Chain Of Hydrogen Atoms Research Articles

Constructing Hubbard models for the hydrogen chain using sliced-basis density matrix renormalization group

Sliced-basis DMRG(sb-DMRG) is used to simulate a chain of hydrogen atoms and to construct low-energy effective Hubbard-like models. The downfolding procedure first involves a change of basis to a set of atom-centered Wannier functions constructed from the natural orbitals of the exact DMRG one-particle density matrix. The Wannier function model is then reduced to a fewer-parameter Hubbard-like model, whose parameters are determined by minimizing the expectation value of the Wannier Hamiltonian in the ground state of the Hubbard Hamiltonian. This indirect variational procedure not only yields compact and simple models for the hydrogen chain, but also allows us to explore the importance of constraints in the effective Hamiltonian, such as the restricting the range of the single-particle hopping and two-particle interactions, and to assess the reliability of more conventional downfolding. The entanglement entropy for a model's ground state, cut in the middle, is an important property determining the ability of DMRG and tensor networks to simulate the model, and we study its variation with the range of the interactions. Counterintuitively, we find that shorter ranged interactions often have larger entanglement.

Symmetry-Projected Jastrow Mean-Field Wave Function in Variational Monte Carlo.

We extend our low-scaling variational Monte Carlo (VMC) algorithm to optimize the symmetry-projected Jastrow mean-field (SJMF) wave functions. These wave functions consist of a symmetry-projected product of a Jastrow and a general broken-symmetry mean-field reference. Examples include Jastrow antisymmetrized geminal power, Jastrow-Pfaffians, and resonating valence bond states, among others, all of which can be treated with our algorithm. We will demonstrate using benchmark systems including the nitrogen molecule, a chain of hydrogen atoms, and 2-D Hubbard model that a significant amount of correlation can be obtained by optimizing the energy of the SJMF wave function. This can be achieved at a relatively small cost when multiple symmetries including spin, particle number, and complex conjugation are simultaneously broken and projected. We also show that reduced density matrices can be calculated using the optimized wave functions, which allows us to calculate other observables such as correlation functions and will enable us to embed the VMC algorithm in a complete active-space self-consistent field calculation.

Ultrafast ab Initio Quantum Chemistry Using Matrix Product States.

Ultrafast dynamics in chemical systems provide a unique access to fundamental processes at the molecular scale. A proper description of such systems is often very challenging because of the quantum nature of the problem. The concept of matrix product states (MPS), however, has proven its performance in describing such correlated quantum systems in recent years for a wide range of applications. In this work, we continue the development of the MPS approach to study ultrafast electron dynamics in quantum chemical systems. The method combines time evolution schemes, such as fourth-order Runge-Kutta and Krylov space time evolution, with MPS, in order to solve the time-dependent Schrödinger equation efficiently. This allows for describing electron dynamics in molecules on a full configurational interaction (CI) level for a few femtoseconds after excitation. As a benchmark, we compare MPS-based calculations to full CI calculations for a chain of hydrogen atoms and for the water molecule. Krylov space time evolution is in particular suited for the MPS approach, as it provides a wide range of opportunities to be adjusted to the reduced MPS dimension case. Finally, we apply the MPS approach to describe charge migration effects in iodoacetylene and find direct agreement between our results and experimental observations.

Improved Speed and Scaling in Orbital Space Variational Monte Carlo.

