Nonspherical model density matrices for Rung 3.5 density functionals

"Rung 3.5" exchange-correlation functionals for Kohn-Sham density functional theory depend linearly on the nonlocal one-particle density matrix of the noninteracting Kohn-Sham reference system. Rung 3.5 functionals also require a semilocal model for the one-particle density matrix. This work presents new model density matrices for Rung 3.5 functionals. The resulting functionals give reasonable predictions for total energies, molecular thermochemistry and kinetics, odd-electron bonds, and conjugated polymer bandgaps. Global-hybrid-like combinations of semilocal and Rung 3.5 exchange, and empirical density matrix models, also show promise.     

Rung 3.5 density functionals: Another step on Jacob's ladder

Applications of density functional theory (DFT) to computational chemistry and solid‐state physics rely on a “Jacob's Ladder” of progressively more complicated approximations to the many‐body exchange‐correlation (XC) density functional. Accurate, computationally tractable DFT calculations on large and periodic systems remain challenging for existing XC functionals. Simple XC functionals on the three lowest rungs of Jacob's Ladder are insufficiently accurate for many properties, while fourth‐rung hybrid functionals incorporating nonlocal information can be prohibitively expensive. This perspective presents our work toward a compromise, a new class of “Rung 3.5” functionals that incorporate a linear dependence on the nonlocal one‐particle density matrix. This work reviews these functionals' formal underpinning, numerical performance, and prospects for modeling solids and surfaces. © 2012 Wiley Periodicals, Inc.     

Comparing modern density functionals for conjugated polymer band structures: screened hybrid, Minnesota, and Rung 3.5 approximations

Semiconducting polymers with π-conjugated backbones show promise in fields such as photovoltaics. Practical applications of conjugated polymers require precise control over the polymer's electronic band structure. Several new classes of density functional approximation, including screened hybrids, semilocal Minnesota functionals, and Rung 3.5 functionals, show potential for improved predictions of conjugated polymer band structures. This work compares these methods to standard global hybrid density functionals for bandgaps and band structures of representative conjugated polymers. The new methods exhibit particular promise for modeling three-dimensionally periodic bulk polymers, which can be problematic for global hybrids.     

Magnetic exchange couplings from constrained density functional theory: an efficient approach utilizing analytic derivatives

We introduce a method for evaluating magnetic exchange couplings based on the constrained density functional theory (C-DFT) approach of Rudra, Wu, and Van Voorhis [J. Chem. Phys. 124, 024103 (2006)]. Our method shares the same physical principles as C-DFT but makes use of the fact that the electronic energy changes quadratically and bilinearly with respect to the constraints in the range of interest. This allows us to use coupled perturbed Kohn-Sham spin density functional theory to determine approximately the corrections to the energy of the different spin configurations and construct a priori the relevant energy-landscapes obtained by constrained spin density functional theory. We assess this methodology in a set of binuclear transition-metal complexes and show that it reproduces very closely the results of C-DFT. This demonstrates a proof-of-concept for this method as a potential tool for studying a number of other molecular phenomena. Additionally, routes to improving upon the limitations of this method are discussed.     

The importance of middle-range Hartree-Fock-type exchange for hybrid density functionals

Hybrid functionals are responsible for much of the utility of modern Kohn-Sham density functional theory. When rigorously applied to solid-state metallic and small band gap systems, however, the slow decay of their nonlocal Hartree-Fock-type exchange makes hybrids computationally challenging and introduces unphysical effects. This can be remedied by using a range-separated hybrid which only keeps short-range nonlocal exchange, as in the functional of Heyd et al. [J. Chem. Phys. 118, 8207 (2003)]. On the other hand, many molecular properties require full long-range nonlocal exchange, which can also be included by means of a range-separated hybrid such as the recently introduced LC-omegaPBE functional [O. A. Vydrov and G. E. Scuseria, J. Chem. Phys. 125, 234109 (2006)]. In this paper, we show that a three-range hybrid which mainly includes middle-range Hartree-Fock-type exchange and neglects long- and short-range Hartree-Fock-type exchange yields excellent accuracy for thermochemistry, barrier heights, and band gaps, emphasizing that the middle-range part of the 1/r potential seems crucial to accurately model these properties.     

Short-range exchange and correlation energy density functionals: beyond the local-density approximation

We propose approximations which go beyond the local-density approximation for the short-range exchange and correlation density functionals appearing in a multideterminantal extension of the Kohn-Sham scheme. A first approximation consists of defining locally the range of the interaction in the correlation functional. Another approximation, more conventional, is based on a gradient expansion of the short-range exchange-correlation functional. Finally, we also test a short-range generalized-gradient approximation by extending the Perdew-Burke-Ernzerhof exchange-correlation functional to short-range interactions.     

