Correcting the Charge Delocalization Error of Density Functional Theory.

The charge delocalization error, besides nondynamic correlation, has been a major challenge to density functional theory. Contemporary functionals undershoot the dissociation of symmetric charged dimers A2+, a simple but stringent test, predict a spurious barrier, and improperly delocalize charges for charged molecular clusters. We extend a functional designed for nondynamic correlation to treat the charge delocalization error by modifying the nondynamic correlation for parallel spins. The modified functional eliminates those problems and reduces the multielectron self-interaction error. Furthermore, its results are the closest to those of CCSD(T) in the whole range of the dissociation compared with contemporary functionals. It correctly localizes the net positive charge in (CH4)n+ clusters and predicts a nearly constant ionization potential as a result. Testing of the SIE4x4 set shows that the new functional outperforms a wide variety of functionals assessed for this set in the literature. Overall, we show the feasibility of treating charge delocalization together with nondynamic correlation.

Model DFT exchange holes and the exact exchange hole: Similarities and differences.

Exchange and correlation holes are unique quantum concepts for understanding the nature of electron interactions based on quantum conditional probabilities. Among those, the exact exchange hole is of special interest since it is derived rigorously from first principles without approximations and is often modeled by approximate exchange expressions of density functional theory. In this work, the algorithm for the computation of the spherically averaged exact exchange hole for a given reference point is developed and implemented for molecular orbitals in Gaussian basis functions. The formulas include a novel recursive relation for the spherical average of the Bessel function of the first kind and the asymptotic expressions when the exponential factor of the Bessel function becomes large. This new capability is used to explore the extent to which current popular model exchange holes resemble or differ from the exact exchange hole. Point-wise accuracy of the exchange holes for isolated atoms is important in local hybrid schemes, real-space models of static correlation, and others. We find in this vein that among the models tested here, only the BR89 exchange hole seems more or less suitable for that purpose, while better approximations are still very much on demand. Analyzing the deviations of model exchange holes from the exact exchange hole in molecules such as H2 and Cr2 upon bond stretching reveals new aspects of the left-right static correlation.

Performance of new density functionals of nondynamic correlation on chemical properties.

Becke'13 (B13) is a general density functional theory (DFT) designed for strongly correlated molecules. Kong-Proynov'16/B13 (KP16/B13) is a modification of B13 that treats the correlation kinetic term through modeling the adiabatic connection and contains fewer parameters. A question remains whether functionals designed for extremely strong correlations can handle correlation of normal strength. In this work, both functionals are benchmarked with a variety of molecular systems from the Minnesota-2015 datasets and compared with the mainstream functionals such as B3LYP and a spectrum of other types of functionals, as well as Becke'05 (B05), the predecessor of B13. Our calculations, made possible with an efficient self-consistent-field implementation of those new functionals, show that B05, B13, and KP16/B13, based on full HF exchange, recover a majority of the correlation where it is very significant (i.e., where HF has large errors). They perform very well for reaction barriers. On the other hand, their performance on multireference systems is mixed. Overall, B05, B13, and KP16/B13 are competitive to B3LYP, and B05 is even competitive to heavily parameterized functionals for most sets despite containing 100% HF exchange. KP16/B13, in particular, contains only three empirical parameters, demonstrating the possibility of treating correlation of all strength with single-determinant Kohn-Sham DFT.

Long-range-corrected Rung 3.5 density functional approximations.

