Canonical Molecular Orbitals Research Articles

Frozen Virtual Natural Orbitals for Coupled-Cluster Linear-Response Theory.

The frozen-virtual natural-orbital (NO) approach, whereby the unoccupied-orbital space is constructed using a correlated density such as that from many-body perturbation theory, has proven to yield compact wave functions for determining ground-state correlation energies and associated properties, with corresponding occupation numbers providing a guide to the truncation of the virtual space. In this work this approach is tested for the first time for the calculation of higher-order response properties, particularly frequency-dependent dipole polarizabilities using coupled-cluster theory. We find that such properties are much more sensitive to the truncation of virtual space in the NO basis than in the original canonical molecular orbital (CMO) basis, with truncation errors increasing linearly with respect to the number of frozen virtual NOs. The reasons behind this poor performance include the more diffuse nature of NOs with low occupation numbers as well as the reduction in sparsity of the perturbed singles amplitudes in the NO basis and the neglect of orbital response. We tested a number of approaches to improve the performance of the NO space, including the use of a field-perturbed density to define the virtual orbitals and various external-space corrections. The truncation of the CMO space, on the other hand, yields errors in coupled-cluster dipole polarizabilities of less than 2% even after removing as much as 50% of the full virtual space. We find that this positive performance of the CMO space results from a cancellation of errors due to the truncation of the unperturbed and perturbed amplitudes, as well as sparsity of the singles amplitudes. We introduce a simple criterion called a dipole amplitude to use as a threshold for truncating the CMO basis for such property calculations.

The aromaticity of dicupra[10]annulenes.

An extensive theoretical investigation of the electronic structure of a tested fair model dicupra[10]annulene compound, based on the analysis of atom-pair delocalization indices, Bader's molecular graph, the inspection of the canonical molecular orbitals, the z components of their Nuclear Independent Chemical Shifts, NICS(0)zz, and the normalized Giambiagi multicenter delocalization indices, concludes that the perimeter aromaticity of the dicupra[10]annulene ring is consistent with both 10 and 14 π-electron Hückel aromatic 10-membered rings. In either case, the 10-membered ring encloses two 6 π-electron aromatic inner rings, hinged at the Cu-Cu bond. This work demonstrates that the aromaticity of dicupra[10]annulenes closely resembles that of naphthalene. Hence, they are best regarded as metalla-polyacenes, which could make the building blocks of extended structures such as metalated nanotubes.

Sb4Au4Sb4](2-): A designer all-metal aromatic sandwich.

We report on the computational design of an all-metal aromatic sandwich, [Sb4Au4Sb4](2-). The triple-layered, square-prismatic sandwich complex is the global minimum of the system from Coalescence Kick and Minima Hopping structural searches. Following a standard, qualitative chemical bonding analysis via canonical molecular orbitals, the sandwich complex can be formally described as [Sb4](+)[Au4](4-)[Sb4](+), showing ionic bonding characters with electron transfers in between the Sb4/Au4/Sb4 layers. For an in-depth understanding of the system, one needs to go beyond the above picture. Significant Sb → Au donation and Sb ← Au back-donation occur, redistributing electrons from the Sb4/Au4/Sb4 layers to the interlayer Sb-Au-Sb edges, which effectively lead to four Sb-Au-Sb three-center two-electron bonds. The complex is a system with 30 valence electrons, excluding the Sb 5s and Au 5d lone-pairs. The two [Sb4](+) ligands constitute an unusual three-fold (π and σ) aromatic system with all 22 electrons being delocalized. An energy gap of ∼1.6 eV is predicted for this all-metal sandwich. The complex is a rare example for rational design of cluster compounds and invites forth-coming synthetic efforts.

