How accurate is density functional theory at predicting dipole moments? An assessment using a new database of 200 benchmark values

Dipole moments are a simple, global measure of the accuracy of the electron density of a polar molecule. Dipole moments also affect the interactions of a molecule with other molecules as well as electric fields. To directly assess the accuracy of modern density functionals for calculating dipole moments, we have developed a database of 200 benchmark dipole moments, using coupled cluster theory through triple excitations, extrapolated to the complete basis set limit. This new database is used to assess the performance of 88 popular or recently developed density functionals. The results suggest that double hybrid functionals perform the best, yielding dipole moments within about 3.6-4.5% regularized RMS error versus the reference values---which is not very different from the 4% regularized RMS error produced by coupled cluster singles and doubles. Many hybrid functionals also perform quite well, generating regularized RMS errors in the 5-6% range. Some functionals however exhibit large outliers and local functionals in general perform less well than hybrids or double hybrids.Comment: Added several double hybrid functionals, most of which turned out to be better than any functional from Rungs 1-4 of Jacob's ladder and are actually competitive with CCS

CHEAP AND RELIABLE OPTIMIZATION OF EXCITED STATE ORBITALS WITH THE SQUARE GRADIENT MINIMIZATION (SGM) APPROACH.

Linear response (LR) protocols like time dependent density functional theory (TDDFT) or equation of motion coupled cluster (EOM-CC) are often used to compute energies of electronic excited states. While effective for valence excitations, LR methods are susceptible to catastrophic failure for problems like core excitations or charge-transfer states, where the optimal excited state orbitals differ considerably from the ground state reference. Orbital optimized (OO) methods are more effective for such problems, as they permit relaxation of excited state orbitals beyond linear response. Widespread usage of OO methods has however been hindered by their propensity to collapse to the ground state instead of the desired excited state. This is a direct consequence of excited states typically being unstable saddle points in orbital space.% which makes the performance of standard convergers like DIIS considerably less robust. We present a orbital optimization protocol that reliably converges to the closest stationary point to the initial guess, by minimizing the square of the energy gradient instead of explicitly attempting to extremize the energy. The computational cost of this square gradient minimization (SGM) method is only between 2-3 times the cost of ground state orbital optimization (per iteration). SGM+DFT therefore can be readily applied to large systems. We subsequently demonstrate the utility of SGM by application to doubly excited states, core excitations and charge-transfer states. Specifically, we show that cheap DFT based OO approaches can predict energies of doubly excited states to significantly greater accuracy than expensive, LR coupled cluster approaches (that often have $>$ 1 eV error). Similarly, we demonstrate that a DFT based protocol employing SGM predicts core excitation energies (at both the K and L edges) to $10$ eV error). Finally, we demonstrate prediction of charge transfer excitation energies to low error with DFT/SGM -- in stark contrast to standard TDDFT. Time permitting, we would also discuss use of SGM with methods like complete active space self-consistent field (CASSCF) to tackle strongly correlated excited states and model conical intersections. \textbf{References:} Hait, D. and Head-Gordon M. J. Chem. Theory Comput., ASAP (2020); Hait, D. and Head-Gordon M. J. Phys. Chem. Lett. 11, 3, 775-786 (2020)

Excited state orbital optimization via minimizing the square of the gradient: General approach and application to singly and doubly excited states via density functional theory

We present a general approach to converge excited state solutions to any quantum chemistry orbital optimization process, without the risk of variational collapse. The resulting Square Gradient Minimization (SGM) approach only requires analytic energy/Lagrangian orbital gradients and merely costs 3 times as much as ground state orbital optimization (per iteration), when implemented via a finite difference approach. SGM is applied to both single determinant $\Delta$SCF and spin-purified Restricted Open-Shell Kohn-Sham (ROKS) approaches to study the accuracy of orbital optimized DFT excited states. It is found that SGM can converge challenging states where the Maximum Overlap Method (MOM) or analogues either collapse to the ground state or fail to converge. We also report that $\Delta$SCF/ROKS predict highly accurate excitation energies for doubly excited states (which are inaccessible via TDDFT). Singly excited states obtained via ROKS are also found to be quite accurate, especially for Rydberg states that frustrate (semi)local TDDFT. Our results suggest that orbital optimized excited state DFT methods can be used to push past the limitations of TDDFT to doubly excited, charge-transfer or Rydberg states, making them a useful tool for the practical quantum chemist's toolbox for studying excited states in large systems

Delocalization errors in density functional theory are essentially quadratic in fractional occupation number

Approximate functionals used in practical density functional theory (DFT) deviate from the piecewise linear behavior of the exact functional for fractional charges. This deviation causes excess charge delocalization, which leads to incorrect densities, molecular properties, barrier heights, band gaps and excitation energies. We present a simple delocalization function for characterizing this error and find it to be almost perfectly linear vs the fractional electron number for systems spanning in size from the H atom to the C$_{12}$H$_{14}$ polyene. This causes the delocalization energy error to be a quadratic polynomial in the fractional electron number, which permits us to assess the comparative performance of 47 popular and recent functionals through the curvature. The quadratic form further suggests that information about a single fractional charge is sufficient to eliminate the principal source of delocalization error. Generalizing traditional two-point information like ionization potentials or electron affinities to account for a third, fractional charge based data point could therefore permit fitting/tuning of functionals with lower delocalization error.Comment: Discussion about fractional binding issues in anions have been added, with other minor fixes/elaboration

