Recent developments in the PySCF program package

Accuracy of the diffusion quantum Monte Carlo method on dissociation curves of van der Waals systems with the single-Slater-Jastrow trial wavefunction

Non-covalent interactions, particularly van der Waals (vdW) interactions, are crucial in chemistry, physics, and life sciences. Accurate calculation of both the repulsive and attractive parts of vdW potentials is essential for a reliable description of dissociation curves in vdW systems. The diffuse quantum Monte Carlo (DMC) method is known to be able to calculate vdW interactions accurately at equilibrium distances. However, its performance on vdW interaction energies at non-equilibrium distances has not been investigated. In this work, dissociation curves for ten vdW interacting systems with dispersion interactions, dipole-induced dipole interactions and dipole-dipole interactions are calculated using the DMC and some low-level methods, i.e., MP2 and CCSD, to illustrate their accuracy for the attractive and repulsion potentials of vdW potentials. Our results indicate that while MP2 and CCSD provide reliable descriptions of the attractive potential in vdW interactions, they exhibit larger errors in the repulsive region. On the other hand, DMC employing a single-Slater-Jastrow trial wavefunction with the DLA approximation for the non-local part of pseudopotentials demonstrates high accuracy in both repulsive and attractive potentials. Additionally, reliable DMC results can be achieved with T-moves if the Jastrow factor is optimized carefully. These results emphasize the capability of the DMC with DLA to accurately compute vdW interaction energies at non-equilibrium distances.

Double ionization potential equation-of-motion coupled-cluster approach with full inclusion of 4-hole-2-particle excitations and three-body clusters

The double ionization potential (DIP) equation-of-motion (EOM) coupled-cluster (CC) method with a full treatment of 4-hole-2-particle (4h-2p) correlations and triply excited clusters, abbreviated as DIP-EOMCCSDT(4h-2p), and its approximate form called DIP-EOMCCSD(T)(a)(4h-2p) have been formulated and implemented in the open-source CCpy package available on GitHub. The resulting codes work with both nonrelativistic and spin-free scalar-relativistic Hamiltonians. By examining the DIPs of a few small molecules, for which accurate reference data are available, we demonstrate that the DIP-EOMCCSDT(4h-2p) and DIP-EOMCCSD(T)(a)(4h-2p) approaches improve the results obtained using the DIP-EOMCC methods truncated at 3h-1p or 4h-2p excitations on top of the CC calculations with singles and doubles.

On the entanglement of chromophore and solvent orbitals

Among various types of chromophore-solvent interactions, the entanglement of chromophore and solvent orbitals, when significant, can cause the chromophore frontier orbitals to spread over to nearby solvent molecules, introducing partial charge-transfer character to the lowest excitations of the chromophore and lowering the excitation energies. While highly intuitive, the physical details of such orbital entanglement effects on the excitation energies of chromophores have yet to be fully explored. Here, using two well-known biochromophores (oxyluciferin and p-hydroxybenzyledene imidazolinone) as examples, we show that the chromophore-solvent orbital entanglements can be elucidated using two quantum mechanical embedding schemes: density matrix embedding theory and absolutely localized molecular orbitals. However, there remains a great challenge to incorporate the orbital entanglement effect in combined quantum mechanical molecular mechanical (QM/MM) calculations, and we hope that our findings will stimulate the development of new methods in that direction.

An Implicit Solvation Model for Binding Free Energy Estimation in Nonaqueous Solution

Implicit solvation models permit the approximate description of solute-solvent interactions, where water is the most often considered solvent due to its relevance in biological systems. The use of other solvents is less common but is relevant for applications such as in nuclear magnetic resonance (NMR) or chromatography. As an example, chloroform is commonly used in anisotropic NMR to measure residual dipolar couplings (RDCs) of chiral analytes weakly aligned by an alignment medium. They can be calculated from molecular dynamics (MD) simulations with explicit solvent, but it is computationally expensive, because tens of microseconds-long MD trajectories should be collected. Here, we develop a computational protocol and numerical implementation for binding free energies of rigid organic molecules to poly-γ-benzyl-l-glutamate using an implicit solvation model of chloroform. The model parameters are fit to alchemical binding free energies obtained from MD simulations in explicit chloroform and compared to the MD results and another implicit solvation model. Possible applications of the method are docking or Monte Carlo simulations based on a physically meaningful scoring function for the fast prediction of interaction poses of ligands for selective binding or the alignment of analytes.

Capturing Strong Correlation in Molecules with Phaseless Auxiliary-Field Quantum Monte Carlo Using Generalized Hartree-Fock Trial Wave Functions

Generalized Hartree-Fock (GHF) is a long-established electronic structure method that can lower the energy (compared to spin-restricted variants) by breaking physical wave function symmetries, namely S ^ 2 and S ^ z . After an exposition of GHF theory, we assess the use of GHF trial wave functions in phaseless auxiliary field quantum Monte Carlo (ph-AFQMC-G) calculations of strongly correlated molecular systems including symmetrically stretched hydrogen rings, carbon dioxide, and dioxygen. Imaginary time propagation is able to restore S ^ 2 symmetry and yields energies of comparable or better accuracy than CCSD(T) with unrestricted HF and GHF references, and consistently smooth dissociation curves─a remarkable result given the relative scalability of ph-AFQMC-G to larger system sizes. The present exploration of model strongly correlated systems marks a promising starting point for future studies of more chemically relevant molecules, and demonstrates that ph-AFQMC-G provides a highly accurate (and, in contrast to active-space-based trials, relatively black box and always size-consistent) description of challenging systems exhibiting, e.g., antiferromagnetic coupling and/or geometric spin frustration.

Minimum Energy Conical Intersection Optimization Using DFT/MRCI(2)

The combined density functional theory and multireference configuration interaction (DFT/MRCI) method is a semiempirical electronic structure approach that is both computationally efficient and has predictive accuracy for the calculation of electronic excited states and for the simulation of electronic spectroscopies. However, given that the reference space is generated via a selected-CI procedure, a challenge arises in the construction of smooth potential energy surfaces. To address this issue, we treat the local discontinuities that arise as noise within the Gaussian progress regression framework and learn the surfaces by explicitly incorporating and optimizing a white-noise kernel. The characteristic polynomial coefficient surfaces of the potential matrix, which are smooth functions of nuclear coordinates even at conical intersections, are learned in place of the adiabatic energies and are used to optimize the DFT/MRCI(2) minimum energy conical intersection geometries for representative intersection motifs in the molecules ethylene, butadiene, and fulvene. One consequence of explicitly treating the noise in the surfaces is that the energy difference cannot be made arbitrarily small at points of nominal intersection. Despite the limitations, however, we find the structures as well as the branching spaces to compare well with ab initio MRCI and conclude that this approach is a viable method to learn a smooth representation of DFT/MRCI(2) surfaces.

