Constricted Variational Density Functional Theory Approach to the Description of Excited States.

We review the theoretical foundation of constricted variational density functional theory and illustrate its scope through applications.

Applications of time-dependent and time-independent density functional theory to Rydberg transitions.

We have benchmarked the performance of time-independent density functional theory (ΔSCF and RSCF-CV-DFT) in studies on Rydberg transitions employing five different standard functionals and a diffuse basis. Our survey is based on 71 triplet or singlet Rydberg transitions distributed over nine different species: CO (7), CH2O (8), C2H2 (8), H2O (10), C2H4 (13), Be (6), Mg (6), and Zn (8). The best performance comes from the long-range corrected functional LCBP86 (ω = 0.4.) with an average root-mean-square deviation (RMSD) of 0.23 eV. Of similar accuracy are LDA and B3LYP, both with a RMSD of 0.24 eV. The largest RMSD of 0.32 eV comes from BP86 and LCBP86* (ω = 0.75). The performance of ΔSCF is considerably better than that of adiabatic time-dependent density functional theory (ATDDFT) and matches that of highly optimized long-range corrected functionals. However, it is not as accurate as ATDDFT based on highly tuned functionals. The reasonable success of ΔSCF is based on its well-documented ability to afford good estimates of ionization potentials (IP) and electron affinities (EA) even for simple local functionals after orbital relaxation has been taken into account. In ATDDFT based on semilocal functionals, both IP and -EA are poorly described, with errors of up to 5 eV. In the transition energy (ΔE = IP - EA), these errors are canceled to some degree. However, ΔE still carries an error exceeding 1 eV.

Analysis of the bonding between two M(µ-NAr(#)) monomers in the dimeric metal(II) imido complexes {M(µ-NAr(#))}2 [M = Si, Ge, Sn, Pb; Ar(#) = C6H3-2,6-(C6H2-2,4,6-R3)2]. The stabilizing role played by R = Me and iPr.

The nature of the bonding between the two M(µ-NAr(#)) imido monomers [M = Si, Ge, Sn, Pb; Ar(#) = C6H3-2,6-(C6H2-2,4,6-R3)2; R = Me, iPr] in the {M(µ-NAr(#))}2 dimer is investigated with the help of a newly developed energy and density decomposition scheme as well as molecular dynamics. The approach combines the extended transition state energy decomposition method with the natural orbitals for chemical valence density decomposition scheme within the same theoretical framework. The dimers are kept together by two σ bonds and two π bonds. The σ bonding has two major contributions. The first is a dative transfer of charge from nitrogen to M. It amounts to -188 kcal/mol for {Si(µ-NAr(#))}2, -152 kcal/mol for {Ge(µ-NAr(#))}2 with -105 kcal/mol for {Sn(µ-NAr(#))}2, and -79 kcal/mol for {Pb(µ-NAr(#))}2. The second is a charge buildup within the ring made up of the two dimers. It amounts to -82 kcal/mol for M = Si with -61 kcal/mol for M = Ge and ∼-50 kcal/mol for M = Sn and Pb. We finally have π bonding with a donation of charge from M to nitrogen. It has a modest contribution of ∼-30 kcal/mol. The presence of isopropyl (iPr) groups is further shown to stabilize{M(µ-NAr(#))}2 [M = Si, Ge, Sn, Pb; Ar(#) = C6H3-2,6-(C6H2-2,4,6-iPr3)2] compared to the methylated derivatives (R = Me) through attractive van der Waals dispersion interactions.

Activation of H(2) oxidation at sulphur-exposed Ni surfaces under low temperature SOFC conditions.

