Coupled Cluster and Perturbation Theories Based on a Cluster Mean-Field Reference Applied to Strongly Correlated Spin Systems.

We introduce perturbation and coupled-cluster theories based on a cluster mean-field reference for describing the ground state of strongly correlated spin systems. In cluster mean-field, the ground state wave function is written as a simple tensor product of optimized cluster states. The cluster language and the mean-field nature of the ansatz allow for a straightforward improvement which uses perturbation theory and coupled-cluster to account for intercluster correlations. We present benchmark calculations on the 1D chain and 2D square J1-J2 Heisenberg model, using cluster mean-field, perturbation theory, and coupled-cluster. We also present an extrapolation scheme that allows us to compute thermodynamic limit energies accurately. Our results indicate that, with sufficiently large clusters, the correlated methods (cPT2, cPT4, and cCCSD) can provide a relatively accurate description of the Heisenberg model in the regimes considered, which suggests that the methods presented can be used for other strongly correlated systems. Some ways to improve upon the methods presented in this work are discussed.

Prediction of Fluorophore Brightness in Designed Mini Fluorescence Activating Proteins.

The de novo computational design of proteins with predefined three-dimensional structure is becoming much more routine due to advancements both in force fields and algorithms. However, creating designs with functions beyond folding is more challenging. In that regard, the recent design of small beta barrel proteins that activate the fluorescence of an exogenous small molecule chromophore (DFHBI) is noteworthy. These proteins, termed mini fluorescence activating proteins (mFAPs), have been shown to increase the brightness of the chromophore more than 100-fold upon binding to the designed ligand pocket. The design process created a large library of variants with different brightness levels but gave no rational explanation for why one variant was brighter than another. Here, we use quantum mechanics and molecular dynamics simulations to investigate how molecular flexibility in the ground and excited states influences brightness. We show that the ability of the protein to resist dihedral angle rotation of the chromophore is critical for predicting brightness. Our simulations suggest that the mFAP/DFHBI complex has a rough energy landscape, requiring extensive ground-state sampling to achieve converged predictions of excited-state kinetics. While computationally demanding, this roughness suggests that mFAP protein function can be enhanced by reshaping the energy landscape toward conformations that better resist DFHBI bond rotation.

Approaching the full configuration interaction ground state from an arbitrary wavefunction with gradient descent and quasi-Newton algorithms.

We consider gradient descent and quasi-Newton algorithms to optimize the full configuration interaction (FCI) ground state wavefunction starting from an arbitrary reference state |0⟩. We show that the energies obtained along the optimization path can be evaluated in terms of expectation values of |0⟩, thus avoiding explicit storage of intermediate wavefunctions. This allows us to find the energies after the first few steps of the FCI algorithm for systems much larger than what standard deterministic FCI codes can handle at present. We show an application of the algorithm with reference wavefunctions constructed as linear combinations of non-orthogonal determinants.

Hyperfine-Coupling Tensors from Projected Hartree-Fock Theory.

We assess the calculation of hyperfine coupling (HFC) tensors by different variants of Projected Hartree-Fock (PHF) theory. For a set of small main-group S = 1/2 radicals (BO, CO+, CN, AlO, vinyl, methyl, ethynyl), spin-symmetry as well as complex-conjugation and point-group symmetry are first broken in a reference determinant, and then variationally restored, in the frame of the modern formulation of PHF theory. Historically, PHF theory was basically restricted to the restoration of spin symmetry from an unrestricted HF determinant (conserving Sz symmetry). This afforded unsatisfactory HFCs. We obtain far better results for isotropic (and anisotropic) HFCs when the variational energy is further lowered by working with generalized determinants that completely break spin symmetry, and when additional symmetries are used. Specifically, complex-conjugation projection recovers a substantial fraction of the dynamical correlation energy in small molecules, and the detailed equations for combined complex-conjugation, spin- and point-group projection in the density-matrix/diagonalization formulation of PHF theory are here reported for the first time. The compact representation of the PHF wave function allows for a straightforward evaluation of the spin-density matrix and of HFC tensors with little effort. The promising performance of PHF theory may motivate the application of post-PHF methods to the calculation of HFC tensors.

