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Variational and Diffusion Quantum Monte Carlo Calculations with the CASINO Code

R. J. Needs, M. D. Towler, N. D. Drummond, P. Lopez Rios, J. R. Trail

arXiv:2003.06506v1physics.comp-phcond-mat.mtrl-sci

TL;DR

Finite simulation cells complicate calculations intended to describe the thermodynamic limit, and DMC also faces scaling and correlation challenges. The paper reviews CASINO’s QMC methods and applications, summarizing both their capabilities and limitations.

  • Problem

    QMC calculations for extended systems use finite cells, so results intended for the thermodynamic limit require addressing finite-size effects.

    Source: §1 · p. 16

  • Method

    The paper reviews CASINO’s QMC methods and practical developments alongside applications to model and real systems.

    Source: §1 · p. 4, §VI · p. 39

  • Results

    QMC is successful for van der Waals systems and accurate band-gap calculations, while DMC faces difficulty at several-thousand-electron scales and strong static correlation remains challenging.

    Source: §VI · p. 39

  • Takeaways & Limitations

    QMC continues to play a fundamental role among first-principles computational methods by providing data that expand their accuracy and reach.

    Source: §VI · p. 39

    Useful DMC calculations for systems with several thousand electrons will be difficult with current algorithms, and condensed-matter applications depend on finite-size analysis or thermodynamic-limit extrapolation.

    Source: §VI · p. 39

Abstract

from arXiv · show

We present an overview of the variational and diffusion quantum Monte Carlo methods as implemented in the CASINO program. We particularly focus on developments made in the last decade, describing state-of-the-art quantum Monte Carlo algorithms and software and discussing their strengths and their weaknesses. We review a range of recent applications of CASINO.

I. INTRODUCTION

The introduction presents continuum quantum Monte Carlo as a high-accuracy approach to interacting-electron problems, then explains VMC sampling and why correlated samples require care in estimating uncertainty.

I. INTRODUCTION

  • QMC offers high-accuracy calculations for systems beyond the reach of smaller-system methods, while retaining quantifiable random error and positive systematic bias.The introduction notes that systems of up to 2,000 electrons are accessible, including periodic cells for condensed-matter calculations.Source: §I · p. 2
  • VMC samples electron configurations from |Ψ|^2 to estimate operator expectations, whereas DMC projects out the ground-state component through drift, diffusion, and branching processes.For real wave functions, DMC enforces fermionic antisymmetry by pinning the nodal surface to that of the trial wave function.Source: §I · p. 2, §1 · p. 3
  • The article reviews CASINO’s VMC and DMC methods, recent developments, applications, strengths, weaknesses, and future directions.The overview focuses on CASINO and topics relevant to its users.Source: §I · p. 3

1. Sampling a trial wave function

  • Electron-by-electron Metropolis sampling explores configuration space more efficiently than proposing whole-configuration moves, with Gaussian proposals tuned for roughly 50% acceptance.More elaborate gradient-based proposals can increase travel per step, but their derivative-evaluation cost outweighs the efficiency gain in the authors’ experience.Source: §1 · p. 4
  • Blocking-length 1 underestimates the standard error; it rises to a plateau once block averages are effectively independent, while excessively large blocks make the estimate noisy.Reblocking successively averages pairs of data points to reduce serial correlation.Source: §1 · p. 4, Figure 1 · p. 4

2. Slater-Jastrow trial wave functions

The section explains how Slater-Jastrow wave functions describe electronic structure and correlation, and how cusp corrections and backflow address distinct sources of error.

2. Slater-Jastrow trial wave functions

  • A Slater-Jastrow wave function combines a Slater part for the electronic state with a positive Jastrow factor for correlations, which leaves the Slater nodes unchanged.The Jastrow terms describe homogeneous and inhomogeneous electron correlations and can restore the one-electron density altered by the homogeneous two-body term.Source: §2 · pp. 4–5
  • Errors in nodal-surface topology cause static correlation effects that Jastrow factors and backflow cannot fix; multideterminant or pairing wave functions are needed to address them.For many long-range or low-energy energy differences, static correlation effects largely cancel, leaving dynamical correlation as the main accuracy requirement.Source: §2 · p. 5
  • Electron-electron cusp conditions lower local-energy variance, while electron-nucleus cusps are better imposed by correcting cuspless orbitals than by a short-ranged cusp-enforcing function.The local energy varies less near the carbon nucleus with orbital correction than with the short-ranged Jastrow cusp approach.Source: §2 · p. 5, Figure 2 · p. 6
  • In periodic systems, Jastrow truncation lengths must respect Wigner-Seitz-cell limits; cosine expansions provide long-range correlations with simulation-cell symmetry.The one-body cosine term is rarely used, though it may help describe a single charged defect in a simulation supercell.Source: §2 · p. 6
  • Backflow transforms electron coordinates to move Slater nodes; for the homogeneous electron gas it recovers over half the DMC fixed-node error, with a smaller but significant fraction expected in inhomogeneous systems.The transformation uses parameterized displacements dependent on other particles’ positions.Source: §2 · pp. 6–7

