Backflow Transformations Research Articles

Ab-initio variational wave functions for the time-dependent many-electron Schrödinger equation

Understanding the real-time evolution of many-electron quantum systems is essential for studying dynamical properties in condensed matter, quantum chemistry, and complex materials, yet it poses a significant theoretical and computational challenge. Our work introduces a variational approach for fermionic time-dependent wave functions, surpassing mean-field approximations by accurately capturing many-body correlations. We employ time-dependent Jastrow factors and backflow transformations, enhanced through neural networks parameterizations. To compute the optimal time-dependent parameters, we employ the time-dependent variational Monte Carlo technique and introduce a new method based on Taylor-root expansions of the propagator, enhancing the accuracy of our simulations. The approach is demonstrated in three distinct systems. In all cases, we show clear signatures of many-body correlations in the dynamics. The results showcase the ability of our variational approach to accurately describe the time evolution, providing insight into quantum dynamical effects in interacting electronic systems, beyond the capabilities of mean-field.

X marks the spot: Accurate energies from intersecting extrapolations of continuum quantum Monte Carlo data.

We explore the application of an extrapolative method that yields very accurate total and relative energies from variational and diffusion quantum Monte Carlo (VMC and DMC) results. For a trial wave function consisting of a small configuration interaction (CI) wave function obtained from full CI quantum Monte Carlo and reoptimized in the presence of a Jastrow factor and an optional backflow transformation, we find that the VMC and DMC energies are smooth functions of the sum of the squared coefficients of the initial CI wave function and that quadratic extrapolations of the non-backflow VMC and backflow DMC energies intersect within uncertainty of the exact total energy. With adequate statistical treatment of quasi-random fluctuations, the extrapolate and intersect with polynomials of order two method is shown to yield results in agreement with benchmark-quality total and relative energies for the C2, N2, CO2, and H2O molecules, as well as for the C2 molecule in its first electronic singlet excited state, using only small CI expansion sizes.

Matrix product states with backflow correlations

By taking inspiration from the backflow transformation for correlated systems, we introduce a novel tensor network ansatz which extend the well-established Matrix Product State representation of a quantum-many body wave function. This new structure provides enough resources to ensure that states in dimension larger or equal than one obey an area law for entanglement. It can be efficiently manipulated to address the ground-state search problem by means of an optimization scheme which mixes tensor-network and variational Monte-Carlo algorithms. We benchmark the new ansatz against spin models both in one and two dimensions, demonstrating high accuracy and precision. We finally employ our approach to study the challenging $S=1/2$ two dimensional $J_1 - J_2$ model, demonstrating that it is competitive with the state of the art methods in 2D.

Development of anharmonic vibrational structure theory using backflow transformation

A backflow (BF) transformation was applied to a vibrational self-consistent field wavefunction, which is the mean-field approximation in an anharmonic vibrational structure theory. We tested its effectiveness on the H2O molecule using quantum Monte Carlo method. Each normal mode coordinate was transformed into collective coordinates using the correlation function depending on whole coordinates. The vibrational frequencies with the BF transformation reasonably agree with the vibrational configuration interaction (VCI) results. The vibrational probability density to which the BF transformation was applied was also consistent with the VCI results and showed that the BF transformation appropriately take into account the inter-mode couplings.

Quantum Monte Carlo simulations using Slater-Jastrow-backflow wave function

Diffusion Quantum Monte-Carlo computations of the ground state energies of the Li, Li−, Be, Be+, B and B+ species are reported. We use trial wave function of four types: single determinant Slater-Jastrow, single determinant Slater-Jastrow-Backflow, multi-determinant Slater-Jastrow and multi-determinant Slater-Jastrow-Backflow. The single determinant wave functions are computed by the HF method and the multi-determinant wave functions are computed by the CI and CASSCF methods. Up to 99.9% and 100% of the correlation energy are recovered for the studied species without and with the backflow transformation respectively.

Machine learning many-electron wave functions via backflow transformations

1. Backflow Transformations via Neural Networks for Quantum Many-Body Wave-Functions Authors: D. Luo and B. K. Clark Phys. Rev. Lett. 122, 226401 (2019); DOI:10.1103/PhysRevLett.122.226401 arXiv:1807.10770 2. Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks Authors: D. Pfau, J. S. Spencer, A. G. de G. Matthews, and W. M. C. Foulkes arXiv:1909.02487 3. […]

Development of accurate anharmonic vibrational state theory using a novel vibrational coordinate based on backflow transformation

SynopsisThe anharmonic vibrational state theory based on quantum Monte Carlo method is one of the most accurate methods to solve a vibrational Schrödinger equation of molecular systems. In anharmonic vibrational state analyses, one should take account of the inter-mode couplings, which cannot be included by the conventional normal mode coordinates only. In this study, thus, we have developed a novel vibrational coordinate based on backflow transformation which enable us to take correlation effects in quantum many-body systems directly.

