A quantum chemical calculation on Fe(CO)5 revealing the operation of the Dewar–Chatt–Duncanson model

A nonstandard computational scheme has been applied to calculate Fe(CO)₅ with the aim to illustrate the operation of the Dewar–Chatt–Duncanson model by computation. A full configuration interaction (CI) calculation in an active space has been performed. The active space is built from naturally localized molecular orbitals (NLMOs) localized in bond regions or forming lone pairs. For selecting this active space, Weinhold's perturbation theory formulated in the natural bond orbital (NBO) space has been applied. Bonding, lone pair, and antibond NBOs exhibiting large interaction energies serve to define the active space. The actually applied active space, however, comprises NLMOs that are close in shape to the NBOs indicated by perturbation theory. Thus, a CI calculation with localized orbitals has been performed meeting the classical reasoning of chemists that is often based on local bonding concepts. The computational scheme yields the Lewis structure for Fe(CO)₅ whose energy is identical to the Hartree–Fock energy. The Lewis energy comprises CO → Fe σ‐electron transfer (ET) and CO ← Fe electron back donation (BD). This Lewis energy gets lowered by localized correlation energy contributions caused by ET processes where electrons are back donated from the Fe d‐lone pairs into the CO ligands. Thus, electron correlation within the selected active space is dominated by electron BD. Energies and electron populations of the NBOs support the notion that electrons are preferentially back donated into the equatorial CO ligands. Weights of local Slater determinants, determining the correlation energy, also point to a predominant BD into the equatorial CO ligands. Correlation energy increments resulting from electron BD into single antibond orbitals of the CO ligands have been calculated. These energy quantities also demonstrate that BD into the equatorial CO ligands is more energy lowering than BD into the axial CO ligands. © 2012 Wiley Periodicals, Inc.     

Pipek–Mezey localization of occupied and virtual orbitals

Recent advances in orbital localization algorithms are used to minimize the Pipek–Mezey localization function for both occupied and virtual Hartree–Fock orbitals. Virtual Pipek–Mezey orbitals for large molecular systems have previously not been considered in the literature. For this work, the Pipek–Mezey (PM) localization function is implemented for both the Mulliken and a Löwdin population analysis. The results show that the standard PM localization function (using either Mulliken or Löwdin population analyses) may yield local occupied orbitals, although for some systems the occupied orbitals are only semilocal as compared to state‐of‐the‐art localized occupied orbitals. For the virtual orbitals, a Löwdin population analysis shows improvement in locality compared to a Mulliken population analysis, but for both Mulliken and Löwdin population analyses, the virtual orbitals are seen to be considerably less local compared to state‐of‐the‐art localized orbitals. © 2013 Wiley Periodicals, Inc.     

Correlation regions within a localized molecular orbital approach

A hybrid scheme for the computation of reaction energies in large molecular systems is proposed. The approach is based on localized orbitals and allows for the treatment of different parts of a molecule at different computational levels. The localized orbitals are assigned to regions, and then different local correlation methods, such as local MP2 or local CCSD(T), can be applied to different regions. In contrast to previous hybrid schemes, the molecule does not have to be split into parts and, therefore, it is not necessary to saturate dangling bonds using link atoms. For fixed region sizes, the cost of the high-level calculation becomes independent of the molecular size, and it is demonstrated that O(1) scaling can be achieved. Illustrative applications are presented and the convergence of the results with respect to the size of the regions is investigated for reaction energies, barrier heights, and weakly bound complexes.     

Transient bonds and chemical reactivity of molecules

Electron delocalization between the reagent and reactant molecules is the principal driving force of chemical reactions. It brings about the formation of new bonds and the cleavage of old bonds. By taking the aromatic substitution reaction as an example, we have shown the orbitals participating in electron delocalization. The interacting orbitals obtained are localized around the reaction sites, showing the chemical bonds that should be generated and broken transiently along the reaction path. By projecting a reference orbital function that has been chosen to specify the bond being formed on to the MOs of the reactant molecules, the reactive orbitals that are very similar to the interacting orbital have been obtained. The local potential of the reaction site for electron donation estimated for substituted benzene molecules by using these projected orbitals shows a fair correlation with the experimental scale of the electron-donating and -withdrawing strength of substituent groups. The reactivity is shown to be governed by local electronegativity and local chemical hardness and also by the localizability of interaction on the reaction site. © 1996 John Wiley & Sons, Inc.     

