An open‐source framework for analyzing N‐electron dynamics. II. Hybrid density functional theory/configuration interaction methodology

In this contribution, we extend our framework for analyzing and visualizing correlated many‐electron dynamics to non‐variational, highly scalable electronic structure method. Specifically, an explicitly time‐dependent electronic wave packet is written as a linear combination of N‐electron wave functions at the configuration interaction singles (CIS) level, which are obtained from a reference time‐dependent density functional theory (TDDFT) calculation. The procedure is implemented in the open‐source Python program det CI@ORBKIT, which extends the capabilities of our recently published post‐processing toolbox (Hermann et al., J. Comput. Chem. 2016, 37, 1511). From the output of standard quantum chemistry packages using atom‐centered Gaussian‐type basis functions, the framework exploits the multideterminental structure of the hybrid TDDFT/CIS wave packet to compute fundamental one‐electron quantities such as difference electronic densities, transient electronic flux densities, and transition dipole moments. The hybrid scheme is benchmarked against wave function data for the laser‐driven state selective excitation in LiH. It is shown that all features of the electron dynamics are in good quantitative agreement with the higher‐level method provided a judicious choice of functional is made. Broadband excitation of a medium‐sized organic chromophore further demonstrates the scalability of the method. In addition, the time‐dependent flux densities unravel the mechanistic details of the simulated charge migration process at a glance. © 2017 Wiley Periodicals, Inc.     

Time‐dependent density functional calculations on the electronic spectra of the neutral nickel complex [Ni(LISQ)2] (LISQ = 3,5‐di‐tert‐butyl‐o‐diiminobenzosemiquinonate(1−)) and its monoanion and dication

The electronic spectrum of the neutral nickel complex [Ni(L^ISQ)₂] (L^ISQ = 3,5‐di‐tert‐butyl‐o‐diiminobenzosemiquinonate(1^?)) and the spectra of its anion and dication have been calculated by means of time‐dependent density functional theory. The electronic ground state of the neutral complex exhibits an open shell singlet diradical character. The mandatory multireference problem for this electronic ground state has been treated approximately by using the unrestricted and spin symmetry broken Kohn‐Sham Slater determinant as the wave function for the noninteracting reference system in the time‐dependent density functional calculations. A reasonable agreement with observed transition energies and band intensities has been achieved. This holds also for the long wavelength transitions that are shown to be of charge transfer type. The charge distributions in the electronic ground state and the corresponding low lying excited states, however, are rather similar. Thus, the known failure of standard time‐dependent density functional theory to describe improperly long range charge transfer transitions is absent in this work. The applied computational scheme might be adequate for calculating electronic spectra of transition metal complexes with noninnocent ligands. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Evaluation of molecular integral of Cartesian Gaussian type basis function with complex‐valued center coordinates and exponent via the McMurchie–Davidson recursion formula,and its application to electron dynamics

McMurchie–Davidson recursion formula is extended to derive the ab initio molecular integrals with higher angular quantum number complex Gaussian type basis function which has complex‐valued center coordinates and a complex‐valued exponent. Using the analytical recursion formulae, some calculations of electronic dynamics after beta decay of tritium hydride molecular ion HT⁺ are performed by a quantum wave packet method with thawed Gaussian basis functions of s‐ and p‐type. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Necessary conditions for the N‐representability of pair distribution functions

A necessary condition for the N‐representability of the electron pair density proposed by one of the authors (E. R. D.) is generalized. This shows a link between this necessary condition and other, more widely known, N‐representability conditions for the second‐order density matrix. The extension to spin‐resolved electron pair densities is considered, as is the extension to higher‐order distribution functions. Although quantum mechanical systems are our primary focus, the results are also applicable to classical systems, where they reduce to an inequality originally derived by Garrod and Percus. As a simple application, bounds to the average angle between an electron pair are derived. It is shown that computational methods based on variational minimization of the energy with respect to the electron pair density can give extremely poor results unless robust N‐representability constraints are considered. For reference, constraints for the N‐representability of the pair density are summarized. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Interesting properties of Thomas–Fermi kinetic and Parr electron–electron‐repulsion DFT energy functional generated compact one‐electron density approximation for ground‐state electronic energy of molecular systems

