Hybrid Functional and Plane Waves based Ab Initio Molecular Dynamics Study of the Aqueous Fe2+/Fe3+ Redox Reaction**

Kohn-Sham density functional theory and plane wave basis set based ab initio molecular dynamics (AIMD) simulation is a powerful tool for studying complex reactions in solutions, such as electron transfer (ET) reactions involving Fe²⁺/Fe³⁺ ions in water. In most cases, such simulations are performed using density functionals at the level of Generalized Gradient Approximation (GGA). The challenge in modelling ET reactions is the poor quality of GGA functionals in predicting properties of such open-shell systems due to the inevitable self-interaction error (SIE). While hybrid functionals can minimize SIE, standard plane-wave based AIMD at that level of theory is typically 150 times slower than GGA for systems containing ∼100 atoms. Among several approaches reported to speed-up AIMD simulations with hybrid functionals, the noise-stabilized MD (NSMD) procedure, together with the use of localized orbitals to compute the required exchange integrals, is an attractive option. In this work, we demonstrate the application of the NSMD approach for studying the Fe²⁺/Fe³⁺ redox reaction in water. It is shown here that long AIMD trajectories at the level of hybrid density functionals can be obtained using this approach. Redox properties of the aqueous Fe²⁺/Fe³⁺ system computed from these simulations are compared with the available experimental data for validation.     

An application of double exponential formula to radial quadrature grid in density functional calculation

We report an application of the double exponential formula to the numerical integration of the radial electron distribution function for atomic and diatomic molecular systems with a quadrature grid. Three types of mapping transformation in the double exponential formula are introduced into the radial quadrature scheme to generate new radial grids. The double exponential grids are examined for the electron-counting integrals of He, Ne, Ar, and Kr atoms which include occupied orbitals up to the 4p shell. The performance of radial grid is compared for the double exponential formula and the formulas proposed in earlier studies. We mainly focus our attention on the behavior of accuracy by the quadrature estimation for each radial grid with varying the mapping parameter and the number of grid points. The convergence behavior of the radial grids with high accuracy for atomic system are also examined for the electron-counting integrals of LiH, NaH, KH, Li₂, Na₂, K₂, HF, HCl, HBr, F₂, Cl₂, Br₂, LiF, NaCl, KBr, [ScH]⁺, [MnH]⁺, and [CuH]⁺ molecules. The results reveal that fast convergence of the integrated values to the exact value is achieved by applying the double exponential formula. It is demonstrated that the double exponential grids show similar or higher accuracies than the other grids particularly for the Kr atom, Br₂ molecule, alkali metal hydrides, alkali metal halogenides, and transition metal hydride cations, suggesting that the double exponential transformations have potential ability to improve the reliability and efficiency of the numerical integration for energy functionals.     

Accurate nonrelativistic energies for 2Po states of the Li isoelectronic series

With large Hylleraas-configuration interaction (CI) basis sets highly accurate upper bounds for the lowest ²P^o states of the Li isoelectronic series up to Ne are given. The corresponding Hamiltonian H and the operator Σ_{i < j}, ▿_i ▿_j are transformed into nonorthogonal coordinates expressed in interparticle and angular terms. The evaluation of the occurring integrals is reduced to the calculation of well-known auxiliary integrals. Furthermore some expectation values and isotope energies are calculated. The isotope energies are obtained using perturbation theory in first-order approximation. © 1997 John Wiley & Sons, Inc.     

Influence of the exchange‐correlation functional in all‐electron calculations of the vibrational frequencies of corundum (α‐Al2O3)

The equilibrium structural parameters, high‐ and low‐frequency dielectric tensors, Born effective charges, and Γ‐point vibrational frequencies of bulk Al₂O₃ corundum are calculated by using the periodic, ab initio program CRYSTAL, which adopts an all‐electron Gaussian‐type basis set. The effect of basis set and the performance of three different functionals, i.e., LDA, PW91, and B3LYP, are discussed. The mean absolute deviation from the measured frequencies is as small as 7 cm^?1 for both the LDA and B3LYP functionals, indicating that these functionals perform extremely well in this case. The mean absolute deviation increases to 18 cm^?1 when the PW91 functional is used. All three functionals reproduce the equilibrium geometry of corundum to a high level of accuracy, with LDA and B3LYP outperforming PW91 slightly. The comparison of the current all‐electron calculations with previous plane‐wave, pseudo‐potential calculations shows an overall similar performance. The results of isotopic substitution for both Al and O are also presented. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Slater orbital molecular integrals with numerical fourier transform methods. I. (coplanar) multicenter exchange integrals over 1s orbitals