Estimating local bonding/antibonding character of canonical molecular orbitals from their energy derivatives. The case of coordinating lone pair orbitals

According to Koopmans theorem, the derivative of the energy of a canonical molecular orbital (MO) with respect to nuclear coordinates quantifies its bonding/antibonding character. This quantity allows predictions of bond length variation on ionisation in a panel of 19 diatomic species. In polyatomic molecules, the derivative of a MO energy with respect to a given bond length reveals the nature and the degree of the bonding/antibonding contribution of this MO with respect to this bond. Accordingly, the HOMO “lone pairs” of CO and CN− and the HOMO-2 of CH3CN are found to be antibonding with respect to the CX bond (X = N, O), whereas the HOMO of N2 is found to be bonding. With the same approach, the variation of the bonding character in the MOs of CO and CH3CN on interaction with an electron acceptor (modeled through the approach of a proton) or by applying an electric field was studied. © 2016 Wiley Periodicals, Inc.

Impact of the Kohn-Sham Delocalization Error on the 4f Shell Localization and Population in Lanthanide Complexes.

The extent of ligand to metal donation bonding and mixing of 4f (and 5d) orbitals with ligand orbitals is studied by Kohn-Sham (KS) calculations for LaX3 (X = F, Cl, Br, I), GdX3, and LuX3 model complexes, CeCl6(2-), YbCp3, and selected lanthanide complexes with larger ligands. The KS delocalization error (DE) is quantified via the curvature of the energy for noninteger electron numbers. The extent of donation bonding and 4f-ligand mixing correlates well with the DE. For Lu complexes, the DE also correlates with the extent of mixing of ligand and 4f orbitals in the canonical molecular orbitals (MOs). However, the localized set of MOs and population analyses indicate that the closed 4f shell is localized. Attempts to create situations where mixing of 4f and ligand orbitals occurs due to a degeneracy of fragment orbitals were unsuccessful. For La(III) and, in particular, for Ce(IV), Hartree-Fock, KS, and coupled cluster singles and doubles calculations are in agreement in that excess 4f populations arise from ligand donation, along with donation into the 5d shell. Likewise, KS calculations for all systems with incompletely filled 4f shells, even those with "optimally tuned" functionals affording a small DE, produce varying degrees of excess 4f populations which may be only partially attributed to 5d polarization.

Performance of Shannon-entropy compacted N-electron wave functions for configuration interaction methods

The coefficients of full configuration interaction wave functions (FCI) for N-electron systems expanded in N-electron Slater determinants depend on the orthonormal one-particle basis chosen although the total energy remains invariant . Some bases result in more compact wave functions, i.e. result in fewer determinants with significant expansion coefficients. In this work, the Shannon entropy, as a measure of information content, is evaluated for such wave functions to examine whether there is a relationship between the FCI Shannon entropy of a given basis and the performance of that basis in truncated CI approaches. The results obtained for a set of randomly picked bases are compared to those obtained using the traditional canonical molecular orbitals, natural orbitals, seniority minimising orbitals and a basis that derives from direct minimisation of the Shannon entropy. FCI calculations for selected atomic and molecular systems clearly reflect the influence of the chosen basis. However, it is found that there is no direct relationship between the entropy computed for each basis and truncated CI energies.

Ionization of vitamin C in gas phase: Theoretical study

In this work, the gas phase ionization energies and photoelectron spectra of four important conformers of vitamin C were calculated. Symmetry adapted cluster/configuration interaction methodology employing the single and double excitation operators (SAC-CI SD-R) along with D95++(d,p) basis set were used for the calculations. Thermochemistry calculations were also performed on all possible conformers of vitamin C to find the relative stability of conformers in the gas phase. The calculated ionization bands of each conformer were assigned by calculating the contribution of natural bonding orbital (NBO) in the calculated canonical molecular orbitals involved in the ionization. SAC-CI calculations showed that the first ionization band of vitamin C is related to the π electrons of CC bond of the ring of molecule although, there is the lone electron pairs of oxygen atoms and π electrons of CO bond in the molecule.

On the nature of chemical bonding in the all-metal aromatic [Sb3Au3Sb3](3-) sandwich complex.

