Analytic first-order derivatives of CASPT2 with IPEA shift.

Complete active space second-order perturbation theory (CASPT2) is useful for accurately predicting properties of complex electronic structures, but it is well known that it systematically underestimates excitation energies. The underestimation can be corrected using the ionization potential-electron affinity (IPEA) shift. In this study, analytic first-order derivatives of CASPT2 with the IPEA shift are developed. CASPT2-IPEA is not invariant with respect to rotations among active molecular orbitals, and two additional constraint conditions are necessary in the CASPT2 Lagrangian to formulate analytic derivatives. The method developed here is applied to methylpyrimidine derivatives and cytosine, and minimum energy structures and conical intersections are located. By comparing energies relative to the closed-shell ground state, we find that the agreement with experiments and high-level calculations is indeed improved by the inclusion of the IPEA shift. The agreement of geometrical parameters with high-level calculations may also be improved in some cases.

Analytic First-Order Derivatives of (X)MS, XDW, and RMS Variants of the CASPT2 and RASPT2 Methods.

Crossings between states involve complex electronic structures, making the accurate characterization of the crossing point difficult. In this study, the analytic derivatives of three complete active space second-order perturbation theory (CASPT2) variants as well as an extension of the restricted active space (RASPT2) are developed. These variants are applied to locating minimum energy conical intersections. Our results demonstrate that the three CASPT2 variants predict qualitatively similar results, but a recently developed variant, the rotated multistate CASPT2 (RMS-CASPT2), is least sensitive to the number of states considered in the calculation. We demonstrate that CASPT2 and the reference self-consistent field calculations predict qualitatively different energetics and bond lengths.

Analytic gradients for restricted active space second-order perturbation theory (RASPT2).

The computational cost of analytic derivatives in multireference perturbation theory is strongly affected by the size of the active space employed in the reference self-consistent field calculation. To overcome previous limits on the active space size, the analytic gradients of single-state restricted active space second-order perturbation theory (RASPT2) and its complete active space second-order perturbation theory (CASPT2) have been developed and implemented in a local version of OpenMolcas. Similar to previous implementations of CASPT2, the RASPT2 implementation employs the Lagrangian or Z-vector method. The numerical results show that restricted active spaces with up to 20 electrons in 20 orbitals can now be employed for geometry optimizations.

The fragment molecular orbital method combined with density-functional tight-binding and periodic boundary conditions.

The density-functional tight-binding (DFTB) formulation of the fragment molecular orbital method is combined with periodic boundary conditions. Long-range electrostatics and dispersion are evaluated with the Ewald summation technique. The first analytic derivatives of the energy with respect to atomic coordinates and lattice parameters are formulated. The accuracy of the method is established in comparison to numerical gradients and DFTB without fragmentation. The largest elementary cell in this work has 1631 atoms. The method is applied to elucidate the polarization, charge transfer, and interactions in the solution.

Multicolor polymeric carbon dots: synthesis, separation and polyamide-supported molecular fluorescence.

Multicolor carbon dots (CDs) have been developed recently and demonstrate great potential in bio-imaging, sensing, and LEDs. However, the fluorescence mechanism of their tunable colors is still under debate, and efficient separation methods are still challenging. Herein, we synthesized multicolor polymeric CDs through solvothermal treatment of citric acid and urea in formamide. Automated reversed-phase column separation was used to achieve fractions with distinct colors, including blue, cyan, green, yellow, orange and red. This work explores the physicochemical properties and fluorescence origins of the red, green, and blue fractions in depth with combined experimental and computational methods. Three dominant fluorescence mechanism hypotheses were evaluated by comparing time-dependent density functional theory and molecular dynamics calculation results to measured characteristics. We find that blue fluorescence likely comes from embedded small molecules trapped in carbonaceous cages, while pyrene analogs are the most likely origin for emission at other wavelengths, especially in the red. Also important, upon interaction with live cells, different CD color fractions are trafficked to different sub-cellular locations. Super-resolution imaging shows that the blue CDs were found in a variety of organelles, such as mitochondria and lysosomes, while the red CDs were primarily localized in lysosomes. These findings significantly advance our understanding of the photoluminescence mechanism of multicolor CDs and help to guide future design and applications of these promising nanomaterials.

