The Bonding Nature of the Canonical Molecular Orbitals in Simple Diatomic Molecules

The bonding nature of the canonical molecular orbitals 2σg, 2σu and 3σg in the molecules N₂,O₂, F₂ and the related analogous molecular orbitals in the molecules P₂ and CO, is analysed using Weinhold's natural bond orbital set. When the canonical molecular orbitals can be well localized into natural bond orbitals, the covalent bond can be completely attributed to the bonding type natural bond orbitals. The decomposition of canonical molecular orbitals into the natural bond orbital basis then gives the weighted bond order as the component of the bonding portion in the canonical molecular orbital. The weighted bond order results match the photoelectron spectroscopy assignment quite satisfactorily.     

Analysis of chemical bonding in C2 using Dyson orbitals

Multireference configuration interaction wave functions with single and double excitations were calculated for the ¹Σ⁺_g ground state of the C₂ molecule and the excited states of C⁺₂ with symmetries ²Σ⁺_g, ²Σ^-_u, ²Π_u, and ²Π_g. The corresponding σ_g, σ_u, π_u, and π_g valence Dyson orbitals were calculated. Most of the density due to the valence electrons is accounted for by three σ_g, one σ_u, and one degenerate pair of π_u Dyson orbitals. Electron correlation plays an important role in the bond strength of C₂ by increasing the occupation of the σ_g valence orbitals and decreasing the occupation of the σ_u and π_u valence orbitals. © 1996 John Wiley & Sons, Inc.     

Photoionization of carbon disulfide: theoretical studies of outer-valence-shell partial cross sections

Studies in vertical-electronic, static-exchange approximation are reported of the outer-valence-shell (2π_g⁻¹, 2π_u⁻¹, 5σ_u⁻¹, 6σ_g⁻¹) partial-channel photoionization cross sections of CS₂. The origins of strong features present in the calculated profiles, and in corresponding experimental cross sections, are attributed to the 6σ_u(σ¹) and 7σ_g (σ¹) orbitals of Mulliken on basis of Stieltjes orbital diagnostics.     

Time‐Dependent Density Functional Theory Molecular Dynamics Simulations of Liquid Water Radiolysis

The early stages of the Coulomb explosion of a doubly ionized water molecule immersed in liquid water are investigated with time‐dependent density functional theory molecular dynamics (TD–DFT MD) simulations. Our aim is to verify that the double ionization of one target water molecule leads to the formation of atomic oxygen as a direct consequence of the Coulomb explosion of the molecule. To that end, we used TD–DFT MD simulations in which effective molecular orbitals are propagated in time. These molecular orbitals are constructed as a unitary transformation of maximally localized Wannier orbitals, and the ionization process was obtained by removing two electrons from the molecular orbitals with symmetry 1B₁, 3A₁, 1B₂ and 2A₁ in turn. We show that the doubly charged H₂O²⁺ molecule explodes into its three atomic fragments in less than 4 fs, which leads to the formation of one isolated oxygen atom whatever the ionized molecular orbital. This process is followed by the ultrafast transfer of an electron to the ionized molecule in the first femtosecond. A faster dissociation pattern can be observed when the electrons are removed from the molecular orbitals of the innermost shell. A Bader analysis of the charges carried by the molecules during the dissociation trajectories is also reported.     

Pseudosymmetry analysis of molecular orbitals

We introduce a pseudosymmetry analysis of molecular orbitals by means of the newly proposed irreducible representation measures. To do that we define first what we consider as molecular pseudosymmetry and the relationships of this concept with those of approximate symmetry and quasisymmetry. We develop a general algorithm to quantify the pseudosymmetry content of a given object within the framework of the finite group algebra. The obtained mathematical expressions are able to decompose molecular orbitals by means of the irreducible representations of any reference symmetry point group. The implementation and usefulness of the pseudosymmetry analysis of molecular orbitals is demonstrated in the study of σ and π orbitals in planar and nonplanar polycyclic aromatic hydrocarbons and the t_2g and e_g character of the d‐orbitals in the [FeH₆]^3? anion in its high spin state along the Bailar twist pathway. © 2013 Wiley Periodicals, Inc.     