In a recent communication, an all-metal aromatic sandwich [Sb3Au3Sb3](3-) was synthesized and characterized. We report herein a density-functional theory (DFT) study on the chemical bonding of this unique cluster, which makes use of a number of computational tools, including the canonical molecular orbital (CMO), adaptive natural density partitioning (AdNDP), Wiberg bond index, and orbital composition analyses. The 24-electron, triangular prismatic sandwich is intrinsically electron-deficient, being held together via six Sb-Sb, three Au-Au, and six Sb-Au links. A standard, qualitative bonding analysis suggests that all CMOs are primarily located on the three Sb3/Au3/Sb3 layers, three Au 6s based CMOs are fully occupied, and the three extra charges are equally shared by the two cyclo-Sb3 ligands. This bonding picture is referred to as the zeroth order model, in which the cluster can be formally formulated as [Sb3(1.5+)Au3(3-)Sb3(1.5+)](3-) or [Sb3(0)Au3(3-)Sb3(0)]. However, the system is far more complex and covalent than the above picture. Seventeen CMOs out of 33 in total involve remarkable Sb → Au electron donation and Sb ← Au back-donation, which are characteristic of covalent bonding and effectively redistribute electrons from the Sb3 and Au3 layers to the interlayer edges. This effect collectively leads to three Sb-Au-Sb three-center two-electron (3c-2e) σ bonds as revealed in the AdNDP analyses, despite the fact that not a single such bond can be identified from the CMOs. Orbital composition analyses for the 17 CMOs allow a quantitative understanding of how electron donation and back-donation redistribute the charges within the system from the formal Sb3(0)/Au3(3-) charge states in the zeroth order model to the effective Sb3(1.5-)/Au3(0) charge states, the latter being revealed from the natural bond orbital analysis.

Concentric dual π aromaticity in bowl-like B30 cluster: an all-boron analogue of corannulene.

A chemical bonding model is presented for the bowl-like C5v B30 global-minimum cluster with a central pentagonal hole. The B30 cluster is composed of three concentric boron rings: first B5, second B10, and third B15. The first and second B rings constitute an inner double-chain ribbon and support a delocalized π sextet. The second and third rings form an outer double-chain ribbon, where 14π delocalized electrons are situated. The unique π systems lead to concentric dual π aromaticity for B30, a concept established from concerted computational data on the bases of canonical molecular orbital (CMO) analysis, adaptive natural density partitioning (AdNDP), nucleus-independent chemical shifts (NICS), and natural charge calculations. A proposal is put forward that the bowl-like B30 cluster is an exact all-boron analogue of corannulene (C20H10), a fragment of C60 fullerene. The bonding nature of corannulene is revisited and fully elucidated herein. A comparison of the bonding patterns in bowl-like C5v B30 cluster and two other structural isomers (Cs and C1) unravels the mechanism as to why the defective hole prefers to be positioned at the center.

DFT study of the 1JCF coupling constant in XCHFX fragments (X=O and S)

Abstract The fluorine (reverse) Perlin-like effect is a coupling constant-related phenomenon that can be useful to investigate the stereochemistry and molecular properties of organofluorine compounds. While 1,3-dioxane is known to exhibit the Perlin effect ( 1 J C-Hax 1 J C-Heq ) and 1,3-dithiane experiences the reverse Perlin effect ( 1 J C-Hax > 1 J C-Heq ), both their fluorinated analogues 2-fluoro-1,3-dioxane ( 1 ) and 2-fluoro-1,3-dithiane ( 2 ) showed the reverse fluorine Perlin-like effect (| 1 J C-Fax | > | 1 J C-Feq |). There has been some recent discussion in the literature about the origin of these effects, either as being due to hyperconjugation or dipolar interactions, but our findings indicate that while the angular dependence of 1 J C-Fax in 2-fluorotetrahydropyran is consistent with the molecular dipole moment (ruled by the orientation of local dipole vectors from C F bond and oxygen electron lone pairs), the reverse fluorine Perlin-like effect in 1 and 2 is due to a combination of hyperconjugative, steric and electrostatic interactions. The role of C X and n X orbitals on the Fermi contact term for the 1 J C F coupling constant has also been analyzed on the basis of canonical molecular orbitals.

Efficiency of perturbation-selection and its orbital dependence in the SAC-CI calculations for valence excitations of medium-size molecules.

