Natural J-coupling analysis: interpretation of scalar J-couplings in terms of natural bond orbitals.

The natural J-coupling (NJC) method presented here analyzes the Fermi contact portion of J-coupling in the framework of finite perturbation theory applied to ab initio/density function theory (DFT) wave functions, to compute individual and pairwise orbital contributions to the net J-coupling. The approach is based on the concepts and formalisms of natural bond orbital (NBO) methods. Computed coupling contributions can be classified as Lewis (individual orbital contributions corresponding to the natural Lewis structure of the molecule), delocalization (resulting from pairwise donor-acceptor interactions), and residual repolarization (corresponding to correlation-like interactions). This approach is illustrated by an analysis of the angular and distance dependences of the contributions to vicinal (3)J(HH) couplings in ethane and to the long-range (6)J(HH) couplings in pentane. The results indicate that approximately 70% or more of the net J-coupling is propagated by steric exchange antisymmetry interactions between Lewis orbitals (predominantly sigma bonding orbitals). Hyperconjugative sigma to sigma delocalization interactions account for the remainder of the coupling. Calculated pairwise-steric and hyperconjugative-delocalization energies provide a means for relating coupling mechanisms to molecular energetics. In this way, J-coupling contributions can be related directly to the localized features of the molecular electronic structure in order to explain measured J-coupling patterns and to predict J-coupling trends that have yet to be measured.     

Illumination of the effect of the overlap of lone-pairs on indirect nuclear spin-spin coupling constants

The effect of electron lone-pairs on the Fermi-contact (FC) contribution to indirect nuclear spin-spin coupling constants is analyzed using new tools for their interpretation. In particular, visualization of spin-spin coupling pathways using the coupling deformation density (CDD) has been employed. Furthermore, the recently developed perturbation-stable localization procedure has been applied for decomposition of CDD and the calculated value of couplings into contributions from localized molecular orbitals (LMOs). Correlation between the overlap of densities of LMOs representing lone-pairs and the Fermi-contact contribution to spin-spin coupling constants has been demonstrated. A new way for analyzing spin-spin couplings using the expansion of CDD as a linear combination of the products of molecular orbitals has been suggested. The considered examples include two- and three-bond phosphor-phosphor couplings. Significance of the obtained insight is not restricted to spin-spin couplings of nuclei possessing lone-pairs, as demonstrated in the example of vicinal hydrogen-hydrogen coupling in ethane.     

Intrinsic local constituents of molecular electronic wave functions. I. Exact representation of the density matrix in terms of chemically deformed and oriented atomic minimal basis set orbitals

A coherent, intrinsic, basis-set-independent analysis is developed for the invariants of the first-order density matrix of an accurate molecular electronic wavefunction. From the hierarchical ordering of the natural orbitals, the zeroth-order orbital space is deduced, which generates the zeroth-order wavefunction, typically an MCSCF function in the full valence space. It is shown that intrinsically embedded in such wavefunctions are elements that are local in bond regions and elements that are local in atomic regions. Basis-set-independent methods are given that extract and exhibit the intrinsic bond orbitals and the intrinsic minimal-basis quasi-atomic orbitals in terms of which the wavefunction can be exactly constructed. The quasi-atomic orbitals are furthermore oriented by a basis-set independent method (viz. maximization of the sum of the fourth powers of all off-diagonal density matrix elements) so as to exhibit clearly the chemical interactions. The unbiased nature of the method allows for the adaptation of the localized and directed orbitals to changing geometries. Contribution of the Mark S. Gordon 65th Birthday Fegtschrift Issue.     

Multiconfiguration optimized effective potential method for a density-functional treatment of static correlation

An approach to treat static correlation within a density-functional framework is presented. To that end, a multiconfiguration optimized effective potential (MCOEP) method is derived. In contrast to standard multiconfiguration self-consistent field (MCSCF) methods and previous combinations of MCSCF procedures with density-functional theory, the MCOEP method yields well-defined physically meaningful orbital and eigenvalue spectra. In addition to the electronic ground state also excited electronic states can be described. The MCOEP method is implemented invoking the localized Hartree-Fock approximation, leading to a multiconfiguration localized Hartree-Fock approach. Applications of the new method to the dissociation of the hydrogen molecule and the isomerization of ethene and cyclobutadiene show that it is capable of describing situations that are characterized by strong static correlation.     