Slater orbital r₁₂^?1 integrals are calculated with a numerical Fourier-transform method based on a formulation first given by Bonham, Peacher and Cox. Spherical wave expansions are introduced that decouple the Feynman integrations for the charge distribution Fourier transforms. The Feynman integrals are evaluated semianalytically, and their properties are analyzed in detail. The final computational step involves a numerical integration over charge distribution quantities. Results for (coplanar) multicenter exchange integrals over 1s orbitals are given. As long as the charge distributions are overlapping considerably, the method gives good results, even when these distributions are highly asymmetric. The method as presently implemented fails when highly disconnected charge distributions are involved.     

Electric and magnetic multipole integrals in STO basis sets: an analytical approach

A new method for the evaluation of one- and two-centre magnetic and electric multipole integrals for Slater-type functions is presented. The method is strictly analytical in that no approximations of any kind are involved. Two simple functions, ℐ₁ ^aug and ℐ₂ ^aug, are introduced, which employ only functions that are well known in electronic structure theory. With the use of augmentation exponents these functions apply to multipole integrals as well as other one-electron integrals, such as nuclear attraction integrals. The proposed method includes the analytic determination of derivatives of the integrals with respect to atomic displacements. Some illustrative test calculations are presented and compared to results from the literature. Received: 20 April 1998 / Accepted: 13 October 1998 / Published online: 1 February 1999     

The Boron conundrum: the case of cationic clusters B n + with n?=?2–20

We investigate the molecular and electronic structure and thermochemical properties of the cationic boron clusters B _n ⁺ with n?=?2–20, using both MO and DFT methods. Several functionals are used along with the MP2, G3, G3B3, G4, and CCSD(T)/CBS methods. The latter is the high accuracy reference. While the TPSS, TPSSh, PW91, PB86, and PBE functionals show results comparable to high-accuracy MO methods, both BLYP and B3LYP functionals are not accurate enough for three-dimensional (3D) structures. A negligible difference is observed between the B3LYP, MP2, and CCSD(T) geometries. A transition between 2D and 3D structures occurs for this series at the B₁₆ ⁺–B₁₉ ⁺ sizes. While smaller clusters B _n ⁺ with n?≤?15 are planar or quasi-planar, a structural competition takes place in the intermediate sizes of B _16–19 ⁺ . The B₂₀ ⁺ cation has a 3D tubular shape. The standard heats of formation are determined and used to evaluate the cluster stability. The average binding energy tends to increase with increasing size toward a limit. All closed-shell species B _n ⁺ has an aromatic character, but an enhanced stability is found for B₅ ⁺ and B₁₃ ⁺ whose aromaticity and electron delocalization are analyzed using the LOL technique.     

On the hyperfine structure in the configurations 4f n 6s 2, 4f n 5d6s, 4f n 6s6p,and 4f n−15d6s 2 of the lanthanides

A survey of hyperfine analyses in the low-lying configurations 4f ⁿ 6s ², 4f ⁿ 5d6s, 4f ⁿ 6s6p, and 4f ^n?1 5d6s ² of the lanthanides is given. Experimental hf radial integrals 〈r ^?3〉 _nl ^kj are indicated for the configurations under investigation. From a comparison of experimental and theoretical hf radial integrals configuration-interaction contributions Δ _nl ^kj to the hfs have been evaluated for the configurations for which corresponding integrals were available. With nuclear electric quadrupole moments from muonic x-ray investigations and radial integrals 〈r ^?3〉 _nl ⁰¹ and 〈r ^?3〉 _nl ⁰² from experimental hfs quadrupole shielding correctionsR _nl ⁰¹ have been obtained. The variation of hyperfine radial integrals, configuration-interaction contributions and quadrupole shielding corrections over the 4f shell in the configurations under study is discussed. Trends of these quantities in different configurations are compared.     

