Advances in vibrational configuration interaction theory - part 2: Fast screening of the correlation space.

For larger molecules, the computational demands of configuration selective vibrational configuration interaction theory (cs-VCI) are usually dominated by the configuration selection process, which commonly is based on second order vibrational Møller-Plesset perturbation (VMP2) theory. Here we present two techniques, which lead to substantial accelerations of such calculations while retaining the desired high accuracy of the final results. The first one introduces the concept of configuration classes, which allows for a highly efficient exploitation of the analogs of the Slater-Condon rules in vibrational structure calculations with large correlation spaces. The second approach uses a VMP2 like vector for augmenting the targeted vibrational wavefunction within the selection of configurations and thus avoids any intermediate diagonalization steps. The underlying theory is outlined and benchmark calculations are provided for highly correlated vibrational states of several molecules.

VCI Calculations Based on Canonical and Localized Normal Coordinates for Non-Abelian Molecules: Accurate Assignment of the Vibrational Overtones of Allene.

Vibrational configuration interaction calculations (VCI) have been performed to study the impact of the nature of the underlying coordinate systems, i.e., canonical vs localized normal coordinates, on accurate vibrational structure calculations for non-Abelian molecules. Once the correlation space is represented by real-valued primitive Hartree products, the assignment of vibrational states beyond the fundamentals is usually a tedious task and is further complicated by the use of non-symmetry-adapted coordinates. Our recently presented approach based on sparse grid integration of overlap integrals of the VCI wave function with the corresponding harmonic oscillator wave function has been used to determine and assign all fundamentals and vibrational overtones of allene and its deuterated isotopologue. Excellent agreement with available experimental results is observed in all cases.

Account of non-Condon effects in time-independent Raman wavefunction theory: Calculation of the S1← S0 vibronic absorption spectrum of formaldehyde.

The time-independent eigenstate-free Raman wavefunction approach for calculating anharmonic vibronic spectra has been extended for the calculation of Herzberg-Teller contributions on the basis of an n-mode expansion of the transition electric dipole moment surface. This allows for the efficient simulation of Franck-Condon dark vibronic spectra. In addition, vibrational angular momentum terms have been implemented into this formalism, as they are important for an accurate description of vibrational wavefunctions spanning double-well potentials. This approach has been used to compute the FC-forbidden vibronic spectrum of the n → π* transition of formaldehyde based on a potential energy and transition dipole moment surfaces obtained from explicitly correlated multi-reference configuration interaction calculations. An extensive analysis of the resulting vibronic structure is provided, which allows for a detailed assignment and interpretation of the experimental spectrum.

Refined analysis of the X̃ 2A2←X̃ 1A1 photoelectron spectrum of furan.

The X̃ 2A2←X̃ 1A1 photoelectron spectrum of furan has been studied by a time-independent eigenstate-free Raman wave function approach based on multi-dimensional potential energy surfaces obtained from explicitly correlated distinguishable clusters calculations. Individual vibronic transitions with the most significant Franck-Condon factors were determined by our recently developed residual-based algorithm for the calculation of eigenpairs in conjunction with the formalism of contracted invariant Krylov subspaces. The account of anharmonic and temperature effects allowed us to explain most bands in an experimental high-resolution zero kinetic energy photoelectron spectrum. This led to the reassignment of many spectral features, as well as a refined interpretation of the intensity mechanism for the corresponding transitions.

A General Approach for Calculating Strongly Anharmonic Vibronic Spectra with a High Density of States: The X̃2B1 ← X̃1A1 Photoelectron Spectrum of Difluoromethane.

Due to a low-lying fragmentation channel, the X̃2B1 ← X̃1A1 photoelectron spectrum of difluoromethane is dominated by strong anharmonicity effects. We have used a time-independent eigenstate-free Raman wave function approach (RWF) to calculate the entire spectrum. Vibronic transitions with the most significant Franck-Condon factors were determined by employing our recently developed residual-based algorithm for the calculation of eigenpairs (RACE). An analysis of the factors controlling the accuracy of the predicted band shape is provided. The calculated spectrum is in very close agreement with experimental results.

A new efficient method for the calculation of interior eigenpairs and its application to vibrational structure problems.

