Recent developments in the PySCF program package

Conservation laws in coupled cluster dynamics at finite temperature

We extend the finite-temperature Keldysh non-equilibrium coupled cluster theory (Keldysh-CC) [A. F. White and G. K.-L. Chan, J. Chem. Theory Comput. 15, 6137-6253 (2019)] to include a time-dependent orbital basis. When chosen to minimize the action, such a basis restores local and global conservation laws (Ehrenfest's theorem) for all one-particle properties while remaining energy conserving for time-independent Hamiltonians. We present the time-dependent Keldysh orbital-optimized coupled cluster doubles method in analogy with the formalism for zero-temperature dynamics, extended to finite temperatures through the time-dependent action on the Keldysh contour. To demonstrate the conservation property and understand the numerical performance of the method, we apply it to several problems of non-equilibrium finite-temperature dynamics: a 1D Hubbard model with a time-dependent Peierls phase, laser driving of molecular H₂, driven dynamics in warm-dense silicon, and transport in the single impurity Anderson model.

Efficient Evaluation of Two-Center Gaussian Integrals in Periodic Systems

By using Poisson's summation formula, we calculate periodic integrals over Gaussian basis functions by partitioning the lattice summations between the real and reciprocal space, where both sums converge exponentially fast with a large exponent. We demonstrate that the summation can be performed efficiently to calculate two-center Gaussian integrals over various kernels including overlap, kinetic, and Coulomb. The summation in real space is performed using an efficient flavor of the McMurchie-Davidson recurrence relation. The expressions for performing summation in the reciprocal space are also derived and implemented. The algorithm for reciprocal space summation allows us to reuse several terms and leads to a significant improvement in efficiency when highly contracted basis functions with large exponents are used. We find that the resulting algorithm is only between a factor of 5 and 15 slower than that for molecular integrals, indicating the very small number of terms needed in both the real and reciprocal space summations. An outline of the algorithm for calculating three-center Coulomb integrals is also provided.

Decomposed Mean-Field Simulations of Local Properties in Condensed Phases

The present work demonstrates a robust protocol for probing localized electronic structure in condensed-phase systems, operating in terms of a recently proposed theory for decomposing the results of Kohn-Sham density functional theory in a basis of spatially localized molecular orbitals. In an initial application to liquid, ambient water and the assessment of the solvation energy and the embedded dipole moment of H₂O in solution, we find that both properties are amplified on average-in accordance with expectation-and that correlations are indeed observed to exist between them. However, the simulated solvent-induced shift to the dipole moment of water is found to be significantly dampened with respect to typical literature values. The local nature of our methodology has further allowed us to evaluate the convergence of bulk properties with respect to the extent of the underlying one-electron basis set, ranging from single-ζ to full (augmented) quadruple-ζ quality. Albeit a pilot example, our work paves the way toward future studies of local effects and defects in more complex phases, e.g., liquid mixtures and even solid-state crystals.

Molecular vibrational frequencies from analytic Hessian of constrained nuclear-electronic orbital density functional theory

Nuclear quantum effects are important in a variety of chemical and biological processes. The constrained nuclear-electronic orbital density functional theory (cNEO-DFT) has been developed to include nuclear quantum effects in energy surfaces. Herein, we develop the analytic Hessian for cNEO-DFT energy with respect to the change in nuclear (expectation) positions, which can be used to characterize stationary points on energy surfaces and compute molecular vibrational frequencies. This is achieved by constructing and solving the multicomponent cNEO coupled-perturbed Kohn-Sham (cNEO-CPKS) equations, which describe the response of electronic and nuclear orbitals to the displacement of nuclear (expectation) positions. With the analytic Hessian, the vibrational frequencies of a series of small molecules are calculated and compared to those from conventional DFT Hessian calculations as well as those from the vibrational second-order perturbation theory (VPT2). It is found that even with a harmonic treatment, cNEO-DFT significantly outperforms DFT and is comparable to DFT-VPT2 in the description of vibrational frequencies in regular polyatomic molecules. Furthermore, cNEO-DFT can reasonably describe the proton transfer modes in systems with a shared proton, whereas DFT-VPT2 often faces great challenges. Our results suggest the importance of nuclear quantum effects in molecular vibrations, and cNEO-DFT is an accurate and inexpensive method to describe molecular vibrations.

