Standard Density Functional Theory Methods Research Articles

ωB97X-3c: A composite range-separated hybrid DFT method with a molecule-optimized polarized valence double-ζ basis set.

A new composite density functional theory (DFT) method is presented. It is based on ωB97X-V as one of the best-performing density functionals for the GMTKN55 thermochemistry database and completes the family of "3c" methods toward range-separated hybrid DFT. This method is consistently available for all elements up to Rn (Z = 1-86). Its further key ingredients are a polarized valence double-ζ (vDZP) Gaussian basis set, which was fully optimized in molecular DFT calculations, in combination with large-core effective core potentials and a specially adapted D4 dispersion correction. Unlike most existing double-ζ atomic orbital sets, vDZP shows only small basis set superposition errors (BSSEs) and can compete with standard sets of triple-ζ quality. Small residual BSSE effects are efficiently absorbed by the D4 damping scheme, which overall eliminates the need for an explicit treatment or empirical corrections for BSSE. Thorough tests on a variety of thermochemistry benchmark sets show that the new composite method, dubbed ωB97X-3c, is on par with or even outperforms standard hybrid DFT methods in a quadruple-zeta basis set at a small fraction of the computational cost. Particular strengths of this method are the description of non-covalent interactions and barrier heights, for which it is among the best-performing density functionals overall.

Point defect formation energies in graphene from diffusion quantum Monte Carlo and density functional theory

Density functional theory (DFT) is widely used to study defects in monolayer graphene with a view to applications ranging from water filtration to electronics to investigation of radiation damage in graphite moderators. To assess the accuracy of DFT in such applications, we report diffusion quantum Monte Carlo (DMC) calculations of the formation energies of some common and important point defects in monolayer graphene: monovacancies, Stone-Wales defects, and silicon substitutions. We find that standard DFT methods underestimate monovacancy formation energies by around 1 eV. The disagreement between DFT and DMC is somewhat smaller for Stone-Wales defects and silicon substitutions. We examine vibrational contributions to the free energies of formation for these defects, finding that vibrational effects are non-negligible. Finally, we compare the DMC atomization energies of monolayer graphene, monolayer silicene, and bulk silicon, finding that bulk silicon is significantly more stable than monolayer silicene by 0.7522(5) eV per atom.

Range-separated hybrid functionals for mixed dimensional heterojunctions: Application to phthalocyanines/MoS2

We analyze the electronic structure and level alignment of transition-metal phthalocyanine (MPc) molecules adsorbed on two-dimensional MoS2 employing density functional theory (DFT) calculations. We develop a procedure for multi-objective optimal tuning of parameters of range-separated hybrid functionals in these mixed-dimensional systems. Using this procedure, which leads to the asymptotically correct exchange–correlation potential between molecules and two-dimensional materials, we obtain electronic structures consistent with experimental photoemission results for both energy level alignment and electronic bandgaps, representing a significant advance compared to standard DFT methods. We elucidate the MoS2 valence resonance with the transition-metal phthalocyanine non-frontier 3d orbitals and its dependence on the transition-metal atomic number. Based on our calculations, we derive parameter-free model self-energy corrections that quantitatively account for the effects of the heterogeneous dielectric environment on the electronic structure of these mixed-dimensional heterojunctions.

Influence of Surface Passivation on Indium Arsenide Nanowire Band Gap Energies

The interplay between surface chemistry and quantum confinement on the band gap energies of indium arsenide (InAs) nanowires is investigated by first principle computations as the surface-to-volume ratio increases with decreasing cross section. Electronic band structures are presented as determined by both density functional and hybrid density functional theory (DFT) calculations; the latter are used to provide improved band gap energy estimates over those from standard approximate DFT methods. Different monovalent chemical species with varying electron affinity are used to eliminate surface states to enable direct comparison between surface chemistry and quantum confinement. The influence of these effects on energy band gaps and electron effective masses is highlighted. It is found that many desirable properties in terms of electronic properties and the elimination of surface states for nanoscale field effect transistors fabricated using [100]-oriented InAs can be achieved.

