Quantum Monte Carlo study of the first-row atoms and ions

Beyond Single-Reference Fixed-Node Approximation in Ab Initio Diffusion Monte Carlo Using Antisymmetrized Geminal Power Applied to Systems with Hundreds of Electrons

Diffusion Monte Carlo (DMC) is an exact technique to project out the ground state (GS) of a Hamiltonian. Since the GS is always bosonic, in Fermionic systems, the projection needs to be carried out while imposing antisymmetric constraints, which is a nondeterministic polynomial hard problem. In practice, therefore, the application of DMC on electronic structure problems is made by employing the fixed-node (FN) approximation, consisting of performing DMC with the constraint of having a fixed, predefined nodal surface. How do we get the nodal surface? The typical approach, applied in systems having up to hundreds or even thousands of electrons, is to obtain the nodal surface from a preliminary mean-field approach (typically, a density functional theory calculation) used to obtain a single Slater determinant. This is known as single reference. In this paper, we propose a new approach, applicable to systems as large as the C60 fullerene, which improves the nodes by going beyond the single reference. In practice, we employ an implicitly multireference ansatz (antisymmetrized geminal power wave function constraint with molecular orbitals), initialized on the preliminary mean-field approach, which is relaxed by optimizing a few parameters of the wave function determining the nodal surface by minimizing the FN-DMC energy. We highlight the improvements of the proposed approach over the standard single-reference method on several examples and, where feasible, the computational gain over the standard multireference ansatz, which makes the methods applicable to large systems. We also show that physical properties relying on relative energies, such as binding energies, are affordable and reliable within the proposed scheme.

Potential Energy Curves of the Low-Lying Five 1Σ+ and 1Π States of a CH+ Molecule Based on the Free Complement - Local Schrödinger Equation Theory and the Chemical Formula Theory

The potential energy curves (PECs) of the low-lying five 1Σ+ and 1Π states (X1Σ+, C1Σ+, 31Σ+, A1Π, and D1Π states) of a CH+ molecule, an important interstellar molecule, were calculated by the free complement (FC) - local Schrödinger equation (LSE) theory with the direct local sampling scheme. The FC wave functions were constructed based on the chemical formula theory (CFT), whose local characters correspond to the covalent dissociations: C+(2P°(s2p))) + H(2S) of the X1Σ+ and A1Π states and the ionic dissociations: C(1D(s2p2)) + H+ of the C1Σ+ and D1Π states. All the calculated PECs were obtained with satisfying the chemical accuracy, i.e., error less than 1 kcal/mol, as absolute total energy of the Schrödinger equation without any energy shift. The spectroscopic data calculated from the PECs agreed well with both experimental and other accurate theoretical references. We also analyzed the wave functions using the inverse overlap weights proposed by Gallup et al. with the CFT configurations. For the X1Σ+ and A1Π states, the covalent C+(sp2) and C+(p3) configurations played important roles for bond formation. In the small internuclear distances of the C1Σ+, D1Π, and 31Σ+ states, the covalent character was also dominant as a result of the electron charge transfer from C to H+. Thus, the present FC-LSE results not only are accurate but also can provide chemical understanding according to the CFT.

Tail-regression estimator for heavy-tailed distributions of known tail indices and its application to continuum quantum Monte Carlo data

Standard statistical analysis is unable to provide reliable confidence intervals on expectation values of probability distributions that do not satisfy the conditions of the central limit theorem. We present a regression-based estimator of an arbitrary moment of a probability distribution with power-law heavy tails that exploits knowledge of the exponents of its asymptotic decay to bypass this issue entirely. Our method is applied to synthetic data and to energy and atomic force data from variational and diffusion quantum Monte Carlo calculations, whose distributions have known asymptotic forms [J. R. Trail, Phys. Rev. E 77, 016703 (2008)PLEEE81539-375510.1103/PhysRevE.77.016703; A. Badinski et al., J. Phys.: Condens. Matter 22, 074202 (2010)JCOMEL0953-898410.1088/0953-8984/22/7/074202]. We obtain convergent, accurate confidence intervals on the variance of the local energy of an electron gas and on the Hellmann-Feynman force on an atom in the all-electron carbon dimer. In each of these cases the uncertainty on our estimator is 45% and 60 times smaller, respectively, than the nominal (ill-defined) standard error.