Ni-YSZ (yttria-stabilized zirconia) cermets are known to be very good anodes in solid oxide fuel cells (SOFCs), which are typically operated at 700-1000 °C. However, they are expected to be increasingly degraded as the operating temperature is lowered in the presence of H2S (5-10 ppm) in the H2 fuel stream. However, at 500 to 600 °C, a temperature range rarely examined for sulphur poisoning, but of great interest for next generation SOFCs, we report that H2S-exposed Ni-YSZ anodes are catalytic towards the H2 oxidation reaction, rather than poisoned. By analogy with bulk Ni3S2/YSZ anodes, shown previously to enhance H2 oxidation kinetics, it is proposed that a thin layer of Ni sulphide, akin to Ni3S2, is forming, at least at the triple point boundary (TPB) region under our conditions. To explain why Ni3S2/YSZ is so active, it is shown from density functional theory (DFT) calculations that the O(2-) anions at the Ni3S2/YSZ TPB are more reactive towards hydrogen oxidation than is O(2-) at the Ni/YSZ TPB. This is accounted for primarily by structural transformations of Ni3S2 during H2 oxidation, rather than by the electronic properties of this interface. To understand why a thin layer of Ni3S2 could form when a single monolayer of sulphur on the Ni surface is the predicted surface phase under our conditions, it is possible that the reaction of H2 with O(2-), forming water, prevents sulphur from re-equilibrating to H2S. This may then promote Ni sulphide formation, at least in the TPB region.

Introducing constricted variational density functional theory in its relaxed self-consistent formulation (RSCF-CV-DFT) as an alternative to adiabatic time dependent density functional theory for studies of charge transfer transitions.

We have applied the relaxed and self-consistent extension of constricted variational density functional theory (RSCF-CV-DFT) for the calculation of the lowest charge transfer transitions in the molecular complex X-TCNE between X = benzene and TCNE = tetracyanoethylene. Use was made of functionals with a fixed fraction (α) of Hartree-Fock exchange ranging from α = 0 to α = 0.5 as well as functionals with a long range correction (LC) that introduces Hartree-Fock exchange for longer inter-electronic distances. A detailed comparison and analysis is given for each functional between the performance of RSCF-CV-DFT and adiabatic time-dependent density functional theory (TDDFT) within the Tamm-Dancoff approximation. It is shown that in this particular case, all functionals afford the same reasonable agreement with experiment for RSCF-CV-DFT whereas only the LC-functionals afford a fair agreement with experiment using TDDFT. We have in addition calculated the CT transition energy for X-TCNE with X = toluene, o-xylene, and naphthalene employing the same functionals as for X = benzene. It is shown that the calculated charge transfer excitation energies are in as good agreement with experiment as those obtained from highly optimized LC-functionals using adiabatic TDDFT. We finally discuss the relation between the optimization of length separation parameters and orbital relaxation in the RSCF-CV-DFT scheme.

A theoretical analysis of supported quintuple and quadruple chromium-chromium bonds.

The extended transition state (ETS) energy decomposition scheme has been combined with the natural orbitals for chemical valence (NOCV) density decomposition method (ETS-NOCV) in a study on the shortest, fully supported metal-metal bond (Cr-Cr = 1.73 Å) in Cr2[Ar'NC(NMe2)NAr']2 [Ar' = C6H3-2,6(C6H3-2,6-Pr(i)2)2]. The scope of the ETS-NOCV method is further demonstrated by a metal-metal bond analysis of the paddlewheel M2(O2CCH3)4 (M = Cr, Mo, W) complexes. The influence of axial ligands as well as R' goups on the bridging ligands is also analyzed. In addition to the quintuple bonding components (σ(2), π(4), δ(4)) for Cr2[Ar'NC(NMe2)NAr']2 and quadruple components (σ(2), π(4), δ(2)) for the paddlewheel complexes, we notice additional stability (17-27 kcal/mol) introduced to the metal-metal bond from participation of the lone pairs residing on the π-systems of the bridging X-C-X (X = N, O) ligand. This is to our knowledge the first time that the strength of the metal-metal bonding components has been determined in a supported metal-metal bond by an energy decomposition scheme.

Role played by isopropyl substituents in stabilizing the putative triple bond in Ar'EEAr' [E = Si, Ge, Sn; Ar' = C6H3-2,6-(C6H3-2,6-Pr(i)2)2] and Ar*PbPbAr* [Ar* = C6H3-2,6-(C6H2-2,4,6-Pr(i)3)2].