The divide-expand-consolidate family of coupled cluster methods: numerical illustrations using second order Møller-Plesset perturbation theory

Previously, we have introduced the linear scaling coupled cluster (CC) divide-expand-consolidate (DEC) method, using an occupied space partitioning of the standard correlation energy. In this article, we show that the correlation energy may alternatively be expressed using a virtual space partitioning, and that the Lagrangian correlation energy may be partitioned using elements from both the occupied and virtual partitioning schemes. The partitionings of the correlation energy leads to atomic site and pair interaction energies which are term-wise invariant with respect to an orthogonal transformation among the occupied or the virtual orbitals. Evaluating the atomic site and pair interaction energies using local orbitals leads to a linear scaling algorithm and a distinction between Coulomb hole and dispersion energy contributions to the correlation energy. Further, a detailed error analysis is performed illustrating the error control imposed on all components of the energy by the chosen energy threshold. This error control is ultimately used to show how to reduce the computational cost for evaluating dispersion energy contributions in DEC.     

Charge fluctuations and correlation strength in chemical bonds: First-row homonuclear diatomic molecules

We investigate, by means of ab initio calculations, the strength of electron correlations within covalent bonds: the first-row homonuclear diatomics serve as test cases. As an appropriate measure of the correlation strength, we introduce the reduction of the mean-square deviations of the electronic charges in localized orbitals forming a bond. A recently developed population analysis in terms of local operators derived from localized molecular orbitals is thereby used. The correlation-strength parameter depends only weakly on dynamical correlations as test calculations demonstrate. Therefore, the full-valence complete active space self-consistent field (CASSCF) approximation is applied in order to study the changes in the correlation strength with changing bond length for different types of bonds. A number of simple rules emerge from this discussion. In addition, we show that charge fluctuations are not only a reliable measure of intrabond correlation effects, but also can be used to monitor intraatomic quasi-degeneracy effects as well as the interdependence within multiple bonds. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 157–173, 1998     

The perturbation theory of the extended Hartree–Fock approximation for two-electron atoms

The first-order 1/Z perturbation theory of the extended Hartree–Fock approximation for two-electron atoms is described. A number of unexpected features emerge: (a) it is proved that the orbitals must be expanded in powers of Z^?1/2, rather than in Z^?1 as expected; (b) it is shown that the restricted Hartree–Fock and correlation parts of the orbitals can be uncoupled to first order, so that second-order energies are additive; (c) the equation describing the first-order correlation orbital has an infinite number of solutions of all angular symmetries in general, rather than only one of a single symmetry as expected; (d) the first-order correlation equation is a homogeneous linear eigenvalue-type equation with a non-local potential. It involves a parameter μ and an eigenvalue ω(μ) which may be interpreted as the probability amplitude and energy of a virtual correlation state. The second-order correlation energy is 2μ²ω. Numerical solutions for the first-order correlation orbitals, obtained variationally, are presented. The approximate second-order correlation energy is nearly 90% of the exact value. The first-order 1/Z perturbation theory of the natural-orbital expansion is described, and the coupled first-order integro-differential perturbation equations are obtained. The close relationship between the first-order extended Hartree–Fock correlation orbitals and the first-order natural correlation orbitals is discussed. A comparison of the numerical results with those of Kutzelnigg confirms the similarity.     