Rung 3.5 functionals are a new class of approximations for density functional theory. They provide a flexible intermediate between exact (Hartree-Fock, HF) exchange and semilocal approximations for exchange. Existing Rung 3.5 functionals inherit semilocal functionals' limitations in atomic cores and density tails. Here we address those limitations using range-separated admixture of HF exchange. We present three new functionals. LRC-ωΠLDA combines long-range HF exchange with short-range Rung 3.5 ΠLDA exchange. SLC-ΠLDA combines short- and long-range HF exchange with middle-range ΠLDA exchange. LRC-ωΠLDA-AC incorporates a combination of HF, semilocal, and Rung 3.5 exchange in the short range, based on an adiabatic connection. We test these in a new Rung 3.5 implementation including up to analytic fourth derivatives. LRC-ωΠLDA and SLC-ΠLDA improve atomization energies and reaction barriers by a factor of 8 compared to the full-range ΠLDA. LRC-ωΠLDA-AC brings further improvement approaching the accuracy of standard long-range corrected schemes LC-ωPBE and SLC-PBE. The new functionals yield highest occupied orbital energies closer to experimental ionization potentials and describe correctly the weak charge-transfer complex of ethylene and dichlorine and the hole-spin distribution created by an Al defect in quartz. This study provides a framework for more flexible range-separated Rung 3.5 approximations.

Practical Density Functionals beyond the Overdelocalization-Underbinding Zero-Sum Game.

Density functional theory (DFT) uses a density functional approximation (DFA) to add electron correlation to mean-field electronic structure calculations. Standard strategies (generalized gradient approximations GGAs, meta-GGAs, hybrids, etc.) for building DFAs, no matter whether based on exact constraints or empirical parametrization, all face a zero-sum game between overdelocalization (fractional charge error, FC) and underestimation of covalent bonding (fractional spin error, FS). This work presents an alternative strategy. Practical "Rung 3.5" ingredients are used to implement insights from hyper-GGA DFAs that reduce both FS and FC errors. Prototypes of this strategy qualitatively improve FS and FC error over 40 years of standard DFAs while maintaining low cost and practical evaluation of properties. Numerical results ranging from transition metal thermochemistry to absorbance peaks and excited-state geometry optimizations highlight this strategy's promise and indicate areas requiring further development.

Describing a Strongly Correlated Model System with Density Functional Theory.

The linear chain of hydrogen atoms, a basic prototype for the transition from a metal to Mott insulator, is studied with a recent density functional theory model functional for nondynamic and strong correlation. The computed cohesive energy curve for the transition agrees well with accurate literature results. The variation of the electronic structure in this transition is characterized with a density functional descriptor that yields the atomic population of effectively localized electrons. These new methods are also applied to the study of the Peierls dimerization of the stretched even-spaced Mott insulator to a chain of H2 molecules, a different insulator. The transitions among the two insulating states and the metallic state of the hydrogen chain system are depicted in a semiquantitative phase diagram. Overall, we demonstrate the capability of studying strongly correlated materials with a mean-field model at the fundamental level, in contrast to the general pessimistic view on such a feasibility.

Testing Exact Upper Bounds to Exact Exchange.

The exact exchange energy and its energy density are useful but computationally expensive ingredients in density functional approximations for Kohn-Sham density functional theory. We present detailed tests of some exact nonempirical upper bounds to exact exchange. These "Rung 3.5" upper bounds contract the Kohn-Sham one-particle density matrix with model density matrices used to construct semilocal model exchange holes and invoke the Cauchy-Schwarz inequality. The contraction automatically eliminates the computationally expensive long-range component of the exact exchange hole. Numerical tests show that the exchange upper bounds underestimate total exchange energies while predicting other properties with accuracy approaching standard hybrid approximations.

Accurate alkynyl radical structures from density functional calculations without Hartree-Fock exchange.

Density functional approximations (DFAs) often suffer from self-repulsion and delocalization errors which are reduced by exact (Hartree-Fock-like) exchange admixture. Oyeyemi and co-workers recently showed that several DFAs with little exact exchange incorrectly predict bent alkynyl radical geometries, giving errors in ab initio composite methods using density functional theory geometries [V. B. Oyeyemi et al., J. Phys. Chem. Lett. 3, 289 (2012)]. We show that the simple Hartree-Fock-Slater and Xα DFAs, which have substantial delocalization error, predict linear alkynyl radical geometries without incorporating exact exchange. Our Rung 3.5 DFAs, and rescaled generalized gradient approximations, can give either linear σ, bent σ-π, or nearly linear π radicals, all without incorporating exact exchange. This highlights the complexity of delocalization error, the utility of accurate empirical DFA geometries for ab initio composite methods, and the insights to be gained from Rung 3.5 DFAs.

