Configuration Interaction Results Research Articles

Classical and quantum trial wave functions in auxiliary-field quantum Monte Carlo applied to oxygen allotropes and a CuBr2 model system.

In this work, we test a recently developed method to enhance classical auxiliary-field quantum Monte Carlo (AFQMC) calculations with quantum computers against examples from chemistry and material science, representative of classes of industry-relevant systems. As molecular test cases, we calculate the energy curve of H4 and the relative energies of ozone and singlet molecular oxygen with respect to triplet molecular oxygen, which is industrially relevant in organic oxidation reactions. We find that trial wave functions beyond single Slater determinants improve the performance of AFQMC and allow it to generate energies close to chemical accuracy compared to full configuration interaction or experimental results. In the field of material science, we study the electronic structure properties of cuprates through the quasi-1D Fermi-Hubbard model derived from CuBr2, where we find that trial wave functions with both significantly larger fidelities and lower energies over a mean-field solution do not necessarily lead to AFQMC results closer to the exact ground state energy.

Calculation of molecular g-tensors by sampling spin orientations of generalised Hartree-Fock states

The variational inclusion of spin–orbit coupling in self-consistent field (SCF) calculations affords single-determinant wave functions that completely break spin symmetry. The individual components of the molecular g-tensor are commonly obtained from separate SCF solutions that align the magnetic moment along one of the three principal tensor axes. However, this strategy raises the question if energy differences between solutions are relevant, or how convergence is achieved if the principal axis system is not determined by molecular symmetry. The present work resolves these issues by a simple two-step procedure akin to the generator coordinate method (GCM). First, the orientation of the magnetic moment is constrained in generalised Hartree-Fock (GHF) calculations. Then, non-orthogonal configuration interaction between GHF determinants yields a Kramers doublet for the calculation of the complete g-tensor. Alternatively, diagonalisation in a basis spanned by spin rotations of a single GHF determinant results in qualitatively correct g-tensors by eliminating errors related to spin contamination. For small first-row molecules, these approaches are evaluated against experimental data and full configuration interaction results. It is further demonstrated for two fictitious tetrahedral and species that a GCM strategy can describe the spin–orbit splitting of orbitally-degenerate ground states, which may lead to negative g-values.

Development of anharmonic vibrational structure theory using backflow transformation

A backflow (BF) transformation was applied to a vibrational self-consistent field wavefunction, which is the mean-field approximation in an anharmonic vibrational structure theory. We tested its effectiveness on the H2O molecule using quantum Monte Carlo method. Each normal mode coordinate was transformed into collective coordinates using the correlation function depending on whole coordinates. The vibrational frequencies with the BF transformation reasonably agree with the vibrational configuration interaction (VCI) results. The vibrational probability density to which the BF transformation was applied was also consistent with the VCI results and showed that the BF transformation appropriately take into account the inter-mode couplings.

Benchmarking angular-momentum projected Hartree–Fock as an approximation

We benchmark angular-momentum projected-after-variation Hartree–Fock calculations as an approximation to full configuration-interaction results in a shell model basis. For such a simple approximation we find reasonably good agreement between excitation spectra, including for many odd-A and odd–odd nuclides. We frequently find shape coexistence, in the form of multiple Hartree–Fock minima; mixing in shape coexistence, the first step beyond single-reference projected Hartree–Fock, demonstrably improves the spectrum in the sd- and pf-shells. The complex spectra of germanium isotopes present a challenge: for even A the spectra are only moderately good and those of odd A bear little resemblance to the configuration-interaction results. Despite this failure we are able to broadly reproduce the odd–even staggering of ground state binding energies, save for germanium isotopes with N > 40. To illustrate potential applications, we compute the spectrum of the recently measured dripline nuclide 40Mg. All in all, projected Hartree–Fock often provides a better description of low-lying nuclear spectra than one might expect. Key to this is the use of gradient descent and unrestricted shapes.

Excited States from State-Specific Orbital-Optimized Pair Coupled Cluster.

