Algebraic modifications to second quantization for non‐Hermitian complex scaled hamiltonians with application to a quadratically convergent multiconfigurational self‐consistent field method

The algebraic structure for creation and annihilation operators defined on orthogonal orbitals is generalized to permit easy development of bound‐state techniques involving the use of non‐Hermitian Hamiltonians arising from the use of complex‐scaling or complex‐absorbing potentials in the treatment of electron scattering resonances. These extensions are made possible by an orthogonal transformation of complex biorthogonal orbitals and states as opposed to the customary unitary transformation of real orthogonal orbitals and states and preserve all other formal and numerical simplicities of existing bound‐state methods. The ease of application is demonstrated by deriving the modified equations for implementation of a quadratically convergent multiconfigurational self‐consistent field (MCSCF) method for complex‐scaled Hamiltonians but the generalizations are equally applicable for the extension of other techniques such as single and multireference coupled cluster (CC) and many‐body perturbation theory (MBPT) methods for their use in the treatment of resonances. This extends the domain of applicability of MCSCF, CC, MBPT, and methods based on MCSCF states to an accurate treatment of resonances while still using L² real basis sets. Modification of all other bound‐state methods and codes should be similarly straightforward. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach

A density matrix formulation of the super-CI MCSCF method is presented. The MC expansion is assumed to be complete in an active subset of the orbital space, and the corresponding CI secular problem is solved by a direct scheme using the unitary group approach. With a density matrix formulation the orbital optimization step becomes independent of the size of the CI expansion. It is possible to formulate the super-CI in terms of density matrices defined only in the small active subspace; the doubly occupied orbitals (the inactive subspace) do not enter. Further, in the unitary group formalism it is straightforward and simple to obtain the necessary density matrices from the symbolic formula list. It then becomes possible to treat very long MC expansions, the largest so far comprising 726 configurations. The method is demonstrated in a calculation of the potential curves for the three lowest states (¹Σ⁺_g, ³Σ⁺_u and ³Π_g) of the N₂ molecule, using a medium-sized gaussian basis set. Seven active orbitals were used yielding the following results: D_e: 8.76 (9.90), 2.43 (3.68) and 3.39 (4.90) eV; r_e: 1.108 (1.098), 1.309 (1.287) and 1.230 (1.213) Å; ω_e: 2333 (2359), 1385 (1461) and 1680 (1733) cm^?1, for the three states (experimental values within parentheses). The results of these calculations indicate that it is important to consider not only the dissociation limit but also the united atom limit in partitioning the occupied orbital space into an active and an inactive part.     

Performance of dynamically weighted multiconfiguration self-consistent field and spin-orbit coupling calculations of diatomic molecules of Group 14 elements

The efficacy of several multiconfiguration self-consistent field (MCSCF) methods in the subsequent spin-orbit coupling calculations was studied. Three MCSCF schemes to generate molecular orbitals were analyzed: state-specific, state-averaged, and dynamically weighted MCSCF. With Sn(2)(+) as the representative case, we show that the state-specific MCSCF orbitals lead to discontinuities in potential energy curves when avoided crossings of electronic states occur; this problem can be solved using the state-averaged or dynamically weighted MCSCF orbitals. The latter two schemes are found to give similar results when dynamic electron correlation is considered, which we calculated at the level of multiconfigurational quasidegenerate perturbation theory (MCQDPT). We employed the recently developed Douglas-Kroll spin-orbit adapted model core potential, ZFK3-DK3, and the dynamically weighted MCSCF scheme to calculate the spectroscopic constants of the mono-hydrides and compared them to the results obtained using the older set of potentials, MCP-TZP. We also showed that the MCQDPT tends to underestimate the dissociation energies of the hydrides and discussed to what extent coupled-cluster theory can be used to improve results.     

Atomic structure of the highest molecular orbitals of small tetra-heme cytochrome <Emphasis Type="Italic">c</Emphasis> 1M1P

The atomic structure of the highest molecular orbitals (MO) of small tetra-heme cytochrome (STC) c 1M1P is studied in large-scale ab initio all-electrons Hartree–Fock calculations. It is shown that the highest MOs of STC are mainly formed by atomic orbitals of negatively charged amino acid atoms whose types and corresponding numbers are determined. The results obtained permit the conclusion that these amino acids can be considered as possible active centers in the electron transfer reaction between STC and an external electron acceptor.     