Analytic first-order derivatives of partially contracted n-electron valence state second-order perturbation theory (PC-NEVPT2).

A balanced treatment of dynamic and static electron correlation is important in computational chemistry, and multireference perturbation theory (MRPT) is able to do this at a reasonable computational cost. In this paper, analytic first-order derivatives, specifically gradients and dipole moments, are developed for a particular MRPT method, state-specific partially contracted n-electron valence state second-order perturbation theory (PC-NEVPT2). Only one linear equation needs to be solved for the derivative calculation if the Z-vector method is employed, which facilitates the practical application of this approach. A comparison of the calculated results with experimental geometrical parameters of O3 indicates excellent agreement although the calculated results for O3 - are slightly outside the experimental error bars. The 0-0 transition energies of various methylpyrimidines and trans-polyacetylene are calculated by performing geometry optimizations and seminumerical second-order geometrical derivative calculations. In particular, the deviations of 0-0 transition energies of trans-polyacetylene from experimental values are consistently less than 0.1 eV with PC-NEVPT2, indicating the reliability of the method. These results demonstrate the importance of adding dynamic electron correlation on top of methods dominated by static electron correlation and of developing analytic derivatives for highly accurate methods.

Time-Dependent Long-Range-Corrected Density-Functional Tight-Binding Method Combined with the Polarizable Continuum Model.

In this study, excited-state free energies and geometries were efficiently evaluated using a linear-response time-dependent long-range-corrected density-functional tight-binding method integrated with the polarizable continuum model (TD-LC-DFTB2/PCM). Although the LC-DFTB method required the evaluation of the exchange-type term, which was moderately computationally expensive, a single evaluation of the excited-state gradient for a system consisting of more than 1000 atoms in a vacuum was completed within 30 min using one CPU core. Benchmark calculations were conducted for 3-hydroxyflavone, which exhibits dual emission: the absorption and enol-form emission wavelengths calculated by TD-LC-DFTB2/PCM agreed well with those predicted based on the density functional theory using a long-range corrected functional; however, there was a large error in the predicted keto-form emission wavelength. Further benchmark calculations for more than 20 molecules indicated that the conventional TD-DFTB method underestimated the absorption and 0-0 transition energies compared with those which were measured experimentally, whereas the TD-LC-DFTB2 method systematically overestimated these metrics. Nevertheless, the agreement of the results of the TD-LC-DFTB2 method with those obtained by the CAM-B3LYP method demonstrates the potential of the TD-LC-DFTB2/PCM method. Moreover, changing the range separation parameter to 0.15 minimized this deviation.

The Fragment Molecular Orbital Method Based on Long-Range Corrected Density-Functional Tight-Binding.

The presently available linear scaling approaches to density-functional tight-binding (DFTB) based on the fragment molecular orbital (FMO) method are severely impacted by the problem of artificial charge transfer due to the self-interaction error (SIE), which hampers the simulation of zwitterionic systems such as biopolymers or ionic liquids. Here we report an extension of FMO-DFTB where we included a long-range corrected (LC) functional designed to mitigate the DFTB SIE, called the FMO-LC-DFTB method, resulting in a robust method which succeeds in simulating zwitterionic systems. Both energy and analytic gradient are developed for the gas phase and the polarizable continuum model of solvation. The scaling of FMO-LC-DFTB with system size N is shown to be almost linear, O( N1.13-1.28), and its numerical accuracy is established for a variety of representative systems including neutral and charged polypeptides. It is shown that pair interaction energies between fragments for two mini-proteins are in excellent agreement with results from long-range corrected density functional theory. The new method was employed in long time scale (1 ns) molecular dynamics simulations of the tryptophan cage protein (PDB: 1L2Y ) in the gas phase for four different protonation states and in stochastic global minimum structure searches for 1-ethyl-3-methylimidazolium nitrate ionic liquid clusters containing up to 2300 atoms.

