Physical nature of the chemical bond II. Valence atomic orbital and energy partitioning studies of linear nitriles

The valence atomic orbitals (VAO 's) of several linear nitriles are determined using non-empirical SCF –LCAO –MO wave functions expanded in a minimal (CN^?, HCN, FCN, C₂N₂), double-zeta (CN^?, HCN), or double-zeta + polarization (HCN) basis of Slater atomic orbitals (AO 's). The molecular energy of each system (except the double-zeta + polarization HCN system) is partitioned according to the procedure of Ruedenberg to obtain numerical values of nitrile C and N atomic and C?N bond components of the energy. In addition, the nitrile results are compared with minimal AO basis results obtained previously by other authors for homonuclear diatomics, diatomic hydrides and H₂O. The numerical data are used to test the internal self-consistency of the various definitions entering the partitioning method, i.e. whether or not analogous quantities assume similar values in chemically similar situations. The analysis of nitrile SCF –MO wave functions in terms of the set of VAO 's characteristic of the system under consideration is shown to be a promising approach to the problem of extracting useful information from the wave functions. In general, numerical results for the nitrile systems studied are fairly consistent with the concepts on which the partitioning method is based: promotion, quasi-classical interaction, sharing penetration, sharing interference and charge transfer. However, the VAO expansions for several energy components need to be investigated further and possibly revised.     

Ab initio MRD–CI calculations on cubane (neutral,carbocation, carboanion) and dissociation of nitrocubanes based on localized orbitals

Ab initio MRD –CI calculations based on localized orbitals were carried out for cubane (neutral, carbocation, carboanion) both in our customary MODPOT basis set and in an all-electron 4–31G basis set. The calculated MRD –CI charge distributions on C1 (the skeletal atom from which the H^? or H⁺ was removed) (ab initio MODPOT neutral 4.221, carbocation 3.796, carboanion 4.282; all-electron 4–31G neutral 6.171, carbocation 5.717, carboanion 6.078) indicate that the + or - charge does not remain localized on C1 but redistributes itself. This has significant implications for preparative reactions of energetically substituted cubanes. The MRD –CI population analyses differ somewhat from the SCF population analyses, especially in the calculated total overlap populations. To investigate this effect on electrostatic molecular potential contour (EMPC ) maps generated from SCF or MRD –CI wave functions, we wrote additional routines to calculate EMPC maps from MRD –CI wave functions. The EMPC maps generated from SCF or MRD –CI wave functions are different if the molecule needs an MRD –CI multideterminant wave function to describe it adequately. The EMPC map is a one-electron property. One-electron properties are derived from the 1-matrix. The 1-matrix is different for SCF or MRD –CI wave functions. Thus, all the one-electron properties (EMPC maps, population analyses, bond deviation indices, etc.) are different when calculated from SCF or MRD –CI wave functions if MRD –CI wave functions are necessary to describe a system properly. We calculate these one-electron properties from the 1-matrix from the final natural orbitals. Our preliminary calculations for the dissociation pathway indicate it takes more energy to dissociate a bond in 1-nitrocubane than in octanitrocubane. Even in their ground electronic states at equilibrium geometry, both 1-nitrocubane and octanitrocubane require MRD –CI wave functions to describe them properly. The c² of the single determinant SCF wave function is only 0.8401 for 1-nitrocubane and 0.8300 for octanitrocubane. There are contributions from skeletal excitations as there are for cubane itself as well as excitations involving the nitrogroup. As the bond in nitrocubane is dissociated to 8.00 bohrs, the c² of the SCF contribution drops to only 0.4606 (1-nitrocubane) and 0.4445 (octanitrocubane). At this C₁? N₁ intermolecular distance, the largest excitations are in the C₁? N₁ bond: (C₁? N₁)² → (C₁? N₁*)², (C₁? N₁) → (C₁? N₁*). We also calculated the first electronically excited state for the dissociation pathway for selected points for both 1-nitrocubane and octanitrocubane.     

Ab initio restricted Hartree—Fock and configuration interaction calculations for Li3

Calculations of the energies for various geometries of Li₃, involving linear and perpendicular approaches of a Li atom to a Li₂ molecule, are described. The basis set was of double zeta plus polarization contracted gaussian orbitals, and both a single configuration restricted Hartree—Fock function, and a five configuration function were used. The perpendicular approach was found to give the lowest energy, and the most stable form of Li₃ is estimated to be about 10 kcal/mole lower in energy than Li + Li₂.     

Potential energy surface of the system CO + O: An ab initio study on the O?CO bond formation

Quantum mechanical computations on the potential energy surface of the system CO + O at an SCF –MO ab initio level are presented and discussed. The calculations are performed on a minimal basis set of atomic functions. Comparison with results obtained with extended basis sets are also presented.     