Approximate MCSCF optimization for multiconfigurational spin‐tensor electron propagator method (MCSTEP): The vertical ionization potentials of CO,HCN, HNC,H2CO,and O3

The multiconfigurational spin tensor electron propagator method (MCSTEP) was developed as an implementation of electron propagator/single particle Green's function methods. MCSTEP was specifically designed for open shell and highly correlated (nondynamically correlated) initial states. The initial state used in MCSTEP is typically a small complete active space (CAS) with multiconfigurational self‐consistent field (MCSCF) state. In some cases, because of our use of a small CAS in MCSTEP, the Lagrangian eigenvalues of the MCSCF reference state are in an undesired order (u). The desired order (d) can usually be obtained by excluding one or more orbital rotations in MCSCF optimization between the doubly occupied and partially occupied orbitals. We systematically examine several cases where the undesired order occurs for the low‐lying vertical MCSTEP ionization potentials (IPs) of the molecules CO, HCN, HNC, H₂CO, and O₃ with our recently established CAS choices for MCSCF/MCSTEP. By excluding one or more orbital rotations between the partially and doubly occupied orbitals, an approximate MCSCF reference state with the same CAS choice is obtained for use in standard MCSTEP calculations that, in general, gives more reliable vertical MCSTEP IPs. © 2007 Wiley Periodicals, Inc. J Quantum Chem, 2008     

Potential magnetic properties of nanotubes (n, 0) with Klein and Fujita edges

Analytical solutions for localized states of zigzag-type nanotube (NT) fragments with various combinations of Klein and Fujita borders are considered using the Hückel approach. It is shown that the equations for determining molecular orbitals (MOs) in systems with two Klein edges are similar to equations for systems with two Fujita edges. An analytical formula for the energies of all ?? MOs is obtained for systems that have a Klein edge on one side and a Fujita edge on the other. It is established that these systems have n orbitals with energy ?? that are localized on the Fujita and Klein edges in dependence on the MO symmetry. The degeneracy of edge orbitals indicates that there is a tendency toward single occupancy of them and to the appearance of spin (magnetic) properties. In addition, the energies of the states of different multiplicity for NT fragments (8, 0) are calculated using the CASSCF approach. It is shown that the ground state has a multiplicity of 9, as was also indicated by estimates obtained using the density functional method (B3LYP). It is concluded that zigzag-type NTs with asymmetric edges have a tendency to exhibit spin properties. It is noted that the construction of nanoscale magnetic materials based on them is very promising.     

Octacarbonyl Anion Complexes of Group Three Transition Metals [TM(CO)8]− (TM=Sc,Y, La) and the 18‐Electron Rule

We report the gas‐phase synthesis of stable 20‐electron carbonyl anion complexes of group 3 transition metals, TM(CO)₈^? (TM=Sc, Y, La), which are studied by mass‐selected infrared (IR) photodissociation spectroscopy. The experimentally observed species, which are the first octacarbonyl anionic complexes of a TM, are identified by comparison of the measured and calculated IR spectra. Quantum chemical calculations show that the molecules have a cubic (O_h) equilibrium geometry and a singlet (¹A_1g) electronic ground state. The 20‐electron systems TM(CO)₈^? are energetically stable toward loss of one CO ligand, yielding the 18‐electron complexes TM(CO)₇^? in the ¹A₁ electronic ground state; these exhibit a capped octahedral structure with C_3v symmetry. Analysis of the electronic structure of TM(CO)₈^? reveals that there is one occupied valence molecular orbital with a_2u symmetry, which is formed only by ligand orbitals without a contribution from the metal atomic orbitals. The adducts of TM(CO)₈^? fulfill the 18‐electron rule when only those valence electrons that occupy metal–ligand bonding orbitals are considered.     

