A challenge for density functional theory: the XONO and XNO2 (X=F, Cl, and Br) molecules

The equilibrium geometries, harmonic frequencies, dipole moments, infrared intensities, and relative energies of the cis-XONO, trans-XONO, and XNO₂ (X=F, Cl, and Br) have been investigated using four functionals in common use in Kohn-Sham density functional theory (DFT) calculations. Two of the functionals include non-local or gradient correction terms, while the other two also incorporate some exact Hartree-Fock exchange and are labeled hybrid functionals. The quality of the results obtained from the functionals is determined by comparison to previously published high-level coupled-cluster calculations. The hybrid functionals perform better for prediction of the equilibrium geometries, where the two gradient corrected functionals yield qualitatively incorrect molecular structures for cis-FONO and cis-ClONO. None of the functionals perform well in predicting all six harmonic frequencies, showing that the correlation between equilibrium geometries and harmonic frequencies is not as strong for these DFT methods as it is for conventional wavefunction ab initio methods, such as coupled-cluster theory. Results from the various functionals generally come into better agreement with each other and also with the coupled-cluster results moving down the periodic table. Received: 12 February 1997 / Accepted: 25 March 1997     

Generation of ground-state structures and electronic properties of ternary AlxTiyNiz clusters (x + y + z = 6) with a two-stage density functional theory global search approach

The structural and electronic properties of ternary Al_xTi_yNi_z clusters, where x, y, and z are integers and x + y + z = 6 , are investigated. Both Slater, Vosko, Wilks, and Nusair and B3LYP exchange-correlation (XC) functionals are employed in a two-stage density functional theory (DFT) calculations to generate these clusters. In the first stage, a minimum energy cluster structure is generated by an unbiased global search algorithm coupled with a DFT code using a light XC functional and small basis sets. In the second stage, the obtained cluster structure is further optimized by another round of global minimization search coupled with a DFT calculator using a heavier XC functional and more costly basis set. Electronic properties of the structures are illustrated in the form of a ternary diagram. Our DFT calculations find that the thermodynamic stability of the clusters increases with the increment in the number of constituent nickel atoms. These results provide a new insight to the structure, stability, chemical order, and electronic properties for the ternary alloy nanoclusters.     

Accurate prediction of vertical electronic transitions of Ni(II) coordination compounds via time dependent density functional theory

Time dependent density functional theory calculations are completed for five Ni(II) complexes formed by polydentate peptides to predict the electronic absorption spectrum. The ligands examined were glycyl‐glycyl‐glycine (GGG), glycyl‐glycyl‐glycyl‐glycine (GGGG), glycyl‐glycyl‐histidine (GGH), glycyl‐glycyl‐cysteine (GGC), and triethylenetetramine (trien). Fifteen functionals and two basis sets were tested. On the basis of the mean absolute percent deviation (MAPD), the ranking among the functionals is: HSE06 ∼ MPW1PW91 ∼ PBE0 > ω‐B97x‐D ∼ B3P86 ∼ B3LYP ∼ CAM‐B3LYP > PBE ∼ BLYP ∼ BP86 > TPSS > TPSSh > BHandHLYP > M06 ≫ M06‐2X. Concerning the basis sets, the triple‐ζ def2‐TZVP performs better than the double‐ζ LANL2DZ. With the functional HSE06 and basis set def2‐TZVP the MAPD with respect to the experimental λ_max is 1.65% with a standard deviation of 1.26%. The absorption electronic spectra were interpreted in terms of vertical excitations between occupied and virtual MOs based on Ni‐d atomic orbitals. The electronic structure of the Ni(II) species is also discussed.     

NMR shielding constants of CuX,AgX, and AuX (X = F,Cl, Br,and I) investigated by density functional theory based on the Douglas–Kroll–Hess Hamiltonian