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.     

On the choice of active space orbitals in MCSCF calculations

We describe a procedure which may be used to aid selection of the active space in multiconfigurational self-consistent field (MCSCF) calculations for general chemical systems. Starting from a restricted Hartree-Fock calculation, we define a hierarchy of interacting virtual orbitals for every occupied orbital. The most strongly interacting orbitals are then taken to constitute the active space in a configuration interaction (CI) calculation. The natural orbital occupation numbers obtained from the CI calculation are then used to choose the active space to be used in a subsequent MCSCF calculation. We illustrate our method on a number of systems (Li₂, B₂, C₂, carbonyl oxide and the transition state for oxidation of H₂S by dioxirane). In all these cases, ‘intuitive’ active spaces are inadequate, as are active spaces derived from the natural orbitals of unrestricted Hartree-Fock calculations.     

Extended charge decomposition analysis and its application for the investigation of electronic relaxation

A general and comprehensive molecular orbital method for the investigation of the electronic relaxation contribution to redox processes is presented. This method is based on the population analysis of the molecular orbitals of the final electronic state in terms of the occupied and unoccupied molecular orbitals of the Koopmans’ state. The DFT calculations for oxidation and reduction of transition-metal species indicate a dramatic magnitude of electronic relaxation in these systems. The passive molecular orbitals play a more significant role in electronic relaxation than the redox-active molecular orbital that directly participates in the redox process. The mechanism of electronic relaxation in the oxidation of Fe^II and Cu^I species varies from the ligand to metal 3d charge transfer (LMCT) interactions to the ligand to metal 4s,4p LMCT. For systems with significant electronic delocalization, electronic relaxation becomes smaller leading to much smaller contributions to the redox processes. Dedication: This contribution is to celebrate Philip Stephen’s seminal contributions to theory and experiment. An erratum to this article can be found at     

A MCSCF method for ground and excited states based on full optimizations of successive Jacobi rotations

A new multiconfigurational self-consistent field (MCSCF) method based on successive optimizations of Jacobi rotation angles is presented. For given one- and two-particle density matrices and an initial set of corresponding integrals, a technique is developed for the determination of a Jacobi angle for the mixing of two orbitals, such that the exact energy, written as a function of the angle, is fully minimized. Determination of the energy-minimizing orbitals for given density matrices is accomplished by successive optimization and updating of Jacobi angles and integrals. The total MCSCF energy is minimized by alternating between CI and orbital optimization steps. Efficiency is realized by optimizing CI and orbital vectors quasi-simultaneously by not fully optimizing each in each improvement step. On the basis of the Jacobi-rotation based approach, a novel MCSCF procedure is formulated for excited states, which avoids certain shortcomings of traditional excited-state MCSCF methods. Applications to specific systems show the practicability of the developed methods.     

Study of localized molecular orbitals using group theory methods and its approach to the many-electron correlation problem. IV. The symmetry-adaptation of many-center integrals and hamiltonian matrix elements in MCSCF calculations

Group theoretic methods are presented for the transformations of integrals and the evaluation of matrix elements encountered in multiconfigurational self-consistent field (MCSCF) and configuration interaction (CI) calculations. The method has the advantages of needing only to deal with a symmetry unique set of atomic orbitals (AO) integrals and transformation from unique atomic integrals to unique molecular integrals rather than with all of them. Hamiltonian matrix element is expressed by a linear combination of product terms of many-center unique integrals and geometric factors. The group symmetry localized orbitals as atomic and molecular orbitals are a key feature of this algorithm. The method provides an alternative to traditional method that requires a table of coupling coefficients for products of the irreducible representations of the molecular point group. Geometric factors effectively eliminate these coupling coefficients. The saving of time and space in integral computations and transformations is analyzed. © 1994 by John Wiley & Sons, Inc.     