Octacarbonyl Anion Complexes of Group Three Transition Metals [TM(CO)8]− (TM=Sc,Y, La) and the 18‐Electron Rule

We report the gas‐phase synthesis of stable 20‐electron carbonyl anion complexes of group 3 transition metals, TM(CO)₈⁻ (TM=Sc, Y, La), which are studied by mass‐selected infrared (IR) photodissociation spectroscopy. The experimentally observed species, which are the first octacarbonyl anionic complexes of a TM, are identified by comparison of the measured and calculated IR spectra. Quantum chemical calculations show that the molecules have a cubic (O_h) equilibrium geometry and a singlet (¹A_1g) electronic ground state. The 20‐electron systems TM(CO)₈⁻ are energetically stable toward loss of one CO ligand, yielding the 18‐electron complexes TM(CO)₇⁻ in the ¹A₁ electronic ground state; these exhibit a capped octahedral structure with C_3v symmetry. Analysis of the electronic structure of TM(CO)₈⁻ reveals that there is one occupied valence molecular orbital with a_2u symmetry, which is formed only by ligand orbitals without a contribution from the metal atomic orbitals. The adducts of TM(CO)₈⁻ fulfill the 18‐electron rule when only those valence electrons that occupy metal–ligand bonding orbitals are considered.     

A study of the molecular orbital localization into an extended Hartree—Fock approach: Application to the BeH2 ground state

In this article, we present a study of the localization and properties of the molecular orbitals (MOs) of polyatomic systems by using a comprehensive version of the G1 model. In this version, the wave function is written as a DODS product of univocally determined spin orbitals (MOs), “projected” on the singlet ground state. A procedure for determining the MOs is given and applied to the BeH₂ ground state. Equivalent split shell and localized MOs are found. The Be orbitals are seen to exhibit sp hybridization and the localized valence MOs are found to produce − 13.7 kcal/mol localization energy. Multistructural calculations are carried out and show that the present approach is able to describe localized and well-oriented bonds whenever the molecule under study presents only a single well-defined nonresonant chemical structure. © 1996 John Wiley & Sons, Inc.     

Equivalent orbitals for multiconfigurational spin‐tensor electron propagator method (MCSTEP): The vertical ionization potentials of B,NO, CF,and OF

The multiconfigurational spin tensor electron propagator method (MCSTEP) was developed as an implementation of electron propagator/single particle Green's function methods for ionization potentials (IPs) and electron affinities (EAs). MCSTEP was specifically designed for open shell and highly correlated (nondynamically correlated) initial states. For computational efficiency the initial state used in MCSTEP is typically a small complete active space (CAS) multiconfigurational self‐consistent field (MCSCF) state. If in a molecule there are some degenerate orbitals which are not fully or half occupied, usual MCSCF calculations will make these orbitals inequivalent, i.e., the occupied ones will be different from the nonoccupied ones, so that the degeneracy is broken. In this article, we use a state averaged MCSCF method to get equivalent orbitals for the initial state and import the integrals into the subsequent MCSTEP calculations. This gives, in general, more reliable MCSTEP vertical IPs. © 2008 Wiley Periodicals, Inc., 2008     

DFT study on the selective complexation of B12N12 nanocage with alkali metal ions

Density functional theory (DFT) calculations were applied at the M05-2X/6-311++G(d,p) level of the theory to investigate the interaction of the B₁₂N₁₂ nanocage (BN) and alkali metal ions (Li⁺, Na⁺, K⁺, Rb⁺ and Cs⁺) in the gas phase and in water. On the basis of the results, BN nanocage is able to form a selective complex with Li⁺. Water, as a solvent, reduces the stability of the metal ion-BN complexes in comparison with the gas phase. Natural bond orbital (NBO) and quantum theory of atoms in molecules (QTAIM) analyses, reveal that the electrostatic interaction between the BN and metal ions can be considered as the driving force for complex formation in which the role of water is of significance. Density of states (DOSs) analysis of the BN nanocage structure in the presence of different metal ions showed a noticeable change in the frontier orbitals, especially in the gas phase, and Fermi level shifting toward the lower values.     

On the lowest electronic states of the C2F radical

The adiabatic energy surfaces of the lowest three electronic states [²(²A′ and ²A′)] and ²Σ⁺[²A′] of the C₂F radical were investigated by the Hartree-Fock multiconfiguration self-consistent field (HF—MCSCF) ab initio method using a large set of atomic natural orbitals (ANO) and an extended configuration space, and the results were shown to be in agreement with the predictions of valence theory for this radical. The electronic ground state was found to have a bent equilibrium structure, hence contradicting the Walsh rule which predicts for the isoelectronic molecules a ² linear state. The three states were found to be nearly degenerate and the potential energy surfaces of the two lowest electronic states exhibit an avoided crossing at an energy ∼2000 cm⁻¹ above the ground-state minimum, lower than the highest vibrational fundamental. The strong adiabatic interaction which is responsible for the ordering of the electronic states and their equilibrium geometry involves not only the bending coordinate as normally found for Renner-Teller pairs of states, but also the C—C stretching coordinate, due to the near degeneracy of the ²Σ⁺ and the ² lowest electronic states at linear geometries. © 1996 John Wiley & Sons, Inc.     

