Environmental Effects with Frozen-Density Embedding in Real-Time Time-Dependent Density Functional Theory Using Localized Basis Functions.

Frozen-density embedding (FDE) represents a versatile embedding scheme to describe the environmental effect on electron dynamics in molecular systems. The extension of the general theory of FDE to the real-time time-dependent Kohn-Sham method has previously been presented and implemented in plane waves and periodic boundary conditions [Pavanello, M.; J. Chem. Phys. 2015, 142, 154116]. In the current paper, we extend our recent formulation of the real-time time-dependent Kohn-Sham method based on localized basis set functions and developed within the Psi4NumPy framework to the FDE scheme. The latter has been implemented in its "uncoupled" flavor (in which the time evolution is only carried out for the active subsystem, while the environment subsystems remain at their ground state), using and adapting the FDE implementation already available in the PyEmbed module of the scripting framework PyADF. The implementation was facilitated by the fact that both Psi4NumPy and PyADF, being native Python API, provided an ideal framework of development using the Python advantages in terms of code readability and reusability. We employed this new implementation to investigate the stability of the time-propagation procedure, which is based on an efficient predictor/corrector second-order midpoint Magnus propagator employing an exact diagonalization, in combination with the FDE scheme. We demonstrate that the inclusion of the FDE potential does not introduce any numerical instability in time propagation of the density matrix of the active subsystem, and in the limit of the weak external field, the numerical results for low-lying transition energies are consistent with those obtained using the reference FDE calculations based on the linear-response TDDFT. The method is found to give stable numerical results also in the presence of a strong external field inducing nonlinear effects. Preliminary results are reported for high harmonic generation (HHG) of a water molecule embedded in a small water cluster. The effect of the embedding potential is evident in the HHG spectrum reducing the number of the well-resolved high harmonics at high energy with respect to the free water. This is consistent with a shift toward lower ionization energy passing from an isolated water molecule to a small water cluster. The computational burden for the propagation step increases approximately linearly with the size of the surrounding frozen environment. Furthermore, we have also shown that the updating frequency of the embedding potential may be significantly reduced, much less than one per time step, without jeopardizing the accuracy of the transition energies.

Orbital Decomposition of the Carbon Chemical Shielding Tensor in Gold(I) N-Heterocyclic Carbene Complexes.

The good performance of N-heterocyclic carbenes (NHCs), in terms of versatility and selectivity, has called the attention of experimentalists and theoreticians attempting to understand their electronic properties. Analyses of the Au(I)-C bond in [(NHC)AuL]+/0 (L stands for a neutral or negatively charged ligand), through the Dewar-Chatt-Duncanson model and the charge displacement function, have revealed that NHC is not purely a σ-donor but may have a significant π-acceptor character. It turns out, however, that only the σ-donation bonding component strongly correlates with one specific component of the chemical shielding tensor. Here, in extension to earlier works, a current density analysis, based on the continuous transformation of the current density diamagnetic zero approach, along a series of [(NHC)AuL]+/0 complexes is presented. The shielding tensor is decomposed into orbital contributions using symmetry considerations together with a spectral analysis in terms of occupied to virtual orbital transitions. Analysis of the orbital transitions shows that the induced current density is largely influenced by rotational transitions. The orbital decomposition of the shielding tensor leads to a deeper understanding of the ligand effect on the magnetic response properties and the electronic structure of (NHC)-Au fragments. Such an orbital decomposition scheme may be extended to other magnetic properties and/or substrate-metal complexes.

BERTHA: Implementation of a four-component Dirac-Kohn-Sham relativistic framework.

In this paper, we present and review the most recent computational advances in the BERTHA code. BERTHA can be regarded as the state of the art in fully relativistic four-component Dirac-Kohn-Sham (DKS) software. Thanks to the implementation of various parallelization and memory open-ended distribution schemes in combination with efficient "density fitting" algorithms, it greatly reduces the computational burden of four-component DKS calculations. We also report the newly developed OpenMP version of the code, that, together with the berthmod Python module, provides a significant leap forward in terms of usability and applicability of the BERTHA software. Some applications of the recently developed natural orbitals for chemical valence/charge displacement bonding analysis and the real-time time dependent DKS implementation are also reported.

