Minimum-Energy Conical Intersections by Compressed Multistate Pair-Density Functional Theory.

Compressed multistate pair-density functional theory (CMS-PDFT) is a multistate version of multiconfiguration pair-density functional theory that can capture the correct topology of coupled potential energy surfaces (PESs) around conical intersections. In this work, we develop interstate coupling vectors (ISCs) for CMS-PDFT in the OpenMolcas and PySCF/mrh electronic structure packages. Yet, the main focus of this work is using ISCs to calculate minimum-energy conical intersections (MECIs) by CMS-PDFT. This is performed using the projected constrained optimization method in OpenMolcas, which uses ISCs to restrain the iterations to the conical intersection seam. We optimize the S1/S0 MECIs for ethylene, butadiene, and benzene and show that CMS-PDFT gives smooth PESs in the vicinities of the MECIs. Furthermore, the CMS-PDFT MECIs are in good agreement with the MECI calculated by the more expensive XMS-CASPT2 method.

Distinguishing homolytic vs heterolytic bond dissociation of phenylsulfonium cations with localized active space methods.

Modeling chemical reactions with quantum chemical methods is challenging when the electronic structure varies significantly throughout the reaction and when electronic excited states are involved. Multireference methods, such as complete active space self-consistent field (CASSCF), can handle these multiconfigurational situations. However, even if the size of the needed active space is affordable, in many cases, the active space does not change consistently from reactant to product, causing discontinuities in the potential energy surface. The localized active space SCF (LASSCF) is a cheaper alternative to CASSCF for strongly correlated systems with weakly correlated fragments. The method is used for the first time to study a chemical reaction, namely the bond dissociation of a mono-, di-, and triphenylsulfonium cation. LASSCF calculations generate smooth potential energy scans more easily than the corresponding, more computationally expensive CASSCF calculations while predicting similar bond dissociation energies. Our calculations suggest a homolytic bond cleavage for di- and triphenylsulfonium and a heterolytic pathway for monophenylsulfonium.

Understanding Antiferromagnetic and Ligand Field Effects on Spin Crossover in a Triple-Decker Dimeric Cr(II) Complex.

Two possible explanations for the temperature dependence of spin-crossover (SCO) behavior in the dimeric triple-decker Cr(II) complex ([(η5-C5Me5)Cr(µ2:η5-P5)Cr(η5-C5Me5)]+) have been offered. One invokes variations in antiferromagnetic interactions between the two Cr(II) ions, whereas the other posits the development of a strong ligand-field effect favoring the low-spin ground state. We perform multireference electronic structure calculations based on the multiconfiguration pair-density functional theory to resolve these effects. We find quintet, triplet, and singlet electronic ground states, respectively, for the experimental geometries at high, intermediate, and low temperatures. The ground-state transition from quintet to triplet at an intermediate temperature derives from increased antiferromagnetic interactions between the two Cr(II) ions. By contrast, the ground-state transition from triplet to singlet at low temperature can be attributed to increased ligand-field effects, which dominate with continued variations in antiferromagnetic coupling. This study provides quantitative detail for the degree to which these two effects can act in concert for the observed SCO behavior in this complex and others subject to temperature-dependent variations in geometry.

Analytic gradients for multiconfiguration pair-density functional theory with density fitting: Development and application to geometry optimization in the ground and excited states.

Density fitting reduces the computational cost of both energy and gradient calculations by avoiding the computation and manipulation of four-index electron repulsion integrals. With this algorithm, one can efficiently optimize the geometries of large systems with an accurate multireference treatment. Here, we present the derivation of multiconfiguration pair-density functional theory for energies and analytic gradients with density fitting. Six systems are studied, and the results are compared to those obtained with no approximation to the electron repulsion integrals and to the results obtained by complete active space second-order perturbation theory. With the new approach, there is an increase in the speed of computation with a negligible loss in accuracy. Smaller grid sizes have also been used to reduce the computational cost of multiconfiguration pair-density functional theory with little effect on the optimized geometries and gradient values.

Analytic gradients for state-averaged multiconfiguration pair-density functional theory.

