Walsh Diagrams Research Articles

Configurational inversion of the tetrahedral molecules CH 4, SiH 4, and GeH 4

We discuss from computed density functional theory energies as well as Hartree–Fock, Kohn–Sham, and their hybrid combinations how the configurational inversion of the tetrahedral AH 4 molecules CH 4, SiH 4, and GeH 4 takes place. In contrast to methane, the configurational inversion of which can occur via a C s transition state, both silane and germane can undergo configurational inversion via a square-planar D 4h transition state. The activation energy for the inversion is 109.4, 88.6, and 93.7 kcal/mol in methane, silane, and germane, respectively, at the B3LYP/6-311G** level of theory. Walsh diagrams are analyzed to increase our knowledge with respect to the configurational inversion of these tetrahedral molecules in terms of one-electron orbitals. The way of configurational inversion is strictly determined by which of the “nonbonding” a 2u and b 1g orbitals in the D 4h structures is occupied by a pair of electrons.

Microwave spectroscopic study of the SiF3 radical: Spin-rotation interaction and molecular structure

The trifluorosilyl radical has been produced by glow discharge in hexafluorodisilane in a free space cell. The rotational spectrum of the radical has been measured from the 330 (N=22−21) GHz region down to the 90 (N=6−5) and 75 (N=5−4) GHz regions. In the lower N transitions the K=1 hyperfine components showed a distinct splitting. From the splitting of the K=1 lines it was concluded that the spin-rotation interaction constant of trifluorosilyl radical has a positive sign, which is different from that of trifluoromethyl. This difference is discussed in terms of the electronic states. The simplified discussion based on the Mulliken–Walsh diagram of the energy level is used to rationalize the difference in the sign of the interaction constants in the two radicals. The Si–F bond length is determined from the rotational constant B0 to be 1.565 Å when the bond angle F–Si–F is assumed to be 109.9° deduced in the matrix infrared spectroscopy. The spin density on the fluorine atoms is derived from the hyperfine coupling constants. The obtained molecular structure is compared with those of related molecules.

The bite angle makes the difference: a practical ligand parameter for diphosphine ligands

Over the past twenty years, a correlation between the P–M–P bite angle in diphosphine complexes and selectivity has been observed in various catalytic reactions such as hydroformylation, hydrocyanation and cross coupling. The large number of examples indicates that this correlation is not fortuitous. In order better to understand the underlying principles of the bite angle effect, we have first analysed crystal structures available in the Cambridge Crystallographic Database. Systematic searches indicate that for many bidentate diphosphine ligands the P–M–P angles concentrate in surprisingly small ranges, even if complexes of different metals in various oxidation states are considered. Several examples in the literature show that continuous electronic changes associated with changing bite angles cannot only be verified by different spectroscopic techniques, but also explained on a theoretical level (Walsh diagrams). The ligand bite angle is a useful parameter for the explanation of observed rates and selectivities and likewise for the design of ligands for new catalytic reactions.

Walsh diagram and the linear combination of bond orbital method

The slopes of the orbital energy curves in a Walsh diagram can be accounted for by the bonding behavior of the canonical molecular orbitals. The canonical molecular orbital can be characterized by decomposing it into the localized bond orbital components. The orbital energy of the canonical molecular orbital can also be decomposed into the orbital energies of the bond orbital components. For AH 2 and AH 3 systems, the p-content can account for the slopes of the bond orbitals for the lone pair ( n z ) and the directed bond ( σ AH ) types. The valence canonical molecular orbitals of a 1 species are the mixtures of n z and σ AH components. The positive slope of the a 1 type highest occupied molecular orbital is accounted for by the strong interaction between the two bond orbital components.

