Reaction path Hamiltonian and the unified reaction valley approach

Beyond Cavity Born-Oppenheimer: On Nonadiabatic Coupling and Effective Ground State Hamiltonians in Vibro-Polaritonic Chemistry

The emerging field of vibro-polaritonic chemistry studies the impact of light-matter hybrid states known as vibrational polaritons on chemical reactivity and molecular properties. Here, we discuss vibro-polaritonic chemistry from a quantum chemical perspective beyond the cavity Born-Oppenheimer (CBO) approximation and examine the role of electron-photon correlation in effective ground state Hamiltonians. We first quantitatively review ab initio vibro-polaritonic chemistry based on the molecular Pauli-Fierz Hamiltonian in dipole approximation and a vibrational strong coupling (VSC) Born-Huang expansion. We then derive nonadiabatic coupling elements arising from both "slow" nuclei and cavity modes compared to "fast" electrons via the generalized Hellmann-Feynman theorem, discuss their properties, and reevaluate the CBO approximation. In the second part, we introduce a crude VSC Born-Huang expansion based on adiabatic electronic states, which provides a foundation for widely employed effective Pauli-Fierz Hamiltonians in ground state vibro-polaritonic chemistry. Those do not strictly respect the CBO approximation but an alternative scheme, which we name crude CBO approximation. We argue that the crude CBO ground state misses electron-photon correlation relative to the CBO ground state due to neglected cavity-induced nonadiabatic transition dipole couplings to excited states. A perturbative connection between both ground state approximations is proposed, which identifies the crude CBO ground state as a first-order approximation to its CBO counterpart. We provide an illustrative numerical analysis of the cavity Shin-Metiu model with a focus on nonadiabatic coupling under VSC and electron-photon correlation effects on classical activation barriers. We finally discuss the potential shortcomings of the electron-polariton Hamiltonian when employed in the VSC regime.

A Structure-Based Gaussian Expansion for Quantum Reaction Dynamics in Molecules: Application to Hydrogen Tunneling in Malonaldehyde

We developed an approximate method for quantum reaction dynamics simulations, namely, a structure-based Gaussian (SBG) expansion approach, where SBG bases for the expansion of the wave function Ψ, expressed by a product of single-atom Cartesian Gaussians centered at the positions of respective nuclei, are mainly placed around critical structures on reaction pathways such as on the intrinsic reaction coordinate (IRC) through a transition state. In the present approach, the "pseudo-lattice points" at which SBGs are deployed are selected in a perturbative manner so as to make moderate the expansion length. We first applied the SBG idea to a two-dimensional quadruple-well model and obtained accurate tunneling splitting values between the lowest four states. We then applied it to hydrogen tunneling in malonaldehyde and achieved a tunneling splitting of 27.1 cm^-1 with only 875 SBGs at the MP2/6-31G(d,p) level of theory, in good agreement with 25 cm^-1 by the more elaborate multiconfiguration time-dependent Hartree method. Reasonable results were also obtained for singly and doubly deuterated malonaldehyde. We analyzed the tunneling states by utilizing expansion coefficients of individual SBGs and found that 40-45% of the SBGs in Ψ are nonplanar structures and SBGs away from the IRC contribute a little to hydrogen transfer.

Mechanistic Details of the Sharpless Epoxidation of Allylic Alcohols—A Combined URVA and Local Mode Study

In this work, we investigated the catalytic effects of a Sharpless dimeric titanium (IV)–tartrate–diester catalyst on the epoxidation of allylalcohol with methyl–hydroperoxide considering four different orientations of the reacting species coordinated at the titanium atom (reactions R1–R4) as well as a model for the non-catalyzed reaction (reaction R0). As major analysis tools, we applied the URVA (Unified Reaction Valley Approach) and LMA (Local Mode Analysis), both being based on vibrational spectroscopy and complemented by a QTAIM analysis of the electron density calculated at the DFT level of theory. The energetics of each reaction were recalculated at the DLPNO-CCSD(T) level of theory. The URVA curvature profiles identified the important chemical events of all five reactions as peroxide OO bond cleavage taking place before the TS (i.e., accounting for the energy barrier) and epoxide CO bond formation together with rehybridization of the carbon atoms of the targeted CC double bond after the TS. The energy decomposition into reaction phase contribution phases showed that the major effect of the catalyst is the weakening of the OO bond to be broken and replacement of OH bond breakage in the non-catalyzed reaction by an energetically more favorable TiO bond breakage. LMA performed at all stationary points rounded up the investigation (i) quantifying OO bond weakening of the oxidizing peroxide upon coordination at the metal atom, (ii) showing that a more synchronous formation of the new CO epoxide bonds correlates with smaller bond strength differences between these bonds, and (iii) elucidating the different roles of the three TiO bonds formed between catalyst and reactants and their interplay as orchestrated by the Sharpless catalyst. We hope that this article will inspire the computational community to use URVA complemented with LMA in the future as an efficient mechanistic tool for the optimization and fine-tuning of current Sharpless catalysts and for the design new of catalysts for epoxidation reactions.