Analytic gradients are important for efficient calculations of stationary points on potential energy surfaces, for interpreting spectroscopic observations, and for efficient direct dynamics simulations. For excited electronic states, as are involved in UV-Vis spectroscopy and photochemistry, analytic gradients are readily available and often affordable for calculations using a state-averaged complete active space self-consistent-field (SA-CASSCF) wave function. However, in most cases, a post-SA-CASSCF step is necessary for quantitative accuracy, and such calculations are often too expensive if carried out by perturbation theory or configuration interaction. In this work, we present the analytic gradients for multiconfiguration pair-density functional theory based on SA-CASSCF wave functions, which is a more affordable alternative. A test set of molecules has been studied with this method, and the stationary geometries and energetics are compared to values in the literature as obtained by other methods. Excited-state geometries computed with state-averaged pair-density functional theory have similar accuracy to those from complete active space perturbation theory at the second-order.

Recent developments in the PySCF program package.

PySCF is a Python-based general-purpose electronic structure platform that supports first-principles simulations of molecules and solids as well as accelerates the development of new methodology and complex computational workflows. This paper explains the design and philosophy behind PySCF that enables it to meet these twin objectives. With several case studies, we show how users can easily implement their own methods using PySCF as a development environment. We then summarize the capabilities of PySCF for molecular and solid-state simulations. Finally, we describe the growing ecosystem of projects that use PySCF across the domains of quantum chemistry, materials science, machine learning, and quantum information science.

Merging symmetry projection methods with coupled cluster theory: Lessons from the Lipkin model Hamiltonian.

Coupled cluster and symmetry projected Hartree-Fock are two central paradigms in electronic structure theory. However, they are very different. Single reference coupled cluster is highly successful for treating weakly correlated systems but fails under strong correlation unless one sacrifices good quantum numbers and works with broken-symmetry wave functions, which is unphysical for finite systems. Symmetry projection is effective for the treatment of strong correlation at the mean-field level through multireference non-orthogonal configuration interaction wavefunctions, but unlike coupled cluster, it is neither size extensive nor ideal for treating dynamic correlation. We here examine different scenarios for merging these two dissimilar theories. We carry out this exercise over the integrable Lipkin model Hamiltonian, which despite its simplicity, encompasses non-trivial physics for degenerate systems and can be solved via diagonalization for a very large number of particles. We show how symmetry projection and coupled cluster doubles individually fail in different correlation limits, whereas models that merge these two theories are highly successful over the entire phase diagram. Despite the simplicity of the Lipkin Hamiltonian, the lessons learned in this work will be useful for building an ab initio symmetry projected coupled cluster theory that we expect to be accurate in the weakly and strongly correlated limits, as well as the recoupling regime.

A Computational Re-examination of the Criegee Intermediate-Sulfur Dioxide Reaction.

The atmospheric oxidation of sulfur dioxide by the parent and dimethyl Criegee intermediates (CIs) may be an important source of sulfuric acid aerosol, which has a large impact on radiative forcing and therefore upon climate. A number of computational studies have considered how the CH2OOS(O)O heteroozonide (HOZ) adduct formed in the CI + SO2 reaction converts SO2 to SO3. In this work we use the CBS-QB3 quantum chemical method along with equation-of-motion spin-flip CCSD(dT) and MCG3 theories to reveal new details regarding the formation and decomposition of the endo and exo conformers of the HOZ. Although ∼75% of the parent CI + SO2 reaction is initiated by formation of the exo HOZ, hyperconjugation preferentially stabilizes many of the endo intermediates and transition structures by 1-5 kcal mol(-1). Our quantum chemical calculations, in conjunction with statistical rate theory models, predict a rate coefficient for the parent CI + SO2 reaction of 3.68 × 10(-11) cm(3) molecule(-1) s(-1), in good agreement with recent experimental measurements. RRKM/master equation simulations based on our quantum chemical data predict a prompt carbonyl + SO3 yield of >95% for the reaction of both the parent and dimethyl CI with SO2. The existence of concerted cycloreversion transition structures 10-15 kcal mol(-1) higher in energy than the HOZ accounts for most of the predicted SO3 formation.

Finite-temperature coupled-cluster, many-body perturbation, and restricted and unrestricted Hartree-Fock study on one-dimensional solids: Luttinger liquids, Peierls transitions, and spin- and charge-density waves.