Vibrational configuration interaction theory is a common method for calculating vibrational levels and associated IR and Raman spectra of small and medium-sized molecules. When combined with appropriate configuration selection procedures, the method allows the treatment of configuration spaces with up to 1010 configurations. In general, this approach pursues the construction of the eigenstates with significant contributions of physically relevant configurations. The corresponding eigenfunctions are evaluated in the subspace of selected configurations. However, it can easily reach the dimension which is not tractable for conventional eigenvalue solvers. Although Davidson and Lanczos methods are the methods of choice for calculating exterior eigenvalues, they usually fall into stagnation when applied to interior states. The latter are commonly treated by the Jacobi-Davidson method. This approach in conjunction with matrix factorization for solving the correction equation (CE) is prohibitive for larger problems, and it has limited efficiency if the solution of the CE is based on Krylov's subspace algorithms. We propose an iterative subspace method that targets the eigenvectors with significant contributions to a given reference vector and is based on the optimality condition for the residual norm corresponding to the error in the solution vector. The subspace extraction and expansion are modified according to these principles which allow very efficient calculation of interior vibrational states with a strong multireference character in different vibrational structure problems. The convergence behavior of the method and its performance in comparison with the aforementioned algorithms are investigated in a set of benchmark calculations.

Time-independent eigenstate-free calculation of vibronic spectra beyond the harmonic approximation.

The calculation of vibronic spectra and resonance Raman intensities can be performed on the basis of the Raman wavefunction (RWF) formalism. In general, the well-known sum-over-states (SOS) and time-dependent methods can be applied for calculating the RWF. We present an alternative route in which the RWF is determined pointwise in a spectral range on the basis of the inhomogeneous Schrödinger equation using an iterative subspace method, in which explicit state-by-state calculations of vibrational eigenstates are bypassed. We study this approach within the framework of vibrational configuration interaction theory in conjunction with high-level electronic structure calculations for the multidimensional Born-Oppenheimer potential energy surface. The method benefits from an implicit account of interference effects between vibrational states, so that its computational cost correlates with the required resolution in the spectra. The accuracy and efficiency of the method with respect to comparable SOS calculations are tested for the simulation of the photoelectron spectra of ClO2, HS2 (-), ZnOH(-), and Zn(H2O)(+).

Magnetic circular dichroism and computational study of mononuclear and dinuclear iron(IV) complexes.

High-valent iron(IV)-oxo species are key intermediates in the catalytic cycles of a range of O2-activating iron enzymes. This work presents a detailed study of the electronic structures of mononuclear ([FeIV(O)(L)(NCMe)]2+, 1, L = tris(3,5-dimethyl-4-methoxylpyridyl-2-methyl)amine) and dinuclear ([(L)FeIV(O)(µ-O)FeIV(OH)(L)]3+, 2) iron(IV) complexes using absorption (ABS), magnetic circular dichroism (MCD) spectroscopy and wave-function-based quantum chemical calculations. For complex 1, the experimental MCD spectra at 2-10 K are dominated by a broad positive C-term band between 12000 and 18000 cm-1. As the temperature increases up to ~20 K, this feature is gradually replaced by a derivative-shaped signal. The computed MCD spectra are in excellent agreement with experiment, which reproduce not only the excitation energies and the MCD signs of key transitions but also their temperature-dependent intensity variations. To further corroborate the assignments suggested by the calculations, the individual MCD sign for each transition is independently determined from the corresponding electron donating and accepting orbitals. Thus, unambiguous assignments can be made for the observed transitions in 1. The ABS/MCD data of complex 2 exhibit ten features that are assigned as ligand-field transitions or oxo- or hydroxo-to-metal charge transfer bands, based on MCD/ABS intensity ratios, calculated excitation energies, polarizations, and MCD signs. In comparison with complex 1, the electronic structure of the FeIV=O site is not significantly perturbed by the binding to another iron(IV) center. This may explain the experimental finding that complexes 1 and 2 have similar reactivities toward C-H bond activation and O-atom transfer.

Efficient and automatic calculation of optical band shapes and resonance Raman spectra for larger molecules within the independent mode displaced harmonic oscillator model.