In this work, we introduce three algorithmic improvements to reduce the cost and improve the scaling of orbital space variational Monte Carlo (VMC). First, we show that, by appropriately screening the one- and two-electron integrals of the Hamiltonian, one can improve the efficiency of the algorithm by several orders of magnitude. This improved efficiency comes with the added benefit that the cost of obtaining a constant error per electron scales as the second power of the system size O( N2), down from the fourth power O( N4). Using numerical results, we demonstrate that the practical scaling obtained is, in fact, O( N1.5) for a chain of hydrogen atoms. Second, we show that, by using the adaptive stochastic gradient descent algorithm called AMSGrad, one can optimize the wave function energies robustly and efficiently. Remarkably, AMSGrad is almost as inexpensive as the simple stochastic gradient descent but delivers a convergence rate that is comparable to that of the Stochastic Reconfiguration algorithm, which is significantly more expensive and has a worse scaling with the system size. Third, we introduce the use of the rejection-free continuous time Monte Carlo (CTMC) to sample the determinants. Unlike the first two improvements, CTMC does come at an overhead that the local energy must be calculated at every Monte Carlo step. However, this overhead is mitigated to a large extent because of the reduced scaling algorithm, which ensures that the asymptotic cost of calculating the local energy is equal to that of updating the walker. The resulting algorithm allows us to calculate the ground state energy of a chain of 160 hydrogen atoms using a wave function containing ∼2 × 105 variational parameters with an accuracy of 1 mEh/particle at a cost of just 25 CPU h, which when split over 2 nodes of 24 processors each amounts to only about half hour of wall time. This low cost coupled with embarrassing parallelizability of the VMC algorithm and great freedom in the forms of usable wave functions, represents a highly effective method for calculating the electronic structure of model and ab initio systems.

Charge density wave in hydrogen at high pressure

We present extensive molecular dynamics (MD) simulations investigating numerous candidate crystal structures for hydrogen in conditions around the present experimental frontier (400 GPa). Spontaneous phase transitions in the simulations reveal a new structure candidate comprising twofold coordinated chains of hydrogen atoms. We explain the electronic structure of this phase in terms of a charge density wave and calculate its experimental signature. In detailed tests of the accuracy of our calculation, we find that k-point sampling is far more important in MD than in static calculations, because of the freedom it give the atoms to rearrange themselves optimally for the given sampling.

Towards the Solution of the Many-Electron Problem in Real Materials: Equation of State of the Hydrogen Chain with State-of-the-Art Many-Body Methods

We present numerical results for the equation of state of an infinite chain of hydrogen atoms. A variety of modern many-body methods are employed, with exhaustive cross-checks and validation. Approaches for reaching the continuous space limit and the thermodynamic limit are investigated, proposed, and tested. The detailed comparisons provide a benchmark for assessing the current state of the art in many-body computation, and for the development of new methods. The ground-state energy per atom in the linear chain is accurately determined versus bondlength, with a confidence bound given on all uncertainties.

Describing a Strongly Correlated Model System with Density Functional Theory.

The linear chain of hydrogen atoms, a basic prototype for the transition from a metal to Mott insulator, is studied with a recent density functional theory model functional for nondynamic and strong correlation. The computed cohesive energy curve for the transition agrees well with accurate literature results. The variation of the electronic structure in this transition is characterized with a density functional descriptor that yields the atomic population of effectively localized electrons. These new methods are also applied to the study of the Peierls dimerization of the stretched even-spaced Mott insulator to a chain of H2 molecules, a different insulator. The transitions among the two insulating states and the metallic state of the hydrogen chain system are depicted in a semiquantitative phase diagram. Overall, we demonstrate the capability of studying strongly correlated materials with a mean-field model at the fundamental level, in contrast to the general pessimistic view on such a feasibility.

Pure density functional for strong correlation and the thermodynamic limit from machine learning

We use density-matrix renormalization group, applied to a one-dimensional model of continuum Hamiltonians, to accurately solve chains of hydrogen atoms of various separations and numbers of atoms. We train and test a machine-learned approximation to $F[n]$, the universal part of the electronic density functional, to within quantum chemical accuracy. Our calculation (a) bypasses the standard Kohn-Sham approach, avoiding the need to find orbitals, (b) includes the strong correlation of highly-stretched bonds without any specific difficulty (unlike all standard DFT approximations) and (c) is so accurate that it can be used to find the energy in the thermodynamic limit to quantum chemical accuracy.

Calculation of space localized properties in correlated quantum Monte Carlo methods with reweighting: the nonlocality of statistical uncertainties.

We study the efficiency of quantum Monte Carlo (QMC) methods in computing space localized ground state properties (properties which do not depend on distant degrees of freedom) as a function of the system size N. We prove that for the commonly used correlated sampling with reweighting method, the statistical fluctuations σ2(N) do not obey the locality property. σ2(N) grow at least linearly with N and with a slope that is related to the fluctuations of the reweighting factors. We provide numerical illustrations of these tendencies in the form of QMC calculations on linear chains of hydrogen atoms.