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     

Two-dimensional Raman and infrared vibrational spectroscopy for a harmonic oscillator system nonlinearly coupled with a colored noise bath

Multidimensional vibrational response functions of a harmonic oscillator are reconsidered by assuming nonlinear system-bath couplings. In addition to a standard linear-linear (LL) system-bath interaction, we consider a square-linear (SL) interaction. The LL interaction causes the vibrational energy relaxation, while the SL interaction is mainly responsible for the vibrational phase relaxation. The dynamics of the relevant system are investigated by the numerical integration of the Gaussian-Markovian Fokker-Planck equation under the condition of strong couplings with a colored noise bath, where the conventional perturbative approach cannot be applied. The response functions for the fifth-order nonresonant Raman and the third-order infrared (or equivalently the second-order infrared and the seventh-order nonresonant Raman) spectra are calculated under the various combinations of the LL and the SL coupling strengths. Calculated two-dimensional response functions demonstrate that those spectroscopic techniques are very sensitive to the mechanism of the system-bath couplings and the correlation time of the bath fluctuation. We discuss the primary optical transition pathways involved to elucidate the corresponding spectroscopic features and to relate them to the microscopic sources of the vibrational nonlinearity induced by the system-bath interactions. Optical pathways for the fifth-order Raman spectroscopies from an "anisotropic" medium were newly found in this study, which were not predicted by the weak system-bath coupling theory or the standard Brownian harmonic oscillator model.     

Lone-pair orbital interactions in polythiaadamantanes

The electronic structures of a series of polythiaadamantanes from thiaadamantane through 2,4,6,8,9,10-hexathiaadamantane (HTA) have been analyzed using density functional theory calculations in conjunction with Hückel and natural bond orbital analysis. The effects of multiple sulfur p-type lone-pair orbital interactions on ionization potentials, hole mobilities, and electronic coupling have been determined. An overall increase in the average energy of the lone-pair orbitals as the number of sulfur atoms increases is predicted, with the exact positioning of the HOMO depending on specific lone-pair interactions. Separation of through-bond (TB) and through-space (TS) interactions between intramolecular sulfur atoms has been performed using localized molecular orbitals and model systems based on interacting hydrogen sulfide molecules. TB interations were found to reduce orbital splitting, while TS interactions were found to increase orbital splitting. TS interactions were more or less constant from one polythiaadamantane to the next, and the contributions of TB effects to individual orbital energies vary depending on the relative orientation of sulfur atoms as determined by the sigma molecular framework. Electronic coupling between intermolecular sulfur lone-pair orbitals was determined by investigating unique dimer pairs observed in the crystal structure of HTA. Electronic coupling is not as strong as expected given the short intermolecular S-S distances observed in the crystal structure. In general, B3LYP/6-31G(d) and B3LYP/6-311+G(d,p) give very similar orbital energies and splittings.     

Theoretical design of the biradical character in 1,3-diphosphacyclobutanediyl and homologous structures

The electronic nature of 1,3-diphosphacyclobutane-2,4-diyl is explored with wavefunction based and density functional methods. According to MCSCF calculations the singlet state of the title compound is a biradicaloid with closed shell character, the number of unpaired electrons, assigned upon the analysis of the natural orbitals, is close to one. The participation of closed shell contributions in the overall wavefunction arises from a strong mixing of canonical structures, which emphasizes (a) the phosphorane type of bonding as well as (b) π-delocalization within the ring system. The bonding situation changes when σ-attracting substituents, e.g. amino groups, are attached to the phosphorus atoms. They inhibit possible cyclic π-delocalization and enhance the biradical character within the ring system.     

Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals

A method is presented for expressing the occupied self-consistent-field (SCF) orbitals of a molecule exactly in terms of chemically deformed atomic minimal-basis-set orbitals that deviate as little as possible from free-atom SCF minimal-basis orbitals. The molecular orbitals referred to are the exact SCF orbitals, the free-atom orbitals referred to are the exact atomic SCF orbitals, and the formulation of the deformed "quasiatomic minimal-basis-sets" is independent of the calculational atomic orbital basis used. The resulting resolution of molecular orbitals in terms of quasiatomic minimal basis set orbitals is therefore intrinsic to the exact molecular wave functions. The deformations are analyzed in terms of interatomic contributions. The Mulliken population analysis is formulated in terms of the quasiatomic minimal-basis orbitals. In the virtual SCF orbital space the method leads to a quantitative ab initio formulation of the qualitative model of virtual valence orbitals, which are useful for calculating electron correlation and the interpretation of reactions. The method is applicable to Kohn-Sham density functional theory orbitals and is easily generalized to valence MCSCF orbitals.     