Bayesian Flow Network Framework for Chemistry Tasks

In this work, we introduce ChemBFN, a language model that handles chemistry tasks based on Bayesian flow networks working with discrete data. A new accuracy schedule is proposed to improve sampling quality by significantly reducing reconstruction loss. We show evidence that our method is appropriate for generating molecules with satisfied diversity, even when a smaller number of sampling steps is used. A classifier-free guidance method is adapted for conditional generation. It is also worthwhile to point out that after generative training, our model can be fine-tuned on regression and classification tasks with state-of-the-art performance, which opens the gate of building all-in-one models in a single module style. Our model has been open sourced at https://github.com/Augus1999/bayesian-flow-network-for-chemistry.

Introducing GPU Acceleration into the Python-Based Simulations of Chemistry Framework

We introduce the first version of GPU4PySCF, a module that provides GPU acceleration of methods in PySCF. As a core functionality, this provides a GPU implementation of two-electron repulsion integrals (ERIs) for contracted basis sets comprising up to g functions using the Rys quadrature. As an illustration of how this can accelerate a quantum chemistry workflow, we describe how to use the ERIs efficiently in the integral-direct Hartree-Fock build and nuclear gradient construction. Benchmark calculations show a significant speedup of 2 orders of magnitude with respect to the multithreaded CPU Hartree-Fock code of PySCF and the performance comparable to other open-source GPU-accelerated quantum chemical packages, including GAMESS and QUICK, on a single NVIDIA A100 GPU.

Self-consistent Quantum Linear Response with a Polarizable Embedding Environment

Quantum computing presents a promising avenue for solving complex problems, particularly in quantum chemistry, where it could accelerate the computation of molecular properties and excited states. This work focuses on computing excitation energies with hybrid quantum-classical algorithms for near-term quantum devices, combining the quantum linear response (qLR) method with a polarizable embedding (PE) environment. We employ the self-consistent operator manifold of quantum linear response (q-sc-LR) on top of a unitary coupled cluster (UCC) wave function in combination with a Davidson solver. The latter removes the need to construct the entire electronic Hessian, improving computational efficiency when going toward larger molecules. We introduce a new superposition-state-based technique to compute Hessian-vector products and show that this approach is more resilient toward noise than our earlier gradient-based approach. We demonstrate the performance of the PE-UCCSD model on systems such as butadiene and para-nitroaniline in water and find that PE-UCCSD delivers comparable accuracy to classical PE-CCSD methods on such simple closed-shell systems. We also explore the challenges posed by hardware noise and propose simple error mitigation techniques to maintain accurate results on noisy quantum computers.

Quantum Computing Approach to Fixed-Node Monte Carlo Using Classical Shadows

Quantum Monte Carlo (QMC) methods are powerful approaches for solving electronic structure problems. Although they often provide high-accuracy solutions, the precision of most QMC methods is ultimately limited by the trial wave function that must be used. Recently, an approach has been demonstrated to allow the use of trial wave functions prepared on a quantum computer [Huggins et al., Unbiasing fermionic quantum Monte Carlo with a quantum computer. Nature 2022, 603, 416] in the auxiliary-field QMC (AFQMC) method using classical shadows to estimate the required overlaps. However, this approach has an exponential post-processing step to construct these overlaps when performing classical shadows obtained using random Clifford circuits. Here, we study an approach to avoid this exponential scaling step by using a fixed-node Monte Carlo method based on full configuration interaction quantum Monte Carlo. This method is applied to the local unitary cluster Jastrow ansatz. We consider H4, ferrocene, and benzene molecules using up to 12 qubits as examples. Circuits are compiled to native gates for typical near-term architectures, and we assess the impact of circuit-level depolarizing noise on the method. We also provide a comparison of AFQMC and fixed-node approaches, demonstrating that AFQMC is more robust to errors, although extrapolations of the fixed-node energy reduce this discrepancy. Although the method can be used to reach chemical accuracy, the sampling cost to achieve this is high even for small active spaces, suggesting caution about the prospect of outperforming conventional QMC approaches.

Improving Exchange-Correlation Potentials of Standard Density Functionals with the Optimized-Effective-Potential Method for Higher Accuracy of Excitation Energies

We present a general scheme to improve the exchange-correlation potential of standard Kohn-Sham methods, like the PBE (Perdew, Burke, Ernzerhof) or PBE0 method, by enforcing exact conditions the exchange-correlation potential has to obey during their calculation. The required modifications of the potentials are enabled by generating the potentials within the optimized-effective-potential (OEP) framework instead of directly taking the functional derivative with respect to the electron density on a real-space grid as usual. We generalize a condition for the exact exchange potential that involves the eigenvalues of the highest occupied molecular orbital such that it is applicable to arbitrary approximate exchange potentials. The new approach yields strongly improved exchange-correlation potentials which lead to qualitatively and quantitatively improved KS orbital and eigenvalue spectra containing a Rydberg series as required and obeying much better the Kohn-Sham ionization energy theorem. If the resulting orbitals and eigenvalues are used as input quantities in time-dependent density-functional theory (TDDFT) to calculate excitation energies then the accuracy of the latter is drastically improved, e.g., for TDDFT with the PBE functional the accuracy of excitation energies is improved by a factor of roughly three. This make the introduced approach highly attractive for generating input orbitals and eigenvalues for TDDFT but potentially also for high-rung correlation functionals that are typically evaluated in a post-SCF (post self-consistent-field) manner. We apply the new approach to calculate exchange-correlation potentials to the PBE and PBE0 functionals but the approach is generally applicable to any functional.

Accurate Neural Network Fine-Tuning Approach for Transferable Ab Initio Energy Prediction across Varying Molecular and Crystalline Scales

Existing machine learning models attempt to predict the energies of large molecules by training small molecules, but eventually fail to retain high accuracy as the errors increase with system size. Through an orbital pairwise decomposition of the correlation energy, a pretrained neural network model on hundred-scale data containing small molecules is demonstrated to be sufficiently transferable for accurately predicting large systems, including molecules and crystals. Our model introduces a residual connection to explicitly learn the pairwise energy corrections, and employs various low-rank retraining techniques to modestly adjust the learned network parameters. We demonstrate that with as few as only one larger molecule retraining the base model originally trained on only small molecules of (H2O)6, the MP2 correlation energy of the large liquid water (H2O)64 in a periodic supercell can be predicted at chemical accuracy. Similar performance is observed for large protonated clusters and periodic poly glycine chains. A demonstrative application is presented to predict the energy ordering of symmetrically inequivalent sublattices for distinct hydrogen orientations in the ice XV phase. Our work represents an important step forward in the quest for cost-effective, highly accurate and transferable neural network models in quantum chemistry, bridging the electronic structure patterns between small and large systems.

