Many-Body-Expansion Based on Variational Quantum Eigensolver and Deflation for Dynamical Correlation.

In this work, we utilize the framework of many-body expansion (MBE) to decompose electronic structures into fragments by incrementing virtual orbitals, aiming to accurately solve the ground and excited state energies of each fragment using the variational quantum eigensolver and deflation algorithms. While our approach is primarily based on unitary coupled cluster singles and doubles (UCCSD) and its generalization, we also introduce modifications and approximations to conserve quantum resources in MBE by partially generalizing the UCCSD operator and neglecting the relaxation of the reference states. As a proof of concept, we investigate the potential energy surfaces for the bond-breaking processes of the ground state of two molecules (H2O and N2) and calculate the ground and excited state energies of three molecules (LiH, CH+, and H2O). The results demonstrate that our approach can, in principle, provide reliable descriptions in all the tests including strongly correlated systems when appropriate approximations are chosen. Additionally, we perform model simulations to investigate the impact of shot noise on the total MBE energy and show that precise energy estimation is crucial for lower-order MBE fragments.

Nonunitary projective transcorrelation theory inspired by the F12 ansatz.

An alternative nonunitary transcorrelation, inspired by the F12 ansatz, is investigated. In contrast to the Jastrow transcorrelation of Boys-Handy, the effective Hamiltonian of this projective transcorrelation features: 1. a series terminating formally at four-body interactions. 2. no spin-contamination within the non-relativistic framework. 3. simultaneous satisfaction of the singlet and triplet first-order cusp conditions. 4. arbitrary choices of pairs for correlation including frozen-core approximations. We discuss the connection between the projective transcorrelation and F12 theory with applications to small molecules, to show that the cusp conditions play an important role to reduce the uncertainty arising from the nonunitary transformation.

Quantum Inverse Algorithm via Adaptive Variational Quantum Linear Solver: Applications to General Eigenstates.

We propose a quantum inverse algorithm (QInverse) to directly determine general eigenstates by repeatedly applying the inverse power of a shifted Hamiltonian to an arbitrary initial state. To properly deal with the strongly entangled inverse power states and the resultant excited states, we solved the underlying linear equation, both variationally and adaptively, to obtain a faithful inverse power state with a shallow quantum circuit. QInverse is singularity-free and successfully obtains the target excited states with an energy closest to the shift ω, which is difficult to reach using variational methods. We also propose a subspace expansion approach to accelerate convergence and show that it is helpful to determine the two nearest eigenvalues when they are equally close to ω. These approaches were compared with the folded-spectrum method, which aims to generate excited states through variational optimization. It is shown that, whereas the folded-spectrum approach often fails to predict the target state by falling into a local minimum owing to its variational features, the success rate and accuracy of our algorithms are systematically improvable.

Improved Algorithms of Quantum Imaginary Time Evolution for Ground and Excited States of Molecular Systems.

Quantum imaginary time evolution (QITE) is a recently proposed quantum-classical hybrid algorithm that is guaranteed to reach the lowest state of systems. In this study, we present several improvements on QITE, mainly focusing on molecular applications. We analyze the derivation of the underlying QITE equation order-by-order and suggest a modification that is theoretically well founded. Our results clearly indicate the soundness of the here-derived equation, enabling a better approximation of the imaginary time propagation by a unitary. We also discuss how to accurately estimate the norm of an imaginary-time-evolved state and applied it to excited-state calculations using the quantum Lanczos algorithm. Finally, we propose the folded-spectrum QITE scheme as a straightforward extension of QITE for general excited-state simulations. The effectiveness of all these developments is illustrated by simulations with or without noise effect, offering further insights into quantum algorithms for imaginary time evolution.

Binary dopant segregation enables hematite-based heterostructures for highly efficient solar H2O2 synthesis.

Dopant segregation, frequently observed in ionic oxides, is useful for engineering materials and devices. However, due to the poor driving force for ion migration and/or the presence of substantial grain boundaries, dopants are mostly confined within a nanoscale region. Herein, we demonstrate that core-shell heterostructures are formed by oriented self-segregation using one-step thermal annealing of metal-doped hematite mesocrystals at relatively low temperatures in air. The sintering of highly ordered interfaces between the nanocrystal subunits inside the mesocrystal eliminates grain boundaries, leaving numerous oxygen vacancies in the bulk. This results in the efficient segregation of dopants (~90%) on the external surface, which forms their oxide overlayers. The optimized photoanode based on hematite mesocrystals with oxide overlayers containing Sn and Ti dopants realises high activity (~0.8 µmol min-1 cm-2) and selectivity (~90%) for photoelectrochemical H2O2 production, which provides a wide range of application for the proposed concept.