Outer valence orbital momentum distributions of carbon dioxide by binary (e,2e) spectroscopy

The outer valence orbital momentum distributions of CO₂ have been reinvestigated using a high momentum resolution (0.1 a_o^?1 fwhm) binary (e,2e) spectrometer operated at 1200 eV impact energy under the non-coplanar symmetric scattering condition. Generally good agreement of the measured momentum distributions with theoretical momentum distributions calculated using literature SCF double-zeta quality wavefunctions has been obtained for the 1π_g, (1π_u + 3σ_u) and 4σ_g orbitals. Although there is a reasonable agreement of the measured momentum distributions with earlier low momentum resolution (0.4 a_o^?1 fwhm) non-coplanar measurements at 400 eV impact energy reported by Cook and Brion, given the large differences in the momentum resolutions much more definitive results are obtained in the present study. In particular, the significantly higher momentum resolution clearly shows the mixed s-p character of the 4σ_g orbital. The present study also gives a much better agreement with theory in the case of the 4σ_g momentum distribution. For each orbital the calculated and where possible the experimentally determined spherically averaged momentum distributions are compared and contrasted with their respective two-dimensional momentum and position density maps. These together with three-dimensional surface plots at selected constant density values of the four outermost orbitals are used to provide a detailed comparison of momentum-space bonding and orbital properties with their more familiar position-space counterparts in the CO₂ triatomic molecule. The calculated momentum-space density contour maps of the core orbitals exhibit rather large density oscillations and the feasibility of future experiments is discussed.     

Electron correlation effects on the shape of the Kohn–Sham molecular orbital

Even systems in which strong electron correlation effects are present, such as the large near-degeneracy correlation in a dissociating electron pair bond exemplified by stretched H₂, are represented in the Kohn–Sham (KS) model of non-interacting electrons by a determinantal wavefunction built from the KS molecular orbitals. As a contribution to the discussion on the status and meaning of the KS orbitals we investigate, for the prototype system of H₂ at large bond distance, and also for a one-dimensional molecular model, how the electron correlation effects show up in the shape of the KS σ _g orbital. KS orbitals φ^HL and φ^FCI obtained from the correlated Heitler-London and full configuration interaction wavefunctions are compared to the orbital φ^LCAO, the traditional linear combination of atomic orbitals (LCAO) form of the (approximate) Hartree-Fock orbital. Electron correlation manifests itself in an essentially non-LCAO structure of the KS orbitals φ^HL and φ^FCI around the bond midpoint, which shows up particularly clearly in the Laplacian of the KS orbital. There are corresponding features in the kinetic energy density t _s of the KS system (a well around the bond midpoint) and in the one-electron KS potential v _s (a peak). The KS features are lacking in the Hartree-Fock orbital, in a minimal LCAO approximation as well as in the exact one. Received: 11 December 1996 / Accepted: 10 January 1997     

Orbital Energy Order and the Through Bond Interaction in Homonuclear Diatomic Molecules

The homonuclear diatomic molecules are the simplest systems having both the σ framework and the lone pair orbitals n_a and _b for investigating their through space and through bond interaction. The striking orbital energy order n_g～ n_a+ n_b > n_n ～ n_a - n_b has been accounted for by the through bond interaction. However, when the p-content in the lone pair orbitals n_a and n_b decreases, one may have the reverse orbital energy order: n_g < n_g. A reverse orbital energy order has been found in F₂ and Cl₂, whose n_a and n_b are almost pure s-type atomic orbitals. The reverse order also occurs in molecule N₂ when the internuclear distance is larger man 1.5 Å. It is also found that the detail through space and through bond interaction and the eventual orbital energy order for n_g and n_u can be accounted for by the Fock operator within the localized molecular orbital space.     

Variation of calculated single bond lengths in saturated hydrocarbon and silicon hydride molecules

A series of saturated X_nH_m (X being either C or Si) molecules is studied using the MINDO /3 and MNDO quantum-chemistry procedures. We find that the proximity of H atoms tends to shorten X? X bonds, and that this effect is (especially for MINDO /3) much larger than a similar trend in experimental values. The connection between this and hybridization values calculated from localized orbitals is investigated. It is found that the hybridization model works well for X? H bonds, whereas for X? X bonds non-σ-bond effects also play a significant role. It is shown that the changes in electronic structure may be directly connected to differences in X and H atomic levels, so that similar bond length differences may be expected in other molecular orbital calculations.     