Magnetic exchange couplings in the single-molecule magnet of Mn(12)-Ac

Four types of isotropic exchange interactions of Mn(12)-Ac are obtained by using the classical Monte Carlo simulations. The equilibrium susceptibilities are well reproduced in the temperature range between 10 and 100 K. The calculated effective spin at 0.1 K coincides with the ground-state spin. Our results show that J(1) and J(2) are strong antiferromagnetic, but the magnitude of J(2) is much smaller than that of J(1). Both J(3) and J(4) favor weakly antiferromagnetic couplings. The effects of the exchange couplings on the magnetic properties and ground-state spin are investigated too. The magnetic susceptibilities below 100 K depend more on J(2) rather than on the stronger J(1). The weak exchange couplings J(3) and J(4) have significant frustration effects on the ground-state configuration.     

Time-dependent exchange-correlation current density functionals with memory

Most present applications of time-dependent density functional theory use adiabatic functionals, i.e., the effective potential at time t is determined solely by the density at the same time. This paper discusses a method that aims to go beyond this approximation, by incorporating "memory" effects: the derived exchange-correlation potential will depend not only on present densities but also on the past. In order to ensure the potentials are causal, we formulate the action on the Keldysh contour for electrons in electromagnetic fields, from which we derive suitable Kohn-Sham equations. The exchange-correlation action is now a functional of the electron density and velocity field. A specific action functional is constructed which is Galilean invariant and yields a causal exchange-correlation vector potential for the Kohn-Sham equations incorporating memory effects. We show explicitly that the net exchange-correlation Lorentz force is zero. The potential is consistent with known dynamical properties of the homogeneous electron gas (in the linear response limit).     

About the difference between density functionals defined by energy criterion and density functionals defined by density criterion: Exchange functionals

The difference between density functionals defined by energy criterion and density functionals defined by density criterion is studied for the exchange functional. It is shown that Slater potentials are exact exchange potentials in the sense that they yield the Hartree–Fock electron density if all operators are given by local expressions. © 2016 Wiley Periodicals, Inc.     

Improved density functionals for water

The accuracy of existing density functional methods for describing the noncovalent interaction energies in small water clusters is investigated by testing 25 density functionals against a data set of 28 water dimers and 8 water trimers whose structures are taken from the literature and from simulations. The most accurate functionals are found to be PW6B95 with a mean unsigned error of 0.13 kcal/mol and MPWB1K and B98 with mean unsigned errors of 0.15 kcal/mol; the best functional with no Hartree-Fock exchange is mPWLYP, which is a GGA with a mean unsigned error of 0.28 kcal/mol. In comparison, the most popular GGA functionals, PBE and BLYP, have mean unsigned errors of 0.52 and 1.03 kcal/mol, respectively. Since GGAs are very cost efficient for both condensed-phase simulations and electronic structure calculations on large systems, we optimized four new GGAs for water. The best of these, PBE1W and MPWLYP1W, have mean unsigned errors of 0.12 and 0.17 kcal/mol, respectively. These new functionals are well suited for use in condensed-phase simulations of water and ice.     

General performance of density functionals

The density functional theory (DFT) foundations date from the 1920s with the work of Thomas and Fermi, but it was after the work of Hohenberg, Kohn, and Sham in the 1960s, and particularly with the appearance of the B3LYP functional in the early 1990s, that the widespread application of DFT has become a reality. DFT is less computationally demanding than other computational methods with a similar accuracy, being able to include electron correlation in the calculations at a fraction of time of post-Hartree-Fock methodologies. In this review we provide a brief outline of the density functional theory and of the historic development of the field, focusing later on the several types of density functionals currently available, and finishing with a detailed analysis of the performance of DFT across a wide range of chemical properties and system types, reviewed from the most recent benchmarking studies, which encompass several well-established density functionals together with the most recent efforts in the field. Globally, an overall picture of the level of performance of the plethora of currently available density functionals for each chemical property is drawn, with particular attention being dedicated to the relative performance of the popular B3LYP density functional.     

