Self-Consistent Constricted Variational Theory RSCF-CV(∞)-DFT and Its Restrictions To Obtain a Numerically Stable ΔSCF-DFT-like Method: Theory and Calculations for Triplet States.

In this paper, the relaxed self-consistent field infinite order constricted variational density functional theory (RSCF-CV(∞)-DFT) for triplet calculations is presented. Here, we focus on two main features of our implementation. First, as an extension of our previous work by Krykunov and Ziegler ( J. Chem. Theory Comput. 2013 , 9 , 2761 ), the optimization of the transition matrix representing the orbital transition is implemented and applied for vertical triplet excitations. Second, restricting the transition matrix, we introduce RSCF-CV(∞)-DFT-based numerically stable ΔSCF-DFT-like methods, the most general of them being SVD-RSCF-CV(∞)-DFT. The reliability of the different methods, RSCF-CV(∞)-DFT and its restricted versions, is examined using the benchmark test set of Silva-Junior et al. ( J. Chem. Phys. 2008 , 129 , 104103 ). The obtained excitation energies validate our approach and implementation for RSCF-CV(∞)-DFT and also show that SVD-RSCF-CV(∞)-DFT mimics very well ΔSCF-DFT, as the root-mean-square deviations between these methods are less than 0.1 eV for all functionals examined.

[A rare cause of severe gastrointestinal bleeding]. / Seltene Ursache einer transfusionspflichtigen gastrointestinalen Blutung.

Upper gastrointestinal hemorrhage represents a common symptom in the internal medical practice. This Case Report depicts a severe gastrointestinal bleeding from an angiosarcoma that required multiple blood transfusion. As shown in this case, the actual cause may initially remain unclear. Repeated tissue sampling is mandatory to confirm a diagnosis and provide adequate treatment. Primary gastrointestinal angiosarcomas are extremely rare and highly aggressive malignant tumors that tend to grow multifocally and metastasise early. The only curative treatment option is radical surgery, thus an early stage diagnosis is crucial. Palliative care for metastatic angiosarkoma is currently a topic for clinical studies.

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 Electronic Transitions in Tetrahedral d(0) Metal Oxides.

We present benchmark calculations on excitation energies based on time-dependent density functional theory (TDDFT) as well as orbital relaxed self-consistent and constricted variational DFT (RSCF-CV-DFT) with and without use of the Tamm-Dancoff approximation. The compilation contains results for the 3d complexes MnO4⁻, CrO4²â», and VO4³â», as well as the 4d congeners RuO4, TcO4⁻, and MoO4²â», and 5d homologues OsO4, ReO4⁻, and WO4²â». Considerations have been given to the local density approximation (LDA) and the functionals BP86 and PBE based on the generalized gradient approximation (GGA), as well as the hybrids B3LYP, BHLYP, and PBE0 and the length corrected functional LCBP86. We find for the 3d complexes that RSCF-CV-DFT fares better than TDDFT. Thus, in the case of RSCF-CV-DFT, the average root-mean-square deviations (RMSDs) are 0.25-0.3 eV for GGAs, 0.1 eV for B3LYP, and 0.45 eV for BHLYP. TDDFT affords RMSDs that on average range from 0.3 eV for local functionals to 0.7 eV for BHLYP with the largest fraction of Hartree-Fock (HF) exchange. TDDFT is seen to fare better among the heavier tetraoxo systems. For the 4d and 5d systems, the three functionals B3LYP, PBE0 with an intermediate fraction of HF exchange, and LCBP86 have the lowest RMSD of 0.2 eV, whereas the local functionals (LDA, BP86, BPE) and BHLYP with the highest HF fraction and LCBP86* have a somewhat larger RMSD of 0.3 eV. Nearly the same performance is observed for RSCF-CV-DFT with respect to the different functionals in the case of the 4d and 5d systems. Thus, for the heavier tetraoxo systems, the two DFT schemes are comparable in accuracy.

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.

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.

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.

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.

Applications of Time Dependent and Time Independent Density Functional Theory to the First π to π* Transition in Cyanine Dyes.

The first π → π* transition in a number of cyanine dyes was studied using both time dependent and time independent density functional methods using a coupled cluster (CC2) method as the benchmark scheme. On the basis of 10 different functionals, it was concluded that adiabatic time dependent density functional theory (ATDDFT) almost independently of the functional gives rise to a singlet-triplet separation that is too large by up to 1 eV, leading to too high singlet energies and too low triplet energies. This trend is even clearer when the Tamm-Dancoff (TD) approximation is introduced and can in ATDDFT/TD be traced back to the representation of the singlet-triplet separation by a HF-type exchange integral between π and π*. The time independent DFT methods (ΔSCF and RSCF-CV-DFT) afford triplet energies that are functional independent and close to those obtained by ATDDFT. However, both the singlet energies and the singlet-triplet separations increases with the fraction α of HF exchange. This trend can readily be explained in terms of the modest magnitude of a KS-exchange integral between π and π* in comparison to the much larger HF-exchange integral. It was shown that a fraction α of 0.5 affords good estimates of both the singlet energies and the singlet-triplet separations in comparison to several ab initio benchmarks.