Two‐component relativistic density functional theory (DFT) with the second‐order Douglas–Kroll–Hess (DKH2) one‐electron Hamiltonian was applied to the calculation of nuclear magnetic resonance (NMR) shielding constant. Large basis set dependence was observed in the shielding constant of Xe atom. The DKH2‐DFT‐calculated shielding constants of I and Xe in HI, I₂, CuI, AgI, and XeF₂ agree well with those obtained by the four‐component relativistic theory and experiments. The Au NMR shielding constant in AuF is extremely more positive than in AuCl, AuBr, and AuI, as reported recently. This extremely positive shielding constant arises from the much larger Fermi contact (FC) term of AuF than in others. Interestingly, the absolute values of the paramagnetic and the FC terms are considerably larger in CuF and AuF than in others. The large paramagnetic term of AuF arises from the large d‐components in the Au d_π –F p_π and Au sd_σ–F p_σ molecular orbitals (MOs). The large FC term in AuF arises from the small energy difference between the Au sd_σ + F p_σ and Au sd_σ–F p_σ MOs. The second‐order magnetically relativistic effect, which is the effect of DKH2 magnetic operator, is important even in CuF. This effect considerably improves the overestimation of the spin‐orbit effect calculated by the Breit–Pauli magnetic operator. © 2013 Wiley Periodicals, Inc.     

The nature of transitions in electronic absorption spectra of radical cations of dipyrroles with phenylene bridging groups

The electronic absorption spectra of radical cations of dipyrroles with a phenylene bridge were studied by laser flash photolysis and quantum chemical methods. Intense absorption bands of the radical cations in the visible region (λ_max ≈ 500 nm, ε_max > 2 · 10⁴ L mol⁻¹ cm⁻¹) are caused by excitation of electrons from single occupied MOs to the LUMO. In the near IR region, calculations predict additional, relatively intense (f≈ 0.27–0.29) electronic transitions associated with excitation of electrons from low-lying MOs to the single occupied MO.     

Perturbative corrections to basis incompleteness in molecular SCF calculations

A unified summary is presented of the mathematical approach developed by McDowell for employing perturbation theory to correct for basis-set incompleteness in ab initio SCF calculations. Revised expressions for the corrections to the wavefunction both in terms of orbitals and spin-orbitals are presented with explicit incorporation of the spin variables. Employing H₂O as an example, we show that this approach is considerably more powerful for computing molecular energies with standard basis sets than was indicated by previous work. In particular at the higher levels of approximation it accurately reproduces the effect of polarization functions in sets such as 6-31G^* and 6-31G^**. The equilibrium molecular structure of H₂O was also computed by this approach and found to give good accuracy. In each case perturbing functions coupled to both occupied and virtual orbitals are required for acceptable results.     

New Cu(II) complex with acetylpyridine benzoyl hydrazone: experimental and theoretical analysis

This work describes the synthesis and the crystal structure of a copper(II) complex – [Cu(HL)(NO₃)₂]·H₂O – based on 2-acetylpyridine-benzoylhydrazone ligand (HL). In the complex, copper(II) is five-coordinate with two nitrate ligands and a tridentate NNO-donor. The copper center has square pyramidal geometry with an axial nitrate and the equatorial positions occupied by another nitrate and the hydrazone. Besides X-ray studies, the complete structural characterization includes elemental analysis, IR, and UV–vis spectroscopy. Plane wave and localized basis set calculations support the distorted square pyramidal geometry. Theoretical calculations using several DFT functionals were used to study the performance of functionals for this complex. Solvent effect was studied on optimized geometries allowing better support to its spectroscopic data. Noncovalent π?π stacking interactions were also analyzed.     

The wavefunction of non-stationary states in terms of complex coordinates and a related variational principle

The wavefunction of a decaying state is expressed in terms of complex coordinates as ψ(p) = a(θ)ψ _o(p) + b(θ)x(p), where the square integrable θ_o(p) and x(p) contain the localized and asymptotic information respectively. From Fano theory, we derive the relationship of a(?) and b(?) to the energy and width of the resonance state. This forms the basis for a new variatfonal principle for optimization of trial functions in the complex coordinate—energy plane.     