The efficiency and accuracy of the perturbation-selection used in the symmetry-adapted cluster-configuration interaction (SAC-CI) calculations are investigated for several low-lying valence excited states of 21 medium-size molecules, including typical chromophores with heterocyclic macrocycles (free-base porphine, coumarin, indole, and BODIPY), nucleobases, amino acids (tyrosine and tryptophan), polycyclic aromatic hydrocarbons, and organometallics (ferrocene and Re(bpy)(CO)4+1). Benchmark SAC-CI calculations with up to 110 million operators are performed. The efficiency of the perturbation-selection depends on the molecular orbitals (MOs); therefore, the canonical MO and localized MO (LMO) obtained by Pipek-Mezey's method are examined. Except for the highly symmetric molecules, using LMOs improves the efficiency and accuracy of the perturbation-selection. With using LMOs and perturbation-selection, sufficiently reliable results can be obtained in less than 10% of the computational costs required for the full-dimensional calculations. The perturbation-selection with LMOs is suggested to be a promising method for excited states in larger molecular systems.

Symmetry adapted cluster–configuration interaction calculation of the photoelectron spectra of famous biological active steroids

The photoelectron spectra of some famous steroids, important in biology, were calculated in the gas phase. The selected steroids were 5α-androstane-3,11,17-trione, 4-androstane-3,11,17-trione, cortisol, cortisone, corticosterone, dexamethasone, estradiol and cholesterol. The calculations were performed employing symmetry-adapted cluster/configuration interaction (SAC–CI) method using the 6-311++G(2df,pd) basis set. The population ratios of conformers of each steroid were calculated and used for simulating the photoelectron spectrum of steroid. It was found that more than one conformer contribute to the photoelectron spectra of some steroids. To confirm the calculated photoelectron spectra, they compared with their corresponding experimental spectra. There were no experimental gas phase HeI photoelectron spectra for some of the steroids of this work in the literature and their calculated spectra can show a part of intrinsic characteristics of this molecules in the gas phase. The canonical molecular orbitals involved in the ionization of each steroid were calculated at the HF/6-311++g(d,p) level of theory. The spectral bands of each steroid were assigned by natural bonding orbital (NBO) calculations. Knowing the electronic structures of steroids helps us to understand their biological activities and find which sites of steroid become active when a modification is performing under a biological pathway.

Localization of molecular orbitals: from fragments to molecule.

Conspectus Localized molecular orbitals (LMO) not only serve as an important bridge between chemical intuition and molecular wave functions but also can be employed to reduce the computational cost of many-body methods for electron correlation and excitation. Therefore, how to localize the usually completely delocalized canonical molecular orbitals (CMO) into confined physical spaces has long been an important topic: It has a long history but still remains active to date. While the known LMOs can be classified into (exact) orthonormal and nonorthogonal, as well as (approximate) absolutely localized MOs, the ways for achieving these can be classified into two categories, a posteriori top-down and a priori bottom-up, depending on whether they invoke the global CMOs (or equivalently the molecular density matrix). While the top-down approaches have to face heavy tasks of minimizing or maximizing a given localization functional typically of many adjacent local extrema, the bottom-up ones have to invoke some tedious procedures for first generating a local basis composed of well-defined occupied and unoccupied subsets and then maintaining or resuming the locality when solving the Hartree-Fock/Kohn-Sham (HF/KS) optimization condition. It is shown here that the good of these kinds of approaches can be combined together to form a very efficient hybrid approach that can generate the desired LMOs for any kind of gapped molecules. Specifically, a top-down localization functional, applied to individual small subsystems only, is minimized to generate an orthonormal local basis composed of functions centered on the preset chemical fragments. The familiar notion for atomic cores, lone pairs, and chemical bonds emerges here automatically. Such a local basis is then employed in the global HF/KS calculation, after which a least action is taken toward the final orthonormal localized molecular orbitals (LMO), both occupied and virtual. This last step is very cheap, implying that, after the CMOs, the LMOs can be obtained essentially for free. Because molecular fragments are taken as the basic elements, the approach is in the spirit of "from fragments to molecule". Two representatives of highly conjugated molecules, that is, C12H2 and C60, are taken as showcases for demonstrating the success of the proposed approach. The use of the so-obtained LMOs will lead naturally to low-order scaling post-HF/KS methods for electron correlation or excitation. In addition, the underlying fragment picture allows for easy and pictorial interpretations of the correlation/excitation dynamics.