A theoretical study of the bonding in ArEEAr (where E = Si, Ge, Sn, Pb; Ar = terphenyl ligand) revealed for the first time why bulky isopropyl substituents electronically are required in order to isolate stable ArEEAr species. This was accomplished by combining the natural orbitals for chemical valence (NOCV) method with the extended transition state (ETS) scheme. The NOCV-ETS analysis was based on two ArE fragments in their doublet ground state with the configuration σ(2)π(1). For E = Si, Ge, and Sn, it revealed one π-bond perpendicular to the CEEC plane and two σ/π-type bonds in the plane, whereas the ArPbPbAr system was found to have a single σ bond with a C-Pb-Pb trans-bent angle close to 90°. While similar bonding pictures have been obtained in previous model studies with Ar = H and CH3, the NOCV-ETS scheme was able to obtain quantitative estimates for the strength of various σ/π components without artificial truncations or twisting of the system. More importantly, NOCV-ETS analysis was able to show that the electronic influence of the isopropyl substituents on the σ/π components differs little from that found in a system where they are replaced by hydrogen. Instead, the favorable role of the isopropyl substituents is due to dispersive van der Waals attractions between Pr(i) groups on aryl rings attached to different E atoms as well as hyperconjugation involving donation into σ* orbitals on Pr(i). Dispersive interaction amounts to -27.5 kcal/mol (Si), -29.1 kcal/mol (Ge), -26.2 kcal/mol (Sn), and -44.0 kcal/mol (Pb). The larger dispersive stabilization for Pb reflects the fact that the longer Pb-Pb and Pb-C bonds sterically allow for more isopropyl groups with Ar = C6H3-2,6-(C6H2-2,4,6-Pr(i)3)2. This is compared to the other elements where Ar = C6H3-2,6-(C6H3-2,6-Pr(i)2)2. It is finally concluded from the analysis that real ArEEAr systems reveal little character of the EE bond in contrast to the findings of previous studies on model systems.

Analysis of the putative Cr-Cr quintuple bond in Ar'CrCrAr' (Ar' = C6H3-2,6(C6H3-2,6-Pr(i)2)2 based on the combined natural orbitals for chemical valence and extended transition state method.

The nature of the putative Cr-Cr quintuple bond in Ar'CrCrAr' (Ar' = C(6)H(3)-2,6(C(6)H(3)-2,6-Pr(i)(2))(2)) is investigated with the help of a newly developed energy and density decomposition scheme. The new approach combines the extended transition state (ETS) energy decomposition method with the natural orbitals for chemical valence (NOCV) density decomposition scheme within the same theoretical framework. The results show that in addition to the five bonding components (σ(2)π(2)π'(2)δ(2)δ'(2)) of the Cr-Cr bond, the quintuple bond is augmented by secondary Cr-C interactions involving the Cr-ipso-carbon of the flanking aryl rings. The presence of isopropyl groups (Pr(i)) is further shown to stabilize Ar'CrCrAr' by 20 kcal/mol compared to the two Ar'Cr monomers through stabilizing van der Waals dispersion interactions.

Molecular and vibrational structure of tetroxo d0 metal complexes in their excited states. a study based on time-dependent density functional calculations and Franck-Condon theory.

We have applied time dependent density functional theory to study excited state structures of the tetroxo d(0) transition metal complexes MnO(4)(-), TcO(4)(-), RuO(4), and OsO(4). The excited state geometry optimization was based on a newly implemented scheme [Seth et al. Theor. Chem. Acc. 2011, 129, 331]. The first excited state has a C(3v) geometry for all investigated complexes and is due to a "charge transfer" transition from the oxygen based HOMO to the metal based LUMO. The second excited state can uniformly be characterized by "charge transfer" from the oxygen HOMO-1 to the metal LUMO with a D(2d) geometry for TcO(4)(-), RuO(4), and OsO(4) and two C(2v) geometries for MnO(4)(-). It is finally found that the third excited state of MnO(4)(-) representing the HOMO to metal based LUMO+1 orbital transition has a D(2d) geometry. On the basis of the calculated excited state structures and vibrational modes, the Franck-Condon method was used to simulate the vibronic structure of the absorption spectra for the tetroxo d(0) transition metal complexes. The Franck-Condon scheme seems to reproduce the salient features of the experimental spectra as well as the simulated vibronic structure for MnO(4)(-) generated from an alternative scheme [Neugebauer J. J. Phys. Chem. A 2005, 109, 1168] that does not apply the Franck-Condon approximation.

