Stochastic effective core potentials, improving efficiency using a spin-dependent core definition.

Numerically cheap single-core subsamplings have been used to build improved estimators for molecular properties in the variational Monte Carlo framework. The resulting estimators depend only on the valence electron positions and can be thought of as an exact effective core potential for the total energy. We are proposing a spin-dependent core definition which enables exploiting these single-core subsamplings (or sidewalks) not only to decrease the variance of the estimators but also to restrict the main variational Monte Carlo dynamics to the valence region. This results mainly in a simplification of the algorithm and additionally in a gain in efficiency as illustrated on alkane chains and silicon clusters. An evaluation of the efficiency on transition metal systems is done using cobalt clusters, a gain of up to two orders of magnitude is achieved compared to a standard all-electron calculation.

Curing basis-set convergence of wave-function theory using density-functional theory: A systematically improvable approach.

The present work proposes to use density-functional theory (DFT) to correct for the basis-set error of wave-function theory (WFT). One of the key ideas developed here is to define a range-separation parameter which automatically adapts to a given basis set. The derivation of the exact equations are based on the Levy-Lieb formulation of DFT, which helps us to define a complementary functional which corrects uniquely for the basis-set error of WFT. The coupling of DFT and WFT is done through the definition of a real-space representation of the electron-electron Coulomb operator projected on a one-particle basis set. Such an effective interaction has the particularity to coincide with the exact electron-electron interaction in the limit of a complete basis set, and to be finite at the electron-electron coalescence point when the basis set is incomplete. The non-diverging character of the effective interaction allows one to define a mapping with the long-range interaction used in the context of range-separated DFT and to design practical approximations for the unknown complementary functional. Here, a local-density approximation is proposed for both full-configuration-interaction (FCI) and selected configuration-interaction approaches. Our theory is numerically tested to compute total energies and ionization potentials for a series of atomic systems. The results clearly show that the DFT correction drastically improves the basis-set convergence of both the total energies and the energy differences. For instance, a sub kcal/mol accuracy is obtained from the aug-cc-pVTZ basis set with the method proposed here when an aug-cc-pV5Z basis set barely reaches such a level of accuracy at the near FCI level.

Ab initio lifetime correction to scattering states for time-dependent electronic-structure calculations with incomplete basis sets.

We propose a method for obtaining effective lifetimes of scattering electronic states for avoiding the artificial confinement of the wave function due to the use of incomplete basis sets in time-dependent electronic-structure calculations of atoms and molecules. In this method, using a fitting procedure, the lifetimes are extracted from the spatial asymptotic decay of the approximate scattering wave functions obtained with a given basis set. The method is based on a rigorous analysis of the complex-energy solutions of the Schrödinger equation. It gives lifetimes adapted to any given basis set without using any empirical parameters. The method can be considered as an ab initio version of the heuristic lifetime model of Klinkusch et al. [J. Chem. Phys. 131, 114304 (2009)]. The method is validated on H and He atoms using Gaussian-type basis sets for the calculation of high-harmonic-generation spectra.

Simple formalism for efficient derivatives and multi-determinant expansions in quantum Monte Carlo.

We present a simple and general formalism to compute efficiently the derivatives of a multi-determinant Jastrow-Slater wave function, the local energy, the interatomic forces, and similar quantities needed in quantum Monte Carlo. Through a straightforward manipulation of matrices evaluated on the occupied and virtual orbitals, we obtain an efficiency equivalent to algorithmic differentiation in the computation of the interatomic forces and the optimization of the orbital parameters. Furthermore, for a large multi-determinant expansion, the significant computational gain afforded by a recently introduced table method is here extended to the local value of any one-body operator and to its derivatives, in both all-electron and pseudopotential calculations.

Systematic lowering of the scaling of Monte Carlo calculations by partitioning and subsampling.

We propose to compute physical properties by Monte Carlo calculations using conditional expectation values. The latter are obtained on top of the usual Monte Carlo sampling by partitioning the physical space in several subspaces or fragments, and subsampling each fragment (i.e., performing side walks) while freezing the environment. No bias is introduced and a zero-variance principle holds in the limit of separability, i.e., when the fragments are independent. In practice, the usual bottleneck of Monte Carlo calculations-the scaling of the statistical fluctuations as a function of the number of particles N-is relieved for extensive observables. We illustrate the method in variational Monte Carlo on the two-dimensional Hubbard model and on metallic hydrogen chains using Jastrow-Slater wave functions. A factor O(N) is gained in numerical efficiency.

Chaotic versus nonchaotic stochastic dynamics in Monte Carlo simulations: a route for accurate energy differences in N-body systems.