Too big, too small or just right? A benchmark assessment of density functional theory for predicting the spatial extent of the electron density of small chemical systems

Multipole moments are the first order responses of the energy to spatial derivatives of the electric field strength. The quality of density functional theory (DFT) prediction of molecular multipole moments thus characterizes errors in modeling the electron density itself, as well as the performance in describing molecules interacting with external electric fields. However, only the lowest non-zero moment is translationally invariant, making the higher order moments origin-dependent. Therefore, instead of using the $3 \times 3$ quadrupole moment matrix, we utilize the translationally invariant $3 \times 3$ matrix of second cumulants (or spatial variances) of the electron density as the quantity of interest (denoted by $\mathcal{K}$). The principal components of ${\mathcal{K}}$ are the square of the spatial extent of the electron density along each axis. A benchmark dataset of the prinicpal components of ${\mathcal{K}}$ for 100 small molecules at the coupled cluster singles and doubles with perturbative triples (CCSD(T)) at the complete basis set (CBS) limit is developed, resulting in 213 independent ${\mathcal{K}}$ components. The performance of 47 popular and recent density functionals is assessed against this Var213 dataset. Several functionals, especially double hybrids, and also SCAN and SCAN0 yield reliable second cumulants, although some modern, empirically parameterized functionals yield more disappointing performance. The H and Be atoms in particular are challenging for nearly all methods, indicating that future functional development could benefit from inclusion of their density information in training or testing protocols

Modern Approaches to Exact Diagonalization and Selected Configuration Interaction with the Adaptive Sampling CI Method.

Recent advances in selected configuration interaction methods have made them competitive with the most accurate techniques available and, hence, creating an increasingly powerful tool for solving quantum Hamiltonians. In this work, we build on recent advances from the adaptive sampling configuration interaction (ASCI) algorithm. We show that a useful paradigm for generating efficient selected CI/exact diagonalization algorithms is driven by fast sorting algorithms, much in the same way iterative diagonalization is based on the paradigm of matrix vector multiplication. We present several new algorithms for all parts of performing a selected CI, which includes new ASCI search, dynamic bit masking, fast orbital rotations, fast diagonal matrix elements, and residue arrays. The ASCI search algorithm can be used in several different modes, which includes an integral driven search and a coefficient driven search. The algorithms presented here are fast and scalable, and we find that because they are built on fast sorting algorithms they are more efficient than all other approaches we considered. After introducing these techniques, we present ASCI results applied to a large range of systems and basis sets to demonstrate the types of simulations that can be practically treated at the full-CI level with modern methods and hardware, presenting double- and triple-ζ benchmark data for the G1 data set. The largest of these calculations is Si2H6 which is a simulation of 34 electrons in 152 orbitals. We also present some preliminary results for fast deterministic perturbation theory simulations that use hash functions to maintain high efficiency for treating large basis sets

Relativistic Orbital Optimized Density Functional Theory for Accurate Core-Level Spectroscopy

Core-level spectra of 1s electrons (K-edge) of elements heavier than Ne show significant relativistic effects. We combine recent advances in orbital optimized density functional theory (OO-DFT) with the spin-free exact two-component (X2C) model for scalar relativistic effects, to study K-edge spectra of elements in the third period of the periodic table. OO-DFT/X2C is found to be quite accurate at predicting energies, yielding $\sim 0.5$ eV root mean square error (RMSE) vs experiment with the local SCAN functional and the related SCANh hybrid functional. This marks a signficant improvement over the $>50$ eV deviations that are typical for the popular time-dependent DFT (TDDFT) approach. Consequently, experimental spectra are quite well reproduced by OO-DFT/X2C, without any need for empirical shifts for alignment between the two. OO-DFT/X2C therefore is a promising route for computing core-level spectra of third period elements, as it combines high accuracy with ground state DFT cost. We also explored K and L edges of 3d transition metals to identify possible limitations of the OO-DFT/X2C approach and discuss what additional features would be needed for accurately modeling the spectra of such electrons

What Levels of Coupled Cluster Theory Are Appropriate for Transition Metal Systems? A Study Using Near-Exact Quantum Chemical Values for 3d Transition Metal Binary Compounds.

Transition metal compounds are traditionally considered to be challenging for standard quantum chemistry approximations like coupled cluster (CC) theory, which are usually employed to validate lower level methods like density functional theory (DFT). To explore this issue, we present a database of bond dissociation energies (BDEs) for 74 spin states of 69 diatomic species containing a 3d transition metal atom and a main group element, in the moderately sized def2-SVP basis. The presented BDEs appear to have an (estimated) 3σ error less than 1 kJ/mol relative to the exact solutions to the nonrelativistic Born-Oppenheimer Hamiltonian. These benchmark values were used to assess the performance of a wide range of standard single reference CC models, as the results should be beneficial for understanding the limitations of these models for transition metal systems. We find that interactions between metals and monovalent ligands like hydride and fluoride are well described by CCSDT. Similarly, CCSDTQ appears to be adequate for bonds between metals and nominally divalent ligands like oxide and sulfide. However, interactions with polyvalent ligands like nitride and carbide are more challenging, with even CCSDTQ(P)Λ yielding errors on the scale of a few kJ/mol. We also find that many perturbative and iterative approximations to higher order terms either yield disappointing results or actually worsen the performance relative to the baseline low level CC method, indicating that complexity does not always guarantee accuracy