3. Optimization of wave-function parameters

The section compares variance-based and energy-based wave-function optimization, describing where each works well and where local-energy variance becomes problematic.

3. Optimization of wave-function parameters

  • Local-energy variance minimization optimizes parameters by repeatedly sampling |Ψ|^2 and reducing the spread of sampled local energies, typically converging in 1–3 cycles.The energy is quadratic and its variance quartic in parameters that enter linearly in the Jastrow exponent, enabling rapid minimization along those directions.Source: §3 · p. 7
  • Variance minimization is robust for Jastrow parameters but can converge slowly for nodal parameters because moving a node through sampled configurations makes local-energy variance diverge.The mean absolute deviation from the median can be advantageous for parameters that affect the nodal surface.Source: §3 · p. 7
  • For an all-electron neon atom, minimizing mean absolute deviation from the median local energy performs relatively well in VMC energy, especially for parameters affecting the nodal surface.The comparison is reported for optimization methods based on different measures of local-energy spread.Source: §3 · p. 7
  • Energy minimization transforms stochastic optimization into a generalized eigenproblem over the initial wave function and its parameter derivatives, iteratively updating the parameters toward lower energy.The method works for both Jastrow and Slater parameters, and importance-sampled DMC is maximally efficient when the VMC energy is minimized.Source: §3 · p. 7

B. Diffusion quantum Monte Carlo

The section explains how fixed-phase DMC projects a trial wave function toward a constrained ground state and approximates that evolution with a population of walkers. It also describes the resulting time-step and population-control biases and the conditions used to manage them.

1. Fixed-phase approximation

  • Fixing the trial wave function’s phase while allowing its modulus to vary defines the fixed-phase approximation, a complex generalization of fixed-node DMC.With an antisymmetric trial wave function, the fixed-phase energy is variational for the lowest-energy antisymmetric state; equality with the fermionic ground-state energy requires the exact phase.Source: §1 · p. 8

2. Imaginary-time Schr¨odinger equation

  • With a nonzero ground-state component initially present, imaginary-time evolution exponentially suppresses excited-state components and projects out the fixed-phase ground state.The ground-state component is selected in the long-time limit when the reference energy is set to the ground-state energy.Source: §2 · p. 8

3. Propagation in imaginary time

  • DMC represents the mixed distribution with walkers that drift, diffuse, and branch or die, approximating evolution through short time steps.The drift-diffusion Green’s function assumes a constant drift velocity between configurations; branching adjusts walker weights while avoiding domination by a single walker.Source: §3 · pp. 9–10
  • Finite time steps introduce an O(τ) error in the mixed distribution, and the energy bias vanishes only as the time step approaches zero.A Metropolis-style accept/reject step reduces the magnitude of Green’s-function time-step errors in practice, although it does not change their formal scaling.Source: §3 · p. 10
  • Hard limiting of the branching local energy restores size consistency and greatly reduces time-step errors, while population control introduces a positive DMC energy bias.The hard-limiting scheme is presented as an alternative to a limiting scheme that produces size-inconsistent energies at fixed time step and substantial time-step errors.Source: §3 · pp. 10–11
  • DMC implicitly enforces the fixed-phase approximation by representing the mixed distribution as a real, nonnegative probability density.Source: §3 · p. 11
  • The mixed energy estimate equals the fixed-phase ground-state energy, which is an upper bound on the fermionic ground-state energy.For operators that do not commute with the Hamiltonian, the mixed estimate differs from the pure expectation value; an extrapolated estimate for local operators has second-order error in the trial-wave-function error.Source: §3 · p. 11
  • Population-control bias is roughly independent of system size per electron and decreases with better trial wave functions, but eliminating it at large sizes can require exponentially growing populations.The bias is typically small and removable by linear extrapolation for moderate systems; the authors advise good trial wave functions and restricting DMC to fewer than 1000–2000 electrons to keep it at bay.Source: Figure 4 · p. 11, §3 · p. 11

4. Time step and decorrelation period

The section relates DMC’s time-step bias and equilibration period to the shortest and longest length scales in a system. It describes extrapolation and equilibration choices for controlling these errors, while noting that shortened equilibration requires case-by-case testing.