Water-oxidation mechanism of cobalt phosphate co-catalyst in artificial photosynthesis: a theoretical study.

The initial water-oxidation reaction mechanism of the hydrated cobalt phosphate (CoPi) co-catalyst, which is consistent with conventional experimental findings, is investigated for O-O bond and OOH formation in this study. Theoretical calculations of hydrated CoPi cluster models, which are validated by vibrational spectrum calculations, elucidate the roles of phosphate as a source of oxygen and deliverer of protons, both of which result in the spontaneous formation of an O-O bond after the release of two electrons and two protons. The calculations also show that OOH formation proceeds subsequently depending on the spin electronic states of the hydrated CoPi surface, and O2 formation then spontaneously progresses after the release of two electrons and two protons. By theoretically tracing these processes, the initial water-oxidation reaction mechanism of the hydrated CoPi co-catalyst is proposed.

Improved Description and Efficient Implementation of Spin-Projected Perturbation Theory for Practical Applications.

In this study, we continue to develop the recently proposed second-order perturbation theory for the spin-projected Hartree-Fock method [Tsuchimochi, T.; Ten-no, S. L. J. Chem. Theory Comput. 2019, 15, 6688] in various aspects. A new, stable imaginary level-shift scheme is derived to obtain a well-conditioned equation, enabling a significantly faster convergence. To achieve a further speed-up, we propose a preconditioning scheme considering the pair character on a spin-projected basis. We also eliminate the computational memory bottleneck in solving the linear equation for large systems using a distributed memory parallel implementation. Finally, for the description of open-shell molecules, several modified zeroth-order Hamiltonians are introduced and tested using the Mn2O2(NHCHCO2)4 complex. These developments enable practical calculations of a second-order perturbation theory with improved accuracy at a reduced computational cost.

Quadratically convergent self-consistent field of projected Hartree-Fock.

We report on a quadratically convergent self-consistent field (QC-SCF) algorithm for the spin-projected unrestricted Hartree-Fock (SUHF) to mitigate the slow convergence of SUHF due to the presence of small eigenvalues in the orbital Hessian matrix. The new QC-SCF is robust and stable, allowing us to obtain the SUHF solutions very quickly. To demonstrate the applicability of the method, we present results for test systems with abundant non-dynamic correlation in comparison with the Roothaan repeated diagonalization, Pople extrapolation, and direct inversion of iterative subspace.

Towards Near-Exact Solutions of Molecular Electronic Structure: Full Coupled-Cluster Reduction with a Second-Order Perturbative Correction.

We introduce a new augmented adaptation of the recently developed full coupled-cluster reduction (FCCR) with a second-order perturbative correction, abbreviated as FCCR(2). FCCR is a selected coupled-cluster expansion aimed at optimally reducing the excitation manifold and commutator expansions for high-rank excitations for obtaining accurate solutions of the electronic Schödinger equation in a size-extensive manner. The present FCCR(2) enables estimating the residual correlation of FCCR by the second-order perturbative correction E(2) from the complementary space of the FCCR projection manifold. The linear relationship between E(2) and the energy of FCCR(2) allows accurate estimates of near-exact energies for a wide variety of molecules with strong electron correlation. The potential of the method is demonstrated using challenging cases, the ground-state electronic energy of the benzene molecule in equilibrium and stretched geometries, and the isomerization energy of the transition metal complex [Cu(NH3)]2O22+.

The Ground State Electronic Energy of Benzene.

We report on the findings of a blind challenge devoted to determining the frozen-core, full configuration interaction (FCI) ground-state energy of the benzene molecule in a standard correlation-consistent basis set of double-ζ quality. As a broad international endeavor, our suite of wave function-based correlation methods collectively represents a diverse view of the high-accuracy repertoire offered by modern electronic structure theory. In our assessment, the evaluated high-level methods are all found to qualitatively agree on a final correlation energy, with most methods yielding an estimate of the FCI value around -863 mEH. However, we find the root-mean-square deviation of the energies from the studied methods to be considerable (1.3 mEH), which in light of the acclaimed performance of each of the methods for smaller molecular systems clearly displays the challenges faced in extending reliable, near-exact correlation methods to larger systems. While the discrepancies exposed by our study thus emphasize the fact that the current state-of-the-art approaches leave room for improvement, we still expect the present assessment to provide a valuable community resource for benchmark and calibration purposes going forward.

