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Multi-Determinant Wave-functions in Quantum Monte Carlo
M. A. Morales, J. McMinis, B. K. Clark, J. Kim, G. Scuseria
TL;DR
QMC has low scaling for electronic systems, but controlling fixed-node error remains difficult. This Perspective applies large multi-determinant expansions to QMC and benchmarks them on first-row dimers and the G1 molecules. The reported QMC results outperform several traditional QC methods and approach explicitly correlated CCSD(T) accuracy.
Problem
Fixed-node errors can be large and difficult to control, limiting QMC accuracy despite its promise for electronic systems.
Source: §I · p. 3Method
The paper applies systematically expanded multi-determinant trial wave functions in QMC, using an efficient scheme for calculations with thousands of determinants.
Source: §II · p. 4Results
QMC was more accurate than MP2, CCSD(T), and various DFT approximations for first-row dimers and G1 molecules; the G1 atomization-energy MAE was approximately 0.8 kcal/mol.
Source: §V · pp. 12–13Takeaways & Limitations
Increasing determinant count improves results and offers extrapolation and convergence checks, giving more control over fixed-node error than single-shot QMC calculations.
Source: §V · p. 13For first-row systems, errors increase with atomic number, partly because core-region basis quality matters more and the SOCI truncation is not size extensive.
Source: §III · p. 8
Abstract
from arXiv · showhide
Quantum Monte Carlo (QMC) methods have received considerable attention over the last decades due to their great promise for providing a direct solution to the many-body Schrodinger equation in electronic systems. Thanks to their low scaling with number of particles, QMC methods present a compelling competitive alternative for the accurate study of large molecular systems and solid state calculations. In spite of such promise, the method has not permeated the quantum chemistry community broadly, mainly because of the fixed-node error, which can be large and whose control is difficult. In this Perspective, we present a systematic application of large scale multi-determinant expansions in QMC, and report on its impressive performance with first row dimers and the 55 molecules of the G1 test set. We demonstrate the potential of this strategy for systematically reducing the fixed-node error in the wave function and for achieving chemical accuracy in energy predictions. When compared to traditional quantum chemistry methods like MP2, CCSD(T), and various DFT approximations, the QMC results show a marked improvement over all of them. In fact, only the explicitly-correlated CCSD(T) method with a large basis set produces more accurate results. Further developments in trial wave functions and algorithmic improvements appear promising for rendering QMC as the benchmark standard in large electronic systems.
I. INTRODUCTION
Quantum Monte Carlo offers a low-scaling stochastic alternative for electronic systems, but fixed-node errors from trial wave functions limit its accuracy. This work uses large multi-determinant expansions to improve trial wave functions and benchmark QMC against established quantum-chemistry methods.
I. INTRODUCTION
- Exact fermionic Schrödinger-equation methods scale exponentially with system size, so larger systems of practical interest require approximate schemes.The introduction contrasts this challenge with the use of stochastic methods for quantum problems.Source: §I · p. 2
- CCSD(T) is the QC “gold standard” for molecular ground states but is most reliable for weakly correlated systems; multi-reference CC methods for strong correlations are less developed and limited to relatively small systems.The passage associates strong correlations with degeneracies or near-degeneracies in the reference determinant.Source: §I · p. 2
- Fixed-node QMC accuracy depends on the trial wave function, and a simple ansatz can yield results that do not compare satisfactorily with experiment.The fixed-node approximation is used to make QMC practical despite the fermion sign problem.Source: §I · p. 3
- Large multi-determinant QMC calculations address accuracy by improving trial wave functions, with benchmarks on first-row molecules and comparisons to SOCI and CCSD(T)-R12.The authors describe the expansion as a systematically improvable parameter and note that QMC determinants combined with a bosonic Jastrow factor are more powerful than the same number of QC determinants.Source: §I · p. 3
II. TRIAL WAVE FUNCTION AND OPTIMIZATION METHOD
The method uses a multi-determinant trial wave function multiplied by a Jastrow factor, with its parameters optimized using linear optimization. Efficient evaluation makes expansions of thousands of determinants practical.