In this work, an improved method for the efficient automatic simulation of optical band shapes and resonance Raman (rR) intensities within the "independent mode displaced harmonic oscillator" is described. Despite the relative simplicity of this model, it is able to account for the intensity distribution in absorption (ABS), fluorescence, and rR spectra corresponding to strongly dipole allowed electronic transitions with high accuracy. In order to include temperature-induced effects, we propose a simple extension of the time dependent wavepacket formalism developed by Heller which enables one to derive analytical expressions for the intensities of hot bands in ABS and rR spectra from the dependence of the wavepacket evolution on its initial coordinate. We have also greatly optimized the computational procedures for numerical integration of complicated oscillating integrals. This is important for efficient simulations of higher-order rR spectra and excitation profiles, as well as for the fitting of experimental spectra of large molecules. In particular, the multimode damping mechanism is taken into account for efficient reduction of the upper time limit in the numerical integration. Excited state energy gradient as well as excited state geometry optimization calculations are employed in order to determine excited state dimensionless normal coordinate displacements. The gradient techniques are highly cost-effective provided that analytical excited state derivatives with respect to nuclear displacements are available. Through comparison with experimental spectra of some representative molecules, we illustrate that the gradient techniques can even outperform the geometry optimization method if the harmonic approximation becomes inadequate.

Key hydride vibrational modes in [NiFe] hydrogenase model compounds studied by resonance Raman spectroscopy and density functional calculations.

Hydrogenase proteins catalyze the reversible conversion of molecular hydrogen to protons and electrons. While many enzymatic states of the [NiFe] hydrogenase have been studied extensively, there are multiple catalytically relevant EPR-silent states that remain poorly characterized. Analysis of model compounds using new spectroscopic techniques can provide a framework for the study of these elusive states within the protein. We obtained optical absorption and resonance Raman (RR) spectra of (dppe)Ni(µ-pdt)Fe(CO)(3) and [(dppe)Ni(µ-pdt)(µ-H)Fe(CO)(3)][BF(4)], which are structural and functional model compounds for the EPR-silent Ni-SI and Ni-R states of the [NiFe] hydrogenase active site. The studies presented here use RR spectroscopy to probe vibrational modes of the active site, including metal-hydride stretching vibrations along with bridging ligand-metal and Fe-CO bending vibrations, with isotopic substitution used to identify key metal-hydride modes. The metal-hydride vibrations are essentially uncoupled and represent isolated, localized stretching modes; the iron-hydride vibration occurs at 1530 cm(-1), while the nickel-hydride vibration is observed at 945 cm(-1). The significant discrepancy between the metal-hydride vibrational frequencies reflects the slight asymmetry in the metal-hydride bond lengths. Additionally, time-dependent density functional theory (TD-DFT) calculations were carried out to obtain theoretical RR spectra of these compounds. On the basis of the detailed comparison of theory and experiment, the dominant electronic transitions and significant normal modes probed in the RR experiments were assigned; the primary transitions in the visible wavelengths represent metal-to-metal and metal-to-ligand charge transfer bands. Inherent properties of metal-hydride vibrational modes in resonance Raman spectra and DFT calculations are discussed together with the prospects of observing such vibrational modes in metal-hydride-containing proteins. Such a combined theoretical and experimental approach may be valuable for characterization of analogous redox states in the [NiFe] hydrogenases.

A step beyond the Feltham-Enemark notation: spectroscopic and correlated ab initio computational support for an antiferromagnetically coupled M(II)-(NO)- description of Tp*M(NO) (M = Co, Ni).

Multiple spectroscopic and computational methods were used to characterize the ground-state electronic structure of the novel {CoNO}(9) species Tp*Co(NO) (Tp* = hydro-tris(3,5-Me(2)-pyrazolyl)borate). The metric parameters about the metal center and the pre-edge region of the Co K-edge X-ray absorption spectrum were reproduced by density functional theory (DFT), providing a qualitative description of the Co-NO bonding interaction as a Co(II) (S(Co) = 3/2) metal center, antiferromagnetically coupled to a triplet NO(-) anion (S(NO) = 1), an interpretation of the electronic structure that was validated by ab initio multireference methods (CASSCF/MRCI). Electron paramagnetic resonance (EPR) spectroscopy revealed significant g-anisotropy in the S = ½ ground state, but the linear-response DFT performed poorly at calculating the g-values. Instead, CASSCF/MRCI computational studies in conjunction with quasi-degenerate perturbation theory with respect to spin-orbit coupling were required for obtaining accurate modeling of the molecular g-tensor. The computational portion of this work was extended to the diamagnetic Ni analogue of the Co complex, Tp*Ni(NO), which was found to consist of a Ni(II) (S(Ni) = 1) metal center antiferromagnetically coupled to an S(NO) = 1 NO(-). The similarity between the Co and Ni complexes contrasts with the previously studied Cu analogues, for which a Cu(I) bound to NO(0) formulation has been described. This discrepancy will be discussed along with a comparison of the DFT and ab initio computational methods for their ability to predict various spectroscopic and molecular features.

