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WIREs Comput Mol Sci
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Ab initio computations of molecular systems by the auxiliary‐field quantum Monte Carlo method

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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
Pictorial illustration of the auxiliary‐field quantum Monte Carlo (AFQMC) algorithm, and of its beyond‐mean‐field nature. Left: Independent‐electron methods (e.g., Hartree‐Fock [HF] or density‐functional theory [DFT]) provide an approximation of the ground state wave function through a deterministic evolution in a manifold (gray surface) of Slater determinants (SDs) (solid black line: given an initial condition, the deterministic evolution brings it to ΨMF; dotted black lines: potential convergence trajectories to ΨMF from different initial conditions). Right: AFQMC represents the ground state as a stochastic linear combination of SDs by mapping the electron–electron interaction onto a fluctuating external potential, and the imaginary‐time evolution onto an open‐ended random walk in (colored curves departing from ΨI)
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Example of statistical analysis of auxiliary‐field quantum Monte Carlo (AFQMC) data. After removal of the equilibration phase (first 200 samples), data are reblocked until convergence of the standard deviation is reached (shown in the inset, and attained for L ≃ 125)
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Left: Complete basis set (CBS)‐extrapolated potential‐energy curve of Cr2 from ph‐auxiliary‐field quantum Monte Carlo (AFQMC), free‐projection AFQMC (red circles, blue diamonds) (Purwanto et al., ) compared to experiment (Casey & Leopold, ; Simard et al., ) (solid green line), UHF, UCASSCF(12,12) and UCCSD(T). Right: CBS‐extrapolated potential‐energy curve of Mo2 from ph‐AFQMC (red circles) (Purwanto et al., ) compared to experiment (Kraus, Lorenz, & Bondybey, )
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Potential‐energy curve of C2 at 6‐31G* level from auxiliary‐field quantum Monte Carlo (AFQMC) (red points, blue squares, green diamonds) using a truncated CASSCF(8,16) trial wave function, FCI (red dashed line, green dotted line, blue dot‐dashed line) and RCCSD(T) (solid orange line)
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Elimination of spin contamination in auxiliary‐field quantum Monte Carlo (AFQMC) calculations. (Reprinted with permission from Purwanto et al. (). Copyright 2008 American Institute of Physics) The trial wave function |ΨT〉 is UHF. Using ROHF as |ΨI〉 to initialize the population restores spin symmetry in the walkers. The AFQMC energies have an overall better accuracy and a more systematic behavior, without increasing the cost of the simulation. The system is F2 at cc‐pVDZ level, and RCCSDTQ results are from Musial and Bartlett ()
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Left: Energy per particle of H10 in the complete basis set (CBS) limit, versus the distance R between two consecutive H atoms, from a recent benchmark study (Motta et al., ). Inset: Comparison of the H10 energies per particle, using MRCI+Q+F12 results at CBS as reference (Motta et al., ). The excellent agreement between auxiliary‐field quantum Monte Carlo (AFQMC) and MRCI+Q, MRCI+Q+F12 confirms the uniform and overall high accuracy of the methodology. Right: Equation of state (potential‐energy curve per atom) of the H chain, extrapolated to the joint CBS and thermodynamic limit (Motta et al., )
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Comparison between exact or experimental (line) and auxiliary‐field quantum Monte Carlo (AFQMC; filled symbols) binding energies for several molecules. AFQMC calculations with plane‐waves and pseudopotentials (pw+psp), Gaussian AOs (AO+ae, AO+ecp for all‐electron and effective‐core potential calculations) are marked with filled squares and circles respectively. Green, blue, purple, and red symbols refer to first‐ and second‐row post‐d elements (Al‐Saidi, Krakauer, & Zhang, ), sp‐bonded materials (Al‐Saidi, Krakauer, & Zhang, , ; Al‐Saidi, Zhang, & Krakauer, ; Purwanto et al., ; Purwanto, Zhang, & Krakauer, ; Zhang & Krakauer, ), transition‐metal oxides (Al‐Saidi, Krakauer, & Zhang, ), and dimers respectively (Zhang et al., ). The AFQMC takes as its initial and trial wave function a single determinant from HF or density‐functional theory (DFT); the corresponding results from these independent‐electron calculations are also shown (HF: empty stars, DFT‐LDA: empty triangles, DFT‐GGA: empty diamonds)
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The ratio Nγ/M between Cholesky vectors and basis functions, for atoms He to Kr and dimers He2 to Kr2 at bondlength R = 2aB, using STO‐6G and cc‐pVxZ bases with x = 2–6 and threshold δ = 10−6. Asymptotically Nγ ≃ 7 M. Inset: Dependence of the auxiliary‐field quantum Monte Carlo (AFQMC) energy Eδ on the threshold δ for Cl2 at experimental equilibrium bondlength, R = 3.7566aB. δ ≤ 10−5 yield energies within 1 mEHa of converged value
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Left: Illustration of the downfolding approach for solids (Ma et al., ) using the idea of active space in a molecular system. A set of MOs are first obtained, for example, with density‐functional theory (DFT) or ROHF. A certain number (Mfrozen = 3 in the figure) of orbitals can be frozen and then a certain number of high‐energy orbitals (M − Mactive = 4) can be discarded. Matrix elements for the effective Hamiltonian in the remaining active space are obtained for use in auxiliary‐field quantum Monte Carlo (AFQMC). Right: Computed total energy in the Kr atom relative to the reference value. Red points: Versus the size of the frozen core (cc‐pVDZ basis); blue diamonds: versus the size of truncation to the active space (cc‐pV5Z basis). In both cases RHF orbitals are used as basis of the one‐electron Hilbert space
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Illustration of the back‐propagation algorithm: evolution of the dipole moment is shown versus back‐propagation time using phaseless formalism (BP‐PhL), path restoration (BP‐PRes) and free‐projection (BP, free‐projection) to obtain results of increasing quality. (Reprinted with permission from Motta and Zhang (). Copyright 2017 American Chemical Society) The system is NH3 (STO‐3G level, trigonal pyramid geometry, RNH = 1.07 Å, θHNH = 100.08°)
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Left: Time‐step extrapolation of free‐projection and phaseless auxiliary‐field quantum Monte Carlo (AFQMC) energies for H2O (cc‐pVDZ level, triangular geometry with ROH = 1.8434aB, θHOH = 110.6°), using a population of Nw = 2 × 104 walkers (inset: extrapolation of ph‐AFQMC energies vs. the inverse population size, illustrating population control bias at small Nw. A time step of was used here.) Right: Emergence of the phase problem in free‐projection AFQMC calculations of ethane (STO‐6G basis). (inset: comparison between free‐projection and ph‐AFQMC for short projection time)
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Computer and Information Science > Computer Algorithms and Programming
Electronic Structure Theory > Ab Initio Electronic Structure Methods
Structure and Mechanism > Computational Materials Science

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