Martensitic transition in Fe via Bain path at finite temperatures: A comprehensive first-principles study

Due to the magnetic nature of Fe, various phenomena during structural transitions in Fe-based alloys, including martensitic transition (MT), cannot be accurately interpreted even by the state-of-the-art first-principles methods based on density functional theory (DFT), which is mostly limited to zero Kelvin. In the present work, thermodynamics and kinetics of Bain transition in pure Fe, i.e. the simplest model for fcc/bcc transition, are studied by analyzing the minimum energy path (MEP) at finite temperatures. Energies of various lattices and magnetic configurations at ground state are calculated by the standard DFT methods, which are further fitted by the Birch-Murnaghan equation of state (EOS) to obtain the ground state properties. By combing the quasi-harmonic Debye-Grüneisen model with the magnetic partition function approach (PFA), the Helmholtz energies for the body-centered tetragonal lattices with fixed c/a ratio and volume (V) are calculated, where the PFA accounts for the fluctuations of the magnetic configurations. Using free energy surface in the {c/a, V} space, the MEP is searched and a correlation between driving force and energy barrier for the fcc/bcc transition is observed. Further combined with previous heterogeneous nucleation models for MT, the correlation shown in the present work is found to be ubiquitous of MTs, and thus governing the formation of martensite.

Influence of Hubbard U Parameter in Simulating Adsorption and Reactivity on CuO: Combined Theoretical and Experimental Study

Transition metal oxides are an important class of catalytic materials widely used in the chemical manufacturing and processing industry, owing to their low cost, high surface area, low toxicity, and easily tunable surface and structural properties. For these strongly correlated transition metal oxides, standard approximations in the density functional theory (DFT) exchange-correlation functional fail to describe the electron localization accurately due to the intrinsic errors arising from electron self-interactions. DFT+U method is a widely used extension of DFT, where the Hubbard U term is an onsite potential which puts a penalty on electron delocalization, successfully describing such systems at only slightly higher computational cost than standard DFT methods. The U-value is usually chosen based on its accuracy in reproducing bulk properties like lattice parameters and band structure. However, chemical reactions on transition metal oxide surfaces involve complex surface–adsorbate interactions, and usin...

Comparison of standard DFT and Hubbard-DFT methods in structural and electronic properties of TiO2 polymorphs and H-titanate ultrathin sheets for DSSC application

Abstract The structural and electronic properties of several TiO2 polymorphs and hydrogen titanate surfaces are modeled and studied by density functional theory (DFT). By implementing the Hubbard parameter “U” in our calculations more realistic results of electronic properties are obtained, paying with a small deviation in geometric optimization. Lattice parameters difference is found to be less than 6.2%, as well as some changes in surface energy are found, but the reactivity tendency of surfaces is maintained. Calculated work function is less energetic for faces (001) for anatase, (101) for rutile and (001) for TiO2-B.

Strong correlation in surface chemistry

Density functional theory (DFT) simulations of surface chemistry have emerged as a valuable complement to experiment. However, standard DFT methods do not always accurately model the ‘strong’ electron correlation effects seen in stretched covalent bonds. Such systems’ ground-state wavefunctions are not well-described by single molecular orbital configurations. I review some of the challenges of strong correlation, and some methods used to simulate it in surface chemistry. I also use the electron delocalisation range function EDR(), which quantifies the extent to which electrons at point delocalise over distance d, to highlight how a nearby metal cluster affects strong correlation in a dissociating covalent bond.

