Electronic reorganization triggered by electron transfer: The intervalence charge transfer of a Fe3+/Fe2+ bimetallic complex

The key role of the molecular orbitals in describing electron transfer processes is put in evidence for the intervalence charge transfer (IVCT) of a synthetic nonheme binuclear mixed‐valence Fe³⁺/Fe²⁺ compound. The electronic reorganization induced by the IVCT can be quantified by controlling the adaptation of the molecular orbitals to the charge transfer process. We evaluate the transition energy and its polarization effects on the molecular orbitals by means of ab initio calculations. The resulting energetic profile of the IVCT shows strong similarities to the Marcus' model, suggesting a response behaviour of the ensemble of electrons analogue to that of the solvent. We quantify the extent of the electronic reorganization induced by the IVCT process to be 11.74 eV, a very large effect that induces the crossing of states reducing the total energy of the transfer to 0.89 eV. © 2015 Wiley Periodicals, Inc.     

用分子片轨道逐步剔除法研究电子离域

估计分子中不同类型的电子离域作用（如p-π → d-π和p-π → σ~*）的相 对强弱对理解分子中化学键的本质有关键的作用。剔除某一分子片轨道（d-π或σ ~*）后分子体系能量的改变可用来计算电子离域到该轨道的离域能。由于轨道之间 的相互作用,使离域能的计算与轨道剔除的次序有关。为克服这种不确定性,可以 逐步轮流增加某一对特定轨道（d-π和σ~*）的库仑积分,以使这对轨道在分子波 函数中的比重逐步减少,即将这对轨道轮流逐步剔除。这样可将轨道间的相互影响 减小以至消除,从而得到各轨道的精确的离域能。以H_3PO中P-O键为例说明了轨道 逐步剔除方法的应用。     

Extremely localized molecular orbitals: theory and applications

Orbitals that are extremely localized on molecular fragments represent a powerful tool for a number of purposes: to cite a few examples, they allow to reduce strongly the complexity of calculations on large systems and are easily transferable from one molecule to another, providing a suitable and efficient way to build up the electronic structure of large molecules. Recently, we have developed efficient algorithms to determine extremely localized molecular orbitals (ELMOs), which will be reviewed in this paper. As a rigorous localization is strictly connected to a reduction in the number of variational parameters, which reflects into an increased value of the associated energy with respect to the Hartree Fock value, we have developed a number of strategies to relax the wavefunction built up using transferred localized orbitals. The extreme localization has also been exploited in connection with the “Divide and Conquer” technique to determine the electron densities of large polypeptides assembled from orbitals computed on small model molecules. Moreover, we will discuss the recent application of the ELMOs in the framework of the hybrid QM/MM methods to describe the frontier region. We will also show that the ELMOs can be used to extract chemical interpretations from numerical results. A variety of applications will be presented.     

The Fock Matrix Analysis for Atomic Orbitals in Molecular Orbitals II. The Electronic Structure of N2 Molecules

Weinhold's natural hybrid orbitals can be chosen as the molecular adapted atomic orbitals to build the canonical molecular orbitals of N₂ molecules. The molecular Fock matrix expanded in the natural hybrid orbitals can reveal deeper insight of the electronic structure and reaction of the N₂ molecule. For example, the magnitude of F_ab can signify the bonding character of the paired electrons as well as the diradical character of the unpaired electrons for both σ‐ and π‐types. Discarding the concept of the overlap between non‐orthogonal atomic orbitals, the different orbitals for different spins in the unrestricted Hartree‐Fock wavefunction reveal that there are three pairs of opposite spin density flows between two atoms, which proceed until the bonding molecular orbitals form.     