The pair coupled cluster doubles (pCCD) method (where the excitation manifold is restricted to electron pairs) has a series of interesting features. Among others, it provides ground-state energies very close to what is obtained with doubly occupied configuration interaction (DOCI), but with a polynomial cost (compared with the exponential cost of the latter). Here, we address whether this similarity holds for excited states by exploring the symmetric dissociation of the linear H4 molecule. When ground-state Hartree–Fock (HF) orbitals are employed, pCCD and DOCI excited-state energies do not match, a feature that is assigned to the poor HF reference. In contrast, by optimizing the orbitals at the pCCD level (oo-pCCD) specifically for each excited state, the discrepancies between pCCD and DOCI decrease by 1 or 2 orders of magnitude. Therefore, the pCCD and DOCI methodologies still provide comparable energies for excited states, but only if suitable, state-specific orbitals are adopted. We also assessed whether a pCCD approach could be used to directly target doubly excited states, without having to resort to the equation-of-motion (EOM) formalism. In our Δoo-pCCD model, excitation energies are extracted from the energy difference between separate oo-pCCD calculations for the ground state and the targeted excited state. For a set comprising the doubly excited states of CH+, BH, nitroxyl, nitrosomethane, and formaldehyde, we found that Δoo-pCCD provides quite accurate excitation energies, with root-mean-square deviations (with respect to full configuration interaction results) lower than those of CC3 and comparable to those of EOM-CCSDT, two methods with a much higher computational cost.

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.

A stochastic approach to unitary coupled cluster.

Unitary coupled cluster (UCC), originally developed as a variational alternative to the popular traditional coupled cluster method, has seen a resurgence as a functional form for use on quantum computers. However, the number of excitors present in the Ansatz often presents a barrier to implementation on quantum computers. Given the natural sparsity of wavefunctions obtained from quantum Monte Carlo methods, we consider here a stochastic solution to the UCC problem. Using the coupled cluster Monte Carlo framework, we develop cluster selection schemes that capture the structure of the UCC wavefunction, as well as its Trotterized approximation, and use these to solve the corresponding projected equations. Due to the fast convergence of the equations with order in the cluster expansion, this approach scales polynomially with the size of the system. Unlike traditional UCC implementations, our approach naturally produces a non-variational estimator for the energy in the form of the projected energy. For unitary coupled cluster singles and doubles (UCCSD) in small systems, we find that this agrees well with the expectation value of the energy and, in the case of two electrons, with full configuration interaction results. For the larger N2 system, the two estimators diverge, with the projected energy approaching the coupled cluster result, while the expectation value is close to results from traditional UCCSD.

Simulating Periodic Systems on a Quantum Computer Using Molecular Orbitals.

The variational quantum eigensolver (VQE) is one of the most appealing quantum algorithms to simulate electronic structure properties of molecules on near-term noisy intermediate-scale quantum devices. In this work, we generalize the VQE algorithm for simulating periodic systems. However, the numerical study of a one-dimensional (1D) infinite hydrogen chain using existing VQE algorithms shows a remarkable deviation of the ground-state energy with respect to the exact full configuration interaction (FCI) result. Here, we present two schemes to improve the accuracy of quantum simulations for periodic systems. The first one is a modified VQE algorithm, which introduces a unitary transformation of Hartree-Fock orbitals to avoid the complex wave function. The second one is combining VQE with the quantum subspace expansion approach (VQE/QSE). Numerical benchmark calculations demonstrate that both of the two schemes provide an accurate description of the potential energy curve of the 1D hydrogen chain. In addition, excited states computed with the VQE/QSE approach also agree very well with FCI results.

Unitary coupled cluster ground- and excited-state molecular properties.

A scheme for the calculation of molecular properties within the framework of unitary coupled-cluster (UCC) theory in both the electronic ground and excited states is presented. The scheme is based on an expectation-value ansatz, similar to the equation-of-motion coupled-cluster method or the intermediate state representation (ISR) approach of the algebraic-diagrammatic construction (ADC) scheme. Due to the UCC ansatz, the resulting equations cannot be given by closed-form expressions but need to be approximated. Explicit expressions for the expectation value of a general one-particle operator correct through second order in perturbation theory have been derived and coded for the electronic ground state as well as for excited states of predominant single-excitation character. The resulting equations are shown to be equivalent to those of the second-order ADC/ISR procedure. As first computational tests, the second-order UCC method (UCC2) and the one employing third-order amplitudes (also eigenvectors) together with the second-order density matrix, denoted as UCC3(2), are applied to the calculation of dipole moments for a series of small closed- and open-shell systems as well as 4-cyanoindole and 2,3-benzofuran and compared to full configuration interaction or experimental results. For the aromatic organic molecules, the UCC2 method is shown to be sufficient for the ground-state dipole moment, whereas the UCC3(2) scheme is superior for excited-state dipole moments.

Intermediate state representation approach to physical properties of molecular electron-detached states. II. Benchmarking.

The third-order algebraic-diagrammatic construction method for studies of electron detachment processes within the electron propagator framework [IP-ADC(3)] was extended to treat the properties of molecular states with a detached electron using the intermediate state representation (ISR) formalism. The second-order ISR(2) equations for the one-particle (transition) density matrix have been derived and implemented as an extension of the IP-(U)ADC(3) method available in the Q-CHEM program. As a first systematic test of the present IP-(U)ADC(3)/ISR(2) method, the dipole moments of various electronic states of closed- and open-shell molecules have been computed and compared to full configuration interaction (FCI) results. The present study employing FCI benchmarks also provides the first rigorous estimates for the accuracy of electron detachment energies obtained using the IP-ADC(3) method.