Calculation of exchange coupling constants in triply-bridged dinuclear Cu(II) compounds based on spin-flip constricted variational density functional theory.

The performance of the second-order spin-flip constricted variational density functional theory (SF-CV(2)-DFT) for the calculation of the exchange coupling constant (J) is assessed by application to a series of triply bridged Cu(II) dinuclear complexes. A comparison of the J values based on SF-CV(2)-DFT with those obtained by the broken symmetry (BS) DFT method and experiment is provided. It is demonstrated that our methodology constitutes a viable alternative to the BS-DFT method. The strong dependence of the calculated exchange coupling constants on the applied functionals is demonstrated. Both SF-CV(2)-DFT and BS-DFT affords the best agreement with experiment for hybrid functionals.

The implementation of a self-consistent constricted variational density functional theory for the description of excited states.

We present here the implementation of a self-consistent approach to the calculation of excitation energies within regular Kohn-Sham density functional theory. The method is based on the n-order constricted variational density functional theory (CV(n)-DFT) [T. Ziegler, M. Seth, M. Krykunov, J. Autschbach, and F. Wang, J. Chem. Phys. 130, 154102 (2009)] and its self-consistent formulation (SCF-CV(∞)-DFT) [J. Cullen, M. Krykunov, and T. Ziegler, Chem. Phys. 391, 11 (2011)]. A full account is given of the way in which SCF-CV(∞)-DFT is implemented. The SCF-CV(∞)-DFT scheme is further applied to transitions from occupied π orbitals to virtual π(∗) orbitals. The same series of transitions has been studied previously by high-level ab initio methods. We compare here the performance of SCF-CV(∞)-DFT to that of time dependent density functional theory (TD-DFT), CV(n)-DFT and ΔSCF-DFT, with the ab initio results as a benchmark standard. It is finally demonstrated how adiabatic TD-DFT and ΔSCF-DFT are related through different approximations to SCF-CV(∞)-DFT.

First principle simulation of the temperature dependent magnetic circular dichroism of a trinuclear copper complex in the presence of zero field splitting.

We present a test of a recently developed density functional theory (DFT) based methodology for the calculation of magnetic circular dichroism (MCD) spectra in the presence of zero-field splitting (ZFS). The absorption and MCD spectra of the trinuclear copper complex µ(3)O ([Cu(3)(L)(µ(3)-O)](4+)), which models the native intermediate produced in the catalytic cycle of the multicopper oxidases, have been simulated from first principle within the framework of adiabatic time dependent density functional theory. The effects of the ZFS of the quartet (4)A(2) ground state on the theoretical MCD spectrum of µ(3)O have been analyzed. The simulated spectra are consistent with the experimental ones. The theoretical assignments of the MCD spectra are based on direct simulation as well as a detailed analysis of the molecular orbitals in µ(3)O. Some of the assignments differ from those given in previous studies. The ZFS effects in the presence of a strong external magnetic field (7 T) prove negligible. The change of the sign of the ZFS changes systematically the intensity of the MCD bands of the z-polarized excitations. The effect of the ZFS on the x,y-polarized excitations is not uniform.

Calculation of the exchange coupling constants of copper binuclear systems based on spin-flip constricted variational density functional theory.

We have recently developed a methodology for the calculation of exchange coupling constants J in weakly interacting polynuclear metal clusters. The method is based on unrestricted and restricted second order spin-flip constricted variational density functional theory (SF-CV(2)-DFT) and is here applied to eight binuclear copper systems. Comparison of the SF-CV(2)-DFT results with experiment and with results obtained from other DFT and wave function based methods has been made. Restricted SF-CV(2)-DFT with the BH&HLYP functional yields consistently J values in excellent agreement with experiment. The results acquired from this scheme are comparable in quality to those obtained by accurate multi-reference wave function methodologies such as difference dedicated configuration interaction and the complete active space with second-order perturbation theory.

