Zero-variance zero-bias principle for observables in quantum Monte Carlo: Application to forces

Assaraf, Roland; Caffarel, Michel

doi:10.1063/1.1621615

Cited by 115 publications

(117 citation statements)

References 42 publications

(76 reference statements)

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“…42,[53][54][55][56] To tackle this problem, in a recent work Sorella and co-workers 56 proposed a combination of the reweighting method, 57 the correlated sampling technique, 58 and the space warp coordinate transformation. [59][60][61] With the help of the algorithmic adjoint differentiation, all the components of the ionic VMC forces can be calculated with and without pseudopotentials in a computational time that is only about 4 times that of an ordinary energy calculation. 56 Vibrational properties, which require the calculation of second derivatives of the energy, to the best of our knowledge have never been calculated at the full QMC level for systems larger than diatomic molecules.…”

Section: Introductionmentioning

confidence: 99%

Optimized Structure and Vibrational Properties by Error Affected Potential Energy Surfaces

Zen

Zhelyazov

Guidoni

2012

J. Chem. Theory Comput.

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The precise theoretical determination of the geometrical parameters of molecules at the minima of their potential energy surface and of the corresponding vibrational properties are of fundamental importance for the interpretation of vibrational spectroscopy experiments. Quantum Monte Carlo techniques are correlated electronic structure methods promising for large molecules, which are intrinsically affected by stochastic errors on both energy and force calculations, making the mentioned calculations more challenging with respect to other more traditional quantum chemistry tools. To circumvent this drawback in the present work, we formulate the general problem of evaluating the molecular equilibrium structures, the harmonic frequencies, and the anharmonic coefficients of an error affected potential energy surface. The proposed approach, based on a multidimensional fitting procedure, is illustrated together with a critical evaluation of systematic and statistical errors. We observe that the use of forces instead of energies in the fitting procedure reduces the statistical uncertainty of the vibrational parameters by 1 order of magnitude. Preliminary results based on variational Monte Carlo calculations on the water molecule demonstrate the possibility to evaluate geometrical parameters and harmonic and anharmonic coefficients at this level of theory with an affordable computational cost and a small stochastic uncertainty (<0.07% for geometries and <0.7% for vibrational properties).

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Section: Introductionmentioning

confidence: 99%

Optimized Structure and Vibrational Properties by Error Affected Potential Energy Surfaces

Zen

Zhelyazov

Guidoni

2012

J. Chem. Theory Comput.

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“…This focus has been the subject of considerable interest for the QMC community in general and has posed a considerable challenge for decades. [17][18][19][20][21][22][23][24][25][26][27][28][29] These properties, which include static correlation functions and entropy estimators as well as the forces, multipole moments, and polarisabilities considered here, may be deduced from the effect of a perturbation from the corresponding operator,P, upon the Hamiltonian,…”

Section: Introductionmentioning

confidence: 99%

Analytic nuclear forces and molecular properties from full configuration interaction quantum Monte Carlo

Thomas

Opalka

Overy

et al. 2015

The Journal of Chemical Physics

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Articles you may be interested in A deterministic alternative to the full configuration interaction quantum Monte Carlo method J. Chem. Phys. 145, 044112 (2016) Unbiased stochastic sampling of the one-and two-body reduced density matrices is achieved in full configuration interaction quantum Monte Carlo with the introduction of a second, "replica" ensemble of walkers, whose population evolves in imaginary time independently from the first and which entails only modest additional computational overheads. The matrices obtained from this approach are shown to be representative of full configuration-interaction quality and hence provide a realistic opportunity to achieve high-quality results for a range of properties whose operators do not necessarily commute with the Hamiltonian. A density-matrix formulated quasi-variational energy estimator having been already proposed and investigated, the present work extends the scope of the theory to take in studies of analytic nuclear forces, molecular dipole moments, and polarisabilities, with extensive comparison to exact results where possible. These new results confirm the suitability of the sampling technique and, where sufficiently large basis sets are available, achieve close agreement with experimental values, expanding the scope of the method to new areas of investigation. C 2015 AIP Publishing LLC. [http://dx