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.

Iterative multireference configuration interaction

Iterative multireference configuration interaction (IMRCI) is proposed. It is exploited to compute the electronic energies of H2O and CH2 (singlet and triplet states) at equilibrium and non-equilibrium geometries. The potential energy curves of H2O, CH2 (singlet and triplet states) and N2 have also been calculated with IMRCI as well as the Møller Plesset perturbation theory (MP2, MP3, and MP4), the coupled cluster method with single and double substitutions (CCSD), and CCSD with perturbative triples correction (CCSD(T)). These calculations demonstrate that IMRCI results are independent of the initial guess of configuration functions in the reference space and converge quickly to the results of the full configuration interaction. The IMRCI errors relative to the full configuration interaction results are at the order of magnitude of 10−5 hartree within just 2–4 iterations. Further, IMRCI provides an efficient way to find on the potential energy surface the leading electron configurations which, as correct reference states, will be very helpful for the single-reference and multireference theoretical models to obtain accurate results.

Stochastic perturbation theory in a limited configuration space.

A general-order stochastic perturbation algorithm is obtained from the order-by-order expansion of the imaginary-time evolution of a configuration interaction wave function. A truncation of configuration space that is required for the practical treatment of the perturbative corrections, however, does not preserve size-consistency as is the case for a truncated configuration interaction. To circumvent this problem, we formulate a linked variant of stochastic perturbation theory based on the coupled-cluster ansatz. The implementation based on the linearized coupled-cluster is compared with several full configuration interaction results. We also compare the results with those obtained from deterministic coupled-cluster and many-body perturbation theories.

Spin-symmetry adaptation to the Monte Carlo correction configuration interaction wave functions.

We propose a method to adapt the spin-symmetry to the Monte Carlo correction configuration interaction (MC3I) wave function which is expanded by the selected Slater determinants (SDs). The spin-symmetry of the MC3I wave function is usually broken because the Monte Carlo method is used to select the SDs, and this problem becomes worse as the electron correlation becomes stronger. In the present method, the S^2 operator is applied to the set of the SDs in the MC3I wave function iteratively until the set becomes closed under S^2. The spin-symmetry adapted MC3I wave functions are calculated by diagonalization of the Hamiltonian matrix which is spanned by the converged set of SDs. The present method is tested by the application to the excited states of C2 in the bond dissociation region and the 100 lowest states of [Fe2S2(SCH3)4]3-. The deviations of S (total spin angular momentum) of some states were too large to assign the electronic states in the original MC3I calculations, while all states have the correct S after spin-symmetry adaptation and become comparable with the full configuration interaction and density matrix renormalization group results. With the present spin-symmetry adaptation, the MC3I method becomes applicable to strong electron correlation systems.

Lattice density-functional theory for quantum chemistry

We propose a lattice density-functional theory for {\it ab initio} quantum chemistry or physics as a route to an efficient approach that approximates the full configuration interaction energy and orbital occupations for molecules with strongly-correlated electrons. We build on lattice density-functional theory for the Hubbard model by deriving Kohn-Sham equations for a reduced then full quantum chemistry Hamiltonian, and demonstrate the method on the potential energy curves for the challenging problem of modelling elongating bonds in a linear chain of six hydrogen atoms. Here the accuracy of the Bethe-ansatz local-density approximation is tested for this quantum chemistry system and we find that, despite this approximate functional being designed for the Hubbard model, the shapes of the potential curves generally agree with the full configuration interaction results. Although there is a discrepancy for very stretched bonds, this is lower than when using standard density-functional theory with the local-density approximation.

Linear-Response Density Cumulant Theory for Excited Electronic States.

We present a linear-response formulation of density cumulant theory (DCT) that provides a balanced and accurate description of many electronic states simultaneously. In the original DCT formulation, only information about a single electronic state (usually, the ground state) is obtained. We discuss the derivation of linear-response DCT, present its implementation for the ODC-12 method (LR-ODC-12), and benchmark its performance for excitation energies in small molecules (N2, CO, HCN, HNC, C2H2, and H2CO), as well as challenging excited states in ethylene, butadiene, and hexatriene. For small molecules, LR-ODC-12 shows smaller mean absolute errors in excitation energies than equation-of-motion coupled cluster theory with single and double excitations (EOM-CCSD), relative to the reference data from EOM-CCSDT. In a study of butadiene and hexatriene, LR-ODC-12 correctly describes the relative energies of the singly excited 11Bu and the doubly excited 21Ag states, in excellent agreement with highly accurate semistochastic heat-bath configuration interaction results, while EOM-CCSD overestimates the energy of the 21Ag state by almost 1 eV. Our results demonstrate that linear-response DCT is a promising theoretical approach for excited states of molecules.

