Full Configuration Interaction Calculations Research Articles

Assessing the global natural orbital functional approximation on model systems with strong correlation.

In the past decade, natural orbital functional (NOF) approximations have emerged as prominent tools for characterizing electron correlation. Despite their effectiveness, these approaches, which rely on natural orbitals and their associated occupation numbers, often require hybridization with other methods to fully account for all correlation effects. Recently, a global NOF (GNOF) has been proposed [Piris, Phys. Rev. Lett. 127, 233001 (2021)] to comprehensively address both dynamic and static correlations. This study evaluates the performance of GNOF on strongly correlated model systems, including comparisons with highly accurate Full Configuration Interaction calculations for hydrogen atom clusters in one, two, and three dimensions. Additionally, the investigation extends to a BeH2 reaction, involving the insertion of a beryllium atom into a hydrogen molecule along a C2v pathway. According to the results obtained using GNOF, consistent behavior is observed across various correlation regions, encompassing a range of occupations and orbital schemes. Furthermore, distinctive features are identified when varying the dimensionality of the system.

Ground state property calculations of LiHn complexes using IBM Qiskit’s quantum simulator

In this study, the variational quantum eigensolver (VQE) on a quantum simulator is used in calculating ground state electronic structure properties of the LiHn, n = 1–3, complexes including their singly charged ions. Results calculated using classical electronic structure algorithms are also included. We investigate the use of the unitary coupled cluster with singles and doubles (UCCSD) Ansatz using VQE within Qiskit and compare results to full configuration interaction (FCI) calculations. Computed ground state energies, electron affinities, ionization potentials, and dipole moments are considered. We report the first-of-its-kind simulated quantum computing results of selected LiHn species and use the parity orbital to qubit mapping scheme. We find that VQE/UCCSD results are comparable to classical coupled clusters with singles and doubles for all considered systems with respect to FCI. A VQE calculation cost evaluation is included in which we evaluate performance using both Jordan–Wigner and parity orbital to qubit mapping schemes. We also discuss some of the current limitations of utilizing VQE for the study of chemical systems.

Distributed Implementation of Full Configuration Interaction for One Trillion Determinants.

Full configuration interaction (FCI) can provide an exact molecular ground-state energy within a given basis set and serve as a benchmark for approximate methods in quantum chemical calculations, including the emerging variational quantum eigensolver. However, its exponential computational and memory requirements easily exceed the capability of a single server and limit its applicability to large molecules. In this paper, we present a distributed FCI implementation employing a hybrid parallelization scheme with multithreading and multiprocessing to expand FCI's applicability. We optimize this scheme to minimize the bottlenecks arising from interprocess communications and interthread data management. Our implementation achieves higher scalability than the naive combination of prior works and successfully calculates the exact energy of C3H8/STO-3G with 1.31 trillion determinants, which is the largest FCI calculation to the best of our knowledge. Furthermore, we provide a comprehensive list of FCI results with 136 combinations of molecules and basis sets for future evaluation and development of approximate methods.

Accurate and efficient calculations of Hellmann-Feynman forces for quantum computation.

First-order derivatives of energies with respect to atomic coordinates are widely computed and used in quantum chemistry simulations. The rapidly emerging technology of quantum computing offers a new paradigm for solving relevant quantum chemistry equations. In this work, we have achieved analytical calculations of atomic forces based on the Hellmann-Feynman theorem within the framework of the variational quantum eigensolver. The accuracy of the approach is demonstrated by calculating the atomic forces of H2, LiH, H2O, and NH3 molecules, which are in excellent agreement with values obtained from full configuration interaction calculations. In particular, for systems with degenerate molecular orbitals, the analytical approach has a significant accuracy advantage over finite-difference-based methods and will not involve additional computational effort on a quantum computer. The calculated forces are further used to optimize the geometries of NH3 and CH4 molecules and to perform abinitio molecular dynamics simulations for the umbrella inversion of NH3, demonstrating the feasibility of the approach in practical quantum chemistry simulations.