A radial probability density function for analysis of canonical molecular orbitals

A one‐dimensional probability density function, analogous to the atomic radial density for the hydrogen atom, r²R_nl(r), is defined for an arbitrary three‐dimensional density. It is obtained numerically by taking the derivative of a cumulative probability distribution with respect to the cubic root of the volume enclosed by each in a series of isosurfaces. Each point in the function is associated with a unique isosurface, and the isosurface associated with the maximum of the defined function represents the most probable isosurface with respect to the putative radius. This function therefore provides an objective selection criterion for a single isosurface to represent a three‐dimensional density. This technique is applied to set of canonical molecular orbitals. The selected threshold value varies from orbital to orbital, but the enclosed probability falls in the range of 20% to 55% for the reported orbitals. In all cases, the enclosed probability is much smaller than the common choices found in the literature. The concomitant smaller volume often makes possible a more localized interpretation and helps to clarify the conventional delocalized interpretation of molecular orbitals. For example, the isosurface plots selected by this method distinguish the formally bonding orbital in He₂ from the true bonding orbital in H₂. Examples from N₂, F₂, HF, H₂O, C₂H₆, and Ni(CO)₄ are also presented. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 310–321, 2000     

Photoelectron Spectra of Unsaturated Oxides. I. 1,4-Dioxin and related systems

The He (Iα) photoelectron spectra of the four unsaturated oxides 3,4-dihydropyran ( 6 ), γ-pyran ( 7 ), 2, 3-dihydro-1, 4-dioxin ( 9 ) and 1, 4-dioxin ( 10 ) are reported and analysed. Band assignments are based on ab-initio calculations, using the STO-3G basis set. The proposed orbital sequences (with reference to the coordinate systems given in Table 1) are, for the top three orbitals: 6 , π, nσ, nπ; 7 , 3b₁(π), 1a₂(π), 11a₁(σ); 9 , 11b(π), 12a(σ), 11a(π); 10 , 2b_3u(π), 1b_1g(π), 6a_g(σ). Finally the (almost) localized π-orbitals have been computed by the Foster-Boys localization procedure.     

Unrestricted CNDO-MO calculations. II. MnO 4 2− and CrO 4 3−

The electronic structures of the tetrahedral molecule ions MnO ₄ ^2– and CrO ₄ ^3– have been investigated within an unrestricted CNDO-MO approximation [Theoret. Chim. Acta (Berl.)20, 317 (1971)]. Calculations assuming the unpaired electron occupies the 3a ₁, 2e, and 4t2 molecular orbitals indicate that the 3a ₁ and2e orbitals have similar orbital energies and that the 4t ₂ orbital is at a higher energy. The experimentally indicated2e orbital for the unpaired electron is obtained with expanded O^1– type atomic orbitals for oxygen and valence metal orbitals of the expanded 3d and plus one ion 4p types. The metal 4s orbitals must be held to the neutral atom type. The optimum valence orbitals above with a slightly contracted 4s type metal orbitals yield the minimum total energy and places the unpaired electron in the 3a ₁ orbital. Since the contracted 4s metal orbital produces results that are not in agreement with experimental data, the method used apparently does not adequately take into account the increased electron-electron repulsions that contracted 4s orbitals produce.     

The ionization energies of bridged [14]annulenes and of dicyclohepta[cd,gh] pentalene

The ionization energies J_J, of 1,6;8,13-alkanediylidene-[14]annulenes ( 2 to 5 ) and of dicyclohepta[cd,gh]pentalene ( 1 ) have been determined by photoelectron spectroscopy, using HeI radiation. The data are interpreted in terms of Koopmans' theorem (J_J = ?ε_J) on the basis of correlation diagrams and with the help of simple molecular orbital models. If the bridge is an ethane-, propane- or butane-diylidene group, the π-orbital sequence, in descending order of orbital energies, is (in C_2v): b₁, b₂, a₂, a₁. The sequence is due to a complicated and not uniquely definable interplay of inductive, conjugative and homoconjugative effects. A detailed analysis of these effects suggests that the effective angle of twist between two consecutive basis-AOs 2p_μ, 2p_ν of the peripheral π-system should be smaller than the twist angles θ_μν determined by X-ray analysis, i.e. that the pi;-ribbon adjusts elastically and is no longer locally orthogonal to the σ-frame. In the non-alternant hydrocarbon 1 of symmetry D_2h, the sequence is 2b_2g, 3b_1u, 2b_3g, 1a_u, 2b_1u. The sequence 3b_1u above 2b_3g, _i.e. the reverse of b₂ above a₁ in the bridged [14]annulenes, is explained as being due to the interaction of the semilocalized perimeter orbitals b_1u and b_3g with the bonding (π(B_1u))and antibonding (π*(B_3g)) orbital of the central double bond. In 2 the replacement of the two latter orbitals by the Walsh-orbitals of the cyclopropane moiety leads to the sequence b₁, b₂, a₁, a₂. From the data observed for 1 to 5 and for 1,6-methano-[10]annulene [11], a crude estimate for the orbital energies of the hypothetical all-cis D_10h-[10]- and D_14h-[14]annulenes can be derived.     