Understanding Why Sodium Intercalated Graphite Is Unstable

Sodium-ion batteries offer a potential solution to the problem of limited lithium resources, although developing compatible, high-performance cathode, anode, and electrolyte materials for such devices presents a more difficult challenge. In conventional lithium ion batteries, graphite is used as the anode, but it has been known for more than 40 years from experiment that, unlike the other alkali metals (Li, K, Rb, and Cs), intercalation of Na into graphite is thermodynamically unfavorable, so that modification of the anode material or addition of solvating species is needed if intercalation is to occur. First-principles calculations by Nobuhara et al. [2] and Wang et al. [3] showed that even at low Na densities NaC x compounds are energetically unstable. However, the underlying mechanisms responsible for the endothermic formation energies are still uncertain. To investigate this phenomenon in more detail we performed first-principles calculations of the intercalation behavior of alkali metals into graphite. A van der Waals correction was added to the standard density functional theory methods to better reproduce the laminated structures of graphite intercalated compounds. Calculated formation energies of intercalated systems XC6 (X = Li, Na, K, Rb, or Cs) are shown in Fig. 1(a). As the atomic number decreases from Cs to Na, the magnitude of the formation energy decreases, becoming positive (endothermic) in the case of Na. This tendency can be explained in terms of ionic bonding energies, which decrease as the atomic number of the alkali metal decreases because electronegativity increases as the alkali metal group is ascended. The reason for the negative formation energy in the case of LiC6 can be understood by examining the electron density and density of states, plotted in Fig. 1(b), which show greater covalent bonding between Li and C than the other alkali metals.

Resolution of the Band Gap Prediction Problem for Materials Design

An important property with any new material is the band gap. Standard density functional theory methods grossly underestimate band gaps. This is known as the band gap problem. Here, we show that the hybrid B3PW91 density functional returns band gaps with a mean absolute deviation (MAD) from experiment of 0.22 eV over 64 insulators with gaps spanning a factor of 500 from 0.014 to 7 eV. The MAD is 0.28 eV over 70 compounds with gaps up to 14.2 eV, with a mean error of -0.03 eV. To benchmark the quality of the hybrid method, we compared the hybrid method to the rigorous GW many-body perturbation theory method. Surprisingly, the MAD for B3PW91 is about 1.5 times smaller than the MAD for GW. Furthermore, B3PW91 is 3-4 orders of magnitude faster computationally. Hence, B3PW91 is a practical tool for predicting band gaps of materials before they are synthesized and represents a solution to the band gap prediction problem.

Construction and application of a new dual-hybrid random phase approximation.

The direct random phase approximation (dRPA) combined with Kohn-Sham reference orbitals is among the most promising tools in computational chemistry and applicable in many areas of chemistry and physics. The reason for this is that it scales as N(4) with the system size, which is a considerable advantage over the accurate ab initio wave function methods like standard coupled-cluster. dRPA also yields a considerably more accurate description of thermodynamic and electronic properties than standard density-functional theory methods. It is also able to describe strong static electron correlation effects even in large systems with a small or vanishing band gap missed by common single-reference methods. However, dRPA has several flaws due to its self-correlation error. In order to obtain accurate and precise reaction energies, barriers and noncovalent intra- and intermolecular interactions, we construct a new dual-hybrid dRPA (hybridization of exact and semilocal exchange in both the energy and the orbitals) and test the performance of this new functional on isogyric, isodesmic, hypohomodesmotic, homodesmotic, and hyperhomodesmotic reaction classes. We also use a test set of 14 Diels-Alder reactions, six atomization energies (AE6), 38 hydrocarbon atomization energies, and 100 reaction barrier heights (DBH24, HT-BH38, and NHT-BH38). For noncovalent complexes, we use the NCCE31 and S22 test sets. To test the intramolecular interactions, we use a set of alkane, cysteine, phenylalanine-glycine-glycine tripeptide, and monosaccharide conformers. We also discuss the delocalization and static correlation errors. We show that a universally accurate description of chemical properties can be provided by a large, 75% exact exchange mixing both in the calculation of the reference orbitals and the final energy.

Determination of structure and properties of molecular crystals from first principles.

