Scheme for adding electron–nucleus cusps to Gaussian orbitals

We have calculated the ground state energy of the hydrogen molecule using the quantum Monte Carlo (QMC) method of solving the Schrödinger equation, without the use of the Born–Oppenheimer or any other adiabatic approximations. The wave function sampling was carried out in the full 12-dimensional configuration space of the four particles (two electrons and two protons). Two different methods were employed: the diffusion quantum Monte Carlo (DQMC) method and the Green’s function quantum Monte Carlo (GFQMC) method. This computation is very demanding because the configurations must be evolved on the time scale of the electronic motion, whereas the finite nuclear mass effects are resolved accurately only after equilibration on the much slower time scale of the nuclear motion. Thus, a very large number of iterations is required. The calculations were performed on the CM-2 Connection Machine computer, a massively parallel supercomputer. The enormous speedup afforded by the massive parallelism allowed us to complete the computation in a reasonable amount of time. The total energy from the DQMC calculations is −1.163 97±0.000 05 a.u. A more accurate result was obtained from the GFQMC calculations of −1.164 024±0.000 009 a.u. Expressed as a dissociation energy, the GFQMC result is 36 117.9±2.0 cm−1, including the corrections for relativistic and radiative effects. This result is in close agreement with accurate nonadiabatic-relativistic dissociation energies from variational calculations (corrected for radiative effects) in the range of 36 117.9–36 118.1 cm−1 and with the best experimentally determined dissociation energy of McCormack and Eyler 36 118.1±0.2 cm−1.

We report a study of the electronic dissociation energy of the water dimer using quantum Monte Carlo techniques. We have performed variational quantum Monte Carlo and diffusion quantum Monte Carlo (DMC) calculations of the electronic ground state of the water monomer and dimer using all-electron and pseudopotential approaches. We have used Slater-Jastrow trial wave functions with B3LYP type single-particle orbitals, into which we have incorporated backflow correlations. When backflow correlations are introduced, the total energy of the water monomer decreases by about 4-5 mhartree, yielding a DMC energy of -76.428 30(5) hartree, which is only 10 mhartree above the experimental value. In our pseudopotential DMC calculations, we have compared the total energies of the water monomer and dimer obtained using the locality approximation with those from the variational scheme recently proposed by Casula [Phys. Rev. B 74, 161102(R) (2006)]. The time step errors in the Casula scheme are larger, and the extrapolation of the energy to zero time step always lies above the result obtained with the locality approximation. However, the errors cancel when energy differences are taken, yielding electronic dissociation energies within error bars of each other. The dissociation energies obtained in our various all-electron and pseudopotential calculations range between 5.03(7) and 5.47(9) kcalmol and are in good agreement with experiment. Our calculations give monomer dipole moments which range between 1.897(2) and 1.909(4) D and dimer dipole moments which range between 2.628(6) and 2.672(5) D.

We report on the results of both the diffusion quantum Monte Carlo (DMC) and reptation quantum Monte Carlo (RMC) methods on the potential energy curve of the helium dimer. We show that it is possible to obtain a highly accurate description of the helium dimer. An improved stochastic reconfiguration technique is employed to optimize the many-body wave function, which is the starting point for highly accurate simulations based on the DMC and RMC methods. We find that the results of these methods are in excellent agreement with the best theoretical results at short range, especially the recently developed RMC method, yield particularly accurate results with reduced statistical error, which gives very excellent agreement across the whole potential curve. For the equilibrium internuclear distance of 5.6 bohrs, the calculated total energy with RMC method is -5.807 483 599+/-0.000 000 016 hartree and the corresponding well depth is -11.003+/-0.005 K.

ArC 2H 2: a challenging system for ab initio calculations

Atomic forces are calculated for first-row monohydrides and carbon monoxide within electronic quantum Monte Carlo (QMC). Accurate and efficient forces are achieved by using an improved method for moving variational parameters in variational QMC. Newton's method with singular value decomposition (SVD) is combined with steepest-descent (SD) updates along directions rejected by the SVD, after initial SD steps. Dissociation energies in variational and diffusion QMC agree well with the experiment. The atomic forces agree quantitatively with potential-energy surfaces, demonstrating the accuracy of this force procedure. The harmonic vibrational frequencies and anharmonicity constants, derived from the QMC energies and atomic forces, also agree well with the experimental values.

We studied the low-pressure (0–10 GPa) phase diagram of crystalline benzene using quantum Monte Carlo and density functional theory (DFT) methods. We performed diffusion quantum Monte Carlo (DMC) calculations to obtain accurate static phase diagrams as benchmarks for modern van der Waals density functionals. Using density functional perturbation theory, we computed the phonon contributions to the free energies. Our DFT enthalpy-pressure phase diagrams indicate that the Pbca and P21/c structures are the most stable phases within the studied pressure range. The DMC Gibbs free-energy calculations predict that the room temperature Pbca to P21/c phase transition occurs at 2.1(1) GPa. This prediction is consistent with available experimental results at room temperature. Our DMC calculations give 50.6 ± 0.5 kJ/mol for crystalline benzene lattice energy.

