Axial-equatorial equilibrium in substituted cyclohexanes: a DFT perspective on a small but complex problem.

In Chemistry, complexity is not necessarily associated to large systems, as illustrated by the textbook example of axial-equatorial equilibrium in mono-substituted cyclohexanes. The difficulty in modelling such a simple isomerization is related to the need for reproducing the delicate balance between two forces, with opposite effects, namely the attractive London dispersion and the repulsive steric interactions. Such balance is a stimulating challenge for density-functional approximations and it is systematically explored here by considering 20 mono-substituted cyclohexanes. In comparison to highly accurate CCSD(T) reference calculations, their axial-equatorial equilibrium is studied with a large set of 48 exchange-correlation approximations, spanning from semilocal to hybrid to more recent double hybrid functionals. This dataset, called SAV20 (as Steric A-values for 20 molecules), allows to highlight the difficulties encountered by common and more original DFT approaches, including those corrected for dispersion with empirical potentials, the 6-31G*-ACP model, and our cost-effective PBE-QIDH/DH-SVPD protocol, in modeling these challenging interactions. Interestingly, the performance of the approaches considered in this contribution on the SAV20 dataset does not correlate with that obtained with other more standard datasets, such as S66, IDISP or NC15, thus indicating that SAV20 covers physicochemical features not already considered in previous noncovalent interaction benchmarks.

Modeling Multi-Step Organic Reactions: Can Density Functional Theory Deliver Misleading Chemistry?

Many organic reactions are characterized by a complex mechanism with a variety of transition states and intermediates of different chemical natures. Their correct and accurate theoretical characterization critically depends on the accuracy of the computational method used. In this work, we study a complex ambimodal cycloaddition with five transition states, two intermediates, and three products, and we ask whether density functional theory (DFT) can provide a correct description of this type of complex and multifaceted reaction. Our work fills a gap in that most systematic benchmarks of DFT for chemical reactions have considered much simpler reactions. Our results show that many density functionals not only lead to seriously large errors but also differ from one another in predicting whether the reaction is ambimodal. Only a few of the available functionals provide a balanced description of the complex and multifaceted reactions. The parameters varied in the tested functionals are the ingredients, the treatment of medium-range and nonlocal correlation energy, and the inclusion of Hartree-Fock exchange. These results show a clear need for more benchmarks on the mechanisms of large molecules in complex reactions.

M11pz: A Nonlocal Meta Functional with Zero Hartree-Fock Exchange and with Broad Accuracy for Chemical Energies and Structures.

The accuracy of Kohn-Sham density functional theory depends strongly on the approximation to the exchange-correlation functional. In this work, we present a new exchange-correlation functional called M11pz (M11 plus rung-3.5 terms with zero Hartree-Fock exchange) that is built on the M11plus functional with the goal of using its rung-3.5 terms without a Hartree-Fock exchange term, especially to improve the accuracy for strongly correlated systems. The M11pz functional is optimized with the same local and rung-3.5 ingredients that are used in M11plus but without any percentage of Hartree-Fock exchange. The performance of M11pz is compared with eight local functionals, and M11pz is found to be in top three when the errors or ranks are averaged over eight grouped and partially overlapping databases: AME418/22, atomic and molecular energies; MGBE172, main-group bond energies; TMBE40, transition-metal bond energies; SR309, single-reference systems; MR54, multireference systems; BH192, barrier heights; NC579, noncovalent interaction energies; and MS20, molecular structures. For calculations of band gaps of solids, M11pz is the second best of the nine tested functionals that have zero Hartree-Fock exchange.

Comparing Density Functional Theory Metal-Ligand Bond Dissociation Enthalpies with Experimental Solution-Phase Enthalpies of Activation for Bond Dissociation.

The predictive ability of density functional theory is fundamental to its usefulness in chemical applications. Recent work has compared solution-phase enthalpies of activation for metal-ligand bond dissociation to enthalpies of reaction for bond dissociation, and the present work continues those comparisons for 43 density functional methods. The results for ligand dissociation enthalpies of 30 metal-ligand complexes tested in this work reveal significant inadequacies of some functionals as well as challenges from the dispersion corrections to some functionals. The analysis presented here demonstrates the excellent performance of a recent density functional, M11plus, which contains nonlocal rung-3.5 correlation. We also find a good agreement between theory and experiment for some functionals without empirical dispersion corrections such as M06, r2SCAN, M06-L, and revM11, as well as good performance for some functionals with added dispersion corrections such as ωB97X-D (which always has a correction) and BLYP, B3LYP, CAM-B3LYP, and PBE0 when the optional dispersion corrections are added.