4. Time step and decorrelation period

  • In the small-time-step regime, DMC time-step bias is linear in τ, relatively small per particle, and largely cancels in energy differences for similarly accurate trial wave functions.The bias does not become more severe in larger systems; accurate total energies require a sufficiently small step or extrapolation to zero time step.Source: §4 · p. 11, Figure 5 · p. 12
  • DMC is especially challenging when length scales are widely separated: the shortest sets the time step, while the longest sets the decorrelation period.The RMS distance diffused per step should not exceed the shortest scale, whereas equilibration must allow diffusion over the longest relevant scale.Source: §4 · pp. 11–12
  • The fixed-phase equilibration timescale is set by the reciprocal fixed-phase gap, which is at least the true gap and can make DMC equilibrate faster than the actual material gap suggests.A length-scale estimate based on diffusion across the longest relevant scale provides a cautious equilibration period.Source: §4 · p. 12
  • In metals and narrow-gap semiconductors, equilibration steps and decorrelation time increase as the square of the simulation-cell linear size.The cell’s finite periodic size provides an upper bound on the longest length scale in these systems.Source: §4 · p. 12
  • Equilibration shorter than the theoretical decorrelation period may work, but the shortcut cannot be assumed and should be tested case by case.The authors note possible bias cancellation in energy differences and small excited-state components for accurate trial wave functions as reasons shorter periods may work.Source: §4 · p. 12
  • For linear zero-time-step extrapolation, the recommended time-step ratio is 1:4, with eight times as much data gathered at the smaller step.The passage also recommends target populations in a 4:1 ratio and statistics-accumulation steps in a 2:1 ratio.Source: §4 · pp. 12–13

C. Overview of the CASINO software

CASINO is a Fortran 2003 code parallelized with MPI and OpenMP, designed for portability and straightforward porting. Its distribution includes utilities, documentation, examples, and regression tests to support workflows and analysis.

C. Overview of the CASINO software

  • CASINO is written in Fortran 2003 and parallelized with MPI and OpenMP, with strict adherence to the standard except for optional Cray-pointer functionality.It includes its libraries, permits optimized BLAS and LAPACK replacements, and can be compiled serially with a dummy MPI library.Source: C. Overview of the CASINO software · p. 13
  • The program is designed for portability, and porting CASINO to a new machine is usually extremely straightforward.Source: C. Overview of the CASINO software · p. 13
  • CASINO includes utilities for job management and data analysis, plus a manual, examples, and automated regression tests.The utilities are written in Fortran 2003, Bash, C, C++, and Python; tests compare outputs against library files.Source: C. Overview of the CASINO software · p. 13

III. SOME RECENT DEVELOPMENTS IN QMC AND CASINO

CASINO supports flexible Jastrow correlations and compressed multideterminant expansions, with reported gains in variational energy and expansion efficiency. The developments include higher-order correlation terms and a compression method whose size grows sublinearly for the tested C2 expansion.

1. Generalized Jastrow factors

  • CASINO’s Jastrow framework supports terms involving any number of electrons and nuclei, with automatic symmetry and cusp constraints on linear parameters.Higher-order terms can be constructed by increasing the number of electrons or nuclei in a term; these are useful in some systems beyond the generally adequate DTN factor.Source: §1 · p. 13
  • Extrapolation to zero VMC variance accounts for 99.94(2)% of Ps2’s binding energy, compared with 99.53(2)% for the lowest VMC energy.The exact wave function is nodeless, and DMC is exact for this system.Source: §1 · p. 13, Figure 6 · p. 14
  • Adding an electron-electron-nucleus Jastrow term lowers the all-electron silicon total energy by 6.4(9) mHa.The term distinguishes electron configurations on the same or opposite sides of a nucleus, which isotropic functions have difficulty distinguishing.Source: §1 · p. 14

2. Multideterminant wave functions

  • The multideterminant compression method groups determinants differing by a single column and minimizes the compressed expansion through a set-covering problem.Exact set-covering scales nonpolynomially with the original determinant count, but expansions of tens of thousands of determinants typically compress in seconds; a greedy polynomial-time heuristic is also available.Source: §2 · pp. 14–15
  • For C2 at equilibrium, the compressed expansion scales as Ncomp ∝ Ndet^0.895, and compression speedup is close to Ndet/Ncomp.Evaluating orbital linear combinations adds only a tiny fraction of computer time; the reported exponent is α = 0.895 < 1.Source: Figure 8 · p. 15, §2 · p. 15

3. Geminal and pairing wave functions

CASINO implements geminal and pairing wave functions for describing electronic and electron-hole correlation. The section also outlines supercell approaches to finite-size effects, including twist averaging and corrections based on single-particle energies.