Development of spin-orbit coupling for stochastic configuration interaction techniques.

To perform spin-orbit coupling calculations on atoms and molecules, good zeroth-order wavefunctions are necessary. Here, we present the software development of the Monte Carlo Configuration Interaction (MCCI) method, to enable calculation of such properties, where MCCI iteratively constructs a multireference wavefunction using a stochastic procedure. In this initial work, we aim to establish the efficacy of this technique in predicting the splitting of otherwise degenerate energy levels on a range of atoms and small diatomic molecules. It is hoped that this work will subsequently act as a gateway toward using this method to investigate singlet-triplet interactions in larger multireference molecules. We show that MCCI can generate very good results using highly compact wavefunctions compared to other techniques, with no prior knowledge of important orbitals. Higher-order relativistic effects are neglected and spin-orbit coupling effects are incorporated using first-order degenerate perturbation theory with the Breit-Pauli Hamiltonian and effective nuclear charges in the one-electron operator. Results are obtained and presented for B, C, O, F, Si, S, and Cl atoms and OH, CN, NO, and C2 diatomic radicals including spin-orbit coupling constants and the relative splitting of the lowest energy degenerate state for each species. Convergence of MCCI to the full configuration interaction result is demonstrated on the multireference problem of stretched OH. We also present results from the singlet-triplet interaction between the X3Σg- and both the a1Δg and b1Σg+ states of the O2 molecule. © 2017 Wiley Periodicals, Inc.

A Jeziorski-Monkhorst fully uncontracted multi-reference perturbative treatment. I. Principles, second-order versions, and tests on ground state potential energy curves.

The present paper introduces a new multi-reference perturbation approach developed at second order, based on a Jeziorski-Mokhorst expansion using individual Slater determinants as perturbers. Thanks to this choice of perturbers, an effective Hamiltonian may be built, allowing for the dressing of the Hamiltonian matrix within the reference space, assumed here to be a CAS-CI. Such a formulation accounts then for the coupling between the static and dynamic correlation effects. With our new definition of zeroth-order energies, these two approaches are strictly size-extensive provided that local orbitals are used, as numerically illustrated here and formally demonstrated in the Appendix. Also, the present formalism allows for the factorization of all double excitation operators, just as in internally contracted approaches, strongly reducing the computational cost of these two approaches with respect to other determinant-based perturbation theories. The accuracy of these methods has been investigated on ground-state potential curves up to full dissociation limits for a set of six molecules involving single, double, and triple bond breaking together with an excited state calculation. The spectroscopic constants obtained with the present methods are found to be in very good agreement with the full configuration interaction results. As the present formalism does not use any parameter or numerically unstable operation, the curves obtained with the two methods are smooth all along the dissociation path.

Multi-Configuration Dirac–Hartree–Fock (MCDHF) Calculations for B-Like Ions

Relativistic configuration interaction results are presented for several B-like ions (Ge XXVIII, Rb XXXIII, Sr XXXIV, Ru XL, Sn XLVI, and Ba LII) using the multi-configuration Dirac–Hartree–Fock (MCDHF) method. The calculations are carried out in the active space approximation with the inclusion of the Breit interaction, the finite nuclear size effect, and quantum electrodynamic corrections. Results for fine structure energy levels for 1s22s22p and 2s2p2 configurations relative to the ground state are reported. The transition wavelengths, transition probabilities, line strengths, and absorption oscillator strengths for 2s22p–2s2p2 electric dipole (E1) transitions are calculated. Both valence and core-valence correlation effects were accounted for through single-double multireference (SD-MR) expansions to increasing sets of active orbitals. Comparisons are made with the available data and good agreement is achieved. The values calculated using core–valence correlation are found to be very close to other theoretical and experimental values. The behavior of oscillator strengths as a function of nuclear charge is studied. We believe that our results can guide experimentalists in identifying the fine-structure levels in their future work.

Positronic molecule calculations using Monte Carlo configuration interaction

We modify the Monte Carlo configuration interaction procedure to model atoms and molecules combined with a positron. We test this method with standard quantum chemistry basis sets on a number of positronic systems and compare results with the literature and full configuration interaction when appropriate. We consider positronium hydride, positronium hydroxide, lithium positride and a positron interacting with lithium, magnesium or lithium hydride. We demonstrate that we can capture much of the full configuration interaction results, but often require less than 10% of the configurations of these multireference wavefunctions. The effect of the number of frozen orbitals is also discussed.