Adaptive frozen orbital treatment for the fragment molecular orbital method combined with density-functional tight-binding.

The exactly analytic gradient is derived and implemented for the fragment molecular orbital (FMO) method combined with density-functional tight-binding (DFTB) using adaptive frozen orbitals. The response contributions which arise from freezing detached molecular orbitals on the border between fragments are computed by solving Z-vector equations. The accuracy of the energy, its gradient, and optimized structures is verified on a set of representative inorganic materials and polypeptides. FMO-DFTB is applied to optimize the structure of a silicon nano-wire, and the results are compared to those of density functional theory and experiment. FMO accelerates the DFTB calculation of a boron nitride nano-ring with 7872 atoms by a factor of 406. Molecular dynamics simulations using FMO-DFTB applied to a 10.7 µm chain of boron nitride nano-rings, consisting of about 1.2 × 106 atoms, reveal the rippling and twisting of nano-rings at room temperature.

Theoretical Elucidation of Potential Enantioselectivity in a Pd-Catalyzed Aromatic C-H Coupling Reaction.

The mechanism of an aromatic C-H coupling reaction between heteroarenes and arylboronic acids using a Pd catalyst was theoretically and experimentally investigated. We identified the C-B transmetalation as the rate-determining step. The (S)-catalyst-reactant complex was found to be stabilized by hyperconjugation between π-orbitals on the tolyl group and the S-O σ* antibonding orbital in the catalyst ligand. Our findings suggest routes for the design of new, improved Pd catalysts with higher stereoselectivity.

Analytic hyperpolarizability and polarizability derivative with fractional occupation numbers for large extended systems.

The analytic hyperpolarizability and polarizability derivative with fractional occupation numbers are derived using Wigner's 2n + 1 rule. The formulation contains no terms that blow up for quasi-degenerate systems. The density-functional tight-binding method is used for implementation, which makes it possible to compute these third-order derivatives for systems containing up to one thousand atoms within 8 h using 24 CPU cores. A comparison between analytic and numerical non-resonance Raman activity spectra indicates that the numerical differentiation approach can give a significant deviation unless the strength of perturbative electric field is carefully chosen. With extremely high electronic temperatures, the polarizability and hyperpolarizability should converge to zero.

Three-body expansion of the fragment molecular orbital method combined with density-functional tight-binding.

The three-body fragment molecular orbital (FMO3) method is formulated for density-functional tight-binding (DFTB). The energy, analytic gradient, and Hessian are derived in the gas phase, and the energy and analytic gradient are also derived for polarizable continuum model. The accuracy of FMO3-DFTB is evaluated for five proteins, sodium cation in explicit solvent, and three isomers of polyalanine. It is shown that FMO3-DFTB is considerably more accurate than FMO2-DFTB. Molecular dynamics simulations for sodium cation in water are performed for 100 ps, yielding radial distribution functions and coordination numbers. © 2017 Wiley Periodicals, Inc.

Analytic second derivative of the energy for density-functional tight-binding combined with the fragment molecular orbital method.

The analytic second derivative of the energy is developed for the fragment molecular orbital (FMO) method combined with density-functional tight-binding (DFTB), enabling simulations of infrared and Raman spectra of large molecular systems. The accuracy of the method is established in comparison to full DFTB without fragmentation for a set of representative systems. The performance of the FMO-DFTB Hessian is discussed for molecular systems containing up to 10 041 atoms. The method is applied to the study of the binding of α-cyclodextrin to polyethylene glycol, and the calculated IR spectrum of an epoxy amine oligomer reproduces experiment reasonably well.

The fragment molecular orbital method combined with density-functional tight-binding and the polarizable continuum model.