Efficient Open-Source Implementations of Linear-Scaling Polarizable Embedding: Use Octrees to Save the Trees

We present open-source implementations of the linear-scaling fast multipole method (FMM) within the polarizable embedding (PE) model for efficient treatment of large polarizable environments in computational spectroscopy simulations. The implementations are tested for accuracy, efficiency, and usability on model systems as well as more realistic biomolecular systems. We explain how FMM parameters affect the calculation of molecular properties and show that PE calculations employing FMM can be carried out in a black-box manner. The efficiency of the linear-scaling approach is demonstrated by simulating the UV/vis spectrum of a chromophore in an environment of more than 1 million polarizable sites. Our implementations are interfaced to several open-source quantum chemistry programs, making computational spectroscopy simulations within the PE model and FMM available to a large variety of methods and a broad user base.

Metal Oxo-Fluoride Molecules OnMF2 (M = Mn and Fe; n = 1-4) and O2MnF: Matrix Infrared Spectra and Quantum Chemistry

On reacting laser-ablated manganese or iron difluorides with O₂ or O₃ during codeposition in solid neon or argon, infrared absorptions of several new metal oxo-fluoride molecules, including OMF₂, (η¹-O₂)MF₂, (η²-O₃)MF₂, (η¹-O₂)₂MF₂ (M = Mn and Fe), and O₂MnF, have been observed. Quantum chemical density functional and multiconfiguration wavefunction calculations have been applied to characterize these new products by their geometric and electronic structures, vibrations, charges, and bonding. The assignment of the main vibrational absorptions as dominant symmetric or antisymmetric M-F or M-O stretching modes is confirmed by oxygen isotopic shifts and quantum chemical calculations of frequencies and thermal stabilities. The tendency of Fe to form polyoxygen complexes in lower oxidation states than the preceding element Mn is affirmed experimentally and supported theoretically. The M-F stretching frequencies of the isolated metal oxo-fluorides may provide a scale for the local charge on the MF₂ sites in active energy conversion systems. The study of these species provides insights for understanding the trend of oxidation state changes across the transition-metal series.

Correlation-driven phenomena in periodic molecular systems from variational two-electron reduced density matrix theory

Correlation-driven phenomena in molecular periodic systems are challenging to predict computationally not only because such systems are periodically infinite but also because they are typically strongly correlated. Here, we generalize the variational two-electron reduced density matrix (2-RDM) theory to compute the energies and properties of strongly correlated periodic systems. The 2-RDM of the unit cell is directly computed subject to necessary N-representability conditions such that the unit-cell 2-RDM represents at least one N-electron density matrix. Two canonical but non-trivial systems, periodic metallic hydrogen chains and periodic acenes, are treated to demonstrate the methodology. We show that while single-reference correlation theories do not capture the strong (static) correlation effects in either of these molecular systems, the periodic variational 2-RDM theory predicts the Mott metal-to-insulator transition in the hydrogen chains and the length-dependent polyradical formation in acenes. For both hydrogen chains and acenes, the periodic calculations are compared with previous non-periodic calculations with the results showing a significant change in energies and increase in the electron correlation from the periodic boundary conditions. The 2-RDM theory, which allows for much larger active spaces than are traditionally possible, is applicable to studying correlation-driven phenomena in general periodic molecular solids and materials.