CONSPECTUS: Until recently, it had been impossible to predict structures of molecular crystals just from the knowledge of the chemical formula for the constituent molecule(s). A solution of this problem has been achieved using intermolecular force fields computed from first principles. These fields were developed by calculating interaction energies of molecular dimers and trimers using an ab initio method called symmetry-adapted perturbation theory (SAPT) based on density-functional theory (DFT) description of monomers [SAPT(DFT)]. For clusters containing up to a dozen or so atoms, interaction energies computed using SAPT(DFT) are comparable in accuracy to the results of the best wave function-based methods, whereas the former approach can be applied to systems an order of magnitude larger than the latter. In fact, for monomers with a couple dozen atoms, SAPT(DFT) is about equally time-consuming as the supermolecular DFT approach. To develop a force field, SAPT(DFT) calculations are performed for a large number of dimer and possibly also trimer configurations (grid points in intermolecular coordinates), and the interaction energies are then fitted by analytic functions. The resulting force fields can be used to determine crystal structures and properties by applying them in molecular packing, lattice energy minimization, and molecular dynamics calculations. In this way, some of the first successful determinations of crystal structures were achieved from first principles, with crystal densities and lattice parameters agreeing with experimental values to within about 1%. Crystal properties obtained using similar procedures but empirical force fields fitted to crystal data have typical errors of several percent due to low sensitivity of empirical fits to interactions beyond those of the nearest neighbors. The first-principles approach has additional advantages over the empirical approach for notional crystals and cocrystals since empirical force fields can only be extrapolated to such cases. As an alternative to applying SAPT(DFT) in crystal structure calculations, one can use supermolecular DFT interaction energies combined with scaled dispersion energies computed from simple atom-atom functions, that is, use the so-called DFT+D approach. Whereas the standard DFT methods fail for intermolecular interactions, DFT+D performs reasonably well since the dispersion correction is used not only to provide the missing dispersion contribution but also to fix other deficiencies of DFT. The latter cancellation of errors is unphysical and can be avoided by applying the so-called dispersionless density functional, dlDF. In this case, the dispersion energies are added without any scaling. The dlDF+D method is also one of the best performing DFT+D methods. The SAPT(DFT)-based approach has been applied so far only to crystals with rigid monomers. It can be extended to partly flexible monomers, that is, to monomers with only a few internal coordinates allowed to vary. However, the costs will increase relative to rigid monomer cases since the number of grid points increases exponentially with the number of dimensions. One way around this problem is to construct force fields with approximate couplings between inter- and intramonomer degrees of freedom. Another way is to calculate interaction energies (and possibly forces) "on the fly", i.e., in each step of lattice energy minimization procedure. Such an approach would be prohibitively expensive if it replaced analytic force fields at all stages of the crystal predictions procedure, but it can be used to optimize a few dozen candidate structures determined by other methods.

On asymptotic behavior of density functional theory

The performance of several previously proposed as well as of some novel approaches for correcting the asymptotic behavior of electron densities in density functional theory (DFT) is evaluated. The comparisons are made for molecular properties that are known to be sensitive to the asymptotic behavior of densities such as polarizabilities, energies of excitations to Rydberg states, and interaction energies (computed using symmetry-adapted perturbation theory). We find that whereas DFT-based methods without asymptotic corrections predict the investigated properties with errors often as large as a dozen or so percent relative to the best known values, the best performing asymptotically corrected hybrid functionals reduce these errors to below 2%. In many cases, the errors are just a fraction of one percent and in a few cases DFT reproduces benchmark values to all figures. These errors are also very close to those produced by the coupled-cluster methods at the highest levels of electron excitations applicable in practice. Such performance is unprecedented for any applications of DFT and is due to high sensitivity of the investigated properties to tails of the electron densities. Range-separated hybrid (RSH) functionals are also examined and found to perform comparably to the asymptotically corrected hybrid functionals for excitation energies and only somewhat worse for polarizabilities. However, very surprisingly, RSH functionals fail completely in interaction energy calculations. We explain the latter problem by showing that, despite some expectations in the literature reflected by the alternative name, long-range corrected, used for the RSH functionals, these methods give densities that are not better in the asymptotic region than those produced by asymptotically uncorrected standard DFT methods. We further show that this failure can be corrected by cutting, displacing, and splicing the exchange-correlation potentials of RSH methods such that these potentials approach the correct value at infinity.

Assessment of a Nonlocal Correction Scheme to Semilocal Density Functional Theory Methods.