3. Geminal and pairing wave functions

  • When norb exceeds the number of spin-up electrons, AGP pairing builds static correlation into the wave function; MAGP combines AGPs with independent parameters.For the HEG, MAGP with backflow produced DMC energies competitive with FCIQMC, albeit at significant computational expense.Source: §3 · p. 15
  • Electron-hole pairing can describe both a two-component fluid and an excitonic fluid, and provides a particularly good description near their phase transition.The implemented pairing orbital includes plane-wave terms and an optimizable polynomial component; when that component is zero and plane-wave count matches particles per spin, the wave function factorizes into electron and hole determinants.Source: §3 · pp. 15–16
  • For a single hole in an electron gas, CASINO implements a specialized pairing orbital with an orbital-specific Jastrow-like prefactor and electron-hole backflow transformation.The orbital is essentially a plane wave of the electron-hole distance.Source: §3 · p. 16

1. Single-particle finite-size effects: momentum quantization

  • Periodic supercells avoid surface effects and restore translational symmetry, but introduce spurious interactions between periodic images; nondiagonal cells can reduce finite-size effects in energy differences.Nondiagonal supercells also allow a broader choice of system sizes.Source: §1 · p. 16
  • Finite supercells replace continuous wavevector integrals with discrete k-point sums, causing quasirandom size-dependent energy fluctuations that are especially large in metals.Twist averaging samples offsets to the k-point grid; DFT energies at matching k points can be used as a covariate to reduce residual bias and noise.Source: §1 · pp. 16–17

2. Long-range finite-size effects

Long-range correlations and the periodic Coulomb interaction produce finite-size errors in QMC energies. The section describes corrections for the Coulomb exchange-correlation energy and for kinetic-energy bias from the truncated long-range Jastrow factor, with some corrections depending on system symmetry.

2. Long-range finite-size effects

  • The finite-cell Ewald interaction differs from 1/r by O(r2/N), producing an O(N^-1) error in Coulomb exchange-correlation energy per particle.The missing long-range exchange-correlation-hole tail contributes a smaller O(N^-2) error per particle.Source: §2 · p. 18
  • A model periodic Coulomb interaction removes the leading-order finite-size error in Coulomb exchange-correlation energy, leaving O(N^-2) residual error from truncating the exchange-correlation hole.The Hartree energy retains the Ewald potential while the Coulomb exchange-correlation energy uses 1/r within the simulation cell.Source: §2 · p. 18
  • A structure-factor finite-size correction uses the small-k behavior of the spherically averaged structure factor, which can be evaluated from QMC data.For cubic systems, the remaining error is O(N^-2); for noncubic systems, spherical averaging does not fully remove the leading-order error.Source: §2 · p. 18
  • Truncating the long-range two-body Jastrow factor in a finite cell biases kinetic energy, and CASINO corrects this by interpolating its Fourier transform.In 3D, the small-k form is fitted to optimized Jastrow terms at the first few nonzero reciprocal-lattice stars; the leading kinetic-energy error is O(N^-1).Source: §2 · pp. 18–19
  • Finite-size LDA corrects single-particle and long-range finite-size errors together and can be combined with twist averaging.The available parameterization neglects long-range effects due to the shape of the simulation cell.Source: §2 · p. 19

3. Finite-size effects in properties other than the total energy

Twist averaging can reduce single-particle finite-size effects in many observables, though some quantities have distinct limitations. For correlated QMC, CASINO describes correlated-electron pseudopotentials and alternative treatments of nonlocal pseudopotential bias, while the eCEPP set discussed is restricted to certain atoms.