We present a method to efficiently evaluate small energy differences of two close N-body systems by employing stochastic processes having a stability versus chaos property. By using the same random noise, energy differences are computed from close trajectories without reweighting procedures. The approach is presented for quantum systems but can be applied to classical N-body systems as well. It is exemplified with diffusion Monte Carlo simulations for long chains of hydrogen atoms and molecules for which it is shown that the long-standing problem of computing energy derivatives is solved.

Stochastic Effective Core Potentials, toward Efficient Quantum Monte Carlo Simulations of Molecules with Large Atomic Numbers.

We propose a Monte Carlo method which exploits that core regions are physically independent in a molecule to almost remove their contribution to the numerical cost. The method is tantamount to computing an effective core potential on the fly, by efficiently subsampling the core regions with independent sidewalks. The removal of fluctuations in the core region enables also the dynamic in the valence region to be accelerated using a process with two time steps. As a function of the total number of electrons N the numerical overhead O(N) is negligible in comparison to the overall scaling O(N3) (due to the evaluation of determinants). Tests are presented on atoms, alkane chains, and clusters of silicons. We report a transferability of the parameters of the method from atoms to molecules, enabling a calibration using only single atoms. These tests display a gain in numerical efficiency between one and two orders of magnitude for large N.

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.

Improved Monte Carlo estimators for the one-body density.

An alternative Monte Carlo estimator for the one-body density rho(r) is presented. This estimator has a simple form and can be readily used in any type of Monte Carlo simulation. Comparisons with the usual regularization of the delta-function on a grid show that the statistical errors are greatly reduced. Furthermore, our expression allows accurate calculations of the density at any point in space, even in the regions never visited during the Monte Carlo simulation. The method is illustrated with the computation of accurate variational Monte Carlo electronic densities for the Helium atom (one-dimensional curve) and for the water dimer (three-dimensional grid containing up to 51x51x51=132,651 points).

Optimizing the Energy with Quantum Monte Carlo: A Lower Numerical Scaling for Jastrow-Slater Expansions.

We present an improved formalism for quantum Monte Carlo calculations of energy derivatives and properties (e.g., the interatomic forces), with a multideterminant Jastrow-Slater function. As a function of the number Ne of Slater determinants, the numerical scaling of O(Ne) per derivative we have recently reported is here lowered to O(Ne) for the entire set of derivatives. As a function of the number of electrons N, the scaling to optimize the wave function and the geometry of a molecular system is lowered to O(N3) + O(NNe), the same as computing the energy alone in the sampling process. The scaling is demonstrated on linear polyenes up to C60H62 and the efficiency of the method is illustrated with the structural optimization of butadiene and octatetraene with Jastrow-Slater wave functions comprising as many as 200 000 determinants and 60 000 parameters.

Computing physical properties with quantum Monte Carlo methods with statistical fluctuations independent of system size.

We show that the recently proposed correlated sampling without reweighting procedure extends the locality (asymptotic independence of the system size) of a physical property to the statistical fluctuations of its estimator. This makes the approach potentially vastly more efficient for computing space-localized properties in large systems compared with standard correlated methods. A proof is given for a large collection of noninteracting fragments. Calculations on hydrogen chains suggest that this behavior holds not only for systems displaying short-range correlations, but also for systems with long-range correlations.

Calculation of space localized properties in correlated quantum Monte Carlo methods with reweighting: the nonlocality of statistical uncertainties.

We study the efficiency of quantum Monte Carlo (QMC) methods in computing space localized ground state properties (properties which do not depend on distant degrees of freedom) as a function of the system size N. We prove that for the commonly used correlated sampling with reweighting method, the statistical fluctuations σ2(N) do not obey the locality property. σ2(N) grow at least linearly with N and with a slope that is related to the fluctuations of the reweighting factors. We provide numerical illustrations of these tendencies in the form of QMC calculations on linear chains of hydrogen atoms.

Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density.

We construct improved quantum Monte Carlo estimators for the spherically and system-averaged electron pair density (i.e., the probability density of finding two electrons separated by a relative distance u), also known as the spherically averaged electron position intracule density I(u), using the general zero-variance zero-bias principle for observables, introduced by Assaraf and Caffarel. The calculation of I(u) is made vastly more efficient by replacing the average of the local delta-function operator by the average of a smooth nonlocal operator that has several orders of magnitude smaller variance. These new estimators also reduce the systematic error (or bias) of the intracule density due to the approximate trial wave function. Used in combination with the optimization of an increasing number of parameters in trial Jastrow-Slater wave functions, they allow one to obtain well converged correlated intracule densities for atoms and molecules. These ideas can be applied to calculating any pair-correlation function in classical or quantum Monte Carlo calculations.