A nonlocal correction method to (semi)local density-functional theory (DFT) methods is derived that is based on a partitioning of the correlation-energy density into atom-atom contributions. Nonlocal interaction contributions, which are absent in standard DFT methods, are introduced in this method by using atomic weight functions that do not vanish exponentially as the atomic densities but with the inverse sixth power of the atomic distances. The parameters contained in these weight functions were fitted to reproduce intermolecular interaction energies for a range a small dimer systems. The new functional has then been tested both for intermolecular interactions, using the S22, S66 × 8, and IonHB databases from Hobza et al., and for other thermodynamical properties using a subset of 14 databases of the GMTKN30 database of Grimme et al. It is found that for intermolecular interactions the accuracy of the method is often higher than with standard DFT+D methods while for other properties, such as reaction energies or relative conformation energies of medium sized organic molecules, the accuracy is similar to hybrid-DFT+D methods. The nonlocal correction method has been tested also to predict the interaction energy of the water-graphene system yielding an estimated interaction energy of -2.87 kcal/mol, which is in line with previous theoretical investigations.

Multinuclear solid-state NMR of square-planar platinum complexes — Cisplatin and related systems

Multinuclear solid-state nuclear magnetic resonance (SSNMR) experiments have been performed on cisplatin and four related square-planar compounds. The wideband uniform rate smooth truncation – Carr–Purcell–Meiboom–Gill (WURST–CPMG) pulse sequence was utilized in NMR experiments to acquire 195Pt, 14N, and 35Cl ultra-wideline NMR spectra of high quality. Standard Hahn-echo and magic-angle spinning 195Pt NMR experiments are also performed to refine extracted chemical shielding (CS) tensor parameters. Platinum magnetic shielding (MS) tensor orientations are calculated using both plane-wave density functional theory (DFT) and standard DFT methods. The tensor orientations are shown to be highly constrained by molecular symmetry elements, but also influenced to some degree by intermolecular interactions. 14N WURST–CPMG experiments were performed on three compounds and electric field gradient (EFG) parameters (the quadrupolar coupling constant, CQ, and the asymmetry parameter, ηQ) are reported. First principles calculations of the 14N EFG tensor parameters and orientations and affirm their dependence on the local hydrogen bonding environment. 35Cl WURST–CPMG experiments on cisplatin and transplatin are reported, using two different static magnetic fields to extract EFG and CS tensor parameters, and 35Cl EFG tensor magnitudes and orientations are predicted using first principles calculations. Transverse (T2) relaxation data for all nuclei are used to investigate heteronuclear dipolar relaxation mechanisms, as well as the nature of the local hydrogen bonding environments.

First-Principles-Based Dispersion Augmented Density Functional Theory: From Molecules to Crystals

Standard implementations of density functional theory (DFT) describe well strongly bound molecules and solids but fail to describe long-range van der Waals attractions. We propose here first-principles-based augmentation to DFT that leads to the proper long-range 1/R6 attraction of the London dispersion while leading to low gradients (small forces) at normal valence distances so that it preserves the accurate geometries and thermochemistry of standard DFT methods. The DFT-low gradient (DFT-lg) formula differs from previous DFT-D methods by using a purely attractive dispersion correction while not affecting valence bond distances. We demonstrate here that the DFT-lg model leads to good descriptions for graphite, benzene, naphthalene, and anthracene crystals, using just three parameters fitted to reproduce the full potential curves of high-level ab initio quantum mechanics [CCSD(T)] on gas-phase benzene dimers. The additional computational costs for this DFT-lg formalism are negligible.

Approximate quantum mechanical method for describing excitations and related properties of finite single-walled carbon nanotubes