Periodic plane-wave electronic structure calculations on quantum computers

A procedure for defining virtual spaces, and the periodic one-electron and two-electron integrals, for plane-wave second quantized Hamiltonians has been developed, and it was validated using full configuration interaction (FCI) calculations, as well as executions of variational quantum eigensolver (VQE) circuits on Quantinuum’s ion trap quantum computers accessed through Microsoft’s Azure Quantum service. This work is an extension to periodic systems of a new class of algorithms in which the virtual spaces were generated by optimizing orbitals from small pairwise CI Hamiltonians, which we term as correlation optimized virtual orbitals with the abbreviation COVOs. In this extension, the integration of the first Brillouin zone is automatically incorporated into the two-electron integrals. With these procedures, we have been able to derive virtual spaces, containing only a few orbitals, that were able to capture a significant amount of correlation. The focus in this manuscript is on comparing the simulations of small molecules calculated with plane-wave basis sets with large periodic unit cells at the Gamma-point, including images, to results for plane-wave basis sets with aperiodic unit cells. The results for this approach were promising, as we were able to obtain good agreement between periodic and aperiodic results for an LiH molecule. Calculations performed on the Quantinuum H1-1 quantum computer produced surprisingly good energies, in which the error mitigation played a small role in the quantum hardware calculations and the (noisy) quantum simulator results. Using a modest number of circuit runs (500 shots), we reproduced the FCI values for the 1 COVO Hamiltonian with an error of 11 milliHartree, which is expected to improve with a larger number of circuit runs.

Quantum-Compute Algorithm for Exact Laser-Driven Electron Dynamics in Molecules.

In this work, we investigate the capability of known quantum computing algorithms for fault-tolerant quantum computing to simulate the laser-driven electron dynamics of excitation and ionization processes in small molecules such as lithium hydride, which can be benchmarked against the most accurate time-dependent full configuration interaction (TD-FCI) calculations. The conventional TD-FCI wave packet propagation is reproduced using the Jordan-Wigner transformation for wave function and operators and the Trotter product formula for expressing the propagator. In addition, the time-dependent dipole moment, as an example of a time-dependent expectation value, is calculated using the Hadamard test. To include non-Hermitian operators in the ionization dynamics, a similar approach to the quantum imaginary time evolution (QITE) algorithm is employed to translate the propagator, including a complex absorption potential, into quantum gates. The computations are executed on a quantum computer simulator. By construction, all quantum computer algorithms, except for the QITE algorithm used only for ionization but not for excitation dynamics, would scale polynomially on a quantum computer with fully entangled qubits. In contrast, TD-FCI scales exponentially. Hence, quantum computation holds promises for substantial progress in the understanding of electron dynamics of excitation processes in increasingly large molecular systems, as has already been witnessed in electronic structure theory.

Nucleon-pair truncation of the shell model for medium-heavy nuclei

Background: Establishing computationally tractable models of atomic nuclei is a long-time goal of nuclear structure physics. A flexible framework which easily includes excited states and many-body correlations is the configuration-interaction shell model (SM), but the exponential growth of the basis means one needs an efficient truncation scheme, ideally one that includes both deformation and pairing correlations.Purpose: We propose an efficient truncation scheme of the SM: starting from a pair condensate variationally defined by Hartree-Fock single-particle states and the particle-number conserved Bardeen-Cooper-Schrieffer (NBCS) approximation, we carry out projection of states with good angular momentum.Methods: After generating Hartree-Fock single-particle states with Kramers degeneracy in a SM space, we optimize the pair amplitudes in the NBCS by minimizing the energy, and then use linear algebra projection (LAP) of states with good angular momentum. Both NBCS and LAP are computationally fast.Results: Our calculations yield good agreement with full configuration-interaction SM calculations for low-lying states of transitional and rotational nuclei with axially symmetric and triaxial deformation in medium- and heavy-mass regions: $^{44,46,48}\mathrm{Ti}$, $^{48,50}\mathrm{Cr}$, $^{52}\mathrm{Fe}$, $^{60,62,64}\mathrm{Zn}$, $^{66,68}\mathrm{Ge}$, $^{68}\mathrm{Se}$, and $^{108,110}\mathrm{Xe}$. We predict low-lying states of $^{112--114}\mathrm{Ba}$ and $^{116--120}\mathrm{Ce}$, nuclei difficult to reach by large-scale SM calculations.Conclusions: Both pair correlation and the configuration mixing between different intrinsic states play a key role in reproducing collectivity and shape coexistence, demonstrating the utility of this truncation scheme of the SM to study transitional and deformed nuclei.

