Rapidly convergent quantum Monte Carlo using a Chebyshev projector.

The multireference coupled-cluster Monte Carlo (MR-CCMC) algorithm is a determinant-based quantum Monte Carlo (QMC) algorithm that is conceptually similar to Full Configuration Interaction QMC (FCIQMC). It has been shown to offer a balanced treatment of both static and dynamic correlation while retaining polynomial scaling, although application to large systems with significant strong correlation remained impractical. In this paper, we document recent algorithmic advances that enable rapid convergence and a more black-box approach to the multireference problem. These include a logarithmically scaling metric-tree-based excitation acceptance algorithm to search for determinants connected to the reference space at the desired excitation level and a symmetry-screening procedure for the reference space. We show that, for moderately sized reference spaces, the new search algorithm brings about an approximately 8-fold acceleration of one MR-CCMC iteration, while the symmetry screening procedure reduces the number of active reference space determinants with essentially no loss of accuracy. We also introduce a stochastic implementation of an approximate wall projector, which is the infinite imaginary time limit of the exponential projector, using a truncated expansion of the wall function in Chebyshev polynomials. Notably, this wall-Chebyshev projector can be used to accelerate any projector-based QMC algorithm. We show that it requires significantly fewer applications of the Hamiltonian to achieve the same statistical convergence. We benchmark these acceleration methods on the beryllium and carbon dimers, using initiator FCIQMC and MR-CCMC with basis sets up to cc-pVQZ quality.

Corrected density functional theory and the random phase approximation: Improved accuracy at little extra cost.

We recently introduced an efficient methodology to perform density-corrected Hartree-Fock density functional theory [DC(HF)-DFT] calculations and an extension to it we called "corrected" HF DFT [C(HF)-DFT] [Graf and Thom, J. Chem. Theory Comput. 19 5427-5438 (2023)]. In this work, we take a further step and combine C(HF)-DFT, augmented with a straightforward orbital energy correction, with the random phase approximation (RPA). We refer to the resulting methodology as corrected HF RPA [C(HF)-RPA]. We evaluate the proposed methodology across various RPA methods: direct RPA (dRPA), RPA with an approximate exchange kernel, and RPA with second-order screened exchange. C(HF)-dRPA demonstrates very promising performance; for RPA with exchange methods, on the other hand, we often find over-corrections.

A hybrid stochastic configuration interaction-coupled cluster approach for multireference systems.

The development of multireference coupled cluster (MRCC) techniques has remained an open area of study in electronic structure theory for decades due to the inherent complexity of expressing a multiconfigurational wavefunction in the fundamentally single-reference coupled cluster framework. The recently developed multireference-coupled cluster Monte Carlo (mrCCMC) technique uses the formal simplicity of the Monte Carlo approach to Hilbert space quantum chemistry to avoid some of the complexities of conventional MRCC, but there is room for improvement in terms of accuracy and, particularly, computational cost. In this paper, we explore the potential of incorporating ideas from conventional MRCC-namely, the treatment of the strongly correlated space in a configuration interaction formalism-to the mrCCMC framework, leading to a series of methods with increasing relaxation of the reference space in the presence of external amplitudes. These techniques offer new balances of stability and cost against accuracy, as well as a means to better explore and better understand the structure of solutions to the mrCCMC equations.

Periodicity of Single-Molecule Magnet Behaviour of Heterotetranuclear Lanthanide Complexes across the Lanthanide Series: A Compendium.

Acetato-bridged palladium-lanthanide tetranuclear heterometallic complexes of the form [Pd2 Ln2 (H2 O)2 (CH3 COO)10 ]⋅2 CH3 COOH [Ln2 =Ce2 (1), Pr2 (2), Nd2 (3), Sm2 (4), Tb2 (5), Dy2 (6), Dy0.2 Y1.8 (6''), Ho2 (7), Er2 (8), Er0.24 Y1.7 (8''), Tm2 (9), Yb2 (10), Y2 (11)] were synthesised and characterised by experimental and theoretical techniques. All complexes containing Kramers lanthanide ions [Ln3+ =Ce (1), Nd (3), Sm (4), Dy (6), DyY (6''), Er (8), ErY (8''), Yb (10)] showed field-induced slow magnetic relaxation, characteristic of single-molecule magnetism and purely of molecular origin. In contrast, all non-Kramers lanthanide ions [Ln3+ =Pr (2), Tb (5), Ho (7), Tm (9), Y3+ (11) is diamagnetic and non-lanthanide] did not show any slow magnetic relaxation. The variation in the electronic structure and accompanying consequences across the complexes representing all Kramers and non-Kramers lanthanide ions were investigated. The origin of the magnetic properties and the extent to which the axial donor-acceptor interaction involving the lanthanide ions and an electron-deficient d z 2 orbital of palladium affects the observed magnetic and electronic properties across the lanthanide series are presented. Unique consistent electronic and magnetic properties of isostructural complexes spanning the lanthanide series with properties dependent on whether the ions are Kramers or non-Kramers are reported.