One-dimensional (1D) solids exhibit a number of striking electronic structures including charge-density wave (CDW) and spin-density wave (SDW). Also, the Peierls theorem states that at zero temperature, a 1D system predicted by simple band theory to be a metal will spontaneously dimerize and open a finite fundamental bandgap, while at higher temperatures, it will assume the equidistant geometry with zero bandgap (a Peierls transition). We computationally study these unique electronic structures and transition in polyyne and all-trans polyacetylene using finite-temperature generalizations of ab initio spin-unrestricted Hartree-Fock (UHF) and spin-restricted coupled-cluster doubles (CCD) theories, extending upon previous work [He et al., J. Chem. Phys. 140, 024702 (2014)] that is based on spin-restricted Hartree-Fock (RHF) and second-order many-body perturbation (MP2) theories. Unlike RHF, UHF can predict SDW as well as CDW and metallic states, and unlike MP2, CCD does not diverge even if the underlying RHF reference wave function is metallic. UHF predicts a gapped SDW state with no dimerization at low temperatures, which gradually becomes metallic as the temperature is raised. CCD, meanwhile, confirms that electron correlation lowers the Peierls transition temperature. Furthermore, we show that the results from all theories for both polymers are subject to a unified interpretation in terms of the UHF solutions to the Hubbard-Peierls model using different values of the electron-electron interaction strength, U/t, in its Hamiltonian. The CCD wave function is shown to encompass the form of the exact solution of the Tomonaga-Luttinger model and is thus expected to describe accurately the electronic structure of Luttinger liquids.

Second-order many-body perturbation theory: an eternal frontier.

Second-order many-body perturbation theory [MBPT(2)] is the lowest-ranked member of a systematic series of approximations convergent at the exact solutions of the Schrödinger equations. It has served and continues to serve as the testing ground for new approximations, algorithms, and even theories. This article introduces this basic theory from a variety of viewpoints including the Rayleigh-Schrödinger perturbation theory, the many-body Green's function theory based on the Dyson equation, and the related Feynman-Goldstone diagrams. It also explains the important properties of MBPT(2) such as size consistency, its ability to describe dispersion interactions, and divergence in metals. On this basis, this article surveys three major advances made recently by the authors to this theory. They are a finite-temperature extension of MBPT(2) and the resolution of the Kohn-Luttinger conundrum, a stochastic evaluation of the correlation and self-energies of MBPT(2) using the Monte Carlo integration of their Laplace-transformed expressions, and an extension to anharmonic vibrational zero-point energies and transition frequencies based on the Dyson equation.

Normal-ordered second-quantized Hamiltonian for molecular vibrations.

A normal-ordered second-quantized form of the Hamiltonian is derived for quantum dynamics in a bound potential energy surface expressed as a Taylor series in an arbitrary set of orthogonal, delocalized coordinates centered at an arbitrary geometry. The constant, first-, and second-order excitation amplitudes of this Hamiltonian are identified as the ground-state energy, gradients, and frequencies, respectively, of the size-extensive vibrational self-consistent field (XVSCF) method or the self-consistent phonon method. They display the well-defined size dependence of V(1-n/2), where V is the volume and n is the number of coordinates associated with the amplitudes. It is used to rapidly derive the equations of XVSCF and vibrational many-body perturbation methods with the Møller-Plesset partitioning of the Hamiltonian.

Stochastic many-body perturbation theory for anharmonic molecular vibrations.

A new quantum Monte Carlo (QMC) method for anharmonic vibrational zero-point energies and transition frequencies is developed, which combines the diagrammatic vibrational many-body perturbation theory based on the Dyson equation with Monte Carlo integration. The infinite sums of the diagrammatic and thus size-consistent first- and second-order anharmonic corrections to the energy and self-energy are expressed as sums of a few m- or 2m-dimensional integrals of wave functions and a potential energy surface (PES) (m is the vibrational degrees of freedom). Each of these integrals is computed as the integrand (including the value of the PES) divided by the value of a judiciously chosen weight function evaluated on demand at geometries distributed randomly but according to the weight function via the Metropolis algorithm. In this way, the method completely avoids cumbersome evaluation and storage of high-order force constants necessary in the original formulation of the vibrational perturbation theory; it furthermore allows even higher-order force constants essentially up to an infinite order to be taken into account in a scalable, memory-efficient algorithm. The diagrammatic contributions to the frequency-dependent self-energies that are stochastically evaluated at discrete frequencies can be reliably interpolated, allowing the self-consistent solutions to the Dyson equation to be obtained. This method, therefore, can compute directly and stochastically the transition frequencies of fundamentals and overtones as well as their relative intensities as pole strengths, without fixed-node errors that plague some QMC. It is shown that, for an identical PES, the new method reproduces the correct deterministic values of the energies and frequencies within a few cm(-1) and pole strengths within a few thousandths. With the values of a PES evaluated on the fly at random geometries, the new method captures a noticeably greater proportion of anharmonic effects.