INDO-Type calculations on the ground state and various ionic states of transition metal tricarbonyls

By means of the ΔSCF and transition operator (TO) methods based on a recently developed INDO extension to the first transition metal series, the first ionization potentials of benzene—chromium tricarbonyl ( I ), cyclopentadienyl manganese tricarbonyl ( II ), the iron—tricarbonyl complexes with trimethylenemethane ( III ), and cyclobutadiene ( IV ) have been calculated and compared with experimental data. It is shown that the electronic structure of I to IV can be rationalized by Hoffmann's fragment approach in both the ground state and the cationic hole states. Within the series I—IV there are remarkable energy differences in the ground state for MOs derived from the 1a₁ and 1e orbitals of the M(CO)₃ fragment. The observation that only one band is associated with the ionization events from MOs predominantly localized at the metal site is traced back to large relaxation effects. In the cationic hole states the split of the M(CO)₃ fragment orbitals 1a₁ and 1e is minute in all four compounds.     

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.     

Dyson orbitals for ionization from the ground and electronically excited states within equation-of-motion coupled-cluster formalism: theory, implementation, and examples

Implementation of Dyson orbitals for coupled-cluster and equation-of-motion coupled-cluster wave functions with single and double substitutions is described and demonstrated by examples. Both ionizations from the ground and electronically excited states are considered. Dyson orbitals are necessary for calculating electronic factors of angular distributions of photoelectrons, Compton profiles, electron momentum spectra, etc, and can be interpreted as states of the leaving electron. Formally, Dyson orbitals represent the overlap between an initial N-electron wave function and the N-1 electron wave function of the corresponding ionized system. For the ground state ionization, Dyson orbitals are often similar to the corresponding Hartree-Fock molecular orbitals (MOs); however, for ionization from electronically excited states Dyson orbitals include contributions from several MOs and their shapes are more complex. The theory is applied to calculating the Dyson orbitals for ionization of formaldehyde from the ground and electronically excited states. Partial-wave analysis is employed to compute the probabilities to find the ejected electron in different angular momentum states using the freestanding and Coulomb wave representations of the ionized electron. Rydberg states are shown to yield higher angular momentum electrons, as compared to valence states of the same symmetry. Likewise, faster photoelectrons are most likely to have higher angular momentum.     

Electronic structure and vertical excitation spectrum of methylene amidogen CH2N

Ab initio electronic structure calculations are reported for five electronic states of the methylene amidogen radical. Structure parameters for the ground electronic state are predicted by RHF and D -MBPT (4) calculations. Vertical excitation energies were determined using four different theoretical chemical models: complete active space (CAS ) MCSCF , CAS /MCSCF plus singles and doubles Cl, fourth-order many-body perturbation theory SDQ -MBPT (4), and coupled-cluster theory.     

Structure and dynamics of room temperature ionic liquids with bromide anion: results from 81Br NMR spectroscopy

We report the results of a comprehensive ⁸¹Br NMR spectroscopic study of the structure and dynamics of two room temperature ionic liquids (RTILs), 1‐butyl‐3‐methylimidazolium bromide ([C₄mim]Br) and 1‐butyl‐2,3‐dimethylimidazolium bromide ([C₄C₁mim]Br), in both liquid and crystalline states. NMR parameters in the gas phase are also simulated for stable ion pairs using quantum chemical calculations. The combination of ⁸¹Br spin‐lattice and spin‐spin relaxation measurements in the motionally narrowed region of the stable liquid state provides information on the correlation time of the translational motion of the cation. ⁸¹Br quadrupolar coupling constants (C_Q) of the two RTILs were estimated to be 6.22 and 6.52 MHz in the crystalline state which were reduced by nearly 50% in the liquid state, although in the gas phase, the values are higher and span the range of 7–53 MHz depending on ion pair structure. The C_Q can be correlated with the distance between the cation–anion pairs in all the three states. The ⁸¹Br C_Q values of the bromide anion in the liquid state indicate the presence of some structural order in these RTILs, the degree of which decreases with increasing temperature. On the other hand, the ionicity of these RTILs is estimated from the combined knowledge of the isotropic chemical shift and the appropriate mean energy of the excited state. [C₄C₁mim]Br has higher ionicity than [C₄mim]Br in the gas phase, while the situation is reverse for the liquid and the crystalline states. Copyright © 2015 John Wiley & Sons, Ltd.     