Quantum Monte Carlo calculations of energy gaps from first principles

We review the use of continuum quantum Monte Carlo (QMC) methods for the calculation of energy gaps from first principles, and present a broad set of excited-state calculations carried out with the variational and fixed-node diffusion QMC methods on atoms, molecules, and solids. We propose a finite-size-error correction scheme for bulk energy gaps calculated in finite cells subject to periodic boundary conditions. We show that finite-size effects are qualitatively different in two-dimensional materials, demonstrating the effect in a QMC calculation of the band gap and exciton binding energy of monolayer phosphorene. We investigate the fixed-node errors in diffusion Monte Carlo gaps evaluated with Slater-Jastrow trial wave functions by examining the effects of backflow transformations, and also by considering the formation of restricted multideterminant expansions for excited-state wave functions. For several molecules, we examine the importance of structural relaxation in the excited state in determining excited-state energies. We study the feasibility of using variational Monte Carlo with backflow correlations to obtain accurate excited-state energies at reduced computational cost, finding that this approach can be valid. We find that diffusion Monte Carlo gap calculations can be performed with much larger time steps than are typically required to converge the total energy, at significantly diminished computational expense, but that in order to alleviate fixed-node errors in calculations on solids the inclusion of backflow correlations is sometimes necessary.

Iterative backflow renormalization procedure for many-body ground-state wave functions of strongly interacting normal Fermi liquids

We show how a ground state trial wavefunction of a Fermi liquid can be systematically improved introducing a sequence of renormalized coordinates through an iterative backflow transformation. We apply this scheme to calculate the ground state energy of liquid $^3$He in two dimensions at freezing density using variational and fixed-node diffusion Monte Carlo. Comparing with exact transient estimate results for systems with small number of particles, we find that variance extrapolations provide accurate results for the true ground state together with stringent lower bounds. For larger systems these bounds can in turn be used to quantify the systematic bias of fixed-node calculations. These wave functions are size consistent and the scaling of their computational complexity with the number of particles is the same as for standard backflow wave functions.

Quantum Monte Carlo studies of 13-atom simple metallic clusters

We have performed diffusion quantum Monte Carlo calculations for nine isomers of 13-atom clusters of the $sp$-bonded elements Li, Na, Be, Mg, and Al and compared the energies with density functional theory (DFT) results. Taking the diffusion Monte Carlo (DMC) data as the benchmark, we find that the local spin-density approximation (LSDA) and Perdew-Burke-Ernzerhof (PBE) semilocal generalized gradient approximation functionals give good descriptions of the energy differences between the clusters for each of the elements and their cations. The errors in the energy differences between clusters are a little larger for the PBE functional, but on average the errors for both functionals are smaller than 10 meV per atom, except for Be${}_{13}$ clusters, which yield errors of 17 and 22 meV per atom from LSDA and PBE, respectively. We have, in addition, investigated using an inhomogeneous backflow transformation to improve the nodal surface of the trial wave functions used in the DMC calculations. We find that backflow lowers the DMC energies but does not affect the relative stabilities of the clusters.

Diffusion Monte Carlo Study of Bond Dissociation Energies for BH2, B(OH)2, BCl2, and BCl

On basis of bond dissociation energies (BDEs) for BH2, B(OH)2, BCl2, and BCl, the diffusion Monte Carlo (DMC) method is applied to explore the BDEs of HB-H, HOB-OH, ClB-Cl, and B-Cl. The effect of the choice of orbitals, as well as the backflow transformation, is studied. The Slater-Jastrow DMC algorithm gives BDEs of 359.1±0.12 kJ/mol for HB-H, 410.5±0.50 kJ/mol for HOB-OH, 357.8±1.46 kJ/mol for ClB-Cl, and 504.5±0.96 kJ/mol for B-Cl using B3PW91 orbitals and similar BDEs when B3LYP orbitals are used. DMC with backflow corrections (BF-DMC) gives a HB-H BDE of 369.9±0.12 kJ/mol which is close to one of the available experimental value (375.8 kJ/mol). In the case of HOB-OH BDE, the BF-DMC calculation is 446.0±1.84 kJ/mol that is closer to the experimental BDE. The BF-DMC BDE for ClB-Cl is 343.2±2.34 kJ/mol and the BF-DMC B-Cl BDE is 523.3±0.33 kJ/mol, which are close to the experimental BDEs, 341.9 and 530.0 kJ/mol, respectively.

