A new quantum Monte Carlo (QMC) method is used to calculate exact, full configuration-interaction (FCI) energies of the neutral and cationic elements from Li to Mg, in a family of commonly used basis sets. Annihilation processes between positive and negative walkers enable the exact N-electron wave function to emerge as a linear superposition of the (factorially large) space of Slater determinants, with individual determinants being stochastically sampled. As a result, extremely large spaces (exceeding 10(15) determinants) become accessible for FCI calculations. No fixed-node approximation is necessary, and the only remaining source of error is the one-electron basis set, which can be systematically reduced by enlargement of the basis set. We have investigated the family of aug-cc-pVXZ Dunning basis sets up to X=5. The resulting ionization potentials are--with one exception (Na)--consistently accurate to within chemical accuracy. The anomalous case of Na suggests that its basis set may be improvable. Extrapolation schemes are examined as a way of further improving the values obtained, and although an improvement is seen in the mean-absolute error, the results of extrapolation are not uniformly better than the largest basis set calculations reported. More generally, these results demonstrate the utility of the QMC method to provide FCI energies for realistic systems and basis sets.
A new quantum Monte Carlo (QMC) method is used to calculate exact, full configuration-interaction (FCI) energies of the neutral and cationic elements from Li to Mg, in a family of commonly used basis sets. Annihilation processes between positive and negative walkers enable the exact N-electron wave function to emerge as a linear superposition of the (factorially large) space of Slater determinants, with individual determinants being stochastically sampled. As a result, extremely large spaces (exceeding 10(15) determinants) become accessible for FCI calculations. No fixed-node approximation is necessary, and the only remaining source of error is the one-electron basis set, which can be systematically reduced by enlargement of the basis set. We have investigated the family of aug-cc-pVXZ Dunning basis sets up to X=5. The resulting ionization potentials are--with one exception (Na)--consistently accurate to within chemical accuracy. The anomalous case of Na suggests that its basis set may be improvable. Extrapolation schemes are examined as a way of further improving the values obtained, and although an improvement is seen in the mean-absolute error, the results of extrapolation are not uniformly better than the largest basis set calculations reported. More generally, these results demonstrate the utility of the QMC method to provide FCI energies for realistic systems and basis sets.
By performing a stochastic dynamic in a space of Slater determinants, the full configuration interaction quantum Monte Carlo (FCIQMC) method has been able to obtain energies which are essentially free from systematic error to the basis set correlation energy, within small and systematically improvable error bars. However, the weakly exponential scaling with basis size makes converging the energy with respect to basis set costly and in larger systems, impossible. To ameliorate these basis set issues, here we use perturbation theory to couple the FCIQMC wavefunction to an explicitly correlated strongly orthogonal basis of geminals, following the [2](R12) approach of Valeev et al. The required one- and two-particle density matrices are computed on-the-fly during the FCIQMC dynamic, using a sampling procedure which incurs relatively little additional computation expense. The F12 energy corrections are shown to converge rapidly as a function of sampling, both in imaginary time and number of walkers. Our pilot calculations on the binding curve for the carbon dimer, which exhibits strong correlation effects as well as substantial basis set dependence, demonstrate that the accuracy of the FCIQMC-F12 method surpasses that of all previous FCIQMC calculations, and that the F12 correction improves results equivalent to increasing the quality of the one-electron basis by two cardinal numbers.
The full configuration interaction quantum Monte Carlo (FCIQMC) method, as well as its "initiator" extension (i-FCIQMC), is used to tackle the complex electronic structure of the carbon dimer across the entire dissociation reaction coordinate, as a prototypical example of a strongly correlated molecular system. Various basis sets of increasing size up to the large cc-pVQZ are used, spanning a fully accessible N-electron basis of over 10(12) Slater determinants, and the accuracy of the method is demonstrated in each basis set. Convergence to the FCI limit is achieved in the largest basis with only O[10(7)] walkers within random errorbars of a few tenths of a millihartree across the binding curve, and extensive comparisons to FCI, CCSD(T), MRCI, and CEEIS results are made where possible. A detailed exposition of the convergence properties of the FCIQMC methods is provided, considering convergence with elapsed imaginary time, number of walkers and size of the basis. Various symmetries which can be incorporated into the stochastic dynamic, beyond the standard abelian point group symmetry and spin polarisation are also described. These can have significant benefit to the computational effort of the calculations, as well as the ability to converge to various excited states. The results presented demonstrate a new benchmark accuracy in basis-set energies for systems of this size, significantly improving on previous state of the art estimates.
