Full Configuration Interaction Method Research Articles

Transformer-Based Neural-Network Quantum State Method for Electronic Band Structures of Real Solids.

Recent advancements in neural networks have led to significant progress in addressing many-body electron correlations in small molecules and various physical models. In this work, we propose QiankunNet-Solid, which incorporates periodic boundary conditions into the neural network quantum state (NNQS) framework based on generative Transformer architecture along with a batched autoregressive sampling (BAS) method, enabling the effective ab initio calculation of real solid materials. The accuracy of this method is demonstrated in one-, two-, and three-dimensional periodic systems, with results comparable to those of full configuration interaction and coupled-cluster method, even in the strongly correlated regime. Furthermore, we compute the band structures and density of states for silicon crystal. The successful incorporation of periodic boundary conditions into the NNQS framework through QiankunNet-Solid opens up new possibilities for the accurate and efficient study of electronic structure properties in solid-state physics.

MBE-CASSCF Approach for the Accurate Treatment of Large Active Spaces.

We present a novel implementation of the complete active space self-consistent field (CASSCF) method that makes use of the many-body expanded full configuration interaction (MBE-FCI) method to incrementally approximate electronic structures within large active spaces. On the basis of a hybrid first-order algorithm employing both Super-CI and quasi-Newton strategies for the optimization of molecular orbitals, we demonstrate both computational efficacy and high accuracy of the resulting MBE-CASSCF method. We assess the performance of our implementation on a set of established numerical tests before applying MBE-CASSCF in the investigation of the triplet-quintet spin gap of iron(II) porphyrin with active spaces as large as 50 electrons in 50 orbitals.

Benchmarking boron cluster calculations: Establishing reliable geometrical and energetic references for Bn (n = 1-4).

Using full configuration interaction (FCI) and multi-reference configuration interaction methods (MRCI), reliable geometrical and energetic references for Bn (n = 1-4) clusters were established. The accuracy of the computed results was confirmed by comparison with available experimental data. Benchmark calculations indicated that B97D3, B97D, VSXC, HCTH407, BP86 and CCSD(T) methods provided reasonable results for structural parameters, with mean absolute error (MAEs) within 0.020 Å. Among the tested density functional theory (DFT) methods, the VSXC functional showed the best performance in predicting the relative energies of B1 B4 with a MAE of 12.8 kJ mol-1 . Besides, B1B95, B971, TPSS, B3LYP, and BLYP functionals exhibited reasonable performance with MAE values of less than 15.0 kJ mol-1 . T1 diagnostic values between 0.035 and 0.109 at the CCSD(T) level revealed strong correlations in B2 B4 clusters, highlighting the need for caution in using CCSD(T) as an energy reference for small boron clusters. The methods of CCSDT, CCSDT(Q) and CCSDT[Q], which incorporate three-electron and four-electron excitations, effectively improved the accuracy of the energy calculations.

Electronic Free Energy Surface of the Nitrogen Dimer Using First-Principles Finite Temperature Electronic Structure Methods.

We use full configuration interaction and density matrix quantum Monte Carlo methods to calculate the electronic free energy surface of the nitrogen dimer within the free-energy Born-Oppenheimer approximation. As the temperature is raised from T = 0, we find a temperature regime in which the internal energy causes bond strengthening. At these temperatures, adding in the entropy contributions is required to cause the bond to gradually weaken with increasing temperature. We predict a thermally driven dissociation for the nitrogen dimer between 22,000 to 63,200 K depending on symmetries and basis set. Inclusion of more spatial and spin symmetries reduces the temperature required. The origin of these observations is explored using the structure of the density matrix at various temperatures and bond lengths.

Collision-induced three-body polarizability of helium.