3. Finite-size effects in properties other than the total energy

  • Twist averaging reduces single-particle finite-size effects in many expectation values, but momentum density uses the chosen simulation-cell Bloch wavevector to define its sampled k points.Metallic momentum densities can retain slowly decaying finite-size errors near the Fermi edge, while charge densities and pair-correlation functions also have errors from commensurate Friedel oscillations.Source: §3 · p. 19

C. Pseudopotentials for correlated methods

  • Pseudopotentials reduce the effective atomic number and make QMC calculations feasible for all atoms, but applying DFT- or HF-generated pseudopotentials in many-body methods introduces uncontrolled error that is difficult to estimate.For QMC accuracy, pseudopotentials must reproduce scattering and/or excitation properties of the many-body Hamiltonian.Source: C. Pseudopotentials for correlated… · p. 19
  • eCEPPs reproduce valence-core scattering, atomic ionization and excitation energies, and long-range polarization interactions; better than chemical accuracy was demonstrated on a moderately large set of small molecules with CCSD(T).The reported accuracy required care with basis-set error; standard contracted all-electron CCSD(T) Gaussian basis sets were insufficient.Source: C. Pseudopotentials for correlated… · p. 20
  • The eCEPPs discussed achieve better than chemical accuracy for all tested cases but are available only for first-row and transition-metal atoms.The paper notes that further research could improve QMC pseudopotential theory.Source: C. Pseudopotentials for correlated… · p. 20
  • The locality approximation’s error is system-dependent and nonvariational, and can be large relative to fixed-node error.The approximation is exact if the trial wave function equals the many-body wave function.Source: C. Pseudopotentials for correlated… · p. 20
  • The T-move scheme is variational and removes negative local-energy divergences that can cause DMC population explosions; determinant locality also improves stability and resilience to Jastrow-optimization noise.Both schemes are available in CASINO.Source: C. Pseudopotentials for correlated… · p. 20

1. Quasiparticle and excitonic gaps

The section explains how QMC represents quasiparticle and excitonic gaps and treats finite-cell errors, then describes intraband applications and CASINO’s parallel scaling.

1. Quasiparticle and excitonic gaps

  • Quasiparticle energies are obtained by adding or removing an electron, while the excitonic gap is a neutral excitation and lies below the quasiparticle gap by the exciton binding energy.For a quasiparticle gap, the leading finite-size correction is the screened supercell Madelung constant; corrected gaps can then be extrapolated to infinite size.Source: §1 · pp. 21–22, §2 · pp. 22–23

2. Finite-size effects in gaps

  • Finite-cell quasiparticle gaps require subtracting the screened Madelung constant; residual errors can be reduced by extrapolating corrected gaps to infinite system size.For 3D materials the uncorrected leading error falls off as N^-1/3, while the suggested extrapolation assumes a 1/N error after correction.Source: §2 · pp. 22–23

3. Intraband excitations

  • QMC can also study intraband excitations, including electron-mass renormalization by interactions in the 2D homogeneous electron gas.Source: §3 · p. 23

1. Parallelization of QMC algorithms

  • CASINO’s DMC calculations show strong parallel scaling to at least 10,000 cores for silicon and bilayer graphene; short-run checkpointing can bottleneck larger-core tests.The silicon test used 216 electrons; its checkpointing bottleneck was an artifact of the short test runs.Source: §1 · pp. 23–24, Figure 11 · p. 24, Figure 12 · p. 24

2. Scaling of QMC with system size

QMC’s practical system-size scaling reflects both the cost of sampling and the need to equilibrate DMC walker populations. Equilibration and branching impose limits that increasing processor count alone cannot remove.

2. Scaling of QMC with system size

  • With negligible equilibration and O(N^2) walker-move cost, QMC’s cost scales as O(N^3) for a fixed error bar on total energy.Walker moves include electron interactions, Jastrow terms, and orbital evaluations; localized orbitals help avoid an extra factor of N.Source: §2 · pp. 24–25
  • DMC equilibration in metallic systems can take many days above about 800 electrons with one thread per MPI process, while 16 threads raises the estimated limit to about 2000 electrons.These estimates assume a 0.01 Ha^-1 time step and allow larger time steps, incomplete equilibration, or localized-electron systems as possible exceptions.Source: §2 · p. 25
  • For the stated 800-electron, one-thread setup and a 2 × 10^-6 Ha per-electron target error, the maximum useful process count is 6000 and total runtime is eight days.Source: §2 · p. 25
  • DMC’s O(N^3) scaling eventually breaks down around 1000–2000 electrons as branching correlations reduce the effective walker population and produce an exponential scaling wall.Excellent processor scaling does not overcome this limit; accurate condensed-matter results therefore require careful finite-size analysis below 2000 electrons.Source: §2 · p. 26

IV. SOME RECENT APPLICATIONS OF CASINO

The applications include charge-carrier complexes relevant to semiconductor photoluminescence and high-density ferromagnetic electron-gas energies. The latter combine finite-size corrections with FCIQMC data to estimate thermodynamic-limit energies and fixed-node errors.