A theoretical study on the exciton circular dichroism of propeller-like metal complexes of bipyridine and tripodal tris(2-pyridylmethyl)amine derivatives.

Time-dependent density functional theory (TD-DFT) has been employed to simulate the circular dichroism (CD) spectra of bipyridyl ruthenium(II) complexes as well as zinc(II) and copper(II) complexes containing tris(2-pyridylmethyl)amine (TPA) derivatives. A qualitative model is used to account for the mechanism by which the bis- and tris-bipyridine complexes (or analogous systems) exhibit exciton CD. The model is further used to predict the sign of the exciton CD bands. The predictions are in agreement with experiment and DFT calculations. A comprehensive analysis is presented of the subtle differences in the CD spectra of this series of related complexes.

Anionic N-heterocyclic carbenes with N,N'-bis(fluoroaryl) and N,N'-bis(perfluoroaryl) substituents.

A series of rhodium complexes, [Rh(cod)(NHC-F(x))(OH(2))] (cod = 1,5-cyclooctadiene; NHC = N-heterocyclic carbene), incorporating anionic N-heterocyclic carbenes with 2-tert-butylmalonyl backbones and 2,6-dimethylphenyl (x = 0), 2,6-difluorophenyl (x = 4), 2,4,6-trifluorophenyl (x = 6), and pentafluorophenyl (x = 10) N,N'-substituents, respectively, has been prepared by deprotonation of the corresponding zwitterionic precursors with potassium hexamethyldisilazide, followed by immediate reaction of the resulting potassium salts with [{RhCl(cod)}(2)]. These complexes could be converted to the related carbonyl derivatives [Rh(CO)(2)(NHC-F(x))(OH(2))] by displacement of the COD ligand with CO. IR and NMR spectroscopy demonstrated that the degree of fluorination of the N-aryl substituents has a considerable influence on the σ-donating and π-accepting properties of the carbene ligands and could be effectively used to tune the electronic properties of the metal center. The carbonyl groups on the carbene ligand backbone provided a particularly sensitive probe for the assessment of the metal-to-ligand π donation. The ortho-fluorine substituents on the N-aryl groups in the carbene ligands interacted with the other ligands on rhodium, determining the conformation of the complexes and creating a pocket suitable for the coordination of water to the metal center. Computational studies were used to explain the influence of the fluorinated N-substituents on the electronic properties of the ligand and evaluate the relative contribution of the σ- and π-interactions to the ligand-metal interaction.

Electronic structure and circular dichroism of tris(bipyridyl) metal complexes within density functional theory.

Time-dependent density functional theory (TD-DFT) has been employed to calculate the electronic circular dichroism (CD) spectra of tris-bidentate iron group complexes [M(L-L)(3)](2+) (M = Fe, Ru, Os; L-L = 2,2'-bipyridine). The simulated CD spectra are compared with the experiment, and reasonably good agreement is obtained. In this study, much effort has been made to interpret the exciton CD arising from the long-axis-polarized pi --> pi* excitations in the ligands of the complexes. Metal-ligand orbital interactions as well as the origin of the optical activity of the exciton transitions have been elucidated in connection with the detailed analysis of the TD-DFT results within a general model that is applicable to similar chiral compounds.

Density functional theory study of the electron paramagnetic resonance parameters and the magnetic circular dichroism spectrum for model compounds of dimethyl sulfoxide reductase.

We report a density functional theory (DFT) study of electron paramagnetic resonance (EPR) parameters for complexes modeling the paramagnetic center Mo(V) of the molybdoenzyme dimethyl sulfoxide reductase. We pay special attention to the Mo-OH link to find the most likely geometry and orientation of the metal center in the enzyme and provide an analysis of the physical origin of the g-values in terms of magnetically induced orbital mixing. We also present a study of the magnetic circular dichroism (MCD) spectrum for a complex that models the Mo(V) center of the enzyme. The calculation of the MCD-parameters that give rise to the spectrum was performed using a newly implemented method based on time-dependent DFT. On the basis of the theoretical calculations, it was possible to give a full assignment of the bands of the MCD spectrum for the enzyme.