Stochastic algorithm for size-extensive vibrational self-consistent field methods on fully anharmonic potential energy surfaces.

A stochastic algorithm based on Metropolis Monte Carlo (MC) is presented for the size-extensive vibrational self-consistent field methods (XVSCF(n) and XVSCF[n]) for anharmonic molecular vibrations. The new MC-XVSCF methods substitute stochastic evaluations of a small number of high-dimensional integrals of functions of the potential energy surface (PES), which is sampled on demand, for diagrammatic equations involving high-order anharmonic force constants. This algorithm obviates the need to evaluate and store any high-dimensional partial derivatives of the potential and can be applied to the fully anharmonic PES without any Taylor-series approximation in an intrinsically parallelizable algorithm. The MC-XVSCF methods reproduce deterministic XVSCF calculations on the same Taylor-series PES in all energies, frequencies, and geometries. Calculations using the fully anharmonic PES evaluated on the fly with electronic structure methods report anharmonic effects on frequencies and geometries of much greater magnitude than deterministic XVSCF calculations, reflecting an underestimation of anharmonic effects in a Taylor-series approximation to the PES.

Core Binding Energy Calculations: A Scalable Approach with the Quantum Embedding-Based Equation-of-Motion Coupled-Cluster Method.

We investigated the use of density matrix embedding theory to facilitate the computation of core ionization energies (IPs) of large molecules at the equation-of-motion coupled-cluster singles doubles with perturbative triples (EOM-CCSD*) level in combination with the core-valence separation (CVS) approximation. The unembedded IP-CVS-EOM-CCSD* method with a triple-ζ basis set produced ionization energies within 1 eV of experiment with a standard deviation of ∼0.2 eV for the core65 data set. The embedded variant contributed very little systematic error relative to the unembedded method, with a mean unsigned error of 0.07 eV and a standard deviation of ∼0.1 eV, in exchange for accelerating the calculations by many orders of magnitude. By employing embedded EOM-CC methods, we computed the core ionization energies of the uracil hexamer, doped fullerene, and chlorophyll molecule, utilizing up to ∼4000 basis functions within 1 eV from experimental values. Such calculations are not currently possible with the unembedded EOM-CC method.

Automatic State Interaction with Large Localized Active Spaces for Multimetallic Systems.

The localized active space self-consistent field method factorizes a complete active space wave function into an antisymmetrized product of localized active space wave function fragments. Correlation between fragments is then reintroduced through localized active space state interaction (LASSI), in which the Hamiltonian is diagonalized in a model space of LAS states. However, the optimal procedure for defining the LAS fragments and LASSI model space is unknown. We here present an automated framework to explore systematically convergent sets of model spaces, which we call LASSI[r, q]. This method requires the user to select only r, the number of electron hops from one fragment to another, and q, the number of fragment basis functions per Hilbert space, which converges to CASCI in the limit of r, q → ∞. Numerical tests of this method on the trimetal oxo-centered complexes [Fe(III)Al(III)Fe(II)(µ3-O)(HCOO)6] and [Fe(III)2Fe(II)(µ3-O)(HCOO)6] show efficient convergence to the CASCI limit with 4-10 orders of magnitude fewer states than CASCI.

Analytic Nuclear Gradients for Complete Active Space Linearized Pair-Density Functional Theory.

Accurately modeling photochemical reactions is difficult due to the presence of conical intersections and locally avoided crossings, as well as the inherently multiconfigurational character of excited states. As such, one needs a multistate method that incorporates state interaction in order to accurately model the potential energy surface at all nuclear coordinates. The recently developed linearized pair-density functional theory (L-PDFT) is a multistate extension of multiconfiguration PDFT, and it has been shown to be a cost-effective post-MCSCF method (as compared to more traditional and expensive multireference many-body perturbation methods or multireference configuration interaction methods) that can accurately model potential energy surfaces in regions of strong nuclear-electronic coupling in addition to accurately predicting Franck-Condon vertical excitations. In this paper, we report the derivation of analytic gradients for L-PDFT and their implementation in the PySCF-forge software, and we illustrate the utility of these gradients for predicting ground- and excited-state equilibrium geometries and adiabatic excitation energies for formaldehyde, s-trans-butadiene, phenol, and cytosine.

