Orthogonal natural atomic orbitals form an appropriate one-electron basis for expanding CASSCF wave functions into localized bonding schemes and their weights

Localized bonding schemes and their weights have been obtained for the pi-electron system of nitrone by expanding complete active space self-consistent field wave functions into a set of Slater determinants composed of orthogonal natural atomic orbitals (NAOs) of Weinhold and Landis (Valency and Bonding: A Natural Bond Orbital Donor-Acceptor Perspective, 2005). Thus, the derived bonding schemes are close to orthogonal valence bond structures. The calculated sequence of bonding scheme weights accords with the sequence of genuine resonance structure weights derived previously by Ohanessian and Hiberty (Chem Phys Lett 1987, 137, 437), who employed nonorthogonal atomic orbitals. This accord supports the notion that NAOs form an appropriate orthogonal one-electron basis for expanding complete active space self-consistent field wave functions into meaningful bonding schemes and their weights.     

On obtaining localized bonding schemes from delocalized molecular-orbital wave functions

A straightforward procedure is proposed for expanding a molecular orbital determinantal wave function into a set of determinantal wave functions composed of atomic orbitals localized at the atoms of a molecule. By employing this method, atomic orbital determinants and their weights can be derived for a molecule from the computed molecular-orbital wave function. The procedure permits the interpretation of a molecular orbital determinantal wave function in terms of bonding schemes related to the classic resonance structures used by organic chemists. By using the unrestricted molecular orbital determinant, bonding schemes and their weights are obtained for butadiene, the butadiene radical cation and the acrylonitrile radical anion. Their dominant bonding schemes are in accord with the relevant resonance structures for these molecules. For the butadiene radical cation and the acrylonitrile anion they are shown to be compatible with the accepted mechanisms of the electrochemical coupling reactions of butadiene and acrylonitrile. Received: 7 August 1996 / Accepted: 18 March 1997     

The photochemistry of 1,3-butadiene rationalized by means of theoretical resonance structures and their weights

A complete active-space self-consistent-field wave function for the pi-electron part of s-trans-1,3-butadiene has been expanded into a set of localized bonding schemes and their weights. These bonding schemes are close to the resonance structures used in organic chemistry. The expansion technique has been applied to both the electronic ground state and the electronically first-excited singlet and triplet pi,pi* states. The manifolds of large-weight bonding schemes represent approximate resonance hybrids for the ground and the singlet and triplet pi,pi* states of s-trans-1,3-butadiene. These resonance hybrids, obtained by theory alone, permit a qualitative rationalization of a significant part of the known singlet and triplet photochemistry.     

The problem of hole localization in inner-shell states of N2 and CO2 revisited with complete active space self-consistent field approach

Potential energy curves for inner-shell states of nitrogen and carbon dioxide molecules are calculated by inner-shell complete active space self-consistent field (CASSCF) method, which is a protocol, recently proposed, to obtain specifically converged inner-shell states at multiconfigurational level. This is possible since the collapse of the wave function to a low-lying state is avoided by a sequence of constrained optimization in the orbital mixing step. The problem of localization of K-shell states is revisited by calculating their energies at CASSCF level based on both localized and delocalized orbitals. The localized basis presents the best results at this level of calculation. Transition energies are also calculated by perturbation theory, by taking the above mentioned MCSCF function as zeroth order wave function. Values for transition energy are in fairly good agreement with experimental ones. Bond dissociation energies for N(2) are considerably high, which means that these states are strongly bound. Potential curves along ground state normal modes of CO(2) indicate the occurrence of Renner-Teller effect in inner-shell states.     

New relativistic atomic natural orbital basis sets for lanthanide atoms with applications to the Ce diatom and LuF3

New basis sets of the atomic natural orbital (ANO) type have been developed for the lanthanide atoms La-Lu. The ANOs have been obtained from the average density matrix of the ground and lowest excited states of the atom, the positive ions, and the atom in an electric field. Scalar relativistic effects are included through the use of a Douglas-Kroll-Hess Hamiltonian. Multiconfigurational wave functions have been used with dynamic correlation included using second-order perturbation theory (CASSCF/CASPT2). The basis sets are applied in calculations of ionization energies and some excitation energies. Computed ionization energies have an accuracy better than 0.1 eV in most cases. Two molecular applications are included as illustration: the cerium diatom and the LuF3 molecule. In both cases it is shown that 4f orbitals are not involved in the chemical bond in contrast to an earlier claim for the latter molecule.     