We present the first-principles determination of the three-body polarizability and the third dielectric virial coefficient of helium. Coupled-cluster and full configuration interaction methods were used to perform electronic structure calculations. The mean absolute relative uncertainty of the trace of the polarizability tensor, resulting from the incompleteness of the orbital basis set, was found to be 4.7%. Additional uncertainty due to the approximate treatment of triple and the neglect of higher excitations was estimated at 5.7%. An analytic function was developed to describe the short-range behavior of the polarizability and its asymptotics in all fragmentation channels. We calculated the third dielectric virial coefficient and its uncertainty using the classical and semiclassical Feynman-Hibbs approaches. The results of our calculations were compared with experimental data and with recent Path-Integral Monte Carlo (PIMC) calculations [Garberoglio et al., J. Chem. Phys. 155, 234103 (2021)] employing the so-called superposition approximation of the three-body polarizability. For temperatures above 200K, we observed a significant discrepancy between the classical results obtained using superposition approximation and the ab initio computed polarizability. For temperatures from 10K up to 200K, the differences between PIMC and semiclassical calculations are several times smaller than the uncertainties of our results. Except at low temperatures, our results agree very well with the available experimental data but have much smaller uncertainties. The data reported in this work eliminate the main accuracy bottleneck in the optical pressure standard [Gaiser et al., Ann. Phys. 534, 2200336 (2022)] and facilitate further progress in the field of quantum metrology.

Three-body potential and third virial coefficients for helium including relativistic and nuclear-motion effects.

The non-additive three-body interaction potential for helium was computed using the coupled-cluster theory and the full configuration interaction method. The obtained potential comprises an improved nonrelativistic Born-Oppenheimer energy and the leading relativistic and nuclear-motion corrections. The mean absolute uncertainty of our calculations due to the incompleteness of the orbital basis set was determined employing complete-basis-set extrapolation techniques and was found to be 1.2%. For three helium atoms forming an equilateral triangle with the side length of 5.6 bohr - a geometry close to the minimum of the total potential energy surface - our three-body potential amounts to -90.6 mK, with an estimated uncertainty of 0.5 mK. An analytic function, developed to accurately fit the computed three-body interaction energies, was chosen to correctly describe the asymptotic behavior of the three-body potential for trimer configurations corresponding to both the three-atomic and the atom-diatom fragmentation channels. For large triangles with sides r12, r23, and r31, the potential takes correctly into account all angular terms decaying as r-l12 r-m23 r-n21 with l + m + n ≤ 14 for the nonrelativistic Born-Oppenheimer energy and l + m + n ≤ 9 for the post-Born-Oppenheimer corrections. We also developed a short-range analytic function describing the local behavior of the total uncertainty of the computed three-body interaction energies. Using both fits we calculated the third pressure and acoustic virial coefficients for helium and their uncertainties for a wide range of temperatures. The results of these calculations were compared with available experimental data and with previous theoretical determinations. The estimated uncertainties of present calculations are 3-5 times smaller than those reported in the best previous works.

Valleytronic full configuration-interaction approach: Application to the excitation spectra of Si double-dot qubits

The influence of strong electron-electron interactions and Wigner-molecule (WM) formation on the spectra of $2e$ singlet-triplet double-dot Si qubits is presented based on a full configuration interaction (FCI) approach that incorporates the valley degree of freedom (VDOF) in the context of the continuous (effective mass) description of semiconductor materials. Our FCI treats the VDOF as an isospin in addition to the regular spin. Our treatment is able to assign to each energy curve in the qubit's spectrum a complete set of good quantum numbers for both the spin and the valley isospin. This reveals an underlying SU(4) $\supset$ SU(2) $\times$ SU(2) group-chain organization in the Si double-dot spectra. With parameters in the range of actual experimental situations, we demonstrate in a double-dot qubit that, in the (2,0) charge configuration and compared to the expected large, and dot-size determined, single-particle (orbital) energy gap, the strong $e-e$ interactions drastically quench the spin-singlet$-$spin-triplet energy gap, $E_{\rm ST}$, within the same valley, making it competitive to the small energy gap, $E_V$, between the two valleys. We present results for both the $E_{\rm ST} < E_V$ and $E_{\rm ST} > E_V$ cases. We investigate the spectra as a function of detuning and demonstrate the strengthening of the avoided crossings due to a lowering of the interdot barrier and/or the influence of valley-orbit coupling. We further demonstrate, as a function of an applied magnetic field, the emergence of avoided crossings in the (1,1) charge configuration due to the spin-valley coupling. The valleytronic FCI formulated here, and implementeded for two electrons confined in a tunable double quantum dot, offers also a most effective tool for analyzing the spectra of Si qubits with more than two wells and/or more than two electrons.

