Dirac-Coulomb-Breit Molecular Mean-Field Exact-Two-Component Relativistic Equation-of-Motion Coupled-Cluster Theory.

We present a relativistic equation-of-motion coupled-cluster with single and double excitation formalism within the exact two-component framework (X2C-EOM-CCSD), where both scalar relativistic effects and spin-orbit coupling are variationally included at the reference level. Three different molecular mean-field treatments of relativistic corrections, including the one-electron, Dirac-Coulomb, and Dirac-Coulomb-Breit Hamiltonian, are considered in this work. Benchmark calculations include atomic excitations and fine-structure splittings arising from spin-orbit coupling. Comparison with experimental values and relativistic time-dependent density functional theory is also carried out. The computation of the oscillator strength using the relativistic X2C-EOM-CCSD approach allows for studies of spin-orbit-driven processes, such as the spontaneous phosphorescence lifetime.

Single Reference Treatment of Strongly Correlated H4 and H10 Isomers with Richardson-Gaudin States.

Richardson-Gaudin (RG) states are employed as a variational wave function ansatz for strongly correlated isomers of H4 and H10. In each case, a single RG state describes the seniority-zero sector quite well. Simple natural orbital functionals offer a cheap and reasonable approximation of the outstanding weak correlation in the seniority-zero sector, while systematic improvement is achieved by performing a configuration interaction in terms of RG states.

N-representability violations in truncated equation-of-motion coupled-cluster methods.

One-electron reduced density matrices (1RDMs) from equation-of-motion (EOM) coupled-cluster with single and double excitations (CCSD) calculations are analyzed to assess their N-representability (i.e., whether they are derivable from a physical N-electron state). We identify EOM-CCSD stationary states whose 1RDMs violate either ensemble-state N-representability conditions or pure-state conditions known as generalized Pauli constraints. As such, these 1RDMs do not correspond to any physical N-electron state. Unphysical states are also encountered in the course of time-dependent EOM-CC simulations; when an external field drives transitions between a pair of stationary states with pure-state N-representable 1RDMs, the 1RDM of the time-dependent state can violate ensemble-state conditions. These observations point to potential challenges in interpreting the results of time-dependent EOM-CCSD simulations.

Time-dependent equation-of-motion coupled-cluster simulations with a defective Hamiltonian.

Simulations of laser-induced electron dynamics in a molecular system are performed using time-dependent (TD) equation-of-motion (EOM) coupled-cluster (CC) theory. The target system has been chosen to highlight potential shortcomings of truncated TD-EOM-CC methods [represented in this work by TD-EOM-CC with single and double excitations (TD-EOM-CCSD)], where unphysical spectroscopic features can emerge. Specifically, we explore driven resonant electronic excitations in magnesium fluoride in the proximity of an avoided crossing. Near the avoided crossing, the CCSD similarity-transformed Hamiltonian is defective, meaning that it has complex eigenvalues, and oscillator strengths may take on negative values. When an external field is applied to drive transitions to states exhibiting these traits, unphysical dynamics are observed. For example, the stationary states that make up the time-dependent state acquire populations that can be negative, exceed one, or even complex-valued.

Data-Driven Refinement of Electronic Energies from Two-Electron Reduced-Density-Matrix Theory.

The exponential computational cost of describing strongly correlated electrons can be mitigated by adopting a reduced-density matrix (RDM)-based description of the electronic structure. While variational two-electron RDM (v2RDM) methods can enable large-scale calculations on such systems, the quality of the solution is limited by the fact that only a subset of known necessary N-representability constraints can be applied to the 2RDM in practical calculations. Here, we demonstrate that violations of partial three-particle (T1 and T2) N-representability conditions, which can be evaluated with knowledge of only the 2RDM, can serve as physics-based features in a machine-learning (ML) protocol for improving energies from v2RDM calculations that consider only two-particle (PQG) conditions. Proof-of-principle calculations demonstrate that the model yields substantially improved energies relative to reference values from configuration-interaction-based calculations.

Assessing the Effects of Orbital Relaxation and the Coherent-State Transformation in Quantum Electrodynamics Density Functional and Coupled-Cluster Theories.