Optical properties of three kinds of zigzag (5,0), (13,0), and (9,0) single-walled carbon nanotubes (SWCNTs) are studied using an approximate quantum mechanical method named complete neglect of differential overlap, which distinguishes basis atomic orbitals with different azimuthal $l$ quantum numbers (CNDOL). This method models the electron energy transitions and excited state charge distributions through a configuration interaction of singly (CIS) excited determinants allowing the direct understanding of properties related with the total electronic wave function of nanoscopic systems, projecting a reliable quantum mechanical understanding to real life objects. The finite SWCNT's structures were obtained by replicating the unit cells of periodic SWCNTs and saturating the edge dangling bonds with hydrogens. The unit cell was previously relaxed using standard density functional theory methods. The behavior of these SWCNTs were interpreted in the framework of the CNDOL scheme by increasing the lengths of the tubes above 3 nm. As the nanotubes grow in length, the position of excited states for each SWCNT evolve differently: in contrast with (9,0) SWCNT, which exhibits favorable conditions for photoexcitation, the (13,0) and (5,0) SWCNTs do not show a lowering of the lowest excited states. This behavior is discussed by taking into account electron---electron interactions as considered in the framework of the CIS procedure. Furthermore, the (13,0) and (5,0) SWCNTs present forbidden transitions for the lowest excitations and its first dipole-allowed transitions are at 0.9--1.0 and 1.4--1.6 eV, respectively. In contrast, (9,0) SWCNT allows excitations by photon at less than 0.4 eV as the length of the nanotube tends to infinite. Excitons appear more bounded, energetically and spatially, in the (13,0) than in the (9,0) and (5,0) SWCNTs.

Accurate dispersion interactions from standard density-functional theory methods with small basis sets

B971, PBE and PBE1 density functionals with 6-31G(d) basis sets are shown to accurately describe the binding in dispersion bound dimers. This is achieved through the use of dispersion-correcting potentials (DCPs) in conjunction with counterpoise corrections. DCPs resemble and are applied like conventional effective core potentials that can be used with most computational chemistry programs without code modification. Rather, DCPs are implemented by simple appendage to the input files for these types of programs. Binding energies are predicted to within ca. 11% and monomer separations to within ca. 0.06 A of high-level wavefunction data using B971/6-31G(d)-DCP. Similar results are obtained for PBE and PBE1 with the 6-31G(d) basis sets and DCPs. Although results found using the 3-21G(d) are not as impressive, they never-the-less show promise as a means of initial study for a wide variety of dimers, including those dominated by dispersion, hydrogen-bonding and a mixture of interactions. Notable improvement is found in comparison to M06-2X/6-31G(d) data, e.g., mean absolute deviations for the S22-set of dimers of ca. 13.6 and 16.5% for B971/6-31G(d)-DCP and M06-2X, respectively. However, it should be pointed out that the latter data were obtained using a larger integration grid size since a smaller grid results in different binding energies and geometries for simple dispersion-bound dimers such as methane and ethene.

Corrections to the density-functional theory electronic spectrum: copper phthalocyanine

A method for improving the electronic spectrum of standard Density-Functional Theory (DFT) calculations (i.e., LDA or GGA approximations) is presented, and its application is discussed for the case of the copper phthalocyanine (CuPc) molecule. The method is based on a treatment of exchange and correlation in a many-body Hamiltonian, and it leads to easy-to-evaluate corrections to the DFT eigenvalues. Self-interaction is largely corrected, so that the modified energy levels do not suffer from spurious crossings, as often encountered for CuPc in DFT, and they remedy the standard underestimation of the gap. As a specific example we study the sequence and position of the CuPc molecular orbitals, which are wrongly calculated by standard DFT, and show that they are correctly reproduced after our corrections are included. The suggested method is fast and simple and, while not as accurate as hybrid or semiempirical functionals for molecular levels, it can be easily applied to any local-orbital DFT approach, improving on several important limitations of standard DFT methods.

Accurate treatment of van der Waals interactions using standard density functional theory methods with effective core-type potentials: Application to carbon-containing dimers

Carbon atom-centered effective core-type potentials were developed to compensate for the incorrect long-range behaviour of density functionals and for limitations in basis set in treating van der Waals binding in dimers containing carbon atoms. Three method-specific potentials (PBE, PW91, B971) were designed. B971/6-311+G(2d,2p) was found to give binding energies for 22 dimers with mean absolute deviation from high-level results of 0.25 kcal/mol (13.0%). The potentials add little to calculation times and are useable with most computational chemistry programs. It is expected that such deficiencies in B86-based functionals can be largely corrected in the manner presented.