Structure and magnetism of [M3](6/7+) metal chain complexes from density functional theory: analysis for copper and predictions for silver

The ground-state electronic structure of the trinuclear complex Cu3(dpa)4Cl2 (1), where dpa is the anion of di(2-pyridyl)amine, has been investigated within the framework of density functional theory (DFT) and compared with that obtained for other known M3(dpa)4Cl2 complexes (M = Cr, Co, Ni) and for the still hypothetical Ag3(dpa)4Cl2 compound. Both coinage metal compounds display three singly occupied x2-y2-like (delta) orbitals oriented toward the nitrogen environment of each metal atom, generating antibonding M-(N4) interactions. All other metal orbital combinations are doubly occupied, resulting in no delocalized metal-metal bonding. This is at variance with the other known symmetric M3(dpa)4Cl2 complexes of the first transition series, which all display some delocalized bonding through the metal backbone, with formal bond multiplicity decreasing in the order Cr > Co > Ni. An antiferromagnetic coupling develops between the singly occupied MOs via a superexchange mechanism involving the bridging dpa ligands. This magnetic interaction can be considered as an extension to the three aligned Cu(II) atoms of the well-documented exchange coupling observed in carboxylato-bridged dinuclear copper compounds. Broken-symmetry calculations with approximate spin projection adequately reproduce the coupling constant observed for 1. Oxidation of 1 removes an electron from the magnetic orbital located on the central Cu atom and its ligand environment; 1+ displays a much weaker antiferromagnetic interaction coupling the terminal Cu-N4 moieties via four ligand pathways converging through the x2-y2 orbital of the central metal. The silver homologues of 1 and 1+ display similar electronic ground states, but the calculated magnetic couplings are stronger by factors of about 3 and 4, respectively, resulting from a better overlap between the metal centers and their equatorial ligand environment within the magnetic orbitals.     

PCILO calculation of intermolecular energies: The water dimer

The PCILO (perturbative configuration interaction using localized orbitals) method for approximating the electronic structure of molecules has been used with some success for calculating intramolecular interactions in large molecules where intramolecular hydrogen bonding is involved. In this note we show that the PCILO method may be used to calculate the energy of interaction between two water molecules in selected configurations.     

Magnetic coupling in dinuclear Gd complexes

A spin density functional (SDFT) study of carboxylate-bridged and diazenido-bridged dinuclear gadolinium compounds is presented. Calculated magnetic coupling constants for the carboxylate-bridged structures are in good agreement with experimental data, confirming the ability of the broken symmetry approach used in this work to predict magnetic behavior in such compounds. The systematic trend wherein symmetrically bridged complexes are antiferromagnetically coupled and asymmetrically bridged are ferromagnetically coupled is reproduced by the SDFT calculations. The mechanism underlying magnetic coupling in closed- and open-shell dinuclear complexes is described using a perturbative molecular orbital model that focuses the influence of the 4f(7)-5d exchange interaction on molecular orbitals with significant 5d-orbital character for the complex [[[(Me(3)Si)(2)N](2)(thf)Gd](2)(N(2))]. Open-shell electronic configurations facilitate strong ferromagnetic coupling, whereas in closed-shell systems antiferromagnetic coupling is usually preferred.     

The use of localized molecular orbitals and the polarization propagator to identify transmission mechanisms in nuclear spin-spin couplings

An extension of the IPPP (inner projections of the polarization propagator) method to theoretically analyze transmission mechanisms of indirect nuclear spin-spin couplings is presented. The localization technique used is modified so that all the canonical molecular orbitals of a compound may be localized to represent chemical bonds, lone pairs, and the corresponding antibonding molecular orbitals. These localized molecular orbitals, together with the polarization propagator, are used to obtain an intuitive picture of how a coupling is generated as a sum of terms, each one consisting of two particle-hole single excitations. This picture can be used to identify underlying transmission mechanisms and quantitatively evaluate their importance toward the total coupling. The paramagnetic spin-orbit and the spin-dipole interactions are studied in detail.