Singling Out Dynamic and Nondynamic Correlation

All-Purpose Measure of Electron Correlation for Multireference Diagnostics

We present an analytical relationship between two natural orbital occupancy-based indices, I N D ¯ and INDmax, and two established electron correlation metrics: the leading term of a configuration interaction expansion, c0, and the D2 diagnostic. Numerical validation revealed that I N D ¯ and INDmax can effectively substitute for c0 and D2, respectively. These indices offer three distinct advantages: (i) they are universally applicable across all electronic structure methods, (ii) their interpretation is more intuitive, and (iii) they can be readily incorporated into the development of hybrid electronic structure methods. Additionally, we draw a distinction between correlation measures and correlation diagnostics, establishing MP2 and CCSD numerical thresholds for INDmax, which are to be used as a multireference diagnostic. Our findings further demonstrate that establishing thresholds for other electronic structure methods can be easily accomplished using small data sets.

Are Accelerated and Enhanced Wave Function Methods Accurate to Compute Static Linear and Nonlinear Optical Properties?

Key components of organic-based electro-optic devices are challenging to design or optimize because they exhibit nonlinear optical responses, which are difficult to model or rationalize. Computational chemistry furnishes the tools to investigate extensive collections of molecules in the quest for target compounds. Among the electronic structure methods that provide static nonlinear optical properties (SNLOPs), density functional approximations (DFAs) are often preferred because of their low cost/accuracy ratio. However, the accuracy of the SNLOPs critically depends on the amount of exact exchange and electron correlation included in the DFA, precluding the reliable calculation of many molecular systems. In this scenario, wave function methods such as MP2, CCSD, and CCSD(T) constitute a reliable alternative to compute SNLOPs. Unfortunately, the computational cost of these methods significantly restricts the size of molecules to study, a limitation that hampers the identification of molecules with significant nonlinear optical responses. This paper analyzes various flavors and alternatives to MP2, CCSD, and CCSD(T) methods that either drastically reduce the computational cost or improve their performance but were scarcely and unsystematically employed to compute SNLOPs. In particular, we have tested RI-MP2, RIJK-MP2, RIJCOSX-MP2 (with GridX2 and GridX4 setups), LMP2, SCS-MP2, SOS-MP2, DLPNO-MP2, LNO-CCSD, LNO-CCSD(T), DLPNO-CCSD, DLPNO-CCSD(T0), and DLPNO-CCSD(T1). Our results indicate that all these methods can be safely employed to calculate the dipole moment and the polarizability with average relative errors below 5% with respect to CCSD(T). On the other hand, the calculation of higher-order properties represents a challenge for LNO and DLPNO methods, which present severe numerical instabilities in computing the single-point field-dependent energies. RI-MP2, RIJK-MP2, or RIJCOSX-MP2 are cost-effective methods to compute first and second hyperpolarizabilities with a marginal average error with respect to canonical MP2 (up to 5% for β and up to 11% for γ). More accurate hyperpolarizabilities can be obtained with DLPNO-CCSD(T1); however, this method cannot be employed to obtain reliable second hyperpolarizabilities. These results open the way to obtain accurate nonlinear optical properties at a computational cost that can compete with current DFAs.

1-Matrix functional for long-range interaction energy of two hydrogen atoms

The leading terms in the large-R asymptotics of the functional of the one-electron reduced density matrix for the ground-state energy of the H₂ molecule with the internuclear separation R are derived thanks to the solution of the phase dilemma at the R → ∞ limit. At this limit, the respective natural orbitals (NOs) are given by symmetric and antisymmetric combinations of "half-space" orbitals with the corresponding natural amplitudes having the same amplitudes but opposite signs. Minimization of the resulting explicit functional yields the large-R asymptotics for the occupation numbers of the weakly occupied NOs and the C₆ dispersion coefficient. The highly accurate approximates for the radial components of the p-type "half-space" orbitals and the corresponding occupation numbers (that decay like R^-6), which are available for the first time thanks to the development of the present formalism, have some unexpected properties.