BF3–Catalyzed Diels–Alder Reaction between Butadiene and Methyl Acrylate in Aqueous Solution—An URVA and Local Vibrational Mode Study

In this study we investigate the Diels–Alder reaction between methyl acrylate and butadiene, which is catalyzed by BF3 Lewis acid in explicit water solution, using URVA and Local Mode Analysis as major tools complemented with NBO, electron density and ring puckering analyses. We considered four different starting orientations of methyl acrylate and butadiene, which led to 16 DA reactions in total. In order to isolate the catalytic effects of the BF3 catalyst and those of the water environment and exploring how these effects are synchronized, we systematically compared the non-catalyzed reaction in gas phase and aqueous solution with the catalyzed reaction in gas phase and aqueous solution. Gas phase studies were performed at the B3LYP/6-311+G(2d,p) level of theory and studies in aqueous solution were performed utilizing a QM/MM approach at the B3LYP/6-311+G(2d,p)/AMBER level of theory. The URVA results revealed reaction path curvature profiles with an overall similar pattern for all 16 reactions showing the same sequence of CC single bond formation for all of them. In contrast to the parent DA reaction with symmetric substrates causing a synchronous bond formation process, here, first the new CC single bond on the CH2 side of methyl acrylate is formed followed by the CC bond at the ester side. As for the parent DA reaction, both bond formation events occur after the TS, i.e., they do not contribute to the energy barrier. What determines the barrier is the preparation process for CC bond formation, including the approach diene and dienophile, CC bond length changes and, in particular, rehybridization of the carbon atoms involved in the formation of the cyclohexene ring. This process is modified by both the BF3 catalyst and the water environment, where both work in a hand-in-hand fashion leading to the lowest energy barrier of 9.06 kcal/mol found for the catalyzed reaction R1 in aqueous solution compared to the highest energy barrier of 20.68 kcal/mol found for the non-catalyzed reaction R1 in the gas phase. The major effect of the BF3 catalyst is the increased mutual polarization and the increased charge transfer between methyl acrylate and butadiene, facilitating the approach of diene and dienophile and the pyramidalization of the CC atoms involved in the ring formation, which leads to a lowering of the activation energy. The catalytic effect of water solution is threefold. The polar environment leads also to increased polarization and charge transfer between the reacting species, similar as in the case of the BF3 catalyst, although to a smaller extend. More important is the formation of hydrogen bonds with the reaction complex, which are stronger for the TS than for the reactant, thus stabilizing the TS which leads to a further reduction of the activation energy. As shown by the ring puckering analysis, the third effect of water is space confinement of the reacting partners, conserving the boat form of the six-member ring from the entrance to the exit reaction channel. In summary, URVA combined with LMA has led to a clearer picture on how both BF3 catalyst and aqueous environment in a synchronized effort lower the reaction barrier. These new insights will serve to further fine-tune the DA reaction of methyl acrylate and butadiene and DA reactions in general.

On the formation of CN bonds in Titan's atmosphere-a unified reaction valley approach study