Hybridization: A Chemical Bonding Nature of Atoms

Both the bonding mode and geometry can serve as the chemical bonding nature of central cation, which is essentially determined by the atomic orbital‐hybridization. In this work, we focus on the possible chemical bonding scheme of central cations on the basis of a quantitative analysis of electron domain of an atom. Starting from the hybridization of outer atomic orbitals that are occupied by valence electrons, we studied the possible orbital hybridization scheme of atoms in the periodic table and the corresponding coordination number as well as possible molecular geometries. According to distinct hybrid orbital sets, the chemical bonding of central cations can be classified into three typical types, resulting in the cations with a variety of coordination numbers ranging from 2 to 16. Owing to different hybridization modes, the highest coordination number of cations in IA and IIA groups is larger than that in IB‐VIIIB groups, and the coordination number of lanthanide elements is most abundant. We also selected NaNO₃ , Fe(NO₃ )₃•9H₂O , Zn(NO₃ )₂•6H₂O , Y(NO₃ )₃•3H₂O , and La(NO₃ )₃•6H₂O as examples to confirm the direct relationship between chemical bonding characteristics and orbital hybrid set by IR spectra. The present study opens the door to reveal the chemical bonding nature of atoms on the basis of hybridization and will provide theoretical guides in structural design at an atomic level.     

Valence-shell momentum distributions and ionization energies of carbon disulfide

Valence-shell binding energy spectra and momentum distributions of CS₂ have been measured using non-coplanar symmetric binary (e,2e) spectroscopy. The present measurements are compared with previously published binding energy spectra calculated using the many body 2ph-TDA Green's function (GF) method and the symmetry-adapted cluster configuration-interaction (SAC CI) method. The measured and the calculated binding energy spectra both show extensive population splittings particularly above 20 eV, confirming a significant breakdown of independent particle ionization picture. A relatively strong-outer valence many-body state at 17.0 eV is shown to be satellite of the (2π₀)^?1 state, in accord with earlier conclusions of photoelectron studies. Momentum distributions measured at several carefully chosen binding energies are compared with the corresponding molecular orbital momentum distributions calculated using small and extended gaussian basis sets. The good qualitative agreement between momentum distributions measured in the inner-valence region wth theoretical 4σ_m and 5σ_g orbital momentum distributions confirms the qualitative predictions of satellite parentages by GF and SAC CI calculations. Momentum and position density contour maps of individual orbitals are used to interpret the shapes and atomic characters of the experimental momentum distributions. Momentum densities of the valence orbitals of CS₂ are compared with those of the respective valence isoelectronic species CO₂     

Density functional and molecular orbital study of physical process of inversion of nitrogen trifluoride (NF3) molecule