Accurate and Efficient Quantum Computations of Molecular Properties Using Daubechies Wavelet Molecular Orbitals: A Benchmark Study against Experimental Data

Although quantum computation (QC) is regarded as a promising numerical method for computational quantum chemistry, current applications of quantum-chemistry calculations on quantum computers are limited to small molecules. This limitation can be ascribed to technical problems in building and manipulating more qubits and the associated complicated operations of quantum gates in a quantum circuit when the size of the molecular system becomes large. As a result, reducing the number of required qubits is necessary to make QC practical. Currently, the minimal STO-3G basis set is commonly used in benchmark studies because it requires the minimum number of spin orbitals. Nonetheless, the accuracy of using STO-3G is generally low and thus cannot provide useful predictions. We propose to adopt Daubechies wavelet functions as an accurate and efficient method for QCs of molecular electronic properties. We demonstrate that a minimal basis set constructed from Daubechies wavelet basis can yield accurate results through a better description of the molecular Hamiltonian, while keeping the number of spin orbitals minimal. With the improved Hamiltonian through Daubechies wavelets, we calculate vibrational frequencies for H$_2$ and LiH using quantum-computing algorithm to show that the results are in excellent agreement with experimental data. As a result, we achieve quantum calculations in which accuracy is comparable with that of the full configuration interaction calculation using the cc-pVDZ basis set, whereas the computational cost is the same as that of a STO-3G calculation. Thus, our work provides a more efficient and accurate representation of the molecular Hamiltonian for efficient QCs of molecular systems, and for the first time demonstrates that predictions in agreement with experimental measurements are possible to be achieved with quantum resources available in near-term quantum computers.

Molecular formations and spectra due to electron correlations in three-electron hybrid double-well qubits

We show that systematic full configuration-interaction (FCI) calculations enable prediction of the energy spectra and the intrinsic spatial and spin structures of the many-body wave functions as a function of the detuning parameter for the case of three-electron hybrid qubits based on GaAs asymmetric double quantum dots. Specifically, in comparison with the case of weak interactions and treating the entire three-electron double-dot hybrid qubit as an integral unit, it is shown that the predicted spectroscopic patterns, originating from strong electron correlations, manifest the formation of Wigner molecules (WMs). Signatures of WM formation include: (1) a strong suppression of the energy gaps relative to the non-interacting-electrons modeling, and (2) the appearance of a pair of avoided crossings arising between states associated with two-electron occupancies in the left and right wells. The Wigner molecule is a physical entity associated with electron localization within each well and it cannot be captured by the previously employed independent-particle or two-site-Hubbard theoretical modeling of the hybrid qubits. The emergence of strong WMs is investigated in depth through the concerted use of FCI-adapted diagnostic tools like charge and spin densities, as well as conditional probability distributions. Furthermore, the energy spectrum as a function of the strength of the Coulomb repulsion (at constant detuning) is calculated in order to complement the thorough analysis of the factors contributing to WM emergence. We report remarkable agreement with recent experimental measurements. The present FCI methodology for multi-well quantum dots can be straightforwardly extended to treat valleytronic two-band Si/SiGe hybrid qubits, where the central role of the WMs was confirmed recently.

Reduced density matrix approach to ultracold few-fermion systems in one dimension

The variational determination of the two-fermion reduced density matrix (2-RDM) is described for harmonically trapped, ultracold few-fermion systems in one dimension with equal spin populations. This is accomplished by formulating the problem as a semi-definite program, with the 2-RDM being subject to well-known N-representability conditions. The ground-state energies, as well as the density, pair-correlation function, and lower-order eigenvalues of the 2-RDM of various fermionic systems are found by utilising an augmented Lagrangian method for semi-definite programming. The ground-state energies are found to match well to those determined by full-configuration interaction and coupled-cluster calculations and the density, pair-correlation function, and eigenvalue results demonstrate that the salient features of these systems are well-described by this method. These results collectively demonstrate the utility of the reduced density matrix method firstly in describing strong correlation arising from short-range interactions, suggesting that the well-known N-representability conditions are sufficient to model ultracold fermionic systems, and secondly in illustrating the prospect of treating larger systems currently out of the reach of established methods.

How Valley-Orbit States in Silicon Quantum Dots Probe Quantum Well Interfaces.

The energies of valley-orbit states in silicon quantum dots are determined by an as yet poorly understood interplay between interface roughness, orbital confinement, and electron interactions. Here, we report measurements of one- and two-electron valley-orbit state energies as the dot potential is modified by changing gate voltages, and we calculate these same energies using full configuration interaction calculations. The results enable an understanding of the interplay between the physical contributions and enable a new probe of the quantum well interface.