Exploring Intrinsic Bond Properties with the Fukui Matrix from Conceptual Density Matrix Functional Theory

We extend the traditional conceptual density functional theory (CDFT) to conceptual density matrix functional theory (CDMFT) by replacing the external potential v(r) by the one-electron integral hrs in the energy functional. This approach provides a new path for investigating intrinsic bond properties such as bond reactivity. The derivation of the Fukui matrix, i.e., derivative of the density matrix P with respect to the number of electrons N, is elucidated, and the result is illustrated in a case study on H2O. The matrix is shown to play a crucial role in quantifying changes of bond strength for electron removal or addition processes via the bond order derivative ( ∂ B ∂ N ) - . Using the Mayer bond order and different atoms-in-molecules partitioning methods, we show that as a first-order response quantity, the bond order derivative agrees well with the finite difference bond order changes. The bond order derivative (bond Fukui function) is a bond reactivity descriptor. We demonstrate this by predicting the regioselectivity of a classical electrophilic addition reaction (the bromination of alkenes) and predicting the initial electron-driven bond cleavage in mass spectrometry. Specifically, the bond order derivative captures all of the major signals from the experimental mass spectra for a series of small molecules with a variety of functional groups.

Adapting hybrid density functionals with machine learning

Exact exchange contributions significantly affect electronic states, influencing covalent bond formation and breaking. Hybrid density functional approximations, which average exact exchange admixtures empirically, have achieved success but fall short of high-level quantum chemistry accuracy due to delocalization errors. We propose adaptive hybrid functionals, generating optimal exact exchange admixture ratios on the fly using data-efficient quantum machine learning models with negligible overhead. The adaptive Perdew-Burke-Ernzerhof hybrid density functional (aPBE0) improves energetics, electron densities, and HOMO-LUMO gaps in QM9, QM7b, and GMTKN55 benchmark datasets. A model uncertainty-based constraint reduces the method smoothly to PBE0 in extrapolative regimes, ensuring general applicability with limited training. By tuning exact exchange fractions for different spin states, aPBE0 effectively addresses the spin gap problem in open-shell systems such as carbenes. We also present a revised QM9 (revQM9) dataset with more accurate quantum properties, including stronger covalent binding, larger bandgaps, more localized electron densities, and larger dipole moments.

Simulating Ionized States in Realistic Chemical Environments with Algebraic Diagrammatic Construction Theory and Polarizable Embedding

Theoretical simulations of electron detachment processes are vital for understanding chemical redox reactions, semiconductor and electrochemical properties, and high-energy radiation damage. However, accurate calculations of ionized electronic states are very challenging due to their open-shell nature, importance of electron correlation effects, and strong interactions with chemical environment. In this work, we present an efficient approach based on algebraic diagrammatic construction theory with polarizable embedding that allows to accurately simulate ionized electronic states in condensed-phase or biochemical environments (PE-IP-ADC). We showcase the capabilities of PE-IP-ADC by computing the vertical ionization energy (VIE) of thymine molecule solvated in bulk water. Our results show that the second- and third-order PE-IP-ADC methods combined with the basis of set of triple-ζ quality yield a solvent-induced shift in VIE of -0.92 and -0.93 eV, respectively, in an excellent agreement with experimental estimate of -0.9 eV. This work demonstrates the power of PE-IP-ADC approach for simulating charged electronic states in realistic chemical environments and motivates its further development.

Constrained Nuclear-Electronic Orbital Transition State Theory Using Energy Surfaces with Nuclear Quantum Effects

Hydrogen-atom transfer is crucial in a myriad of chemical and biological processes, yet the accurate and efficient description of hydrogen-atom transfer reactions and kinetic isotope effects remains challenging due to significant quantum effects on hydrogenic motion, especially tunneling and zero-point energy. In this paper, we combine transition state theory (TST) with the recently developed constrained nuclear-electronic orbital (CNEO) theory to propose a new transition state theory denoted CNEO-TST. We use CNEO-TST with CNEO density functional theory (CNEO-DFT) to predict reaction rate constants for two prototypical gas-phase hydrogen-atom transfer reactions and their deuterated isotopologic reactions. CNEO-TST is similar to conventional TST except that it employs constrained minimized energy surfaces to include zero-point energy and shallow tunneling effects in the effective potential. We find that the new theory predicts reaction rates quite accurately at room temperature. The effective potential surface must be generated by CNEO theory rather than by ordinary electronic structure theory, but because of the favorable computational scaling of CNEO-DFT, the cost is economical even for large systems. Our results show that dynamics calculations with this approach achieve accuracy comparable to variational TST with a semiclassical multidimensional tunneling transmission coefficient at and above room temperature. Therefore, CNEO-TST can be a useful tool for rate prediction, even for reactions involving highly quantal motion, such as many chemical and biochemical reactions involving transfers of hydrogen atoms, protons, or hydride ions.

Alchemical harmonic approximation based potential for iso-electronic diatomics: Foundational baseline for Δ-machine learning

We introduce the alchemical harmonic approximation (AHA) of the absolute electronic energy for charge-neutral iso-electronic diatomics at fixed interatomic distance d0. To account for variations in distance, we combine AHA with this ansatz for the electronic binding potential, E(d)=(Eu-Es)Ec-EsEu-Esd/d0+Es, where Eu, Ec, Es correspond to the energies of the united atom, calibration at d0, and the sum of infinitely separated atoms, respectively. Our model covers the two-dimensional electronic potential energy surface spanned by distances of 0.7-2.5 Å and differences in nuclear charge from which only one single point (with elements of nuclear charge Z1, Z2, and distance d0) is drawn to calibrate Ec. Using reference data from pbe0/cc-pVDZ, we present numerical evidence for the electronic ground-state of all neutral diatomics with 8, 10, 12, and 14 electrons. We assess the validity of our model by comparison to legacy interatomic potentials (harmonic oscillator, Lennard-Jones, and Morse) within the most relevant range of binding (0.7-2.5 Å) and find comparable accuracy if restricted to single diatomics and significantly better predictive power when extrapolating to the entire iso-electronic series. We also investigated Δ-learning of the electronic absolute energy using our model as a baseline. This baseline model results in a systematic improvement, effectively reducing training data needed for reaching chemical accuracy by up to an order of magnitude from ∼1000 to ∼100. By contrast, using AHA+Morse as a baseline hardly leads to any improvement and sometimes even deteriorates the predictive power. Inferring the energy of unseen CO converges to a prediction error of ∼0.1 Ha in direct learning and ∼0.04 Ha with our baseline.

Impact of Dipole Self-Energy on Cavity-Induced Nonadiabatic Dynamics

The coupling of matter to the quantized electromagnetic field of a plasmonic or optical cavity can be harnessed to modify and control chemical and physical properties of molecules. In optical cavities, a term known as the dipole self-energy (DSE) appears in the Hamiltonian to ensure gauge invariance. The aim of this work is twofold. First, we introduce a method, which has its own merits and complements existing methods, to compute the DSE. Second, we study the impact of the DSE on cavity-induced nonadiabatic dynamics in a realistic system. For that purpose, various matrix elements of the DSE are computed as functions of the nuclear coordinates and the dynamics of the system after laser excitation is investigated. The cavity is known to induce conical intersections between polaritons, which gives rise to substantial nonadiabatic effects. The DSE is shown to slightly affect these light-induced conical intersections and, in particular, break their symmetry.