The reduction of the electronic Schrodinger equation or its calculating algorithm from 4N‐dimensions to a (nonlinear, approximate) density functional of three spatial dimension one‐electron density for an N‐electron system, which is tractable in the practice, is a long desired goal in electronic structure calculation. If the Thomas‐Fermi kinetic energy (～∫ρ^5/3d r ₁) and Parr electron–electron repulsion energy (～∫ρ^4/3d r ₁) main‐term functionals are accepted, and they should, the later described, compact one‐electron density approximation for calculating ground state electronic energy from the 2nd Hohenberg–Kohn theorem is also noticeable, because it is a certain consequence of the aforementioned two basic functionals. Its two parameters have been fitted to neutral and ionic atoms, which are transferable to molecules when one uses it for estimating ground‐state electronic energy. The convergence is proportional to the number of nuclei (M) needing low disc space usage and numerical integration. Its properties are discussed and compared with known ab initio methods, and for energy differences (here atomic ionization potentials) it is comparable or sometimes gives better result than those. It does not reach the chemical accuracy for total electronic energy, but beside its amusing simplicity, it is interesting in theoretical point of view, and can serve as generator function for more accurate one‐electron density models. © 2008 Wiley Periodicals, Inc. J Comput Chem 2009     

The self‐consistent charge density functional tight‐binding theory study of carbon adatoms using tuned Hubbard U parameters

The self‐consistent charge density functional tight‐binding (DFTB) theory is a useful tool for realizing the electronic structures of large molecular complex systems. In this study, the electronic structure of C₆₁ formed by fullerene C₆₀ with a carbon adatom is analyzed, using the fully localized limit and pseudo self‐interaction correction methods of DFTB to adjust the Hubbard U parameter (DFTB + U). The results show that both the methods used to adjust U can significantly reduce the molecular orbital energy of occupied states localized on the defect carbon atom and improve the gap between highest occupied molecular orbital(HOMO) and lowest unoccupied molecular orbital(LUMO) of C₆₁. This work will provide a methodological reference point for future DFTB calculations of the electronic structures of carbon materials.     

Tunable Electronic Properties of a Proton‐Responsive N,N‐Dimethylaminoazobenzene Fullerene (C60) Dyad

A basic N,N‐dimethylaminoazobenzene–fullerene (C₆₀) dyad molecular skeleton is modelled and synthesized. In spite of the myriad use of azobenzene as a photo‐ and electrochromic moiety, the idea presented herein is to adopt a conceptually different path by using it as a bridge in a donor–bridge–acceptor single‐molecular skeleton, connecting the electron acceptor N‐methylfulleropyrrolidine with an electron donor N,N‐dimethylaniline. Addition of trifluoroacetic acid (TFA) results in a drastic colour change of the dyad from yellow to pink in dichloromethane (DCM). The structure of the protonated species are established from electronic spectroscopy and time‐dependent density functional theory (TD‐DFT) calculations. UV/Vis spectroscopic investigations reveal the disappearance of the 409 nm ¹(π→π*) transition with appearance of new features at 520 and 540 nm, attributed to protonated β and α nitrogens, respectively, along with a finite weight of the C₆₀ pyrrolidinic nitrogen. Calculations reveal intermixing of n_(N?N)→π*_(N?N) and charge transfer (CT) transitions in the neutral dyad, whereas, the n_(N?N)→π*_(N?N) transition in the protonated dyad is buried under the dominant ¹(π →π*) feature and is red‐shifted upon Gaussian deconvolution. The experimental binding constants involved in the protonation of N,N‐dimethylanilineazobenzene and the dyad imply an almost equal probability of existence of both α‐ and β‐protonated forms. Larger binding constants for the protonated dyads imply more stable dyad complexes than for the donor counterparts. One of the most significant findings upon protonation resulted in frontier molecular orbital (FMO) switching with the dyad LUMO located on the donor part, evidenced from electrochemical investigations. The appearance of a new peak, prior to the first reduction potential of N‐methylfulleropyrrolidine, clearly indicates location of the first incoming electron on the donor‐centred LUMO of the dyad, corroborated by unrestricted DFT calculations performed on the monoanions of the protonated dyad. The protonation of the basic azo nitrogens thus enables a rational control over the energetics and location of the FMOs, indispensable for electron transport across molecular junctions in realizing futuristic current switching devices.     