Efficient time-dependent density functional theory approximations for hybrid density functionals: analytical gradients and parallelization.

In this paper, we present the implementation of efficient approximations to time-dependent density functional theory (TDDFT) within the Tamm-Dancoff approximation (TDA) for hybrid density functionals. For the calculation of the TDDFT/TDA excitation energies and analytical gradients, we combine the resolution of identity (RI-J) algorithm for the computation of the Coulomb terms and the recently introduced "chain of spheres exchange" (COSX) algorithm for the calculation of the exchange terms. It is shown that for extended basis sets, the RIJCOSX approximation leads to speedups of up to 2 orders of magnitude compared to traditional methods, as demonstrated for hydrocarbon chains. The accuracy of the adiabatic transition energies, excited state structures, and vibrational frequencies is assessed on a set of 27 excited states for 25 molecules with the configuration interaction singles and hybrid TDDFT/TDA methods using various basis sets. Compared to the canonical values, the typical error in transition energies is of the order of 0.01 eV. Similar to the ground-state results, excited state equilibrium geometries differ by less than 0.3 pm in the bond distances and 0.5° in the bond angles from the canonical values. The typical error in the calculated excited state normal coordinate displacements is of the order of 0.01, and relative error in the calculated excited state vibrational frequencies is less than 1%. The errors introduced by the RIJCOSX approximation are, thus, insignificant compared to the errors related to the approximate nature of the TDDFT methods and basis set truncation. For TDDFT/TDA energy and gradient calculations on Ag-TB2-helicate (156 atoms, 2732 basis functions), it is demonstrated that the COSX algorithm parallelizes almost perfectly (speedup ~26-29 for 30 processors). The exchange-correlation terms also parallelize well (speedup ~27-29 for 30 processors). The solution of the Z-vector equations shows a speedup of ~24 on 30 processors. The parallelization efficiency for the Coulomb terms can be somewhat smaller (speedup ~15-25 for 30 processors), but their contribution to the total calculation time is small. Thus, the parallel program completes a Becke3-Lee-Yang-Parr energy and gradient calculation on the Ag-TB2-helicate in less than 4 h on 30 processors. We also present the necessary extension of the Lagrangian formalism, which enables the calculation of the TDDFT excited state properties in the frozen-core approximation. The algorithms described in this work are implemented into the ORCA electronic structure system.

Probing valence orbital composition with iron Kbeta X-ray emission spectroscopy.

A systematic study of 12 ferric and ferrous Kbeta X-ray emission spectra (XES) is presented. The factors contributing to the Kbeta main line and the valence to core region of the spectra are experimentally assessed and quantitatively evaluated. While the Kbeta main line spectra are dominated by spin state contributions, the valence to core region is shown to have greater sensitivity to changes in the chemical environment. A density functional theory (DFT) based approach is used to calculate the experimental valence spectra and to evaluate the contributions to experimental intensities and energies. The spectra are found to be dominated by iron np to 1s electric dipole allowed transitions, with pronounced sensitivity to spin state, ligand identity, ligand ionization state, hybridization state, and metal-ligand bond lengths. These findings serve as an important calibration for future applications to iron active sites in biological and chemical catalysis. Potential applications to Compound II heme derivatives are highlighted.

Square planar bis{3,6-bis(trimethylsilyl)benzene-1,2-dithiolato}metal complexes of Cr(II), Co(III), and Rh(II): an experimental and density functional theoretical study.