FragPT2: Multifragment Wave Function Embedding with Perturbative Interactions

Embedding techniques allow the efficient description of correlations within localized fragments of large molecular systems while accounting for their environment at a lower level of theory. We introduce FragPT2: a novel embedding framework that addresses multiple interacting active fragments. Fragments are assigned separate active spaces, constructed by localizing canonical molecular orbitals. Each fragment is then solved with a multireference method, self-consistently embedded in the mean field from other fragments. Finally, interfragment correlations are reintroduced through multireference perturbation theory. Our framework provides an exhaustive classification of interfragment interaction terms, offering a tool to analyze the relative importance of various processes such as dispersion, charge transfer, and spin exchange. We benchmark FragPT2 on challenging test systems, including N2 dimers, multiple aromatic dimers, and butadiene. We demonstrate that our method can be successful even for fragments defined by cutting through a covalent bond.

Accurate QM/MM Molecular Dynamics for Periodic Systems in GPU4PySCF with Applications to Enzyme Catalysis

We present an implementation of the quantum mechanics/molecular mechanics (QM/MM) method for periodic systems using GPU accelerated QM methods, a distributed multipole formulation of the electrostatics, and a pseudobond treatment of the QM/MM boundary. We demonstrate that our method has well-controlled errors, stable self-consistent QM convergence, and energy-conserving dynamics. We further describe an application to the catalytic kinetics of chorismate mutase. Using an accurate hybrid functional reparametrized to coupled cluster energetics, our QM/MM simulations highlight the sensitivity in the calculated rate to the choice of quantum method, quantum region selection, and local protein conformation. Our work is provided through the open-source PySCF package using acceleration from the GPU4PySCF module.

Enhancing the Optically Detected Magnetic Resonance Signal of Organic Molecular Qubits

In quantum information science and sensing, electron spins are often purified into a specific polarization through an optical-spin interface, a process known as optically detected magnetic resonance (ODMR). Diamond-NV centers and transition metals are both excellent platforms for these so-called color centers, while metal-free molecular analogues are also gaining popularity for their extended polarization lifetimes, milder environmental impacts, and reduced costs. In our earlier attempt at designing such organic high-spin π-diradicals, we proposed to spin-polarize by shelving triplet M S = ±1 populations as singlets. This was recently verified by experiments albeit with low ODMR contrasts of <1% at temperatures above 5 K. In this work, we propose to improve the ODMR signal by moving singlet populations back into the triplet M S = 0 sublevel, designing a true carbon-based molecular analogue to the NV center. Our proposal is based upon transition-orbital and group-theoretical analyses of beyond-nearest-neighbor spin-orbit couplings, which are further confirmed by ab initio calculations of a realistic trityl-based radical dimer. Microkinetic analyses point toward high ODMR contrasts of around 30% under experimentally feasible conditions, a stark improvement from previous works. Finally, in our quest toward ground-state optically addressable molecular spin qubits, we exemplify how our symmetry-based design avoids Zeeman-induced singlet-triplet mixings, setting the scene for realizing electron spin qubit gates.

Efficient Polarizable QM/MM Using the Direct Reaction Field Hamiltonian with Electrostatic Potential Fitted Multipole Operators

Electronic polarization and dispersion are decisive actors in determining interaction energies between molecules. These interactions have a particularly profound effect on excitation energies of molecules in complex environments, especially when the excitation involves a significant degree of charge reorganization. The direct reaction field (DRF) approach, which has seen a recent revival of interest, provides a powerful framework for describing these interactions in quantum mechanics/molecular mechanics (QM/MM) models of systems, where a small subsystem of interest is described using quantum chemical methods and the remainder is treated with a simple MM force field. In this paper we show how the DRF approach can be combined with the electrostatic potential fitted (ESPF) multipole operator description of the QM region charge density, which significantly improves the efficiency of the method, particularly for large MM systems, and for typical calculations effectively eliminates the dependence on MM system size. We also show how the DRF approach can be combined with fluctuating charge descriptions of the polarizable environment, as well as previously used atom-centered dipole-polarizability based models. We further show that the ESPF-DRF method provides an accurate description of molecular interactions in both ground and excited electronic states of the QM system and apply it to predict the gas to aqueous solution solvatochromic shifts in the UV/visible absorption spectrum of acrolein.

Atomic Decompositions of Periodic Electronic-Structure Simulations

We present a new theory for partitioning simulations of periodic and solid-state systems into physically sound atomic contributions at the level of Kohn-Sham density functional theory. Our theory is based on spatially localized linear combinations of crystalline Gaussian-type orbitals and, as such, capable of exposing local features within periodic electronic structures in a more intuitive and robust manner than alternatives based on the spatial distribution of atomic basis functions alone. Early decomposed cohesive energies of both molecular polymers and different crystalline polymorphs demonstrate how the atomic properties yielded by our theory convincingly align with the expected charge polarization in these systems, also whenever partial charges and Madelung energies may lend themselves somewhat ambiguous to interpretation.

Understanding and mitigating noise in molecular quantum linear response for spectroscopic properties on quantum computers

The promise of quantum computing to circumvent the exponential scaling of quantum chemistry has sparked a race to develop chemistry algorithms for quantum architecture. However, most works neglect the quantum-inherent shot noise, let alone the effect of current noisy devices. Here, we present a comprehensive study of quantum linear response (qLR) theory obtaining spectroscopic properties on simulated fault-tolerant quantum computers and present-day near-term quantum hardware. This work introduces novel metrics to analyze and predict the origins of noise in the quantum algorithm, proposes an Ansatz-based error mitigation technique, and reveals the significant impact of Pauli saving in reducing measurement costs and noise in subspace methods. Our hardware results using up to cc-pVTZ basis set serve as proof of principle for obtaining absorption spectra on quantum hardware in a general approach with the accuracy of classical multi-configurational methods. Importantly, our results exemplify that substantial improvements in hardware error rates and measurement speed are necessary to lift quantum computational chemistry from proof of concept to an actual impact in the field.

SHARC meets TEQUILA: mixed quantum-classical dynamics on a quantum computer using a hybrid quantum-classical algorithm

Recent developments in quantum computing are highly promising, particularly in the realm of quantum chemistry. Due to the noisy nature of currently available quantum hardware, hybrid quantum-classical algorithms have emerged as a reliable option for near-term simulations. Mixed quantum-classical dynamics methods effectively capture nonadiabatic effects by integrating classical nuclear dynamics with quantum chemical computations of the electronic properties. However, these methods face challenges due to the high computational cost of the quantum chemistry part. To mitigate the computational demand, we propose a method where the required electronic properties are computed through a hybrid quantum-classical approach that combines classical and quantum hardware. This framework employs the variational quantum eigensolver and variational quantum deflation algorithms to obtain ground and excited state energies, gradients, nonadiabatic coupling vectors, and transition dipole moments. These quantities are used to propagate the nonadiabatic molecular dynamics using the Tully's fewest switches surface hopping method, although the implementation is also compatible with other molecular dynamics approaches. The approach, implemented by integrating the molecular dynamics program package SHARC with the TEQUILA quantum computing framework, is validated by studying the cis-trans photoisomerization of methanimine and the electronic relaxation of ethylene. The results show qualitatively accurate molecular dynamics that align with experimental findings and other computational studies. This work is expected to mark a significant step towards achieving a "quantum advantage" for realistic chemical simulations.