Accurate Quantum‐Chemical Prediction of Enthalpies of Formation of Small Molecules in the Gas Phase

The coupled‐cluster approach, including single and double excitations and perturbative corrections for triple excitations, is capable of predicting molecular electronic energies and enthalpies of formation of small molecules in the gas phase with very high accuracy (specifically, with error bars less than 5 kJ mol^?1), provided that the electronic wavefunction is dominated by the Hartree–Fock configuration. This capability is illustrated by calculations on molecules containing O–H and O–F bonds, namely OH, FO, H₂O, HOF, and F₂O. To achieve this very high accuracy, it is imperative to account for electron‐correlation effects in a quantitative manner, either by using explicitly correlated two‐particle basis functions (R12 functions) or by extrapolating to the limit of a complete basis. Besides taking into account harmonic zero‐point vibrational energies, it is also necessary to account for anharmonic corrections to the zero‐point vibrational energies, to include the core orbitals into the coupled‐cluster calculations, and to account for spin–orbit corrections and scalar relativistic effects. These additional corrections constitute small but significant contributions in the range of 1–4 kJ mol^?1 to the enthalpies of formation of the aforementioned molecules. The highly accurate coupled‐cluster results, obtained by employing R12 functions and by including various corrections, are compared with standard Kohn–Sham density‐functional calculations as well as with the Gaussian‐2 and complete‐basis‐set model chemistries.     

An Insight of the Results Provided by Color‐Tuning Studies Made on Ir(III) Complexes: A pi‐Neutral CF3 Group Viewed by Adjusting Energies of pi‐type Molecular Orbitals

The pi‐nature of a CF₃ group can be understood through analysis of its bond orbitals (BOs) mixed into the pi‐type molecular orbitals of CF₃‐substituted Ir(ppy)₂MDPA⁺ complexes (ppy=2‐phenyl‐pyridine and MDPA=methylated 2,2′‐dipyridyl amine). It has been found that, through this natural bond orbital analysis, the parent’s molecular orbitals (MOs) can be stabilized by χρ^*_CF BO via negative hyperconjugation and, simultaneously, destabilized by electron lp(F) BO. Since these two competing pi‐effects are virtually counterbalanced as indicated by the vanishing values of crystal orbital overlap populations, the chemical substitution strategy originated from lowering of HOMO by using this electron‐withdrawing CF₃ group has been found effective in color‐tuning to blue region. Based on reduced shielding effect due to de‐ creased χρ‐electron density, the reported position dependent CF₃‐substitution effects on pi‐type MOs can also be understood through HOMO/LUMO wavefunction analysis.     

Ab initio and density functional theory study of the electronic structure of rhenium complexes with noninnocent dioxolene ligands: Localized vs delocalized valence states

The dioxolene type ligands (Diox) derived from ortho-quinones are the most widely studied redox noninnocent ligands existing in the dianionic (Cat), anion radical (SQ) or neutral (Q) forms although a highly delocalized electronic structure is also possible. For [ReO(Diox)₂PPh₃]⁻ ( 2 ) and [ReCl₃(Diox)PPh₃] ( 3 ) complexes, the Re^V-Cat₂ and Re^IV-SQ localized valence states were proposed on the basis of their XRD structures. To understand in detail the electronic structure of these complexes, we performed a series of the all-electron calculations at the DKH2-CASSCF/CASPT2 and DKH2-CASSCF/NEVPT2 levels taking into account scalar relativistic and spin-orbit effects. All calculations predicted that 2 has a singlet ground state with a predominant contribution of a single electronic configuration with doubly occupied molecular orbitals being pure o-quinone LUMOs of both Diox ligands that corresponds to the Re^V-Cat₂ valence state. Complex 3 has a triplet ground state with four electronic configurations contributing mainly into its wavefunction and differing by the occupation of bonding and antibonding combinations of the o-quinone LUMO and rhenium d-AO with nearly equal contributions. This leads to the empirical “metrical oxidation state” of dioxolene ligand being −1 that is usually referred to the Re^IV-SQ oxidation state. However, in fact, the negative charge on the Diox ligand is mainly provided by a pair of electrons on the bonding MO. The standard DFT calculations entirely fail to correctly predict the ground state multiplicity for 3 .     

One‐bond 29Si‐1H spin‐spin coupling constants in the series of halosilanes: benchmark SOPPA and DFT calculations,relativistic effects,and vibrational corrections

A number of most representative second order polarization propagator approach (SOPPA) based wavefunction methods, SOPPA, SOPPA(CC2) and SOPPA(CCSD), and density functional theory (DFT) based methods, B3LYP, PBE0, KT2, and KT3, have been benchmarked in the calculation of the one‐bond ²⁹Si‐¹H spin‐spin coupling constants in the series of halosilanes SiH_nX_4?n (X = F, Cl, Br, I), both at the non‐relativistic and full four‐parameter Dirac's relativistic levels taking into account vibrational corrections. At the non‐relativistic level, the wavefunction methods showed much better results as compared with those of DFT. At the DFT level, out of four tested functionals, the Perdew, Burke, and Ernzerhof's PBE0 showed best performance. Taking into account, relativistic effects and vibrational corrections noticeably improves wavefunction methods results, but generally worsens DFT results. Copyright © 2013 John Wiley & Sons, Ltd.     