Calculations on the structure and spectral properties of cytochrome c551 using DFT and ONIOM methods

The active site geometry of cytochrome (Cyt) c(551) and its mutated form containing Fe(II) and Fe(III) ions have been calculated using density functional theory (DFT)-based Becke's three-parameter hybrid exchange and Lee-Yang-Parr correlation (B3LYP) method. In addition, calculations have also been carried out using hybrid meta DFT-based M06 functional. The effect of the protein milieu on the active site geometry has also been probed using two-layer via our own N-layered integrated molecular orbital + molecular mechanics (ONIOM) method. Evidence from the calculations reveal that the active site geometry is not significantly affected by the oxidation state of metal ion. The difference in the geometry of the active site and that of the same with the entire protein environment is only minimal, which shows that the protein milieu does not influence the structure of the active site. The calculated electronic transition energies from the time-dependent DFT (TDDFT) calculations are in close agreement with the experimental values. Although there are no significant variations in the active site geometry upon oxidation, the changes in the electronic transition energies have been attributed to the reduction in the overlap of metal ion with the ligand orbitals. In addition, it is found that mutation does not influence the active site geometry and the electronic transition energies. Nevertheless, mutation leads to the formation of more compact structure than the native Cyt c(551).     

Local Hartree–Fock orbitals using a three‐level optimization strategy for the energy

Using the three‐level energy optimization procedure combined with a refined version of the least‐change strategy for the orbitals—where an explicit localization is performed at the valence basis level—it is shown how to more efficiently determine a set of local Hartree–Fock orbitals. Further, a core–valence separation of the least‐change occupied orbital space is introduced. Numerical results comparing valence basis localized orbitals and canonical molecular orbitals as starting guesses for the full basis localization are presented. The results show that the localization of the occupied orbitals may be performed at a small computational cost if valence basis localized orbitals are used as a starting guess. For the unoccupied space, about half the number of iterations are required if valence localized orbitals are used as a starting guess compared to a canonical set of unoccupied Hartree–Fock orbitals. Different local minima may be obtained when different starting guesses are used. However, the different minima all correspond to orbitals with approximately the same locality. © 2013 Wiley Periodicals, Inc.     

Molecular MC–SCF calculations

A practical method for finding multi-configurational SCF wave functions is proposed. The basic equation is equivalent to the Brillouin theorem; comparison with the usual SCF equations obtained through effective hamiltonians gives an interpretation of the offdiagonal Lagrange multipliers. Numerical applications to Formaldehyde in a minimum Slater-type orbital basis with four different variational wave functions are reported. The molecular orbitals found in these calculations are localized on the chemical bonds. The largest contributions to the energy are obtained from π-π and dispersion-type σ-π correlation.     

Charge‐transfer‐to‐solvent absorption spectra of I−(H2O)3–5 at a finite temperature via simulation

Ground‐state equilibrium Born–Oppenheimer molecular dynamics on I^?(H₂O)_3–5 clusters at ～200 K are performed to sample configurations for calculating the charge‐transfer‐to‐solvent (CTTS) absorption spectra for these clusters. When there are more water molecules in clusters, the calculated CTTS spectra are found to become more intense with the absorption maxima shifting to higher energies, which is in agreement with experimental results. In addition, compared with the findings for optimized structures, the absorption energies of the iodide 5p orbitals are red‐shifted at ～200 K because, on average, the distances between the iodide and the dangling hydrogen atoms are increased at finite temperatures which weakens the interactions between the iodide and water molecules in the clusters. Moreover, the number of ionic hydrogen bonds in the clusters are also reduced. However, it is found that all dangling hydrogen atoms must be considered to obtain a good correlation between the CTTS excitation energy and the average distance between the iodide and the dangling hydrogen atoms, which indicates the existence of the strong interactions of the CTTS electron with all of the dangling hydrogen atoms.     

Extended Hartree–Fock calculations for the helium ground state

The extended Hartree–Fock (EHF) wave function of an n-electron system is defined (Löwdin, Phys. Rev. 97 , 1509 (1955)) as the best Slater determinant built on one-electron spin orbitals having a complete flexibility and projected onto an appropriate symmetry subspace. The configuration interaction equivalent to such a wavefunction for the ¹S state of a two-electron atom is discussed. It is shown that there is in this case an infinite number of solutions to the variational problem with energies lower than that of the usual Hartree–Fock function, and with spin orbitals satisfying all the extremum conditions. Two procedures for obtaining EHF spin orbitals are presented. An application to the ground state of Helium within a basic set made up of 4(s), 3(p₀), 2(d₀) and 1 (f₀) Slater orbitals has produced 90% of the correlation energy.     