A benchmark quantum Monte Carlo study of the ground state chromium dimer

AbstractWe report calculations of the ground state energy and binding curve of the chromium dimer using the variational and diffusion quantum Monte Carlo (VMC and DMC) methods. We examined various single‐determinant and multideterminant wavefunctions multiplied by a Jastrow factor as a trial/guiding wavefunction for VMC/DMC. The molecular orbitals in the single determinants were calculated using restricted or unrestricted Hartree–Fock or density functional theory (DFT) calculations where five commonly used local (SVWN5), semilocal (PW91 and BLYP), and hybrid (B1LYP and B3LYP) functionals were examined. The multideterminant expansions were obtained from the generalized valence bond and (truncated) unrestricted configuration interaction with single and double excitations (UCISD) methods. We also examined a UCISD wavefunction in which UCISD expansions were added to the UB3LYP single‐determinant reference, and their coefficients were optimized at the VMC level. In addition to the wavefunction dependence, the effects of pseudopotentials and backflow transformation were also investigated. The UB3LYP single‐determinant and multideterminant wavefunctions were found to give the variationally best DMC energies within the framework of single‐determinant and multideterminants, respectively, though both the DMC energies were higher than twice the DMC atomic energy. Some of the VMC binding curves show a flat or quite shallow well bottom, which gets recovered deeper by DMC. All the DMC binding curves have a minimum indicating a bound state, but the unrestricted ones overestimate the equilibrium bond length. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

A Variational Monte Carlo Study of Positronic Compounds Using Inhomogeneous Backflow Transformations

Abstract A variational Monte Carlo study using inhomogeneous backflow transformations is reported for the [H−;e+] and [Li−;e+] positronic compounds. The backflow transformations greatly improve the total energies, positron affinities, and two-photon annihilation rates. The Slater–Jastrow-backflow trial wave function is well suited to the theoretical analysis of positronic compounds.

The quantum Monte Carlo method—electron correlation from random numbers (abstract only)

The fixed-node diffusion quantum Monte Carlo (DMC) method is the most accuratemethod known for calculating the energies of large many-particle quantum systems.The key element of the method is the development of accurate trial many-bodywavefunctions which control the statistical efficiency of the calculations and theaccuracy obtained. Accurate wavefunctions can be obtained by building correlationeffects on top of mean field descriptions such as density functional theory. Thewavefunctions can be improved by introducing multi-determinants, pairing functions,and backflow transformations. The calculations are expensive, but the methodscales well with system size and calculations on 1000 particles are possible. Somerecent applications of the DMC method to atoms, molecules and solids will bepresented.

Energies of the first row atoms from quantum Monte Carlo

All-electron variational and diffusion quantum Monte Carlo calculations of the ground state energies of the first row atoms (from Li to Ne) are reported. The authors use trial wave functions of four types: single-determinant Slater-Jastrow wave functions, multideterminant Slater-Jastrow wave functions, single-determinant Slater-Jastrow wave functions with backflow transformations, and multideterminant Slater-Jastrow wave functions with backflow transformations. At the diffusion quantum Monte Carlo level and using their multideterminant Slater-Jastrow wave functions with backflow transformations, they recover 99% or more of the correlation energies for Li, Be, B, C, N, and Ne, 97% for O, and 98% for F.

Inhomogeneous backflow transformations in quantum Monte Carlo calculations

An inhomogeneous backflow transformation for many-particle wave functions is presented and applied to electrons in atoms, molecules, and solids. We report variational and diffusion quantum Monte Carlo (VMC and DMC) energies for various systems and study the computational cost of using backflow wave functions. We find that inhomogeneous backflow transformations can provide a substantial increase in the amount of correlation energy retrieved within VMC and DMC calculations. The backflow transformations significantly improve the wave functions and their nodal surfaces.

Thrombosis of the popliteal vein.

Among 3,307 consecutive patients (3,556 legs) with deep venous thrombosis, 54 (1.5%) showed an isolated thrombus of the popliteal vein on phlebography. The majority of those had a history of "effort" or long lasting flexion during air or bus travel. Forty-four percent suffered from pulmonary embolism as the first sign of deep venous thrombosis. Functional phlebography demonstrated the primary site of thrombosis at folds forming in the vein wall at flexion. In order to further elucidate the pathogenetic mechanism, 158 popliteal veins were examined phlebographically in different functional states revealing age-related characteristic wall patterns of rings and folds in flexion causing transient impairment of flow. Complementary morphological studies of 120 popliteal veins during autopsy showed a transverse rippling of the vein wall caused by intimal fibrosis and partial atrophy of the media corresponding to the phlebographic findings. It is concluded that microtrauma during effort in combination with impaired venous backflow and fibrotic transformation of the venous wall can lead to thrombus formation in the popliteal vein.