The three square planar complexes [N(n-Bu)(4)](2)[Cr(II)L(2)] x 4 CH(3)CN (S = 2; 1), [N(n-Bu)(4)][Co(III)L(2)] (S = 1; 3), and [N(n-Bu)(4)](2)[Rh(II)L(2)] x 4 CH(3)CN (S = 1/2; 4) have been prepared and structurally characterized by X-ray crystallography (L(2-) represents the 3,6-bis(trimethylsilyl)benzene-1,2-dithiolate(2-)). Aerial oxidation of CH(2)Cl(2) solutions of 1 produced purple crystals of [N(n-Bu)(4)][Cr(V)OL(2)] x 2 CH(2)Cl(2) (S = 1/2; 2), the structure of which has also been determined by X-ray crystallography. The electro- and magnetochemistry of all species has been studied. Their electronic structures have been experimentally investigated by recording their electronic and EPR spectra; solid-state temperature-dependent magnetic susceptibility measurements have been performed. Density functional theoretical calculations at the ZORA-B3LYP level have been carried out for all four species in order to obtain a better understanding of the electronic structure of the square planar complexes and the one-electron reduced [Rh(I)L(2)](3-) species. The reactivity of the latter with CH(3)I has been studied, and two products have been identified: cis-[Rh(III)I(2)(L'')(L')] and cis-[Rh(III)(CH(3))(I)(L'')(L')] where (L')(-) represents the S-methylated monoanion of L(2-) and (L'')(0) is the neutral, twice-S-methylated form of (L)(2-).

Magnetic and spectroscopic properties of mixed valence manganese(III,IV) dimers: a systematic study using broken symmetry density functional theory.

Exchange coupling parameters and isotropic (55)Mn hyperfine couplings of fourteen mixed-valence Mn(III)-Mn(IV) dimers are determined using broken-symmetry density functional theory (DFT) and spin projection techniques. A systematic evaluation of density functional approaches shows that the TPSSh functional yields the best exchange coupling constants among all investigated methods, with deviations from experiment of the order of approximately 10-15%. For the prediction of (55)Mn hyperfine couplings the deficiencies of DFT in the description of core-level spin-polarization and the neglect of scalar relativistic effects lead to systematic deviations between theory and experiment that can be compensated through the use of a universal scaling factor. We determine this scaling factor to be 1.49 and demonstrate that the (55)Mn hyperfine couplings in mixed-valence Mn(III,IV) dimers can be successfully and systematically predicted with the TPSSh functional and the proposed spin projection techniques. The dependence of isotropic (55)Mn hyperfine couplings on the Mn(III) zero-field splitting values is studied in detail using a dimer for which the strong exchange approximation breaks down. In this case we apply a rigorous form of our spin projection technique that incorporates zero-field splitting contributions to the site spin expectation values. These results form the basis for future studies that aim at deducing unknown structures on the basis of computed spectroscopic parameters.

Structure of the oxygen-evolving complex of photosystem II: information on the S(2) state through quantum chemical calculation of its magnetic properties.

Twelve structural models for the S(2) state of the oxygen-evolving complex (OEC) of photosystem II are evaluated in terms of their magnetic properties. The set includes ten models based on the 'fused twist' core topology derived by polarized EXAFS spectra and two related models proposed in recent mechanistic investigations. Optimized geometries and spin population analyses suggest that Mn(iii), which is most often identified with the manganese ion at site D, is always associated with a penta-coordinate environment, unless a chloride is directly ligated to the metal. Exchange coupling constants were determined by broken-symmetry density functional theory calculations and the complete spectrum of magnetic sublevels was obtained by direct diagonalization of the Heisenberg Hamiltonian. Seven models display a doublet ground state and are considered spectroscopic models for the ground state corresponding to the multiline signal (MLS) of the S(2) state of the OEC, whereas the remaining five models display a sextet ground state and could be related to the g = 4.1 signal of the S(2) state. It is found that the sign of the exchange coupling constant between the Mn centres at positions A and B of the cluster is directly related to the ground state multiplicity, implying that interconversion between the doublet and sextet can be induced by only small structural perturbations. The recently proposed quantum chemical method for the calculation of (55)Mn hyperfine coupling constants is subsequently applied to the S(2) MLS state models and the quantities that enter into the individual steps of the procedure (site-spin expectation values, intrinsic site isotropic hyperfine parameters and projected (55)Mn isotropic hyperfine constants) are analyzed and discussed in detail with respect to the structural and electronic features of each model. The current approach performs promisingly. It reacts sensitively to structural distortions and hence may be able to distinguish between different structural proposals. Thus it emerges as a useful contributor to the ongoing efforts that aim at establishing correlations between the body of spectroscopic data available for the various S(i) states of the OEC and their actual geometric features.

