78 research outputs found
Efficient Monte Carlo Calculations of 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 (1D curve) and
for the water dimer (3D grid containing up to 51x51x51=132651 points).Comment: 12 pages with 3 postscript figure
Fixed-Node Diffusion Monte Carlo potential energy curve of the fluorine molecule F2 using selected configuration interaction trial wavefunctions
The potential energy curve of the F molecule is calculated with
Fixed-Node Diffusion Monte Carlo (FN-DMC) using Configuration Interaction
(CI)-type trial wavefunctions. To keep the number of determinants reasonable
(the first and second derivatives of the trial wavefunction need to be
calculated at each step of FN-DMC), the CI expansion is restricted to those
determinants that contribute the most to the total energy. The selection of the
determinants is made using the so-called CIPSI approach (Configuration
Interaction using a Perturbative Selection made Iteratively). Quite remarkably,
the nodes of CIPSI wavefunctions are found to be systematically improved when
increasing the number of selected determinants. To reduce the non-parallelism
error of the potential energy curve a scheme based on the use of a
-dependent number of determinants is introduced. Numerical results show that
improved FN-DMC energy curves for the F molecule are obtained when
employing CIPSI trial wavefunctions. Using the Dunning's cc-pVDZ basis set the
FN-DMC energy curve is of a quality similar to that obtained with FCI/cc-pVQZ.
A key advantage of using selected CI in FN-DMC is the possibility of improving
nodes in a systematic and automatic way without resorting to a preliminary
multi-parameter stochastic optimization of the trial wavefunction performed at
the Variational Monte Carlo level as usually done in FN-DMC.Comment: 16 pages, 15 figure
Zero-Variance Zero-Bias Principle for Observables in quantum Monte Carlo: Application to Forces
A simple and stable method for computing accurate expectation values of
observable with Variational Monte Carlo (VMC) or Diffusion Monte Carlo (DMC)
algorithms is presented. The basic idea consists in replacing the usual
``bare'' estimator associated with the observable by an improved or
``renormalized'' estimator. Using this estimator more accurate averages are
obtained: Not only the statistical fluctuations are reduced but also the
systematic error (bias) associated with the approximate VMC or (fixed-node) DMC
probability densities. It is shown that improved estimators obey a
Zero-Variance Zero-Bias (ZVZB) property similar to the usual Zero-Variance
Zero-Bias property of the energy with the local energy as improved estimator.
Using this property improved estimators can be optimized and the resulting
accuracy on expectation values may reach the remarkable accuracy obtained for
total energies. As an important example, we present the application of our
formalism to the computation of forces in molecular systems. Calculations of
the entire force curve of the H,LiH, and Li molecules are presented.
Spectroscopic constants (equilibrium distance) and (harmonic
frequency) are also computed. The equilibrium distances are obtained with a
relative error smaller than 1%, while the harmonic frequencies are computed
with an error of about 10%
Quantum Monte Carlo for large chemical systems: Implementing efficient strategies for petascale platforms and beyond
Various strategies to implement efficiently QMC simulations for large
chemical systems are presented. These include: i.) the introduction of an
efficient algorithm to calculate the computationally expensive Slater matrices.
This novel scheme is based on the use of the highly localized character of
atomic Gaussian basis functions (not the molecular orbitals as usually done),
ii.) the possibility of keeping the memory footprint minimal, iii.) the
important enhancement of single-core performance when efficient optimization
tools are employed, and iv.) the definition of a universal, dynamic,
fault-tolerant, and load-balanced computational framework adapted to all kinds
of computational platforms (massively parallel machines, clusters, or
distributed grids). These strategies have been implemented in the QMC=Chem code
developed at Toulouse and illustrated with numerical applications on small
peptides of increasing sizes (158, 434, 1056 and 1731 electrons). Using 10k-80k
computing cores of the Curie machine (GENCI-TGCC-CEA, France) QMC=Chem has been
shown to be capable of running at the petascale level, thus demonstrating that
for this machine a large part of the peak performance can be achieved.