Leading Interaction Components in the Structure and Reactivity of Noble Gases Compounds.

The nature, strength, range and role of the bonds in adducts of noble gas atoms with both neutral and ionic partners have been investigated by exploiting a fine-tuned integrated phenomenological-theoretical approach. The identification of the leading interaction components in the noble gases adducts and their modeling allows the encompassing of the transitions from pure noncovalent to covalent bound aggregates and to rationalize the anomalous behavior (deviations from noncovalent type interaction) pointed out in peculiar cases. Selected adducts affected by a weak chemical bond, as those promoting the formation of the intermolecular halogen bond, are also properly rationalized. The behavior of noble gas atoms excited in their long-life metastable states, showing a strongly enhanced reactivity, has been also enclosed in the present investigation.

PyBERTHART: A Relativistic Real-Time Four-Component TDDFT Implementation Using Prototyping Techniques Based on Python.

We present a real-time time-dependent four-component Dirac-Kohn-Sham (RT-TDDKS) implementation based on the BERTHA code. This new implementation takes advantage of modern software engineering, including the prototyping techniques. The software design follows a three step approach: (i) the prototype implementation of a time-propagation algorithm in nonrelativistic real-time TDDFT within the Psi4NumPy framework, which provides a suitable environment for the creation of a clear, readable, and easy to test reference code in Python, (ii) the design of an original Python application programming interface for the relativistic four-component code BERTHA (PyBERTHA), which has an efficient computational kernel for relativistic integrals written in FORTRAN, and (iii) the porting of the time-propagation scheme enveloped within the Psi4NumPy framework to PyBERTHA. The propagation scheme consequently resides in a single readable Python computer code that is easy to maintain and in which the key quantities, such as the Dirac-Kohn-Sham and dipole matrices, can be accessed directly from the PyBERTHA module. For linear algebra operations (matrix-matrix multiplications and diagonalization) we use the highly optimized procedures implemented in the popular NumPy library. The overhead introduced by the Python interface to BERTHA is almost negligible (less than 1% evaluated on the SCF procedure), and the interoperability between different programming languages (FORTRAN, C, and Python) does not affect the numerical stability of the time-propagation scheme. Our new RT-TDDKS implementation has been employed to investigate the stability of the time-propagation procedure in combination with a density-fitting algorithm (both for the Coulomb and for the exchange-correlation matrix construction), which are employed in BERTHA to speed up the Dirac-Kohn-Sham matrix evaluation. On the basis of systematic calculations, employing several density-fitting basis sets of increasing accuracy, we showed that quantitative agreement can be achieved in combination with extended-fitting basis sets, with an error in the Coulomb energy below 1 µ-hartree. Convergence of the transition energies increasing of quality of the fitting basis sets has been also observed. Our data suggest that the error in the Coulomb energy may also represent a good estimate of the fitting basis set quality for real-time electron dynamics simulations. Further, we study the applicability of the RT-TDDKS method in combination with both weak- and extreme strong-field regime. Numerical results of excited-state transitions for the Group 12 atoms are reported and compared with a previous real-time Dirac-Kohn-Sham implementation (Repisky et al. J. Chem. Theory Comput. 2015, 11, 980-991). Finally, calculations of high harmonic generation in the hydrogen molecule and Au dimer have been also carried out. We were able to generate high harmonics with relatively well-defined peaks up to the 21st and 13th order in the case of H2 and Au2, respectively. Our findings show that the four-component structure of the Dirac-Kohn-Sham Hamiltonian provides a suitable theoretical framework, with no intrinsic unfavorable features, to study molecules in the strong-field regime.

The Halogen-Bond Nature in Noble Gas-Dihalogen Complexes from Scattering Experiments and Ab Initio Calculations.

In order to clarify the nature of the halogen bond (XB), we considered the prototype noble gas-dihalogen molecule (Ng-X2) systems, focusing on the nature, range, and strength of the interaction. We exploited data gained from molecular beam scattering experiments with the measure of interference effects to obtain a suitable formulation of the interaction potential, with the support of high-level ab initio calculations, and charge displacement analysis. The essential interaction components involved in the Ng-X2 adducts were characterized, pointing at their critical balance in the definition of the XB. Particular emphasis is devoted to the energy stability of the orientational Ng-X2 isomers, the barrier for the X2 hindered rotation, and the influence of the X2 electronic state. The present integrated study returns reliable force fields for molecular dynamic simulations in Ng-X2 complexes that can be extended to systems with increasing complexity and whose properties depend on the selective formation of XB.