Third-order Epstein-Nesbet perturbative correction to the initiator approximation of configuration space quantum Monte Carlo.

We implement Epstein-Nesbet perturbative corrections in the third-order for the initiator approximation of the configuration space quantum Monte Carlo. An improved sampling algorithm is proposed to address the stochastic noise of the corrections. The stochastic error for the perturbative corrections is considerably larger than that for the reference energy, and it fails to provide reasonable results unless a very long imaginary time integration is performed. The new sampling algorithm accumulates rejected walkers from multiple independent steps to cover a larger portion of the secondary space. The performance of the perturbative corrections is demonstrated for small molecules.

Second-Order Perturbation Theory with Spin-Symmetry-Projected Hartree-Fock.

We propose two different schemes for second-order perturbation theory with spin-projected Hartree-Fock. Both schemes employ the same ansatz for the first-order wave function, which is a linear combination of spin-projected configurations. The first scheme is based on the normal-ordered projected Hamiltonian, which is partitioned into the Fock-like component and the remaining two-particle-like contribution. In the second scheme, the generalized Fock operator is used to construct a spin-free zeroth-order Hamiltonian. To avoid the intruder state problem, we adopt the level-shift techniques frequently used in other multireference perturbation theories. We describe both real and imaginary shift schemes and compare their performances on small systems. Our results clearly demonstrate the superiority of the second perturbation scheme with an imaginary shift over other proposed approaches in various aspects, giving accurate potential energy curves, spectroscopic constants, and singlet-triplet splitting energies. We also apply these methods to the calculation of spin gaps of transition-metal complexes as well as the potential energy curve of the chromium dimer.

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.

Effect of Molecular Orientational Correlations on Solvation Free Energy Computed by Reference Interaction Site Model Theory.

The effect of molecular orientational correlations on the solvation free energy (SFE) of one-dimensional and three-dimensional reference interaction site models (1D- and 3D-RISM) is investigated. The repulsive bridge correction (RBC) and the partial wave (PW) expansion are representative approaches for accounting for the orientational correlation partially lacking in original 1D- and 3D-RISM. The SFEs of 1D- and 3D-RISM for a set of small organic molecules are compared with the simulation results. Accordingly, the SFE expressions, based on RBC and PW, provide more accurate results than those of the uncorrected HNC or KH SFE expressions, which indicates that accounting for molecular orientational dependencies significantly contributes to the improvement of the SFE. The SFE component analysis indicates that the nonpolar component mainly contributes to the correction. The dependence of the error in the RISM SFE on the number of solute sites is examined. In addition, we discuss the differences between 1D- and 3D-RISM through the effect of these corrections.

Extending spin-symmetry projected coupled-cluster to large model spaces using an iterative null-space projection technique.

Recently, we introduced an orbital-invariant approximate coupled-cluster (CC) method in the spin-projection manifold. The multi-determinantal property of spin-projection means that the parametrization in the spin-extended CC (ECC) ansatz is nonorthogonal and overcomplete. Therefore, the linear dependencies must be removed by an orthogonalization procedure to obtain meaningful solutions. Multi-reference methods often achieve this by diagonalizing a metric of the equation system, but this is not feasible with ECC because of the enormous size of the metric, a consequence of the incomplete active space of the spin-projected Hartree-Fock reference. As a result, the applicability of ECC has been limited to small benchmark systems, for which the ansatz was shown to be superior to the configuration interaction and linearized approximations. In this article, we provide a solution to this problem that completely avoids the metric diagonalization by iteratively projecting out its null-space from the working equations. As the additional computational cost required for this iterative projection is only marginal, it greatly expands the application range of ECC. We demonstrate the potential of approximate ECC by studying the complete basis set limit of F2 and transition metal complexes such as NiO, Mn2 , and [Cu2 O2 ]2+ , which have all been hindered by the prohibitively large metric size. We also identify the potential inadequacy of the molecular orbitals given by spin-projected Hartree-Fock in some cases, and propose possible solutions. © 2018 Wiley Periodicals, Inc.