Exploiting Ligand Additivity for Transferable Machine Learning of Multireference Character across Known Transition Metal Complex Ligands

Accurate virtual high-throughput screening (VHTS) of transition metal complexes (TMCs) remains challenging due to the possibility of high multireference (MR) character that complicates property evaluation. We compute MR diagnostics for over 5,000 ligands present in previously synthesized octahedral mononuclear transition metal complexes in the Cambridge Structural Database (CSD). To accomplish this task, we introduce an iterative approach for consistent ligand charge assignment for ligands in the CSD. Across this set, we observe that the MR character correlates linearly with the inverse value of the averaged bond order over all bonds in the molecule. We then demonstrate that ligand additivity of the MR character holds in TMCs, which suggests that the TMC MR character can be inferred from the sum of the MR character of the ligands. Encouraged by this observation, we leverage ligand additivity and develop a ligand-derived machine learning representation to train neural networks to predict the MR character of TMCs from properties of the constituent ligands. This approach yields models with excellent performance and superior transferability to unseen ligand chemistry and compositions.

Dynamical electron correlation and the chemical bond. I. Covalent bonds in AH and AF (A = B-F)

Dynamical electron correlation has a major impact on the computed values of molecular properties and the energetics of molecular processes. The present study focused on the effect of dynamical electron correlation on the spectroscopic constants, ( Re , ωe , De ) and potential energy curves, Δ E( R), of the covalently-bound AH and AF molecules, A = B-F. The changes in the spectroscopic constants, (Δ Re , Δωe , Δ De ) caused by dynamical correlation are erratic and, at times, even surprising. These changes could be understood based on the dependence of the dynamical electron correlation energies of the AH and AF molecules as a function of the bond distance, i.e., Δ EDEC( R). At large R, the magnitude of Δ EDEC( R) increases nearly exponentially with decreasing R, but this increase slows as R continues to decrease and, in many cases, even reverses at very short R. The changes in Δ EDEC( R) in the region around Re were as unexpected as they were surprising, e.g., distinct minima and maxima were found in the curves of Δ EDEC( R) for the most polar molecules. The variations in Δ EDEC( R) for R ≲ Re are directly correlated with major changes in the electronic structure of the molecules as revealed by a detailed analysis of the SCGVB wave function. The results reported here indicate that we have much to learn about the nature of dynamical electron correlation and its effect on the chemical bond and molecular properties and processes.

Coupling Natural Orbital Functional Theory and Many-Body Perturbation Theory by Using Nondynamically Correlated Canonical Orbitals

We develop a new family of electronic structure methods for capturing at the same time the dynamic and nondynamic correlation effects. We combine the natural orbital functional theory (NOFT) and many-body perturbation theory (MBPT) through a canonicalization procedure applied to the natural orbitals to gain access to any MBPT approximation. We study three different scenarios: corrections based on second-order Møller-Plesset (MP2), random-phase approximation (RPA), and coupled-cluster singles doubles (CCSD). Several chemical problems involving different types of electron correlation in singlet and multiplet spin states have been considered. Our numerical tests reveal that RPA-based and CCSD-based corrections provide similar relative errors in molecular dissociation energies (D_e) to the results obtained using a MP2 correction. With respect to the MP2 case, the CCSD-based correction improves the prediction, while the RPA-based correction reduces the computational cost.

Extraction of a One-Particle Reduced Density Matrix from a Quantum Monte Carlo Electronic Density: A New Tool for Studying Nondynamic Correlation

In this work, we present a method to build a first order reduced density matrix (1-RDM) of a molecule from variational Quantum Monte Carlo (VMC) computations by means of a given correlated mapping wave function. Such a wave function is modeled on a Generalized Valence Bond plus Complete Active Space Configuration Interaction form and fits at best the density resulting from the Slater-Jastrow wave function of VMC. The accuracy of the method proposed has been proved by comparing the resulting kinetic energy with the corresponding VMC value. This 1-RDM is used to analyze the amount of correlation eventually captured in Kohn-Sham calculations performed in an unrestricted approach (UKS-DFT) and with different energy functionals. We performed test calculations on a selected set of molecules that show a significant multireference character. In this analysis, we compared both local and global indicators of nondynamic and dynamic correlation. Moreover, following the natural orbital decomposition of the 1-RDM, we also compared the effective temperatures of the corresponding Fermi-like distributions. Although there is a general agreement between UKS-DFT and VMC, we found the best match with the functional LC-BLYP.

On the inclusion of one double within CIS and TDDFT

We present an improved approach for generating a set of optimized frontier orbitals (HOMO and LUMO) that minimizes the energy of one double configuration. We further benchmark the effect of including such a double within a rigorous configuration interaction singles or a parameterized semi-empirical time-dependent density functional theory Hamiltonian for a set of test cases. Although we cannot quite achieve quantitative accuracy, the algorithm is quite robust and routinely delivers an enormous qualitative improvement to standard single-reference electronic structure calculations.