Pseudo second-order Jahn-Teller effects and symmetry considerations in transition metal polyhydride complexes

Pseudo second-order Jahn-Teller distortions are proposed to explain why the equatorial hydride ligands in a series of cyclopentadienyl metal polyhydrides (Cp ∗ReH 6, Cp ∗OsH 5, Cp ∗IrH 4) bend away from the Cp ring at such large angles. Restricted Hartree-Fuck (RHF) geometry optimizations on models of these complexes where the Cp* ring has been replaced by a hydride reproduce these large distortions. Walsh diagrams and Mulliken populations have been used to show that the stability provided by distortion results from maximizing the overlap population between the hydride s orbitals and the metal d orbitals. This study also includes a comparison of previous work on the structures of do MH6 complexes with our work on d 2 MH6 complexes. We found that an occupied d orbital has a significant effect on the geometry of this type of molecule due to the change in hybridization on the metal from sd 5 to spd 4. Symmetry arguments are demonstrated as a means to predict the low energy geometries of these complexes.

Stationary points on the lowest doublet and quartet hypersurfaces of the N3 radical: A comparison of molecular orbital and density functional approaches

Stationary points on the doublet and quartet hypersurfaces of the N3 system are studied using the single-configurational and multi-configurational Hartree–Fock approaches and the methods which include the dynamic correlation effects on a perturbational way or through the density functional theory (DFT). A general structure of both the hypersurfaces within the 𝒞2v symmetry is explained on the basis of the Walsh diagram and studied by a simplified configuration interaction (CI) based on a small complete active space (CAS) of five valence orbitals. The minima found in this way are studied in detail by the other methods mentioned above without the symmetry constraints. The stability of minima and existence of energetic barriers are explained by the changes of the bonding — antibonding character of some valence orbitals, observed in the Walsh diagrams for both the orbital and the Kohn–Sham energies. All the methods applied indicate, that a stable ring N3 conformer should exist with the 2B1 ground state. Alternative mechanisms for a deactivation of the linear excited states of N3 are proposed.

Molecular Structures of 10-Valence Electron Species. 4. Potential Energy Surfaces and Properties of HSiX+ (X = O, S, Se)

Potential energy surface (PES) searches of HSiX+ (X = O, S, Se) ions were performed using HF and MP2 computational methods. Split-valence triple-ζ isotropic basis sets augmented by sets of p and d polarization functions on hydrogen atoms and sets of d and f polarization functions on heavy atoms (TZP(2d,2p) and TZP(2df,2pd)) were used. Unlike those of the linear form of HOSi+, the strongly bent HSSi+ and HSeSi+ global minimum-energy structures are predicted at both HF and correlated levels of theory, in contradiction to Mulliken−Walsh's rules. Isomeric HSiS+ and HSiSe+ forms are higher-energy local minima on the studied PESs.

Molecular structures and properties of HGeX + and HXGe + (X=O, S, Se) species

Ab initio molecular orbital calculations were carried out on HGeX + and HXGe + ions (X=O, S, Se). Equilibrium geometries of the studied species were located at the HF and MP2 levels using valence triple-ζ basis set with two sets of p-polarization functions on hydrogens and two sets of d-polarization functions on heavier elements (TZP basis set). At both levels bent conformations of HXGe + ions are found to be the global minimum structures and HGeX + ions are local minimum species. The predicted bent conformations for HXGe + ions are in contradiction to the Walsh's rules, which require linear structures for the HXY molecules possessing ten valence electrons. The MP4/TZP/MP2/TZP level of calculations for the possible dissociation products (GeX, H + and GeX +, H) predicts that all title compounds are thermodynamically stable and should be observed by experimentalists. To assist such experimental studies on these experimentally elusive ions, we furnish reliable theoretical molecular parameters, rotational constants, vibrational frequencies and intensities.

Model potential method in molecular calculation: application to SrX 2 and BaX 2 (X = F, Cl, Br, and I)

Electronic structures of SrX 2 and BaX 2 (X = F, Cl, Br, and I) were calculated in the Hartree-Fock scheme using newly determined model potentials and valence basis sets. Calculated bond lengths agree well with the experimental data. BaF 2, BaCl 2, BaBr 2, and SrF 2 were calculated to be bent. Total energy minimums for the other molecules were found at the apex angle of 180°. However, the potential energy surfaces for bending motions were extremely flat for these molecules. The linear or bent nature of these molecules is discussed in detail with reference to the results of population analysis, Walsh diagram, and contour maps of MOs and of the electron density change upon molecular formation.

