Quantum Monte Carlo studies of density functional theory (original) (raw)
Physical Review B, 2003
We use a variational quantum Monte Carlo realization of the adiabatic connection technique to calculate the most relevant quantities in Hohenberg-Kohn-Sham density functional theory for several strongly inhomogeneous electron-gas systems. Results for the coupling-constant dependence of the exchange-correlation energy, the pair-correlation function, the exchange-correlation hole, and the exchange and correlation energy densities are presented. Comparisons are made with the interaction strength interpolation ͑ISI͒ approximation, the local density approximation ͑LDA͒, the gradient expansion approximation ͑GEA͒, the generalized gradient approximation ͑GGA͒, and the weighted density approximation ͑WDA͒. The coupling-constant dependence of the exchange-correlation energy is accurately described by an ISI model that incorporates information on the strong-interaction limit. Unlike either the LDA or GEA, the WDA is successful in describing the nonlocal structure of the exchange-correlation hole. The LDA errors in the exchange-correlation energy density show a remarkable correlation with the Laplacian of the density. The GGA worsens the error in the integrated exchange-correlation energy as the inhomogeneity of the systems increases. This failure is shared by current meta-GGA functionals and is shown to be caused by the inability of these functionals to describe the LDA overestimation ͑in absolute value͒ of the exchange energy density around density maxima. It is suggested that this effect could be taken into account by including Laplacian terms in semilocal density functionals.
Quantum Monte Carlo analysis of exchange and correlation in the strongly inhomogeneous electron gas
Physical review letters, 2001
We use the variational quantum Monte Carlo method to calculate the density-functional exchange-correlation hole n(xc), the exchange-correlation energy density e(xc), and the total exchange-correlation energy E(xc) of several strongly inhomogeneous electron gas systems. We compare our results with the local density approximation and the generalized gradient approximation. It is found that the nonlocal contributions to e(xc) contain an energetically significant component, the magnitude, shape, and sign of which are controlled by the Laplacian of the electron density.
Monte Carlo implementation of density-functional theory
Physical Review B, 2012
We propose a conceptually easy and relatively straigthforward numerical method for calculating the groundstate properties of many-particle systems based on the Hohenberg-Kohn theorems. In this "density-functional Monte Carlo" method a direct numerical minimization of the energy functional is performed by a Monte Carlo algorithm in which the density is simulated by a distribution of Bernoulli walkers. The total number of particles is conserved by construction, unlike for other implementations of density-functional theory. The feasibility of the method is illustrated by applying it to a nanoshell.
Comments on the quantum Monte Carlo method and the density matrix theory
The Journal of Chemical Physics, 2003
Density matrix theory is implemented in a variational quantum Monte Carlo computation of electronic properties of atoms and molecules. Differences between electronic densities from conventional and density matrix methods are detected. However, calculated properties present similar behavior and partial antisymmetry can be ignored in the cases studied.
Exact-exchange density-functional theory applied to a strongly inhomogeneous electron gas
Physical Review B, 2003
A recently developed quasi two-dimensional exact-exchange formalism within the framework of Density Functional Theory has been applied to a strongly inhomogeneous interacting electron gas, and the results were compared with state-of-the-art Variational Quantum Monte Carlo (VMC) numerical simulations for a three-dimensional electron gas under a strong external potential. The VMC results, extremely demanding from the computational point of view, could be considered as a benchmark for the present theory. We observe a remarkable qualitative and quantitative agreement between both methods from the comparison of the exchange-hole densities, exchange-energy densities, and total exchange-energies per particle. This agreement is increasingly improved with the strength of the external potential when the electron gas becomes quasi-two-dimensional.