A new quantum chemical approach to the magnetic properties of oligonuclear transition-metal complexes: application to a model for the tetranuclear manganese cluster of photosystem II.

The reliable correlation of structural features and magnetic or spectroscopic properties of oligonuclear transition-metal complexes is a critical requirement both for research into innovative magnetic materials and for elucidating the structure and function of many metalloenzymes. We have developed a novel method that for the first time enables the extraction of hyperfine coupling constants (HFCs) from broken-symmetry density functional theory (BS-DFT) calculations on clusters. Using the geometry-optimized tetranuclear manganese complex [Mn(4)O(6)(bpy)(6)](4+/3+) as a model, we first examine in detail the calculation of exchange coupling constants J through the BS-DFT approach. Complications arising from the indeterminacy of experimentally fitted J constants are identified and analyzed. It is found that only the energy levels derived from Hamiltonian diagonalization are a physically meaningful basis for comparing theory and experiment. Subsequently, the proposed theoretical scheme is applied to the calculation of (55)Mn HFCs of the Mn(III,IV,IV,IV) state of the complex, which is similar to the S(2) state of the oxygen-evolving complex (OEC) in photosystem II of oxygenic photosynthesis. The new approach performs reliably and accurately, and yields calculated HFCs that can be directly compared with experimental data. Finally, we carefully examine the dependence of HFC on the J value and draw attention to the sensitivity of the calculated values to the exchange coupling parameters. The proposed strategy extends naturally to hetero-oligonuclear clusters of arbitrary shape and nuclearity, and hence is of general validity and usefulness in the study of magnetic metal clusters. The successful application of the new approach presented here is a first step in the effort to establish correlations between the available spectroscopic information and the structural features of complex metalloenzymes like OEC.

Prediction of iron K-edge absorption spectra using time-dependent density functional theory.

Iron K-edge X-ray absorption pre-edge features have been calculated using a time-dependent density functional approach. The influence of functional, solvation, and relativistic effects on the calculated energies and intensities has been examined by correlation of the calculated parameters to experimental data on a series of 10 iron model complexes, which span a range of high-spin and low-spin ferrous and ferric complexes in O(h) to T(d) geometries. Both quadrupole and dipole contributions to the spectra have been calculated. We find that good agreement between theory and experiment is obtained by using the BP86 functional with the CP(PPP) basis set on the Fe and TZVP one of the remaining atoms. Inclusion of solvation yields a small improvement in the calculated energies. However, the inclusion of scalar relativistic effects did not yield any improved correlation with experiment. The use of these methods to uniquely assign individual spectral transitions and to examine experimental contributions to backbonding is discussed.

Analysis and prediction of absorption band shapes, fluorescence band shapes, resonance Raman intensities, and excitation profiles using the time-dependent theory of electronic spectroscopy.

A general method for the simulation of absorption (ABS) and fluorescence band shapes, resonance-Raman (rR) spectra, and excitation profiles based on the time-dependent theory of Heller is discussed. The following improvements to Heller's theory have been made: (a) derivation of new recurrence relations for the time-dependent wave packet overlap in the case of frequency changes between the ground and electronically excited states, (b) a new series expansion that gives insight into the nature of Savin's preresonance approximation, (c) incorporation of inhomogeneous broadening effects into the formalism at no additional computational cost, and (d) derivation of a new and simple short-time dynamics based equation for the Stokes shift that remains valid in the case of partially resolved vibrational structure. Our implementation of the time-dependent theory for the fitting of experimental spectra and the simulation of model spectra as well as the quantum mechanical calculation of the model parameters is discussed. The implementation covers all electronic structure approaches which are able to deliver ground- and excited-state energies and transition dipole moments. The technique becomes highly efficient if analytic gradients for the excited-state surface are available. In this case, the computational cost for the simultaneous prediction of ABS, fluorescence, and rR spectra is equal to that of a single excited-state geometry optimization step while the limitations of the short-time dynamics approximation are completely avoided. As a test case we discuss the well-known case of the strongly allowed 1 (1)A(g) --> 1 (1)B(u) transition in 1,3,5 trans-hexatriene in detail using method ranging from simple single-reference treatments to elaborate multireference electronic structure approaches. At the highest computational level, the computed spectra show the best agreement that has so far been obtained with quantum chemical methods for this problem.