Electron trajectories in molecular orbitals

The time-dependent Schrödinger equation can be rewritten so that its interpretation is no longer probabilistic. Two well-known and related reformulations are Bohmian mechanics and quantum hydrodynamics. In these formulations, quantum particles follow real, deterministic trajectories influenced by a quantum force. Generally, trajectory methods are not applied to electronic structure calculations as they predict that the electrons in a ground-state, real, molecular wavefunction are motionless. However, a spin-dependent momentum can be recovered from the nonrelativistic limit of the Dirac equation. Therefore, we developed new, spin-dependent equations of motion for the quantum hydrodynamics of electrons in molecular orbitals. The equations are based on a Lagrange multiplier, which constrains each electron to an isosurface of its molecular orbital, as required by the spin-dependent momentum. Both the momentum and the Lagrange multiplier provide a unique perspective on the properties of electrons in molecules.     

An open‐source framework for analyzing N‐electron dynamics. I. Multideterminantal wave functions

The aim of the present contribution is to provide a framework for analyzing and visualizing the correlated many‐electron dynamics of molecular systems, where an explicitly time‐dependent electronic wave packet is represented as a linear combination of N‐electron wave functions. The central quantity of interest is the electronic flux density, which contains all information about the transient electronic density, the associated phase, and their temporal evolution. It is computed from the associated one‐electron operator by reducing the multideterminantal, many‐electron wave packet using the Slater‐Condon rules. Here, we introduce a general tool for post‐processing multideterminant configuration‐interaction wave functions obtained at various levels of theory. It is tailored to extract directly the data from the output of standard quantum chemistry packages using atom‐centered Gaussian‐type basis functions. The procedure is implemented in the open‐source Python program det CI@ORBKIT, which shares and builds on the modular design of our recently published post‐processing toolbox (Hermann et al., J. Comput. Chem. 2016, 37, 1511). The new procedure is applied to ultrafast charge migration processes in different molecular systems, demonstrating its broad applicability. Convergence of the N‐electron dynamics with respect to the electronic structure theory level and basis set size is investigated. This provides an assessment of the robustness of qualitative and quantitative statements that can be made concerning dynamical features observed in charge migration simulations. © 2017 Wiley Periodicals, Inc.     

Natural bond orbital analysis in the ONETEP code: Applications to large protein systems

First principles electronic structure calculations are typically performed in terms of molecular orbitals (or bands), providing a straightforward theoretical avenue for approximations of increasing sophistication, but do not usually provide any qualitative chemical information about the system. We can derive such information via post‐processing using natural bond orbital (NBO) analysis, which produces a chemical picture of bonding in terms of localized Lewis‐type bond and lone pair orbitals that we can use to understand molecular structure and interactions. We present NBO analysis of large‐scale calculations with the ONETEP linear‐scaling density functional theory package, which we have interfaced with the NBO 5 analysis program. In ONETEP calculations involving thousands of atoms, one is typically interested in particular regions of a nanosystem whilst accounting for long‐range electronic effects from the entire system. We show that by transforming the Non‐orthogonal Generalized Wannier Functions of ONETEP to natural atomic orbitals, NBO analysis can be performed within a localized region in such a way that ensures the results are identical to an analysis on the full system. We demonstrate the capabilities of this approach by performing illustrative studies of large proteins—namely, investigating changes in charge transfer between the heme group of myoglobin and its ligands with increasing system size and between a protein and its explicit solvent, estimating the contribution of electronic delocalization to the stabilization of hydrogen bonds in the binding pocket of a drug‐receptor complex, and observing, in situ, the n → π* hyperconjugative interactions between carbonyl groups that stabilize protein backbones. © 2012 Wiley Periodicals, Inc.     

An optimized initialization algorithm to ensure accuracy in quantum Monte Carlo calculations

Quantum Monte Carlo (QMC) calculations require the generation of random electronic configurations with respect to a desired probability density, usually the square of the magnitude of the wavefunction. In most cases, the Metropolis algorithm is used to generate a sequence of configurations in a Markov chain. This method has an inherent equilibration phase, during which the configurations are not representative of the desired density and must be discarded. If statistics are gathered before the walkers have equilibrated, contamination by nonequilibrated configurations can greatly reduce the accuracy of the results. Because separate Markov chains must be equilibrated for the walkers on each processor, the use of a long equilibration phase has a profoundly detrimental effect on the efficiency of large parallel calculations. The stratified atomic walker initialization (STRAW) shortens the equilibration phase of QMC calculations by generating statistically independent electronic configurations in regions of high probability density. This ensures the accuracy of calculations by avoiding contamination by nonequilibrated configurations. Shortening the length of the equilibration phase also results in significant improvements in the efficiency of parallel calculations, which reduces the total computational run time. For example, using STRAW rather than a standard initialization method in 512 processor calculations reduces the amount of time needed to calculate the energy expectation value of a trial function for a molecule of the energetic material RDX to within 0.01 au by 33%.     

