Ground State Wave Function Research Articles

Efficient Implementation of Block-Correlated Coupled Cluster Theory Based on the Generalized Valence Bond Reference for Strongly Correlated Systems

An optimized implementation of block-correlated coupled cluster theory based on the generalized valence bond wave function (GVB-BCCC) for the singlet ground state of strongly correlated systems is presented. The GVB-BCCC method with two-pair correlation (GVB-BCCC2b) or up to three-pair correlation (GVB-BCCC3b) will be the focus of this work. Three major techniques have been adopted to dramatically accelerate GVB-BCCC2b and GVB-BCCC3b calculations. First, the GVB-BCCC2b and GVB-BCCC3b codes are noticeably optimized by removing redundant calculations. Second, independent amplitudes are identified by constraining excited configurations to be pure singlet states and only independent amplitudes need to be solved. Third, an incremental updating scheme for the amplitudes in solving the GVB-BCCC equations is adopted. With these techniques, accurate GVB-BCCC3b calculations are now accessible for systems with relatively large active spaces (50 electrons in 50 orbitals) and GVB-BCCC2b calculations are affordable for systems with much larger active spaces. We have applied GVB-BCCC methods to investigate three typical kinds of systems: polyacenes, pentaprismane, and [Cu2O2]2+ isomers. For polyacenes, we demonstrate that GVB-BCCC3b can capture more than 94% of the total correlation energy even for 12-acene with 50 π electrons. For the potential energy curve of simultaneously stretching 15 C-C bonds in pentaprismane, our calculations show that the GVB-BCCC3b results are quite close to the results from the density matrix renormalization group (DMRG) over the whole range. For two dinuclear copper oxide isomers, their relative energy predicted by GVB-BCCC3b is also in good accord with the DMRG result. All calculations show that the inclusion of three-pair correlation in GVB-BCCC is critical for accurate descriptions of strongly correlated systems.

Rayleigh-Gauss-Newton optimization with enhanced sampling for variational Monte Carlo

Variational Monte Carlo (VMC) is an approach for computing ground-state wave functions that has recently become more powerful due to the introduction of neural network-based wave-function parametrizations. However, efficiently training neural wave functions to converge to an energy minimum remains a difficult problem. In this work, we analyze optimization and sampling methods used in VMC and introduce alterations to improve their performance. First, based on theoretical convergence analysis in a noiseless setting, we motivate a new optimizer that we call the Rayleigh-Gauss-Newton method, which can improve upon gradient descent and natural gradient descent to achieve superlinear convergence at no more than twice the computational cost. Second, to realize this favorable comparison in the presence of stochastic noise, we analyze the effect of sampling error on VMC parameter updates and experimentally demonstrate that it can be reduced by the parallel tempering method. In particular, we demonstrate that RGN can be made robust to energy spikes that occur when the sampler moves between metastable regions of configuration space. Finally, putting theory into practice, we apply our enhanced optimization and sampling methods to the transverse-field Ising and $XXZ$ models on large lattices, yielding ground-state energy estimates with remarkably high accuracy after just 200 parameter updates.

A note on shape invariant potentials for discretized Hamiltonians

Using the method of the “exact discretization” of the Schrödinger equation, we propose a particular discretized version of the [Formula: see text] Supersymmetric Quantum Mechanics (SQM). After defining the corresponding shape invariance condition, we show that the energy spectra and wave functions for discretized Quantum Mechanical systems can be found using the technique of [Formula: see text] Supersymmetric Quantum Mechanics exactly the same way as it is done for their continuous counterparts. As a demonstration of the present method, we find the energy spectrum for a discretized Coulomb potential and its ground state wave function.

Feed-forward neural network based variational wave function for the fermionic Hubbard model in one dimension

We explore the suitability of a feed-forward neural network (FNN) to represent the ground state of the fermionic Hubbard model in one dimension (1D). We consider the model at half-filling, represent the ground state wave function in terms of an FNN and optimize it using the variational Monte Carlo (VMC) method. The results are compared with the exact Bethe Ansatz solution. We find that for lattice sizes which give a ‘filled-shell’ condition for the non-interacting Fermi sea wave function, a simple FNN performs very well at all values of Hubbard interaction U. For lattice sizes where this condition is not obtained, the simple FNN fails and we find a modified network with a ‘sign’ component (sFNN) to work in such cases. On the flip side, though we find the FNN to be successful in providing an unbiased variational wave function for the fermionic many-body system in 1D, the computational cost for the wave function scales up rapidly with lattice size which limits its applicability.