In this work, we investigated the formation of protonated hydrogen cyanide HCNH⁺ and methylene amine cation CH[Formula: see text] (both identified in Titan's upper atmosphere) from three different pathways which stem from the interaction between CH₄ and N⁺(³P). As a mechanistic tool, we used the Unified Reaction Valley Approach (URVA) complemented with the Local Mode Analysis (LMA) assessing the strength of the CN bonds formed in these reactions. Our URVA studies could provide a comprehensive overview on bond formation/cleavage processes relevant to the specific mechanism of eight reactions R1- R8 that occur across the three pathways. In addition, we could explain the formation of CH[Formula: see text] and the appearance of HCNH⁺ and CHNH[Formula: see text] along these paths. Although only smaller molecules are involved in these reactions including isomerization, hydrogen atom abstraction, and hydrogen molecule capture, we found a number of interesting features, such as roaming in reaction R3 or the primary interaction of H₂ with the carbon atom in HCNH⁺ in reaction R8 followed by migration of one of the H₂ hydrogen atoms to the nitrogen which is more cost effective than breaking the HH bond first; a feature often found in catalysis. In all cases, charge transfer between carbon and nitrogen could be identified as a driving force for the CN bond formation. As revealed by LMA, the CN bonds formed in reactions R1-R8 cover a broad bond strength range from very weak to very strong, with the CN bond in protonated hydrogen cyanide HCNH⁺ identified as the strongest of all molecules investigated in this work. Our study demonstrates the large potential of both URVA and LMA to shed new light into these extraterrestrial reactions to help better understand prebiotic processes as well as develop guidelines for future investigations involving areas of complex interstellar chemistry. In particular, the formation of CN bonds as a precursor to the extraterrestrial formation of amino acids will be the focus of future investigations. Formation of CN bonds in Titan's atmosphere visualized via the reaction path curvature.

Interacting Quantum Atoms Analysis of the Reaction Force: A Tool to Analyze Driving and Retarding Forces in Chemical Reactions

The analysis of the reaction force and its topology has provided a wide range of fruitful concepts in the theory of chemical reactivity over the years, allowing to identify chemically relevant regions along a reaction profile. The reaction force (RF), a projection of the Hellmann-Feynman forces acting on the nuclei of a molecular system onto a suitable reaction coordinate, is partitioned using the interacting quantum atoms approach (IQA). The exact IQA molecular energy decomposition is now shown to open a unique window to identify and quantify the chemical entities that drive or retard a chemical reaction. The RF/IQA coupling offers an extraordinarily detailed view of the type and number of elementary processes that take reactants into products, as tested on two sets of simple reactions.

Photodissociation Dynamics of CO-Forming Channels on the Ground-State Surface of Methyl Formate at 248 nm: Direct Dynamics Study and Assessment of Generalized Multicenter Impulsive Models

The photodissociation dynamics of methyl formate in the electronic ground state S₀, initiated by a 248 nm-wavelength laser, is studied by direct dynamics simulations. We analyze five channels, where four of them have as products CH₃OH + CO, one leading to the formation of three fragments, H₂CO + H₂ + CO, and a channel characterized by a roaming transition state. The analysis of energy distribution among the degrees of freedom of the product and the comparison with experimental results previously published by other groups provide the ingredients to distinguish the examined dissociation pathways. The interpretation of the results proves that the characterization of dissociation mechanisms must rely on a dynamics approach involving multiple electronic states, including considerations on the features of the S₁/S₀ conical intersection. Here, we also assess the generalized multicenter impulsive model, GMCIM, that has been designed for dissociation processes with exit barriers, and the energy distribution in the products is predicted on the basis of information from the saddle points and the intrinsic reaction coordinates. Main features, advantages, limits, and future perspectives of the method are reported and discussed.

Modeling Hydrogen Release from Water with Borane and Alane Catalysts: A Unified Reaction Valley Approach

The unified reaction valley approach combined with the local vibrational mode and ring puckering analysis is applied to investigate the hydrogen evolution from water in the presence of small hydrides such as BH₃, metal hydrides as AlH₃, and their derivatives. We studied a series of reactions involving BH₃, AlH₃, B₂H₆, Al₂H₆, and AlH₃BH₃ with one- and two-water molecules, considering multiple reaction paths. In addition, the influence of the aqueous medium was examined. A general reaction mechanism was identified for most of the reactions. Those that deviate could be associated with unusually high reaction barriers with no hydrogen release. The charge transfer along the reaction path suggests that a viable hydrogen release is achieved when the catalyst adopts the role of a charge donor during the chemical processes. The puckering analysis showed that twistboat and boat forms are the predominant configurations in the case of an intermediate six-membered ring formation, which influences the activation barrier. The local mode analysis was used as a tool to detect the H-H bond formation as well as to probe catalyst regenerability. Based on the correlation between the activation energy and the change in the charge separation for cleaving O-H and B(Al)-H bonds, two promising subsets of reactions could be identified along with prescriptions for lowering the reaction barrier individually with electron-donating/withdrawing substituents.