Convergence to the fixed-node limit in deep variational Monte Carlo

Variational quantum Monte Carlo (QMC) is an ab initio method for solving the electronic Schrödinger equation that is exact in principle, but limited by the flexibility of the available Ansätze in practice. The recently introduced deep QMC approach, specifically two deep-neural-network Ansätze PauliNet and FermiNet, allows variational QMC to reach the accuracy of diffusion QMC, but little is understood about the convergence behavior of such Ansätze. Here, we analyze how deep variational QMC approaches the fixed-node limit with increasing network size. First, we demonstrate that a deep neural network can overcome the limitations of a small basis set and reach the mean-field (MF) complete-basis-set limit. Moving to electron correlation, we then perform an extensive hyperparameter scan of a deep Jastrow factor for LiH and H₄ and find that variational energies at the fixed-node limit can be obtained with a sufficiently large network. Finally, we benchmark MF and many-body Ansätze on H₂O, increasing the fraction of recovered fixed-node correlation energy of single-determinant Slater-Jastrow-type Ansätze by half an order of magnitude compared to previous variational QMC results, and demonstrate that a single-determinant Slater-Jastrow-backflow version of the Ansatz overcomes the fixed-node limitations. This analysis helps understand the superb accuracy of deep variational Ansätze in comparison to the traditional trial wavefunctions at the respective level of theory and will guide future improvements of the neural-network architectures in deep QMC.

Sulfur Molecules in Space by X-rays: A Computational Study

X-ray astronomy lacks high resolution spectra of interstellar dust analogues and molecules, severely hampering interstellar medium studies based on upcoming X-ray missions. Various theoretical approaches may be used to address this problem, but they must first be shown to reproduce reliable spectra compared to the experiment. In this work, we calculate the sulfur K edge X-ray absorption spectra of H₂S, SO₂, and OCS, whose spectra are already known from X-ray experiments and predict the X-ray spectrum of CS, which as far as we are aware has not been measured, thereby hampering its detection by X-ray telescopes. We chose these four molecules as the astrochemistry of sulfur is an unsolved problem and as the four molecules are already known to exist in space. We consider three types of methods for modeling the X-ray spectra: more accurate calculations with the algebraic-diagrammatic construction (ADC) and the CC2, CCSD, and CC3 coupled cluster (CC) approaches as well as more affordable ones with transition potential density functional theory (TP-DFT). A comparison of our computational results to previously reported experimental spectra shows that the core-valence separation (CVS) approaches CVS-ADC(2)-x and CVS-CC3 generally yield a good qualitative level of agreement with the experiment, suggesting that they can be used for interpreting measured spectra, while the TP-DFT method is not reliable for these molecules. However, quantitative agreement with the experiment is still outside the reach of the computational methods studied in this work.

Benchmarking Magnetizabilities with Recent Density Functionals

We have assessed the accuracy of the magnetic properties of a set of 51 density functional approximations, including both recently published and already established functionals. The accuracy assessment considers a series of 27 small molecules and is based on comparing the predicted magnetizabilities to literature reference values calculated using coupled-cluster theory with full singles and doubles and perturbative triples [CCSD(T)] employing large basis sets. The most accurate magnetizabilities, defined as the smallest mean absolute error, are obtained with the BHandHLYP functional. Three of the six studied Berkeley functionals and the three range-separated Florida functionals also yield accurate magnetizabilities. Also, some older functionals like CAM-B3LYP, KT1, BHLYP (BHandH), B3LYP, and PBE0 perform rather well. In contrast, unsatisfactory performance is generally obtained with Minnesota functionals, which are therefore not recommended for calculations of magnetically induced current density susceptibilities and related magnetic properties such as magnetizabilities and nuclear magnetic shieldings. We also demonstrate that magnetizabilities can be calculated by numerical integration of magnetizability density; we have implemented this approach as a new feature in the gauge-including magnetically induced current (GIMIC) method. Magnetizabilities can be calculated from magnetically induced current density susceptibilities within this approach even when analytical approaches for magnetizabilities as the second derivative of the energy have not been implemented. The magnetizability density can also be visualized, providing additional information that is not otherwise easily accessible on the spatial origin of magnetizabilities.

Efficient implementation of the single-reference algebraic diagrammatic construction theory for charged excitations: Applications to the TEMPO radical and DNA base pairs

We present an efficient implementation of the second- and third-order single-reference algebraic diagrammatic construction (ADC) theory for electron attachment and ionization energies and spectra [EA/IP-ADC(n), n = 2, 3]. Our new EA/IP-ADC program features spin adaptation for closed-shell systems, density fitting for efficient handling of the two-electron integral tensors, and vectorized and parallel implementation of tensor contractions. We demonstrate capabilities of our efficient implementation by applying the EA/IP-ADC(n) (n = 2, 3) methods to compute the photoelectron spectrum of the (2,2,6,6-tetramethylpiperidin-1-yl)oxyl (TEMPO) radical, as well as the vertical and adiabatic electron affinities of TEMPO and two DNA base pairs (guanine-cytosine and adenine-thymine). The spectra and electron affinities computed using large diffuse basis sets with up to 1028 molecular orbitals are found to be in good agreement with the best available results from the experiment and theoretical simulations.