Configuration interaction wave functions: a seniority number approach.

This work deals with the configuration interaction method when an N-electron Hamiltonian is projected on Slater determinants which are classified according to their seniority number values. We study the spin features of the wave functions and the size of the matrices required to formulate states of any spin symmetry within this treatment. Correlation energies associated with the wave functions arising from the seniority-based configuration interaction procedure are determined for three types of molecular orbital basis: canonical molecular orbitals, natural orbitals, and the orbitals resulting from minimizing the expectation value of the N-electron seniority number operator. The performance of these bases is analyzed by means of numerical results obtained from selected N-electron systems of several spin symmetries. The comparison of the results highlights the efficiency of the molecular orbital basis which minimizes the mean value of the seniority number for a state, yielding energy values closer to those provided by the full configuration interaction procedure.

Interpretation of Electron Delocalization in Benzene, Cyclobutadiene, and Borazine Based on Visualization of Individual Molecular Orbital Contributions to the Induced Magnetic Field

The magnetic response of the valence molecular orbitals (MOs) of benzene, cyclobutadiene, and borazine to an external magnetic field has been visualized by calculating the chemical shielding in two-dimensional grids of points on the molecular plane and on a plane perpendicular to it, using gauge-including atomic orbitals (GIAOs). The visualizations of canonical MO contributions to the induced magnetic field (CMO-IMF) provide a clear view of the spatial extension, the shape, and the magnitude of shielding and deshielding areas within the vicinity of the molecule, originating from the induced currents of each valence orbital. The results are used to investigate the delocalization of each valence MO and to evaluate its contribution to the aromatic character of systems under study. The differentiation of the total magnetic response among the three molecules originates exclusively from π-HOMO orbitals because the magnetic response of the subsets of the remaining MOs is found to be almost identical. Borazine is classified as nonaromatic as the four electrons that occupy the π-HOMO are found to be strongly localized on nitrogen centers. CMO-IMF can clarify the interpretation of various NICS indexes and can be applied for the investigation of various types of electron delocalization.

A third-generation density-functional-theory-based method for calculating canonical molecular orbitals of large molecules

We used grid-free modified Cholesky decomposition (CD) to develop a density-functional-theory (DFT)-based method for calculating the canonical molecular orbitals (CMOs) of large molecules. Our method can be used to calculate standard CMOs, analytically compute exchange-correlation terms, and maximise the capacity of next-generation supercomputers. Cholesky vectors were first analytically downscaled using low-rank pivoted CD and CD with adaptive metric (CDAM). The obtained Cholesky vectors were distributed and stored on each computer node in a parallel computer, and the Coulomb, Fock exchange, and pure exchange-correlation terms were calculated by multiplying the Cholesky vectors without evaluating molecular integrals in self-consistent field iterations. Our method enables DFT and massively distributed memory parallel computers to be used in order to very efficiently calculate the CMOs of large molecules.

Charge transfer interaction using quasiatomic minimal-basis orbitals in the effective fragment potential method

The charge transfer (CT) interaction, the most time-consuming term in the general effective fragment potential method, is made much more computationally efficient. This is accomplished by the projection of the quasiatomic minimal-basis-set orbitals (QUAMBOs) as the atomic basis onto the self-consistent field virtual molecular orbital (MO) space to select a subspace of the full virtual space called the valence virtual space. The diagonalization of the Fock matrix in terms of QUAMBOs recovers the canonical occupied orbitals and, more importantly, gives rise to the valence virtual orbitals (VVOs). The CT energies obtained using VVOs are generally as accurate as those obtained with the full virtual space canonical MOs because the QUAMBOs span the valence part of the virtual space, which can generally be regarded as "chemically important." The number of QUAMBOs is the same as the number of minimal-basis MOs of a molecule. Therefore, the number of VVOs is significantly smaller than the number of canonical virtual MOs, especially for large atomic basis sets. This leads to a dramatic decrease in the computational cost.