A matrix completion algorithm to recover modes orthogonal to the minimum energy path in chemical reactions

Structured statistical methods are promising for recovering or completing information from noisy and incomplete data with high fidelity. In particular, matrix completion exploits underlying structural properties such as rank or sparsity. Our objective is to employ matrix completion to reduce computational effort associated with the calculation of multiple quantum chemical Hessians, which are necessary for identification of temperature-dependent free energy maxima under canonical variational transition state theory (VTST). We demonstrate proof-of-principle of an algebraic variety-based matrix completion method for recovering missing elements in a matrix of transverse Hessian eigenvalues constituting the minimum energy path (MEP) of a reaction. The algorithm, named harmonic variety-based matrix completion (HVMC), utilizes the fact that the points lying on the MEP of a reaction step constitute an algebraic variety in the reaction path Hamiltonian representation. We demonstrate that, with as low as 30% random sampling of matrix elements for the largest system in our test set (46 atoms), the complete matrix of eigenvalues can be recovered. We further establish algorithm performance for VTST rate calculations by quantifying errors in zero-point energies and vibrational free energies. Motivated by this success, we outline next steps toward developing a practical HVMC algorithm, which utilizes a gradient-based sampling protocol for low-cost VTST rate computations.

Exploring the Mechanism of Catalysis with the Unified Reaction Valley Approach (URVA)—A Review

The unified reaction valley approach (URVA) differs from mainstream mechanistic studies, as it describes a chemical reaction via the reaction path and the surrounding reaction valley on the potential energy surface from the van der Waals region to the transition state and far out into the exit channel, where the products are located. The key feature of URVA is the focus on the curving of the reaction path. Moving along the reaction path, any electronic structure change of the reacting molecules is registered by a change in their normal vibrational modes and their coupling with the path, which recovers the curvature of the reaction path. This leads to a unique curvature profile for each chemical reaction with curvature minima reflecting minimal change and curvature maxima, the location of important chemical events such as bond breaking/forming, charge polarization and transfer, rehybridization, etc. A unique decomposition of the path curvature into internal coordinate components provides comprehensive insights into the origins of the chemical changes taking place. After presenting the theoretical background of URVA, we discuss its application to four diverse catalytic processes: (i) the Rh catalyzed methanol carbonylation—the Monsanto process; (ii) the Sharpless epoxidation of allylic alcohols—transition to heterogenous catalysis; (iii) Au(I) assisted [3,3]-sigmatropic rearrangement of allyl acetate; and (iv) the Bacillus subtilis chorismate mutase catalyzed Claisen rearrangement—and show how URVA leads to a new protocol for fine-tuning of existing catalysts and the design of new efficient and eco-friendly catalysts. At the end of this article the pURVA software is introduced. The overall goal of this article is to introduce to the chemical community a new protocol for fine-tuning existing catalytic reactions while aiding in the design of modern and environmentally friendly catalysts.

Construction of Two-Dimensional Potential Energy Surfaces of Reactions with Post-Transition-State Bifurcations

Reactions with post-transition-state bifurcations (PTSBs) involve initial ambimodal transition-state structures followed by an unstable region leading to two possible products. PTSBs are seen in many organic, organometallic, and biosynthetic reactions, but analyzing the origins of selectivity for these reactions is challenging, in large part due to the complex nature of the potential energy surfaces involved, which precludes analyses based on single intrinsic reaction coordinate (IRC; steepest-descent path in mass-weighted coordinate). While selectivity can be predicted using molecular dynamics simulation, connecting results from such calculations to the topography of potential energy surfaces is difficult. In the present work, a method for generating two-dimensional potential energy surfaces for PTSBs is described. The first dimension starts with the IRC for the first transition-state structure, followed by a modified reaction coordinate that reaches the second transition-state structure, which interconverts the two products of a bifurcating reaction path. The IRC for the second transition-state structure constitutes the second dimension. In addition, a method for mapping trajectories from Born-Oppenheimer molecular dynamics simulations onto these surfaces is described. Both approaches are illustrated with representative examples from the field of organic chemistry. The 2D-PESs for five asymmetric cases tested have clear tilted topography after the first transition-state structure, and the tilted direction correlates well with the selectivity observed from previous dynamic simulation. Instead of selecting reaction coordinates by chemical intuition, our method provides a general means to construct two-dimensional potential energy surfaces for reactions with post-transition-state bifurcations.