Theory of variational calculation with a scaling correct moment functional to solve the electronic schrödinger equation directly for ground state one‐electron density and electronic energy

The reduction of the electronic Schrodinger equation or its calculating algorithm from 4N‐dimensions to a nonlinear, approximate density functional of a three spatial dimension one‐electron density for an N electron system which is tractable in practice, is a long‐desired goal in electronic structure calculation. In a seminal work, Parr et al. (Phys. Rev. A 1997, 55, 1792) suggested a well behaving density functional in power series with respect to density scaling within the orbital‐free framework for kinetic and repulsion energy of electrons. The updated literature on this subject is listed, reviewed, and summarized. Using this series with some modifications, a good density functional approximation is analyzed and solved via the Lagrange multiplier device. (We call the attention that the introduction of a Lagrangian multiplier to ensure normalization is a new element in this part of the related, general theory.) Its relation to Hartree–Fock (HF) and Kohn–Sham (KS) formalism is also analyzed for the goal to replace all the analytical Gaussian based two and four center integrals (∫g_i( r ₁)g_k( r ₂)rd r ₁d r ₂, etc.) to estimate electron‐electron interactions with cheaper numerical integration. The KS method needs the numerical integration anyway for correlation estimation. © 2012 Wiley Periodicals, Inc.     

A quantitative analysis of light‐driven charge transfer processes using voronoi partitioning of time dependent DFT‐derived electron densities

An analytical method is presented that provides quantitative insight into light‐driven electron density rearrangement using the output of standard time‐dependent density functional theory (TD‐DFT) computations on molecular compounds. Using final and initial electron densities for photochemical processes, the subtraction of summed electron density in each atom‐centered Voronoi polyhedron yields the electronic charge difference, Q ^VECD. This subtractive method can also be used with Bader, Mulliken and Hirshfeld charges. A validation study shows Q ^VECD to have the most consistent performance across basis sets and good conservation of charge between electronic states. Besides vertical transitions, relaxation processes can be investigated as well. Significant electron transfer is computed for isomerization on the excited state energy surface of azobenzene. A number of linear anilinepyridinium donor‐bridge‐acceptor chromophores was examined using Q ^VECD to unravel the influence of its pi‐conjugated bridge on charge separation. Finally, the usefulness of the presented method as a tool in optimizing charge transfer is shown for a homologous series of organometallic pigments. The presented work allows facile calculation of a novel, relevant quantity describing charge transfer processes at the atomic level. © 2017 The Authors Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Quantum chemistry aspects of the solvent effects on the ene reaction of 1‐Phenyl‐1,3,4‐triazolin‐2,5‐dione and 2‐methyl‐2‐butene

A density functional theory study was used to investigate the quantum aspects of the solvent effects on the kinetic and mechanism of the ene reaction of 1‐phenyl‐1,3,4‐triazolin‐2,5‐dione and 2‐methyl‐2‐butene. Using the B3LYP/6–311++ G(d,p) level of the theory, reaction rates have been calculated in the various solvents and good agreement with the experimental data has been obtained. Natural bond orbital analysis has been applied to calculate the stabilization energy of N18? H19 bond during the reaction. Topological analysis of quantum theory of atom in molecule (QTAIM) studies for the electron charge density in the bond critical point (BCP) of N18? H19 bond of the transition states (TSs) in different solvents shows a linear correlation with the interaction energy. It is also seen form the QTAIM analysis that increase in the electron density in the BCP of N18? H19, raises the corresponding vibrational frequency. Average calculated ratio of 0.37 for kinetic energy density to local potential energy density at the BCPs as functions of N18? H19 bond length in different media confirmed covalent nature of this bond. Using the concepts of the global electrophilicity index, chemical hardness and electronic chemical potentials, some correlations with the rate constants and interaction energy have been established. Mechanism and kinetic studies on 1‐phenyl‐1,3,4‐triazolin‐2,5‐dione and 2‐methyl‐2‐butene ene reaction suggests that the reaction rate will boost with interaction energy enhancement. Interaction energy of the TS depends on the solvent nature and is directly related to electron density of the bonds involved in the reaction proceeding, global electrophilicity index and electronic chemical potential. However, the chemical hardness relationship is reversed. Finally, an interesting and direct correlation between the imaginary vibrational frequency of the N18? H19 critical bond and its electron density at the TS has been obtained. © 2014 Wiley Periodicals, Inc.     