The “Inverted Bonds” Revisited: Analysis of “In Silico” Models and of [1.1.1]Propellane by Using Orbital Forces

This article dwells on the nature of “inverted bonds”, which refer to the σ interaction between two sp hybrids by their smaller lobes, and their presence in [1.1.1]propellane. Firstly, we study H₃C−C models of C−C bonds with frozen H-C-C angles reproducing the constraints of various degrees of “inversion”. Secondly, the molecular orbital (MO) properties of [1.1.1]propellane and [1.1.1]bicyclopentane are analyzed with the help of orbital forces as a criterion of bonding/antibonding character and as a basis to evaluate bond energies. Triplet and cationic states of [1.1.1]propellane species are also considered to confirm the bonding/antibonding character of MOs in the parent molecule. These approaches show an essentially non-bonding character of the σ central C−C interaction in propellane. Within the MO theory, this bonding is thus only due to π-type MOs (also called “banana” MOs or “bridge” MOs) and its total energy is evaluated to approximately 50 kcal mol⁻¹. In bicyclopentane, despite a strong σ-type repulsion, a weak bonding (15–20 kcal mol⁻¹) exists between both central C−C bonds, also due to π-type interactions, though no bond is present in the Lewis structure. Overall, the so-called “inverted” bond, as resulting from a σ overlap of the two sp hybrids by their smaller lobes, appears highly questionable.     

An Insight of the Results Provided by Color‐Tuning Studies Made on Ir(III) Complexes: A pi‐Neutral CF3 Group Viewed by Adjusting Energies of pi‐type Molecular Orbitals

The pi‐nature of a CF₃ group can be understood through analysis of its bond orbitals (BOs) mixed into the pi‐type molecular orbitals of CF₃‐substituted Ir(ppy)₂MDPA⁺ complexes (ppy=2‐phenyl‐pyridine and MDPA=methylated 2,2′‐dipyridyl amine). It has been found that, through this natural bond orbital analysis, the parent’s molecular orbitals (MOs) can be stabilized by χρ^*_CF BO via negative hyperconjugation and, simultaneously, destabilized by electron lp(F) BO. Since these two competing pi‐effects are virtually counterbalanced as indicated by the vanishing values of crystal orbital overlap populations, the chemical substitution strategy originated from lowering of HOMO by using this electron‐withdrawing CF₃ group has been found effective in color‐tuning to blue region. Based on reduced shielding effect due to de‐ creased χρ‐electron density, the reported position dependent CF₃‐substitution effects on pi‐type MOs can also be understood through HOMO/LUMO wavefunction analysis.     

Localization of molecular orbitals on fragments

A non‐iterative algorithm for the localization of molecular orbitals (MOs) from complete active space self consistent field (CASSCF) and for single‐determinantal wave functions on predefined moieties is given. The localized fragment orbitals can be used to analyze chemical reactions between fragments and also the binding of fragments in the product molecule with a fragments‐in‐molecules approach by using a valence bond expansion of the CASSCF wave function. The algorithm is an example of the orthogonal Procrustes problem, which is a matrix optimization problem using the singular value decomposition. It is based on the similarity of the set of MOs for the moieties to the localized MOs of the molecule; the similarity is expressed by overlap matrices between the original fragment MOs and the localized MOs. For CASSCF wave functions, localization is done independently in the space of occupied orbitals and active orbitals, whereas, the space of virtual orbitals is mostly uninteresting. Localization of Hartree–Fock or Kohn–Sham density functional theory orbitals is not straightforward; rather, it needs careful consideration, because in this case some virtual orbitals are needed but the space of virtual orbitals depends on the basis sets used and causes considerable problems due to the diffuse character of most virtual orbitals. © 2012 Wiley Periodicals, Inc.