The energy and its analytic gradient are formulated for the fragment molecular orbital (FMO) method combined with density-functional tight-binding (DFTB) and the polarizable continuum model (PCM). The accuracy is demonstrated in comparison with unfragmented calculations and numerical gradients. The instability in the description of proteins using density functional theory (DFT) and DFTB is analyzed for both unfragmented and FMO methods. The cause of the instability is shown to be charged residues, and the problem is particularly severe in the gas phase when long-range functionals are not used. Adding solvent effects considerably increases the gap between occupied and virtual orbitals and stabilizes convergence. The pair interaction energies calculated using FMO-DFT and FMO-DFTB in solution are shown to correlate, whereas the latter method is 4840 times faster than the former for a protein consisting of 1961 atoms. The structures of five proteins (containing up to 3578 atoms) optimized using FMO-DFTB/PCM agree reasonably well with experiment.

DFTB/PCM Applied to Ground and Excited State Potential Energy Surfaces.

Accounting for solvent effects in quantum chemical calculations is vital for the accurate description of potential energy surfaces in solution. In this study, we derive a formulation of the analytical first-order geometrical derivative of ground- and excited-state energies within the time-dependent density-functional tight-binding (TD-DFTB) method with the polarizable continuum model (PCM), TD-DFTB/PCM. The performance of this is then evaluated for a series of halogen-exchange SN2 reactions. DFTB/PCM reproduces DFT results well for isolated monohalogenated methanes, but its agreement for transition structures significantly depends on the halogen element. The performance of TD-DFTB/PCM is evaluated for the excited-state intramolecular proton transfer (ESIPT) reaction of 3-hydroxyflavone (3HF) in ethanol. TD-DFTB/PCM reproduces the barrier height of the ESIPT reaction in terms of geometry and energy relatively well, but it fails to reproduce the experimental absorption and fluorescence energies as a consequence of the absence of long-range corrections. Computational timings with and without PCM show that the additional cost of PCM for C500H502 is only 10% greater than the corresponding calculation in vacuum. Furthermore, the potential applications of TD-DFTB/PCM are highlighted by applying it to a double-stranded DNA complexed with dye (PDB ID 108D ). We conclude that TD-DFTB/PCM single-point calculations and geometry optimizations for systems consisting of more than 1000 and 500 atoms, respectively, is now manageable and that properties predicted with TD-DFTB must be interpreted with care.

Large-Scale Quantum-Mechanical Molecular Dynamics Simulations Using Density-Functional Tight-Binding Combined with the Fragment Molecular Orbital Method.

The fully analytic gradient is developed for density-functional tight-binding (DFTB) combined with the fragment molecular orbital (FMO) method (FMO-DFTB). The response terms arising from the coupling of the electronic state to the embedding potential are derived, and the gradient accuracy is demonstrated on water clusters and a polypeptide. The radial distribution functions (RDFs) obtained with FMO-DFTB are found to be similar to those from conventional DFTB, while the computational cost is greatly reduced; for 256 water molecules one molecular dynamics (MD) step takes 73.26 and 0.68 s with full DFTB and FMO-DFTB, respectively, showing a speed-up factor of 108. FMO-DFTB/MD is applied to 100 ps MD simulations of liquid hydrogen halides and is found to reproduce experimental RDFs reasonably well.

Time-dependent density-functional tight-binding method with the third-order expansion of electron density.

We develop a formalism for the calculation of excitation energies and excited state gradients for the self-consistent-charge density-functional tight-binding method with the third-order contributions of a Taylor series of the density functional theory energy with respect to the fluctuation of electron density (time-dependent density-functional tight-binding (TD-DFTB3)). The formulation of the excitation energy is based on the existing time-dependent density functional theory and the older TD-DFTB2 formulae. The analytical gradient is computed by solving Z-vector equations, and it requires one to calculate the third-order derivative of the total energy with respect to density matrix elements due to the inclusion of the third-order contributions. The comparison of adiabatic excitation energies for selected small and medium-size molecules using the TD-DFTB2 and TD-DFTB3 methods shows that the inclusion of the third-order contributions does not affect excitation energies significantly. A different set of parameters, which are optimized for DFTB3, slightly improves the prediction of adiabatic excitation energies statistically. The application of TD-DFTB for the prediction of absorption and fluorescence energies of cresyl violet demonstrates that TD-DFTB3 reproduced the experimental fluorescence energy quite well.