Theoretical study on the structure and UV–visible spectrum of pyridine–chloranil charge-transfer complex

The ground-state structure of the charge-transfer complex formed by pyridine (Py) as electron donor and chloranil (CA) as acceptor has been studied by full geometry optimization at the MP2 and DFT levels of theory. Binding energies were calculated and counterpoise corrections were used to correct the BSSE. Both MP2 and DFT indicate that the pyridine binds with chloranil to form an inclined T-shape structure, with the pyridine plane perpendicular to the chloranil. The CP and ZPE corrected binding energies were calculated to be 14.21 kJ/mol by PBEPBE/6-31G(d) and 23.21 kJ/mol by MP2/6-31G(d). The charge distribution of the ground state Py–CA complex was evaluated with the natural population analysis, showing a net charge transfer from Py to CA. Analysis of the frontier molecular orbitals reveals a σ–π interaction between CA and Py, and the binding is reinforced by the attraction of the O₇ atom of CA with the H₂₃ atom of Py. TD-DFT calculations have been performed to analyze the UV–visible spectrum of Py–CA complex, revealing both the charge transfer transitions and the weak symmetry-relieved chloranil π–π^* transition in the UV–visible region.     

Calculation of wave‐functions with frozen orbitals in mixed quantum mechanics/molecular mechanics methods. II. Application of the local basis equation

The application of the local basis equation (Ferenczy and Adams, J. Chem. Phys. 2009 , 130, 134108) in mixed quantum mechanics/molecular mechanics (QM/MM) and quantum mechanics/quantum mechanics (QM/QM) methods is investigated. This equation is suitable to derive local basis nonorthogonal orbitals that minimize the energy of the system and it exhibits good convergence properties in a self‐consistent field solution. These features make the equation appropriate to be used in mixed QM/MM and QM/QM methods to optimize orbitals in the field of frozen localized orbitals connecting the subsystems. Calculations performed for several properties in divers systems show that the method is robust with various choices of the frozen orbitals and frontier atom properties. With appropriate basis set assignment, it gives results equivalent with those of a related approach [G. G. Ferenczy previous paper in this issue] using the Huzinaga equation. Thus, the local basis equation can be used in mixed QM/MM methods with small size quantum subsystems to calculate properties in good agreement with reference Hartree–Fock–Roothaan results. It is shown that bond charges are not necessary when the local basis equation is applied, although they are required for the self‐consistent field solution of the Huzinaga equation based method. Conversely, the deformation of the wave‐function near to the boundary is observed without bond charges and this has a significant effect on deprotonation energies but a less pronounced effect when the total charge of the system is conserved. The local basis equation can also be used to define a two layer quantum system with nonorthogonal localized orbitals surrounding the central delocalized quantum subsystem. © 2013 Wiley Periodicals, Inc.     

Computational Analysis of 47/49Ti NMR Shifts and Electric Field Gradient Tensors of Half‐Titanocene Complexes: Structure–Bonding–Property Relationships

Metal NMR shielding and electric‐field gradient (EFG) tensors are examined by quantum‐chemical calculations for a set of 14 titanium(IV) complexes. Benchmarks are performed for titanocene chlorides that have been characterized recently by solid‐state NMR experiments, focusing on the dependence of Ti^IV NMR parameters on the computational model in terms of the choice of the density functional, and considering molecular clusters versus infinite‐periodic solid. Nearest‐neighbor and long‐range effects in the solid state are found to influence NMR parameters in systems without spatially extended ligands. Bulky ligands increase the fraction of local structure and bonding information encoded in the EFG tensors by reducing intermolecular interactions. Next, Ti shielding constants and EFG tensors for a variety of olefin (co)polymerization catalysts are analyzed in terms of contributions from localized molecular orbitals representing Lewis bonds and lone pairs. Direct links between the observed theoretical trends and the local bonding environment around the Ti metal center are found. A specific dependence of the Ti EFG tensors on the exact arrangement and type of surrounding bonds is demonstrated, providing a basis for further studies on solid‐supported titanium catalytic systems.     