Implementation of large-scale QMC simulations for future exascale platforms
with a comparable level of efficiency is expected to be feasible
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
() with a non-negligible weight in the expansion is
of order . We show that a careful implementation
of the calculation of the -dependent contributions can make this
step negligible enough so that in practice the algorithm scales as the total
number of unique spin-specific determinants, , over a wide range of total number of determinants (here,
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.Comment: 9 pages, 3 figure
Spin density distribution in open-shell transition metal systems: A comparative post-Hartree-Fock, Density Functional Theory and quantum Monte Carlo study of the CuCl2 molecule
We present a comparative study of the spatial distribution of the spin
density (SD) of the ground state of CuCl2 using Density Functional Theory
(DFT), quantum Monte Carlo (QMC), and post-Hartree-Fock wavefunction theory
(WFT). A number of studies have shown that an accurate description of the
electronic structure of the lowest-lying states of this molecule is
particularly challenging due to the interplay between the strong dynamical
correlation effects in the 3d shell of the copper atom and the delocalization
of the 3d hole over the chlorine atoms. It is shown here that qualitatively
different results for SD are obtained from these various quantum-chemical
approaches. At the DFT level, the spin density distribution is directly related
to the amount of Hartree-Fock exchange introduced in hybrid functionals. At the
QMC level, Fixed-node Diffusion Monte Carlo (FN-DMC) results for SD are
strongly dependent on the nodal structure of the trial wavefunction employed
(here, Hartree-Fock or Kohn-Sham with a particular amount of HF exchange) : in
the case of this open-shell system, the 3N -dimensional nodes are mainly
determined by the 3-dimensional nodes of the singly occupied molecular orbital.
Regarding wavefunction approaches, HF and CASSCF lead to strongly localized
spin density on the copper atom, in sharp contrast with DFT. To get a more
reliable description and shed some light on the connections between the various
theoretical descriptions, Full CI-type (FCI) calculations are performed. To
make them feasible for this case a perturbatively selected CI approach
generating multi-determinantal expansions of reasonable size and a small
tractable basis set are employed. Although semi-quantitative, these near-FCI
calculations allow to clarify how the spin density distribution evolves upon
inclusion of dynamic correlation effects. A plausible scenario about the nature
of the SD is proposed.Comment: 13 pages, 12 Figure
Deterministic construction of nodal surfaces within quantum Monte Carlo: the case of FeS
In diffusion Monte Carlo (DMC) methods, the nodes (or zeroes) of the trial
wave function dictate the magnitude of the fixed-node (FN) error. Within
standard DMC implementations, they emanate from short multideterminant
expansions, \textit{stochastically} optimized in the presence of a Jastrow
factor. Here, following a recent proposal, we follow an alternative route by
considering the nodes of selected configuration interaction (sCI) expansions
built with the CIPSI (Configuration Interaction using a Perturbative Selection
made Iteratively) algorithm. In contrast to standard implementations, these
nodes can be \textit{systematically} and \textit{deterministically} improved by
increasing the size of the sCI expansion. The present methodology is used to
investigate the properties of the transition metal sulfide molecule FeS. This
apparently simple molecule has been shown to be particularly challenging for
electronic structure theory methods due to the proximity of two low-energy
quintet electronic states of different spatial symmetry. In particular, we show
that, at the triple-zeta basis set level, all sCI results --- including those
extrapolated at the full CI (FCI) limit --- disagree with experiment, yielding
an electronic ground state of symmetry. Performing FN-DMC
simulation with sCI nodes, we show that the correct ground state
is obtained if sufficiently large expansions are used. Moreover, we show that
one can systematically get accurate potential energy surfaces and reproduce the
experimental dissociation energy as well as other spectroscopic constants.Comment: 8 pages, 2 figure and 4 table
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