Becke's Exchange Research Articles

Assessment of the Handy–Cohen optimized exchange density functional for organic reactions

We have investigated the performance of the new optimized exchange functional (OPTX) developed by Handy and Cohen [Mol. Phys. 99, 403 (2001)] for predicting geometries, heats of reaction, and barrier heights for twelve organic reactions (six closed-shell and six radical). OPTX has been used in conjunction with, among others, the well-known Lee–Yang–Parr (LYP) correlational functional to form two new functionals, OLYP and O3LYP. These are similar to the well-established BLYP and B3LYP functionals, respectively, with OPTX replacing the standard Becke exchange functional, B88. Our results strongly support claims made by their developers that OLYP is superior to BLYP, and essentially renders it obsolete. The computed OLYP heats of reaction, barrier heights, and even molecular geometries (with larger basis sets), are comparable with, if not better than, the corresponding B3LYP values. The O3LYP functional is overall better than B3LYP, albeit not by much. Both OLYP and O3LYP are among the best functionals currently available; the performance of OLYP in particular is noteworthy given that this functional includes no exact exchange.

Implicit and Explicit Coverage of Multi-reference Effects by Density Functional Theory

Multi-reference effects can be covered by density functional theory (DFT) either implicitly via the exchange-correlation functional or explicitly via the form of the Kohn-Sham wave function. With the help of the exchange hole it is shown that the self-interaction error of the exchange functional will mimic long-range electron correlation effects if restricted Kohn-Sham theory is used. Functionals based on Slater or Becke exchange have a relatively large self-interaction error and, therefore, lead to a relatively large implicit coverage of long-range correlation, which, because of the possibility of doublecounting of electron correlation, has to be considered when using these functionals in connection with two- or multi-configurational descriptions based on ensemble DFT methods such as REKS (spin-Restricted Ensemble-referenced KS-DFT). Arguments are given that a REKS description of a multireference problem avoids a double-counting of long-range correlation effects, in particular as in this situation the self-interaction error of the exchange functional simulates more short- rather than longrange correlation effects. There is, however, no guarantee that the short-range effects are not doublecounted, namely once via the exchange and once via the correlation functional. Therefore, one should use hybrid functionals such as B3LYP in connection with multi-reference DFT methods because for hybrid functionals the self-interaction error and by this the implicit coverage of long(short)-range correlation effects is reduced due to the admixture of exact exchange. This rule applies also to broken-symmetry UDFT, which performs better with hybrid rather than GGA functionals. A way of avoiding the implicit coverage of multi-reference effects is given by the combination of wave function theory and DFT methods. The advantages and disadvantages of CAS-DFT are discussed and it is shown that an effective reduction of a double-counting of correlation effects is possible within this method.

Density functional theory study of vibrational spectra of acridine and phenazine

Density functional theory (DFT) calculations using Becke's exchange in conjunction with Lee–Yang–Parr's correlation functionals (BLYP), Becke's three-parameter hybrid DFT/HF method using Lee–Yang–Parr's correlation functionals (B3LYP) and ab initio Hartree–Fock (HF) method have been carried out to investigate the structure and vibrational spectra of acridine and phenazine. Structural parameters obtained by B3LYP/6-31G* geometry optimization are in good agreement with available experimental data. The raw BLYP non-CH stretching frequencies approximate the experimental results much better than the HF results with the mean absolute deviation about 16 cm −1. The scaled B3LYP frequencies are more reliable than that of the BLYP and HF methods with the mean absolute deviation about 17 cm −1. On the basis of the comparison between calculated and experimental results, assignments of fundamental vibrational modes are examined. Also the structure and vibrational frequencies are compared with those of anthracene, pyridine and benzene to study the similarities and differences.

Some thoughts about the stability and reliability of commonly used exchange?correlation functionals ? coverage of dynamic and nondynamic correlation effects

The self-interaction error (SIE) of commonly used density functional theory (DFT) exchange functionals mimics long-range (nondynamic) pair correlation effects in an unspecified way. Slater exchange suffers from a larger SIE and, therefore, covers more nondynamic correlation effects than Becke exchange, which is the reason why exchange–correlation (XC) functionals based on Slater exchange lead to stabler restricted DFT solutions than those based on Becke exchange. However, the stability of an XC functional does not guarantee higher accuracy. On the contrary, if system-specific nondynamic correlation effects have to be introduced via the form of the wave function, these will be suppressed by nondynamic correlation effects already covered by the exchange functional. Hybrid functionals suffer less from the SIE and, therefore, cover a smaller number of nondynamic electron correlation effects. Accordingly, they are better suited when nondynamic electron correlation has to be introduced by the form of the wave function. It is shown that, for example, broken-symmetry unrestricted DFT calculations are more accurate when carried out with B3LYP than BLYP contrary to claims made in the literature.

