Variational Optimization of the Second-Order Density Matrix Corresponding to a Seniority-Zero Configuration Interaction Wave Function

A variance-based optimization for determining ground and excited N-electron wave functions within the doubly occupied configuration interaction scheme

This work describes optimizations of N-electron system wave functions by means of the simulated annealing technique within the doubly occupied configuration interaction framework. Using that technique, we minimize the energy variance of a Hamiltonian, providing determinations of wave functions corresponding to ground or excited states in an identical manner. The procedure that allows us to determine electronic spectra can be performed using treatments of restricted or unrestricted types. The results found in selected systems, described in terms of energy, spin, and wave function, are analyzed, showing the performance of each method. We also compare these results with those arising from more traditional approaches that minimize the energy, in both restricted and unrestricted versions, and with those obtained from the full configuration interaction treatment.

Determination of electronic excitation energies within the doubly occupied configuration interaction space by means of the Hermitian operator method

In this work, we formulate the equations of motion corresponding to the Hermitian operator method in the framework of the doubly occupied configuration interaction space. The resulting algorithms turn out to be considerably simpler than the equations provided by that method in more conventional spaces, enabling the determination of excitation energies in N-electron systems under an affordable polynomial computational cost. The implementation of this technique only requires to know the elements of low-order reduced density matrices of an N-electron reference state, which can be obtained from any approximate method. We contrast our procedure against the reduced Bardeen-Cooper-Schrieffer and Richardson-Gaudin-Kitaev integrable models, pointing out the reliability of our proposal.

Data-Driven Refinement of Electronic Energies from Two-Electron Reduced-Density-Matrix Theory

The exponential computational cost of describing strongly correlated electrons can be mitigated by adopting a reduced-density matrix (RDM)-based description of the electronic structure. While variational two-electron RDM (v2RDM) methods can enable large-scale calculations on such systems, the quality of the solution is limited by the fact that only a subset of known necessary N-representability constraints can be applied to the 2RDM in practical calculations. Here, we demonstrate that violations of partial three-particle (T1 and T2) N-representability conditions, which can be evaluated with knowledge of only the 2RDM, can serve as physics-based features in a machine-learning (ML) protocol for improving energies from v2RDM calculations that consider only two-particle (PQG) conditions. Proof-of-principle calculations demonstrate that the model yields substantially improved energies relative to reference values from configuration-interaction-based calculations.

Quantum Many-Body Theory from a Solution of the N-Representability Problem

Here we present a many-body theory based on a solution of the N-representability problem in which the ground-state two-particle reduced density matrix (2-RDM) is determined directly without the many-particle wave function. We derive an equation that re-expresses physical constraints on higher-order RDMs to generate direct constraints on the 2-RDM, which are required for its derivation from an N-particle density matrix, known as N-representability conditions. The approach produces a complete hierarchy of 2-RDM constraints that do not depend explicitly upon the higher RDMs or the wave function. By using the two-particle part of a unitary decomposition of higher-order constraint matrices, we can solve the energy minimization by semidefinite programming in a form where the low-rank structure of these matrices can be potentially exploited. We illustrate by computing the ground-state electronic energy and properties of the H_{8} ring.

Reducing the Quantum Many-Electron Problem to Two Electrons with Machine Learning

An outstanding challenge in chemical computation is the many-electron problem where computational methodologies scale prohibitively with system size. The energy of any molecule can be expressed as a weighted sum of the energies of two-electron wave functions that are computable from only a two-electron calculation. Despite the physical elegance of this extended "aufbau" principle, the determination of the distribution of weights─geminal occupations─for general molecular systems has remained elusive. Here we introduce a new paradigm for electronic structure where approximate geminal-occupation distributions are "learned" via a convolutional neural network. We show that the neural network learns the N-representability conditions, constraints on the distribution for it to represent an N-electron system. By training on hydrocarbon isomers with only 2-7 carbon atoms, we are able to predict the energies for isomers of octane as well as hydrocarbons with 8-15 carbons. The present work demonstrates that machine learning can be used to reduce the many-electron problem to an effective two-electron problem, opening new opportunities for accurately predicting electronic structure.