Hellmann–Feynman theorem near the threshold

The features of the spectrum structure are considered for situations where some parameter λ of the N‐electron Hamiltonian reaches the threshold value η under which the discrete energy level falls into the continuous spectrum. The electron density properties are also studied. It is proved that for a sequence of the wave functions converging in energy to the lower bound of the continuous spectrum as λ approaches η the corresponding sequence of the electron densities converges to the density of the (N ? 1)‐electron ground state. The results generalize the Hellmann–Feynman theorem for the cases where only the one‐side energy derivatives exist or there is no limiting wave function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Rational Design for Building Blocks of DNA‐Based Conductive Nanowires through Multi‐Copper Incorporation into Mismatched Base Pairs

Metal‐modified DNA base pairs, which possess potential electrical conductivity and can serve as conductive nanomaterials, have recently attracted much attention. Inspired by our recent finding that multicopper incorporation into natural DNA base pairs could improve the electronic properties of base pairs, herein, we designed two novel multi‐copper‐mediated mismatched base pairs (G_3CuT and A_2CuC), and examined their structural and electronic properties by means of density functional theory calculations. The results reveal that these multi‐Cu‐mediated mismatched base pairs still have planar geometries that are thermodynamically favorable to stability, and their binding energies are close to those of multi‐Cu‐mediated normal base pairs (G_3CuC and A_2CuT). Their HOMO–LUMO gaps and ionization potentials decrease significantly compared to the corresponding natural base pairs. As evidenced by the charge transfer excitation transitions, transverse electronic communication of G_3CuT and A_2CuC is remarkably enhanced, suggesting that they facilitate electron migration along the DNA wires upon incorporation. Further examinations also clarify the possibility to build promising DNA helices using the G_3CuT and/or A_2CuC base pairs. The calculated electronic properties of the three‐layer‐stacked multi‐Cu‐mediated mismatched base pairs illustrate that the Cu_m‐DNA have better conductivity. This work provides perspectives for the development and application of DNA nanowires.     

Grid‐based density functional calculations of many‐electron systems

Exploratory variational pseudopotential density functional calculations are performed for the electronic properties of many‐electron systems in the 3D cartesian coordinate grid (CCG). The atom‐centered localized gaussian basis set, electronic density, and the two‐body potentials are set up in the 3D cubic box. The classical Hartree potential is calculated accurately and efficiently through a Fourier convolution technique. As a first step, simple local density functionals of homogeneous electron gas are used for the exchange‐correlation potential, while Hay‐Wadt‐type effective core potentials are employed to eliminate the core electrons. No auxiliary basis set is invoked. Preliminary illustrative calculations on total energies, individual energy components, eigenvalues, potential energy curves, ionization energies, and atomization energies of a set of 12 molecules show excellent agreement with the corresponding reference values of atom‐centered grid as well as the grid‐free calculation. Results for three atoms are also given. Combination of CCG and the convolution procedure used for classical Coulomb potential can provide reasonably accurate and reliable results for many‐electron systems. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Evaluation of the electron momentum density of crystalline systems from ab initio linear combination of atomic orbitals calculations

Alternative techniques are presented for the evaluation of the electron momentum density (EMD) of crystalline systems from ab initio linear combination of atomic‐orbitals calculations performed in the frame of one‐electron self‐consistent‐field Hamiltonians. Their respective merits and drawbacks are analyzed with reference to two periodic systems with very different electronic features: the fully covalent crystalline silicon and the ionic lithium fluoride. Beyond one‐electron Hamiltonians, a post‐Hartree–Fock correction to the EMD of crystalline materials is also illustrated in the case of lithium fluoride. © 2012 Wiley Periodicals, Inc.     