Influence of the N-representability conditions on the variational determination of two-electron reduced density matrices for ground and excited N-electron states in the doubly occupied configuration interaction space.

In this work, we analyze the effectiveness of different sets of well-known necessary N-representability conditions that are imposed on the variational determination of two-electron reduced density matrices corresponding to the ground and excited states of N-electron Hamiltonians within doubly occupied configuration interaction subspaces. The two-electron reduced density matrix elements are determined so that they are suitably related to those of higher-order reduced density matrices and other associated matrices, up to a given order, constituting positive semidefinite matrices. These conditions up to two-, three-, and four-order have been sequentially imposed on the variational calculation of the expectation values of the dispersion operator corresponding to several Hamiltonian models, obtaining the electronic spectra. These results have been contrasted with those arising from the full configuration interaction method, illuminating the sufficiency/insufficiency of the variational N-representability conditions applied.

High-precision real-space simulation of electrostatically confined few-electron states

In this paper, we present a computational procedure that utilizes real-space grids to obtain high precision approximations of electrostatically confined few-electron states such as those that arise in gated semiconductor quantum dots. We use the full configuration interaction method with a continuously adapted orthonormal orbital basis to approximate the ground and excited states of such systems. We also introduce a benchmark problem based on a realistic analytical electrostatic potential for quantum dot devices. We show that our approach leads to highly precise computed energies and energy differences over a wide range of model parameters. The analytic definition of the benchmark allows for a collection of tests that are easily replicated, thus facilitating comparisons with other computational approaches.

Shallow dopant pairs in silicon: An atomistic full configuration interaction study

The two-electron states and exchange couplings are investigated for a phosphorous donor pair in silicon using an atomistic full configuration interaction method for donor separations spanning from 0.4 to 15 nm. Three distinct donor separation regimes appear from our large basis calculations, from which the validity of simplified methods such as Heitler-London and Hartree-Fock type approaches can be assessed. For bulk donors, the exchange coupling saturates below 5 nm due to excited bonding orbital contributions to the wave functions. Ionic contributions to the two-electron state decrease between 5 and 14 nm, and a fully correlated Heitler-London-like state is reached from 14 nm onwards. Oscillations in exchange couplings can be strongly suppressed by placing the donors in the same $z$ plane and at a small depth, $D$, from the surface. This is a consequence of the $z$-valley terms becoming dominant within the dopant's wave function and of small changes with $x$ and $y$ separations no longer having much effect. We find the depth to be an important parameter in determining the exchange coupling for subsurface dopants, not only through valley repopulation ($D&lt;10$ nm) but also through additional interface effects for ultrashallow depths ($D&lt;2.5$ nm). Our full configuration interaction method provides insights in the exchange interaction for various regimes of donor separation and depths, from the Heitler-London limit at large distances to the 0.4--5 nm range relevant for scanning tunneling microscope based quantum state imaging and spectroscopy experiments. The precise control of electron-electron quantum correlations in such engineered atoms in the solid state is useful to design quantum logic gates and quantum simulators.

The many-body electronic interactions of Fe(II)-porphyrin.

Fe(II)-porphyrin complexes exhibit a diverse range of electronic interactions between the metal and macrocycle. Herein, the incremental full configuration interaction method is applied to the entire space of valence orbitals of a Fe(II)-porphyrin model using a modest basis set. A novel visualization framework is proposed to analyze individual many-body contributions to the correlation energy, providing detailed maps of this complex's highly correlated electronic structure. This technique is used to parse the numerous interactions of two low-lying triplet states (3A2g and 3Eg) and to show that strong metal d-d and macrocycle π-π orbital interactions preferentially stabilize the 3A2g state. d-π interactions, on the other hand, preferentially stabilize the 3Eg state and primarily appear when correlating six electrons at a time. Ultimately, the Fe(II)-porphyrin model's full set of 88 valence electrons are correlated in 275 orbitals, showing the interactions up to the 4-body level, which covers the great majority of correlations in this system.