Theoretical Modelling of Photoswitching of Hyperpolarisabilities in Ruthenium Complexes

Static excited‐state polarisabilities and hyperpolarisabilities of three Ru^II ammine complexes are computed at the density functional theory (DFT) and several correlated ab initio levels. Most accurate modelling of the low energy electronic absorption spectrum is obtained with the hybrid functionals B3LYP, B3P86 or M06 for the complex [Ru^II(NH₃)₅(MeQ⁺)]³⁺ (MeQ⁺=N‐methyl‐4,4′‐bipyridinium, 3 ) in acetonitrile. The match with experimental data is less good for [Ru^II(NH₃)₅L]³⁺ (L=N‐methylpyrazinium, 2 ; N‐methyl‐4‐{E,E‐4‐(4‐pyridyl)buta‐1,3‐dienyl}pyridinium, 4 ). These calculations confirm that the first dipole‐ allowed excited state (FDAES) has metal‐to‐ligand charge‐transfer (MLCT) character. Both the solution and gas‐phase results obtained for 3 by using B3LYP, B3P86 or M06 are very similar to those from restricted active‐space SCF second‐order perturbation theory (RASPT2) with a very large basis set and large active space. However, the time‐dependent DFT λ_max predictions from the long‐range corrected functionals CAM‐B3LYP, LC‐ωPBE and wB97XB and also the fully ab initio resolution of identity approximate coupled‐cluster method (gas‐phase only) are less accurate for all three complexes. The ground state (GS) two‐state approximation first hyperpolarisability β_2SA for 3 from RASPT2 is very close to that derived experimentally via hyper‐Rayleigh scattering, whereas the corresponding DFT‐based values are considerably larger. The β responses calculated by using B3LYP, B3P86 or M06 increase markedly as the π‐conjugation extends on moving along the series 2 → 4 , for both the GS and FDAES species. All three functionals predict substantial FDAES β enhancements for each complex, increasing with the π‐conjugation, up to about sevenfold for 4 . Also, the computed second hyperpolarisabilities γ generally increase in the FDAES, but the results vary between the different functionals.     

Application of modified Neumann expansion for analytical evaluation of two-center,two-electron integrals over slater-type orbitals

A modified form of the Neumann expansion in terms of products of orthogonal polynomials for the inverse interelectronic distance r¹₁₂ is proposed. This expansion has been applied in order to derive a unified analytical formula for two-center and two-electron integrals over Slater-type orbitals. The results are equivalent to those given recently by Yasui and Saika, but the expansion itself can be used for building up a realistic algorithm for evaluation of three- and four-electron integrals determined by using correlated variational wave functions.     

Comparative Computational Study of Hydrogen Abstraction Reactions of CY3H + XO− (X,Y = F,Cl, and Br)

The effects of halogen substituents on the reactivity are characterized by the hybrid B3LYP and BHandHLYP functionals of density functional theory using the aug‐cc‐pVDZ basis set. The species XO⁻ and CY₃H, where X, Y = F, Cl, and Br, have been chosen as model reactants in this work. Also, the mechanism of the hydrogen abstraction (HAT) reaction has been used to study the chemical reactivity of these anionic reactions. Our theoretical findings suggest that the relative reactivity of the CY₃H + XO⁻ reactions increases as Y goes from F to Br and decreases as X goes from F to Br. Moreover, among all reactions investigated in this study, the special role of the Y has very dominant effect on activation of the C–H bond in CY₃H when XO⁻ attacks the CY₃H. Again, through the transition state theory the rate constants at 298–1000 K are also evaluated for the HAT reactions, indicating the lower the temperature the faster is the chemical reaction.     

Hyperfine structure in the configuration 4f 115d6s 2 of Er I

Hyperfine constants of low lying levels of the configuration 4f ¹¹5d6s ² in Er I have been measured for the only stable odd isotope,¹⁶⁷Er, using high resolution laseratomic-beam spectroscopy. Hyperfine parameters were evaluated from the experimental data using the effective tensor operator formalism. For this purpose eigenfunctions have been determined from an analysis of the fine structure energies of Er I as well as from ab initio multiconfiguration Dirac-Fock calculations. With the latter method also ab initio hyperfine constants for the levels investigated were evaluated. A comparison of calculated fine structure energies, Landég _J -factors and hyperfine constants with the experimental data allowed a test of the reliability of the fine structure and multiconfiguration Dirac-Fock wavefunctions. Effective nuclear electric quadrupole moments for¹⁶⁷Er have been determined from the experimental hyperfine constants using both calculated relativistic electronic radial integrals and hyperfine parameters for the 4f and 5d electrons in the configuration 4f ¹¹5d6s ² in Er I. From a comparison with the nuclear quadrupole moment measured in the mesic atom Sternheimer shielding factors are calculated. Configuration-interaction contributions to the radial integrals 〈r ^?3〉 _nl ⁰¹ of the 4f and 5d electrons have been determined.     