Minimal Matrix Product States and Generalizations of Mean-Field and Geminal Wave Functions.

Simple wave functions of low computational cost but which can achieve qualitative accuracy across the whole potential energy surface (PES) are of relevance to many areas of electronic structure theory as well as to applications to dynamics. Here, we explore a class of simple wave functions, the minimal matrix product state (MMPS), that generalizes many simple wave functions in common use, such as projected mean-field wave functions, geminal wave functions, and generalized valence bond states. By examining the performance of MMPSs for PESs of some prototypical systems, we find that they yield good qualitative behavior across the whole PES, often significantly improving on the aforementioned ansätze.

Electronic Spectroscopy of 2-Phenyl-1,3,2-benzodioxaborole and Its Derivatives: Important Building Blocks of Covalent Organic Frameworks.

Aryl boronate esters, such as 2-phenyl-1,3,2-benzodioxaborole (1), are important components in the formation of a variety of covalent organic frameworks. The addition of substituents on the aromatic rings of aryl boronate esters has the potential to modify the structure, reactivity, and electronic properties of the resulting materials, and so, it is useful to understand at a more fundamental level the properties of these important compounds. Experimental measurements and computational investigations are presented herein that provide insight regarding the structural and electronic properties of parent aryl boronate ester 1 as well as three substituted derivatives: 2-(o-tolyl)-1,3,2-benzodioxaborole (2), 2-(2,6-dimethylphenyl)-1,3,2-benzodioxaborole (3), and 2-(4-(tert-butyl)phenyl)-1,3,2-benzodioxaborole (4). Electronic spectroscopy combined with excited-state calculations reveal two closely spaced electronic states, S1 and S2, which appear to have excitation primarily localized on the aromatic system of the phenyl substituent or the catecholborane moiety, respectively. Interestingly, the ortho-dimethyl derivative (3) shows a significantly red-shifted electronic origin with an extensive vibronic progression of a low-frequency torsional motion about the C-B bond. Franck-Condon calculations on the ab initio determined ground- and excited-state potentials very accurately reproduce this spectrum, confirming the nonplanar ground state of this compound.

Low-Cost Molecular Excited States from a State-Averaged Resonating Hartree-Fock Approach.

Quantum chemistry methods that describe excited states on the same footing as the ground state are generally scarce. In previous work, ( Gill J. Phys. Chem. A , 112 , 2008 , 13164 and later Sundstrom ; ; Head-Gordon J. Chem. Phys. , 140 , 2014 , 114103 ) considered excited states resulting from a nonorthogonal configuration interaction (NOCI) on stationary solutions of the Hartree-Fock equations. We build upon those contributions and present the state-averaged resonating Hartree-Fock (sa-ResHF) method, which differs from NOCI in that spin-projection and orbital relaxation effects are incorporated from the onset. Our results in a set of small molecules (alanine, formaldehyde, acetaldehyde, acetone, formamide, and ethylene) suggest that sa-ResHF excitation energies are a notable improvement over configuration interaction singles, at a mean-field computational cost. The orbital relaxation in sa-ResHF, carried out in the presence of a spin-projection operator, generally results in excitation energies that are closer to the EOM-CCSD and experimental values than the corresponding NOCI ones.

Efficient Implementation of Variation after Projection Generalized Hartree-Fock.