1. Excitonic complexes

  • Charge-carrier complexes—including excitons, trions, and biexcitons—can produce low-temperature photoluminescence peaks whose positions and persistence depend on their energies and binding energies.Source: §1 · p. 27
  • QMC has been applied to binding energies of excitons, trions, and larger complexes across bulk, quantum-well, quantum-dot, and 2D-material models.Studies also examined low-temperature photoluminescence peaks and inhibition of Bose-Einstein condensation in 2D coupled quantum wells.Source: §1 · p. 27

2. Ground-state energy of the HEG

  • Finite-size corrections suppress quasirandom fluctuations while retaining the leading O(N^-1) behavior, enabling reliable thermodynamic-limit extrapolation of fixed-node energies.The correction uses the Hartree–Fock kinetic and exchange energies; the corrected energies are fitted with a power expansion in N^-1/3.Source: §2 · pp. 27–28, Figure 13 · p. 27
  • FCIQMC energies for HEG systems with N = 15–33 provide estimates of the fixed-node error, which are extrapolated to the thermodynamic limit.Source: §2 · p. 28
  • 1 meV/electron: high-density ferromagnetic HEG energies are estimated accurate to this level, enabling refinement of correlation-energy parameterizations used in DFT.Source: §2 · p. 28, Figure 14 · p. 28

3. Phase diagram of electron-hole bilayers

The review surveys QMC applications to electron-hole bilayers, photoexcited doped semiconductors, and quasiparticle properties of the 2D electron gas. Across these systems, it reports phase behavior, experimentally consistent crossover predictions, and spin-dependent effective-mass trends.

3. Phase diagram of electron-hole bilayers

  • Maezono et al.106 used the condensate fraction to distinguish the two-component fluid from the excitonic BEC and identified a biexcitonic phase at small layer separations and low carrier densities.The pair-correlation function distinguished the biexcitonic phase from the two-component fluid; the DTN Jastrow factor captured exciton-exciton binding.Source: §3 · p. 28
  • The predicted superfluid density range is in order-of-magnitude agreement with experiment despite omitted multivalley effects and the 2D dielectric response.The experimental system has multivalley effects and requires a Rytova-Keldysh interaction rather than the bare Coulomb interaction.Source: §3 · p. 28
  • Electron-hole bilayer superfluid parameters resemble those of non-Coulomb systems across the BCS, BCS-BEC crossover, and BEC regimes, while the BCS stability regime is nearly suppressed.108,176,177The authors suggest this similarity reflects universal physical behavior independent of microscopic interaction details.Source: §3 · p. 28

4. Photoexcitation in doped semiconductors

  • For a single hole in a 2D HEG, the low-density limit of the electron-hole correlation energy is the isolated-trion energy, and the crossover to collective exciton behavior agrees with experimental spectra.The model used VMC and DMC for 86 electrons plus one hole; the crossover was reported in a parameter range consistent with absorption and photoluminescence spectra.Source: §4 · p. 29
  • DMC correlation-energy calculations for the 2D HEG enabled a correction to the quasiparticle gap for finite conduction-electron concentration and electron-hole correlation.The correlation contribution to the energy per particle was calculated as a function of density.Source: §4 · pp. 29–30

5. Quasiparticle effective mass of the 2D HEG

  • DMC effective-mass finite-size errors scale as O(N^-1) in the whole-band fit, rather than the predicted N^-1/4 scaling near the Fermi surface.The authors attribute the difference presumably to fitting the entire band, which has less severe finite-size errors.Source: §5 · p. 31
  • In paramagnetic 2D HEGs, the effective mass remains close to the bare electron mass, while in ferromagnetic HEGs it decreases as density is lowered.The results suggest that the paramagnetic effective mass does not grow rapidly as density is reduced.Source: §5 · p. 31

6. Positrons immersed in electron gases

The review describes QMC calculations for positrons in electron gases and for van der Waals interactions in layered and molecular systems. These applications provide data that differ from some perturbative or DFT results and, in several cases, agree with experiment or distinguish competing structures.