Density functional theory study of the magnetic circular dichroism spectra of molybdenyl complexes.

We report a density functional theory (DFT) study of the magnetic circular dichroism (MCD) spectra for four molybdenyl complexes: [MoOCl(4)](-), [MoO(S(2)C(2)H(4))(2)](-), [(Tp*)MoO(bdt)], and [(L3S)MoO(bdt)] (Tp* = hydrotris (3,5-dimethyl-1-pyrazolyl) borate; L3S = (2-dimethylethane-thiolate)bis(3,5-dimethylpyrazolyl)-methane; bdt =1,2-benzenedithiolate). The simulation of the temperature dependent MCD-bands (C-terms) that give rise to the spectra was performed using a method based on time-dependent DFT. In this method, the C-parameters are calculated by including spin-orbit perturbations. On the basis of the theoretical calculations, new or additional assignments are made for the MCD spectra of the complexes; specially for [(L3S)MoO(bdt)], for which case only tentative assignments of the excitations have been proposed in recent years.

A magnetic and electronic circular dichroism study of azurin, plastocyanin, cucumber basic protein, and nitrite reductase based on time-dependent density functional theory calculations.

The excitation, circular dichroism, magnetic circular dichroism (MCD) and electron paramagnetic resonance (EPR) spectra of small models of four blue copper proteins are simulated on the TDDFT/BP86 level. X-Ray diffraction geometries are used for the modeling of the blue copper sites in azurin, plastocyanin, cucumber basic protein, and nitrite reductase. Comparison with experimental data reveals that the calculations reproduce most of the qualitative trends of the observed experimental spectra with some discrepancies in the orbital decompositions and the values of the excitation energies, the g( parallel) components of the g tensor, and the components of the A tensor. These discrepancies are discussed relative to deficiencies in the time-dependent density functional theory (TDDFT) methodology, as opposed to previous studies which address them as a result of insufficient model size or poor performance of the BP86 functional. In addition, attempts are made to elucidate the correlation between the MCD and EPR signals.

Theoretical analysis of the resonance assisted hydrogen bond based on the combined extended transition state method and natural orbitals for chemical valence scheme.

We have analyzed hydrogen bonding in a number of species, containing from two to four hydrogen bonds. The examples were chosen in such a way that they would enable us to examine three different hydrogen bonds involving OH-O, NH-O, and NH-N. A common feature of the investigated systems is that they all are expected to exhibit resonance assisted hydrogen bonding (RAHB) in the electronic pi-framework. Our analysis was based on a recently developed method that combines the extended transition state scheme with the theory of natural orbitals for chemical valence (ETS-NOCV). We find that hydrogen bonding is associated with charge rearrangement in both the electronic sigma-framework (Deltarho(sigma)) and the electronic pi-framework (Deltarho(pi)). However the stabilization due to Deltarho(sigma) is four times as important as the stabilization (RAHB) due to Deltarho(pi). Stabilization due to the electrostatic interaction (DeltaE(elstat)) between the two monomers that are brought together to form the hydrogen bonds is also important. However DeltaE(el) cannot alone account for the strength of the hydrogen bonds as it is more than compensated for by the repulsive Pauli repulsion (DeltaE(Pauli)). When N' is part of an aromatic ring, N'H-O and N'H-N bonds are similar in strength to OH-O links involving carboxylic groups. However, NH-O bonds involving amide groups (-NH(2)) are considerably weaker than the OH-O links mentioned above. In systems with different hydrogen bonds, their relative strength is determined collectively in such a way as to optimize the total interaction. This can result in that one of the bonds (OH-O, NH-O, and NH-N) becomes particularly strong or exceptionally weak. Even within the same dimer two X'-HX bonds of the same type can show quite different strength.