The Chemical Bond and s-d Hybridization in Coinage Metal(I) Cyanides.

We present a four-component relativistic density functional theory study of the chemical bond and s-d hybridization in the group 11 cyanides M-CN (M = Cu, Ag, and Au). The analysis is carried out within the charge-displacement/natural orbital for chemical valence (CD-NOCV) scheme, which allows us to single out meaningful contributions to the total charge rearrangement that arises upon bond formation and to quantify the components of the Dewar-Chatt-Duncanson bonding model (the ligand-to-metal donation and metal-to-ligand back-donation). The M-CN bond is characterized by a large donation from the cyanide ion to the metal cation and by two small back-donation components from the metal toward the cyanide anion. The case of gold cyanide elucidates the peculiar role of the relativistic effects in determining the characteristics of the Au-C bond with respect to the copper and silver homologues. In AuCN, the donation and back-donation components are significantly enhanced, and the spin-orbit coupling, removing the degeneracy of the 5d atomic orbitals, induces a substantial split in the back-donation components. A simple spatial analysis of the NOCV-pair density, related to the ligand-to-metal donation component, allows us to quantify, with unprecedented accuracy, the charge rearrangement due to the s-d hybridization occurring at the metal site. The s-d hybridization plays a key role in determining the shape and size of the metal; it removes electron density from the bond axis and induces a significant flattening at the metal site in the position trans to the ligand. The s-d hybridization is present in all noble metal complexes, influencing the bond distances, and its effect is enhanced for Au, which is consistent with the preference of gold to form linear complexes. A comparative investigation of simple complexes [AuL]+/0 of Au+ with different ligands (L = F-, N-heterocyclic carbene, CO, and PH3) shows that the s-d hybridization mechanism is also influenced by the nature of the ligand.

Chemical Bond Mechanism for Helium Revealed by Electronic Excitation.

Helium chemistry is notoriously very impervious. It is therefore certainly no surprise that, for example, beryllium and helium atoms, in their ground state, do not bind. Full configuration-interaction calculations show that the same turns out to be true, save for a long-range shallow attraction, for the Be+ + He system. However, quite astonishingly, when one electron is re-added to Be+ in an excited 2pπ or 3s orbital (Be 1P or 1S), a bound adduct with He is formed, at an interatomic separation as short as 1.5 Å. Understanding why this happens reveals an unsuspected chemical mechanism that stabilizes helium compounds at the molecular level.

Insight into the halogen-bond nature of noble gas-chlorine systems by molecular beam scattering experiments, ab initio calculations and charge displacement analysis.

We have carried out molecular-beam scattering experiments and high-level ab initio investigations on the potential energy surfaces of a series of noble-gas-Cl2 adducts. This effort has permitted the construction of a simple, reliable and easily generalizable analytical model potential formulation, which is based on a few physically meaningful parameters of the interacting partners and transparently shows the origin, strength, and stereospecificity of the various interaction components. The results demonstrate quantitatively beyond doubt that the interaction between a noble-gas (Ng) atom - even He - and Cl2 in a collinear configuration is characterized by weak halogen bond (XB) formation, accompanied by charge transfer (CT) from the Ng to chlorine. This characteristic, which stabilizes the adduct, rapidly disappears on going towards the T-shaped configuration, dominated by pure van der Waals (vdW) forces. Similarly, a pure vdW interaction takes place - with no CT component in any configuration - if Cl2 is present in the lowest πg* → σu* excited state, because the change in electron density that accompanies the excitation eliminates the Cl2 polar flattening and σ hole, making the XB interaction inaccessible.

Selective Emergence of the Halogen Bond in Ground and Excited States of Noble-Gas-Chlorine Systems.