Monte Carlo integration of density functional theory: Fermions in a harmonic well
Chemical Physics Letters, 1988
The first numerical results with a recently developed path integration formulation of electron density functional theory are presented. The many-dimensional integral which provides the fundamental expression for the density is presented in a rearranged form amenable to straightforward Monte Carlo estimation. This method is then applied to the case of independent electrons in a harmonic well. The numerical results satisfactorily reproduce the structured electron densities which reflect fermion level filing for this simple problem. These encouraging results also exemplify the algorithmic problems which must be overcome before the present formulation of electron density functional theory shall be of important practical use.
Density-density functionals and effective potentials in many-body electronic structure calculations
2008
We demonstrate the existence of different density-density functionals designed to retain selected properties of the many-body ground state in a non-interacting solution starting from the standard density functional theory ground state. We focus on diffusion quantum Monte Carlo applications that require trial wave functions with optimal Fermion nodes. The theory is extensible and can be used to understand current practices in several electronic structure methods within a generalized density functional framework. The theory justifies and stimulates the search of optimal empirical density functionals and effective potentials for accurate calculations of the properties of real materials, but also cautions on the limits of their applicability. The concepts are tested and validated with a near-analytic model.
2005
We construct a quantum Monte Carlo algorithm for interacting fermions using the two-body density as the fundamental quantity. The central idea is mapping the interacting fermionic system onto an auxiliary system of interacting bosons. The correction term is approximated using correlated wave functions for the interacting system, resulting in an effective potential that represents the nodal surface. We calculate the properties of 3He and find good agreement with experiment and with other theoretical work. In particular, our results for the total energy agree well with other calculations where the same approximations were implemented but the standard quantum Monte Carlo algorithm was used
Path-Integral Monte Carlo Simulation of the Warm Dense Homogeneous Electron Gas
Physical Review Letters, 2013
We perform calculations of the 3D finite-temperature homogeneous electron gas (HEG) in the warm-dense regime (rs ≡ (3/4πn) 1/3 a −1 B = 1.0−40.0 and Θ ≡ T /TF = 0.0625−8.0) using restricted path integral Monte Carlo (RPIMC). Precise energies, pair correlation functions, and structure factors are obtained. For all densities, we find a significant discrepancy between the ground state parameterized local density approximation (LDA) and our results around TF . These results can be used as a benchmark for improved functionals, as well as input for orbital-free DFT formulations.
Physical Review A, 2011
We present a local density approximation (LDA) for one-dimensional (1D) systems interacting via the soft-Coulomb interaction based on quantum Monte-Carlo calculations. Results for the ground-state energies and ionization potentials of finite 1D systems show excellent agreement with exact calculations, obtained by exploiting the mapping of an NNN-electron system in ddd dimensions, onto a single electron in NtimesdN\times dNtimesd dimensions properly symmetrized by the Young diagrams. We conclude that 1D LDA is of the same quality as its three-dimensional (3D) counterpart, and we infer conclusions about 3D LDA. The linear and non-linear time-dependent responses of 1D model systems using LDA, exact exchange, and the exact solution are investigated and show very good agreement in both cases, except for the well known problem of missing double excitations. Consequently, the 3D LDA is expected to be of good quality beyond linear response. In addition, the 1D LDA should prove useful in modeling the interaction of atoms with strong laser fields, where this specific 1D model is often used.
Quantum Monte Carlo modelling of the spherically averaged structure factor of a many-electron system
2006
The interaction and exchange-correlation contributions to the ground-state energy of an arbitrary many-electron system can be obtained from a spherical average of the wavevector-dependent diagonal structure factor (SF). We model the continuous-k spherically averaged SF using quantum Monte Carlo calculations in finite simulation cells. We thus derive a method that allows to substantially reduce the troublesome Coulomb finite-size errors that are usually present in ground-state energy calculations. To demonstrate this, we perform variational Monte Carlo calculations of the interaction energy of the homogeneous electron gas. The method is, however, equally applicable to arbitrary inhomogeneous systems.