Impact of van der Waals interactions on the structural and nonlinear optical properties of azobenzene switches

The geometrical structures, relative Z-E energies, and second-order nonlinear responses of a collection of azobenzene molecules symmetrically substituted in the meta-position with functional groups of different bulkiness are investigated using various ab initio and density functional approximations. We show that RI-MP2 and RI-CC2 approximations provide very similar geometries and relative energies and evidence that London dispersion interactions existing between bulky meta-substituents stabilize the Z conformer. The ωB97X-D exchange-correlation functional provides an accurate description of these effects and gives a good account of the nonlinear optical response of the molecules. We show that density functional approximations should include no less than 50% of Hartree-Fock exchange to provide accurate hyperpolarizabilities. A property-structure analysis of the azobenzene derivatives reveals that the main contribution to the first hyperpolarizability comes from the azo bond, but phenyl meso-substituents can enhance it.

Variations of the Hartree-Fock fractional-spin error for one electron

Fractional-spin errors are inherent in all current approximate density functionals, including Hartree-Fock theory, and their origin has been related to strong static correlation effects. The conventional way to encode fractional-spin calculations is to construct an ensemble density that scales between the high-spin and low-spin densities. In this article, we explore the variation of the Hartree-Fock fractional-spin (or ghost-interaction) error in one-electron systems using restricted and unrestricted ensemble densities and the exact generalized Hartree-Fock representation. By considering the hydrogen atom and H⁺ ₂ cation, we analyze how the unrestricted and generalized Hartree-Fock schemes minimize this error by localizing the electrons or rotating the spin coordinates. We also reveal a clear similarity between the Coulomb hole of He-like ions and the density depletion near the nucleus induced by the fractional-spin error in the unpolarized hydrogen atom. Finally, we analyze the effect of the fractional-spin error on the Møller-Plesset adiabatic connection, excited states, and functional- and density-driven errors.

How Many Electrons Does a Molecular Electride Hold?

Electrides are very peculiar ionic compounds where electrons occupy the anionic positions. In a crystal lattice, these isolated electrons often form channels or surfaces, furnishing electrides with many traits with promising technological applications. Despite their huge potential, thus far, only a few stable electrides have been produced because of the intricate synthesis they entail. Due to the difficulty in assessing the presence of isolated electrons, the characterization of electrides also poses some serious challenges. In fact, their properties are expected to depend on the arrangement of these electrons in the molecule. Among the criteria that we can use to characterize electrides, the presence of a non-nuclear attractor (NNA) of the electron density is both the rarest and the most salient feature. Therefore, a correct description of the NNA is crucial to determine the properties of electrides. In this paper, we analyze the NNA and the surrounding region of nine molecular electrides to determine the number of isolated electrons held in the electride. We have seen that the correct description of a molecular electride hinges on the electronic structure method employed for the analyses. In particular, one should employ a basis set with sufficient flexibility to describe the region close to the NNA and a density functional approximation that does not suffer from large delocalization errors. Finally, we have classified these nine molecular electrides according to the most likely number of electrons that we can find in the NNA. We believe this classification highlights the strength of the electride character and will prove useful in designing new electrides.

Measuring correlated electron motion in atoms with the momentum-balance density

Three new measures of relative electron motion are introduced: equimomentum, antimomentum, and momentum-balance. The equimomentum is the probability that two electrons have the exact same momentum, whereas the antimomentum is the probability that their momenta are the exact opposite. Momentum-balance (MB) is the difference between the equimomentum and antimomentum and, therefore, indicates if equal or opposite momentum is more probable in a system of electrons. The equimomentum, antimomentum, and MB densities are also introduced, which are the local contribution to each quantity. The MB and MB density of the extrapolated-full configuration interaction wave functions of atoms of the first three rows of the periodic table are analyzed, with a particular focus on contrasting the correlated motion of electrons with opposite-spin and parallel-spin. Coulomb correlation between opposite-spin electrons leads to a higher probability of equimomentum, whereas Fermi correlation between parallel-spin electrons leads to a higher probability of antimomentum. The local contribution to MB, given an electron is present, is a minimum at the nucleus and generally increases as the distance from the nucleus increases. There are also interesting similarities between the effects of Fermi correlation and Coulomb correlation (of opposite-spin electrons) on MB.