Super-reduced polyoxometalates: excellent molecular cluster battery components and semipermeable molecular capacitors.

Theoretical investigations are presented on the molecular and electronic structure changes that occur as α-Keggin-type polyoxometalate (POM(3-)) clusters [PM12O40](3-) (M = Mo, W) are converted toward their super-reduced POM(27-) state during the discharging process in lithium-based molecular cluster batteries. Density functional theory was employed in geometry optimization, and first-principles molecular dynamics simulations were used to explore local minima on the potential energy surface of neutral POM clusters adorned with randomly placed Li atoms as electron donors around the cluster surface. On the basis of structural, electron density, and molecular orbital studies, we present evidence that the super-reduction is accompanied by metal-metal bond formation, beginning from the 12th to 14th excess electron transferred to the cluster. Afterward, the number of metal-metal bonds increases nearly linearly with the number of additionally transferred excess electrons. In α-Keggin-type POMs, metal triangles are a prominently emerging structural feature. The origin of the metal triangle formation during super-reduction stems from the formation of characteristic three-center two-electron bonds in triangular metal atom sites, created under preservation of the POM skeleton via "squeezing out" of oxygen atoms bridging two metal atoms when the underlying metal atoms form covalent bonds. The driving force for this unusual geometrical and electronic structure change is a local Jahn-Teller distortion at individual transition-metal octahedral sites, where the triply degenerate t2 d orbitals become partially filled during reduction and gain energy by distortion of the octahedron in such a way that metal-metal bonds are formed. The bonding orbitals show strong contributions from mixing with metal-oxygen antibonding orbitals, thereby "shuffling away" excess electrons from the cluster center to the outside of the cage. The high density of negatively charged yet largely separated oxygen atoms on the surface of the super-reduced POM(27-) polyanion allows the huge Coulombic repulsion due to the presence of the excess electrons to be counterbalanced by the presence of Li countercations, which partially penetrate into the outer oxygen shell. This "semiporous molecular capacitor" structure is likely the reason for the effective electron uptake in POMs.

Density-Functional Tight-Binding Combined with the Fragment Molecular Orbital Method.

We developed the energy and its gradient for the self-consistent-charge density-functional tight-binding (DFTB) method, combined with the fragment molecular orbital (FMO) approach, FMO-DFTB, including an optional a posteriori treatment for dispersion interaction, and evaluated its accuracy as well as computational efficiency for a set of representative systems: polypeptides, a DNA segment, and a small protein. The error in the total energy of FMO-DFTB versus full SCC-DFTB was below 1 kcal/mol for the polyalanine system consisting of about 2000 atoms partitioned into fragments containing 2 residues, and the optimized structures had root-mean-square deviations below 0.1 Å. The scaling of FMO-DFTB with the system size N is only marginally larger than linear [O(N(1.2)) in the worst case]. A parallelization efficiency of 94% was achieved using 128 CPU cores, and we demonstrate the applicability of FMO-DFTB for systems containing more than one million atoms by performing a geometry optimization of a fullerite cluster.

Growth of carbon nanotubes via twisted graphene nanoribbons.

Carbon nanotubes have long been described as rolled-up graphene sheets. It is only fairly recently observed that longitudinal cleavage of carbon nanotubes, using chemical, catalytical and electrical approaches, unzips them into thin graphene strips of various widths, the so-called graphene nanoribbons. In contrast, rolling up these flimsy ribbons into tubes in a real experiment has not been possible. Theoretical studies conducted by Kit et al. recently demonstrated the tube formation through twisting of graphene nanoribbon, an idea very different from the rolling-up postulation. Here we report the first experimental evidence of a thermally induced self-intertwining of graphene nanoribbons for the preferential synthesis of (7, 2) and (8, 1) tubes within parent-tube templates. Through the tailoring of ribbon's width and edge, the present finding adds a radically new aspect to the understanding of carbon nanotube formation, shedding much light on not only the future chirality tuning, but also contemporary nanomaterials engineering.