A benchmark study of atomization energies is reported for 22 hydrocarbons using single determinant trial functions in the diffusion Monte Carlo (DMC) variant of the quantum Monte Carlo (QMC) method. The DMC atomization energies are compared to experiment, a complete basis set approach (CBS-Q), density functional theory with the B3LYP functional, and coupled-cluster singles, doubles and perturbative triples, CCSD(T), methods. Comparison of the DMC results to experiment yields a mean absolute deviation of 1.9kcalmol−1, which is comparable to that of the B3LYP/cc-pVQZ (1.7kcalmol−1) level of theory, but less accurate than that of CBS-Q (1.1kcalmol−1). DMC performs similarly for both closed-shell and open-shell molecules with mean absolute deviations of 2.1kcalmol−1 for the former and 1.7kcalmol−1 for the latter systems. The use of experimental zero-point energies (ZPEs), rather than scaled B3LYP ZPEs, is found to have negligible effect on DMC atomization energies. The latter reported here provide a baseline from which further improvement in the calculation of DMC atomization energies, including the use of multi-determinant and other trial function improvements, can be measured.

We use the diffusion quantum Monte Carlo (DMC) method to calculate the ground-state phase diagram of solid molecular hydrogen and examine the stability of the most important insulating phases relative to metallic crystalline molecular hydrogen. We account for finite-size errors by combining the use of twist-averaged boundary conditions with corrections obtained using the Kwee–Zhang–Krakauer functional in density functional theory. To study the closure of the band gap with increasing pressure, we perform quasi-particle many-body calculations using the GW method. In the static approximation, our DMC simulations indicate a transition from the insulating Cmca-12 structure to the metallic Cmca structure at around 375 GPa. The GW band gap of Cmca-12 closes at roughly the same pressure. In the dynamic DMC phase diagram, which includes the effects of zero-point energy in the quasi-harmonic approximation, the Cmca-12 structure remains stable up to 430 GPa, well above the pressure at which the GW band gap closes. Our results predict that the semimetallic state observed experimentally at around 360 GPa (2012 Phys. Rev. Lett. 108, 146402) may correspond to the Cmca-12 structure near the pressure at which the band gap closes. The dynamic DMC phase diagram indicates that the hexagonal-close-packed P63/m structure, which has the largest band gap of the insulating structures considered, is stable up to 220 GPa. This is consistent with recent x-ray data taken at pressures up to 183 GPa (2010 Phys. Rev. B 82 060101), which also reported a hexagonal-close-packed arrangement of hydrogen molecules.

Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic signproblem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.

Using fixed-node diffusion quantum Monte Carlo (DMC) simulation we investigate the structural properties and energetics of the linear and cyclic carbon clusters ${\mathrm{C}}_{n}$ for $n\ensuremath{\le}10$. We calculate the binding energy, the electron correlation energy, the dissociation energy, and the second difference in energy. We also present an analysis of the structural properties of the clusters. It is found that the bond lengths, binding energies, and dissociation energies obtained from the DMC calculations are in excellent agreement with the available experimental results. The electron-correlation contribution to the binding energy indicates that in the case of the linear isomers, the clusters of odd-number size are relatively more favored than their neighbors of even-number size, whereas for the cyclic isomers, we do not observe the oscillation pattern. In the range of cluster size under investigation, we find that the electron-correlation impact in the binding energy of the cyclic clusters is larger than that of the corresponding linear ones, varying from $30%$ to $40%$ of the binding energy values. The electron correlation is also essential to the stability of the clusters enhancing by up to $52%$ their dissociation energy. A comparative analysis of the dissociation energy and second difference in energy indicates that the linear isomers ${\mathrm{C}}_{3}$ and ${\mathrm{C}}_{5}$ are the most stable ones.