Molecular-beam scattering experiments and theoretical calculations prove the nature, strength, and selectivity of the halogen bonds (XB) in the interaction of halogen molecules with the series of noble gas (Ng) atoms. The XB, accompanied by charge transfer from the Ng to the halogen, is shown to take place in, and measurably stabilize, the collinear conformation of the adducts, which thus becomes (in contrast to what happens for other Ng-molecule systems) approximately as bound as the T-shaped form. It is also shown how and why XB is inhibited when the halogen molecule is in the 3 Πu excited state. A general potential formulation fitting the experimental observables, based on few physically essential parameters, is proposed to describe the interaction accurately and is validated by ab initio computations.

Alkyne Activation with Gold(III) Complexes: A Quantitative Assessment of the Ligand Effect by Charge-Displacement Analysis.

A quantitative assessment of the Dewar-Chatt-Duncanson components of the Au(III)-alkyne bond in a series of cationic and dicationic bis- and monocyclometalated gold(III) complexes with 2-butyne via charge-displacement (CD) analysis is reported. Bonding between Au(III) and 2-butyne invariably shows a dominant σ donation component, a smaller, but significant, π back-donation, and a remarkable polarization of the alkyne CC triple bond toward the metal fragment. A very large net electron charge transfer from CC triple bond to the metal fragment results, which turns out to be unexpectedly insensitive to the charge of the complex and more strictly related to the nature of the ancillary ligand. The combination of σ donation, π back-donation, and polarization effects is in fact modulated by the different ligand frameworks, with ligands bearing atoms different from carbon in trans position with respect to the alkyne emerging as especially interesting for both imparting Au(III)-alkyne bond stability and inducing a more effective alkyne activation. A first attempt to figure out a rationale on the bonding/reactivity relationship for Au(III)-alkyne is made by performing a comparative study in a model nucleophilic attack of water to the alkyne triple bond. Smaller π back-donation facilitates alkyne slippage in the transition states, which is energetically less demanding for Au(III) than for Au(I), and suggests a greater propensity of Au(III) to facilitate the nucleophilic attack.

Ligand Effect on Bonding in Gold(III) Carbonyl Complexes.

We quantitatively assess the Dewar-Chatt-Duncanson (DCD) components of the Au(III)-CO bond and the charge density polarization at the CO, in a series of neutral, cationic, and dicationic bis- and monocyclometalated gold(III) complexes via charge-displacement (CD) analysis. A striking feature concerns the very small net electron charge flux from CO to the metal fragment which is unexpectedly stable toward both the charge of the complex and the oxidation state of gold (I, III). All systems exhibit a similar trend for the σ charge rearrangement in the region of the carbonyl bond, where, by contrast, the π back-donation trend variation is large, which is strictly correlated to the change in CO bond distance and the shift in CO stretching frequencies, in close analogy with the gold(I) carbonyl complexes. In the whole series of gold(III) compounds, a large Au(III) ← CO σ donation is measured (from 0.19 to 0.31 electrons), as well as a significant Au(III) → CO π back-donation (from -0.09 to -0.22 electrons), which however is not generally able to completely balance the polarization of the CO π electrons in the direction from oxygen to carbon (C ← O) induced by the presence of the metal fragment [LAu(III)]0/+1/+2. Surprisingly, all the gold(III) complexes in the series are characterized by a very small anisotropy in the Au(III) → CO in-plane and out-of-plane π back-donation components, in sharp contrast with the marked anisotropy found before for the experimentally characterized [(C^N^C)Au(III)CO]+ complex. A first attempt to figure out a rationale on the bonding/reactivity relationship for Au(III)-CO is made by performing a comparative study with an isostructural [(N^N^C)Pt(II)CO]+ complex in a model water-gas shift (WGS) reaction.

Modelling Charge Transfer in Weak Chemical Bonds: Insights from the Chemistry of Helium.

We studied the nature of the interaction of the weakly bound Be-He adduct by means of an integrated theoretical approach based on high-level quantum chemical calculations for the characterization of the potential energy surfaces and charge displaced upon adduct formation, together with the development of a semi-empirical analytical formulation of the interaction potential. Our results show that Be is able to form a stable adduct with He when the Be(1 D) (1s2 2s2 →1s2 2s0 2p2 ) excited state is involved, with a binding energy of as much as 10.2 kcal/mol, an astonishingly large value for He in neutral systems. The analysis of the leading interaction components in the Be*-He adduct proves the relevance of the charge transfer to the overall stability, which contributes to decreasing the intermolecular distance, thus strengthening the induction-energy component.