Model potential method in molecular calculation: application to SrX 2 and BaX 2 (X = F, Cl, Br, and I)

Electronic structures of SrX 2 and BaX 2 (X = F, Cl, Br, and I) were calculated in the Hartree—Fock scheme using newly determined model potentials and valence basis sets. Calculated bond lengths agree well with the experimental data. BaF 2, BaCl 2, BaBr 2, and SrF 2 were calculated to be bent. Total energy minimums for the other molecules were found at the apex angle of 180°. However, the potential energy surfaces for bending motions were extremely flat for these molecules. The linear or bent nature of these molecules is discussed in detail with reference to the results of population analysis, Walsh diagram, and contour maps of MOs and of the electron density change upon molecular formation

Molecular mechanism of photosynthetic oxygen evolution. A theoretical approach

Possible mechanisms for O2 evolution in photosystem I1 (PSII) are studied with a semiempirical one-electron molecular orbital procedure, the extended Hiickel method. The oxygen coupling is examined starting from some known inorganic complexes of manganese. Idealized geometries of four binuclear complexes (two di-p-oxo, one tri-p-oxo and one p-carboxylato-di-1-oxo) are distorted to transform into Oz-bound complexes. The computed Walsh diagrams are analyzed. For these structures the predicted oxidation to dioxygen is energetically unfavorable and would require two two-electron steps forming bound peroxide as an intermediate. An in-plane approach of two oxo ligands has a lower barrier for peroxide bond formation. Four tetranuclear Mn clusters are built combining binuclear complexes, a Mn404 cubane-like core, three (Mn202),(p-02) dimer of dimers cores with p4,t2-02 bound either in or out of the Mn4 plane formed by two parallel Mn202 units, and a planar T combination of two orthogonal MnzO2 units with a Mn3O2 core and p3,q2-O2 bound to three Mn atoms. The assumed pathway leading to peroxide bond formation in each of these is found to have appreciably lower energy for the (Mn202),(p-02) and the planar T models with in-plane 0-0 approach. A slight energetic advantage is calculated in the peroxide to dioxygen step when the oxo ligands are coordinated to at least three Mn ions compared to coordination to only two Mn ions. Qualitative valence bond analysis indicates that release of symmetrically coordinated dioxygen, as in cis- and trans-p-02, requires overcoming an -1-eV electronic barrier for formation of ground-state O2 (triplet). The tetranuclear models are introduced into speculative mechanisms and compared with current knowledge of the photosynthetic water oxidation reaction.

Quantum chemical investigation of the oxidation reactions of the dinuclear low-valent molybdenum carbonyl complexes containing thiolato bridges

The electronic structure and bonding of the dinuclear low-valent molybdenum thiolato-bridged complexes, [Mo 2(μ-SR) 2(CO) 8] n− ( n = 2, 1; n = 0, 2; R = CH 2COOEt), Mo 2(μ-SR) 2(CO) 6(NCCH 3) 2 ( 3) and [Mo 2(μ- S, S-C 6H 4-1,2(CO) 7] 2− ( 4), have been studied using the EHMO method. On the basis of analysis of the MO energies, orbital characteristics, atomic charges and Mulliken bond orders, the possible oxidation of these complexes was discussed. The Walsh diagram shows that further two-electron oxidation of 2 would be very unlikely. However, because 4 contains a non-planar MoS 2Mo bimetallic core, an irreversible one-electron oxidation in solution could take place to form a monoanion complex [Mo 2(μ- S, S-C 6H 4-1,2)(CO) 7(NCH)] 1−.

Electrophilic aromatic substitution and the moments method

The method of moments is used to look at an old problem in organic chemistry, namely the site specificity of electrophilic aromatic substitution. By plotting the energy difference curves between the various Wheland intermediates as a function of electron count, the topological aspects of the problem become very clear. It is shown how the fundamental electronic driving force for the selectivity is related via the moments approach to other important theoretical ideas, such as Walsh's rules and the Woodward-Hoffmann rules.