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.

Theory and implementation of a novel stochastic approach to coupled cluster.

We present a detailed discussion of our novel diagrammatic coupled cluster Monte Carlo (diagCCMC) [Scott et al. J. Phys. Chem. Lett. 10, 925 (2019)]. The diagCCMC algorithm performs an imaginary-time propagation of the similarity-transformed coupled cluster Schrödinger equation. Imaginary-time updates are computed by the stochastic sampling of the coupled cluster vector function: each term is evaluated as a randomly realized diagram in the connected expansion of the similarity-transformed Hamiltonian. We highlight similarities and differences between deterministic and stochastic linked coupled cluster theory when the latter is re-expressed as a sampling of the diagrammatic expansion and discuss details of our implementation that allow for a walker-less realization of the stochastic sampling. Finally, we demonstrate that in the presence of locality, our algorithm can obtain a fixed errorbar per electron while only requiring an asymptotic computational effort that scales quartically with system size, independent of the truncation level in coupled cluster theory. The algorithm only requires an asymptotic memory cost scaling linearly, as demonstrated previously. These scaling reductions require no ad hoc modifications to the approach.

An Organic-Inorganic Hybrid Exhibiting Electrical Conduction and Single-Ion Magnetism.

The first three-dimensional (3D) conductive single-ion magnet (SIM), (TTF)2 [Co(pdms)2 ] (TTF=tetrathiafulvalene and H2 pdms=1,2-bis(methanesulfonamido)benzene), was electrochemically synthesised and investigated structurally, physically, and theoretically. The similar oxidation potentials of neutral TTF and the molecular precursor [HNEt3 ]2 [M(pdms)2 ] (M=Co, Zn) allow for multiple charge transfers (CTs) between the SIM donor [M(pdms)2 ]n- and the TTF.+ acceptor, as well as an intradonor CT from the pdms ligand to Co ion upon electrocrystallisation. Usually TTF functions as a donor, whereas in our system TTF is both a donor and an accepter because of the similar oxidation potentials. Furthermore, the [M(pdms)2 ]n- donor and TTF.+ acceptor are not segregated but strongly interact with each other, contrary to reported layered donor-acceptor electrical conductors. The strong intermolecular and intramolecular interactions, combined with CT, allow for relatively high electrical conductivity even down to very low temperatures. Furthermore, SIM behaviour with slow magnetic relaxation and opening of hysteresis loops was observed. (TTF)2 [Co(pdms)2 ] (2-Co) is an excellent building block for preparing new conductive SIMs.

Complex adiabatic connection: A hidden non-Hermitian path from ground to excited states.

Processes related to electronically excited states are central in many areas of science; however, accurately determining excited-state energies remains a major challenge in theoretical chemistry. Recently, higher energy stationary states of non-linear methods have themselves been proposed as approximations to excited states, although the general understanding of the nature of these solutions remains surprisingly limited. In this letter, we present an entirely novel approach for exploring and obtaining excited stationary states by exploiting the properties of non-Hermitian Hamiltonians. Our key idea centres on performing analytic continuations of conventional quantum chemistry methods. Considering Hartree-Fock theory as an example, we analytically continue the electron-electron interaction to expose a hidden connectivity of multiple solutions across the complex plane, revealing a close resemblance between Coulson-Fischer points and non-Hermitian degeneracies. Finally, we demonstrate how a ground-state wave function can be morphed naturally into an excited-state wave function by constructing a well-defined complex adiabatic connection.

Using SCF metadynamics to extend density matrix embedding theory to excited states.

A new framework based on density matrix embedding theory (DMET) capable of directly targeting excited electronic states is proposed and implemented. DMET has previously been shown to be an effective method of calculating the ground state energies of systems exhibiting strong static correlation but has never been applied to calculate excited state energies. In this work, the Schmidt decomposition is applied directly on excited states, approximated by higher lying self-consistent field solutions. The DMET prescription is applied following this Schmidt decomposition allowing for a direct embedding of excited states. Initial results are obtained for a system of multiple hydrogen dimers and the lithium hydride dissociation. We analyze the nature of each part of the excited state DMET calculation and identify challenges. These challenges to the implementation of excited state DMET are discussed, and potential suggestions moving forward are recommended.

A study of the dense uniform electron gas with high orders of coupled cluster.