Further developments in the local-orbital density-functional-theory tight-binding method

Improvements to the Sankey-Niklewaki method [O. F. Sankey and D. J. Niklewski, Phys. Rev. B 40, 3979 (1989)] for computing total energies and forces, within an ab initio tight-binding formalism, are presented here. In particular, the improved method (called FIREBALL) uses the separable pseudopotential (Hamann or Troullier) and goes beyond the minimal ${\mathrm{sp}}^{2}$ basis set of the Sankey-Niklewski method, allowing for double numerical basis sets with the addition of polarization orbitals and d orbitals to the basis set. A major improvement includes the use of more complex exchange-correlation functionals, such as Becke exchange with the Lee-Yang-Parr correlation. Results for Cu and GaN band structures using d orbitals within the improved method are reported; the results for GaN are greatly improved compared to the minimal basis results. Finally, to demonstrate the flexibility of the method, results for the ${\mathrm{H}}_{2}\mathrm{O}$ dimer system and the energetics of a gas-phase octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine molecule are reported.

Theoretical investigations of the interaction of silver trimer with ethylene molecule

The interaction of silver trimer with ethylene molecule is investigated within the density functional framework using the Becke exchange and Perdew–Wang (91) correlation functionals. Structural forms and electronic properties (binding energy and ionization potential) of the cluster–molecule complex are determined. Ag 3–C 2H 4 forms two isomers. In agreement with the experimental results, the computed ionization potential of the more stable isomer is lower than that of the bare Ag 3 cluster.

Theoretical study of first-row transition metal oxide cations

The equilibrium geometries, dissociation energies, and electronic structures of the ground and low-lying excited states for the first-row transition metal oxide cations, MO+ (M=Sc to Zn), have been studied using the multireference singles and doubles configuration interaction (MR–SDCI) and the multireference second-order Møller–Plesset methods. To investigate the applicability of the density functional theory (DFT) to an electronic structure system with a multiconfigurational character, the Becke exchange functional with the Lee–Yang–Parr correlation functional, the Becke exchange functional with the one-parameter progressive correlation functional (BOP), and the Becke three-parameter hybrid exchange functional with the Lee–Yang–Parr correlation functional (B3LYP) methods have also been applied. The DFT predicts the ground state M–O bond lengths in good agreement with the multireference-based methods except for MnO+ and CuO+, which have a multiconfigurational electronic structure. With respect to the dissociation energies, the B3LYP results are in good agreement with the multireference-based methods, while the DFT with pure functionals overestimates the energetics by about 20 kcal/mol compared to the MR–SDCI method.

Tautomeric equilibria of 2- and 4-thiouracil in gas phase and in solvent: A density functional study

The relative stabilities of the five favored tautomers of 2- and 4-thiouracil in gas phase and in water solution were determined by density functional theory employing the Becke, Lee, Yang, and Parr (B3LYP) exchange–correlation potential and the three 6-31G(d,p), 6-311++G(d,p), and triple-zeta valence (TZVP) basis sets. Zero-point vibrational corrections were also computed. Bulk solvent effects were studied in the framework of the self-consistent reaction field approach by the polarizable continuum model. All calculations indicate that the most stable tautomer for both species, in the gas phase as well as in solution, has the oxo-thione form, in full agreement with the previous ab initio and experimental studies. The tautomeric stability orders obtained in the aqueous solution are sensibly different from that in the gas phase. At B3LYP/6-311++G(d,p) level in the gas phase, the following orders of stability for 2- and 4-thiouracil tautomers were observed, respectively: S2U1>S2U2>S2U4>S2U5>S2U3 and S4U1>S4U2>S4U3>S4U4>S4U5. The corresponding trends in the aqueous phase are S2U1>S2U3>S2U2>S2U5>S2U4 and S4U1>S4U2>S4U3>S4U5>S4U4. On the basis of the computed energy differences we can hypothesize that only the oxo-thione forms of 2- and 4-thiouracil should exist in the gas phase and in water solution. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 44–52, 2001