Kohn-Sham orbitals and potentials from quantum Monte Carlo molecular densities
The Journal of Chemical Physics, 2014
In this work we show the possibility to extract Kohn-Sham orbitals, orbital energies, and exchange correlation potentials from accurate Quantum Monte Carlo (QMC) densities for atoms (He, Be, Ne) and molecules (H 2 , Be 2 , H 2 O, and C 2 H 4 ). The Variational Monte Carlo (VMC) densities based on accurate Jastrow Antisymmetrised Geminal Power wave functions are calculated through different estimators. Using these reference densities, we extract the Kohn-Sham quantities with the method developed by Zhao, Morrison, and Parr (ZMP) [Phys. Rev. A 50, 2138 (1994)]. We compare these extracted quantities with those obtained form CISD densities and with other data reported in the literature, finding a good agreement between VMC and other high-level quantum chemistry methods. Our results demonstrate the applicability of the ZMP procedure to QMC molecular densities, that can be used for the testing and development of improved functionals and for the implementation of embedding schemes based on QMC and Density Functional Theory. © 2014 AIP Publishing LLC.
Physical Review B, 2009
To assess the accuracy of exchange-correlation approximations within density functional theory ͑DFT͒, diffusion quantum Monte Carlo ͑DMC͒ and DFT methods are used to calculate the energies of isomers of three covalently bonded carbon and boron clusters ͑C 20 , B 18 , and B 20 ͒, and three metallic aluminum and copper clusters ͑Al 13 , Al 55 , and Cu 13 ͒. We find that local and semilocal DFT methods predict the same energy ordering as DMC for the metallic clusters but not for the covalent clusters, implying that the DFT functionals are inadequate in such systems. In addition, we find that DFT fails to describe energy reductions arising from Jahn-Teller distortions.
Ab Initio Quantum Monte Carlo Simulation of the Warm Dense Electron Gas in the Thermodynamic Limit
Physical review letters, 2016
We perform ab initio quantum Monte Carlo (QMC) simulations of the warm dense uniform electron gas in the thermodynamic limit. By combining QMC data with the linear response theory, we are able to remove finite-size errors from the potential energy over the substantial parts of the warm dense regime, overcoming the deficiencies of the existing finite-size corrections by Brown et al. [Phys. Rev. Lett. 110, 146405 (2013)]. Extensive new QMC results for up to N=1000 electrons enable us to compute the potential energy V and the exchange-correlation free energy F_{xc} of the macroscopic electron gas with an unprecedented accuracy of |ΔV|/|V|,|ΔF_{xc}|/|F|_{xc}∼10^{-3}. A comparison of our new data to the recent parametrization of F_{xc} by Karasiev et al. [Phys. Rev. Lett. 112, 076403 (2014)] reveals significant deviations to the latter.
Electronic structure quantum Monte Carlo
Acta Physica Slovaca. Reviews and Tutorials, 2000
Quantum Monte Carlo (QMC) is an advanced simulation methodology for studies of manybody quantum systems. The QMC approaches combine analytical insights with stochastic computational techniques for efficient solution of several classes of important many-body problems such as the stationary Schrödinger equation. QMC methods of various flavors have been applied to a great variety of systems spanning continuous and lattice quantum models, molecular and condensed systems, BEC-BCS ultracold condensates, nuclei, etc. In this review, we focus on the electronic structure QMC, i.e., methods relevant for systems described by the electron-ion Hamiltonians. Some of the key QMC achievements include direct treatment of electron correlation, accuracy in predicting energy differences and favorable scaling in the system size. Calculations of atoms, molecules, clusters and solids have demonstrated QMC applicability to real systems with hundreds of electrons while providing 90-95% of the correlation energy and energy differences typically within a few percent of experiments. Advances in accuracy beyond these limits are hampered by the so-called fixed-node approximation which is used to circumvent the notorious fermion sign problem. Many-body nodes of fermion states and their properties have therefore become one of the important topics for further progress in predictive power and efficiency of QMC calculations. Some of our recent results on the wave function nodes and related nodal domain topologies will be briefly reviewed. This includes analysis of few-electron systems and descriptions of exact and approximate nodes using transformations and projections of the highly-dimensional