Assessment of dispersion‐improved exchange‐correlation functionals for the simulation of CO2 binding by alcoholamines

In this study, 12 bound complexes were selected to construct a database for testing 15 dispersion‐improved exchange‐correlation (XC) functionals, including hybrid generalized gradient approximation (GGA), modified using the Grimme's pairwise strategy, and double hybrid XC functionals, for specifically characterizing the CO₂ binding by alcoholamines. Bound complexes were selected based on the characteristics of their hydrogen bonds, dispersion, and electrostatic (particularly between the positive charge of CO₂ and the lone pair of N of alcoholamines) interactions. The extrapolated binding energy from the aug‐cc‐pVTZ (ATZ) to aug‐cc‐pVQZ (AQZ) basis set at the CCSD(T)/CBS(MP2+DZ) level was used as the reference for the XC functional comparison. M06‐2X produced the optimal agreement if the optimized geometries at MP2/ATZ level were chosen for all the test bound complexes. However, M06‐L, ωB97X, and ωB97, and were preferred if the corresponding density functional theory (DFT) optimized geometries were adapted for the benchmark. Simple bimolecular reaction between CO₂ and monoethanolamine simulated using polarizable continuum solvation model confirmed that ωB97, ωB97X, and ωB97XD qualitatively reproduced the energetics of MP2 level. The inconsistent performance of the tested XC functionals, observed when using MP2 or DFT optimized geometries, raised concerns regarding using the single‐point ab initio correction combined with DFT optimized geometry, particularly for determining the nucleophilic attack by alcoholamines to CO₂. © 2014 Wiley Periodicals, Inc.     

Numerical MC SCF and CI calculations of the ground-state potential energy curve of N2

The numerical procedure of McCullough is used in calculations of Hartree-Fock and MC SCF wavefunctions for ground state of N₂. The latter are derived using the complete set of 18 spin and symmetry adapted configurations in the space of MOs that arise from 2p atomic orbitals. An increase in dissociation energy of 0.17 eV is observed when compared to MC SCF calculations in a large basis of Slater-type functions and the same set of configurations. Integrals involving the numerical MC SCF MOs are used in CI calculations in which substitutions involving the 1s and 2s electrons are included. The increase in dissociation energy due to numerical versus basis set valence CI is 0.08 eV. Spectroscopic constants and molecular quadrupole moments are reported.     

Theoretical Simulation of Vibrational Sum‐Frequency Generation Spectra from Density Functional Theory: Application to p‐Nitrothiophenol and 2,4‐Dinitroaniline

The molecular orientation of adsorbed molecules forming self‐assembled monolayers can be determined by combining vibrational sum‐frequency generation (SFG) measurements with quantum chemical calculations. Herein, we present a theoretical methodology used to simulate the SFG spectra for different combinations of polarizations. These simulations are based on calculations of the IR vectors and Raman tensors, which are obtained from density functional theory computations. The dependency of the SFG vibrational signature with respect to the molecular orientation is presented for the molecules p‐nitrothiophenol and 2,4‐dinitroaniline. It is found that a suitable choice of basis set as well as of exchange‐correlation (XC) functional is mandatory to correctly simulate the SFG intensities and consequently provide an accurate estimation of the adsorbed molecule orientation. Comparison with experimental data shows that calculations performed at the B3LYP/6‐311++G(d,p) level of approximation provide good agreement with experimental frequencies, and with IR and Raman intensities. In particular, it is demonstrated that polarization and diffuse functions are compulsory for reproducing the IR and Raman spectra, and consequently vibrational SFG spectra, of systems such as p‐nitrothiophenol. Moreover, the investigated XC functionals reveal their influence on the relative intensities, which show rather systematic variations with the amount of Hartree–Fock exchange. Finally, further aspects of the modeling are revealed by considering the frequency dependence of the Raman tensors.     