Derivation of the RPA (Random Phase Approximation) Equation of ATDDFT (Adiabatic Time Dependent Density Functional Ground State Response Theory) from an Excited State Variational Approach Based on the Ground State Functional.

The random phase approximation (RPA) equation of adiabatic time dependent density functional ground state response theory (ATDDFT) has been used extensively in studies of excited states. It extracts information about excited states from frequency dependent ground state response properties and avoids, thus, in an elegant way, direct Kohn-Sham calculations on excited states in accordance with the status of DFT as a ground state theory. Thus, excitation energies can be found as resonance poles of frequency dependent ground state polarizability from the eigenvalues of the RPA equation. ATDDFT is approximate in that it makes use of a frequency independent energy kernel derived from the ground state functional. It is shown in this study that one can derive the RPA equation of ATDDFT from a purely variational approach in which stationary states above the ground state are located using our constricted variational DFT (CV-DFT) method and the ground state functional. Thus, locating stationary states above the ground state due to one-electron excitations with a ground state functional is completely equivalent to solving the RPA equation of TDDFT employing the same functional. The present study is an extension of a previous work in which we demonstrated the equivalence between ATDDFT and CV-DFT within the Tamm-Dancoff approximation.

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.

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.

Modeling Transition Metal Reactions with Range-Separated Functionals.

The performance of range-separated functionals for the calculation of reaction profiles of organometallic compounds is considered. Sets of high-level computational results are used as reference data for the most part. The benchmark data include a number of reactions involving small molecules reacting with the Pd atom, PdCl(-), PdCl2, and a Ni atom, the reaction of a model Grubbs catalyst, and the ligand binding in a real Grubbs catalyst. Range-separated functionals are found to improve upon most standard local functionals especially if an optimized range-separation parameter is used. They do not represent an improvement upon the better-performed global hybrid functionals or a local functional that includes a larger number of adjustable parameters. Some unusual results for molecule-molecule interaction energies are observed and explained by a detailed analysis of the contributions to the bonding energies. The influence of range separation on the barriers and reaction energies is also investigated.

Self-consistent Formulation of Constricted Variational Density Functional Theory with Orbital Relaxation. Implementation and Applications.

We introduce here a new version of the constricted nth order variational density functional method (CV(n)-DFT) in which the occupied excited state orbitals are allowed to relax in response to the change of both the Coulomb and exchange-correlation potential in going from the ground state to the excited state. The new scheme is termed the relaxed self-consistent field nth order constricted variational density functional (RSCF-CV(n)-DFT) method. We have applied the RSCF-CV(n)-DFT scheme to the nσ→π* transitions in which an electron is moved from an occupied lone-pair orbital nσ to a virtual π* orbital. A total of 34 transitions involving 16 different compounds were considered using the LDA, B3LYP, and BHLYP functionals. The DFT-based results were compared to the "best estimates" (BE) from high level ab initio calculations. With energy terms included to second order in the variational parameters (CV(2)-DFT), our theory is equivalent to the adiabatic version of time dependent DFT . We find that calculated excitation energies for CV(2)-DFT using LDA and BHLYP differ substantially from BE with root-mean-square-deviations (RMSD) of 0.87 and 0.65 eV, respectively, whereas B3LYP affords an excellent fit with BE at RMSD = 0.33 eV. Resorting next to CV(∞)-DFT where energy terms to all orders in the variational parameters are included results for all three functionals in too high excitation energies with RMSD = 1.62, 1.14, and 1.48 eV for LDA, B3LYP, and BHLYP, respectively. Allowing next for a relaxation of the orbitals (nσ,π*) that participate directly in the transition (SCF-CV(n)-DFT) leads to an improvement with RMSD = 0.49 eV (LDA), 0.50 eV (B3LYP), and 1.12 eV (BHLYP). The best results are obtained with full relaxation of all orbitals (RSCF-CV(n)) where now RMSD = 0.61 eV (LDA), 0.32 eV (B3LYP), and 0.52 eV (BHLYP). We discuss finally the relation between RSCF-CV(n) and Slater's ΔSCF method and demonstrate that the two schemes affords quite similar results in those cases where the excitation can be described by a single orbital displacement (nσ→π*).

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.

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.

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.

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.

Range-Separated Exchange Functionals with Slater-Type Functions.

An implementation of range-separated density functionals utilizing the Yukawa potential and Slater-type functions is described. The density-functional part of the range-separated regime is straightforward. The exact exchange part makes use of established methods for evaluating exchange integrals over Slater-type functions but still requires new one- and two-center integrals. Equations for the one-center integrals are derived. The two-center integrals are evaluated through a combination of new equations and techniques taken from procedures for evaluating two-center Coulomb integrals over Slater-type functions. In a first application, the performance of range-separated functionals in the prediction of transition metal thermochemistry is evaluated using a database of average ligand removal energies. The range-separated functionals perform better than a GGA parent and similarly to commonly used hybrid and meta-hybrid functionals. The results were relatively insensitive to the chosen value of the attenuation parameter.