nodal hypersurfaces into the 3D space. Studies of fermion nodes offer new insights into topological properties of eigenstates such as explicit demonstrations that generic fermionic ground states exhibit the minimal number of two nodal domains. Recently proposed trial wave functions based on pfaffians with pairing orbitals are presented and their nodal properties are tested in calculations of first row atoms and molecules. Finally, backflow "dressed" coordinates are introduced as another possibility for capturing correlation effects and for decreasing the fixed-node bias. detailed analysis of theoretical ideas. Indeed, QMC is very much in the line of "it from bit" paradigm, alongside, for example, of substantional computational efforts in quantum chromodynamics which not only predict hadron masses but, at the same time, contribute to the validation of the fundamental theory. Similar simulations efforts exist in other areas of physics as well. Just a few decades ago it was almost unthinkable that one would be able to solve Schrödinger equation for hundreds of electrons in an explicit, many-body wave function framework. Today, such calculations are feasible using available computational resources. At the same time, much more remains to be done, of course, to make the methods more insightful, more efficient and their application less laborious. We hope this overview will contribute to the growing interest in this rapidly developing field of research.
Applications of the Variational Quantum Monte Carlo Method to the Two-Electron Atoms
International Journal of High Energy Physics, 2019
The variational quantum Monte Carlo method was applied to investigate the ground states of the helium atom and helium like ions with atomic number from 1 to 10 and the first four excited states of the helium atom. Furthermore, the investigation of the ground state of helium, Li + , and Be 2+ in a confined impenetrable spherical box. Moreover, the calculation of the ground state of the helium atom in a strong magnetic field using four simple trial wave functions. The trial wave functions consist of usual orbital hydrogen wave functions multiplied by correlation function. Using four different correlation wave functions, we describe the interaction of the two electrons with each other and having a small number of variational parameters.
Physical Review Letters, 2004
We construct a quantum Monte Carlo algorithm for interacting fermions using the two-body density as the fundamental quantity. The central idea is mapping the interacting fermionic system onto an auxiliary system of interacting bosons. The correction term is approximated using correlated wave-functions for the interacting system, resulting in an effective potential that represents the nodal surface. We calculate the properties of 3 He and find good agreement with experiment and with other theoretical work. In particular our results for the total energy agree well with other calculations where the same approximations were implemented but the standard quantum Monte Carlo algorithm was used. PACS numbers: 02.70.Ss,24.10.Cn,05.30.-d Density-functional theory [1, 2] (DFT) and quantum Monte Carlo (QMC) are generally thought of as two distinct approaches to the problem of interacting fermions. DFT is based on the Hohenberg-Kohn theorems [1] (HK) which state that the energy of an interacting fermion system in an external field can be written as a functional of the density, and that minimizing the energy as a functional of the density gives the ground state energy (HK theorems). Applications of DFT are usually based on the Kohn and Sham [2] method, where an auxiliary noninteracting system is invoked. Minimization is achieved with respect to the orbitals of the auxiliary system. QMC also involves minimization of the energy. One way of minimizing, is to propagate a trial wavefunction in imaginary time , so that it asymptotically approaches the ground state.
Quantum Monte Carlo Study of electrons in low dimensions
1999
We report on a diffusion Monte Carlo investigation of model electron systems in low dimensions, which should be relevant to the physics of systems obtainable nowadays in semiconductor heterostructures. In particular, we present results for a one dimensional electron gas, at selected values of the coupling strength and confinement parameter, briefly analyzing the pair correlations and relating them to predictions by Schulz for a Luttinger liquid with long-range interactions. We find no evidence of the the Bloch instability yielded by approximate treatments such as the STLS and DFT schemes.