Hydrogen bonding in the urea dimers and adenine–thymine DNA base pair: anharmonic effects in the intermolecular H-bond and intramolecular H-stretching vibrations

The equilibrium structures, binding energies, vibrational harmonic frequencies, and the anharmonic corrections for two different (cyclic and asymmetric) urea dimers and for the adenine–thymine DNA base pair system have been studied using the second-order Møller–Plesset perturbation theory (MP2) method and different density functional theory (DFT) exchange–correlation (XC) functionals (BLYP, B3LYP, PBE, HCTH407, KMLYP, and BH and HLYP) with the D95V, D95V**, and D95V++** basis sets. The widely used a posteriori Boys–Bernardi or counterpoise correction scheme for basis set superposition error (BSSE) has been included in the calculations to take into account the BSSE effects during geometry optimization (on structure), on binding energies and on the different levels of approximation used for calculating the vibrational frequencies. The results obtained with the ab initio MP2 method are compared with those calculated with different DFT XC functionals; and finally the suitability of these DFT XC functionals to describe intermolecular hydrogen bonds as well as harmonic frequencies and the anharmonic corrections is assessed and discussed.     

Theoretical studies of atmospheric molecules: SCF and correlated energy levels for the NO2, NO 2 + and NO 2 − systems

SCF and MC-SCF/CI calculations were carried out on the low-lying electronic states of NO₂, NO ₂ ⁺ and NO ₂ ^– , using a double-zeta quality basis set of contracted Gaussian functions. The calculations were performed primarily at the equilibrium geometry (R _NO = 2.25 a_o, _ONO=134 °) of theX ² A ₁ state of NO₂. SCF calculations on NO ₂ ⁺ in a linear conformation were also performed. Results are presented and compared with experiment and other calculations.Research supported in part by Air Force Delivery Orders F33615-72-M-5015 and MIPR889474-00117 and Air Force Office of Scientific Research and in part by the United States Energy Research and Development Administration.     

A numerically stable procedure for calculating Møller-Plesset energy derivatives,derived using the theory of Lagrangians

Summary When Møller-Plesset energy derivatives are determined in the canonical Hartree-Fock basis, singularities or instabilities may arise due to degeneracies among the occupied or unoccupied orbitals. If a non-canonical basis is used these singularities disappear. Numerically stable expressions are presented for the molecular gradient and Hessian of the second-order Møller-Plesset energy, obtained by differentiating a fully variational Lagrangian of the energy constructed in a non-canonical representation. By using a non-canonical representation, singularities and instabilities are avoided, and the variational property of the Lagrangian ensures that Wigner's 2n + 1 rule is satisfied for the orbital derivatives and that the multipliers satisfy the stronger 2n + 2 rule. It is shown that the most expensive step in the calculation of the Hessian scales as Mn ₄o, where M is the number of independent Cartesian distortions, n the total number of orbitals, and o the number of occupied orbitals.     

Analysis of approximations and errors in equations of motion method calculations

We analyze a number of fundamental questions associated with the use of a finite one-particle orbital basis in equations of motion (EOM) method calculations of excitation energies etc., of atomic and molecular systems. This approximation yields an approximate n_e-electron ground state and say, N excited states, while there are (N + 1)² different possible basis operators for EOM calculations. We show that sets of at most 2N basis operators can contribute to the EOM calculations. Any set of 2N basis operators, satisfying certain conditions, provides the exact EOM energies which are equivalent to complete configuration interaction results within the same orbital basis. We investigate the use of particle-particle shifting operators which are not employed in EOM calculations in model calculations on He with operator bases smaller than the complete 2V to consider the convergence of the expansion. The dependence of EOM calculations on the quality of the approximate ground state wavefunction is studied through calculations for Be where additional support is provided for the frequent need for multiconfigurational zeroth order reference functions (as corrected perturbatively). Excited state EOM wavefunctions from EOM calculations are shown to not necessarily be orthogonal to either the exact or approximate ground state wavefunction, suggesting implications in the use of EOM methods to evaluate excited state properties. The He and Be examples and a simple two-level problem are also utilized to illustrate questions concerning the use of the EOM equations to obtain an iteratively improved ground state wavefunction.