State Preparation in Quantum Algorithms for Fragment-Based Quantum Chemistry.

State preparation for quantum algorithms is crucial for achieving high accuracy in quantum chemistry and competing with classical algorithms. The localized active space-unitary coupled cluster (LAS-UCC) algorithm iteratively loads a fragment-based multireference wave function onto a quantum computer. In this study, we compare two state preparation methods, quantum phase estimation (QPE) and direct initialization (DI), for each fragment. We test the two state preparation methods on three systems, ranging from a model system, a set of interacting hydrogen molecules, to more realistic chemical problems, like the C-C double bond breaking in transbutadiene and the spin ladder in a bimetallic system. We analyze the impact of QPE parameters, such as the number of ancilla qubits and Trotter steps, on the prepared state. We find a trade-off between the methods, where DI requires fewer resources for smaller fragments, while QPE is more efficient for larger fragments. Our resource estimates highlight the benefits of system fragmentation in state preparation for subsequent quantum chemical calculations. These findings have broad applications for preparing multireference quantum chemical wave functions on quantum circuits that can be used for realistic chemical applications.

First-order Dyson coordinates and geometry.

The mathematical constructs of the Dyson coordinates and geometry are introduced. The former are a unitary transformation of the normal coordinates and the anharmonic vibrational counterpart of the Dyson orbitals in electronic structure theory. The first-order Dyson coordinates bring the sums of the harmonic force constants and their first-order diagrammatic perturbation corrections (the first-order Dyson self-energy) to a diagonal form. The first-order Dyson geometry has no counterpart in electronic structure theory. It is the point on the potential energy surface at which the sums of the energy gradients and their first-order diagrammatic perturbation corrections vanish. It agrees with the vibrationally averaged geometry of vibrational self-consistent field (VSCF) theory in the bulk limit. These constructs provide a unified view of the relationship of VSCF and its diagrammatically size-consistent modifications as well as the self-consistent phonon method widely used in solid-state physics.

Second-order many-body perturbation expansions of vibrational Dyson self-energies.

Second-order many-body perturbation theories for anharmonic vibrational frequencies and zero-point energies of molecules are formulated, implemented, and tested. They solve the vibrational Dyson equation self-consistently by taking into account the frequency dependence of the Dyson self-energy in the diagonal approximation, which is expanded in a diagrammatic perturbation series up to second order. Three reference wave functions, all of which are diagrammatically size consistent, are considered: the harmonic approximation and diagrammatic vibrational self-consistent field (XVSCF) methods with and without the first-order Dyson geometry correction, i.e., XVSCF[n] and XVSCF(n), where n refers to the truncation rank of the Taylor-series potential energy surface. The corresponding second-order perturbation theories, XVH2(n), XVMP2[n], and XVMP2(n), are shown to be rigorously diagrammatically size consistent for both total energies and transition frequencies, yield accurate results (typically within a few cm(-1) at n = 4 for water and formaldehyde) for both quantities even in the presence of Fermi resonance, and have access to fundamentals, overtones, and combinations as well as their relative intensities as residues of the vibrational Green's functions. They are implemented into simple algorithms that require only force constants and frequencies of the reference methods (with no basis sets, quadrature, or matrix diagonalization at any stage of the calculation). The rules for enumerating and algebraically interpreting energy and self-energy diagrams are elucidated in detail.

Density Matrix Embedding Using Multiconfiguration Pair-Density Functional Theory.

We present a quantum embedding method for ground and excited states of extended systems that uses multiconfiguration pair-density functional theory (MC-PDFT) with densities provided by periodic density matrix embedding theory (pDMET). We compute local excitations in oxygen mono- and divacancies on a magnesium oxide (100) surface and find absolute deviations within 0.05 eV between pDMET using the MC-PDFT, denoted as pDME-PDFT, and the more expensive, nonembedded MC-PDFT approach. We further use pDME-PDFT to calculate local excitations in larger supercells for the monovacancy defect, for which the use of nonembedded MC-PDFT is prohibitively costly.