The nature of the chemical bond in UO2

The nature of the chemical bond in UO₂ was analyzed taking into account the X-ray photoelectron spectroscopy (XPS) structure parameters of the valence and core electrons, as well as the relativistic discrete variation electronic structure calculation results for this oxide. The ionic/covalent nature of the chemical bond was determined for the UO₈ (D_4h) cluster, reflecting uranium's close environment in UO₂, and the U₁₃O₅₆ and U₆₃O₂₁₆ clusters, reflecting the bulk of solid uranium dioxide. The bar graph of the theoretical valence band (from 0 to ~35 eV) of XPS spectrum was built such that it was in satisfactory agreement with the experimental spectrum of a UO₂ single crystalline thin film. It was shown that unlike the crystal field theory results, the covalence effects in UO₂ are significant due to the strong overlap of the U 6p and U 5f atomic orbitals with the ligand orbitals, in addition to the U 6d atomic orbital (AO). A quantitative molecular orbital (MO) scheme for UO₂ was built. The contribution of the MO electrons to the chemical bond covalence component was evaluated on the basis of the bond population values. It was found that the electrons of inner valence molecular orbitals (IVMO) weaken the chemical bond formed by the electrons of outer valence molecular orbitals (OVMO) by 32% in UO₈ and by 25% in U₆₃O₂₁₆.     

Usefulness of modified virtual orbitals in multireference CI procedure illustrated by calculations on lithium clusters

The maximization of the exchange interaction between the canonical Hartree–Fock virtual and occupied orbitals leads to a transformed set of virtual orbitals which are well suited as one-electron functions for CI calculations. The procedure, generally known for a long time is seldom applied, despite its simplicity and very low computational demand. However, it is found to be particularly useful in the case of multireference CI, since an improved energy is obtained with a considerable shortening of the CI expansion. Moreover, in the final CI wave function, several configurations appear with considerable weight, thus allowing an easy choice of additional configurations to be inserted in the definition of a new zero-order wave function. The efficiency of the computational procedure is discussed for the case of a Li₆ cluster of D_3h symmetry and for the NaCO and PdCO complexes. Results are reported for the relative stability of four different geometrical arrangements of the Li₆ cluster.     

Natural Bond Orbital Study on the Nature of Binding in the H2 Molecule

We use the natural bond orbital (NBO) method to decompose a MO wavefunction into the intuitive valence bond (VB) structures. At least two natural orbital type MO are required to describe the essential binding of the H₂ molecule at all inter nuclear distances. At first the MO wavefunction is transformed into an unrestricted Hartree-Fock wave-function consisted of non-orthogonal localized orbitals u' and v', and then the NBO method is used to decompose u' and v' into the physical meaningful orthogonal localized orbitals. Our results show that the orbitals u' and v' are decomposed into an atomic and an overlap parts. The latter part gives rise to the conventional ionic structure in the VB picture.     

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.     

Numerical MC SCF and CI calculations of the ground-state potential energy curve of N2

The numerical procedure of McCullough is used in calculations of Hartree-Fock and MC SCF wavefunctions for ground state of N₂. The latter are derived using the complete set of 18 spin and symmetry adapted configurations in the space of MOs that arise from 2p atomic orbitals. An increase in dissociation energy of 0.17 eV is observed when compared to MC SCF calculations in a large basis of Slater-type functions and the same set of configurations. Integrals involving the numerical MC SCF MOs are used in CI calculations in which substitutions involving the 1s and 2s electrons are included. The increase in dissociation energy due to numerical versus basis set valence CI is 0.08 eV. Spectroscopic constants and molecular quadrupole moments are reported.     

Self consistent molecular orbital calculations on polyatomic molecules: Gaussian approximation for two-electron integrals

A MO –LCAO –SCF treatment is performed on water, ammonia and methane using a recently proposed approximate method. The procedure is found capable of predicting total and orbital energies in close agreement with the results of accurate computations using double ζ basis sets of Slater type orbitals. A comparison is made with the results of similar approximate ab initio procedures.     

Rare-earth orthovanadates: Covalency,chemical bonding,and optical spectra

Investigations of electronic structure and optical spectra were made for yttrium orthovanadate, and for rare earth orthovanadates RVO₄, where R = Ce, Nd, Eu, Tb, Dy, Gd, and Yb. The Hartree-Fock-Slater model was used in conjunction with a numerical discrete variational method to calculate energy levels and wavefunctions for molecular clusters (VO₄)^3? and (RO₈)^13? found in the orthovanadate crystal lattice. Analysis of the MO charge and spin densities reveals a significant involvement of rare earth 4f orbitals in chemical bonding, through hybridization of of R-5p and mixing with O-2p atomic orbitals. The MO energy level diagrams provide a satisfactory semiquantitative interpretation of the experimental excitation, reflection, and luminescence spectra. Energy transfer from the vanadate ion to the rare-earth ion is understood in terms of covalent mixing between metal and shared O-2p orbitals for neighboring (VO₄)^3? and (RO₈)^13? clusters. The relative luminescent efficiency of some rare-earth elements is explained on the basis of the calculated energy level diagrams.     