ORBKIT: A modular python toolbox for cross‐platform postprocessing of quantum chemical wavefunction data

ORBKIT is a toolbox for postprocessing electronic structure calculations based on a highly modular and portable Python architecture. The program allows computing a multitude of electronic properties of molecular systems on arbitrary spatial grids from the basis set representation of its electronic wavefunction, as well as several grid‐independent properties. The required data can be extracted directly from the standard output of a large number of quantum chemistry programs. ORBKIT can be used as a standalone program to determine standard quantities, for example, the electron density, molecular orbitals, and derivatives thereof. The cornerstone of ORBKIT is its modular structure. The existing basic functions can be arranged in an individual way and can be easily extended by user‐written modules to determine any other derived quantity. ORBKIT offers multiple output formats that can be processed by common visualization tools (VMD, Molden, etc.). Additionally, ORBKIT possesses routines to order molecular orbitals computed at different nuclear configurations according to their electronic character and to interpolate the wavefunction between these configurations. The program is open‐source under GNU‐LGPLv3 license and freely available at https://github.com/orbkit/orbkit/ . This article provides an overview of ORBKIT with particular focus on its capabilities and applicability, and includes several example calculations. © 2016 Wiley Periodicals, Inc.     

Transition from a Metal‐Localized Mixed‐Valence Compound to a Fully Delocalized and Bridge‐Biased Electrophore in a Ruthenium–Amine–Ruthenium Tricenter System

A series of cyclometalated diruthenium complexes with a redox‐active amine bridge were synthesized. Depending on the terminal ligands of the ruthenium components and the substituent on the amine unit, the one‐electron‐oxidized state can be either in the form of a weakly or strongly coupled mixed‐valence diruthenium complex, a fully charge‐delocalized three‐center system, or a bridge‐biased electrophore. This transition among different electronic forms was supported by electrochemistry, near‐infrared absorption, electron paramagnetic resonance, and density functional theory analysis.     

Theoretical insights into the 1D‐charge transport properties in a series of hexaazatrinaphthylene‐based discotic molecules

Discotic liquid crystal (DLC) materials have attracted considerable attention mainly due to their high charge carrier mobilities in quasi‐one‐dimensional columns. In this article, five hexaazatrinaphthylene‐based DLC molecules were investigated theoretically, and their frontier molecular orbital energy levels, crystal structures, and electron/hole drift mobilities were calculated by combination of density functional theory (DFT) and semiclassical Marcus charge transfer theory. The systems studied in this work include three experimentally reported molecules ( 1 , 2 , and 3 ) and two theoretically designed molecules ( 4 and 5 ). Compared with the 1 – 3 compounds, 4 and 5 have three more extended benzene rings in the π‐conjugated core. The present results show that the orders of the frontier molecular orbital energy levels and electron drift mobilities agree very well with the experiment. For 4 and 5 , the electron/hole reorganization energies are lower than those of compounds 1 – 3 . Furthermore, the calculated electron/hole transfer integral of 5 is the largest among all the five systems, leading to the highest electron and hole mobilities. In addition, the hydrophobicity and solubility were also evaluated by DFT, indicating that compound 5 has good hydrophobicity and good solubility in trichloromethane. As a result, it is expected that compound 5 can be a potential charge transport material in electronic and optoelectronic devices. © 2017 Wiley Periodicals, Inc.     

Time‐dependent density functional theory study on direction‐dependent electron and hole transfer processes in molecular systems

We report on real‐time time‐dependent density functional theory calculations on direction‐dependent electron and hole transfer processes in molecular systems. As a model system, we focus on α‐sulfur. It is shown that time scale of the electron transfer process from a negatively charged S₈ molecule to a neighboring neutral monomer is comparable to that of a strong infrared‐active molecular vibrations of the dimer with one negatively charged monomer. This results in a strong coupling between the electrons and the nuclei motion which eventually leads to S₈ ring opening before the electron transfer process is completed. The open‐ring structure is found to be stable. The similar infrared‐active peak in the case of hole transfer, however, is shown to be very weak and hence no significant scattering by the nuclei is possible. The presented approach to study the charge transfer processes in sulfur has direct applications in the increasingly growing research field of charge transport in molecular systems. © 2017 Wiley Periodicals, Inc.