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“…In practice, this scheme requires extensive optimization and, although promising, it is unclear if it will be viable for more complicated cases. In addition, the value of the force is very sensitive to small errors [16] in the charge density and the optimization within a stochastic technique is probably not sufficiently stable to eliminate these errors.…”

Section: Qmcmentioning

confidence: 99%

“…The semilocal character of the "zero-variance" estimator makes its DMC implementation trickier. To overcome this problem there have been attempts [2,16] to use correction terms similar in nature to the Pulay terms in single-particle approaches. In practice, this scheme requires extensive optimization and, although promising, it is unclear if it will be viable for more complicated cases.…”

Section: Qmcmentioning

confidence: 99%

Accurate, Efficient, and Simple Forces Computed with Quantum Monte Carlo Methods

2005

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Computation of ionic forces using quantum Monte Carlo methods has long been a challenge. We introduce a simple procedure, based on known properties of physical electronic densities, to make the variance of the Hellmann-Feynman estimator finite. We obtain very accurate geometries for the molecules H2, LiH, CH4, NH3, H2O and HF, with a Slater-Jastrow trial wave function. Harmonic frequencies for diatomics are also in good agreement with experiment. An antithetical sampling method is also discussed for additional reduction of the variance.The optimization of molecular geometries and crystal structures and ab initio molecular dynamics simulations are among the most significant achievements of single particle theories. These accomplishments were both possible thanks to the possibility of readily computing forces on the ions within the framework of the BornOppenheimer approximation. The approximate treatment of electron interactions typical of these approaches can, however, lead to quantitatively, and sometimes qualitatively, wrong results. This fact, together with a favorable scaling of the computational cost with respect to the number of particles, has spurred the development of stochastic techniques, i.e. quantum Monte Carlo (QMC) methods. Despite the higher accuracy achievable for many physical properties, the lack of an efficient estimator for forces has prevented, until recently [1,2,3], the use of QMC methods to predict even the simplest molecular geometry. The chief problem is to have a Monte Carlo (MC) estimator for the force with sufficiently small variance. For example, in all-electron calculations, a straightforward application of MC sampling of the HellmannFeynman estimator has infinite variance. This can be easily seen from the definition of the force. For a nucleus of charge Z at the origin, the force can be written, together with its variance, as a function of the charge density ρ(r) asSince the electronic density is finite at the origin, the variance integral diverges. In this paper, we propose a modified form for the force estimator which has finite variance. This estimator is then used to calculate forces and predict equilibrium geometry and vibrational frequencies for a set of small molecules. Without loss of generality we will consider only the z-component of the force on an atom at * Electronic address: chiesa@uiuc.edu † Electronic address: ceperley@uiuc.edu ‡ Electronic address: shiwei@physics.wm.edu the origin. In a QMC calculation based in configuration space, the charge density is a sum of delta functions: ρ(r) ∝ r ′ δ(r − r ′ ), where the sum is over all N e electron positions and all MC samples. We consider separately the electrons within a distance R of the atom and those outside. The contribution to the force from charges outside, F O z , can be calculated directly with the Hellmann-Feynman estimator in Eq. (1). The contribution from inside the sphere is responsible for the large variances in the direct estimator. It is convenient to introduce a "force density" defined as the force arisin...

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Zero-variance zero-bias principle for observables in quantum Monte Carlo: Application to forces

Cited by 115 publications

References 42 publications

Optimized Structure and Vibrational Properties by Error Affected Potential Energy Surfaces

Optimized Structure and Vibrational Properties by Error Affected Potential Energy Surfaces

Analytic nuclear forces and molecular properties from full configuration interaction quantum Monte Carlo

Accurate, Efficient, and Simple Forces Computed with Quantum Monte Carlo Methods

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