Generalized-molecular-orbital theory: Simple multiconfiguration self-consistent-field method

Even after completing a multiconfiguration self-consistent-field (MCSCF ) calculation, one must often include additional configuration interaction (CI ) to obtain quantitative or semiquantitative results. There is some question of whether the prior MCSCF calculation is worthwhile, if additional CI is needed later. We have developed a new MCSCF computational method, which, because of our assumptions about the nature of the configurations, yields one Fock-like operator for all the “filled” orbitals (high occupation numbers) and a second Fock-like operator for all the “virtual” orbitals (low occupation numbers). Since there are only two matrices to build, our method is considerably faster than other MCSCF approaches. Because of these similarities to standard molecular-orbital (MO ) calculations, we have termed our approach generalized-molecular-orbital (GMO ) theory. However, the “virtual” orbitals, unlike those of standard MO theory, are optimized to correlate the “filled” ones and can he used in a subsequent CI calculation. Results are presented for the correlation energy of H₂O, the spectroscopic constants of N₂, the singlet–triplet energy separations in CH₂, and the nature of the chromium–chromium quadruple bond. Although these results are at a very low level of CI , the GMO approach appears to correct for the gross deficiencies of the single-determinant SCF procedure.     

Multiple basis sets in calculations of triples corrections in coupled-cluster theory

Multiple basis sets are used in calculations of perturbational corrections for triples replacements in the framework of single-reference coupled-cluster theory. We investigate a computational procedure, where the triples correction is calculated from a reduced space of virtual orbitals, while the full space is employed for the coupled-cluster singles-and-doubles model. The reduced space is either constructed from a prescribed unitary transformation of the virtual orbitals (for example into natural orbitals) with subsequent truncation, or from a reduced set of atomic basis functions. After the selection of a reduced space of virtual orbitals, the singles and doubles amplitudes obtained from a calculation in the full space are projected onto the reduced space, the remaining set of virtual orbitals is brought into canonical form by diagonalizing the representation of the Fock operator in the reduced space, and the triples corrections are evaluated as usual. The case studies include the determination of the spectroscopic constants of N₂, F₂, and CO, the geometry of O₃, the electric dipole moment of CO, the static dipole polarizability of F⁻, and the Ne⋯Ne interatomic potential. Received: 28 December 1996 / Accepted: 8 April 1997     

One electron orbitals intrinsic to the reduced hamiltonian

One electron orbitals are determined from the reduced hamiltonian by a simple one-step diagonalization. These reduced hamiltonian orbitals (RHO's) are uniquely determined and virtual orbitals obtained in this procedure are on a par with filled orbitals. These RHO's appear well suited for CI calculations. Minimum basis set calculations are presented for H₂O and compared with similar SCF studies.     

The ground-state potential curve for F2

The ground-state potential curve for F₂ has been obtained using large-scale MC SCF and CI methods. MC SCF curves were obtained with the CAS SCF method using a variety of sets of active orbitals. The main conclusion from the CAS SCF calculations is that the 2π_u orbital is important. CI curves were obtained using the contracted CI method. The largest calculations contained 312000 configurations proper spin and space (d_2h) symmetry. The main conclusions from the CI calculations are that the configuration XXX are important, otherwise errors in D_e of 0.3 eV and in r_e of 0.02 Å are found. The remaining errors at the CI level are 0.08 eV for D_e, 0.005 Å for r_e and less than 10 cm^?1 for the lowest vibrational levels.     

Ab initio molecular orbital calculation of fe-porphine with a double zeta basis set

An ab initio LCAO SCF MO calculation was performed on planar Fe-porphine with a double zeta basis set consisting of 300 CGTO 's. SCF wave functions of several states of Fe-porphine and its cation were obtained. The net charge of Fe is in the range of 1.39 to 1.53. The highest occupied orbital is ascertained to be a pure porphine π-MO , 1a_1u. The calculated ionization potentials of the two highest occupied orbitals, 1a_1u and 5a_2u are 5.98 and 6.43 eV, respectively. They are in good agreement with experiments. The role of the porphine macrocycle on the oxidation of Fe is discussed in terms of gross atomic populations and with contour maps of the density difference.     

An ab initio molecular orbital study on the addition reaction of triplet nitrene to ethylene

A minimal basis set, contracted from an extensive set of primitive Gaussian type functions (GTF), was used to expand the molecular orbitals (MO) within the framework of self consistent field (SCF) theory. The results revealed that aziridine is formed in its first excited triplet state (T ₁) when ethylene is reacted with triplet nitrene. The equilibrium geometry of aziridine in its (T ₁) state had a tetrahedral CCN bond angle.     

Orbital interactions in large molecular systems: Adsorption of H2 on Ni surfaces

The concept of orbital interactions is applied to the adsorption of H₂ on to the Ni (110) and (111) surfaces. We calculate first two orbitals of a Ni cluster one of which forms an orbital pair with the σ MO and the other with the σ*MO of a H₂ molecule. Contributions of these paired orbitals of fragments to the density of states of the surface-adsorbate extended system are then examined. It is shown that the orbital of the surface that participates in electron delocalization to σ* of the H₂ molecule is located significantly below the Fermi level both in the (110) and in the (111) adsorption models. The σ MO of H₂ and its counterpart of the surface represent mainly overlap repulsion which is shown to be stronger on the (111) surface than on the (110) surface. It is feasible to understand chemical interactions of large systems by using the paired interacting orbitals.