6. Positrons immersed in electron gases

  • For a positron in a 3D HEG, DMC results for relaxation energy, pair correlation, and annihilating-pair momentum density differ appreciably from diagrammatic perturbation theory and improve two-component DFT predictions against experiment.The calculations used supercells of 54 electrons.Source: §6 · p. 32

1. Binding energies of 2D materials

  • DMC gives bilayer graphene binding energies of 17.7(9) meV per atom for AB stacking and 11.5(9) meV per atom for AA stacking, confirming AB as more favorable.The QMC breathing-mode frequency agrees well with Raman measurements.Source: §1 · p. 32

2. Van der Waals interactions at surfaces and between molecules

  • For molecular crystals including water ices, carbon dioxide, ammonia, benzene, naphthalene, and anthracene, DMC achieves chemical accuracy for lattice-formation energies, unlike RPA and MP2 in those studies.RPA with GW single excitations and CCSD(T) are also reported as high-accuracy methods, but are more expensive than DMC and/or require difficult fragment-decomposition approaches.Source: §2 · p. 32
  • For a water molecule adsorbed on LiH(001), DMC agrees well with high-accuracy quantum-chemistry methods, while van der Waals-corrected DFT methods underbind or overbind significantly.MP2 performs well in this case.Source: §2 · p. 33
  • Water is weakly physisorbed on hBN and graphene, with interaction energy about 15 meV larger on hBN; van der Waals DFT overbinds on both materials.Different first-principles methods consistently reproduce the approximately 15 meV difference.Source: §2 · p. 33
  • DMC predicts that confined water's ambient-pressure hexagonal and pentagonal phases transition to a square structure at higher pressure, and its dispersion studies expose inconsistencies in van der Waals-inclusive DFT.The confined-water results are relevant to transmission electron microscopy studies of water between graphene sheets.Source: §2 · p. 33

C. Solid hydrogen

DMC calculations address disagreement between DFT predictions and experiments on high-pressure hydrogen, while combined FCIQMC–DMC approaches exploit the methods’ complementary strengths. The calculations resolve the metallic-structure disagreement, but phase-transition pressure and fixed-node error remain important qualifications.

C. Solid hydrogen

  • Different finite-size corrections agree, and the resulting phase diagram qualitatively matches experiment, including the phase III–IV transition temperature.The calculated phase II–III transition pressure is 75 GPa above experiment, although its isotope dependence is reproduced.Source: C. Solid hydrogen · p. 34
  • DMC finds the metallic structure strongly favored by DFT is not energetically competitive, resolving an outstanding disagreement between theory and experiment.The motivating DFT phase diagram favored metallic structures up to 400 GPa, contrary to experiment below 400 GPa.Source: C. Solid hydrogen · p. 34
  • DMC identifies Pca21 as one of the most energetically stable candidate structures for phase V; many low-energy compressed-hydrogen structures are layered.The Pca21 model was proposed for phase V based on vibrational results from structure searches.Source: C. Solid hydrogen · p. 34
  • DMC band-gap corrections are nearly system-size independent, while calculations show nuclear quantum effects matter and high-pressure exciton binding energies are below 100 meV.The correction can be applied to the infinite-system DFT gap to obtain the DMC gap in the thermodynamic limit.Source: C. Solid hydrogen · p. 34

A. Synergy with FCIQMC

  • FCIQMC recovers static correlation readily, whereas DMC captures dynamical correlation but cannot improve the trial wave function’s static-correlation description.This complementarity motivates several approaches to combining the methods.Source: A. Synergy with FCIQMC · p. 35

1. FCIQMC wave functions in DMC

  • Selecting high-weight FCIQMC determinants provides a multideterminant trial wave function for DMC.For the example gas, the truncated expansion’s coefficients were reoptimized with the Jastrow factor and, where applicable, backflow parameters.Source: §1 · p. 35
  • Within about 35 µHa per electron of the exact FCIQMC energy, backflow DMC converges for the 14-electron gas with 1,273 determinants.This corresponds to an expansion with 30 optimizable coefficients; determinant coefficients were reoptimized with Jastrow and, where applicable, backflow parameters.Source: Figure 26 · p. 35, §1 · p. 35
  • Very large determinant expansions can reduce fixed-node errors, but are expensive and make wave-function optimization problematic.Different expansion sizes may be needed to control errors in energy differences; extrapolation toward the exact energy remains under research.Source: §1 · p. 35

2. Jastrow factors and similarity-transformed FCIQMC

The discussed developments combine Jastrow factors with FCIQMC, use FCIQMC results to benchmark or assist DMC, and explore related advances in interactions, computing, spin-orbit coupling, and vibrational corrections. The passages also identify computational costs and implementation constraints that shape these approaches.