Spin-Forbidden Reactions: Adiabatic Transition States Using Spin-Orbit Coupled Density Functional Theory.

A spin-forbidden chemical reaction involves a change in the total electronic spin state from reactants to products. The mechanistic study is challenging because such a reaction does not occur on a single diabatic potential energy surface (PES), but rather on two (or multiple) spin diabatic PESs. One possible approach is to calculate the so-called "minimum energy crossing point" (MECP) between the diabatic PESs, which however is not a stationary point. Inclusion of spin-orbit coupling between spin states (SOC approach) allows the reaction to occur on a single adiabatic PES, in which a transition state (TS SOC) as well as activation free energy can be calculated. This Concept article summarizes a previously published application in which, for the first time, the SOC effects, using spin-orbit ZORA Hamiltonian within density functional theory (DFT) framework, are included and account for the mechanism of a spin-forbidden reaction in gold chemistry. The merits of the MECP and TS SOC approaches and the accuracy of the results are compared, considering both our recent calculations on molecular oxygen addition to gold(I)-hydride complexes and new calculations for the prototype spin-forbidden N2 O and N2 Se dissociation reactions.

The ligand effect on the oxidative addition of dioxygen to gold(i)-hydride complexes.

The ligand effect on the recently uncovered feasible oxidative addition reaction of O2 on [LAuH] complexes has been investigated for a series of fifteen ligands. The activation barriers of this spin-forbidden reaction have been estimated at the minimum energy crossing points (MECP, relativistic scalar level) between the adiabatic triplet (reactants spin state) and singlet (product spin state) potential energy surfaces (PES) and calculated at the transition states by including Spin-Orbit Coupling (SOC) effects, as applied for the mechanistic study of this reaction in a previous study by us [Chem. Sci., 2016, 7, 7034-7039]. We find a sizeable effect of the ligand on the activation barriers, and some of the stronger electron donating phosphines are predicted to induce the highest catalyst efficiency. The inclusion of SOC effects lowers the activation barriers by about 3 kcal mol-1 systematically with respect to the MECP values independently of the ligand type. We used the Charge-Displacement (CD) analysis to quantify the net electron charge donation from the ligand L towards the metallic fragment AuH in the [LAuH] series, and surprisingly only a poor correlation was found between the net electron donor character of L and the activation barriers. Application of the CD-NOCV (Natural Orbitals for Chemical Valence) approach, which allows the quantification of the Dewar-Chatt-Duncanson (DCD) L-AuH bond components, suggests that the ligand effect on the activation barriers is not easily predictable on the basis of solely the electronic properties of the ligand and depends significantly on the ligand nature or carbene or phosphine type. We show that for both phosphine and carbene ligand subsets, however, the σ donation component of the L-AuH bond quantitatively accounts for the ligand effect on the activation energy barriers (a larger σ-donor capability of L correlates with a smaller activation barrier), whereas the π back-donation, strongly affected by geometrical rearrangement, is a poor reactivity descriptor (π acceptor properties of the ligand L in the linear [LAuH] complexes are not transferable to the trigonal [LAuH(O2)] transition state structures).

Helium Accepts Back-Donation In Highly Polar Complexes: New Insights into the Weak Chemical Bond.

We studied the puzzling stability and short distances predicted by theory for helium adducts with some highly polar molecules, such as BeO or AuF. On the basis of high-level quantum-chemical calculations, we carried out a detailed analysis of the charge displacement occurring upon adduct formation. For the first time we have unambiguously ascertained that helium is able not only to donate electron density, but also, unexpectedly, to accept electron density in the formation of weakly bound adducts with highly polar substrates. The presence of a large dipole moment induces a large electric field at He, which lowers its 2p orbital energy and enables receipt of π electron density. These findings offer unprecedented important clues toward the design and synthesis of stable helium compounds.

Modulating the Bonding Properties of N-Heterocyclic Carbenes (NHCs): A Systematic Charge-Displacement Analysis.