Projected Hartree-Fock (PHF) theory can restore important symmetries to broken symmetry wave functions. Variation after projection (VAP) implementations make it possible to deliberately break and then restore a given symmetry by directly minimizing the projected energy expression. This technique can be applied to any symmetry that can be broken from relaxing constraints on single Slater determinant wave functions. For instance, generalized Hartree-Fock (GHF) wave functions are eigenfunctions of neither Sz nor S2. By relaxing these constraints, the wave function can explore a larger variational space and can reach lower energies than more constrained HF solutions. We have implemented spin-projected GHF (SGHF), which retains many of the advantages of breaking symmetry while also being a spin eigenfunction, with some notable improvements over previous implementations. Our new algorithm involves the formation of new intermediate matrices not previously discussed in the literature. Discretization of the necessary integration over the rotation group SO(3) is also accomplished much more efficiently using Lebedev grids. A novel scheme to incrementally build rotated Fock matrices is also introduced and compared with more standard approaches.

Magnetic Structure of Density Matrices.

The spin structure of wave functions is reflected in the magnetic structure of the one-particle density matrix. Indeed, for single determinants we can use either one to determine the other. In this work we discuss how one can simply examine the one-particle density matrix to faithfully determine whether the spin magnetization density vector field is collinear, coplanar, or noncoplanar. For single determinants, this test suffices to distinguish collinear determinants which are eigenfunctions of Sn̂ from noncollinear determinants which are not. We also point out the close relationship between noncoplanar magnetism on the one hand and complex conjugation symmetry breaking on the other. Finally, we use these ideas to classify the various ways single determinant wave functions break and respect symmetries of the Hamiltonian in terms of their one-particle density matrix.

Time-Step Targeting Time-Dependent and Dynamical Density Matrix Renormalization Group Algorithms with ab Initio Hamiltonians.

We study the dynamical density matrix renormalization group (DDMRG) and time-dependent density matrix renormalization group (td-DMRG) algorithms in the ab initio context to compute dynamical correlation functions of correlated systems. We analyze the strengths and weaknesses of the two methods in small model problems and propose two simple improved formulations, DDMRG++ and td-DMRG++, that give increased accuracy at the same bond dimension at a nominal increase in cost. We apply DDMRG++ to obtain the oxygen core-excitation energy in the water molecule in a quadruple-zeta quality basis, which allows us to estimate the remaining correlation error in existing coupled cluster results. Further, we use DDMRG++ to compute the local density of states and gaps and td-DMRG++ to compute the complex polarization function, in linear hydrogen chains with up to 50 H atoms, to study metallicity and delocalization as a function of bond length.

A Practical Guide to Density Matrix Embedding Theory in Quantum Chemistry.

Density matrix embedding theory (DMET) (Knizia, G.; Chan, G. K.-L. Phys. Rev. Lett. 2012, 109, 186404) provides a theoretical framework to treat finite fragments in the presence of a surrounding molecular or bulk environment, even when there is significant correlation or entanglement between the two. In this work, we give a practically oriented and explicit description of the numerical and theoretical formulation of DMET. We also describe in detail how to perform self-consistent DMET optimizations. We explore different embedding strategies with and without a self-consistency condition in hydrogen rings, beryllium rings, and a sample SN2 reaction. The source code for the calculations in this work can be obtained from https://github.com/sebwouters/qc-dmet .

Polyradical character and spin frustration in fullerene molecules: an ab initio non-collinear Hartree-Fock study.

Most ab initio calculations on fullerene molecules have been carried out on the basis of the paradigm of the Hückel model. This is consistent with the restricted nature of the independent-particle model underlying such calculations, even in single-reference-based correlated approaches. Notwithstanding, previous works on some of these molecules using model Hamiltonians have clearly indicated the importance of short-range interatomic spin-spin correlations. In this work, we consider ab initio non-collinear Hartree-Fock (HF) solutions for representative fullerene systems: the bowl, cage, ring, and pentagon isomers of C20, and the larger C30, C36, C60, C70, and C84 fullerene cages. In all cases but the ring we find that the HF minimum corresponds to a truly non-collinear solution with a torsional spin density wave. Optimized geometries at the generalized HF (GHF) level lead to fully symmetric structures, even in those cases where Jahn-Teller distortions have been previously considered. The nature of the GHF solutions is consistent with the π-electron space becoming polyradical in nature: each p-orbital remains effectively singly occupied. The spin frustration, induced by the presence of pentagon rings on an otherwise antiferromagnetic background, is minimized at the HF level by aligning the spins with non-collinear arrangements. The long-range magnetic ordering observed is reminiscent of the character of broken symmetry HF solutions in polyacene systems.