Describing Polytopal Rearrangements of Fluxional Molecules with Curvilinear Coordinates Derived from Normal Vibrational Modes: A Conceptual Extension of Cremer-Pople Puckering Coordinates

In this work a new curvilinear coordinate system is presented for the comprehensive description of polytopal rearrangements of N-coordinate compounds (N = 4-7) and systems containing an N-coordinate subunit. It is based on normal vibrational modes and a natural extension of the Cremer-Pople puckering coordinates ( J. Am. Chem. Soc. 1975, 97, 1354) together with the Zou-Izotov-Cremer deformation coordinates ( J. Phys. Chem. A 2011, 115, 8731) for ring structures to N-coordinate systems. We demonstrate that the new curvilinear coordinates are ideal reaction coordinates describing fluxional rearrangement pathways by revisiting the Berry pseudorotation and the lever mechanism in sulfur tetrafluoride, the Berry pseudorotation and two Muetterties' mechanisms in pentavalent compounds, the chimeric pseudorotation in iodine pentafluoride, Bailar and Ray-Dutt twists in hexacoordinate tris-chelates as well as the Bartell mechanism in iodine heptafluoride. The results of our study reveal that this dedicated curvilinear coordinate system can be applied to most coordination compounds opening new ways for the systematic modeling of fluxional processes.

A Continuum from Halogen Bonds to Covalent Bonds: Where Do λ3 Iodanes Fit?

The intrinsic bonding nature of λ 3 -iodanes was investigated to determine where its hypervalent bonds fit along the spectrum between halogen bonding and covalent bonding. Density functional theory with an augmented Dunning valence triple zeta basis set ( ω B97X-D/aug-cc-pVTZ) coupled with vibrational spectroscopy was utilized to study a diverse set of 34 hypervalent iodine compounds. This level of theory was rationalized by comparing computational and experimental data for a small set of closely-related and well-studied iodine molecules and by a comparison with CCSD(T)/aug-cc-pVTZ results for a subset of the investigated iodine compounds. Axial bonds in λ 3 -iodanes fit between the three-center four-electron bond, as observed for the trihalide species IF 2 − and the covalent FI molecule. The equatorial bonds in λ 3 -iodanes are of a covalent nature. We explored how the equatorial ligand and axial substituents affect the chemical properties of λ 3 -iodanes by analyzing natural bond orbital charges, local vibrational modes, the covalent/electrostatic character, and the three-center four-electron bonding character. In summary, our results show for the first time that there is a smooth transition between halogen bonding → 3c–4e bonding in trihalides → 3c–4e bonding in hypervalent iodine compounds → covalent bonding, opening a manifold of new avenues for the design of hypervalent iodine compounds with specific properties.

Correlating the vibrational spectra of structurally related molecules: A spectroscopic measure of similarity

Using catastrophe theory and the concept of a mutation path, an algorithm is developed that leads to the direct correlation of the normal vibrational modes of two structurally related molecules. The mutation path is defined by weighted incremental changes in mass and geometry of the molecules in question, which are successively applied to mutate a molecule into a structurally related molecule and thus continuously converting their normal vibrational spectra from one into the other. Correlation diagrams are generated that accurately relate the normal vibrational modes to each other by utilizing mode-mode overlap criteria and resolving allowed and avoided crossings of vibrational eigenstates. The limitations of normal mode correlation, however, foster the correlation of local vibrational modes, which offer a novel vibrational measure of similarity. It will be shown how this will open new avenues for chemical studies. © 2017 Wiley Periodicals, Inc.

Relating normal vibrational modes to local vibrational modes with the help of an adiabatic connection scheme

Information on the electronic structure of a molecule and its chemical bonds is encoded in the molecular normal vibrational modes. However, normal vibrational modes result from a coupling of local vibrational modes, which means that only the latter can provide detailed insight into bonding and other structural features. In this work, it is proven that the adiabatic internal coordinate vibrational modes of Konkoli and Cremer [Int. J. Quantum Chem. 67, 29 (1998)] represent a unique set of local modes that is directly related to the normal vibrational modes. The missing link between these two sets of modes are the compliance constants of Decius, which turn out to be the reciprocals of the local mode force constants of Konkoli and Cremer. Using the compliance constants matrix, the local mode frequencies of any molecule can be converted into its normal mode frequencies with the help of an adiabatic connection scheme that defines the coupling of the local modes in terms of coupling frequencies and reveals how avoided crossings between the local modes lead to changes in the character of the normal modes.