In view of their intensive use as ligands in many reactions catalyzed by transition-metal complexes, modulation of the bonding properties of N-heterocyclic carbenes (NHCs) on a rational basis is highly desirable, which should enable optimization of current applications or even promote new functions. In this paper, we provide a quantitative analysis of the chemical bond between a metal fragment AuCl and a series of 29 different NHCs in [(NHC)AuCl] complexes. NHCs electronic properties are modified through: i) variation of the groups attached to the NHC nitrogen atoms or backbone; ii) change of unsaturation/size of the NHC ring; iii) inclusion of paracyclophane moieties; or iv) heteroatom substitution on the NHC ring. For evaluating the donation and back-donation components of the Dewar-Chatt-Duncanson (DCD) model in the NHC-AuCl bond, we apply the charge-displacement (CD) analysis within the NOCV (natural orbitals for chemical valence) framework, a methodology that avoids the constraint of using symmetrized structures. We show that modulation of the NHC bonding properties requires substantial modification of their structure, such as, for instance, insertion of two ketone groups into the NHC backbone (which enhances the π back-donation bond component and introduces an effective electronic communication within the NHC ring) or replacement of a nitrogen atom in the ring with an sp3 or sp2 carbon atom (which increases and decreases the π back-donation bond component, respectively). We extend our investigation by quantitatively comparing the NHC electronic structures for a subset of 13 NHCs in [(NHC)PPh] adducts, the 31 P NMR chemical shift values of which are experimentally available. The latter have been considered as a suitable tool for measuring the NHCs π acceptor properties [Bertrand et al., Angew. Chem. Int. Ed. 2013, 52, 2939-2943]. We show that information obtained using the metal fragment can be transferred to the PPh moiety and vice versa. However, the 31 P NMR chemical shift values only qualitatively correlate with the π acceptor properties of the NHCs, with the stronger π acidic carbenes as the most outliners.

The gold(iii)-CO bond: a missing piece in the gold carbonyl complex landscape.

We investigate the Au(iii)-CO bond in the [(C∧N∧C)Au(iii)CO]+ complex via charge-displacement (CD) analysis, revealing surprising peculiarities with respect to the Au(i)-CO bond. It shows a large Au(iii) ← CO σ donation and, unexpectedly, an even larger Au(iii) → CO π back-donation. The overall bonding picture, including a marked anisotropy in the Au(iii) → CO in-plane and out-of-plane π back-donation components, is consistent with a CO carbon atom highly susceptible to nucleophilic attack.

13 C†NMR Spectroscopy of N-Heterocyclic Carbenes Can Selectively Probe σ Donation in Gold(I) Complexes.

The Dewar-Chatt-Duncanson (DCD) model provides a successful theoretical framework to describe the nature of the chemical bond in transition-metal compounds and is especially useful in structural chemistry and catalysis. However, how to actually measure its constituents (substrate-to-metal donation and metal-to-substrate back-donation) is yet uncertain. Recently, we demonstrated that the DCD components can be neatly disentangled and the π back-donation component put in strict correlation with some experimental observables. In the present work we make a further crucial step forward, showing that, in a large set of charged and neutral N-heterocyclic carbene complexes of gold(I), a specific component of the NMR chemical shift tensor of the carbenic carbon provides a selective measure of the σ donation. This work opens the possibility of 1) to characterize unambiguously the electronic structure of a metal fragment (LAu(I)n+/0 in this case) by actually measuring its σ-withdrawing ability, 2) to quickly establish a comparative trend for the ligand trans effect, and 3) to achieve a more rigorous control of the ligand electronic effect, which is a key aspect for the design of new catalysts and metal complexes.

π Activation of Alkynes in Homogeneous and Heterogeneous Gold Catalysis.

The activation of alkynes toward nucleophilic attack upon coordination to gold-based catalysts (neutral and positively charged gold clusters and gold complexes commonly used in homogeneous catalysis) is investigated to elucidate the role of the σ donation and π back-donation components of the Au-C bond (where we consider ethyne as prototype substrate). Charge displacement (CD) analysis is used to obtain a well-defined measure of σ donation and π back-donation and to find out how the corresponding charge flows affect the electron density at the electrophilic carbon undergoing the nucleophilic attack. This information is used to rationalize the activity of a series of catalysts in the nucleophilic attack step of a model hydroamination reaction. For the first time, the components of the Dewar-Chatt-Duncanson model, donation and back-donation, are put in quantitative correlation with the kinetic parameters of a chemical reaction.