A cross-entropy corrected hybrid multiconfiguration pair-density functional theory for complex molecular systems

Hybrid density functionals, such as B3LYP and PBE0, have achieved remarkable success by substantially improving over their parent methods, namely Hartree-Fock and the generalized gradient approximation, and generally outperforming the second-order Møller-Plesset perturbation theory (MP2) that is more expensive. Here, we extend the linear scheme of hybrid multiconfiguration pair-density functional theory (HMC-PDFT) by incorporating a cross-entropy ingredient to balance the description of static and dynamic correlation effects, leading to a consistent improvement on both exchange and correlation energies. The B3LYP-like translated on-top functional (tB4LYP) developed along this line not only surpasses the accuracy of its parent methods, the complete active space self-consistent field (CASSCF) and the original MC-PDFT functionals (tBLYP and tB3LYP), but also outperforms the widely used complete active space second-order perturbation theory (CASPT2). Remarkably, while remaining satisfactory for general purpose, tB4LYP shows superior accuracy for challenging cases like the Cr2 dissociation and the associated low-lying vibrational energies, the ethylene torsional rotation and the ethyne diabatic colinear dissociations, with the significantly lower computational cost than CASPT2.

A modular and extensible CHARMM-compatible model for all-atom simulation of polypeptoids

Peptoids (N-substituted glycines) are a class of sequence-defined synthetic peptidomimetic polymers with applications including drug delivery, catalysis, and biomimicry. Classical molecular simulations have been used to predict and understand the conformational dynamics of single chains and their self-assembly into morphologies including sheets, tubes, spheres, and fibrils. The CGenFF-NTOID model based on the CHARMM General Force Field has demonstrated success in accurate all-atom molecular modeling of peptoid structure and thermodynamics. Extension of this force field to new peptoid side chains has historically required reparameterization of side chain bonded interactions against ab initio data. This fitting protocol improves the accuracy of the force field but is also burdensome and precludes modular extensibility of the model to arbitrary peptoid sequences. In this work, we develop and demonstrate a Modular Side Chain CGenFF-NTOID (MoSiC-CGenFF-NTOID) as an extension of CGenFF-NTOID employing a modular decomposition of the peptoid backbone and side chain parameterizations, wherein arbitrary side chains within the large family of substituted methyl groups (i.e., -CH3, -CH2R, -CHRR', and -CRR'R″) are directly ported from CGenFF. We validate this approach against ab initio calculations and experimental data to develop a MoSiC-CGenFF-NTOID model for all 20 natural amino acid side chains along with 13 commonly used synthetic side chains and present an extensible paradigm to efficiently determine whether a novel side chain can be directly incorporated into the model or whether refitting of the CGenFF parameters is warranted. We make the model freely available to the community along with a tool to perform automated initial structure generation.

Assessment of electron-proton correlation functionals for vibrational spectra of shared-proton systems by constrained nuclear-electronic orbital density functional theory

Proton transfer plays a crucial role in various chemical and biological processes. A major theoretical challenge in simulating proton transfer arises from the quantum nature of the proton. The constrained nuclear-electronic orbital (CNEO) framework was recently developed to efficiently and accurately account for nuclear quantum effects, particularly quantum nuclear delocalization effects, in quantum chemistry calculations and molecular dynamics simulations. In this paper, we systematically investigate challenging proton transfer modes in a series of shared-proton systems using CNEO density functional theory (CNEO-DFT), focusing on evaluating existing electron-proton correlation functionals. Our results show that CNEO-DFT accurately describes proton transfer vibrational modes and significantly outperforms conventional DFT. The inclusion of the epc17-2 electron-proton correlation functional in CNEO-DFT produces similar performance to that without electron-proton correlations, while the epc17-1 functional yields less accurate results, comparable with conventional DFT. These findings hold true for both asymmetrical and symmetrical shared-proton systems. Therefore, until a more accurate electron-proton correlation functional is developed, we currently recommend performing vibrational spectrum calculations using CNEO-DFT without electron-proton correlation functionals.

ADAPT-QSCI: Adaptive Construction of an Input State for Quantum-Selected Configuration Interaction

We present a quantum-classical hybrid algorithm for calculating the ground state and its energy of the quantum many-body Hamiltonian by proposing an adaptive construction of a quantum state for the quantum-selected configuration interaction (QSCI) method. QSCI allows us to select important electronic configurations in the system to perform configuration interaction (CI) calculation (subspace diagonalization of the Hamiltonian) by sampling measurement for a proper input quantum state on a quantum computer, but how we prepare a desirable input state remains a challenge. We propose an adaptive construction of the input state for QSCI in which we run QSCI repeatedly to grow the input state iteratively. We numerically illustrate that our method, dubbed ADAPT-QSCI, can yield accurate ground-state energies for small molecules, including a noisy situation for eight qubits where error rates of two-qubit gates and the measurement are both as large as 1%. ADAPT-QSCI serves as a promising method to take advantage of current noisy quantum devices and pushes forward its application to quantum chemistry.

Reducing Numerical Precision Requirements in Quantum Chemistry Calculations

The abundant demand for deep learning compute resources has created a renaissance in low-precision hardware. Going forward, it will be essential for simulation software to run on this new generation of machines without sacrificing scientific fidelity. In this paper, we examine the precision requirements of a representative kernel from quantum chemistry calculations: the calculation of the single-particle density matrix from a given mean-field Hamiltonian (i.e., Hartree-Fock or density functional theory) represented in an LCAO basis. We find that double precision affords an unnecessarily high level of precision, leading to optimization opportunities. We show how an approximation built from an error-free matrix multiplication transformation can be used to potentially accelerate this kernel on future hardware. Our results provide a roadmap for adapting quantum chemistry software for the next generation of high-performance computing platforms.