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.     

The behavior of transition metal nitrido bonds towards protonation rationalized by means of localized bonding schemes and their weights

A new computational scheme is applied to rationalize the different protonation behaviors of the nitrido complexes [L'Mn(V)(N)(acac)](+), [LCr(V)(N)(acac)](+), and [LV(V)(N)(acac)](+). L and L' represent the macrocycles 1,4,7-triazacyclononane and its N-methylated derivative, respectively, and acac is the bidentate monoanion pentane-2,4-dionate. The bonds of the complexes are partitioned into bonds to be investigated and bonds of lesser interest. The investigated bonds are the transition metal nitrido bonds M(V)[triple chemical bond]N| (M = Mn, Cr, and V) and the bonds of lesser interest are located in the ligands. The ligand bonds are described by means of the strongly occupied natural bond orbitals. The electrons in the M(V)[triple chemical bond]N| nitrido bonds, however, are treated more accurately. A full configuration interaction procedure is applied in the space spanned by the strongly occupied natural bond orbitals and their corresponding antibonding orbitals. Localized bonding schemes and their weights are obtained for the d(pi)-p(pi) bonds of interest. This is achieved by representing the two-center natural bond orbitals for a d(pi)-p(pi) bond by the one-center natural hybrid orbitals localized at the bond atoms. The obtained bonding schemes are close to orthogonal valence bond structures. Their weights indicate that the nitrido nitrogen in [LV(V)(N)(acac)](+) is more easily protonated than the nitrido nitrogens in [L'Mn(V)(N)(acac)](+) and [LCr(V)(N)(acac)](+). This result is in good accord with experiment.     

The electronic absorption spectra of some acyl azides molecular orbital treatment

The electronic absorption spectra of benzoyl azide and its derivatives: p-methyl, p-methoxy, p-chloro and p-nitrobenzoyl azide were investigated in different solvents. The observed spectra differ basically from the electronic spectra of aryl azides or alkyl azides. Four intense pi-pi* transitions were observed in the accessible UV region of the spectrum of each of the studied compounds. The contribution of charge transfer configurations to the observed transitions is rather weak. Shift of band maximum with solvent polarity is minute. On the other hand, band intensity is highly dependent on the solvent used. The observed transitions are delocalized rather than localized ones as in the case with aryl and alkyl azides. The attachment of the CO group to the azide group in acyl azides has a significant effect on the electronic structure of the molecule. The arrangements as well as energies of the molecular orbitals are different in acyl azides from those in aryl azides. The first electronic transition in phenyl azide is at 276 nm, whereas that of bezoyle azide is at 251 nm. Ab initio molecular orbital calculations using both RHF/6-311G* and B3LYP/6-31+G* levels were carried out on the ground states of the studied compounds. The wave functions of the excited states were calculated using the CIS and the AM1-CI procedures.     

The T(1) (3)(pi-pi*)/S(0) intersections and triplet lifetimes of cyclic alpha,beta-enones.

The ground and triplet excited states of cycloheptenone, cyclohexenone, and cyclopentenone have been studied using CASSCF calculations. For these three molecules, the difference in energy (DeltaE) between the twisted T(1) (3)(pi-pi*) minimum and T(1) (3)(pi-pi*)/S(0) intersection increases as the flexibility of the ring decreases. A strong positive correlation between DeltaE and the natural logarithm of the experimentally determined triplet lifetimes (ln tau) is found, suggesting that DeltaE predominantly determines the relative radiationless decay rates of T(1).     

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.     