Electronic correlation without double counting via a combination of spin projected Hartree-Fock and density functional theories.

Several schemes to avoid the double counting of correlations in methods that merge multireference wavefunctions with density functional theory (DFT) are studied and here adapted to a combination of spin-projected Hartree-Fock (SUHF) and DFT. The advantages and limitations of the new method, denoted SUHF+fcDFT, are explored through calculations on benchmark sets in which the accounting of correlations is challenging for pure SUHF or DFT. It is shown that SUHF+fcDFT can greatly improve the description of certain molecular properties (e.g., singlet-triplet energy gaps) which are not improved by simple addition of DFT dynamical correlation to SUHF. However, SUHF+fcDFT is also shown to have difficulties dissociating certain types of bonds and describing highly charged ions with static correlation. Possible improvements to the current SUHF+fcDFT scheme are discussed in light of these results.

Analytic energy gradient for the projected Hartree-Fock method.

We derive and implement the analytic energy gradient for the symmetry Projected Hartree-Fock (PHF) method avoiding the solution of coupled-perturbed HF-like equations, as in the regular unprojected method. Our formalism therefore has mean-field computational scaling and cost, despite the elaborate multi-reference character of the PHF wave function. As benchmark examples, we here apply our gradient implementation to the ortho-, meta-, and para-benzyne biradicals, and discuss their equilibrium geometries and vibrational frequencies.

Stability of hemi-bonded vs proton-transferred structures of (H2O)2(+), (H2S)2(+), and (H2Se)2(+) studied with projected Hartree-Fock methods.

Hartree-Fock (HF) is known to suffer from drawbacks in the description of the relative stabilities between the hemi-bonded (HB) and proton-transferred (PT) isomers of the water dimer cation, (H2O)2(+). The energy difference predicted by HF is too large, approximately 27 kcal/mol, which is lowered to 7 kcal/mol when correlation effects are added. The error in HF has been previously attributed to the large dynamic correlation effects in the HB structure as well to the large symmetry breaking this structure exhibits. In this study we use the recently developed projected Hartree-Fock (PHF) methods to study the relative stability of the two isomers of (H2O)2(+) as well as its second and third row analogs, namely, (H2S)2(+) and (H2Se)2(+). In PHF, symmetries are broken and restored in a variation-after-projection approach and thus can deal easily with systems for which HF itself spontaneously breaks symmetry. We use different flavors of PHF (SUHF, KSUHF, SGHF, and KSGHF) to explore their ability in capturing dynamic correlation effects and to compare their performance to different wave function based methods. We study the role of the symmetry-breaking in the above systems, using wave function based methods with unrestricted and restricted wave functions as well as performing a single-shot symmetry restoration (a projection-after-variation scheme).

Multi-component symmetry-projected approach for molecular ground state correlations.

The symmetry-projected Hartree-Fock ansatz for the electronic structure problem can efficiently account for static correlation in molecules, yet it is often unable to describe dynamic correlation in a balanced manner. Here, we consider a multi-component, systematically improvable approach, that accounts for all ground state correlations. Our approach is based on linear combinations of symmetry-projected configurations built out of a set of non-orthogonal, variationally optimized determinants. The resulting wavefunction preserves the symmetries of the original Hamiltonian even though it is written as a superposition of deformed (broken-symmetry) determinants. We show how short expansions of this kind can provide a very accurate description of the electronic structure of simple chemical systems such as the nitrogen and the water molecules, along the entire dissociation profile. In addition, we apply this multi-component symmetry-projected approach to provide an accurate interconversion profile among the peroxo and bis(µ-oxo) forms of [Cu2O2](2+), comparable to other state-of-the-art quantum chemical methods.

Excited electronic states from a variational approach based on symmetry-projected Hartree-Fock configurations.

