Toward a systematic improvement of the fixed-node approximation in diffusion Monte Carlo for solids-A case study in diamond.

While Diffusion Monte Carlo (DMC) is in principle an exact stochastic method for ab initio electronic structure calculations, in practice, the fermionic sign problem necessitates the use of the fixed-node approximation and trial wavefunctions with approximate nodes (or zeros). This approximation introduces a variational error in the energy that potentially can be tested and systematically improved. Here, we present a computational method that produces trial wavefunctions with systematically improvable nodes for DMC calculations of periodic solids. These trial wavefunctions are efficiently generated with the configuration interaction using a perturbative selection made iteratively (CIPSI) method. A simple protocol in which both exact and approximate results for finite supercells are used to extrapolate to the thermodynamic limit is introduced. This approach is illustrated in the case of the carbon diamond using Slater-Jastrow trial wavefunctions including up to one million Slater determinants. Fixed-node DMC energies obtained with such large expansions are much improved, and the fixed-node error is found to decrease monotonically and smoothly as a function of the number of determinants in the trial wavefunction, a property opening the way to a better control of this error. The cohesive energy extrapolated to the thermodynamic limit is in close agreement with the estimated experimental value. Interestingly, this is also the case at the single-determinant level, thus, indicating a very good error cancellation in carbon diamond between the bulk and atomic total fixed-node energies when using single-determinant nodes.

ABINIT: Overview and focus on selected capabilities.

abinit is probably the first electronic-structure package to have been released under an open-source license about 20 years ago. It implements density functional theory, density-functional perturbation theory (DFPT), many-body perturbation theory (GW approximation and Bethe-Salpeter equation), and more specific or advanced formalisms, such as dynamical mean-field theory (DMFT) and the "temperature-dependent effective potential" approach for anharmonic effects. Relying on planewaves for the representation of wavefunctions, density, and other space-dependent quantities, with pseudopotentials or projector-augmented waves (PAWs), it is well suited for the study of periodic materials, although nanostructures and molecules can be treated with the supercell technique. The present article starts with a brief description of the project, a summary of the theories upon which abinit relies, and a list of the associated capabilities. It then focuses on selected capabilities that might not be present in the majority of electronic structure packages either among planewave codes or, in general, treatment of strongly correlated materials using DMFT; materials under finite electric fields; properties at nuclei (electric field gradient, Mössbauer shifts, and orbital magnetization); positron annihilation; Raman intensities and electro-optic effect; and DFPT calculations of response to strain perturbation (elastic constants and piezoelectricity), spatial dispersion (flexoelectricity), electronic mobility, temperature dependence of the gap, and spin-magnetic-field perturbation. The abinit DFPT implementation is very general, including systems with van der Waals interaction or with noncollinear magnetism. Community projects are also described: generation of pseudopotential and PAW datasets, high-throughput calculations (databases of phonon band structure, second-harmonic generation, and GW computations of bandgaps), and the library libpaw. abinit has strong links with many other software projects that are briefly mentioned.

Quantum Package 2.0: An Open-Source Determinant-Driven Suite of Programs.

Quantum chemistry is a discipline which relies heavily on very expensive numerical computations. The scaling of correlated wave function methods lies, in their standard implementation, between O(N5) and O(eN) , where N is proportional to the system size. Therefore, performing accurate calculations on chemically meaningful systems requires (i) approximations that can lower the computational scaling and (ii) efficient implementations that take advantage of modern massively parallel architectures. Quantum Package is an open-source programming environment for quantum chemistry specially designed for wave function methods. Its main goal is the development of determinant-driven selected configuration interaction (sCI) methods and multireference second-order perturbation theory (PT2). The determinant-driven framework allows the programmer to include any arbitrary set of determinants in the reference space, hence providing greater methodological freedom. The sCI method implemented in Quantum Package is based on the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm which complements the variational sCI energy with a PT2 correction. Additional external plugins have been recently added to perform calculations with multireference coupled cluster theory and range-separated density-functional theory. All the programs are developed with the IRPF90 code generator, which simplifies collaborative work and the development of new features. Quantum Package strives to allow easy implementation and experimentation of new methods, while making parallel computation as simple and efficient as possible on modern supercomputer architectures. Currently, the code enables, routinely, to realize runs on roughly 2 000 CPU cores, with tens of millions of determinants in the reference space. Moreover, we have been able to push up to 12 288 cores in order to test its parallel efficiency. In the present manuscript, we also introduce some key new developments: (i) a renormalized second-order perturbative correction for efficient extrapolation to the full CI limit and (ii) a stochastic version of the CIPSI selection performed simultaneously to the PT2 calculation at no extra cost.

Communication: Toward an improved control of the fixed-node error in quantum Monte Carlo: The case of the water molecule.

All-electron Fixed-node DiffusionMonte Carlo calculations for the nonrelativistic ground-state energy of the water molecule at equilibrium geometry are presented. The determinantal part of the trial wavefunction is obtained from a selected Configuration Interaction calculation[Configuration Interaction using a Perturbative Selection done Iteratively (CIPSI) method] including up to about 1.4 × 10(6) of determinants. Calculations are made using the cc-pCVnZ family of basis sets, with n = 2 to 5. In contrast with most quantum Monte Carlo works no re-optimization of the determinantal part in presence of a Jastrow is performed. For the largest cc-pCV5Z basis set the lowest upper bound for the ground-state energy reported so far of -76.437 44(18) is obtained. The fixed-node energy is found to decrease regularly as a function of the cardinal numbern and the Complete Basis Set limit associated with exact nodes is easily extracted. The resulting energy of -76.438 94(12) - in perfect agreement with the best experimentally derived value - is the most accurate theoretical estimate reported so far. We emphasize that employing selected configuration interactionnodes of increasing quality in a given family of basis sets may represent a simple, deterministic, reproducible, and systematic way of controlling the fixed-node error in diffusionMonte Carlo.

Quantum Monte Carlo with very large multideterminant wavefunctions.

An algorithm to compute efficiently the first two derivatives of (very) large multideterminant wavefunctions for quantum Monte Carlo calculations is presented. The calculation of determinants and their derivatives is performed using the Sherman-Morrison formula for updating the inverse Slater matrix. An improved implementation based on the reduction of the number of column substitutions and on a very efficient implementation of the calculation of the scalar products involved is presented. It is emphasized that multideterminant expansions contain in general a large number of identical spin-specific determinants: for typical configuration interaction-type wavefunctions the number of unique spin-specific determinants Ndetσ ( σ=↑,↓) with a non-negligible weight in the expansion is of order O(Ndet). We show that a careful implementation of the calculation of the Ndet -dependent contributions can make this step negligible enough so that in practice the algorithm scales as the total number of unique spin-specific determinants,  Ndet↑+Ndet↓, over a wide range of total number of determinants (here, Ndet up to about one million), thus greatly reducing the total computational cost. Finally, a new truncation scheme for the multideterminant expansion is proposed so that larger expansions can be considered without increasing the computational time. The algorithm is illustrated with all-electron fixed-node diffusion Monte Carlo calculations of the total energy of the chlorine atom. Calculations using a trial wavefunction including about 750,000 determinants with a computational increase of ∼400 compared to a single-determinant calculation are shown to be feasible. © 2016 Wiley Periodicals, Inc.