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arXiv:2011.08335 · Physical review. B./Physical review. B

Ab initio electronic density in solids by many-body plane-wave auxiliary-field quantum Monte Carlo calculations

Siyuan Chen, Mário Motta, Fengjie Ma, Shiwei Zhang

Published Feb 22, 2021• 21 citations
arXivHTMLPDF

Abstract

We present accurate many-body results of the electronic densities in several solid materials, including Si, NaCl, and Cu. These results are obtained using the ab initio auxiliary-field quantum Monte Carlo (AFQMC) method working in a plane-wave basis with norm-conserving, multiple-projector pseudopotentials. AFQMC has been shown to be an excellent many-body total energy method. Computation of observables and correlation functions other than the ground-state energy requires back-propagation, whose adaption and implementation in the plane-wave basis AFQMC framework are discussed in the present paper. This development allows us to compute correlation functions, electronic densities, and interatomic forces, paving the way for geometry optimizations and calculations of thermodynamic properties in solids. Finite supercell size effects are considerably more subtle in the many-body framework than in independent-electron calculations. We analyze the convergence of the electronic density, and obtain best estimates for the thermodynamic limit. The densities from several typical density functionals are benchmarked against our near-exact results. The electronic densities we obtained can also be used to help construct improved density functionals.

DOI
10.1103/physrevb.103.075138

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We present accurate many-body results of the electronic densities in several solid materials, including Si, NaCl, and Cu. These results are obtained using the ab initio auxiliary-field quantum Monte Carlo (AFQMC) method working in a plane-wave basis with norm-conserving, multiple-projector pseudopotentials. AFQMC has been shown to be an excellent many-body total energy method. Computation of observables and correlation functions other than the ground-state energy requires back-propagation, whose adaption and implementation in the plane-wave basis AFQMC framework are discussed in the present paper. This development allows us to compute correlation functions, electronic densities, and interatomic forces, paving the way for geometry optimizations and calculations of thermodynamic properties in solids. Finite supercell size effects are considerably more subtle in the many-body framework than in independent-electron calculations. We analyze the convergence of the electronic density, and obtain best estimates for the thermodynamic limit. The densities from several typical density functionals are benchmarked against our near-exact results. The electronic densities we obtained can also be used to help construct improved density functionals.
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.
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The phaseless auxiliary-field quantum Monte Carlo (AF QMC) method [S. Zhang and H. Krakauer, Phys. Rev. Lett. 90, 136401 (2003)] is used to carry out a systematic study of the dissociation and ionization energies of second-row group 3A--7A atoms and dimers: Al, Si, P, S, and Cl. In addition, the P2{\mathrm{P}}_{2}P2​ dimer is compared to the third-row As2{\mathrm{As}}_{2}As2​ dimer, which is also triply bonded. This method projects the many-body ground state by means of importance-sampled random walks in the space of Slater determinants. The Monte Carlo phase problem, due to the electron-electron Coulomb interaction, is controlled via the phaseless approximation, with a trial wave function $\ensuremath{\mid}{\ensuremath{\Psi}}_{T}⟩$. As in previous calculations, a mean-field single Slater determinant is used as $\ensuremath{\mid}{\ensuremath{\Psi}}_{T}⟩$. The method is formulated in the Hilbert space defined by any chosen one-particle basis. The present calculations use a plane wave basis under periodic boundary conditions with norm-conserving pseudopotentials. Computational details of the plane wave AF QMC method are presented. The isolated systems chosen here allow a systematic study of the various algorithmic issues. We show the accuracy of the plane wave method and discuss its convergence with respect to parameters such as the supercell size and plane wave cutoff. The use of standard norm-conserving pseudopotentials in the many-body AF QMC framework is examined.
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.
We present a plane-wave (PW) implementation of the auxiliary-field quantum Monte Carlo (AFQMC) method within the projector augmented-wave (PAW) formalism in the Vienna Simulation Package (VASP). By employing an exact inversion of the PAW overlap operator, our approach maintains cubic scaling while naturally incorporating all excitations defined by the PW cutoff. We benchmark this framework by calculating the equilibrium lattice constants and bulk moduli of C, BN, BP, and Si. Our analysis demonstrates that AFQMC systematically corrects the lack of long-range screening in MP2 and the missing higher-order exchange in RPA. We identify RPA as the optimal reference method because of the rapid convergence of the remaining short-range correlations with respect to supercell size. The resulting lattice constants exhibit a mean absolute relative error of 0.14 % relative to experiment, establishing the method as a rigorous benchmark tool for structural properties of prototypical semiconductors and insulators.