Density Functional Model for Nondynamic and Strong Correlation.

A single-term density functional model for the left-right nondynamic/strong electron correlation is presented based on single-determinant Kohn-Sham density functional theory. It is derived from modeling the adiabatic connection for kinetic correlation energy based on physical arguments, with the correlation potential energy based on the Becke'13 model ( Becke, A.D. J. Chem. Phys . 2013 , 138 , 074109 ). This functional satisfies some known scaling relationships for correlation functionals. The fractional spin error is further reduced substantially with a new density-functional correction. Preliminary tests with self-consistent-field implementation show that the model, with only three empirical parameters, recovers the majority of left-right nondynamic/strong correlation upon bond dissociation and performs reasonably well for atomization energies and singlet-triplet energy splittings. This study also demonstrates the feasibility of developing DFT functionals for nondynamic and strong correlation within the single-determinant KS scheme.

Density-functional approach to the three-body dispersion interaction based on the exchange dipole moment.

We implement and compute the density functional nonadditive three-body dispersion interaction using a combination of Tang-Karplus formalism and the exchange-dipole moment model of Becke and Johnson. The computation of the C9 dispersion coefficients is done in a non-empirical fashion. The obtained C9 values of a series of noble atom triplets agree well with highly accurate values in the literature. We also calculate the C9 values for a series of benzene trimers and find a good agreement with high-level ab initio values reported recently in the literature. For the question of damping of the three-body dispersion at short distances, we propose two damping schemes and optimize them based on the benzene trimers data, and the fitted analytic potentials of He3 and Ar3 trimers fitted to the results of high-level wavefunction theories available from the literature. Both damping schemes respond well to the optimization of two parameters.

Comparison of the performance of exact-exchange-based density functional methods.

How to describe nondynamic electron correlation is still a major challenge to density functional theory (DFT). Recent models designed particularly for this problem, such as Becke'05 (B05) and Perdew-Staroverov-Tao-Scuseria (PSTS) functionals employ the exact-exchange density, the efficient calculation of which is technically quite challenging. We have recently implemented self-consistently the B05 functional based on an efficient resolution-identity (RI) technique. In this study, we report a self-consistent RI implementation of the PSTS functional. In contrast to its original implementation, our version brings no limitation on the choice of the basis set. We have also implemented the Mori-Sanchez-Cohen-Yang-2 (MCY2) functional, another recent DFT method that includes full exact exchange. The performance of PSTS, B05, and MCY2 is validated on thermochemistry, reaction barriers, and dissociation energy curves, with an emphasis on nondynamic correlation effects in the discussion. All three methods perform rather well in general, B05 and MCY2 being on average somewhat better than PSTS. We include also results with other functionals that represent various aspects of the development in this field in recent years, including B3LYP, M06-HF, M06-2X, ωB97X, and TPSSh. The performance of the heavy-parameterized functionals M06-2X and ωB97X is on average better than that of B05, MCY2, and PSTS for standard thermodynamic properties and reactions, while the latter functionals do better in hydrogen abstraction reactions and dissociation processes. In particular, B05 is found to be the only functional that yields qualitatively correct dissociation curves for two-center symmetric radicals like He(2)(+). Finally, we compare the performance of all these functionals on a strongly correlated exemplary case system, the NO dimer. Only PSTS, B05, and MCY2 describe the system qualitatively correctly. Overall, this new type of functionals show good promise of overcoming some of the difficulties DFT encounters for systems with strong nondynamic correlation.

Modified Becke'05 method of nondynamic correlation in density functional theory with self-consistent implementation.

Becke's B05 method for nondynamic correlation is simplified for self-consistent implementation. An alternative form is proposed for the nondynamic correlation factors that do not require solving a complicated nonlinear algebraic equation. The four linear parameters of B05 are re-optimized together with one extra parameter entering a modified expression for the second-order same-spin energy contribution. The latter is co-linear with the exact-exchange energy density and does not require higher moments of the relaxed exchange hole. Preliminary tests of this method show that it leads to a slight improvement over the resolution-of-identity B05 results reported previously for atomization energies, and to a definite improvement for reaction barriers of Hydrogen abstraction.