Challenges for Variational Reduced-Density-Matrix Theory: Total Angular Momentum Constraints

The variational two-electron reduced density matrix (v2RDM) method is generalized for the description of total angular momentum (J) and projection of total angular momentum (M_J) states in atomic systems described by nonrelativistic Hamiltonians, and it is shown that the approach exhibits serious deficiencies. Under ensemble N-representability constraints, v2RDM theory fails to retain the appropriate degeneracies among various J states for fixed spin (S) and orbital angular momentum (L), and for fixed L, S, and J, the manifold of M_J states is not necessarily degenerate. Moreover, a substantial energy error is observed for a system for which the two-electron reduced density matrix is exactly ensemble N-representable; in this case, the error stems from violations in pure-state N-representability conditions. Unfortunately, such violations do not appear to be good indicators of the reliability of energies from v2RDM theory in general. Several states are identified for which energy errors are near zero and yet pure-state conditions are clearly violated.

Variational determination of the two-electron reduced density matrix within the doubly occupied configuration interaction framework: Treatments of triplet N-electron systems

This work performs variational determinations of two-electron reduced density matrices corresponding to open-shell N-electron systems within the framework of the doubly occupied configuration interaction treatment, traditionally limited to studies of closed-shell systems. The procedure has allowed us to describe satisfactorily molecular systems in triplet states following two methods. One of them adds hydrogen atoms at an infinite distance of the triplet system studied, constituting a singlet supersystem. Energies and reduced density matrices of the triplet system are obtained by removing the contributions of the added atoms from the singlet supersystem results. The second procedure determines variationally the two-electron reduced density matrices corresponding to the triplet systems by means of adequate couplings of basis-set functions. Both models have been managed by imposing N-representability conditions on the reduced density matrix calculations. Results obtained from these methods for molecular systems in triplet ground states are reported and compared with those provided by benchmark methods.

Challenges for variational reduced-density-matrix theory with three-particle N-representability conditions

The direct variational optimization of the two-electron reduced density matrix (2RDM) can provide a reference-independent description of the electronic structure of many-electron systems that naturally capture strong or nondynamic correlation effects. Such variational 2RDM approaches can often provide a highly accurate description of strong electron correlation, provided that the 2RDMs satisfy at least partial three-particle N-representability conditions (e.g., the T2 condition). However, recent benchmark calculations on hydrogen clusters [N. H. Stair and F. A. Evangelista, J. Chem. Phys. 153, 104108 (2020)] suggest that even the T2 condition leads to unacceptably inaccurate results in the case of two- and three-dimensional clusters. We demonstrate that these failures persist under the application of full three-particle N-representability conditions (3POS). A variety of correlation metrics are explored in order to identify regimes under which 3POS calculations become unreliable, and we find that the relative squared magnitudes of the cumulant three- and two-particle reduced density matrices correlate reasonably well with the energy error in these systems. However, calculations on other molecular systems reveal that this metric is not a universal indicator for the reliability of the reduced-density-matrix theory with 3POS conditions.

Variational determination of ground and excited-state two-electron reduced density matrices in the doubly occupied configuration space: A dispersion operator approach

This work implements a variational determination of the elements of two-electron reduced density matrices corresponding to the ground and excited states of N-electron interacting systems based on the dispersion operator technique. The procedure extends the previously reported proposal [Nakata et al., J. Chem. Phys. 125, 244109 (2006)] to two-particle interaction Hamiltonians and N-representability conditions for the two-, three-, and four-particle reduced density matrices in the doubly occupied configuration interaction space. The treatment has been applied to describe electronic spectra using two benchmark exactly solvable pairing models: reduced Bardeen-Cooper-Schrieffer and Richardson-Gaudin-Kitaev Hamiltonians. The dispersion operator combined with N-representability conditions up to the four-particle reduced density matrices provides excellent results.