Barrier Heights for Diels-Alder Transition States Leading to Pentacyclic Adducts: A Benchmark Study of Crowded, Strained Transition States of Large Molecules.

Theoretical characterization of reactions of complex molecules depends on providing consistent accuracy for the relative energies of intermediates and transition states. Here we employ the DLPNO-CCSD(T) method with core-valence correlation, large basis sets, and extrapolation to the CBS limit to provide benchmark values for Diels-Alder transition states leading to competitive strained pentacyclic adducts. We then used those benchmarks to test a diverse set of wave function and density functional methods for the absolute and relative barrier heights of these transition states. Our results show that only a few of the tested density functionals can predict the absolute barrier heights satisfactorily, although relative barrier heights are more accurate. The most accurate functionals tested are ωB97M-V, M11plus, ωB97X-V, PBE-D3(0), M11, and MN15 with MUDs from best estimates less than 3.0 kcal. These findings can guide selection of density functionals for future studies of crowded, strained transition states of large molecules.

Exact-Two-Component Multiconfiguration Pair-Density Functional Theory.

Molecules containing late-row elements exhibit large relativistic effects. To account for both relativistic effects and electron correlation in a computationally inexpensive way, we derived a formulation of multiconfiguration pair-density functional theory with the relativistic exact-two-component Hamiltonian (X2C-MC-PDFT). In this new method, relativistic effects are included during variational optimization of a reference wave function by exact-two-component complete active-space self-consistent-field (X2C-CASSCF) theory, followed by an energy evaluation using pair-density functional theory. Benchmark studies of excited-state and ground-state fine-structure splitting of atomic species show that X2C-MC-PDFT can significantly improve the X2C-CASSCF results by introducing additional state-specific electron correlation.

Molecular Dynamics Simulations Enforcing Nonperiodic Boundary Conditions: New Developments and Application to the Solvent Shifts of Nitroxide Magnetic Parameters.

Multiscale methods combining quantum mechanics and molecular mechanics (QM/MM) have become the most suitable and effective strategies for the investigation of the spectroscopic properties of medium-to-large size chromophores in condensed phases. In this context, we are developing a novel workflow aimed at improving the generality, reliability, and ease of use of the available computational tools. In this paper, we report our latest developments with specific reference to a general protocol based on atomistic simulations, carried out under nonperiodic boundary conditions (NPBC). In particular, we add to our in house MD engine a new efficient treatment of mean field electrostatic contributions to energy and forces, together with the capability of performing the simulations either in the canonical (NVT) or in the isothermal-isobaric (NPT) ensemble. Next, we provide convincing evidence that the NBPC approach enhanced by specific tweaks for rigid body propagation, allows for the simulation of solute-solvent systems with a minimum number of degrees of freedom and large integration time step. After its validation, this new approach is applied to the challenging case of solvatochromic effects on the electron paramagnetic resonance (EPR) spectrum of a prototypical nitroxide radical. To this end, we propose and validate also an automated protocol to extract and weight simulation snapshots, making use of a continuous description of the strength of solute-solvent hydrogen bridges. While further developments are being worked on, the effectiveness of our approach, even in its present form, is demonstrated by the accuracy of the results obtained through an unsupervised approach characterized by a strongly reduced computational cost as compared to that of conventional QM/MM models, without any appreciable deterioration of accuracy.

Simple Protocol for Capturing Both Linear-Response and State-Specific Effects in Excited-State Calculations with Continuum Solvation Models.

We present an effective computational protocol (cLR2) to describe both solvatochromism and fluorosolvatochromism. This protocol, which couples the polarizable continuum model to time-dependent density functional theory, simultaneously accounts for both linear-response and state-specific solvation effects. A series of test cases, including solvatochromic and fluorosolvatochromic compounds and excited-state intramolecular proton transfers, are used to highlight that cLR2 is especially beneficial for modeling bright excitations possessing a significant charge-transfer character, as well as cases in which an accurate balance between states of various polarities should be restored.