Optimal virtual orbitals to relax wave functions built up with transferred extremely localized molecular orbitals

Extremely localized molecular orbitals (ELMOs), namely orbitals strictly localized on molecular fragments, are easily transferable from one molecule to another one. Hence, they provide a natural way to set up the electronic structure of large molecules using a data base of orbitals obtained from model molecules. However, this procedure obviously increases the energy with respect to a traditional MO calculation. To gain accuracy, it is important to introduce a partial electron delocalization. This can be carried out by defining proper optimal virtual orbitals that supply an efficient set for nonorthogonal configurations to be employed in VB-like expansions.     

QuanPol: A full spectrum and seamless QM/MM program

The quantum chemistry polarizable force field program (QuanPol) is implemented to perform combined quantum mechanical and molecular mechanical (QM/MM) calculations with induced dipole polarizable force fields and induced surface charge continuum solvation models. The QM methods include Hartree–Fock method, density functional theory method (DFT), generalized valence bond theory method, multiconfiguration self‐consistent field method, Møller–Plesset perturbation theory method, and time‐dependent DFT method. The induced dipoles of the MM atoms and the induced surface charges of the continuum solvation model are self‐consistently and variationally determined together with the QM wavefunction. The MM force field methods can be user specified, or a standard force field such as MMFF94, Chemistry at Harvard Molecular Mechanics (CHARMM), Assisted Model Building with Energy Refinement (AMBER), and Optimized Potentials for Liquid Simulations‐All Atom (OPLS‐AA). Analytic gradients for all of these methods are implemented so geometry optimization and molecular dynamics (MD) simulation can be performed. MD free energy perturbation and umbrella sampling methods are also implemented. © 2013 Wiley Periodicals, Inc.     

A method for the calculation of transition moments between electronic states of molecules using a different one-electron basis set for each state

A state-specific approach to the calculation of transition moments between molecular electronic states requires that the wavefunction for each state is expanded in its optimum one-electron basis and that nonorthonormal basis techniques are used for the calculation of the transition moment integrals. A method has been developed for carrying out such nonorthonormal basis calculations, based on the corresponding orbitals transformation and appropriately defined density matrices, which may be used with configuration interaction (CI ) wavefunctions. Further improvements of the method have resulted in a decrease in the time required for the calculations and thus allow its application with moderately large CI expansions for each state. Nonorthonormal basis calculations on transition moments in H₂O have been carried out using the above method. The results are in agreement with those of large MRD -CI calculations.     

从头计算法分子轨道绘图程序

在前文中,我们报道了以Slater函数(STO)和类氢函数为原子轨道基组的分子轨道绘图程序,文介绍以Gauss函数(GTO)为基组的从头计算分子轨道绘图程序,该程序可绘制各类型(s,p,d等)GTO表示的原子轨道和以STO-GTO系基组构成的各种分子轨道ψ或|ψ|²的截面立体图及等值图,利用从头计算的结果,所绘制的图形能很好的反映从头计算法本身的特点。     

Virtual orbitals for obtaining rapid convergence in configuration interaction calculations

A perturbational method is used to generate virtual orbitals intended to yield rapid convergence in configuration interaction calculations. It is shown that the problem of finding the orbitals and pair functions which maximize the interaction with the Hartree-Fock wavefunction can be solved exactly. The resulting first-order semi-internal virtual orbitals are determined by a simple closed expression. All further semi-internal excitations vanish to first order. As an example, the method of this work is applied to the semi-internal orbitals of the ground state of the carbon +1 ion. The resulting first order orbitals compare very well to corresponding variational virtual orbitals.     

