26 research outputs found
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 of
Slater determinants, the numerical scaling of per derivative we have
recently reported is here lowered to for the entire set of
derivatives. As a function of the number of electrons , the scaling to
optimize the wave function and the geometry of a molecular system is lowered to
, the same as computing the energy alone in the sampling
process. The scaling is demonstrated on linear polyenes up to CH
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 200000 determinants and 60000 parameters
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
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 paramaters. Furthermore, for a large
multi-determinant expansion, the significant computational gain recently
reported for the calculation of the wave function is here improved and extended
to all local properties in both all-electron and pseudopotential calculations.Comment: 15 pages, 3 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%
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 artificially 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 Schr{\"o}dinger 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 the H and He atoms using
Gaussian-type basis sets for calculation of high-harmonic-generation spectra
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 non-local 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.Comment: 13 pages, 9 figures, published versio
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 in 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
Quantum Monte Carlo facing the Hartree-Fock symmetry dilemma: The case of hydrogen rings
When using Hartree-Fock (HF) trial wave functions in quantum Monte Carlo
calculations, one faces, in case of HF instabilities, the HF symmetry dilemma
in choosing between the symmetry-adapted solution of higher HF energy and
symmetry-broken solutions of lower HF energies. In this work, we have examined
the HF symmetry dilemma in hydrogen rings which present singlet instabilities
for sufficiently large rings. We have found that the symmetry-adapted HF wave
function gives a lower energy both in variational Monte Carlo and in fixed-node
diffusion Monte Carlo. This indicates that the symmetry-adapted wave function
has more accurate nodes than the symmetry-broken wave functions, and thus
suggests that spatial symmetry is an important criterion for selecting good
trial wave functions.Comment: 6 pages, 3 figures, 2 tables, to appear in "Advances in Quantum Monte
Carlo", AC