Quantum Information and Algorithms for Correlated Quantum Matter

Discoveries in quantum materials, which are characterized by the strongly quantum-mechanical nature of electrons and atoms, have revealed exotic properties that arise from correlations. It is the promise of quantum materials for quantum information science superimposed with the potential of new computational quantum algorithms to discover new quantum materials that inspires this Review. We anticipate that quantum materials to be discovered and developed in the next years will transform the areas of quantum information processing including communication, storage, and computing. Simultaneously, efforts toward developing new quantum algorithmic approaches for quantum simulation and advanced calculation methods for many-body quantum systems enable major advances toward functional quantum materials and their deployment. The advent of quantum computing brings new possibilities for eliminating the exponential complexity that has stymied simulation of correlated quantum systems on high-performance classical computers. Here, we review new algorithms and computational approaches to predict and understand the behavior of correlated quantum matter. The strongly interdisciplinary nature of the topics covered necessitates a common language to integrate ideas from these fields. We aim to provide this common language while weaving together fields across electronic structure theory, quantum electrodynamics, algorithm design, and open quantum systems. Our Review is timely in presenting the state-of-the-art in the field toward algorithms with nonexponential complexity for correlated quantum matter with applications in grand-challenge problems. Looking to the future, at the intersection of quantum information science and algorithms for correlated quantum matter, we envision seminal advances in predicting many-body quantum states and describing excitonic quantum matter and large-scale entangled states, a better understanding of high-temperature superconductivity, and quantifying open quantum system dynamics.

Reduced Density Matrix-Driven Complete Active Apace Self-Consistent Field Corrected for Dynamic Correlation from the Adiabatic Connection

The recently proposed multireference adiabatic connection (AC) formalism [Pernal, Phys. Rev. Lett. 120, 013001 (2018)] is applied to recover dynamic electron correlation effects lacking in variational two-electron reduced density matrix (v2RDM)-driven complete active space self-consistent field theory (CASSCF). The AC approach is validated by computing potential energy curves for the dissociation of molecular nitrogen and the symmetric double dissociation of H₂O while enforcing two sets of approximate N-representability conditions in the underlying v2RDM-driven CASSCF calculations (either two-particle or two-particle plus partial three-particle conditions). The AC yields smaller absolute errors than second-order N-electron perturbation theory (NEVPT2) at all molecular geometries for both sets of the N-representability conditions considered. The efficacy of the approach for thermochemistry is also assessed for a set of 31 small-molecule reactions. When imposing partial three-particle N-representability conditions, mean and maximum unsigned errors in reaction energies from the AC are superior to those from NEVPT2.

Capturing static and dynamic correlation with ΔNO-MP2 and ΔNO-CCSD

The ΔNO method for static correlation is combined with second-order Møller-Plesset perturbation theory (MP2) and coupled-cluster singles and doubles (CCSD) to account for dynamic correlation. The MP2 and CCSD expressions are adapted from finite-temperature CCSD, which includes orbital occupancies and vacancies, and expanded orbital summations. Correlation is partitioned with the aid of damping factors incorporated into the MP2 and CCSD residual equations. Potential energy curves for a selection of diatomics are in good agreement with extrapolated full configuration interaction results and on par with conventional multireference approaches.

An adiabatic connection for doubly-occupied configuration interaction wave functions

An adiabatic connection (AC) is developed as an electron correlation correction for doubly occupied configuration interaction (DOCI) wave functions. Following the work of Pernal [Phys. Rev. Lett. 120, 013001 (2018)], the working equations of the approach, termed AC-DOCI, are rooted in the extended random phase approximation (ERPA) and require knowledge of only the ground-state two-electron reduced density matrix (2RDM) from the DOCI. As such, the AC is naturally suited to modeling electron correlation in variational 2RDM (v2RDM)-based approximations to the DOCI. The v2RDM-driven AC-DOCI is applied to the dissociation of molecular nitrogen and the double dissociation of water; the approach yields energies that are similar in quality to those from second-order multireference perturbation theory near equilibrium, but the quality of the AC-DOCI energy degrades at stretched geometries. The exact adiabatic connection path suggests the assumption that the one-electron reduced-density matrix is constant along the AC path is invalid at stretched geometries, but this deficiency alone cannot explain the observed behavior. Rather, it appears that the ERPA's single-particle-transition ansatz cannot, in general, provide good approximations to the 2RDM along the AC path. The AC-DOCI is also applied to a set of 45 reaction energies; for these systems, the approach has an average accuracy that is comparable to that of single-reference second-order many-body perturbation theory.