Rapid Detection of Strong Correlation with Machine Learning for Transition-Metal Complex High-Throughput Screening

Despite its widespread use in chemical discovery, approximate density functional theory (DFT) is poorly suited to many targets, such as those containing open-shell, 3d transition metals that can be expected to have strong multireference (MR) character. For discovery workflows to be predictive, we need automated, low-cost methods that can distinguish the regions of chemical space where DFT should be applied from those where it should not. We curate more than 4800 open-shell transition-metal complexes up to hundreds of atoms in size from prior high-throughput DFT studies and evaluate affordable, finite-temperature DFT fractional occupation number (FON)-based MR diagnostics. We show that intuitive measures of strong correlation (i.e., the HOMO-LUMO gap) are not predictive of MR character as judged by FON-based diagnostics. Analysis of independently trained machine learning (ML) models to predict HOMO-LUMO gaps and FON-based diagnostics reveals differences in the metal and ligand sensitivity of the two quantities. We use our trained ML models to rapidly evaluate MR character over a space of ∼187000 theoretical complexes, identifying large-scale trends in spin-state-dependent MR character and finding small HOMO-LUMO gap complexes while ensuring low MR character.

Semi-supervised Machine Learning Enables the Robust Detection of Multireference Character at Low Cost

Multireference (MR) diagnostics are common tools for identifying strongly correlated electronic structure that makes single-reference (SR) methods (e.g., density functional theory or DFT) insufficient for accurate property prediction. However, MR diagnostics typically require computationally demanding correlated wave function theory (WFT) calculations, and diagnostics often disagree or fail to predict MR effects on properties. To overcome these challenges, we introduce a semi-supervised machine learning (ML) approach with virtual adversarial training (VAT) of an MR classifier using 15 WFT and DFT MR diagnostics as inputs. In semi-supervised learning, only the most extreme SR or MR points are labeled, and the remaining point labels are learned. The resulting VAT model outperforms the alternatives, as quantified by the distinct property distributions of SR- and MR-classified molecules. To reduce the cost of generating inputs to the VAT model, we leverage the VAT model's robustness to noisy inputs by replacing WFT MR diagnostics with regression predictions in an MR decision engine workflow that preserves excellent performance. We demonstrate the transferability of our approach to larger molecules and those with distinct chemical composition from the training set. This MR decision engine demonstrates promise as a low-cost, high-accuracy approach to the automatic detection of strong correlation for predictive high-throughput screening.

Performance of Localized Coupled Cluster Methods in a Moderately Strong Correlation Regime: Hückel-Möbius Interconversions in Expanded Porphyrins

Localized orbital coupled cluster theory has recently emerged as a nonempirical alternative to DFT for large systems. Intuitively, one might expect such methods to perform less well for highly delocalized systems. In the present work, we apply both canonical CCSD(T) approximations and a variety of localized approximations to a set of flexible expanded porphyrins—macrocycles that can switch between Hückel, figure-eight, and Möbius topologies under external stimuli. Both minima and isomerization transition states are considered. We find that Möbius(-like) structures have much stronger static correlation character than the remaining structures, and that this causes significant errors in DLPNO-CCSD(T) and even DLPNO-CCSD(T₁) approaches, unless TightPNO cutoffs are employed. If sub-kcal mol^–1 accuracy with respect to canonical relative energies is required even for Möbius-type systems (or other systems plagued by strong static correlation), then Nagy and Kallay’s LNO-CCSD(T) method with “tight” settings is the suitable localized approach. We propose the present POLYPYR21 data set as a benchmark for localized orbital methods or, more broadly, for the ability of lower-level methods to handle energetics with strongly varying degrees of static correlation.

Capturing static and dynamic correlation with ΔNO-MP2 and ΔNO-CCSD

The ΔNO method for static correlation is combined with second-order Møller-Plesset perturbation theory (MP2) and coupled-cluster singles and doubles (CCSD) to account for dynamic correlation. The MP2 and CCSD expressions are adapted from finite-temperature CCSD, which includes orbital occupancies and vacancies, and expanded orbital summations. Correlation is partitioned with the aid of damping factors incorporated into the MP2 and CCSD residual equations. Potential energy curves for a selection of diatomics are in good agreement with extrapolated full configuration interaction results and on par with conventional multireference approaches.

Partition of optical properties into orbital contributions

A new tool to analyze the response property through the partition of nonlinear optical properties in terms of orbital contributions (PNOC), valuable in the assessment of the electronic structure methods in the NLOPs computations, is presented.