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Register a new userAccuracy of Quantum Monte Carlo Methods for Point Defects in Solids
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Quantum Monte Carlo approaches such as the diffusion Monte Carlo (DMC) method are among the most accurate many-body methods for extended systems. Their scaling makes them well suited for defect calculations in solids. We review the various approximations needed for DMC calculations of solids and the results of previous DMC calculations for point defects in solids. Finally, we present estimates of how approximations affect the accuracy of calculations for self-interstitial formation energies in silicon and predict DMC values of 4.4(1), 5.1(1) and 4.7(1) eV for the X, T and H interstitial defects, respectively, in a 16(+1)-atom supercell.
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We present a detailed study of the energetics of water clusters (H$_2$O)$_n$ with $n le 6$, comparing diffusion Monte Carlo (DMC) and approximate density functional theory (DFT) with well converged coupled-cluster benchmarks. We use the many-body decomposition of the total energy to classify the errors of DMC and DFT into 1-body, 2-body and beyond-2-body components. Using both equilibrium cluster configurations and thermal ensembles of configurations, we find DMC to be uniformly much more accurate than DFT, partly because some of the approximate functionals give poor 1-body distortion energies. Even when these are corrected, DFT remains considerably less accurate than DMC. When both 1- and 2-body errors of DFT are corrected, some functionals compete in accuracy with DMC; however, other functionals remain worse, showing that they suffer from significant beyond-2-body errors. Combining the evidence presented here with the recently demonstrated high accuracy of DMC for ice structures, we suggest how DMC can now be used to provide benchmarks for larger clusters and for bulk liquid water.
In these lecture notes some applications of Monte Carlo integration methods in Quantum Field Theory - in particular in Quantum Chromodynamics - are introduced and discussed.
We propose the clock Monte Carlo technique for sampling each successive chain step in constant time. It is built on a recently proposed factorized transition filter and its core features include its O(1) computational complexity and its generality. We elaborate how it leads to the clock factorized Metropolis (clock FMet) method, and discuss its application in other update schemes. By grouping interaction terms into boxes of tunable sizes, we further formulate a variant of the clock FMet algorithm, with the limiting case of a single box reducing to the standard Metropolis method. A theoretical analysis shows that an overall acceleration of ${rm O}(N^kappa)$ ($0 ! leq ! kappa ! leq ! 1$) can be achieved compared to the Metropolis method, where $N$ is the system size and the $kappa$ value depends on the nature of the energy extensivity. As a systematic test, we simulate long-range O$(n)$ spin models in a wide parameter regime: for $n ! = ! 1,2,3$, with disordered algebraically decaying or oscillatory Ruderman-Kittel-Kasuya-Yoshida-type interactions and with and without external fields, and in spatial dimensions from $d ! = ! 1, 2, 3$ to mean-field. The O(1) computational complexity is demonstrated, and the expected acceleration is confirmed. Its flexibility and its independence from the interaction range guarantee that the clock method would find decisive applications in systems with many interaction terms.
We report an accurate study of interactions between Benzene molecules using variational quantum Monte Carlo (VMC) and diffusion quantum Monte Carlo (DMC) methods. We compare these results with density functional theory (DFT) using different van der Waals (vdW) functionals. In our QMC calculations, we use accurate correlated trial wave functions including three-body Jastrow factors, and backflow transformations. We consider two benzene molecules in the parallel displaced (PD) geometry, and find that by highly optimizing the wave function and introducing more dynamical correlation into the wave function, we compute the weak chemical binding energy between aromatic rings accurately. We find optimal VMC and DMC binding energies of -2.3(4) and -2.7(3) kcal/mol, respectively. The best estimate of the CCSD(T)/CBS limit is -2.65(2) kcal/mol [E. Miliordos et al, J. Phys. Chem. A 118, 7568 (2014)]. Our results indicate that QMC methods give chemical accuracy for weakly bound van der Waals molecular interactions, comparable to results from the best quantum chemistry methods.
We analyze the problem of eliminating finite-size errors from quantum Monte Carlo (QMC) energy data. We demonstrate that both (i) adding a recently proposed [S. Chiesa et al., Phys. Rev. Lett. 97, 076404 (2006)] finite-size correction to the Ewald energy and (ii) using the model periodic Coulomb (MPC) interaction [L. M. Fraser et al., Phys. Rev. B 53, 1814 (1996); P. R. C. Kent et al., Phys. Rev. B 59, 1917 (1999); A. J. Williamson et al., Phys. Rev. B 55, 4851 (1997)] are good solutions to the problem of removing finite-size effects from the interaction energy in cubic systems, provided the exchange-correlation (XC) hole has converged with respect to system size. However, we find that the MPC interaction distorts the XC hole in finite systems, implying that the Ewald interaction should be used to generate the configuration distribution. The finite-size correction of Chiesa et al. is shown to be incomplete in systems of low symmetry. Beyond-leading-order corrections to the kinetic energy are found to be necessary at intermediate and high densities, and we investigate the effect of adding such corrections to QMC data for the homogeneous electron gas. We analyze finite-size errors in two-dimensional systems and show that the leading-order behavior differs from that which has hitherto been supposed. We compare the efficiency of different twist-averaging methods for reducing single-particle finite-size errors and we examine the performance of various finite-size extrapolation formulas. Finally, we investigate the system-size scaling of biases in diffusion QMC.
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