Deep-neural-network solution of piezo-phototronic transistor based on GaN/AlN quantum wells

Strain-induced polarization can control the wavelength of intersubband transition in piezo-phototronic transistor based on GaN/AlN quantum wells. Neural networks can be used for solving the wavelength because their hidden layers can accurately approximate any continuous functions. In this study, two feed-forward neural network (FNN) models have been developed for obtaining the wave functions of ground state and first excited state in piezo-phototronic GaN/AlN quantum well. This method provides an effective way of the numerical computation for design and optimization of piezotronic and piezo-phototronic devices.

Intermolecular charge-transfer complexes between chlorothiazide antihypertensive drug against iodine sigma and picric acid pi acceptors: DFT and molecular docking interaction study with Covid-19 protease

The interaction of chlorothiazide (CH) as donor (D) with picric acid (PA) and iodine (I2) as π- and σ-acceptors (A), respectively, gives charge-transfer (CT) complexes as a final products. The reaction of donor and acceptors were studied spectrophotometrically. The complexes are generally of the n-π* and n-σ* types, with the ground state wave function primarily characterized by the non-bonding structure. For the micro determination of chlorothiazide using picric acid and iodine as acceptors, the ideal conditions encouraging the formation of complexes are thoroughly explored. It was discovered that the stoichiometry of the molecular structure is 1:1 (D:A). The equilibrium constant and the molar extinction coefficient were calculated using Benesi-Hildebrand and its modifications. DFT/TD-DFT calculations with B3LYP/LanL2DZ and 6-311G++ level of theory were used to provide comparable theoretical data along with electronic energy gap of HOMO→LUMO. Molecular docking calculations have been performed between CT complexes and Covid-19 protease (6LU7) to study the interaction between them and their inhibitory effect.

Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian: Special Cases

In a previous paper, we used a classification of the self adjoint extensions, also called self-adjoint determinations, of the differential operator −d2/dx2 in order to obtain the whole list of Supersymmetric (SUSY) partners of those selfadjoint determinations for which the ground state has strictly positive energy. The existence of self adjoint determinations with a ground state of zero or even negative energy is a proved fact. In this paper, we analyze the possibility of constructing SUSY partners for those determinations. We also study those cases for which the ground state has a degeneracy, the study of their SUSY partners should be analyzed separately. So far, we have studied those determinations having an exactly solvable eigenvalue problem. On the present study, we also included some comments in relation to determinations not exactly solvable from this point of view. In addition, the use of self adjoint determinations for which the ground state wave function has nodes (zeroes) produces formal SUSY partners with a finite number of eigenvalues or even with a purely continuous spectrum. We give some worked examples of these situations.

Coupled Cluster and Perturbation Theories Based on a Cluster Mean-Field Reference Applied to Strongly Correlated Spin Systems.

We introduce perturbation and coupled-cluster theories based on a cluster mean-field reference for describing the ground state of strongly correlated spin systems. In cluster mean-field, the ground state wave function is written as a simple tensor product of optimized cluster states. The cluster language and the mean-field nature of the ansatz allow for a straightforward improvement which uses perturbation theory and coupled-cluster to account for intercluster correlations. We present benchmark calculations on the 1D chain and 2D square J1-J2 Heisenberg model, using cluster mean-field, perturbation theory, and coupled-cluster. We also present an extrapolation scheme that allows us to compute thermodynamic limit energies accurately. Our results indicate that, with sufficiently large clusters, the correlated methods (cPT2, cPT4, and cCCSD) can provide a relatively accurate description of the Heisenberg model in the regimes considered, which suggests that the methods presented can be used for other strongly correlated systems. Some ways to improve upon the methods presented in this work are discussed.