Cluster-in-Molecule Local Correlation Method with an Accurate Distant Pair Correction for Large Systems

The cluster-in-molecule (CIM) local correlation approach with an accurate distant pair correlation energy correction is presented. For large systems, the inclusion of distant pair correlation energies is essential for the accurate prediction of absolute correlation energies and relative energies. Here, we propose a simple and efficient scheme for evaluating the distant pair correlation energy correction for the CIM approaches. The corrections can be readily extracted from electron correlation calculations of clusters with almost no additional effort. Benchmark calculations show that the improved CIM approach can recover more than 99.94% of the correlation energy calculated by the parent method. By combining the CIM approach with the domain-based local pair natural orbital (DLPNO) local correlation approach, we have provided accurate binding energies at the CIM-DLPNO-CCSD(T) level for a test set consisting of eight weakly bound complexes ranging in size from 200 to 1027 atoms. With these results as the reference data, the accuracy and applicability of other electron correlation methods and a few density functional methods for large systems have been assessed.

Quantum-Inspired Algorithm for the Factorized Form of Unitary Coupled Cluster Theory

The factorized form of unitary coupled cluster theory (UCC) is a promising wave-function ansatz for the variational quantum eigensolver algorithm. Here, we present a quantum-inspired classical algorithm for UCC based on an exact operator identity for the individual UCC factors. We implement this algorithm for calculations of the H₁₀ linear chain and the H₂O molecule with single and double ζ basis sets to provide insights into UCC as a wave-function ansatz. We find that for weakly correlated molecules, the factorized form of the UCC provides similar accuracy to conventional coupled cluster theory (CC); for strongly correlated molecules, where CC often breaks down, UCC significantly outperforms the configuration interaction (CI) ansatz. As a result, the factorized form of the UCC is an accurate, efficient, and reliable electronic structure method in both the weakly and strongly correlated regions. This classical algorithm now allows robust benchmarking of anticipated results from quantum computers and application of coupled-cluster techniques to more strongly correlated molecules.

Full-frequency GW without frequency

Efficient computer implementations of the GW approximation must approximate a numerically challenging frequency integral; the integral can be performed analytically, but doing so leads to an expensive implementation whose computational cost scales as O(N6), where N is the size of the system. Here, we introduce a new formulation of the full-frequency GW approximation by exactly recasting it as an eigenvalue problem in an expanded space. This new formulation (1) avoids the use of time or frequency grids, (2) naturally obviates the need for the common "diagonal" approximation, (3) enables common iterative eigensolvers that reduce the canonical scaling to O(N5), and (4) enables a density-fitted implementation that reduces the scaling to O(N4). We numerically verify these scaling behaviors and test a variety of approximations that are motivated by this new formulation. The new formulation is found to be competitive with conventional O(N4) methods based on analytic continuation or contour deformation. In this new formulation, the relation of the GW approximation to configuration interaction, coupled-cluster theory, and the algebraic diagrammatic construction is made especially apparent, providing a new direction for improvements to the GW approximation.

Regional Embedding Enables High-Level Quantum Chemistry for Surface Science

Compared to common density functionals, ab initio wave function methods can provide greater reliability and accuracy, which could prove useful when modeling adsorbates or defects of otherwise periodic systems. However, the breaking of translational symmetry necessitates large supercells that are often prohibitive for correlated wave function methods. As an alternative, this paper introduces the regional embedding approach, which enables correlated wave function treatments of only a target fragment of interest through small, fragment-localized orbital spaces constructed using a simple overlap criterion. Applications to the adsorption of water on lithium hydride, hexagonal boron nitride, and graphene substrates show that regional embedding combined with focal-point corrections can provide converged CCSD(T) (coupled-cluster) adsorption energies with very small fragment sizes.