Wigner molecules and hybrid qubits

It is demonstrated that exact diagonalization of the microscopic many-body Hamiltonian via systematic full configuration-interaction (FCI) calculations is able to predict the spectra as a function of detuning of three-electron hybrid qubits based on GaAs asymmetric double quantum dots (QDs). It is further shown that, as a result of strong inter-electron correlations, these spectroscopic patterns, including avoided crossings between states associated with different electron occupancies of the left and right wells, are inextricably related to the formation of Wigner molecules (WMs). These physical entities cannot be captured by the previously employed independent-particle or Hubbard-type theoretical modeling of the hybrid qubit. We report remarkable agreement with recent experimental results. Moreover, the present FCI methodology for multi-well QDs can be straightforwardly extended to treat Si/SiGe hybrid qubits, where the central role of WMs was recently experimentally confirmed as well.

Quantum Algorithm for Full Configuration Interaction Calculations without Controlled Time Evolutions.

A quantum phase estimation algorithm allows us to perform full configuration interaction (full-CI) calculations on quantum computers with polynomial costs against the system size under study, but it requires quantum simulation of the time evolution of the wave function conditional on an ancillary qubit, which makes the algorithm implementation on real quantum devices difficult. Here, we discuss an application of the Bayesian phase difference estimation algorithm that is free from controlled time evolution operations to the full-CI calculations.

Coherent Control and Spectroscopy of a Semiconductor Quantum Dot Wigner Molecule.

Semiconductor quantum dots containing more than one electron have found wide application in qubits, where they enable readout and enhance polarizability. However, coherent control in such dots has typically been restricted to only the lowest two levels, and such control in the strongly interacting regime has not been realized. Here we report quantum control of eight different transitions in a silicon-based quantum dot. We use qubit readout to perform spectroscopy, revealing a dense set of energy levels with characteristic spacing far smaller than the single-particle energy. By comparing with full configuration interaction calculations, we argue that the dense set of levels arises from Wigner-molecule physics.

Multihole models for deterministically placed acceptor arrays in silicon

We compute the electronic structure of acceptor clusters in silicon by using three methods to include electron correlations: the full configuration interaction, the Heitler-London approximation, and the unrestricted Hartree-Fock method. We show both the HL approach and the UHF method are good approximations to the ground state of the full CI calculation for a pair of acceptors and for finite linear chains. The total energies for finite linear chains show the formation of a 4-fold degenerate ground state when there is a weak bond at the end of the chain, which is shown to be a manifold of topological edge states. We identify a change in the angular momentum composition of the ground state at a critical pattern of bond lengths and show it is related to a crossing in the Fock matrix eigenvalues. We also test the symmetry of the UHF solution and compare it to the full CI; the symmetry is broken under almost all the arrangements by the formation of a magnetic state in UHF, and we find further broken symmetries for some particular arrangements related to crossings or potential crossings between the Fock-matrix eigenvalues. We also compute the charge distributions across the acceptors obtained from the eigenvectors of the Fock matrix: with weak bonds at the chain ends, two holes are localized at either end of the chain while the others have a nearly uniform distribution over the middle; this implies the existence of the non-trivial edge states. We apply the UHF method to treat an infinite linear chain with periodic boundary conditions. We find the band structures in the UHF approximation and compute the Zak phases for the occupied Fock-matrix eigenvalues; we find they do not correctly predict the topological edge states in this interacting system. We find direct study of the quantum numbers characterising the edge states provides a better insight into their topological nature.

Pressure evolution of the electronic structure of non-centrosymmetric EuRhGe3

Among europium compounds, pressure induced valence transitions and/or intermediate valence states are often observed. In such systems, applying pressure of several GPa can drive a Eu valence from divalent to almost trivalent. Non-centrosymmetric EuRhGe3 possesses magnetic Eu2+ ions and exhibits antiferromagnetic ordering at ∼11 K at ambient pressure. Pressure resistant magnetic ordering and stable divalent Eu state have been reported in EuRhGe3. Here, we study the pressure evolution of the Eu valence of EuRhGe3 by high resolution x-ray absorption spectroscopy using the partial fluorescence yield method. Our study reveals a successive increase of the Eu valence with increasing pressure without any valence transition. The obtained mean Eu valence approaches ∼2.4 around 40 GPa at 300 K. The experimental data are also analyzed by a full multiplet configuration interaction calculation based on the single impurity Anderson model. The analysis reveals a decrease of the Eu 4f orbital occupation by applying pressure. Pressure evolution of the electronic structure studied by density functional theory suggests that the Rh ions have little contribution to the pressure evolution of the Eu valence, while it implies an active involvement of the Ge ions.