Nonunitary Coupled Cluster Enabled by Midcircuit Measurements on Quantum Computers

Many quantum algorithms rely on a quality initial state for optimal performance. Preparing an initial state for specific applications can considerably reduce the cost of probabilistic algorithms such as the well studied quantum phase estimation (QPE). Fortunately, in the application space of quantum chemistry, generating approximate wave functions for molecular systems is well studied, and quantum computing algorithms stand to benefit from importing these classical methods directly into a quantum circuit. In this work, we propose a state preparation method based on coupled cluster (CC) theory, which is a pillar of quantum chemistry on classical computers, by incorporating midcircuit measurements into the circuit construction. Currently, the most well studied state preparation method for quantum chemistry on quantum computers is the variational quantum eigensolver (VQE) with a unitary-CC with single- and double-electron excitation terms (UCCSD) ansatz whose operations are limited to unitary gates. We verify the accuracy of our state preparation protocol using midcircuit measurements by performing energy evaluation and state overlap computation for a set of small chemical systems. We further demonstrate that our approach leads to a reduction of the classical computation overhead, and the number of CNOT and T gates by 28 and 57% on average when compared against the standard VQE-UCCSD protocol.

Acceleration without Disruption: DFT Software as a Service

Density functional theory (DFT) has been a cornerstone in computational chemistry, physics, and materials science for decades, benefiting from advancements in computational power and theoretical methods. This paper introduces a novel, cloud-native application, Accelerated DFT, which offers an order of magnitude acceleration in DFT simulations. By integrating state-of-the-art cloud infrastructure and redesigning algorithms for graphic processing units (GPUs), Accelerated DFT achieves high-speed calculations without sacrificing accuracy. It provides a user-friendly and scalable solution for the increasing demands of DFT calculations in scientific communities. The implementation details, examples, and benchmark results illustrate how Accelerated DFT can significantly expedite scientific discovery across various domains.

A general framework for active space embedding methods with applications in quantum computing

We developed a general framework for hybrid quantum-classical computing of molecular and periodic embedding approaches based on an orbital space separation of the fragment and environment degrees of freedom. We demonstrate its potential by presenting a specific implementation of periodic range-separated DFT coupled to a quantum circuit ansatz, whereby the variational quantum eigensolver and the quantum equation-of-motion algorithm are used to obtain the low-lying spectrum of the embedded fragment Hamiltonian. The application of this scheme to study localized electronic states in materials is showcased through the accurate prediction of the optical properties of the neutral oxygen vacancy in magnesium oxide (MgO). Despite some discrepancies in the position of the main absorption band, the method demonstrates competitive performance compared to state-of-the-art ab initio approaches, particularly evidenced by the excellent agreement with the experimental photoluminescence emission peak.

The diradicaloid electronic structure of dialumenes: a benchmark study at the Full-CI limit

Multiply-bonded main group compounds of groups 13-15 are attracting significant interest not only because they provide fundamental insight into the nature of metal-metal bonding, but also for their potential in small molecule bond activation and catalysis. This includes dialumenes, neutral Al(I) compounds that contain AlAl double bonds, which display high reactivity owing to their intrinsic diradicaloid character. The electronic structure of the simplest dialumene, Al2H2, is here analyzed up to a practical Full-CI limit using DMRG and selected CI methods for the bond dissociation energy (BDE), geometry and properties of the electron density (difference density, ELF). Acquiring Full-CI reference values for the simplest dialumene (but possessing the highest diradical character) allows for a rigorous benchmarking of simpler correlated wavefunction theory (WFT) methods and density functional methods in treating the electronic structure of such systems. Single-reference coupled cluster theory using a RHF reference is found to reliably converge to the Full-CI limit and CCSD(T) is fully capable of capturing the diradical character, while multi-reference methods offer no clear advantages. Density functional methods struggle to fully describe the electronic structure complexity although non-hybrid functionals such as TPSS come close. Solving the inverse Kohn-Sham problem for a Full-CI-quality density revealed minimal density-driven errors in the TPSS-calculated BDE unlike high-percentage hybrids such as M06-2X. No density functional, however, predicts accurate relative energies. Fractional-occupation density plots at the TPSS level correlate well with WFT-based diradical character metrics, a useful result for determining diradical character in larger systems.

Workflow for practical quantum chemical calculations with a quantum phase estimation algorithm: electronic ground and π-π* excited states of benzene and its derivatives

Quantum computers are expected to perform full-configuration interaction calculations with less computational resources compared to classical ones, thanks to the use of quantum phase estimation (QPE) algorithms. However, only a limited number of QPE-based quantum chemical calculations have been reported even for numerical simulations on a classical computer, and the practical workflow for the QPE computation has not yet been established. In this paper, we report the QPE simulations of the electronic ground and the π-π* excited singlet state of benzene and its chloro- and nitro-derivatives as the representative industrially important systems, with the aid of GPGPU acceleration of quantum circuit simulations. We adopted the pseudo-natural orbitals obtained from the MP2 calculation as the basis for the wave function expansion, the CISD calculation within the active space to find the main electronic configurations to be included in the input wave function of the excited state, and the technique to reduce the truncation error of the calculated total energies. The proposed computational workflow is easily applicable to other molecules and can be a standard approach for performing QPE-based quantum chemical calculations of practical molecules.

(Non-) periodic variation of excited-state properties for coinage metal dimers M2 (M = Cu, Ag, Au, Rg)

The impact of relativistic effects on the periodicity of elements has significant implications for the prediction of the properties of atoms and their compounds. In this study, (non-) periodic variations of the properties of Group IB dimers are investigated from the perspective of excited states. The EOM-CCSD and EOM-CCSD(T)(a)* methods along with wave function analysis tools are employed to investigate their excited state. According to our results, the EOM-CCSD(T)(a)* approach with the QZ basis set is required to obtain reasonable results for some states. SOC plays a crucial role in the excited state properties of Au2 and Rg2, and our results show that the ground state of Rg2 is an open-shell 2u state due to considerable SOC splitting in the 3Π state. To rationalize (non-) periodic variations of excited states, ionization potentials and electron affinities of these molecules are obtained to approximate the energies of occupied and virtual orbitals. Low-lying excited states are mainly transitions from occupied orbitals to the LUMO orbital for Cu2, Au2, and Rg2, while they are transitions from the HOMO to virtual orbitals in Ag2. This is due to a large energy difference between the HOMO and HOMO-1 in Ag2. The excited state properties of Au2 are similar to those of Cu2 when SOC is not considered due to scalar relativistic effects. The excited state properties of Rg2 differ from other molecules in the same group, as its LUMO orbital is predominantly composed of d orbitals, while they are primarily s orbitals in the other molecules.

Data Quality in the Fitting of Approximate Models: A Computational Chemistry Perspective

Empirical parametrization underpins many scientific methodologies including certain quantum-chemistry protocols [e.g., density functional theory (DFT), machine-learning (ML) models]. In some cases, the fitting requires a large amount of data, necessitating the use of data obtained using low-cost, and thus low-quality, means. Here we examine the effect of using low-quality data on the resulting method in the context of DFT methods. We use multiple G2/97 data sets of different qualities to fit the DFT-type methods. Encouragingly, this fitting can tolerate a relatively large proportion of low-quality fitting data, which may be attributed to the physical foundations of the DFT models and the use of a modest number of parameters. Further examination using "ML-quality" data shows that adding a large amount of low-quality data to a small number of high-quality ones may not offer tangible benefits. On the other hand, when the high-quality data is limited in scope, diversification by a modest amount of low-quality data improves the performance. Quantitatively, for parametrizing DFT (and perhaps also quantum-chemistry ML models), caution should be taken when more than 50% of the fitting set contains questionable data, and that the average error of the full set is more than 20 kJ mol-1. One may also follow the recently proposed transferability principles to ensure diversity in the fitting set.