The Shape of Full Configuration Interaction to Come

We present a Perspective on what the future holds for full configuration interaction (FCI) theory, with an emphasis on conceptual rather than technical details. Upon revisiting the early history of FCI, a number of its key contemporary approximations are compared on as equal a footing as possible, using a recent blind challenge on the benzene molecule as a testbed [Eriksen et al., J. Phys. Chem. Lett., 2020 11, 8922]. In the process, we review the scope of applications for which FCI continues to prove indispensable, and the required traits in terms of robustness, efficacy, and reliability its modern approximations must satisfy are discussed. We close by conveying a number of general observations on the merits offered by the state-of-the-art alongside some of the challenges still faced to this day. While the field has altogether seen immense progress over the years-the past decade, in particular-it remains clear that our community as a whole has a substantial way to go in enhancing the overall applicability of near-exact electronic structure theory for systems of general composition and increasing size.

Diffusion Monte Carlo method on small boron clusters using single- and multi- determinant-Jastrow trial wavefunctions

Multireference character in some small boron clusters could be significant, and a previous all-electron fixed-node diffusion quantum Monte Carlo (FN-DMC) calculation with the single-determinant-Jastrow (SDJ) trial wavefunction shows that the atomization energy (AE) of B₄ ⁺ is overestimated by about 1.4 eV compared with the coupled cluster method with single, doubles, and perturbative triples [CCSD(T)] results. All-electron FN-DMC calculations and those with the pseudopotential (PP) using SDJ and multi-determinant-Jastrow (MDJ) trial wavefunctions with B3LYP orbitals as well as CC calculations at different levels are carried out on B_n ^Q (n = 1-5, Q = -1, 0, 1) clusters. The obtained FN-DMC energies indicate that the node error of the employed SDJ trial wavefunction in all-electron calculations is different from that with the PP for some clusters. The error of AEs and dissociation energies (DEs) from all-electron FN-DMC calculations is larger than that with the PP when the SDJ trial wavefunction is employed, while errors of CC methods do not depend on whether the PP is used. AEs and DEs of the boron clusters are improved significantly when MDJ trial wavefunctions are used in both all-electron calculations and those with the PP, and their error is similar to that of CCSD(T) compared with CCSDT(Q) results. On the other hand, reasonable adiabatic electron detachment energies (ADEs) and ionization potentials (AIPs) are achieved with FN-DMC using SDJ trial wavefunctions and MDJ is less effective on ADEs and AIPs. Furthermore, the relative energy between two structures of B₉ ^- is predicted reliably with FN-DMC using the SDJ trial wavefunction and the effect of MDJ is negligible, while density functional theory results using different exchange-correlation functionals differ significantly.

Mean-field density matrix decompositions

We introduce new and robust decompositions of mean-field Hartree-Fock and Kohn-Sham density functional theory relying on the use of localized molecular orbitals and physically sound charge population protocols. The new lossless property decompositions, which allow for partitioning one-electron reduced density matrices into either bond-wise or atomic contributions, are compared to alternatives from the literature with regard to both molecular energies and dipole moments. Besides commenting on possible applications as an interpretative tool in the rationalization of certain electronic phenomena, we demonstrate how decomposed mean-field theory makes it possible to expose and amplify compositional features in the context of machine-learned quantum chemistry. This is made possible by improving upon the granularity of the underlying data. On the basis of our preliminary proof-of-concept results, we conjecture that many of the structure-property inferences in existence today may be further refined by efficiently leveraging an increase in dataset complexity and richness.

Intrinsic electronic spectra of cryogenically prepared protoporphyrin IX ions in vacuo – deprotonation-induced Stark shifts

We present electronic spectra containing the Q_x and Q_y absorption bands of singly and doubly deprotonated protoporphyrin IX, prepared as mass selected ions in vacuo at cryogenic temperatures, revealing vibronic structure of both bands.

Quantum simulation of electronic structure with a transcorrelated Hamiltonian: improved accuracy with a smaller footprint on the quantum computer

Molecular quantum computing simulations are currently limited by the use of minimal Gaussian bases, a problem we overcome using a canonical transcorrelated Hamiltonian to accelerate basis convergence, with unitary coupled cluster as an example.