Assessing challenging intra- and inter-molecular charge-transfer excitations energies with double-hybrid density functionals.

We investigate the performance of a set of recently introduced range-separated double-hybrid functionals, namely ωB2-PLYP, ωB2GP-PLYP, RSX-0DH, and RSX-QIDH models for hard-to-calculate excitation energies. We compare with the parent (B2-PLYP, B2GP-PLYP, PBE0-DH, and PBE-QIDH) and other (DSD-PBEP86) double-hybrid models as well as with some of the most widely employed hybrid functionals (B3LYP, PBE0, M06-2X, and ωB97X). For this purpose, we select a number of medium-sized intra- and inter-molecular charge-transfer excitations, which are known to be challenging to calculate using time-dependent density-functional theory (TD-DFT) and for which accurate reference values are available. We assess whether the high accuracy shown by the newest double-hybrid models is also confirmed for those cases too. We find that asymptotically corrected double-hybrid models yield a superior performance, especially for the inter-molecular charge-transfer excitation energies, as compared to standard double-hybrid models. Overall, the PBE-QIDH and its corresponding range-separated RSX-QIDH functional are recommended for general-purpose TD-DFT applications, depending on whether long-range effects are expected to play a significant role.

Calculation of magnetic properties with density functional approximations including rung 3.5 ingredients.

Density functional theory is widely used for modeling the magnetic properties of molecules, solids, and surfaces. Rung-3.5 ingredients, based on the expectation values of nonlocal one-electron operators, are new promising tools for the construction of exchange-correlation functional approximations. We present the formal extension of rung-3.5 ingredients to the calculation of magnetic properties. We add to the underlying nonlocal operators a dependence on the gauge of the magnetic field, and we derive the working equations for rung-3.5 expectation values in basis sets of gauge-including atomic orbitals. We demonstrate that the gauge corrections are significant. We conclude with an initial study of chemical shifts, optical rotatory dispersion, and Raman optical activity spectra predicted by M11plus, a range-separated hybrid meta functional incorporating nonlocal rung-3.5 correlation. M11plus proves to be reasonably accurate, further motivating the incorporation of nonlocal rung-3.5 ingredients in new density functional approximations.

M11plus, a Range-Separated Hybrid Meta Functional Incorporating Nonlocal Rung-3.5 Correlation, Exhibits Broad Accuracy on Diverse Databases.

We present tests of the recent M11plus Minnesota density functional for a broad range of main-group and transition-metal chemistry databases, most of which were not used in in the construction of any of the Minnesota functionals. M11plus is a range-separated hybrid meta functional combining long-range nonlocal Hartree-Fock exchange with nonlocal rung-3.5 correlation. M11plus performs well for main-group thermochemistry, kinetics, and noncovalent interactions and especially well for radical species. It is numerically well behaved, it has a computational cost that is ∼1.2 to 1.5 times that of M11 in realistic calculations, and it is particularly accurate for triplet excited states, which is a difficult challenge for density functional approximations. The results show that nonlocal rung-3.5 correlation is a broadly useful ingredient for improving the performance of density functional approximations.

Double hybrids and time-dependent density functional theory: An implementation and benchmark on charge transfer excited states.

In this paper we present the implementation and benchmarking of a Time Dependent Density Functional Theory approach in conjunction with Double Hybrid (DH) functionals. We focused on the analysis of their performance for through space charge-transfer (CT) excitations which are well known to be very problematic for commonly used functionals, such as global hybrids.Two different families of functionals were compared, each of them containing pure, hybrid and double-hybrid functionals.The results obtained show that, beside the robustness of the implementation, these functionals provide results with an accuracy comparable to that of adjusted range-separated functionals, with the relevant difference that for DHs no parameter is tuned on specific compounds thus making them more appealing for a general use. Furthermore, the algorithm described and implemented is characterized by the same computational cost scaling as that of the ground state algorithm employed for MP2 and double hybrids.

Towards large scale hybrid QM/MM dynamics of complex systems with advanced point dipole polarizable embeddings.