Anomalous nonadditive dispersion interactions in systems of three one-dimensional wires

The non-additive dispersion contribution to the binding energy of three one-dimensional (1D) wires is investigated using wires modelled by (i) chains of hydrogen atoms and (ii) homogeneous electron gases. We demonstrate that the non-additive dispersion contribution to the binding energy is significantly enhanced compared with that expected from Axilrod-Teller-Muto-type triple-dipole summations and follows a different power-law decay with separation. The triwire non-additive dispersion for 1D electron gases scales according to the power law $d^{-\beta}$, where $d$ is the wire separation, with exponents $\beta(r_s)$ smaller than 3 and slightly increasing with $r_s$ from 2.4 at $r_s = 1$ to 2.9 at $r_s=10$, where $r_s$ is the density parameter of the 1D electron gas. This is in good agreement with the exponent $\beta=3$ suggested by the leading-order charge-flow contribution to the triwire non-additivity, and is a significantly slower decay than the $\sim d^{-7}$ behaviour that would be expected from triple-dipole summations.

Strong electronic correlation in the hydrogen chain: A variational Monte Carlo study

In this article, we report a fully ab initio variational Monte Carlo study of the linear, and periodic chain of Hydrogen atoms, a prototype system providing the simplest example of strong electronic correlation in low dimensions. In particular, we prove that numerical accuracy comparable to that of benchmark density matrix renormalization group calculations can be achieved by using a highly correlated Jastrow-antisymmetrized geminal power variational wave function. Furthermore, by using the so-called "modern theory of polarization" and by studying the spin-spin and dimer-dimer correlations functions, we have characterized in details the crossover between the weakly and strongly correlated regimes of this atomic chain. Our results show that variational Monte Carlo provides an accurate and flexible alternative to highly correlated methods of quantum chemistry which, at variance with these methods, can be also applied to a strongly correlated solid in low dimensions close to a crossover or a phase transition.

Chaotic versus Nonchaotic Stochastic Dynamics in Monte Carlo Simulations: A Route for Accurate Energy Differences inN-Body Systems

We present a method to efficiently evaluate small energy differences of two close N-body systems by employing stochastic processes having a stability versus chaos property. By using the same random noise, energy differences are computed from close trajectories without reweighting procedures. The approach is presented for quantum systems but can be applied to classical N-body systems as well. It is exemplified with diffusion Monte Carlo simulations for long chains of hydrogen atoms and molecules for which it is shown that the long-standing problem of computing energy derivatives is solved.

Peculiarities of electron exchange between a negative hydrogen ion and atomic chains

This paper is dedicated to studying electron exchange between a negative hydrogen ion and an atomic chain with subsequent electron transport along the linear chain of hydrogen atoms. This process occurs after the electron transition from the negative hydrogen atom to the atomic chain. In our calculations, a method of wave packet propagation not using perturbation theory was applied. It is shown that the effectiveness of electron transport decreases uniformly as the distance from the ion to the chain increases. The electron propagates along the chain uniformly; in this case, the electron velocity is almost independent of the ion-chain distance.

AMO in linear chains of hydrogen atoms revisited

Detailed numerical studies of the three related methods—restricted Hartree-Fock (RHF), alternant molecular orbital (AMO) with one and several coupling parameters, and unrestricted Hartree-Fock (UHF)—have been made for a linear chain of hydrogen atoms with uniform distances. The long-range electrostatic interactions have been properly included.

Electronic excited-state energies from a linear response theory based on the ground-state two-electron reduced density matrix

Ground-state two-particle reduced density matrices (2-RDMs) are used to calculate excited-state energy spectra. Solving the Schrodinger equation for excited states dominated by single excitations from the ground-state wavefunction requires the ground-state 2- and 3-RDMs. The excited states, however, can be obtained without a knowledge of the ground-state 3-RDM by two methods: (i) cumulant expansion methods which build the 3-RDM from the 2-RDM, and (ii) double commutator methods which eliminate the 3-RDM. Previous work [Mazziotti, Phys. Rev. A 68, 052501 (2003)] examined the accuracy of excited states extracted from ground-state 2-RDMs, which were calculated by full configuration interaction or the variational 2-RDM method. In this work we employ (i) advances in semidefinite programming to treat the excited states of water and hydrogen fluoride and chains of hydrogen atoms, and (ii) the addition of partial three-particle N-representability conditions to compute more accurate ground-state 2-RDMs. With the hydrogen chains we examine the metal-to-insulator transition as measured by the band gap (the difference between the ground-state and the first excited-state energies), which is difficult for excited-state methods to capture.