Monte Carlo explicitly correlated many-body Green's function theory.

A highly scalable stochastic algorithm is proposed and implemented for computing the basis-set-incompleteness correction to the diagonal, frequency-independent self-energy of the second-order many-body Green's function (GF2) theory within the explicitly correlated (F12) formalism. The 6-, 9-, 12-, and 15-dimensional integrals comprising the F12 correction are directly evaluated by the Monte Carlo method using appropriate weight functions for importance sampling. The method is naturally and easily parallelized, involves minimal memory space and no disk I/O, and can use virtually any mathematical form of a correlation factor. Its computational cost to correct all ionization energies (IEs) is observed to increase as the fourth power of system size, as opposed to the fifth power in the case of the deterministic counterparts. The GF2 calculations and their F12 corrections for the first IEs of C60 and C70 were executed on 128 graphical processing units (GF2) and 896 central processing units (F12), respectively, to reach the results with statistical errors of 0.04 eV or less. They showed that the basis-set-incompleteness (from aug-cc-pVDZ) accounts for only 50%-60% of the deviations from experiments, suggesting the significance of higher-order perturbation corrections.

Orbital-invariant spin-extended approximate coupled-cluster for multi-reference systems.

We present an approximate treatment of spin-extended coupled-cluster (ECC) based on the spin-projection of the broken-symmetry coupled-cluster (CC) ansatz. ECC completely eliminates the spin-contamination of unrestricted CC and is therefore expected to provide better descriptions of dynamical and static correlation effects, but introduces two distinct problems. The first issue is the emergence of non-terminating amplitude equations, which are caused by the de-excitation effects inherent in symmetry projection operators. In this study, we take a minimalist approach and truncate the Taylor series of the exponential ansatz at a certain order such that the approximation safely recovers the traditional CC without spin-projection. The second issue is that the nonlinear equations of ECC become underdetermined, although consistent, yielding an infinitude of solutions. This problem arises because of the redundancies in the excitation manifold, as is common in other multi-reference approaches. We remove the linear dependencies in ECC by employing an orthogonal projection manifold. We also propose an efficient solver for our method, in which the components are usually sparse but not diagonal-dominant. It is shown that our approach is rigorously orbital-invariant and provides more accurate results than its configuration interaction and linearized CC analogues for chemical systems.

Partially linearized external models to active-space coupled-cluster through connected hextuple excitations.

Partially linearized external models to active-space coupled-cluster through hextuple excitations, for example, CC{SDtqph}L , CCSD{tqph}L , and CCSD{tqph}hyb, are implemented and compared with the full active-space CCSDtqph. The computational scaling of CCSDtqph coincides with that for the standard coupled-cluster singles and doubles (CCSD), yet with a much large prefactor. The approximate schemes to linearize the external excitations higher than doubles are significantly cheaper than the full CCSDtqph model. These models are applied to investigate the bond dissociation energies of diatomic molecules (HF, F2 , CuH, and CuF), and the potential energy surfaces of the bond dissociation processes of HF, CuH, H2 O, and C2 H4 . Among the approximate models, CCSD{tqph}hyb provides very accurate descriptions compared with CCSDtqph for all of the tested systems. © 2018 Wiley Periodicals, Inc.

Multi-state effective Hamiltonian and size-consistency corrections in stochastic configuration interactions.

Model space quantum Monte Carlo (MSQMC) is an extension of full configuration interaction QMC that allows us to calculate quasi-degenerate and excited electronic states by sampling the effective Hamiltonian in the model space. We introduce a novel algorithm based on the state-selective partitioning for the effective Hamiltonian using left eigenvectors to calculate several electronic states simultaneously at much less computational cost than the original MSQMC with the energy-dependent partitioning. The sampling of walkers in MSQMC is analyzed in the single reference limit using a stochastic algorithm for higher-order perturbation energies by the analogy of the deterministic case utilizing a full configuration interaction program. We further develop size-consistency corrections of the initiator adaptation (i-MSQMC) in three different ways, i.e., the coupled electron pair approximation, a posteriori, and second-order perturbative corrections. It is clearly demonstrated that most of the initiator error is originating from the deficiency of proper scaling of correlation energy due to its truncated CI nature of the initiator approximation and that the greater part of the error can be recovered by the size-consistency corrections developed in this work.