Bond-angle and bond-length dependence of bound-state resonances in inner-shell excitation spectra of SF 4

SF 4 has a pseudo-trigonal bipyramidal (C 2v) structure with a ligand missing at one of the equatorial sites (S-F bond lengths: R(ax) > R(eq)) and an approximate electronic structure of S 4+[(3s) 2(3p) 0]·(F −) 4 with unoccupied S 3pσ orbitals of a 1, b 1(ax) and b 2(eq) symmetry. Contrary to an empirical relationship between bond lengths and σ resonance energies, ab initio Δ SCF calculations for the core-to-valence bound-state resonances predict the ordering b(eq) < a 1 < b (ax), indicating that the bond angle effect is dominant in SF 4. The bond-angle dependence is discussed qualitatively in terms of crystal field theory and Walsh diagrams. The bond-length and bond-angle dependence are discussed quantitatively via the ab initio calculations.

Molecular electronic structure theory in the study of localised defects

Deep-level defect centres occupy a unique position intermediate between isolated molecules and the solid state. Their electronic wavefunctions are often strongly localised about the defect site and resemble small-molecule orbitals; but they are also embedded in an extended host, and they interact with the host matrix. The authors discuss simple concepts and models from molecular electronic structure theory which are equally useful in the study of localised defect centres. First they outline the construction and use of Walsh diagrams, which predict conformations and conformational change as a function of electron number, electronic excitation and atomic orbital energy (electronegativity). Second, they discuss in detail the approximations of the widely used local-hybrid analysis of ESR and ENDOR spectra, point to the situations where those approximations are likely to fail and give an explicit comparison of theoretical calculations at several levels of sophistication for a particular molecular analogue, the silyl radical SiH3. In both cases they concentrate on broken-bond systems of the form AB3, which represent the most important intrinsic defects in silicon and silica. The importance and utility of molecular analogies in the study of defect centres are emphasised throughout.

Small scale variational scheme incorporating local orbitals with fully flexible hybridization: A local orbital reconstruction of Walsh's rules

The idea that chemical bonding is predominantly local, and can be captured at least qualitatively by mixing of orbitals at a single center (hybridization) and between two adjacent atoms (pair-bonding) has served chemistry very well. There are, to be sure, departures from the simplest expectations of bond angles, which we describe as arising from purely geometric constraints, local energy conditions, and long-range strain. To illustrate the flexibility of the one-center hybridization/pairwise interaction description of bonding, we have devised a small variational computation, based on a Lewis bonding scheme refined by constrained unitary transformation of the atomic orbital basis. In its simplest form the model permits full one-center hybridization but only two-center bonding. The model may be extended either by perturbation theoretic corrections for delocalization or by relaxing the constraints on the basis transforms. A simple parameterization of the model permits a demonstration that Walsh rules may be developed from the simplest realization of a local description of molecular electronic structure which can provide a quantitative description of the interplay between atomic energy states, hybridization, and molecular geometry.

MRD-CI study on the isomers SiOH and HSiO

Large scale MRD-CI calculations have been carried out for various low-lying electronic states of SiOH and HSiO. The non-linear ground state equilibrium geometries were optimized at the CI level. The more stable isomer SiOH lies 61·9 ± 4·6 kJ mol-1 below HSiO (without zero point corrections). SiOH is a π-type radical, while HSiO shows σ-character, in agreement with recent E.S.R. measurements. The electronic spectra of the two isomers are quite different. The 1 2 A″ state of SiOH (T vert. = 0·37 eV) is found to be bent, in variance with Walsh's rules, while 2 2 A′ (T vert. = 2·87 eV) is a semidiffuse state. The Rydberg series starts at 4·0 eV. The low-lying X 2 A′ and 1 2 A″ states of HSiO can be derived from two different configurations depending on geometry. The transition X 2 A′ → 1 2 A″ (T vert. = 2·59 eV) is expected to have a complex spectrum since the excited state can relax either to a linear geometry or to a strongly bent structure. The semidiffuse 42 A′ state is placed at 5·00 eV vertically. The l...