Variational reduced density matrix method in the doubly-occupied configuration interaction space using four-particle N-representability conditions: Application to the XXZ model of quantum magnetism

This work deals with the variational determination of the two-particle reduced density matrix (2-RDM) and the energy corresponding to the ground state of N-particle systems within the doubly occupied configuration interaction (DOCI) space. Here, we impose for the first time up to four-particle N-representability constraint conditions in the variational determination of the 2-RDM matrix elements using the standard semidefinite programming algorithms. The energies and 2-RDMs obtained from this treatment and the corresponding computational costs are compared with those arisen from previously reported less restrictive variational methods [D. R. Alcoba et al., J. Chem. Phys. 149, 194105 (2018)] as well as with the exact DOCI values. We apply the different approximations to the one-dimensional XXZ model of quantum magnetism, which has a rich phase diagram with one critical phase and constitutes a stringent test for the method. The numerical results show the usefulness of our treatment to achieve a high degree of accuracy.

Unrestricted treatment for the direct variational determination of the two-electron reduced density matrix for doubly occupied-configuration-interaction wave functions

This work extends to the unrestricted orbital approach the procedure described in our previous report [Alcoba et al., J. Chem. Phys. 148, 024105 (2018)] for determining variationally the two-electron reduced density matrix arising from doubly occupied-configuration-interaction wave functions by imposing two- and three-index N-representability conditions. An analysis of the numerical results obtained in selected systems, from both restricted and unrestricted treatments, allows one to assess the performance of these methodologies as well as to show the influence of the P, Q, G, T1, and T2 positivity conditions. We highlight the satisfactory results obtained within the unrestricted scheme.

Benchmarking the Variational Reduced Density Matrix Theory in the Doubly Occupied Configuration Interaction Space with Integrable Pairing Models

The variational reduced density matrix theory has been recently applied with great success to models within the truncated doubly occupied configuration interaction space, which corresponds to the seniority zero subspace. Conservation of the seniority quantum number restricts the Hamiltonians to be based on the SU(2) algebra. Among them there is a whole family of exactly solvable Richardson-Gaudin pairing Hamiltonians. We benchmark the variational theory against two different exactly solvable models, the Richardson-Gaudin-Kitaev and the reduced BCS Hamiltonians. We obtain exact numerical results for the so-called [Formula: see text] N-representability conditions in both cases for systems that go from 10 to 100 particles. However, when random single-particle energies as appropriate for small superconducting grains are considered, the exactness is lost but still a high accuracy is obtained.

Method for making 2-electron response reduced density matrices approximately N-representable

In methods like geminal-based approaches or coupled cluster that are solved using the projected Schrödinger equation, direct computation of the 2-electron reduced density matrix (2-RDM) is impractical and one falls back to a 2-RDM based on response theory. However, the 2-RDMs from response theory are not N-representable. That is, the response 2-RDM does not correspond to an actual physical N-electron wave function. We present a new algorithm for making these non-N-representable 2-RDMs approximately N-representable, i.e., it has the right symmetry and normalization and it fulfills the P-, Q-, and G-conditions. Next to an algorithm which can be applied to any 2-RDM, we have also developed a 2-RDM optimization procedure specifically for seniority-zero 2-RDMs. We aim to find the 2-RDM with the right properties which is the closest (in the sense of the Frobenius norm) to the non-N-representable 2-RDM by minimizing the square norm of the difference between this initial response 2-RDM and the targeted 2-RDM under the constraint that the trace is normalized and the 2-RDM, Q-matrix, and G-matrix are positive semidefinite, i.e., their eigenvalues are non-negative. Our method is suitable for fixing non-N-representable 2-RDMs which are close to being N-representable. Through the N-representability optimization algorithm we add a small correction to the initial 2-RDM such that it fulfills the most important N-representability conditions.