Effective charges on elements of the first transition period in complex ions

The semiempirical MO method is applied with an angular model to evaluate the overlap integrals, which are used in calculating the Coulomb integrals and the effective charges for elements of the first transition group in 65 compounds with H₂O, NH₃, F^–, Cl^–, and Br^–. The effective charges are found to decrease regularly in the sequence F^–>H₂O>NH₃>Cl^–>Br^–, while the changes with the transition element are found to agree with the known properties of the compounds.     

Numerical integration techniques for quantum chemistry

Korobov theory for multidimensional numerical integration is used to evaluate electronic integrals. This paper shows the important role played by periodization techniques. Singularity (r ₁₂ ^?1 ) in the bielectronic six-dimensional integrals is removed through a twofold three-dimensional integration. Results are presented for atomic integrals involving Slater type atomic orbitals.     

SCF MO INDO calculation of anisotropie hyperfine coupling tensors for σ-type radicals

The anisotropic hyperfine coupling tensors for some -type radicals (HCO, FCO, NO₂, CO ₂ ^– , CN, phenyl, vinyl) were calculated. The calculation was performed with an all-valence-electron approximate open-shell MO method using the INDO approximation and with the dipolar integrals evaluated over Slater-type AO's. The diagonalyzed tensors were in reasonable agreement with the available data of experiment.     

Water exchange rates and mechanisms in tetrahedral [Be(H2O)4]2+ and [Li(H2O)4]+ complexes using DFT methods and cluster‐continuum models

The water exchange reactions in aquated Li⁺ and Be²⁺ ions were investigated with density functional theory calculations performed using the [Li(H₂O)₄]⁺·14H₂O and [Be(H₂O)₄]²⁺·8H₂O systems and a cluster‐continuum approach. A range of commonly used functionals predict water exchange rates several orders of magnitude lower than the experimental ones. This effect is attributed to the overstabilization of coordination number four by these functionals with respect to the five‐coordinated transition states responsible for the associative ( A ) or associative interchange ( I_a ) water exchange mechanisms. However, the M06 and M062X functionals provide results in good agreement with the experimental data: M062X/TZVP calculations yield a concerted I_a mechanism for the water exchange in [Be(H₂O)₄]²⁺·8H₂O that gives an average residence time of water molecules in the first coordination sphere of 260 μs. For [Li(H₂O)₄]⁺·14H₂O the water exchange reaction is predicted to follow an A mechanism with a residence time of inner‐sphere water molecules of 25 ps.     

Insight and performance of LC-DFT vs DFT in the NMR shielding and chemical shift calculations: Case of CHClCH?CF3

Fifteen density functional theory (DFT) methods and fifteen long-range corrected density functional theory (LC-DFT) methods were used in the present work to assess nuclear magnetic resonance parameters such as nuclear shielding constant (NSC), nuclear chemical shift (NCS), and nuclear anisotropic shielding constant (NAS). These different methods were associated with the full basis set 6-311++G(3df,3pd). The gauge-independent atomic orbital was used for the calculation of nuclear shielding tensors of the nuclei contained in the stereoisomers cis- and trans-CHClCHCF₃. Thus, the effects of LC are clearly observed for heavy nuclei (¹³C, ¹⁹F, ³⁵Cl). The results of NSC, NCS, and NAS from DFT are better described than LC-DFT with regard to the KT3 method. Moreover, the results from the LC-DFT are better described than the standard DFT with regard to CCSD(T). Based on the latter method used as the benchmark, the NSCs of nuclei are well fitted by the competitive functionals LC-TPSSTPSS and LC-PKZBPKZB. In the particular case of the trans-isomer, mPWPKZB was found to be the best method. For the NCSs, the more accurate methods include the latter two LC functionals and the non-LC functionals TPSSTPSS and mPWPKZB. The accuracy of NAS depends strongly on the nuclei. Thus, CAM-B3LYP describes it well for ¹⁹F and LC-PKZBPKZB for ³⁵Cl. The rest of nuclei are well fitted by all the methods except ¹³C₁ and ¹³C₂, which are better reproduced by the LC-DFT except the LC-PKZBPKZB, LC-TPSSTPSS, and CAM-B3LYP functionals.