II. TRIAL WAVE FUNCTION AND OPTIMIZATION METHOD
- A computational scheme enables routine QMC calculations with thousands of determinants at a small increase in computational cost.The method is presented as allowing systematic expansion of the determinant component.Source: §II · p. 4
- The trial wave function uses a Jastrow factor and a weighted expansion of determinant configurations, with the Jastrow containing one-, two-, and three-body correlation terms.Its determinant matrices are built from single-particle orbitals selected from an orbital set.Source: §II · p. 4
- The Jastrow parameters and determinant coefficients are optimized by linear optimization, which diagonalizes the Hamiltonian in a subspace formed from linearized parameter derivatives.For a trial function linear in its parameters and without a Jastrow factor, this subspace is complete and the eigenvalue solution gives the lowest energy solution.Source: §II · p. 4
- Typically, fewer than ten iterations are required, with each iteration using an increasing number of Monte Carlo steps.Line minimization or other criteria can be used to rescale the lowest-energy eigenvector when nonlinear parameters are included.Source: §II · pp. 4–5
III. FIRST ROW DIMERS
For first-row dimers, systematically expanding the multi-determinant (MSD) set moves DMC energies toward near-exact results, yielding the most accurate QMC equilibrium-dimer energies reported to date.
III. FIRST ROW DIMERS
- SOCI natural orbitals use all electrons in 10 active orbitals and up to 40 virtual orbitals; coefficient-cutoff-selected CSFs form the expansion.The Jastrow includes one-, two-, and three-body terms, with all parameters optimized simultaneously.Source: §III · p. 5
- At least 99% of the correlation energy is recovered by extrapolated DMC energies across the reported first-row atom and dimer cases.Source: §III · p. 5, Table I · p. 6
- Valence-only FVCAS configurations limit accuracy; higher virtual states contribute to fixed-node-error reduction, while backflow improves energies and variances.Source: §III · pp. 6–7
- Systematic expansion into larger MSD sets converges toward near-exact DMC energies, the most accurate QMC total energies for first-row dimers at equilibrium reported to date.Valence-only FVCAS configurations yield limited accuracy.Source: Figure 2 · p. 7, §III · p. 8
- Errors increase with atomic number as core-region basis quality matters more, while non-size-extensive external single and double excitations weigh more in systems with more electrons.Better all-electron QMC basis sets could reduce the core-basis effect, but were not pursued here.Source: §III · p. 8
IV. G1 SET
For the 55 G1 molecules, the study applies cutoff-controlled multi-determinant DMC and compares total and atomization energies with quantum-chemistry methods. DMC has lower MAEs than the traditional methods considered except CCSD(T)-F12, while the authors identify error cancellation as a concern for atomization energies.
IV. G1 SET
- The calculations use MP2 natural orbitals, cutoff-selected CISDTQ configurations, a 1-, 2-, and 3-body Jastrow factor, and simultaneous parameter optimization.The cutoff is based on the magnitude of the wave-function expansion coefficients.Source: §IV · p. 8
- Across total- and atomization-energy MAEs, DMC outperforms the compared traditional methods; only CCSD(T)-F12 does better.DMC results also have a smaller dependence on basis sets than traditional quantum-chemistry methods.Source: Figure 4 · p. 10, §IV · p. 10, Figure 5 · p. 11
- MAE falls from ∼18 mHa for single-determinant DMC to ∼3 mHa at cutoff 0.001 and ∼2 mHa for extrapolated values.For SO2, the MSD expansion recovers ∼35 mHa, or ∼88% of the fixed-node error; improvements vary across molecules.Source: §IV · p. 10
- DMC scales as N^3 for energy per electron in its basic implementation, with possible improvements to N^2 or linear scaling.The authors see potential for QMC as a community standard in large molecular systems and periodic calculations.Source: §IV · p. 11
- Comparisons with Petruzielo et al. [51] need care for hydrogen-containing molecules because the hydrogen treatments differ; atomization accuracy also relies mainly on error cancellation.The authors say a size-consistency scheme is required for accurate atomization energies, particularly for larger molecules.Source: §IV · p. 12
V. CONCLUSIONS
QMC with an improved multi-Slater-Jastrow wave function produced competitive results on atoms and small molecules, outperforming several traditional QC methods on first-row dimers and G1 molecules. The determinant-count-improvable ansatz offers a way to assess and reduce fixed-node error, while larger-system chemical accuracy remains a future aim.
V. CONCLUSIONS
- For first-row dimers and G1 molecules, QMC was strictly more accurate than MP2, CCSD(T), and various DFT approximations, and competitive with CCSD(T)-R12.Source: §V · p. 12
- For first-row atoms and dimers, QMC recovered more than 99% of the correlation energy in all cases.Source: §V · pp. 12–13
- For the G1 set, the MAE in total energy was approximately 2 mHa and the MAE in atomization energies was approximately 0.8 kcal/mol.Source: §V · p. 13
- Increasing the number of determinants improves results and supports extrapolation, accuracy refinement, and estimates of convergence.Together with QMC's variational upper bound, this provides more control over fixed-node error than single-shot QMC calculations.Source: §V · p. 13
- The authors view chemical accuracy for larger molecules and condensed systems as a longer-term goal, supported by QMC's low scaling and natural parallelization and possible further ansatz improvements.Source: §V · p. 13