Scaled density functional theory correlation functionals

We show that a simple one-parameter scaling of the dynamical correlation energy estimated by the density functional theory (DFT) correlation functionals helps increase the overall accuracy for several local and nonlocal functionals. The approach taken here has been described as the "scaled dynamical correlation" (SDC) method [Ramachandran, J. Phys. Chem. A 2006, 110, 396], and its justification is the same as that of the scaled external correlation (SEC) method of Brown and Truhlar. We examine five local and five nonlocal (hybrid) DFT functionals, the latter group including three functionals developed specifically for kinetics by the Truhlar group. The optimum scale factors are obtained by use of a set of 98 data values consisting of molecules, ions, and transition states. The optimum scale factors, found with a linear regression relationship, are found to differ from unity with a high degree of correlation in nearly every case, indicating that the deviation of calculated results from the experimental values are systematic and proportional to the dynamic correlation energy. As a consequence, the SDC scaling of dynamical correlation decreases the mean errors (signed and unsigned) by significant amounts in an overwhelming majority of cases. These results indicate that there are gains to be realized from further parametrization of several popular exchange-correlation functionals.     

Combination of pseudopotentials and density functionals

Semilocal pseudopotentials have been determined for first–row (Li to Ne), second row (Na to Ar), and third-row atoms (K, Ca). Core–valence correlation is included by adjusting the pseudopotentials to experimental energies of ions with a single valence electron. Correlation within the valence shell is taken into account by using the spin–density functional formalism. The approximations involved in this approach are tested for atomic ionization energies as well as binding energies of monohydrides and alkali diatomics, agreement with experiment is usually satisfactory, but in certain applications density functionals should be already included in the fitting of the local part of the pseudopotential. In addition, 3s/3p and 3s/2p basis sets (for first and second row, respectively), designed for use in connection with our pseudopotentials, are given; it is shown that they yield reasonable results for both SCF and correlation energies.     

Avoiding singularity problems associated with meta-GGA (generalized gradient approximation) exchange and correlation functionals containing the kinetic energy density

Convergence problems of meta-GGA (generalized gradient approximation) XC (exchange and correlation) functionals containing a self-interaction correction term are traced back to a singularity of the latter that occurs at critical points of the electron density. This is demonstrated for the bond critical point of equilibrium and stretched H2. A simple remedy is suggested that cures meta-XC functionals such as VSXC, TPSS, M05, M06, and their derivatives without extra cost.     

Long-range corrected hybrid density functionals with damped atom-atom dispersion corrections

We report re-optimization of a recently proposed long-range corrected (LC) hybrid density functional [J.-D. Chai and M. Head-Gordon, J. Chem. Phys., 2008, 128, 084106] to include empirical atom-atom dispersion corrections. The resulting functional, omegaB97X-D yields satisfactory accuracy for thermochemistry, kinetics, and non-covalent interactions. Tests show that for non-covalent systems, omegaB97X-D shows slight improvement over other empirical dispersion-corrected density functionals, while for covalent systems and kinetics it performs noticeably better. Relative to our previous functionals, such as omegaB97X, the new functional is significantly superior for non-bonded interactions, and very similar in performance for bonded interactions.     

Comparative performance of exchange and correlation density functionals in determining intermolecular interaction potentials of the methane dimer

We have calculated the interaction potentials of the methane dimer for the minimum-energy D(3d) conformation using the density functional theory (DFT) with 90 density functionals chosen from the combinations of nine exchange and 10 correlation functionals. Several hybrid functionals are also considered. While the performance of an exchange functional is related to the large reduced density gradient of the exchange enhancement factor, the correlation energy is determined by the low-density behavior of a correlation enhancement factor. Our calculations demonstrate that the correlation counterpart plays an equally important role as the exchange functional in determining the van der Waals interactions of the methane dimer. These observations can be utilized to better understand the seemingly unsystematic DFT interaction potentials for weakly bound systems.     

On the conjoint gradient correction to the Hartree—Fock kinetic and exchange energy density functionals

The kinetic and the exchange energy functionals are expressed in the form T[ρ] = C_TF∫ drρ^5/3(r)f_t(s) and K[ρ] = C_D∫ drρ^4/3(r)f_K(s), where C_TF = (3/10)(3π²)^2/3 and C_D = −(3/4)(3/π)^4/3 are the Thomas-Fermi and the Dirac coefficients, respectively, and s = |∇ρ(r)|/C_sρ^4/3(r), with C_s = 2(3π²)^1/3. These expressions are used to perform a comparison of f_T(s) and f_K(s) in terms of their generalized gradient expansion approximations. It is shown that f_κ(s) and is congruent to f_T(s) in the range characteristic of the interior regions of atoms and many solids and that the second-order gradient expansion of the kinetic energy provides a rather reasonable approximation to the generalized gradient expansion approximation of both the kinetic and the exchange energy functionals. © 1996 John Wiley & Sons, Inc.