New τ-dependent correlation functional combined with a modified Becke exchange

A new correlation functional is derived within the Kohn–Sham (KS) Density Functional Theory (DFT) involving the electron kinetic energy density τ and the Laplacian of the electron density as key nonlocal variables. The derivation is based on a direct resolution of the adiabatic connection formula and using an analogy with the local thermodynamic approach in DFT, following the Lap3 theory developed previously. Compared to the latter, the new functional involves higher order τ-dependent energy terms in a form suggesting a possible resummation procedure that could be used for further development. It is combined with the nonlocal exchange functional of Becke, by modifying the latter in an empirical fashion to achieve better synchronization between the two energy components. The resulting exchange-correlation scheme (named “Bmτ1”) is validated on several test systems known as difficult for DFT, at least at the Local Spin Density and Generalized Gradient Approximation levels. The recent nonempirical hybrid scheme PBE1PBE (“PBE0”) is included in the comparative tests as a parameter-free benchmark for the hybrid HF-KS DFT approach. Improved results for relative energies, activation barriers and equilibrium geometries are obtained with the Bmτ1 functional, particularly concerning aromatic compounds, systems with weak hydrogen bonds, proton transfer processes and transition-metal carbonyls.

Accurate Density-Functional Calculation of Core-Electron Binding Energies of Some Substituted Benzenes

Abstract The core electron binding energies (CEBE's) of benzene, seven monosubstituted benzenes (Ph-X) and one disubstituted benzene (p-NH2-C6H4-NO2) were calculated using density-functional theory (DFT). The unrestricted generalized transition-state (uGTS) model was employed. The DeMon DFT program with a combined functional of Becke's exchange (B88) with Perdew's correlation (P86) was used. The average absolute deviation of the calculated CEBE's of the title compounds was 0.3 eV when the cc-pVDZ basis set was used. The ``CEBE shift'' of the ring carbon in Ph-X was calculated while taking the CEBE on the ring carbon in Ph-H as a reference. The thus-calculated CEBE shifts agree with experiment within the value of the average absolute deviation, 0.1 eV. The signs and quantitative numerical values of the CEBE shifts are very close to the corresponding Hammett σ constants.

Kinetic and potential energy of isoelectronic atomic ions from density functional theory compared with exact values

The performance of the Fock exchange and the two most common Becke exchange functionals, B88 and B3, in combination with Perdew and Wang's correlation functional (PW91) has been studied for highly charged atomic ions. The study focuses on the accuracy of the components of total energy: kinetic and potential energy. It is found that Becke's exchange functionals introduce a significant error in the kinetic and nuclear attraction energy for most ions studied, and the use of these exchange functionals might lead to inaccurate momentum distributions when their Kohn-Sham orbitals are transformed to momentum space.

Transition Regions in the Cope Rearrangement of 1,5-Hexadiene and Its Cyano Derivatives

Substituent effects on the mechanism of the Cope rearrangement were studied by computing C2h (Cs) cuts through the potential energy surface (PES) for the reaction of 1,5-hexadiene and its di-, tri-, and tetracyano derivatives at the (U)B3LYP/6-31G* and (U)BPW91/6-31G* levels. The stabilization of substituted structures along the cuts is discussed in terms of the energies of isodesmic formation from allyl radicals and acetonitrile molecules. Cyano groups at C1, C3, and C5 provide a nearly additive stabilization of each point along the Cs cut even though their influence on the geometry is competitive. Evaluation of the density of effectively unpaired electrons at various geometries indicates that the radical character of a transition state (TS) is not altered by radical stabilizing substituents as such but depends solely on the interallylic bond length. Although the UB3LYP diyl intermediate for the parent compound is plausible when compared to the lowest triplet PES, neither MRPT2 at the UB3LYP geometries nor UDFT with the original Becke exchange (B) predict any intermediates. Similarly, for each of the three substituted compounds, the most believable (U)BPW91 model gives either one TS or one intermediate. Derivative 1,5-hexadienes with cyano groups at C1, C3, C5 or at C1, C3, C4, C6 rearrange by the same mechanism as the parent, i.e., through a mostly aromatic TS. However, 2,5-dicyano-1,5-hexadiene is found to react through an intermediate. It is suggested that B- rather than B3-type functionals should be used for sigmatropic rearrangements to avoid spurious stationary points.