Diabatic Molecular Orbitals, Potential Energies, and Potential Energy Surface Couplings by the 4-fold Way for Photodissociation of Phenol.

Complete-active-space self-consistent-field (CASSCF) calculations provide useful reference wave functions for configuration interaction or perturbation theory calculations of excited-state potential energy surfaces including dynamical electron correlation. However, the canonical molecular orbitals (MOs) of CASSCF calculations usually have mixed character in regions of strong interaction of two or more electronic states; therefore, they are unsuitable for diabatization using the configurational uniformity approach. Here, CASSCF diabatic MOs for phenol have been obtained by the 4-fold way, and comparison to the CASSCF canonical MOs shows that they are much smoother. Using these smooth CASSCF diabatic MOs, we performed direct diabatization calculations for the three low-lying states ((1)ππ, (1)ππ*, and (1)πσ*) and their diabatic (scalar) couplings at the dynamically correlated multiconfiguration quasidegenerate perturbation theory (MC-QDPT) level. We present calculations along the O-H stretching and C-C-O-H torsion coordinates for the nonadiabatic photodissociation of phenol to the phenoxyl radical and hydrogen atom. The seams of (1)ππ*/(1)πσ* and (1)ππ/(1)πσ* diabatic crossings are plotted as functions of these coordinates. We also present diabatization calculations for displacements along the out-of-plane ring distortion modes 16a and 16b of the phenyl group. The dominant coupling modes of the two conical intersections ((1)ππ*/(1)πσ* and (1)ππ/(1)πσ*) are discussed. The present diabatization method is confirmed to be valid even for significantly distorted ring structures by diabatization calculations along a reaction path connecting the planar equilibrium geometry of phenol to its strongly distorted prefulvenic form. The present work provides insight into the mode specificity of phenol photodissociation and shows that diabatization at the MC-QDPT level employing CASSCF diabatic MOs can be a good starting point for multidimensional dynamics calculations of photochemical reactions.

(B3O3H3)nM]+(n = 1, 2;M = Cu, Ag, Au): a new class of metal-cation complexes

A density functional theory (DFT) investigation into the structures and bonding characteristics of [(B3O3H3)nM](+)(n = 1, 2;M = Cu, Ag, Au) complexes was performed. DFT calculations and natural bond orbital (NBO) analyses indicate that the ΙB metal complexes of boroxine exhibit intriguing bonding characteristics, different from the typical cation-π interactions between ΙB metal-cations and benzene. The complexes of [B3O3H3M](+) and [(B3O3H3)2 M](+) (M = Cu, Ag, and Au) favor the conformation of perfectly planar structures with the C2v and D2h symmetry along one of the threefold molecular axes of boroxine, respectively. Detailed natural resonance theory (NRT) and canonical molecular orbitals (CMOs) analyses show that interaction between the metal cation and the boroxine in [B3O3H3M](+) (M = Cu, Ag, and Au) is mainly ionic, while the ΙB metal-cations←π donation effect is responsible for the binding site. In these complexes, boroxine serves as terminals η(1)-B3O3H3 with one O atom of the B3O3 ring. The infra-red (IR) spectra of [B3O3H3M](+) were simulated to facilitate their future experimental characterization. The complexes all give two IR active modes at about 1,300 and 2,700 cm(-1), which are inactive in pure boroxine. Simultaneously, the B-H stretching modes of the complexes are red-shifted due to the interaction between the metal-cation and boroxine. To explore the possibility of the structural pattern developed in this work forming mesoporous materials, complexes [(B3O3H3M)6](6+) (M = Cu, Ag, and Au) were also studied, which appear to be unique and particular interesting: they are all true minima with D6h symmetries and pore sizes ranging from 12.04 Å to 13.65 Å.

On quadratic bond-order decomposition within molecular orbital space

A simple method of analysing and localization of canonical molecular orbitals for particular chemical bond using the MO-resolved bond-order decompo- sition scheme is presented. An alternative definition of classical bond order orbitals is provided and links to communication theory of the chemical bond are outlined and briefly discussed. The introduced procedure of decomposition of quadratic bond orders allows one to analyse two- as well as three- center chemical bonds within the framework of the same theory.