The physical process of the umbrella inversion of the nitrogen trifluoride molecule has been studied invoking the formalisms of the density functional theory, the frontier orbital theory, and the molecular orbital theory. An intuitive structure and dynamics of evolution of the transition state for the event of inversion is suggested. The physical process of dynamic evolution of the molecular conformations between the equilibrium (C_3v) shape and the planar (D_3h) transition state has been followed by a number of molecular orbital and density functional parameters like the total energy, the eigenvalues of the frontier orbitals, the highest occupied molecular orbital and lowest unoccupied molecular orbital, the (HOMO–LUMO) gap, the global hardness and softness, and the chemical potential. The molecular conformations are generated by deforming the ∠FNF angle through steps of 2° from its equilibrium value, and the cycle is continued till the planar transition state is reached, and the geometry of each conformation is optimized with respect to the length of the N? F bond. The geometry optimization demonstrates that the structural evolution entails an associated slow decrease in the length of the N? F bond. The dipole moment at the equilibrium form is small and that at the transition state is zero and shows a strange behavior with the evolution of conformations. As the molecular structure begins to distort from its equilibrium shape by opening of the ∠FNF angle, the dipole moment starts increasing very sharply, and the trend continues very near to the transition state but abruptly vanishes at the transition state. A rationale of the strange variation of dipole moment as a function of evolution of conformations could be obtained in terms of quantum mechanical hybridization of the lone pair on the N atom. The pattern of charge density reorganization as a function of geometry evolution is a continuous depletion of charge from the F center and piling up of charge on the N center. The continuous shortening of bond length and the pattern of variation of net charge densities on atomic sites with evolution of molecular conformations predicts that the bond moment would decrease continuously. The quantum mechanical hybridization of the lone pair of the central N atom shows that the percentage of s character of the lone‐pair hybrid on the N atom decreases at a very accelerated rate, and the lone pair at the transition state is accommodated in a pure p orbital. The result of the continued destruction of asymmetry of charge distribution in the lone pair on the central N atom due to the elimination of contribution of the s orbital with evolution of molecular conformations is the sharp decrease in lone‐pair moment. The decrease in bond moment is overcompensated by the sharp fall of its offsetting component, the lone‐pair moment, resulting in a net gain in dipole moment with the evolution of molecular geometry. Since the offsetting component decreases very sharply, the net effect is a sharp rise of dipole moment with the evolution of molecular conformations just before the transition state. The lone‐pair moment is zero by virtue of the symmetry of the pure p orbital, the lone pair of the central atom in the transition state, and the sum of the bond moments is zero by symmetry of the geometry. The barrier height is quite high at ～65.45 kcal/mol, which is close to values computed through more sophisticated methods. It is argued that an earlier suggestion regarding the development of high barrier value of NF₃ system seems to be misleading and confronting with the conclusions of the density functional theory. An analysis and a comparative study of the physical components of the one‐ and two‐center energy terms reveals that the pattern of the charge density reorganization has the principal role in deciding the origin and the magnitude of barrier of inversion of the molecule and the barrier originates not from a particular energetic effect localized in a particular region of the molecule, rather the barrier originates from a subtle interplay of one‐ and two‐center components of the total energy. The decomposed energy components show that the F?F nonbonded interaction and N? F bonded interaction favor the formation of transition state, while the one‐center energy terms prohibit the formation of the transition state. The barrier principally develops from the one‐center energy components. The profile of the HOMO is isomorphic and that of the LUMO is homomorphic with the potential energy curve for the physical process of the event of umbrella inversion of the molecule. The variation of the HOMO–LUMO gap, ?ε, the global hardness, η, and the softness, S, as a function of the reaction coordinates of angular deformation of NF₃ molecule are quite consistent with the predictions of the molecular orbital and the density functional theories in connection with the deformation of molecular geometry. The profiles of ?ε, η, and S, as a function of reaction coordinates, mimic the potential energy curve of the molecule. The eigenvalues of the frontier orbitals, and the ?ε, η, S parameters are found to be equally effective theoretical parameters, like the total energy, to monitor the physical process of the inversion of pyramidal molecules. The nature of the variation of the global hardness parameter between the equilibrium shape and the transition state form for the inversion is in accordance with the principle of maximum hardness (PMH). © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2002     

An extended-valence MC SCF procedure: determination of the dissociation energies of C2, N2, O2, and F2

A series of MC SCF calculations have been carried out on C₂, N₂, O₂, and F₂ with the goal of obtaining compact wavefunctions which recover a significant fraction of the electron correlation effects important for bond dissociation. The active orbital space is varied in size, with the largest spaces including the molecular orbitals derived from 2s, 2p, 3s, 3p and 4p atomic orbitals. Several basis sets ranging in size from 5s3p to 5s4p2d1f are investigated to determine the flexibility in the basis set needed with various choices of the active orbital space. The best extended-valence MC SCF (EVMC) dissociation energies are 0.2–0.5 eV less than the experimental values, indicating that further enlargement of the active orbital space is necessary to achieve 0.1 eV accuracy in the computed dissociation energies. The EVMC calculations reveal that, for the calculation of the dissociation energies, inclusion of non-valence orbitals is much more important for O₂ and F₂ than for C₂ and N₂. The EVMC results are compared with the predictions of full fourth-order perturbation theory, coupled cluster theory, and with the best available CI calculations.     

The use of population localised orbitals to interpret molecular orbital calculations

An efficient and general method is derived to calculate population localised molecular orbitals (LMO's) from a given SCF eigenvector matrix, by reduction to an eigenvalue problem. Applications to both localised molecules (NH₃ and C₂H₂) and delocalised ones (B₂H₆, C₆H₆ and butadiene) are discussed in some detail. It is shown that unequal occupation of atomic energy levels leads to non-orthogonal LMO's. The consequences of non-orthogonal atomic hybrid orbitals are discussed, formulas for their overlap in terms of atomic occupation numbers are derived and it is shown that the occupation numbers are connected to LMO atomic orbital coefficients by various sum rules.