsically parallel, AFQMC is able to take full advantage of contemporary high‐performance computing platforms and numerical libraries. In this review, we provide a self‐contained introduction to the exact and constrained variants of AFQMC, with emphasis on its applications to the electronic structure of molecular systems. Representative results are presented, and theoretical foundations and implementation details of the method are discussed. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Structure and Mechanism > Computational Materials Science Computer and Information Science > Computer Algorithms and Programming
The auxiliary-field quantum Monte Carlo (AFQMC) method is a general numerical method for correlated many-electron systems, which is being increasingly applied in lattice models, atoms, molecules, and solids. Here, we introduce the theory and algorithm of the method specialized for real materials and present several recent developments. We give a systematic exposition of the key steps of AFQMC, closely tracking the framework of a modern software library we are developing. The building of a Monte Carlo Hamiltonian, projecting to the ground state, sampling two-body operators, phaseless approximation, and measuring ground state properties are discussed in detail. An advanced implementation for multi-determinant trial wave functions is described, which dramatically speeds up the algorithm and reduces the memory cost. We propose a self-consistent constraint for real materials, and discuss two flavors for its realization, either by coupling the AFQMC calculation to an effective independent-electron calculation or via the natural orbitals of the computed one-body density matrix.