Exploiting the nearsightedness principle within the framework of the anti‐Hermitian contracted Schrödinger equation

AbstractThis work exploits the Kohn nearsightedness principle in the algorithms arising from the anti‐Hermitian contracted Schrödinger equation. The procedure provides the implementation of approximations yielding a considerable saving of computational costs in descriptions of N‐electron molecular systems by means of that technique. We report energies, at correlated level, of linear hydrogen chains designed at different geometries, so that we can analyze the strength of the interactions between first‐, second‐, third‐, fourth‐, and so forth neighbor atoms. Using localized molecular orbital basis sets, we observe rapid decays in the values of the elements of the reduced‐density‐matrix cumulant matrices, beyond a cut‐off radius, which justify the proposed approximations. The results, in terms of energies and reduced density matrices, have been compared with those obtained from the full configuration interaction method, showing the usefulness of the nearsightedness principle in this methodology.

FCI calculations of the adiabatic electronic structure of KRb highlighting the K−Rb+ ionic limit effect

A non-relativistic description of the electronic structure for the heteronuclear molecule KRb was developed through the Full Configuration Interaction (FCI) and pseudo-potential method. Below the ionic limit K−Rb+, the ground and all excited singlet and triplet states of different symmetries Σ+, Π and Δ (in the Hund's case (a): 2s + 1Λ±) were intensively investigated. Several adiabatic results including potential energy curves (PECs), permanent and transition electric dipole moments (PEDM, TEDM) were analyzed in short and long-range of internuclear distances. Tables of these properties as functions of separation are presented. The spectroscopic constants with the vibrational levels were derived using the “Numerov” algorithm. These molecular parameters were compared with those published in the literature, feature a good agreement. In the adiabatic representation, strong couplings between successive 1Σ+ states were reached by series of avoided crossing, yielding to the appearance of the ionic character and the good illustration of charge transfer. This description of the KRb molecule was helpful in the design of the prospect photo-association experiments and the obtaining of the quantum-degenerate gas of the polar molecule.

Full configuration interaction simulations of exchange-coupled donors in silicon using multi-valley effective mass theory

Donor spins in silicon have achieved record values of coherence times and single-qubit gate fidelities. The next stage of development involves demonstrating high-fidelity two-qubit logic gates, where the most natural coupling is the exchange interaction. To aid the efficient design of scalable donor-based quantum processors, we model the two-electron wave function using a full configuration interaction method within a multi-valley effective mass theory. We exploit the high computational efficiency of our code to investigate the exchange interaction, valley population, and electron densities for two phosphorus donors in a wide range of lattice positions, orientations, and as a function of applied electric fields. The outcomes are visualized with interactive images where donor positions can be swept while watching the valley and orbital components evolve accordingly. Our results provide a physically intuitive and quantitatively accurate understanding of the placement and tuning criteria necessary to achieve high-fidelity two-qubit gates with donors in silicon.

Comparison of many-particle representations for selected-CI I: A tree based approach.

The full configuration interaction (FCI) method is only applicable to small molecules with few electrons in moderate size basis sets. One of the main alternatives to obtain approximate FCI energies for bigger molecules and larger basis sets is selected CI. However, due to: (a) the lack of a well-defined structure in a selected CI Hamiltonian, (b) the potentially large number of electrons together with c) potentially large orbital spaces, a computationally and memory efficient algorithm is difficult to construct. In the present series of papers, we describe our attempts to address these issues by exploring tree-based approaches. At the same time, we devote special attention to the issue of obtaining eigenfunctions of the total spin squared operator since this is of particular importance in tackling magnetic properties of complex open shell systems. Dedicated algorithms are designed to tackle the CI problem in terms of determinant, configuration (CFG) and configuration state function many-particle bases by effective use of the tree representation. In this paper we describe the underlying logic of our algorithm design and discuss the advantages and disadvantages of the different many particle bases. We demonstrate by the use of small examples how the use of the tree simplifies many key algorithms required for the design of an efficient selected CI program. Our selected CI algorithm, called the iterative configuration expansion, is presented in the penultimate part. Finally, we discuss the limitations and scaling characteristics of the present approach.