Structures and vibrational frequencies of pyruvic acid: density functional theory study

Density functional theory BLYP (using Becke's exchange and Lee–Yang–Parr's correlation functionals ), ab initio Hartree–Fock (HF) and hybrid DFT/HF B3LYP calculations were carried out to study the structure and vibrational spectra of pyruvic acid. Molecular conformation calculations were made for two possible conformers (eclipsed and staggered of C4O8 bond with respect to the methyl group) of the compound. Calculated results show that the stable conformer of pyruvic acid is the eclipsed one. The raw BLYP and B3LYP frequencies approximate the experimental results much better than the Hartree–Fock results, the scaled B3LYP results are more reliable than that of the BLYP and HF methods with the mean absolute deviation about 12.3 cm −1. On the basis of the comparison between calculated and experimental results, assignments of fundamental vibrational modes are examined.

Density functional study of FeO2, FeO, and FeO

The lowest energy structures of FeO2 were determined by means of density functional theory techniques as implemented in the program DGauss 3.0.1. The calculations performed were of the all-electron type using two levels of theory, namely the local spin density approximation with the use of the Vosko–Wilk–Nusair (VWN) functional and the generalized gradient approximation (GGA) in the form of the Becke (1988) exchange and Perdew (1986) correlation functionals. Results were visualized by means of the program UniChem. Bond distances and angles as well as total energies were calculated for several states of the moieties: Fe(O)2, C2v; Fe(O)2, D∞h; Fe(η2-O2), C2v; Fe(η1-O2), Cs; and Fe(η1-O2), C∞v. Molecular orbital and harmonic vibrational analyses were carried out for these species, in addition to Mulliken population analyses. Singly positive and negative charged species were also considered and fully geometry optimized in a self-consistent field (SCF) gradient method. Accurate ionization potentials and electron affinities (both vertical, v, and adiabatic, a, determinations) were thus able to be computed. The results show the following for the ground state (GS) Fe(O)2, C2v, M=3: ∠OFeO=138.1° (133.6°) [values in parentheses are for local spin density approximation (LSDA)–VWN, while the others are at the GGA-B88/P86 level]. Here, Re Fe–O=1.60 Å (1.57 Å), ET=−1414.2064 au (−1,410.5047 au), EAa=2.47 (2.60) eV, IPa=10.6 (10.5) eV, EAv=2.41 (2.20) eV, IPv=10.67 (10.63) eV, and EAexp=2.349 eV (in agreement with related studies). In the GS the dioxygen molecule is found to be dissociated, compared to those states which have coordination modes where the O2 molecule formally persists. A 3d4sp configuration for the iron atom is found to be especially relevant in Fe–O bond formation. The iron–oxygen and oxygen–oxygen bonds involved are characterized. A direct relationship is observed between these electronic and structural properties, influencing also the total energy for a given molecule. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 307–319, 2000

DFT force fields, vibrational spectra and potential functions for hindered internal rotation of CF 3CH 2CH 2Cl and CCl 3CH 2CH 2Cl

The fully optimized geometries and vibrational frequencies of trans- and gauche-conformers of CF 3CH 2CH 2Cl and CCl 3CH 2CH 2Cl were obtained at the B3LYP and BLYP levels of hybrid functional theory DFT with the different basis sets. Corresponding ab initio calculations were also performed at the HF and MP2 levels of theory. DFT calculations using Becke's exchange and Lee–Yang–Parr correlation functional give a correct picture of the relative stability of different conformers and present rather good agreement of calculated and experimental frequencies. On this basis, the regularized quantum mechanical force field (RQMFF) for both trans- and gauche-conformers of CF 3CH 2CH 2Cl and CCl 3CH 2CH 2Cl were obtained.

Electron affinities of the oxides of aluminum, silicon, phosphorus, sulfur, and chlorine

The adiabatic electron affinities of five second row atoms (Al, Si, P, S, Cl) and their monoxides and dioxides were determined using six different density functional or hybrid Hartree–Fock/density functional methods. The 15 species selected form a convenient closed set for which reliable experimental electron affinities exist for 13 of the species. Zero-point vibrational energy corrected electron affinities are also reported. Equilibrium geometries and vibrational frequencies were determined with each density functional method. The method based on the Becke exchange functional and the Lee–Yang–Parr correlation (BLYP) functional reproduced the experimental electron affinities most accurately, having an average absolute error of 0.15 eV. Using this functional, the electron affinities were predicted for SiO and SiO2, molecules for which electron affinities are not known experimentally, as 0.11 eV and 2.03 eV, respectively. It is concluded that the accuracy observed for density functional theory methods applied to first row atoms and molecules extends to molecules containing second row atoms and that density functional theory continues to provide a computationally affordable means of producing electron affinities reliable to within a few tenths of an eV of definitive experimental values.