In a recent paper, Leung et al. [Phys. Rev. Lett. 83, 2351 (1999)] reported a theoretical study where they calculated the formation energy for a neutral silicon self‐interstitial to be 4.9 eV using the diffusion quantum Monte Carlo (DMC) method, 50% (30%) higher than the LDA (GGA) result of “usual” ab‐initio calculations. In this paper, we show that gap and finite‐size corrections that have been proposed in the past might be able to explain this difference, as long as the defect levels for point defects are well known. We show that B‐interstitial systems are more difficult in that sense, which seriously questions the generality of the approach. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A quantum Monte Carlo (QMC) benchmark study of heats of formation at 298 K and bond dissociation energies (BDEs) of 22 small hydrocarbons is reported. Diffusion Monte Carlo (DMC) results, obtained using a simple product trial wavefunctions consisting of a single determinant and correlation function, are compared to experiment and to other theory including a version of complete basis set theory (CBS‐Q) and density functional theory (DFT) with the B3LYP functional. For heats of formation, the findings are a mean absolute deviation from experiment of 1.2 kcal/mol for CBS‐Q, 2.0 kcal/mol for B3LYP, and 2.2 kcal/mol for DMC. The mean absolute deviation of 31 BDEs is 2.0 kcal/mol for CBS‐Q, 4.2 kcal/mol for B3LYP, and 2.5 kcal/mol for DMC. These findings are for 17 BDEs of closed‐shell molecules that have mean absolute deviations from experiment of 1.7 kcal/mol (CBS‐Q), 4.0 kcal/mol (B3LYP), and 2.2 kcal/mol (DMC). The corresponding results for the 14 BDEs of open‐shell molecules studied are 2.4 kcal/mol (CBS‐Q), 4.3 kcal/mol (B3LYP), and 2.9 kcal/mol (DMC). The DMC results provide a baseline from which improvement using multideterminant trial functions can be measured. © 2005 Wiley Periodicals, Inc. Int J Chem Kinet 37: 583–592, 2005

The applicability of Green's function (GF) and Feynman path-integral quantum Monte Carlo (QMC) methods for the simulation of cyclic networks with (4n + 2) and 4n (n = 1, 2, 3, …) electrons is analysed. Both QMC techniques are employed in simulations on the basis of the simple Huckel Hamiltonian which is exclusively defined by nearest-neighbour hopping elements. In addition we have used the Pariser-Parr-Pople (PPP) Hamiltonian to perform GF QMC simulations. The electronic energies E derived by the QMC methods are compared either with Huckel molecular orbital (HMO) results or exact configuration interaction data where (π) electronic correlations are fully taken into account. A sign problem occurs in QMC simulations of 4n annulenes. This leads to an error in the total energy in the standard formulations of the employed QMC techniques, which is enhanced with decreasing ring size. A simple modification in the QMC formalisms is suggested to avoid the numerical uncertainties caused by the sign problem in 4n annu...

Electronic structure calculations by quantum Monte Carlo methods

This thesis is divided into two parts. In the first part we study quantum phases in bosonic systems in an optical lattice by using numerical unbiased Quantum Monte Carlo (QMC) method. Ultracold atoms experiments in optical lattices provide a possibility to simulate the materials of condensed matter in a clean and well-controled way. We focus on two components bosonic systems where exotic phases such as pair-superfluid and pair-supersolid appear. In the second part we provide numerical methods based on QMC simulations to study the entanglement properties of strong correlated systems. By employing the replica trick, we reconstruct the entanglement spectrum from the Renyi entropies. Furthermore, we study the trace of the power of the partial transposed reduced density matrix, which is related to the entanglement measurement negativity for mixed states. In the following we briefly summarize each chapter. In Chapter 1 we very briefly introduce the cold atom systems and the connection with QMC simulations. In Chapter 2 we review the QMC method based on the path integral (world line) representation. We focus on the directed worm algorithm which is an efficient algorithm with global updtae for bosonic and spin systems. All the following methods in this thesis – the method of two component bosonic systems, the reconstruction of entanglement spectrum and the measuring of partial transposed quantities, are based on directed worm algorithm introduced in this chapter. In Chapter 3 we discuss two component bosons in a square lattice. We show that the interspecies attraction and nearest-neighbor intraspecies repulsion result in the pair-supersolid phase, where a diagonal solid order coexists with an off-diagonal pair-superfluid order. The quantum and thermal transitions out of the pair-supersolid phase are characterized. It is found that there is a direct first-order transition from the pair-supersolid phase to the double-superfluid phase without an intermediate region. Furthermore, the melting of the pair-supersolid occurs in two steps. Upon heating, first the pair-superfluid is destroyed via a Kosterlitz-Thouless transition, then the solid order melts via an Ising transition. In Chapter 4 we represent a new method to reconstruct a subset of the entanglement spectrum of quantum many body systems by QMC, where the method can in principle be applied to two or higher dimension. The approach builds on the replica trick to evaluate particle number resolved traces of the first n of powers of a reduced density matrix. From this information we reconstruct first n entanglement spectrum levels using a polynomial root solver. We illustrate the power and limitations of the method by an application to the extended Bose-Hubbard model in one dimension where we are able to resolve the quasidegeneracy of the entanglement spectrum in the Haldane-insulator phase. In general, the method is able to reconstruct the largest few eigenvalues in each symmetry sector. In Chapter 5 we devise a Quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. In particular, we study several scale invariant combinations of the moments for the 1D hard-core boson model. For two adjacent intervals unusual finite size corrections are present, showing parity effects that oscillate with a filling dependent period. For large chains we find perfect agreement with conformal field theory (CFT) calculations. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. The finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderate large sizes.