Advanced Techniques for High-Performance Fock Matrix Construction on GPU Clusters

This Article presents two optimized multi-GPU algorithms for Fock matrix construction, building on the work of Ufimtsev and Martinez [ J. Chem. Theory Comput. 2009, 5, 1004-1015] and Barca et al. [ J. Chem. Theory Comput. 2021, 17, 7486-7503]. The novel algorithms, opt-UM and opt-Brc, introduce significant enhancements, including improved integral screening, exploitation of sparsity and symmetry, a linear scaling exchange matrix assembly algorithm, and extended capabilities for Hartree-Fock caculations up to f-type angular momentum functions. Opt-Brc excels for smaller systems and for highly contracted triple-ζ basis sets, while opt-UM is advantageous for large molecular systems. Performance benchmarks on NVIDIA A100 GPUs show that our algorithms in the EXtreme-scale Electronic Structure System (EXESS), when combined, outperform all current GPU and CPU Fock build implementations in TeraChem, QUICK, GPU4PySCF, LibIntX, ORCA, and Q-Chem. The implementations were benchmarked on linear and globular systems and average speed ups across three double-ζ basis sets of 1.4×, 8.4×, and 9.4× were observed compared to TeraChem, QUICK, and GPU4PySCF respectively. An increased average speedup of 2.1× over TeraChem is observed when using four A100 GPUs. Strong scaling analysis reveals over 91% parallel efficiency on four GPUs for opt-Brc, making it typically faster for multi-GPU execution. Single-compute-node comparisons with CPU-based software like ORCA and Q-Chem show speedups of up to 42× and 31×, respectively, enhancing power efficiency by up to 18×.

Real-Time TDDFT Using Noncollinear Functionals

Recently, we proposed a method to generalize collinear functionals to noncollinear functionals, called multicollinear approach, which has been applied in density functional theory (DFT) and linear-response time-dependent DFT (TDDFT) for the ground state and excited states calculations, respectively. In this work, we demonstrate the application of this method in real-time TDDFT by simulating electronic absorption spectra, Rabi resonance, and precession of a two-magnetic center system. Thanks to the nonvanishing local exchange-correlation torque provided by multicollinear functionals, research into the torques in the evolution of magnetization vector is carried out, which is useful for the exploration on spin dynamics.

Electron Correlation in 2D Periodic Systems from Periodic Bootstrap Embedding

Given the growing significance of 2D materials in various optoelectronic applications, it is imperative to have simulation tools that can accurately and efficiently describe electron correlation effects in these systems. Here, we show that the recently developed bootstrap embedding (BE) accurately predicts electron correlation energies and structural properties for 2D systems. Without explicit dependence on the reciprocal space sum (k-points) in the correlation calculation, our proof-of-concept calculations shows that BE can typically recover ∼99.5% of the total minimal basis electron correlation energy in 2D semimetal, insulator, and semiconductors. We demonstrate that BE can predict lattice constants and bulk moduli for 2D systems with high precision. Furthermore, we highlight the capability of BE to treat electron correlation in twisted bilayer graphene superlattices with large unit cells containing hundreds of carbon atoms. We find that as the twist angle decreases toward the magic angle, the correlation energy initially decreases in magnitude, followed by a subsequent increase. We conclude that BE is a promising electronic structure method for future applications to 2D materials.

Spectral operator representations

Machine learning in atomistic materials science has grown to become a powerful tool, with most approaches focusing on atomic geometry, typically decomposed into local atomic environments. This approach, while well-suited for machine-learned interatomic potentials, is conceptually at odds with learning complex intrinsic properties of materials, often driven by spectral properties commonly represented in reciprocal space (e.g., band gaps or mobilities) which cannot be readily partitioned in real space. For such applications, methods that represent the electronic rather than the atomic structure could be more promising. In this work, we present a general framework focused on electronic-structure descriptors that take advantage of the natural symmetries and inherent interpretability of physical models. We apply this framework first to material similarity and then to accelerated screening, where a model trained on 217 materials correctly labels 75% of entries in the Materials Cloud 3D database, which meet common screening criteria for promising transparent-conducting materials.

Accelerating Molecular Dynamics Simulations Using Socket-Based Interprocess Communication

Molecular dynamics (MD) simulations are essential for studying the time evolution of molecular systems. Still, their efficiency is often bottlenecked by file-based interprocess communication (IPC) between MD and electronic structure programs. We present a socket-based IPC implementation that dramatically accelerates MD simulations, reducing the computational time by >10-fold compared to those of traditional file-based methods. Our approach, applied to nonadiabatic molecular dynamics with the Newton-X program, eliminates disk read/write overhead, allowing for faster simulations over longer time scales. This method opens the door to more efficient high-throughput simulations, providing new opportunities for exploring complex molecular processes in real time.

Efficient Simulation of Inhomogeneously Correlated Systems Using Block Interaction Product States

The strength of the density matrix renormalization group (DMRG) in handling strongly correlated systems lies in its unbiased and simultaneous treatment of identical sites that are both energetically degenerate and spatially similar, as typically encountered in physical models. However, this very feature becomes a drawback when DMRG is applied to quantum chemistry calculations for large, realistic correlated systems. This is because entangled orbitals often span broad ranges in both energy and space, with their interactions being notably inhomogeneous. In this study, we suggest addressing the strong intrafragment correlations and weak interfragment correlations separately, utilizing a large-scale multiconfigurational calculation framework grounded in the block interaction product state formulation. The strong intrafragment correlation can be encapsulated in several electronic states located on fragments, which are obtained by considering the entanglement between fragments and their environments. Moreover, we incorporate non-Abelian spin-SU(2) symmetry in our work to target the desired states we interested with well-defined particle number and spin, providing deeper insights into the corresponding chemical processes. The described method has been examined in various chemical systems and demonstrates high efficiency in addressing the inhomogeneous effects in strong correlation quantum chemistry.

Simulating Real-Time Molecular Electron Dynamics Efficiently Using the Time-Dependent Density Matrix Renormalization Group

Compared to ground-state electronic structure optimizations, accurate simulations of molecular real-time electron dynamics are usually much more difficult to perform. To simulate electron dynamics, the time-dependent density matrix renormalization group (TDDMRG) has been shown to offer an attractive compromise between accuracy and cost. However, many simulation parameters significantly affect the quality and efficiency of a TDDMRG simulation. So far, it is unclear whether common wisdom from ground-state DMRG carries over to the TDDMRG, and a guideline on how to choose these parameters is missing. Here, in order to establish such a guideline, we investigate the convergence behavior of the main TDDMRG simulation parameters, such as time integrator, the choice of orbitals, and the choice of matrix-product-state representation for complex-valued nonsinglet states. In addition, we propose a method to select orbitals that are tailored to optimize the dynamics. Lastly, we showcase the TDDMRG by applying it to charge migration ionization dynamics in furfural, where we reveal a rapid conversion from an ionized state with a σ character to one with a π character within less than a femtosecond.