In this work, we present a general route to hybrid Quantum Mechanics/Molecular Mechanics (QM/MM) Molecular Dynamics for complex systems using a polarizable embedding. We extend the capabilities of our hybrid framework, combining the Gaussian and Tinker/Tinker-HP packages in the context of the AMOEBA polarizable force field to treat large (bio)systems where the QM and the MM subsystems are covalently bound, adopting pseudopotentials at the boundaries between the two regions. We discuss in detail the implementation and demonstrate the global energy conservation of our QM/MM Born-Oppenheimer molecular dynamics approach using Density Functional Theory. Finally, the approach is assessed on the electronic absorption properties of a 16 500 atom complex encompassing an organic dye embedded in a DNA matrix in solution, extending the hybrid method to a time-dependent Density Functional Theory approach. The results obtained comparing different partitions between the quantum and the classical subsystems also suggest that large QM portions are not necessary if accurate polarizable force fields are used in a variational formulation of the embedding, properly including the QM/MM mutual polarization.

Quantum Calculations in Solution of Energies, Structures, and Properties with a Domain Decomposition Polarizable Continuum Model.

In this work, we present the first implementation of the domain decomposition polarizable continuum model for a solute described at a quantum mechanical level of theory. After briefly recapitulating the theory, we discuss the coupling of ddPCM to a quantum mechanical level of theory based on the self-consistent field approach, i.e., Hartree-Fock, density functional theory, and semiempirical methods. We then present benchmarks of the new implementation, comparing it to a currently available state-of-the-art one, and use it to describe the structure and excitation properties of a large multichromophoric system.

M11plus: A Range-Separated Hybrid Meta Functional with Both Local and Rung-3.5 Correlation Terms and High Across-the-Board Accuracy for Chemical Applications.

The way to improve Kohn-Sham density functional theory is to improve the exchange-correlation functionals, and functionals have been successively improved by adding new ingredients, especially local spin density gradients, nonlocal Hartree-Fock exchange, and local meta terms based on kinetic energy density. Here, we present a new kind of functional obtained by adding rung-3.5 terms to a functional including local gradients, local meta terms, and range-separated Hartree-Fock exchange. A rung-3.5 term has short-range nonlocality designed to account for nondynamic correlation; we add two kinds of rung-3.5 terms, one kind modeled on position-dependent Hartree-Fock exchange and another modeled on the spin density at a point interacting with the opposite-spin exchange hole at the same point. Optimization of the functional yields broad accuracy for both ground states and excited states with especially significant improvement for systems with strong correlation.

Long-range-corrected Rung 3.5 density functional approximations.

Rung 3.5 functionals are a new class of approximations for density functional theory. They provide a flexible intermediate between exact (Hartree-Fock, HF) exchange and semilocal approximations for exchange. Existing Rung 3.5 functionals inherit semilocal functionals' limitations in atomic cores and density tails. Here we address those limitations using range-separated admixture of HF exchange. We present three new functionals. LRC-ωΠLDA combines long-range HF exchange with short-range Rung 3.5 ΠLDA exchange. SLC-ΠLDA combines short- and long-range HF exchange with middle-range ΠLDA exchange. LRC-ωΠLDA-AC incorporates a combination of HF, semilocal, and Rung 3.5 exchange in the short range, based on an adiabatic connection. We test these in a new Rung 3.5 implementation including up to analytic fourth derivatives. LRC-ωΠLDA and SLC-ΠLDA improve atomization energies and reaction barriers by a factor of 8 compared to the full-range ΠLDA. LRC-ωΠLDA-AC brings further improvement approaching the accuracy of standard long-range corrected schemes LC-ωPBE and SLC-PBE. The new functionals yield highest occupied orbital energies closer to experimental ionization potentials and describe correctly the weak charge-transfer complex of ethylene and dichlorine and the hole-spin distribution created by an Al defect in quartz. This study provides a framework for more flexible range-separated Rung 3.5 approximations.

Density-Dependent Formulation of Dispersion-Repulsion Interactions in Hybrid Multiscale Quantum/Molecular Mechanics (QM/MM) Models.