The singlet-triplet splitting of the low-lying electronic states of H 2O 2+ and a comparison with isoelectronic CH 2 and CH 22+

At the MP2/6-31G(d,p) level of theory using complex orbitais, the lowest-lying (bound) singlet state of H 2O 2+ is the linear 1Δ g state 2 with an O-H bond distance of 1.199 Å. 1 is estimated to be 4̃6 kcal mol higher in energy than the lowest-lying 3Σ − g state 2 of H 2O 2+. 1 and 2 are compared to the lowest-lying singlet and triplet states of isoelectronic CH 2 and CHe 2 2+. Using Walsh diagrams, the trend in the singlet/triplet gap and bending angles is explained by the relative electronegativities between the central and substituent atoms.

Analyses of the ‘‘allowed’’ inversion barriers of H2O and NH3: Incompleteness of the Woodward–Hoffmann HOMO–LUMO symmetry ideas due to neglect of 〈φi‖ΔN A‖φi〉 molecular orbital terms

Walsh’s rules correctly attribute the ‘‘bent’’ structures of H2O and NH3 to the occupation of the 1πz→3a1 HOMO not occupied in linear BeH2 and planar BH3. In Walsh’s molecular orbital (MO) diagram E(3a1) decreases sharply with bending angle S. This has always been attributed incorrectly to changes in the 3a1 MO, mainly due to symmetry-allowed mixing with the LUMO, 4a*1. The forbidden bending of BeH2 and BH3 has been similarly ‘‘explained.’’ Using large-basis-set self-consistent field molecular orbital (SCF MO) ψs, we show that the integral Hellmann–Feynman theorem ΔEIHF≂ΔESCF much better than does the analogous second-order perturbation theory λE′′(SE′=0 and λ=S2/2, ΔH≂SH′+λH′′). ΔEIHF=〈ψ0‖ΔNA‖ψ0〉+〈ψ0‖ ΔNA‖Δψ̃〉+ΔNR≂Σni2Δ EIHFi+ΔNR, Δψ̃=(ψ/η)−ψ0, η=〈ψ0‖ψ〉, ΔEIHFi=〈φ0i‖ ΔNA‖φ0i〉+〈φ0i‖ ΔNA‖Δφ̃i〉, Δφ̃i=(φi/ηi)−φ0i, ηi=〈φ0i‖φi〉, ΔNA=ΔH−ΔNR. Both theories show a large negative 〈1πz‖ΔNA‖1πz〉 term and small 〈1πz‖ΔNA‖Δ1π̃z〉 HOMO–UMO mixing term, which is positive in ΔEIHF. The 〈1πz‖SH′‖3σ*g〉 HOMO–LUMO mixing term is small even when 3σ*g is optimized for the excited state. The ΔEIHFis and λE″is give the usual Walsh diagrams for bending of H2O and NH3, with or without MO partitioning of the nuclear repulsion change (ΔNR). However ‘‘decoupling’’ of the φ′is in ψ′ makes the λE″is unreliable. The 〈1πz‖ΔNA‖1πz〉 term acts to create a large allowed barrier to inversion for H2O and CH4, but a strong ΔNR nearly destroys an otherwise large barrier for NH3. 〈1πz‖ΔNA‖1πz〉 acts to bend the linear H2O, planar NH3, and planar CH4, with HOMO–LUMO mixing being ‘‘antibending.’’ We show that understanding of MO correlation diagrams demands consideration of the ‘‘static’’ 〈φ0i‖ΔNA‖φ0i〉 terms as well as the OMO–UMO mixing terms, which has not been appreciated by earlier workers so far as we are aware.

The quantum chemistry of valency

It is shown that the chemist's concept of valency can be put in quantum mechanical language. Such a quantification of valency opens up possibilities of new applications which are briefly reviewed. Valency is related to the equilibrium geometry of the molecule through a maximum valency principle. This principle, combined with the idea of molecular orbital valencies, provides a theoretical foundation for the well known Walsh diagrams. Valency can be used as an index of chemical reactivity and valency changes in a reaction can be used to determine the radical nature of intermediates and transition states as well as to indicate the allowed or forbidden nature of reaction paths. Molecular strain may be related to the loss of sigma valency.