Coupled cluster studies. IV. Analysis of the correlated wavefunction in canonical and localized orbital basis for ethylene,carbon monoxide,and carbon dioxide

An a posteriori analysis of the correlated wavefunctions of three small molecules using canonical and localized orbitals shows that, while more excitations are nearly zero for canonical orbitals than for localized ones, in the latter case a straightforward way exists for a priori selection of negligible excitations. In the case of the larger molecule cytosine the same observation is made. However, in this case 99% of the correlation energy is obtained already with ≈ 10% of the excitations when localized orbitals are used, while ≈ 36% of them are necessary in canonical basis. Furthermore it is shown that, using localized orbitals, the excitations can be split into subsets which can be calculated individually. The results suggest that 80–90% of the correlation energies given by MP2, CCL, or CCD can be obtained from the contributions of individual chemical bonds and their interactions. A simple derivation of the orbital invariant formalism of Pulay and Sæbø for the calculation of MP2 and MP3 correlation energies is given.     

Study on some metal–metal quadruple bonds

In the present paper, the role of (n ? 1)? orbitals in metal–metal quadruple bonds was studied. It was shown by the calculations that the probabilities for finding the σ-, π-, and δ-electrons between two metal atoms, under the influence of the ? orbitals on the metal–metal quadruple bonds, increased while their mean kinetic energy components along the metal bond axis decreased. In addition, the effects of the ? orbitals upon the σ, π, and δ metal–metal bonds were different. In general, σ < π < δ.     

镱硫属化合物的密度泛函理论研究

用密度泛函理论(DFT)研究镱硫属化合物的电子结构和性质,通过与实验比较考察了现有的几种近似密度泛函公式对镧系元素化合物的适用程度和相对论效应的影响.结果表明,用DFT计算的YbO键长对实验值的偏差约为0.002nm;但得到的键能即使在考虑梯度校正和相对论效应之后,仍比实验值高,在定域密度近似基础上引入交换梯度校正使键能计算值减小,其中PW86x使键能计算值减小稍多些,结果更接近实验值;相关梯度校正使键能计算值升高.相对论效应使键长缩短0.004～0.006nm,键能减小约0.5eV.计算结果的分析表明,Yb的5d轨道和配体的np轨道间形成σ键和π键.在所研究的分子体系中,配体原子从O到Te、Yb原子的5d轨道布后数依次减少,同键能减弱的顺序一致.相对论效应使键能减小的主要原因是在成键过程中发生了Yb的6s电子向5d轨道的转移,而相对论效应使该过程能量增加.偶极矩和电荷分布的计算表明,Yb-L键以共价性为主,相对论效应使共价性成份增加.     

Exciting flavins: Absorption spectra and spin–orbit coupling in light–oxygen–voltage (LOV) domains

Flavins are central molecular chromophores for many photobiological processes. In this paper, several aspects of the photophysics and photochemistry of lumiflavin in a (protein) environment will be studied with the help of quantum chemical methods. In a first part of the paper, we present vertical singlet excitation energies for lumiflavin (a molecule of the isoalloxazin type), using time-dependent density functional theory (TD-DFT) in conjunction with the B3LYP hybrid functional. When calculated for isolated species, TD-DFT excitation energies are generally blue-shifted relative to the experimental absorption spectra of isoalloxazines in solution, or in a protein environment. We develop four different models to account for environmental effects, with special emphasis on the LOV1 domain of Chlamydomonas reinhardtii. It is found that the two lowest, allowed singlet excitations are sensitive to the polarizability of an environment, to hydrogen bonds, and to geometrical constraints imposed by the surrounding protein. All of this brings theory and experiment in better agreement.

In the second part of the paper the light-induced adduct formation in LOV domains, between the chromophore and a neighbouring cystein unit is investigated. Energies along a model “reaction path” are calculated on the DFT/B3LYP and MCQDPT2 level of theories. A transition state for a H-transfer between the mercapto (SH–) group of cystein, and the N(5) position of flavin is found. The reaction requires spin–orbit coupling between the S₀ and the T₁ states of the system. Spin–orbit coupling constants between S₀ and T₁ are calculated, and found to be in the range of several tens of cm⁻¹ after the transition state was passed. A biradical intermediate was observed along the reaction path.