Estimating the Euclidean quantum propagator with deep generative modeling of Feynman paths

Feynman path integrals provide an elegant, classically inspired representation for the quantum propagator and the quantum dynamics, through summing over a huge manifold of all possible paths. From computational and simulational perspectives, the ergodic tracking of the whole path manifold is a hard problem. Machine learning can help, in an efficient manner, to identify the relevant subspace and the intrinsic structure residing at a small fraction of the vast path manifold. In this work, we propose the Feynman path generator for quantum mechanical systems, which efficiently generates Feynman paths with fixed endpoints, from a (low-dimensional) latent space and by targeting a desired density of paths in the Euclidean space-time. With such path generators, the Euclidean propagator as well as the ground-state wave function can be estimated efficiently for a generic potential energy. Our work provides an alternative approach for calculating the quantum propagator and the ground-state wave function, paves the way toward generative modeling of quantum mechanical Feynman paths, and offers a different perspective to understand the quantum-classical correspondence through deep learning.

Density Matrix Renormalization Group for Continuous Quantum Systems.

We introduce a versatile and practical framework for applying matrix product state techniques to continuous quantum systems. We divide space into multiple segments and generate continuous basis functions for the many-body state in each segment. By combining this mapping with existing numerical density matrix renormalization group routines, we show how one can accurately obtain the ground-state wave function, spatial correlations, and spatial entanglement entropy directly in the continuum. For a prototypical mesoscopic system of strongly interacting bosons we demonstrate faster convergence than standard grid-based discretization. We illustrate the power of our approach by studying a superfluid-insulator transition in an external potential. We outline how one can directly apply or generalize this technique to a wide variety of experimentally relevant problems across condensed matter physics and quantum field theory.

TeNeS: Tensor network solver for quantum lattice systems

TeNeS (Tensor Network Solver) [1,2] is a free/libre open-source software program package for calculating two-dimensional many-body quantum states based on the tensor network method and the corner transfer matrix renormalization group (CTMRG) method. This package calculates ground-state wavefunctions for user-defined Hamiltonians and evaluates user-defined physical quantities such as magnetization and correlation functions. For certain predefined models and lattices, there is a tool that makes it easy to generate input files. TeNeS uses an OpenMP/MPI hybrid parallelized tensor operation library and thus can perform large-scale calculations using massively parallel machines. Program summaryProgram Title: TeNeSCPC Library link to program files:https://doi.org/10.17632/psm26xxbvd.1Code Ocean capsule:https://codeocean.com/capsule/1510058Licensing provisions: GNU General Public License version 3Programming language:C++ and python3External routines/libraries: mptensorNature of problem: TeNeS calculates the approximate ground states and their properties of user-defined two-dimensional quantum lattice models using user-friendly input files. Numerically exact solutions of such tasks generally require an exponentially diverging computational time, whereas the error in the output of TeNeS is well controlled and can be reduced with a polynomial cost.Solution method: TeNeS implements the tensor networks method based on a tensor-product-state (TPS) wavefunction and the corner transfer matrix renormalization group method. TeNeS also supports massively parallel computing using the library mptensor, which implements OpenMP/MPI hybrid parallelized tensor operations.

Explicitly Correlated Electronic Structure Calculations with Transcorrelated Matrix Product Operators.

In this work, we present the first implementation of the transcorrelated electronic Hamiltonian in an optimization procedure for matrix product states by the density matrix renormalization group (DMRG) algorithm. In the transcorrelation ansatz, the electronic Hamiltonian is similarity-transformed with a Jastrow factor to describe the cusp in the wave function at electron-electron coalescence. As a result, the wave function is easier to approximate accurately with the conventional expansion in terms of one-particle basis functions and Slater determinants. The transcorrelated Hamiltonian in first quantization comprises up to three-body interactions, which we deal with in the standard way by applying robust density fitting to two- and three-body integrals entering the second-quantized representation of this Hamiltonian. The lack of hermiticity of the transcorrelated Hamiltonian is taken care of along the lines of the first work on transcorrelated DMRG [ J. Chem. Phys. 2020, 153, 164115] by encoding it as a matrix product operator and optimizing the corresponding ground state wave function with imaginary-time time-dependent DMRG. We demonstrate our quantum chemical transcorrelated DMRG approach at the example of several atoms and first-row diatomic molecules. We show that transcorrelation improves the convergence rate to the complete basis set limit in comparison to conventional DMRG. Moreover, we study extensions of our approach that aim at reducing the cost of handling the matrix product operator representation of the transcorrelated Hamiltonian.