Determining the N-Representability of a Reduced Density Matrix via Unitary Evolution and Stochastic Sampling

The N-representability problem consists in determining whether, for a given p-body matrix, there exists at least one N-body density matrix from which the p-body matrix can be obtained by contraction, that is, if the given matrix is a p-body reduced density matrix (p-RDM). The knowledge of all necessary and sufficient conditions for a p-body matrix to be N-representable allows the constrained minimization of a many-body Hamiltonian expectation value with respect to the p-body density matrix and, thus, the determination of its exact ground state. However, the number of constraints that complete the N-representability conditions grows exponentially with system size, and hence, the procedure quickly becomes intractable for practical applications. This work introduces a hybrid quantum-stochastic algorithm to effectively replace the N-representability conditions. The algorithm consists of applying to an initial N-body density matrix a sequence of unitary evolution operators constructed from a stochastic process that successively approaches the reduced state of the density matrix on a p-body subsystem, represented by a p-RDM, to a target p-body matrix, potentially a p-RDM. The generators of the evolution operators follow the well-known adaptive derivative-assembled pseudo-Trotter method (ADAPT), while the stochastic component is implemented by using a simulated annealing process. The resulting algorithm is independent of any underlying Hamiltonian, and it can be used to decide whether a given p-body matrix is N-representable, establishing a criterion to determine its quality and correcting it. We apply the proposed hybrid ADAPT algorithm to alleged reduced density matrices from a quantum chemistry electronic Hamiltonian, from the reduced Bardeen-Cooper-Schrieffer model with constant pairing, and from the Heisenberg XXZ spin model. In all cases, the proposed method behaves as expected for 1-RDMs and 2-RDMs, evolving the initial matrices toward different targets.

Interacting-Bath Dynamical Embedding for Capturing Nonlocal Electron Correlation in Solids

Quantitative simulation of electronic structure of solids requires treating local and nonlocal electron correlations on an equal footing. We present a new ab initio formulation of Green's function embedding which, unlike dynamical mean-field theory that uses noninteracting bath, derives bath representation with general two-particle interactions in a systematically improvable manner. The resulting interacting-bath dynamical embedding theory (ibDET) utilizes an efficient real-axis coupled-cluster solver to compute the self-energy, approaching the full system limit at much reduced cost. When combined with the GW theory, GW+ibDET achieves good agreement with experimental spectral properties across a range of semiconducting, insulating, and metallic materials. Our approach also enables quantifying the role of nonlocal electron correlation in determining material properties and addressing the long-standing debate on the bandwidth narrowing of metallic sodium.

Formally Exact and Practically Useful Analytic Solution of Harmonium

We provide a novel exact analytic solution of harmonium with arbitrary Coulomb interaction strength, for ground as well as all the excited states, using our recently developed method for solving Schrödinger equations. By comparing three formally exact analytic representations of the wave function including the one that utilizes biconfluent Heun function, we find that the best and practically useful representation for the ground state is given by an exact factorized form involving a noninteger power pre-exponential factor, an exponentially decaying term and a modulator function. For excited states, additional factors are needed to account for the nodal information. We show that our method is far more efficient than basis-expansion-based methods in representing the wave function. With the exact wave functions, we have also analyzed the evolution trends of the electron density and natural occupation numbers with increasing interaction strength, which gives insight into the interesting physics in the strong correlation limit.

Block-Correlated Coupled Cluster Theory Based on the Generalized Valence Bond Reference for Singlet-Triplet Energy Gaps of Strongly Correlated Systems

A block-correlated coupled cluster (BCCC) method based on the triplet generalized valence bond (GVB) wave function (GVB-BCCC) has been implemented for the first time. By introducing several techniques, we have developed a practical and efficient GVB-BCCC code. The GVB-BCCC3 method (with up to three-pair correlation) can be used to deal with strongly correlated (SC) systems with triplet or singlet ground states, allowing singlet-triplet (S-T) energy gaps in the active space of SC systems computationally available. For selected SC systems, our calculations show that GVB-BCCC3 can always provide correct ground-state spin multiplicity as the complete active space configuration interaction (CASCI) or density matrix renormalization group (DMRG). Furthermore, we found that the S-T energy gaps from GVB-BCCC3 are quite consistent with CASCI or DMRG results. This work demonstrates that GVB-BCCC3 is a promising theoretical tool for describing S-T energy gaps within the active space of SC systems with large active spaces.

Slope of the Delocalization Function Is Proportional to Analytical Hardness

Conceptual Density Functional Theory (CDFT) has been extended beyond its traditional role in elucidating chemical reactivity to the development of density functional theory methods, e.g., the investigation of the delocalization error. This delocalization error causes the dependence of the energy on the number of electrons (N) to deviate from its exact piecewise linear behavior, an error which is the basis of many well-known limitations of commonly used density-functional approximations (DFAs). Following our previous work on the analytical hardness η± for pure functionals, we extend its application to hybrid and range-separated functionals. A comparison is made between the analytical hardness and the slope of the delocalization function introduced by Hait and Head-Gordon. Our results show that there is a linear relationship between its slope and the analytical hardness. An approximate scheme is presented to construct the energy vs N curve without fractional occupation number calculations. The extension to densities is discussed.

Optical Gaps of Ionic Materials from GW/BSE-in-DFT and CC2-in-DFT

This work presents a density functional theory (DFT)-based embedding technique for the calculation of optical gaps in ionic solids. The approach partitions the supercell of the ionic solid and embeds a small molecule-like cluster in a periodic environment using a cluster-in-periodic embedding method. The environment is treated with DFT, and its influence on the cluster is captured by a DFT-based embedding potential. The optical gap is estimated as the lowest singlet excitation energy of the embedded cluster, obtained using a wave function theory method: second-order approximate coupled-cluster singles and doubles (CC2), and a many-body perturbation theory method: GW approximation combined with the Bethe-Salpeter equation (GW/BSE). The calculated excitation energies are benchmarked against the periodic GW/BSE values, equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) results, and experiments. Both CC2-in-DFT and GW/BSE-in-DFT deliver excitation energies that are in good agreement with experimental values for several ionic solids (MgO, CaO, LiF, NaF, KF, and LiCl) while incurring negligible computational costs. Notably, GW/BSE-in-DFT exhibits remarkable accuracy with a mean absolute error (MAE) of just 0.38 eV with respect to experiments, demonstrating the effectiveness of the embedding strategy. In addition, the versatility of the method is highlighted by investigating the optical gap of a 2D MgCl2 system and the excitation energy of an oxygen vacancy in MgO, with results in good agreement with reported values.