Charge noise suppression in capacitively coupled singlet-triplet spin qubits under magnetic field

Charge noise is the main hurdle preventing high-fidelity operation, in particular that of two-qubit gates, of semiconductor-quantum-dot-based spin qubits. While certain sweet spots where charge noise is substantially suppressed have been demonstrated in several types of spin qubits, the existence of one for coupled singlet-triplet qubits is unclear. We theoretically demonstrate, using full configuration-interaction calculations, that a range of nearly sweet spots appear in the coupled singlet-triplet qubit system when a strong enough magnetic field is applied externally. We further demonstrate that ramping to and from the judiciously chosen nearly sweet spot using sequences based on the shortcut to adiabaticity offers maximal gate fidelities under charge noise and phonon-induced decoherence. These results should facilitate realization of high-fidelity two-qubit gates in singlet-triplet qubit systems.

Comparison of Many-Particle Representations for Selected Configuration Interaction: II. Numerical Benchmark Calculations

The present work is the second part in our three-part series on the comparison of many-particle representations for the selected configuration interaction (CI) method. In this work, we present benchmark calculations based on our selected CI program called the iterative configuration expansion (ICE) that is inspired by the CIPSI method of Malrieu and co-workers (MalrieuJ. Chem. Phys.1973, 58, ( (12), ), 5745−5759). We describe the main parameters that enter in this algorithm and perform benchmark calculations on a set of 21 small molecules and compare ground state energies with full configuration interaction (FCI) results (FCI21 test set). The focus is the comparison of the performance of three different types of many-particle basis functions (MPBFs): (1) individual Slater determinants (DETS), (2) individual spin-adapted configuration state functions (CSFs), and (3) all CSFs of a given total spin that can be generated from spatial configurations (CFGs). An analysis of the cost of the calculation in terms of the number of wavefunction parameters and the energy error is evaluated for the DET-, CFG-, and CSF-based ICE. The main differences for the three many-particle basis representations show up in the number of wavefunction parameters and the rate of convergence toward the FCI limit with the thresholds of the ICE. Next, we analyze the best way to extrapolate the ICE energies toward the FCI results as a function of the thresholds. The efficiency of the extrapolation is investigated relative to the FCI21 test set as well as near FCI calculations on three moderately sized hydrocarbon molecules CH4, C2H4, and C4H6. Finally, we comment on the size-inconsistency error for the three many-particle representations and compare it with the error in the total energy. The implication for selected CI implementations with any of the three many-particle representations is discussed.

Magic wavelengths for the helium 23S1→21P1 forbidden transition

A series of magic wavelengths for the forbidden transition of $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}$ in helium are determined by performing large-scale full relativistic configuration interaction calculations based on Dirac-Coulomb-Breit Hamiltonian with the mass shift operator included. QED corrections to magic wavelengths are taken into account by using the perturbation theory in nonrelativistic configuration interaction calculations. The 1335.55(2)-nm magic wavelength for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}(M=\ifmmode\pm\else\textpm\fi{}1)\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}(M=0)$ transition is found to be sensitive to the finite nuclear mass, relativistic, and QED effects, and its measurement might provide a test of atomic structure theory.

Index of multi-determinantal and multi-reference character in coupled-cluster theory.

A full configuration interaction calculation (FCI) ultimately defines the innate molecular orbital description of a molecule. Its density matrix and the natural orbitals obtained from it quantify the difference between having N-dominantly occupied orbitals in a reference determinant for a wavefunction to describe N-correlated electrons and how many of those N-electrons are left to the remaining virtual orbitals. The latter provides a measure of the multi-determinantal character (MDC) required to be in a wavefunction. MDC is further split into a weak correlation part and a part that indicates stronger correlation often called multi-reference character (MRC). If several virtual orbitals have high occupation numbers, then one might argue that these additional orbitals should be allowed to have a larger role in the calculation, as in MR methods, such as MCSCF, MR-CI, or MR-coupled-cluster (MR-CC), to provide adequate approximations toward the FCI. However, there are problems with any of these MR methods that complicate the calculations compared to the uniformity and ease of application of single-reference CC calculations (SR-CC) and their operationally single-reference equation-of-motion (EOM-CC) extensions. As SR-CC theory is used in most of today's "predictive" calculations, an assessment of the accuracy of SR-CC at some truncation of the cluster operator would help to quantify how large an issue MRC actually is in a calculation, and how it might be alleviated while retaining the convenient SR computational character of CC/EOM-CC. This paper defines indices that identify MRC situations and help assess how reliable a given calculation is.