Cavity quantum electrodynamics (QED) generalizations of time-dependent (TD) density functional theory (DFT) and equation-of-motion (EOM) coupled-cluster (CC) theory are used to model small molecules strongly coupled to optical cavity modes. We consider two types of calculations. In the first approach (termed "relaxed"), we use a coherent-state-transformed Hamiltonian within the ground- and excited-state portions of the calculations, and cavity-induced orbital relaxation effects are included at the mean-field level. This procedure guarantees that the energy is origin-invariant in post-self-consistent-field calculations. In the second approach (termed "unrelaxed"), we ignore the coherent-state transformation and the associated orbital relaxation effects. In this case, ground-state unrelaxed QED-CC calculations pick up a modest origin dependence but otherwise reproduce relaxed QED-CC results within the coherent-state basis. On the other hand, a severe origin dependence manifests in ground-state unrelaxed QED mean-field energies. For excitation energies computed at experimentally realizable coupling strengths, relaxed and unrelaxed QED-EOM-CC results are similar, while significant differences emerge for unrelaxed and relaxed QED-TDDFT. First, QED-EOM-CC and relaxed QED-TDDFT both predict that electronic states that are not resonant with the cavity mode are nonetheless perturbed by the cavity. Unrelaxed QED-TDDFT, on the other hand, fails to capture this effect. Second, in the limit of large coupling strengths, relaxed QED-TDDFT tends to overestimate Rabi splittings, while unrelaxed QED-TDDFT underestimates them, given splittings from relaxed QED-EOM-CC as a reference, and relaxed QED-TDDFT generally does the better job of reproducing the QED-EOM-CC results.

Enhanced Diastereocontrol via Strong Light-Matter Interactions in an Optical Cavity.

The enantiopurification of racemic mixtures of chiral molecules is important for a range of applications. Recent work has shown that chiral group-directed photoisomerization is a promising approach to enantioenrich racemic mixtures of BINOL, but increased control of the diasteriomeric excess (de) is necessary for its broad utility. Here we develop a cavity quantum electrodynamics (QED) generalization of time-dependent density functional theory and demonstrate computationally that strong light-matter coupling can alter the de of the chiral group-directed photoisomerization of BINOL. The relative orientation of the cavity mode polarization and the molecules in the cavity dictates the nature of the cavity interactions, which either enhance the de of the (R)-BINOL diasteriomer (from 17% to ≈40%) or invert the favorability to the (S)-BINOL derivative (to ≈34% de). The latter outcome is particularly remarkable because it indicates that the preference in diasteriomer can be influenced via orientational control, without changing the chirality of the directing group. We demonstrate that the observed effect stems from cavity-induced changes to the Kohn-Sham orbitals of the ground state.

Design and Synthesis of Kekulè and Non-Kekulè Diradicaloids via the Radical Periannulation Strategy: The Power of Seven Clar's Sextets.

This work introduces an approach to uncoupling electrons via maximum utilization of localized aromatic units, i.e., the Clar's π-sextets. To illustrate the utility of this concept to the design of Kekulé diradicaloids, we have synthesized a tridecacyclic polyaromatic system where a gain of five Clar's sextets in the open-shell form overcomes electron pairing and leads to the emergence of a high degree of diradical character. According to unrestricted symmetry-broken UCAM-B3LYP calculations, the singlet diradical character in this core system is characterized by the y0 value of 0.98 (y0 = 0 for a closed-shell molecule, y0 = 1 for pure diradical). The efficiency of the new design strategy was evaluated by comparing the Kekulé system with an isomeric non-Kekulé diradical of identical size, i.e., a system where the radical centers cannot couple via resonance. The calculated singlet-triplet gap, i.e., the ΔEST values, in both of these systems approaches zero: -0.3 kcal/mol for the Kekulé and +0.2 kcal/mol for the non-Kekulé diradicaloids. The target isomeric Kekulé and non-Kekulé systems were assembled using a sequence of radical periannulations, cross-coupling, and C-H activation. The diradicals are kinetically stabilized by six tert-butyl substituents and (triisopropylsilyl)acetylene groups. Both molecules are NMR-inactive but electron paramagnetic resonance (EPR)-active at room temperature. Cyclic voltammetry revealed quasi-reversible oxidation and reduction processes, consistent with the presence of two nearly degenerate partially occupied molecular orbitals. The experimentally measured ΔEST value of -0.14 kcal/mol confirms that K is, indeed, a nearly perfect singlet diradical.

Challenges for Variational Reduced-Density-Matrix Theory: Total Angular Momentum Constraints.

The variational two-electron reduced density matrix (v2RDM) method is generalized for the description of total angular momentum (J) and projection of total angular momentum (MJ) states in atomic systems described by nonrelativistic Hamiltonians, and it is shown that the approach exhibits serious deficiencies. Under ensemble N-representability constraints, v2RDM theory fails to retain the appropriate degeneracies among various J states for fixed spin (S) and orbital angular momentum (L), and for fixed L, S, and J, the manifold of MJ states is not necessarily degenerate. Moreover, a substantial energy error is observed for a system for which the two-electron reduced density matrix is exactly ensemble N-representable; in this case, the error stems from violations in pure-state N-representability conditions. Unfortunately, such violations do not appear to be good indicators of the reliability of energies from v2RDM theory in general. Several states are identified for which energy errors are near zero and yet pure-state conditions are clearly violated.