Direct variational determination of the two-electron reduced density matrix for doubly occupied-configuration-interaction wave functions: The influence of three-index N-representability conditions

This work proposes the variational determination of two-electron reduced density matrices corresponding to the ground state of N-electron systems within the doubly occupied-configuration-interaction methodology. The P, Q, and G two-index N-representability conditions have been extended to the T1 and T2 (T2') three-index ones and the resulting optimization problem has been addressed using a standard semidefinite program. We report results obtained from the doubly occupied-configuration-interaction method, from the two-index constraint variational procedure and from the two- and three-index constraint variational treatment. The discussion of these results along with a study of the computational cost demanded shows the usefulness of our proposal.

Pair 2-electron reduced density matrix theory using localized orbitals

Full configuration interaction (FCI) restricted to a pairing space yields size-extensive correlation energies but its cost scales exponentially with molecular size. Restricting the variational two-electron reduced-density-matrix (2-RDM) method to represent the same pairing space yields an accurate lower bound to the pair FCI energy at a mean-field-like computational scaling of O(r³) where r is the number of orbitals. In this paper, we show that localized molecular orbitals can be employed to generate an efficient, approximately size-extensive pair 2-RDM method. The use of localized orbitals eliminates the substantial cost of optimizing iteratively the orbitals defining the pairing space without compromising accuracy. In contrast to the localized orbitals, the use of canonical Hartree-Fock molecular orbitals is shown to be both inaccurate and non-size-extensive. The pair 2-RDM has the flexibility to describe the spectra of one-electron RDM occupation numbers from all quantum states that are invariant to time-reversal symmetry. Applications are made to hydrogen chains and their dissociation, n-acene from naphthalene through octacene, and cadmium telluride 2-, 3-, and 4-unit polymers. For the hydrogen chains, the pair 2-RDM method recovers the majority of the energy obtained from similar calculations that iteratively optimize the orbitals. The localized-orbital pair 2-RDM method with its mean-field-like computational scaling and its ability to describe multi-reference correlation has important applications to a range of strongly correlated phenomena in chemistry and physics.

Spin contamination-free N-electron wave functions in the excitation-based configuration interaction treatment

This work deals with the spin contamination in N-electron wave functions provided by the excitation-based configuration interaction methods. We propose a procedure to ensure a suitable selection of excited N-electron Slater determinants with respect to a given reference determinant, required in these schemes. The procedure guarantees the construction of N-electron wave functions which are eigenfunctions of the spin-squared operator Sˆ(2), avoiding any spin contamination. Our treatment is based on the evaluation of the excitation level of the determinants by means of the expectation value of an excitation operator formulated in terms of spin-free replacement operators. We report numerical determinations of energies and 〈Sˆ(2)〉 expectation values, arising from our proposal as well as from traditional configuration interaction methods, in selected open-shell systems, in order to compare the behavior of these procedures and their computational costs.

Polynomial scaling approximations and dynamic correlation corrections to doubly occupied configuration interaction wave functions

A class of polynomial scaling methods that approximate Doubly Occupied Configuration Interaction (DOCI) wave functions and improve the description of dynamic correlation is introduced. The accuracy of the resulting wave functions is analysed by comparing energies and studying the overlap between the newly developed methods and full configuration interaction wave functions, showing that a low energy does not necessarily entail a good approximation of the exact wave function. Due to the dependence of DOCI wave functions on the single-particle basis chosen, several orbital optimisation algorithms are introduced. An energy-based algorithm using the simulated annealing method is used as a benchmark. As a computationally more affordable alternative, a seniority number minimising algorithm is developed and compared to the energy based one revealing that the seniority minimising orbital set performs well. Given a well-chosen orbital basis, it is shown that the newly developed DOCI based wave functions are especially suitable for the computationally efficient description of static correlation and to lesser extent dynamic correlation.