Configuration interaction trained by neural networks: Application to model polyaromatic hydrocarbons.

The main bottleneck of a stochastic or deterministic configuration interaction method is determining the relative weights or importance of each determinant or configuration, which requires large scale matrix diagonalization. Therefore, these methods can be improved significantly from a computational standpoint if the relative importance of each configuration in the ground and excited states of molecular/model systems can be learned using machine learning techniques such as artificial neural networks (ANNs). We have used neural networks to train the configuration interaction coefficients obtained from full configuration interaction and Monte Carlo configuration interaction methods and have tested different input descriptors and outputs to find the more efficient training techniques. These ANNs have been used to calculate the ground states of one- and two-dimensional Heisenberg spin chains along with Heisenberg ladder systems, which are good approximations of polyaromatic hydrocarbons. We find excellent efficiency of training and the model this trained was used to calculate the variational ground state energies of the systems.

Incremental treatments of the full configuration interaction problem

AbstractThe recent many‐body expanded full configuration interaction (MBE‐FCI) method is reviewed by critically assessing its advantages and drawbacks in the context of contemporary near‐exact electronic structure theory. Besides providing a succinct summary of the history of MBE‐FCI to date within a generalized and unified theoretical setting, its finer algorithmic details are discussed alongside our optimized computational implementation of the theory. A selected few of the most recent applications of MBE‐FCI are revisited, before we close by outlining its future research directions as well as its place among modern near‐exact wave function‐based methods.This article is categorized under: Electronic Structure Theory &gt; Ab Initio Electronic Structure Methods

Ground and excited state first-order properties in many-body expanded full configuration interaction theory.

The recently proposed many-body expanded full configuration interaction (MBE-FCI) method is extended to excited states and static first-order properties different from total, ground state correlation energies. Results are presented for excitation energies and (transition) dipole moments of two prototypical, heteronuclear diatomics-LiH and MgO-in augmented correlation consistent basis sets of up to quadruple-ζ quality. Given that MBE-FCI properties are evaluated without recourse to a sampled wave function and the storage of corresponding reduced density matrices, the memory overhead associated with the calculation of general first-order properties only scales with the dimension of the desired property. In combination with the demonstrated performance, the present developments are bound to admit a wide range of future applications by means of many-body expanded treatments of electron correlation.

Deep-neural-network solution of the electronic Schrödinger equation.

The electronic Schrödinger equation can only be solved analytically for the hydrogen atom, and the numerically exact full configuration-interaction method is exponentially expensive in the number of electrons. Quantum Monte Carlo methods are a possible way out: they scale well for large molecules, they can be parallelized and their accuracy has, as yet, been only limited by the flexibility of the wavefunction ansatz used. Here we propose PauliNet, a deep-learning wavefunction ansatz that achieves nearly exact solutions of the electronic Schrödinger equation for molecules with up to 30 electrons. PauliNet has a multireference Hartree-Fock solution built in as a baseline, incorporates the physics of valid wavefunctions and is trained using variational quantum Monte Carlo. PauliNet outperforms previous state-of-the-art variational ansatzes for atoms, diatomic molecules and a strongly correlated linear H10, and matches the accuracy of highly specialized quantum chemistry methods on the transition-state energy of cyclobutadiene, while being computationally efficient.

Unorthodox Dimensional Interpolations for He, Li, Be Atoms and Hydrogen Molecule

We present a simple interpolation formula using dimensional limits $D=1$ and $D=\infty$ to obtain the $D=3$ ground-state energies of atoms and molecules. For atoms, these limits are linked by first-order perturbation terms of electron-electron interactions. This unorthodox approach is illustrated by ground-states for two, three, and four electron atoms, with modest effort to obtain fairly accurate results. Also, we treat the ground-state of H$_2$ over a wide range of the internuclear distance R, and compares well with the standard exact results from the Full Configuration Interaction method. Similar dimensional interpolations may be useful for complex many-body systems.