Equation-of-motion cavity quantum electrodynamics coupled-cluster theory for electron attachment.

The electron attachment variant of equation-of-motion coupled-cluster theory (EOM-EA-CC) is generalized to the case of strong light-matter coupling within the framework of cavity quantum electrodynamics (QED). The resulting EOM-EA-QED-CC formalism provides an ab initio, correlated, and non-perturbative description of cavity-induced effects in many-electron systems that complements other recently proposed cavity-QED-based extensions of CC theory. Importantly, this work demonstrates that QED generalizations of EOM-CC theory are useful frameworks for exploring particle-non-conserving sectors of Fock space, thereby establishing a path forward for the simultaneous description of both strong electron-electron and electron-photon correlation effects.

Challenges for variational reduced-density-matrix theory with three-particle N-representability conditions.

The direct variational optimization of the two-electron reduced density matrix (2RDM) can provide a reference-independent description of the electronic structure of many-electron systems that naturally capture strong or nondynamic correlation effects. Such variational 2RDM approaches can often provide a highly accurate description of strong electron correlation, provided that the 2RDMs satisfy at least partial three-particle N-representability conditions (e.g., the T2 condition). However, recent benchmark calculations on hydrogen clusters [N. H. Stair and F. A. Evangelista, J. Chem. Phys. 153, 104108 (2020)] suggest that even the T2 condition leads to unacceptably inaccurate results in the case of two- and three-dimensional clusters. We demonstrate that these failures persist under the application of full three-particle N-representability conditions (3POS). A variety of correlation metrics are explored in order to identify regimes under which 3POS calculations become unreliable, and we find that the relative squared magnitudes of the cumulant three- and two-particle reduced density matrices correlate reasonably well with the energy error in these systems. However, calculations on other molecular systems reveal that this metric is not a universal indicator for the reliability of the reduced-density-matrix theory with 3POS conditions.

Short Iterative Lanczos Integration in Time-Dependent Equation-of-Motion Coupled-Cluster Theory.

A time-dependent (TD) formulation of equation-of-motion coupled-cluster (EOM-CC) theory can provide excited-state information over an arbitrarily wide energy window with a reduced memory footprint relative to conventional, frequency-domain EOM-CC theory. However, the floating-point costs of the time-integration required by TD-EOM-CC are generally far larger than those of the frequency-domain form of the approach. This work considers the potential of the short iterative Lanczos (SIL) integration scheme [J. Chem. Phys. 1986, 85, 5870-5876] to reduce the floating-point costs of TD-EOM-CC simulations. Low-energy and K-edge absorption features for small molecules are evaluated using TD-EOM-CC with single and double excitations, with the time-integrations carried out via SIL and fourth-order Runge-Kutta (RK4) schemes. Spectra derived from SIL- and RK4-driven simulations are nearly indistinguishable, and with an appropriately chosen subspace dimension, the SIL requires far fewer floating-point operations than are required by RK4. For K-edge spectra, SIL is the more efficient scheme by an average factor of 7.2.

Cavity-modulated ionization potentials and electron affinities from quantum electrodynamics coupled-cluster theory.

Quantum electrodynamics coupled-cluster (QED-CC) theory is used to model vacuum-field-induced changes to ground-state properties of a series of sodium halide compounds (NaX, X = F, Cl, Br, and I) strongly coupled to an optical cavity. Ionization potentials (IPs) and electron affinities (EAs) are presented, and it is demonstrated that EAs are easily modulated by cavity interactions, while IPs for these compounds are far less sensitive to the presence of the cavity. EAs predicted by QED-CC can be reduced by as much as 0.22 eV (or ≈50%) when considering experimentally accessible coupling parameters.

Real-Time Time-Dependent Electronic Structure Theory.

Real-time electronic structure methods provide an unprecedented view of electron dynamics and ultrafast spectroscopy on the atto- and femtosecond time scale with vast potential to yield new insights into the electronic behavior of molecules and materials. In this Review, we discuss the fundamental theory underlying various real-time electronic structure methods as well as advantages and disadvantages of each. We give an overview of the numerical techniques that are widely used for real-time propagation of the quantum electron dynamics with an emphasis on Gaussian basis set methods. We also showcase many of the chemical applications and scientific advances made by using real-time electronic structure calculations and provide an outlook of possible new directions.

Size-extensive seniority-zero energy functionals derived from configuration interaction with double excitations.