A Periodic Density Functional Theory and Hartree−Fock Study of Alkali Halides with Gaussian Orbitals

Periodic-density functional theory (DFT) and Hartree-Fock calculations have been performed on the alkali halides for the alkali metals Li, Na, K, and Rb and the halogens F, Cl, Br and I. Calculations were done with HF-optimized basis sets and DFT-optimized basis sets, at the HF, local DFT (LDFT), and nonlocal DFT (NLDFT) with the Becke exchange potential and the Perdew-Wang correlation potential. The HF and NLDFT results are similar for the lattice parameters, in general, by overestimating them. The LDFT method always underestimates the lattice parameters. As a consequence, the HF and NLDFT methods predict bulk moduli that are too small and the LDFT method predicts bulk moduli that are too large. The absolute error in the bulk moduli are larger at the LDFT level than at the other levels. Charge distribution maps have been calculated and analyzed. The maps clearly show that the main difference between the HF and DFT densities is in the treatment of exchange and that the nonlocal (gradient) corrections moderate this difference.

An ab initio path integral molecular dynamics study of double proton transfer in the formic acid dimer

Double proton transfer in the formic acid dimer has been investigated with Car–Parrinello ab initio molecular dynamics calculations. The electronic structure of the dimer has been obtained using gradient-corrected density functional theory based on the B-LYP (Becke exchange [Phys. Rev. A 38, 3098 (1988)] and Lee–Yang–Parr correlation [Phys. Rev. B 37, 785 (1988)] functional. The optimized equilibrium and saddle-point geometries, obtained by simulated annealing, are in good agreement with previous ab initio quantum chemical predictions and experiment. Thermal and quantum fluctuations of nuclei along the double proton transfer reaction path have also been investigated at T=300 K. Thermal fluctuations give a broad distribution of nuclei around the minimum energy path on the potential energy surface. Quantum fluctuations, investigated using ab initio path integral molecular dynamics, make the distribution even broader around the equilibrium structure, and cause the distribution to deviate appreciably from the minimum energy path on approaching the reaction barrier. In particular, the system passes through higher energy regions than the geometrical saddle point by tunneling; an observation which is consistent with the conventional understanding of heavy–light–heavy mass combination reactions. While there is asynchronous movement of the two protons around the equilibrium structure, synchronous movement becomes relevant on approaching the reaction barrier.

Density-functional calculation of the inner-shell spectra for two stable enol tautomers: acetylacetone and malonaldehyde

Abstract In this paper we report results from our theoretical studies on two β-diketones which exist as stable enol tautomers: acetylacetone and malonaldehyde. We found that density functional theory (DFT) with Becke's exchange functional and Perdew's correlation functional can accurately predict the core–electron binding energies (CEBEs) of both molecules. The oxygen 1 s →π* inner-shell excitation spectra (ISES) of both tautomers have also been studied using our DFT procedure. The CEBEs for the corresponding carbons and oxygens of the CO and C–OH groups of malonaldehyde are larger than those of acetylacetone. The first term values corresponding to a core-hole on the carbonyl oxygen were found to be larger than those with a core-hole on the enol oxygen. The observation indicates that the effect of a core-hole in the proximity of π* orbital has a stronger effect on increasing the electron affinity of the core-ionized cation.

Density functional theory calculations of magnetic coupling in systems containing organic molecules and transition metal atoms: natural magnetic orbital analysis

Our interest is the study of magnetic properties of organic compounds. In this work, we explore theoretically a simple model based on superposed aniline or benzene molecules and Cu or Zn atoms to study the influence of the nature of the building material on magnetic coupling. We also apply the concept of natural magnetic orbital to analyze the spin state of these molecular systems. Calculations were performed within the non-local spin density approximation with Becke's exchange functional and the gradient-corrected functional of Lee, Yang and Parr (BYLP) using the DMol program. The results obtained show the high sensitivity of magnetic properties on the nature of the molecular components and the possibility of creating molecular magnets by doping purely organic materials with non-magnetic transition metal atoms or ions.