We present a combination of a downfolding many-body approach with auxiliary-field quantum Monte Carlo (AFQMC) calculations for extended systems. Many-body calculations operate on a simpler Hamiltonian which retains material-specific properties. The Hamiltonian is systematically improvable and allows one to dial, in principle, between the simplest model and the original Hamiltonian. As a by-product, pseudopotential errors are essentially eliminated using frozen orbitals constructed adaptively from the solid environment. The computational cost of the many-body calculation is dramatically reduced without sacrificing accuracy. Excellent accuracy is achieved for a range of solids, including semiconductors, ionic insulators, and metals. We apply the method to calculate the equation of state of cubic BN under ultrahigh pressure, and determine the spin gap in NiO, a challenging prototypical material with strong electron correlation effects.
We present a combination of a downfolding many-body approach with auxiliary-field quantum Monte Carlo (AFQMC) calculations for extended systems. Many-body calculations operate on a simpler Hamiltonian which retains material-specific properties. The Hamiltonian is systematically improvable and allows one to dial, in principle, between the simplest model and the original Hamiltonian. As a by-product, pseudopotential errors are essentially eliminated using frozen orbitals constructed adaptively from the solid environment. The computational cost of the many-body calculation is dramatically reduced without sacrificing accuracy. Excellent accuracy is achieved for a range of solids, including semiconductors, ionic insulators, and metals. We apply the method to calculate the equation of state of cubic BN under ultrahigh pressure, and determine the spin gap in NiO, a challenging prototypical material with strong electron correlation effects.
We describe the implementation of the frozen-orbital and downfolding approximations in the auxiliary-field quantum Monte Carlo (AFQMC) method. These approaches can provide significant computational savings, compared to fully correlating all of the electrons. While the many-body wave function is never explicit in AFQMC, its random walkers are Slater determinants, whose orbitals may be expressed in terms of any one-particle orbital basis. It is therefore straightforward to partition the full N-particle Hilbert space into active and inactive parts to implement the frozen-orbital method. In the frozen-core approximation, for example, the core electrons can be eliminated in the correlated part of the calculations, greatly increasing the computational efficiency, especially for heavy atoms. Scalar relativistic effects are easily included using the Douglas-Kroll-Hess theory. Using this method, we obtain a way to effectively eliminate the error due to single-projector, norm-conserving pseudopotentials in AFQMC. We also illustrate a generalization of the frozen-orbital approach that downfolds high-energy basis states to a physically relevant low-energy sector, which allows a systematic approach to produce realistic model Hamiltonians to further increase efficiency for extended systems.
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.
orbitals. Both FC and downfolding provide significant computational savings over fully correlating all the electrons in full plane-wave basis, while retaining excellent transferability and accuracy. We demonstrate the approach by calculating the equation of state of crystalline MnO in antiferromagnetic and ferromagnetic phases. Twist-average boundary condition [2] and a finite-size correction [3] are employed to minimize the eect of finite simulation cell. AFQMC in this basis reproduces the results of the full plane-wave basis many-body calculations, and leads to accurate determination of the magnetic ordering.
The accurate computation of forces and other energy derivatives has been a long-standing challenge for quantum Monte Carlo methods. A number of technical obstacles contribute to this challenge. We discuss how these obstacles can be removed with the auxiliary-field quantum Monte Carlo (AFQMC) approach. AFQMC is a general, high-accuracy, many-body total-energy method for molecules and solids. The implementation of back-propagation for pure estimators allows direct calculation of gradients of the energy via the Hellmann-Feynman theorem. A planewave basis with norm-conserving pseudopotentials is used for the study of periodic bulk materials. Completeness of the planewave basis minimizes the effect of so-called Pulay terms. The ionic pseudopotentials, which can be incorporated in AFQMC in exactly the same manner as in standard independent-electron methods, regulate the force and stress estimators and eliminate any potential divergence of the Monte Carlo variances. The resulting approach allows applications of full geometry optimizations in bulk materials. It also paves the way for many-body computations of the phonon spectrum in solids.
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.
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.