Improved self-consistent and resolution-of-identity approximated Becke'05 density functional model of nondynamic electron correlation.

In a recent letter [E. Proynov, Y. Shao, and J. Kong, Chem. Phys. Lett. 493, 381 (2010)], Becke's B05 model of nondynamic electron correlation in density functional theory was implemented self-consistently with computational efficiency (the "SCF-RI-B05" scheme). Important modifications of the algorithm were done in order to make the self-consistency feasible. In the present work, we give a complete account of the SCF-RI-B05 algorithm, including all the formulae for the analytical representation of the B05 functional and for its self-consistent field (SCF) potential. The average performance of the SCF-RI-B05 method reported in the above letter was somewhat less accurate, compared to the original B05 implementation, mainly because the parameters of the original B05 model were optimized with post-local-spin-density calculations. In this work, we report improved atomization energies with SCF-RI-B05, based on a SCF re-optimization of its four linear parameters. The re-optimized SCF-RI-B05 scheme is validated also on reaction barriers, and on the subtle energetics of NO dimer, an exemplary system of strong nondynamic correlation. It yields both the binding energy and the singlet-triplet splitting of the NO dimer correctly, and close to the benchmarks reported in the literature.

Efficient self-consistent DFT calculation of nondynamic correlation based on the B05 method.

Becke's B05 method of describing nondynamic electron correlation in Density Functional Theory is implemented self-consistently with computational efficiency. Important modifications of the method are proposed in order to make the self-consistency feasible. Resolution-of-identity technique is used to reduce dramatically the cost of the required exact-exchange energy density. The method is briefly validated on a variety of properties. It describes accurately for the first time the subtle energetics of the NO dimer, an exemplary system of strong nondynamic correlation. The efficient algorithm for the exact-exchange energy density can be applied to other functionals that use this quantity.

Analytical representation of the Becke-Roussel exchange functional.

The unique meta-GGA (generalized gradient approximation) exchange functional of Becke and Roussel (BR89) and the correlation functional of Becke related to it (B94) are represented for the first time in an analytical form. All functional derivatives are then obtained analytically, which allows an efficient self-consistent implementation. A brief assessment of this "BR89B94" meta-GGA scheme is made considering molecular atomization energies and equilibrium geometries, with the latter being reported for the first time. The hybrid version of it yields one of the most accurate atomization energies to date, but its bond distances are less satisfactory. Some interesting features of the BR exchange hole are discussed.

Improved meta-GGA Correlation Functional of the Lap Family.

A new correlation functional of the Lap series is derived based on a more elaborated form of correlation wave vectors. Its validation is carried out within two different codes: deMon-KS3 and Q-Chem 3.0. In deMon the implementation in a post-SCF manner is similar to the preceding BLap3 and Bmτ1 schemes. In Q-Chem the new functional is programmed self-consistently using the integration by parts procedure for the matrix elements. The post-SCF version of the functional deals with three fitting parameters; the previous Lap3 and τ1 functionals have four and five parameters, respectively. The SCF implementation requires only two fitting parameters. Preliminary comparative tests are discussed.

Advances in methods and algorithms in a modern quantum chemistry program package.

Advances in theory and algorithms for electronic structure calculations must be incorporated into program packages to enable them to become routinely used by the broader chemical community. This work reviews advances made over the past five years or so that constitute the major improvements contained in a new release of the Q-Chem quantum chemistry package, together with illustrative timings and applications. Specific developments discussed include fast methods for density functional theory calculations, linear scaling evaluation of energies, NMR chemical shifts and electric properties, fast auxiliary basis function methods for correlated energies and gradients, equation-of-motion coupled cluster methods for ground and excited states, geminal wavefunctions, embedding methods and techniques for exploring potential energy surfaces.