Dibenzo-p-dioxin. An ab initio CASSCF/CASPT2 study of the pi-pi* and n-pi* valence excited states

The pi-pi* and n-pi* valence excited states of dibenzo-p-dioxin (DD) were studied via the complete active space SCF and multiconfigurational second-order perturbation theory employing the cc-pVDZ basis set and the full pi-electron active spaces of 16 electrons in 14 active orbitals. The geometry and harmonic vibrational wavenumbers of the ground state correlate well with the experimental and other theoretical data. In particular, significant improvements over previously reported theoretical results are observed for the excitation energies. All of the pi-pi* excited states exhibit planar D(2h)minima. Thus no evidence was found for a C(2v) butterfly-like relaxation, although the wavenumbers of the b(3u) butterfly flapping mode proved exceedingly low in both the ground S(0)((1)A(g)) and the lowest dipole allowed excited S(1)((1)B(2u)) state. The calculations of oscillator strengths established the 2(1)B(2u) <-- 1(1)A(g) and 2(1)B(1u) <-- 1(1)A(g) transitions as by far the most intense, whereas the only allowed of the n-pi* transitions ((1)B(3u)) should possess only a modest intensity. Studies into dependence of the oscillator strengths on the extent of the butterfly-like folding showed that the electronic spectrum is more consistent with a folded equilibrium geometry assumed by DD in solution.     

Conical intersections in thymine

The mechanisms which are responsible for the radiationless deactivation of the npi* and pipi* excited singlet states of thymine have been investigated with multireference ab initio methods (the complete-active-space self-consistent-field (CASSCF) method and second-order perturbation theory with respect to the CASSCF reference (CASPT2)) as well as with the CC2 (approximated singles and doubles coupled-cluster) method. The vertical excitation energies, the equilibrium geometries of the 1npi*and 1pipi* states, as well as their adiabatic excitation energies have been determined. Three conical intersections of the S1 and S0 energy surfaces have been located. The energy profiles of the excited states and the ground state have been calculated with the CASSCF method along straight-line reaction paths leading from the ground-state equilibrium geometry to the conical intersections. All three conical intersections are characterized by strongly out-of-plane distorted geometries. The lowest-energy conical intersection (CI1) arises from a crossing of the lowest 1pipi* state with the electronic ground state. It is found to be accessible in a barrierless manner from the minimum of the 1pipi* state, providing a direct and fast pathway for the quenching of the population of the lowest optically allowed excited states of thymine. This result explains the complete diffuseness of the absorption spectrum of thymine in supersonic jets. The lowest vibronic levels of the optically nearly dark 1npi* state are predicted to lie below CI1, explaining the experimental observation of a long-lived population of dark excited states in gas-phase thymine.     

Molecular mechanics-valence bond method for planar conjugated hydrocarbon cations

We present an extension of the molecular mechanics-valence bond (MMVB) hybrid method to study ground and excited states of planar conjugated hydrocarbon cations. Currently, accurate excited state calculations on these systems are limited to expensive ab initio studies of smaller systems: up to 15 active electrons in 16 pi orbitals with complete active space self-consistent field (CASSCF) theory using high symmetry. The new MMVB extension provides a faster, cheaper treatment to investigate larger cation systems with more than 24 active orbitals. Extension requires both new matrix elements and new parameters: In this paper we present both, for the limited planar case. The scheme is tested for the planar radical cations of benzene, naphthalene, anthracene, and phenanthrene. Calculated MMVB relative energies are in good agreement with CASSCF results for equilibrium geometries on the ground and first excited states, and conical intersections.     

Large systems at ab initio multireference level: a cheap treatment thanks to a division into fragments

Thanks to the use of localized orbitals and the subsequent possibility of neglecting long-range interactions, the linear-scaling methods have allowed to treat large systems at ab initio level. However, the limitation of the number of active orbitals in a complete active space self consistent-field (CASSCF) calculation remains unchanged. The method presented in this paper suggests to divide the system into fragments containing only a small number of active orbitals. Starting from a guess wave function, each orbital is optimized in its corresponding fragment, in the presence of the other fragments. Once all the fragments have been treated, a new set of orbitals is obtained. The process is iterated until convergence. At the end of the calculation, a set of active orbitals is obtained, which is close to the exact CASSCF solution, and an accurate CASSCF energy can be estimated.     