Spherical quantum dot in modified Kratzer–Coulomb potential: study of optical rectification coefficients

We consider the effects of quantum dot radius, confinement potential depth and controllable effective mass on the optical rectification coefficient (ORC) in spherical quantum dots, which is confined with Modified Kratzer–Coulomb Potential (MKCP). Using the Nikiforov–Uvarov method and compact density matrix theory, the ground state energy, ORC and wave function of electrons under the combined action of many factors are calculated. The results show that they affect the optical rectification response from different angles, including the position of peak and formant.

Positron impact elastic scattering by calcium atom using first order distorted wave born approximation with a complex potential

Precise positron scattering cross section data are essential for the development of a wide range of technological fields such as astrophysics, plasma sciences, material sciences and bio-medicine. Thus, lack of complete agreement among the available theoretical data, unavailability of experimental data and the first-order distorted wave born approximation with a complex potential at impact energy range 10-200 eV not having been utilized in such a study; are the reasons that informed this investigation. After formulating the complex potential, with positronium formation channel proficiently incorporated, the radial equation was solved numerically. The extracted phase shifts were used to compute the transition matrix. The Roothaan-Hartree-Fock ground state wavefunctions of calcium were utilized in evaluation of the distorting potential and the transition matrix. The transition matrix was then used to determine the differential cross sections while the optical theorem was used to compute total cross sections. With only static potential, the present distorted wave Born approximation and optical potential models yield results that are in close agreement while the effect of positronium formation channel on the cross sections is insignificant at impact energies beyond 150 eV.

Combined unitary and symmetric group approach applied to low-dimensional Heisenberg spin systems

A novel combined unitary and symmetric group approach is used to study the spin-$\frac{1}{2}$ Heisenberg model and related Fermionic systems in a total spin-adapted representation, using a linearly-parameterised Ansatz for the many-body wave function. We show that a more compact ground-state wave function representation---indicated by a larger leading ground-state coefficient---is obtained when combining the symmetric group ${\mathcal{S}}_{n}$, in the form of permutations of the underlying lattice site ordering, with the cumulative spin coupling based on the unitary group $\mathrm{U}(n)$. In one-dimensional systems the observed compression of the wave function is reminiscent of block-spin renormalization group approaches, and allows us to study larger lattices (here taken up to 80 sites) with the spin-adapted full configuration interaction quantum Monte Carlo method, which benefits from the sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates that emerge from the reordering. We find that in an optimal lattice ordering the configuration state function with highest weight already captures with high accuracy the spin-spin correlation function of the exact ground-state wave function. This feature is found for more general lattice models, such as the Hubbard model, and ab initio quantum chemical models, exemplified by one-dimensional hydrogen chains. We also provide numerical evidence that the optimal lattice ordering for the unitary group approach is not generally equivalent to the optimal ordering obtained for methods based on matrix-product states, such as the density-matrix renormalization group approach.

Bosonic field digitization for quantum computers

Quantum simulation of quantum field theory is a flagship application of quantum computers that promises to deliver capabilities beyond classical computing. The realization of quantum advantage will require methods that can accurately predict error scaling as a function of the resolution and parameters of the model and that can be implemented efficiently on quantum hardware. In this paper, we address the representation of lattice bosonic fields in a discretized field amplitude basis, develop methods to predict error scaling, and present efficient qubit implementation strategies. A low-energy subspace of the bosonic Hilbert space, defined by a boson occupation number cutoff, can be represented with exponentially good accuracy by a low-energy subspace of a finite-size Hilbert space. The finite representation construction and the associated errors are directly related to the accuracy of the Nyquist-Shannon sampling and the finite Fourier transforms of the boson number states in the field and the conjugate-field bases. We analyze the relation between the boson mass, the discretization parameters used for wave function sampling, and the finite representation size. Numerical simulations of small size ${\mathrm{\ensuremath{\Phi}}}^{4}$ problems demonstrate that the boson mass optimizing the sampling of the ground state wave function is a good approximation to the optimal boson mass yielding the minimum low-energy subspace size. However, we find that accurate sampling of general wave functions does not necessarily result in accurate representation. We develop methods for validating and adjusting the discretization parameters to achieve more accurate simulations.