Recent work from our research group has demonstrated that symmetry-projected Hartree-Fock (HF) methods provide a compact representation of molecular ground state wavefunctions based on a superposition of non-orthogonal Slater determinants. The symmetry-projected ansatz can account for static correlations in a computationally efficient way. Here we present a variational extension of this methodology applicable to excited states of the same symmetry as the ground state. Benchmark calculations on the C2 dimer with a modest basis set, which allows comparison with full configuration interaction results, indicate that this extension provides a high quality description of the low-lying spectrum for the entire dissociation profile. We apply the same methodology to obtain the full low-lying vertical excitation spectrum of formaldehyde, in good agreement with available theoretical and experimental data, as well as to a challenging model C2v insertion pathway for BeH2. The variational excited state methodology developed in this work has two remarkable traits: it is fully black-box and will be applicable to fairly large systems thanks to its mean-field computational cost.

Proper and improper zero energy modes in Hartree-Fock theory and their relevance for symmetry breaking and restoration.

We study the spectra of the molecular orbital Hessian (stability matrix) and random-phase approximation (RPA) Hamiltonian of broken-symmetry Hartree-Fock solutions, focusing on zero eigenvalue modes. After all negative eigenvalues are removed from the Hessian by following their eigenvectors downhill, one is left with only positive and zero eigenvalues. Zero modes correspond to orbital rotations with no restoring force. These rotations determine states in the Goldstone manifold, which originates from a spontaneously broken continuous symmetry in the wave function. Zero modes can be classified as improper or proper according to their different mathematical and physical properties. Improper modes arise from symmetry breaking and their restoration always lowers the energy. Proper modes, on the other hand, correspond to degeneracies of the wave function, and their symmetry restoration does not necessarily lower the energy. We discuss how the RPA Hamiltonian distinguishes between proper and improper modes by doubling the number of zero eigenvalues associated with the latter. Proper modes in the Hessian always appear in pairs which do not double in RPA. We present several pedagogical cases exemplifying the above statements. The relevance of these results for projected Hartree-Fock methods is also addressed.

Predicting singlet-triplet energy splittings with projected Hartree-Fock methods.

Hartree-Fock (HF) and density functional theory (DFT) methods are known for having problems in predicting singlet-triplet energy splittings when the system displays significant diradical character. Multireference methods are traditionally advocated to deal with the spin-contamination problem inherent in broken-symmetry mean-field methods. In the present work, spin-contamination is rigorously eliminated by means of a symmetry projection approach, carried out in a variation-after-projection fashion, recently implemented in our research group. We here explore the performance of a variety of projected Hartree-Fock (PHF) approaches (SUHF, KSUHF, SGHF, and KSGHF) in predicting singlet-triplet energy gaps in a broad set of diradical systems: small diatomic molecules, carbenes and silenes, and a few larger molecules (trimethylenemethane and benzyne isomers). For most of these systems, accurate experimental data is available in the literature. Additionally, we assess the quality of the geometrical parameters obtained in SUHF-based optimizations for some of the systems considered. Our results indicate that PHF methods yield high-quality multireference wave functions, providing a good description of the ground state potential surface as well as an accurate singlet-triplet splitting gap, all within a modest mean-field computational cost.

Entanglement and polyradical character of polycyclic aromatic hydrocarbons predicted by projected Hartree-Fock theory.

We study strong correlation effects in a series of fused benzene rings (acenes) of varying length and width using our recently developed projected Hartree-Fock (PHF) method. These molecules, commonly known as polycyclic aromatic hydrocarbons or nanographenes, are very challenging for electronic structure theory because of their strong multireference character. This challenge is here met by PHF at moderate computational cost optimizing a spin eigenfunction obtained by projection of an unrestricted Hartree-Fock (UHF) trial determinant. The resulting method, known as SUHF, predicts that polyradical behavior and orbital entanglement are enhanced with molecular size, especially in systems whose structural motifs are dominated by zigzag edges, like oligoacenes.