QMCPACK: an open source ab initio quantum Monte Carlo package for the electronic structure of atoms, molecules and solids

QMCPACK is an open source quantum Monte Carlo package for ab initio electronic structure calculations. It supports calculations of metallic and insulating solids, molecules, atoms, and some model Hamiltonians. Implemented real space quantum Monte Carlo algorithms include variational, diffusion, and reptation Monte Carlo. QMCPACK uses Slater-Jastrow type trial wavefunctions in conjunction with a sophisticated optimizer capable of optimizing tens of thousands of parameters. The orbital space auxiliary-field quantum Monte Carlo method is also implemented, enabling cross validation between different highly accurate methods. The code is specifically optimized for calculations with large numbers of electrons on the latest high performance computing architectures, including multicore central processing unit and graphical processing unit systems. We detail the program's capabilities, outline its structure, and give examples of its use in current research calculations. The package is available at http://qmcpack.org.

Generic expansion of the Jastrow correlation factor in polynomials satisfying symmetry and cusp conditions

Jastrow correlation factors play an important role in quantum Monte Carlo calculations. Together with an orbital based antisymmetric function, they allow the construction of highly accurate correlation wave functions. In this paper, a generic expansion of the Jastrow correlation function in terms of polynomials that satisfy both the electron exchange symmetry constraint and the cusp conditions is presented. In particular, an expansion of the three-body electron-electron-nucleus contribution in terms of cuspless homogeneous symmetric polynomials is proposed. The polynomials can be expressed in fairly arbitrary scaling function allowing a generic implementation of the Jastrow factor. It is demonstrated with a few examples that the new Jastrow factor achieves 85%-90% of the total correlation energy in a variational quantum Monte Carlo calculation and more than 90% of the diffusion Monte Carlo correlation energy.

Ab initio ground-state potential energy function and vibration-rotation energy levels of imidogen, NH

The accurate ground-state potential energy function of imidogen, NH, has been determined from ab initio calculations using the multireference averaged coupled-pair functional (MR-ACPF) method in conjunction with the correlation-consistent core-valence basis sets up to octuple-zeta quality. The importance of several effects, including electron correlation beyond the MR-ACPF level of approximation, the scalar relativistic, adiabatic, and nonadiabatic corrections were discussed. Along with the large one-particle basis set, all of these effects were found to be crucial to attain "spectroscopic" accuracy of the theoretical predictions of vibration-rotation energy levels of NH.

Compression algorithm for multideterminant wave functions

A compression algorithm is introduced for multideterminant wave functions which can greatly reduce the number of determinants that need to be evaluated in quantum Monte Carlo calculations. We have devised an algorithm with three levels of compression, the least costly of which yields excellent results in polynomial time. We demonstrate the usefulness of the compression algorithm for evaluating multideterminant wave functions in quantum Monte Carlo calculations, whose computational cost is reduced by factors of between about 2 and over 25 for the examples studied. We have found evidence of sublinear scaling of quantum Monte Carlo calculations with the number of determinants when the compression algorithm is used.

Using perturbatively selected configuration interaction in quantum Monte Carlo calculations

Defining accurate and compact trial wavefunctions leading to small statistical and fixed-node errors in quantum Monte Carlo (QMC) calculations is still a challenging problem. Here we propose to make use of selected configuration interaction (CI) expansions obtained by selecting the most important determinants through a perturbative criterion. A major advantage with respect to truncated CASSCF wavefunctions or CI expansions limited to a maximum number of excitations (e.g, CISD) is that much smaller expansions can be considered (many unessential determinants are avoided), an important practical point for efficient QMC calculations. The most important determinants entering first during the selection process (hierarchical construction) the main features of the nodal structure of the wavefunction can be expected to be obtained with a moderate number of determinants. Thanks to this property, the delicate problem of optimizing in a Monte Carlo framework the numerous linear and (or) nonlinear parameters of the determinantal part of the trial wavefunction could be avoided. As a first numerical example, the calculation of the ground-state energy of the oxygen atom is presented. The best DMC value reported so far is obtained.

Framework for constructing generic Jastrow correlation factors

We have developed a flexible framework for constructing Jastrow factors which allows for the introduction of terms involving arbitrary numbers of particles. The use of various three- and four-body Jastrow terms in quantum Monte Carlo calculations is investigated, including a four-body van der Waals-like term, and anisotropic terms. We have tested these Jastrow factors on one- and two-dimensional homogeneous electron gases, the Be, B, and O atoms, and the BeH, H2O, N2, and H2 molecules. Our optimized Jastrow factors retrieve more than 90% of the fixed-node diffusion Monte Carlo correlation energy in variational Monte Carlo for each system studied.