Full configuration interaction quantum Monte Carlo for accurate ab initio nuclear structure calculations: algorithms and calculation details
Full configuration interaction quantum Monte Carlo (FCIQMC) is a stochastic many-body solver that has been widely applied to electronic, molecular, and condensed-matter systems. In this work we apply FCIQMC to ab initio nuclear structure calculations using interactions derived from chiral effective field theory. We describe the algorithm in detail, including imaginary-time propagation, excitation generation, estimator choices, the initiator approximation with adaptive shift correction, and reduced-density-matrix (RDM) sampling. Benchmark calculations in small model spaces, where deterministic full configuration interaction (FCI) results are available, validate the stochastic calculation of energies, radii, and RDM-based pure estimators. For large model spaces, we analyze the residual finite-walker bias through systematic walker-number convergence and infinite-walker extrapolations. We also demonstrate that FCIQMC can be extended beyond ground-state calculations by computing the low-lying spectrum of 6Li.
Development of exponentially scaling methods has seen great progress in tackling larger systems than previously thought possible. One such technique, full configuration interaction quantum Monte Carlo, is a useful algorithm that allows exact diagonalization through stochastically sampling determinants. The method derives its utility from the information in the matrix elements of the Hamiltonian, along with a stochastic projected wave function, to find the important parts of Hilbert space. However, the stochastic representation of the wave function is not required to search Hilbert space efficiently, and here we describe a highly efficient deterministic method that can achieve chemical accuracy for a wide range of systems, including the difficult Cr2 molecule. We demonstrate for systems like Cr2 that such calculations can be performed in just a few cpu hours which makes it one of the most efficient and accurate methods that can attain chemical accuracy for strongly correlated systems. In addition our method also allows efficient calculation of excited state energies, which we illustrate with benchmark results for the excited states of C2.
Development of exponentially scaling methods has seen great progress in tackling larger systems than previously thought possible. One such technique, full configuration interaction quantum Monte Carlo, is a useful algorithm that allows exact diagonalization through stochastically sampling determinants. The method derives its utility from the information in the matrix elements of the Hamiltonian, along with a stochastic projected wave function, to find the important parts of Hilbert space. However, the stochastic representation of the wave function is not required to search Hilbert space efficiently, and here we describe a highly efficient deterministic method that can achieve chemical accuracy for a wide range of systems, including the difficult Cr2 molecule. We demonstrate for systems like Cr2 that such calculations can be performed in just a few cpu hours which makes it one of the most efficient and accurate methods that can attain chemical accuracy for strongly correlated systems. In addition our method also allows efficient calculation of excited state energies, which we illustrate with benchmark results for the excited states of C2.
Highly accurate results for the homogeneous electron gas (HEG) have only been achieved to date within a diffusion Monte Carlo (DMC) framework. Here, we introduce a recently developed stochastic technique, full configuration interaction quantum Monte Carlo (FCIQMC), which samples the exact wave function expanded in plane-wave Slater determinants. Despite the introduction of a basis-set incompleteness error, we obtain a finite-basis energy, which is significantly and variationally lower than any previously published work for the 54-electron HEG at rs= 0.5 a.u., in a Hilbert space of 10108 Slater determinants. At this value of rs, as well as of 1.0 a.u., we remove the remaining basis-set incompleteness error by extrapolation, yielding results comparable to state-of-the-art DMC backflow energies. In doing so, we demonstrate that it is possible to yield highly accurate results with the FCIQMC method in periodic systems.