Model study of the impact of orbital choice on the accuracy of coupled-cluster energies. II. Valence-universal coupled-cluster method

The impact of the choice of molecular orbital sets on the results of the valence-universal coupled cluster method involving up to three-body amplitudes (VU-CCSDT) was studied for the H4 model. This model offers a straightforward way of representing all possible symmetry-adapted orbitals. Moreover, the degree of quasi-degeneracy of its lowest ¹A₁ states can be varied over a wide range by changing its geometry. Calculations were performed both for 13 sets of standard quantum chemical orbitals and for a vast variety of nonstandard orbital sets defined by nodes of a two-dimensional orbital grid. The performance of various standard orbital sets in VU-CCSDT calculations is compared. It is also documented that for every quasi-degeneracy region there exist nonstandard orbital sets which allow one to obtain more accurate VU-CCSDT energies than the standard orbital sets. In an attempt to provide a general interpretation for some of the alternative orbital sets, we defined a set of orbitals which maximize the proximity of the model and target spaces—maximum proximity orbitals (MPO). It is demonstrated that outside the strong quasi-degeneracy region the energies obtained for the VU-CCSDT approach based on the MPOs are more accurate than for the standard orbital sets. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 221–237, 1998     

块定域波函数方法及其应用

提出了块定域波函数方法以定量分析分子内的电子定域现象或分子间的电荷传递效应。对于一个假想的严格定域的分子,我们通过将全部的电子和基轨道配分成几个子空间来构造其相应的波函数。其中每一个分子轨道只对某一个子空间展开,各子空间内的分子轨道相互正交,但不同子空间内的分子轨道间是非正交的。Hartree-Fock波函数和块定域波函数之间的能量之差即为分子内的电子定域能或分子间的电荷传递能。我们应用块定域波函数方法讨论了丁二烯分子中的旋转势垒。     

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.     

Accurate Quantum‐Chemical Prediction of Enthalpies of Formation of Small Molecules in the Gas Phase

The coupled‐cluster approach, including single and double excitations and perturbative corrections for triple excitations, is capable of predicting molecular electronic energies and enthalpies of formation of small molecules in the gas phase with very high accuracy (specifically, with error bars less than 5 kJ mol^?1), provided that the electronic wavefunction is dominated by the Hartree–Fock configuration. This capability is illustrated by calculations on molecules containing O–H and O–F bonds, namely OH, FO, H₂O, HOF, and F₂O. To achieve this very high accuracy, it is imperative to account for electron‐correlation effects in a quantitative manner, either by using explicitly correlated two‐particle basis functions (R12 functions) or by extrapolating to the limit of a complete basis. Besides taking into account harmonic zero‐point vibrational energies, it is also necessary to account for anharmonic corrections to the zero‐point vibrational energies, to include the core orbitals into the coupled‐cluster calculations, and to account for spin–orbit corrections and scalar relativistic effects. These additional corrections constitute small but significant contributions in the range of 1–4 kJ mol^?1 to the enthalpies of formation of the aforementioned molecules. The highly accurate coupled‐cluster results, obtained by employing R12 functions and by including various corrections, are compared with standard Kohn–Sham density‐functional calculations as well as with the Gaussian‐2 and complete‐basis‐set model chemistries.     

Influence of Intermolecular Vibrations on the Electronic Coupling in Organic Semiconductors: The Case of Anthracene and Perfluoropentacene

We have performed classical molecular dynamics simulations and quantum‐chemical calculations on molecular crystals of anthracene and perfluoropentacene. Our goal is to characterize the amplitudes of the room‐temperature molecular displacements and the corresponding thermal fluctuations in electronic transfer integrals, which constitute a key parameter for charge transport in organic semiconductors. Our calculations show that the thermal fluctuations lead to Gaussian‐like distributions of the transfer integrals centered around the values obtained for the equilibrium crystal geometry. The calculated distributions have been plugged into Monte‐Carlo simulations of hopping transport, which show that lattice vibrations impact charge transport properties to various degrees depending on the actual crystal structure.