We investigate the accuracies of different coupled cluster levels in a finite model solid, the 14 electron spin-non-polarised uniform electron gas. For densities between rs = 0.5 a0 and rs = 5 a0, we calculate ground state correlation energies with stochastic coupled cluster ranging from coupled cluster singles and doubles (CCSD) to coupled cluster including all excitations up to quintuples (CCSDTQ5). We find the need to add triple excitations for an accuracy of 0.01 eV/electron beyond rs = 0.5 a0. Quadruple excitations start being significant past rs = 3 a0. At rs = 5 a0, CCSD gives a correlation energy with a 16% error and coupled cluster singles doubles and triples is in error by 2% compared to the CCSDTQ5 result. CCSDTQ5 gives an energy in agreement with full configuration interaction quantum Monte Carlo results.

Stochastic coupled cluster theory: Efficient sampling of the coupled cluster expansion.

We consider the sampling of the coupled cluster expansion within stochastic coupled cluster theory. Observing the limitations of previous approaches due to the inherently non-linear behavior of a coupled cluster wavefunction representation, we propose new approaches based on an intuitive, well-defined condition for sampling weights and on sampling the expansion in cluster operators of different excitation levels. We term these modifications even and truncated selections, respectively. Utilising both approaches demonstrates dramatically improved calculation stability as well as reduced computational and memory costs. These modifications are particularly effective at higher truncation levels owing to the large number of terms within the cluster expansion that can be neglected, as demonstrated by the reduction of the number of terms to be sampled when truncating at triple excitations by 77% and hextuple excitations by 98%.

Developments in stochastic coupled cluster theory: The initiator approximation and application to the uniform electron gas.

We describe further details of the stochastic coupled cluster method and a diagnostic of such calculations, the shoulder height, akin to the plateau found in full configuration interaction quantum Monte Carlo. We describe an initiator modification to stochastic coupled cluster theory and show that initiator calculations can at times be extrapolated to the unbiased limit. We apply this method to the 3D 14-electron uniform electron gas and present complete basis set limit values of the coupled cluster singles and doubles (CCSD) and previously unattainable coupled cluster singles and doubles with perturbative triples (CCSDT) correlation energies for up to r(s) = 2, showing a requirement to include triple excitations to accurately calculate energies at high densities.

Folded Spectrum VQE: A Quantum Computing Method for the Calculation of Molecular Excited States.

The recent developments of quantum computing present novel potential pathways for quantum chemistry as the scaling of the computational power of quantum computers could be harnessed to naturally encode and solve electronic structure problems. Theoretically exact quantum algorithms for chemistry have been proposed (e.g., quantum phase estimation), but the limited capabilities of current noisy intermediate-scale quantum devices motivated the development of less demanding hybrid algorithms. In this context, the variational quantum eigensolver (VQE) algorithm was successfully introduced as an effective method to compute the ground-state energies of small molecules. This study investigates the folded spectrum (FS) method as an extension of the VQE algorithm for the computation of molecular excited states. It provides the possibility of directly computing excited states around a selected target energy using the same quantum circuit as for the ground-state calculation. Inspired by the variance-based methods from the quantum Monte Carlo literature, the FS method minimizes the energy variance, thus, in principle, requiring a computationally expensive squared Hamiltonian to be applied. We alleviate this potentially poor scaling by employing a Pauli grouping procedure to identify sets of commuting Pauli strings that can be evaluated simultaneously. This allows for a significant reduction in the computational cost. We applied the FS-VQE method to small molecules (H2, LiH), obtaining all electronic excited states with chemical accuracy on ideal quantum simulators. Furthermore, we explore the application of quantum error mitigation techniques, demonstrating improved energy accuracy on noisy simulators compared with simulations without mitigation.

Simple and Efficient Route toward Improved Energetics within the Framework of Density-Corrected Density Functional Theory.

The crucial step in density-corrected Hartree-Fock density functional theory (DC(HF)-DFT) is to decide whether the density produced by the density functional for a specific calculation is erroneous and, hence, should be replaced by, in this case, the HF density. We introduce an indicator, based on the difference in noninteracting kinetic energies between DFT and HF calculations, to determine when the HF density is the better option. Our kinetic energy indicator directly compares the self-consistent density of the analyzed functional with the HF density, is size-intensive, reliable, and most importantly highly efficient. Moreover, we present a procedure that makes best use of the computed quantities necessary for DC(HF)-DFT by additionally evaluating a related hybrid functional and, in that way, not only "corrects" the density but also the functional itself; we call that procedure corrected Hartree-Fock density functional theory (C(HF)-DFT).

Studies on the Transcorrelated Method.

We investigate the possibility of using a transcorrelated (TC) Hamiltonian to describe electron correlation. A method to obtain TC wavefunctions was developed based on the mathematical framework of the bi-variational principle. This involves the construction of an effective TC Hamiltonian matrix, which can be solved in a self-consistent manner. This was optimized using a method we call second-order-moment minimization and demonstrate that it is possible to obtain highly accurate energies for some closed-shell atoms and helium-like ions. The effects of certain correlator terms on the description of electron-electron and electron-nuclear cusps were also examined graphically, and some TC wavefunctions were compared against near-exact Hylleraas wavefunctions.