2. Jastrow factors and similarity-transformed FCIQMC

  • Similarity-transformed FCIQMC incorporates a Jastrow factor by transforming the Hamiltonian, but requires six-index integrals and more memory.The transformed Hamiltonian is non-Hermitian; the Jastrow factor reduces the importance of determinants mainly associated with dynamical correlation, potentially yielding compact DMC trial expansions.Source: §2 · pp. 35–36

3. DMC-assisted FCIQMC and benchmarking

  • Combining small-system FCIQMC energies with large-system DMC enables fixed-node-error estimation for the infinite-system correlation energy.For the ferromagnetic electron gas, the fixed-node error is also produced as a byproduct and may inform a density functional correction for inhomogeneous systems.Source: §3 · p. 36

1. Electron-electron pseudopotentials

  • Electron-electron pseudopotentials smooth the Coulomb interaction inside a cutoff while approximating scattering states, with reported speedups up to thirtyfold for homogeneous electron gases.The approach was also demonstrated for lithium and beryllium atoms.Source: §1 · p. 36

2. Current developments in computer architectures

  • Light-touch OpenACC GPU acceleration slows CASINO because small sequential tasks incur data-transfer overhead; efficient GPU QMC requires much of the calculation on the GPU.The authors expect shared-memory CPU–GPU architectures to alleviate some of these issues.Source: §2 · p. 36

1. Spin-orbit coupling

  • A spin-flip DMC scheme treats negative spin-dependent matrix elements exactly to O(τ) and localizes positive ones, producing a positive energy error second order in trial-wave-function error.The approach applies an algorithm similar to the pseudopotential T-move scheme to spin-orbit interactions.Source: §1 · pp. 36–37

2. Inclusion of vibrational effects in ab initio QMC calculations

  • Vibrationally renormalized QMC gaps can sample nuclear configurations from a DFT potential-energy landscape, while proton inclusion requires tailored backflow, Jastrow, and Green’s-function treatments.The proposed gap calculation’s statistical error decreases with the square root of accumulated QMC data, and its cost is not expected to greatly exceed a static-nucleus calculation.Source: §2 · p. 37

D. Atomic forces from QMC calculations

QMC force estimates face fat-tailed distributions whose variance diverges, making standard confidence intervals unusable. Tail regression analytically handles the tails to yield normally distributed estimates, illustrated for LiH forces.

D. Atomic forces from QMC calculations

  • The local force has a |F|−5/2 tail caused by trial-wave-function nodal error, so its expectation is defined but its variance and standard error diverge.Zero-variance force estimators and pseudopotentials alleviate but do not completely solve the problem.Source: D. Atomic forces from QMC calculations · p. 38
  • TRE combines a partial sample mean for the central contribution with analytically computed tail integrals, bypassing the divergent force variance.Its regression coefficients are asymptotically normal, yielding uncertainty proportional to M −1/2.Source: D. Atomic forces from QMC calculations · pp. 38–39
  • −0.54(6) Ha bohr−1 is the TRE force estimate, versus −0.46 Ha bohr−1 from the standard estimator with formally divergent uncertainty (apparent uncertainty 0.18 Ha bohr−1).The example uses VMC forces on lithium in all-electron LiH at a compressed 2-bohr bond length.Source: D. Atomic forces from QMC calculations · p. 38, Figure 27 · p. 39
  • The described TRE requires serially uncorrelated samples, which adds computational time during data accumulation.The authors leave open whether this restriction can be worked around.Source: D. Atomic forces from QMC calculations · p. 39

VI. CONCLUSIONS

  • QMC is successful for van der Waals systems, quasiparticle and excitonic band gaps, and model electron(-hole) systems, including exact solutions for charge-carrier complexes.The conclusions describe QMC as a powerful alternative to many-body GW methods for accurate band gaps.Source: §VI · p. 39
  • Useful DMC calculations for systems with several thousand electrons are expected to remain difficult with current algorithms, and condensed-matter applications depend on finite-size analysis or thermodynamic-limit extrapolation.Strong static correlation also remains challenging because multideterminant expansions introduce arbitrariness into DMC results.Source: §VI · p. 39

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