The doubly occupied configuration interaction (DOCI) approach can provide an accurate black-box description of nondynamic electron correlation at a computational cost that increases combinatorially with the system size. Remarkably, a pair coupled-cluster doubles (pCCD) approach (also known as the antisymmetrized product of one-reference orbital geminals) can reproduce DOCI energies with only a quadratic number of wave function parameters, and, when neglecting the cost associated with any two-electron integral transformations, these parameters can be determined at a cubic computational cost. Other simpler seniority-zero approaches derived from size-extensive modified configuration interaction doubles functionals can also provide approximations to DOCI energies at similar computational costs. We develop seniority-zero formulations of the coupled-electron pair approximation, the averaged coupled-pair functional, averaged quadratic coupled-cluster, and the parametric two-electron reduced density matrix (p2RDM) approach. These methods are Hermitian and thus offer several potential advantages over pCCD theory, including a reduction in the number of variable parameters and simplified definitions of reduced density matrices. Of the methods investigated, only the pair p2RDM (pp2RDM) approach yields energies that are comparable in quality to pCCD and DOCI. For the molecular systems investigated, pp2RDM-derived RDMs are found to be better approximations to DOCI ones than those obtained from pCCD.

Reduced Density Matrix-Driven Complete Active Apace Self-Consistent Field Corrected for Dynamic Correlation from the Adiabatic Connection.

The recently proposed multireference adiabatic connection (AC) formalism [Pernal, Phys. Rev. Lett. 120, 013001 (2018)] is applied to recover dynamic electron correlation effects lacking in variational two-electron reduced density matrix (v2RDM)-driven complete active space self-consistent field theory (CASSCF). The AC approach is validated by computing potential energy curves for the dissociation of molecular nitrogen and the symmetric double dissociation of H2O while enforcing two sets of approximate N-representability conditions in the underlying v2RDM-driven CASSCF calculations (either two-particle or two-particle plus partial three-particle conditions). The AC yields smaller absolute errors than second-order N-electron perturbation theory (NEVPT2) at all molecular geometries for both sets of the N-representability conditions considered. The efficacy of the approach for thermochemistry is also assessed for a set of 31 small-molecule reactions. When imposing partial three-particle N-representability conditions, mean and maximum unsigned errors in reaction energies from the AC are superior to those from NEVPT2.

Psi4 1.4: Open-source software for high-throughput quantum chemistry.

PSI4 is a free and open-source ab initio electronic structure program providing implementations of Hartree-Fock, density functional theory, many-body perturbation theory, configuration interaction, density cumulant theory, symmetry-adapted perturbation theory, and coupled-cluster theory. Most of the methods are quite efficient, thanks to density fitting and multi-core parallelism. The program is a hybrid of C++ and Python, and calculations may be run with very simple text files or using the Python API, facilitating post-processing and complex workflows; method developers also have access to most of PSI4's core functionalities via Python. Job specification may be passed using The Molecular Sciences Software Institute (MolSSI) QCSCHEMA data format, facilitating interoperability. A rewrite of our top-level computation driver, and concomitant adoption of the MolSSI QCARCHIVE INFRASTRUCTURE project, makes the latest version of PSI4 well suited to distributed computation of large numbers of independent tasks. The project has fostered the development of independent software components that may be reused in other quantum chemistry programs.

Global Hybrid Multiconfiguration Pair-Density Functional Theory.

A global hybrid extension of multiconfiguration pair-density functional theory (MC-PDFT) is developed. Using a linear decomposition of the electron-electron repulsion term, a fraction λ of the nonlocal exchange interaction, obtained from variational two-electron reduced-density matrix (v2RDM)-driven complete active-space self-consistent field (CASSCF) theory, is combined with its local counterpart, obtained from an on-top pair-density functional. The resulting scheme (called λ-MC-PDFT) inherits the benefits of MC-PDFT (e.g., its simplicity and the resolution of the symmetry dilemma) and, when combined with the v2RDM approach to CASSCF, requires only polynomially scaling computational effort. As a result, λ-MC-PDFT can efficiently describe static and dynamical correlation effects in strongly correlated systems. The efficacy of the approach is assessed for several challenging multiconfigurational problems, including the dissociation of molecular nitrogen, the double dissociation of a water molecule, and the 1,3-dipolar cycloadditions of ozone to ethylene and ozone to acetylene in the O3ADD6 benchmark set.

Kinetic-energy-based error quantification in Kohn-Sham density functional theory.

We present a basis-independent metric to assess the quality of the electron density obtained from Kohn-Sham (KS) density functional theory (DFT). Given an exact reference density, Levy's constrained search (CS) formalism yields the exact non-interacting kinetic energy. The difference between this value and the kinetic energy obtained from a KSDFT procedure employing an approximate density functional serves as a measure of the density-driven error in the KS solution, which complements other error analyses based solely on the density. The CS also has the nice feature that it provides an estimate of the exact kinetic correlation energy as a byproduct of the procedure.