Mixed multiscale quantum/molecular mechanics (QM/MM) models are widely used to explore the structure, reactivity, and electronic properties of complex chemical systems. Whereas such models typically include electrostatics and potentially polarization in so-called electrostatic and polarizable embedding approaches, respectively, nonelectrostatic dispersion and repulsion interactions are instead commonly described through classical potentials despite their quantum mechanical origin. Here we present an extension of the Tkatchenko-Scheffler semiempirical van der Waals (vdWTS) scheme aimed at describing dispersion and repulsion interactions between quantum and classical regions within a QM/MM polarizable embedding framework. Starting from the vdWTS expression, we define a dispersion and a repulsion term, both of them density-dependent and consistently based on a Lennard-Jones-like potential. We explore transferable atom type-based parametrization strategies for the MM parameters, based on either vdWTS calculations performed on isolated fragments or on a direct estimation of the parameters from atomic polarizabilities taken from a polarizable force field. We investigate the performance of the implementation by computing self-consistent interaction energies for the S22 benchmark set, designed to represent typical noncovalent interactions in biological systems, in both equilibrium and out-of-equilibrium geometries. Overall, our results suggest that the present implementation is a promising strategy to include dispersion and repulsion in multiscale QM/MM models incorporating their explicit dependence on the electronic density.

Excited State Dipole Moments in Solution: Comparison between State-Specific and Linear-Response TD-DFT Values.

We compare different response schemes for coupling continuum solvation models to time-dependent density functional theory (TD-DFT) for the determination of solvent effects on the excited state dipole moments of solvated molecules. In particular, linear-response (LR) and state-specific (SS) formalisms are compared. Using 20 low-lying electronic excitations, displaying both localized and charge-transfer character, this study highlights the importance of applying a SS model not only for the calculation of energies, as previously reported ( J. Chem. Theory Comput. , 2015 , 11 , 5782 , DOI: 10.1021/acs.jctc.5b00679 ), but also for the prediction of excited state properties. Generally, when a range-separated exchange-correlation functional is used, both LR and SS schemes provide very similar dipole moments for local transitions, whereas differences of a few Debye units with respect to LR values are observed for CT transitions. The delicate interplay between the response scheme and the exchange-correlation functional is discussed as well, and we show that using an inadequate functional in a SS framework can yield to dramatic overestimations of the dipole moments.

How are the charge transfer descriptors affected by the quality of the underpinning electronic density?

With the aim of investigating qualitatively and quantitatively the impact of using excited state relaxed or unrelaxed density for the estimation of nature and characteristic of electronic excited states, we analyzed the behavior of 52 exchange correlation functionals for the prediction of density-based indexes such as those recently introduced in literature to evaluate the charge transfer distance (DCT ) (Le Bahers et al. J. Chem. Theory Comput. 2011, 7, 2498) in the case of a prototype family of push-pull dyes. Our results show that while a qualitatively consistent assessment of the nature of the excited states is obtained using either the unrelaxed or the relaxed density, from a quantitative standpoint we observe large discrepancies in the charge transfer distance for electronic transitions having substantial CT character. This behavior is independent of nature of the exchange-correlation functional used. © 2018 Wiley Periodicals, Inc.

Accurate Excited-State Geometries: A CASPT2 and Coupled-Cluster Reference Database for Small Molecules.

We present an investigation of the excited-state structural parameters determined for a large set of small compounds with the dual goals of defining reference values for further works and assessing the quality of the geometries obtained with relatively cheap computational approaches. In the first stage, we compare the excited-state geometries obtained with ADC(2), CC2, CCSD, CCSDR(3), CC3, and CASPT2 and large atomic basis sets. It is found that CASPT2 and CC3 results are generally in very good agreement with one another (typical differences of ca. 3 × 10-3 Å) when all electrons are correlated and when the aug-cc-pVTZ atomic basis set is employed with both methods. In a second stage, a statistical analysis reveals that, on the one hand, the excited-state (ES) bond lengths are much more sensitive to the selected level of theory than their ground-state (GS) counterparts and, on the other hand, that CCSDR(3) is probably the most cost-effective method delivering accurate structures. Indeed, CCSD tends to provide too compact multiple bond lengths on an almost systematic basis, whereas both CC2 and ADC(2) tend to exaggerate these bond distances, with more erratic error patterns, especially for the latter method. The deviations are particularly marked for the polarized CO and CN bonds, as well as for the puckering angle in formaldehyde homologues. In the last part of this contribution, we provide a series of CCSDR(3) GS and ES geometries of medium-sized molecules to be used as references in further investigations.