An experimental and theoretical study of the type C enone rearrangement: mechanistic and exploratory organic photochemistry

We recently described a new photochemical rearrangement which we termed a Type C process. The reaction involves a delta to alpha aryl migration in 5-disubstituted cyclohexenones also having bulky C-3 substituents. In contrast to most cyclohexenone rearrangements, the reaction occurs via a twisted pi-pi excited triplet rather than the usual n-pi state. The electronic nature of the rearrangement was assessed using migration selectivity with p-anisyl and p-cyanophenyl groups. A synthesis of the reactants was elaborated, and the product structures were established by X-ray and NMR analysis. The reaction mechanism was established further with DFT and CASSCF computations. In the latter, localized NBO basis orbitals permitted proper selection of the active space. The nature of the diradical intermediates as well as the transition states was established computationally. Sensitization experiments with regioselectivities the same as those in direct irradiation confirmed the triplet multiplicity of the process.     

CASSCF optimization problem for a group of excited states

A second-order version of the CASSCF approach to the optimization problem for a single (ground or excited) state and a group of excited states (involving, if necessary, also the ground state) is proposed. In contrast to the already existing methods, in the frameworks of our approach, there arises no need in completing the set of states to be optimized to the full basis set of configuration function space. Generation of secondary orbitals in the course of orbital optimization is also not required. All necessary integral transformations are performed only with active orbitals. These certainly attractive features of our approach appear due to employing the Gauss parametrization of average electronic energy domain, which is nonstandard in quantum chemistry. © 1992 John Wiley & Sons, Inc.     

乙基溴及其阳离子的低能激发态从头算研究

应用ANO-S基组以及ECP基组在CASSCF理论水平下计算了乙基溴及其阳离子的低能激发态几何构型, 并应用CASPT2方法对动态相关能进行单点能校正. 根据乙基溴基态能量和相应阳离子电子态的能量差对光电子谱(Photoelectron spectrum)的谱线进行了理论指认. 在乙基溴的基态预测几何下, 进行了谐振频率计算, 对各个振动频率进行了理论指认. 计算结果与实验值符合得较好.     

Coherent superposition of resonance wave function in terms of weighted orthogonalized natural localized configurations

In this research, the projection technique has been applied in order to decompose the electronic wave function into its weighted orthogonalized resonance components. These components have been constructed by determinants whose orbitals are selected among natural bond orbitals. However, the procedure is general and any other localized orbitals can be used as well. Both σ and π delocalize systems have been considered in order to check the reliability of the calculated resonance weights. For π‐systems, the presented procedure could predict significant decrease of weight of certain resonance structures when the molecular planarity was destroyed. Water cyclic clusters were also tested and the results confirmed the existence of strong σ‐delocalization in the clusters. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Quantum chemical modeling of photoadsorption properties of the nitrogen-vacancy point defect in diamond

Quantum chemical calculations of geometric and electronic structure and vertical transition energies for several low-lying excited states of the neutral and negatively charged nitrogen-vacancy point defect in diamond (NV(0) and NV(-)) have been performed employing various theoretical methods and basis sets and using finite model NC(n)H(m) clusters. Unpaired electrons in the ground doublet state of NV(0) and triplet state of NV(-) are found to be localized mainly on three carbon atoms around the vacancy and the electronic density on the nitrogen and rest of C atoms is only weakly disturbed. The lowest excited states involve different electronic distributions on molecular orbitals localized close to the vacancy and their wave functions exhibit a strong multireference character with significant contributions from diffuse functions. CASSCF calculations underestimate excitation energies for the anionic defect and overestimate those for the neutral system. The inclusion of dynamic electronic correlation at the CASPT2 level leads to a reasonable agreement (within 0.25 eV) of the calculated transition energy to the lowest excited state with experiment for both systems. Several excited states for NV(-) are found in the energy range of 2-3 eV, but only for the 1(3)E and 5(3)E states the excitation probabilities from the ground state are significant, with the first absorption band calculated at approximately 1.9 eV and the second lying 0.8-1 eV higher in energy than the first one. For NV(0), we predict the following order of electronic states: 1(2)E (0.0), 1(2)A(2) (approximately 2.4 eV), 2(2)E (2.7-2.8 eV), 1(2)A(1), 3(2)E (approximately 3.2 eV and higher).