Chiral Central Charge from a Single Bulk Wave Function.

A (2+1)-dimensional gapped quantum many-body system can have a topologically protected energy current at its edge. The magnitude of this current is determined entirely by the temperature and the chiral central charge, a quantity associated with the effective field theory of the edge. We derive a formula for the chiral central charge that, akin to the topological entanglement entropy, is completely determined by the many-body ground state wave function in the bulk. According to our formula, nonzero chiral central charge gives rise to a topological obstruction that prevents the ground state wave function from being real valued in any local product basis.

Finite-volume pionless effective field theory for few-nucleon systems with differentiable programming

Finite-volume pionless effective field theory provides an efficient framework for the extrapolation of nuclear spectra and matrix elements calculated at finite volume in lattice QCD to infinite volume, and to nuclei with larger atomic number. In this work, it is demonstrated how this framework may be implemented via a set of correlated Gaussian wave functions optimized using differentiable programming and via solution of a generalized eigenvalue problem. This approach is shown to be significantly more efficient than a stochastic implementation of the variational method based on the same form of correlated Gaussian wave functions, yielding comparably accurate representations of the ground-state wave functions with an order of magnitude fewer terms. The efficiency of representation allows such calculations to be extended to larger systems than in previous work. The method is demonstrated through calculations of the binding energies of nuclei with atomic number $A\ensuremath{\in}{2,3,4}$ in finite volume, matched to lattice QCD calculations at quark masses corresponding to ${m}_{\ensuremath{\pi}}=806\text{ }\text{ }\mathrm{MeV}$, and infinite-volume effective field theory calculations of $A\ensuremath{\in}{2,3,4,5,6}$ systems based on this matching.

Exploring the accuracy and usefulness of semi-empirically scaled ADC schemes by blending second and third order terms.

Different approaches to mixed-order algebraic-diagrammatic construction (ADC) schemes are investigated. The performance of two different strategies for scaling third-order contributions to the ADC secular matrix is evaluated. Both considered schemes employ a single tuning parameter and conserve general properties inherent to all ADC methods, such as hermiticity and size-consistency. The first approach, scaled-matrix ADC[(2) + x(3)], scales all contributions first occurring in ADC(3) equally and leads to an improvement of the accuracy of excitation energies compared to ADC(3) for x = 0.4-0.5. However, with respect to excited state dipole moments, this method provides lower accuracy than ADC(3). The second scaling approach, MP[(1) + x(2)] - ISR(3), scales the second order contributions of the ground-state wavefunction and derives a rigorous ADC scheme via the intermediate state representation formalism. Although the error in excitation energies is not improved, this method provides insight into the relevance of the individual terms of the ADC(3) matrix and indicates that the MP(2) wavefunction is, indeed, the optimal reference wavefunction for deriving a third-order single-reference ADC scheme.

Coulomb scattering in the massless Nelson model IV. Atom–electron scattering

We consider the massless Nelson model with two types of massive particles which we call atoms and electrons. The atoms interact with photons via an infrared regular form-factor and thus they are Wigner-type particles with sharp mass-shells. The electrons have an infrared singular form-factor and thus they are infraparticles accompanied by soft-photon clouds correlated with their velocities. In the weak coupling regime, we construct scattering states of one atom and one electron, and demonstrate their asymptotic clustering into individual particles. The proof relies on the Cook’s argument, clustering estimatesblack, and the non-stationary phase method. The latter technique requires sharp estimates on derivatives of the ground state wave functions of the fiber Hamiltonians of the model, which were proven in the earlier papers of this series. Although we rely on earlier black studies of the atom–atom and electron–photon scattering in the Nelson model, the paper is written in a self-contained manner. A perspective on the open problem of the electron–electron scattering in this model is also given.