Alan Graham MacDiarmid

Alan Graham MacDiarmid

Group functions for the analysis of the electronic structures of polymers

The ab initio calculations of the electronic structures incorporating periodic boundary conditions allow to bring the well-developed standard methods of molecular quantum chemistry into the realm of solid-state physics and chemistry, especially important if the correlated treatment of the electronic structure is necessary. In this paper, we consider the general electron-group functions formalism for the electronic structure calculations as applied to one-dimensional periodic systems. The method for polymers is implemented in a close analogy with the recently proposed method for molecules. It allows us to calculate one-dimensional systems with different distributions of orbitals and electrons among groups which are subject to the strong orthogonality condition. The wave function is optimized both with respect to one- and two-electron density matrices for electron groups as well as the structure of molecular orbitals forming their carrier spaces. The method is applied to model systems: chains of hydrogen atoms, a linear hydrogen fluoride chain, and a lithium hydride chain. The convergence of the energy with respect to the explicitly covered range of interactions and the long-ranged Coulomb interactions is studied. It is shown that the local properties of one-electron states are very important for the whole performance of the method. The flexibility of the wave function based on the choice of electron groups is thoroughly analyzed.

Theoretical studies on magnetic interaction in one‐dimensional spin chains of hydrogen atoms (Hn) and copper bromide (CunBrm)

AbstractTheoretical calculations of ab initio molecular orbital and density functional theory calculations were carried out in order to investigate magnetic interaction in one‐dimensional spin chains. Effective exchange integral (Jab) values are useful parameters in studying magnetic interaction in spin systems. Chemical indices also were evaluated to represent stability of broken symmetry spin. For one set of spin chains, alignment of hydrogen atoms as the most simple organic cluster was employed. Number of H atoms in employed models is two and 50 with interatomic distance 2.1 Å. In addition, copper bromide chains (CunBrm) were also studied. Simple molecules Br2‐Cu‐Br2‐Cu‐Br2 (without apical‐Br) and Br2‐CuBr‐Br2‐CuBr‐Br2 (with apical‐Br) were employed for simple model structure. Four parameters (r, a, b, and c) can fix the structure and study parameter dependency of Jab values. For experimental studies, Sugimoto and his coworkers have reported synthesis and experimental studies for complex of both CunBrm chains and TTF‐derivative donor. Magnetic behavior in only CunBrm chains was investigated in detail by several ab initio methods. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Locality of the reduced-density-matrices: a numerical study

We have considered the expansion of the two-body reduced density matrices (2-RDM) in terms of the two-body cumulant (2-CRDM) which represents pure two-body correlation effects. It possesses also the advantage to be extensive contrarily to the 2-RDM. Thus, the 2-CRDM of a whole system should be well described in a restricted part of the system if one works in a localized basis set of orbitals. We studied the locality of the 2-CRDM, and consequently the correlation energy, for linear chain of hydrogen atoms H N ( N=6,…,12) at different geometries of their dimerization. Our results are encouraging for developing low computational cost ab initio methods using the 1- and 2-RDM formalism.

Bonding and diffusion in hydrogenated amorphous silicon by tight-binding molecular dynamics

The structural and bonding properties of hydrogenated amorphous silicon are investigated by tight-binding molecular dynamics. The formation of clusters and chains of hydrogen atoms is observed when quenching a hydrogenated liquid silicon sample to room temperature. In addition, the high-temperature stability of the amorphous network is investigated as a function of the hydrogen content of the sample. The amorphous silicon lattice is found to be unstable at high temperature when the hydrogen content is above 10%.