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In Situ Measure of Intrinsic Bond Strength in Crystalline Article Cite This: J. Chem. Theory Comput. 2019, 15, 1761−1776 pubs.acs.org/JCTC In Situ Measure of Intrinsic Bond Strength in Crystalline Structures: Local Vibrational Mode Theory for Periodic Systems Yunwen Tao,∥,† Wenli Zou,‡ Daniel Sethio,† Niraj Verma,† Yue Qiu,¶ Chuan Tian,§ Dieter Cremer,⊥,† and Elfi Kraka*,† † Department of Chemistry, Southern Methodist University, 3215 Daniel Avenue, Dallas, Texas 75275-0314, United States ‡ Institute of Modern Physics, Northwest University, Xi’an, Shaanxi 710127, P. R. China ¶ Grimwade Centre for Cultural Materials Conservation, School of Historical and Philosophical Studies, Faculty of Arts, University of Melbourne, Parkville, Victoria 3052, Australia § Department of Chemistry, Stony Brook University, Stony Brook, New York 11794, United States ABSTRACT: The local vibrational mode analysis developed by Konkoli and Cremer has been successfully applied to characterize the intrinsic bond strength via local bond stretching force constants in molecular systems. A wealth of new insights into covalent bonding and weak chemical interactions ranging from hydrogen, halogen, pnicogen, and chalcogen to tetrel bonding has been obtained. In this work we extend the local vibrational mode analysis to periodic systems, i.e. crystals, allowing for the first time a quantitative in situ measure of bond strength in the extended systems of one, two, and three dimensions. We present the study of one- dimensional polyacetylene and hydrogen fluoride chains and two-dimensional layers of graphene, water, and melamine-cyanurate as well as three-dimensional ice Ih and crystalline acetone. Besides serving as a new powerful tool for the analysis of bonding in crystals, a systematic comparison of the intrinsic bond strength in periodic systems and that in isolated molecules becomes possible, providing new details into structure and bonding changes upon crystallization. The potential application for the analysis of solid-state vibrational spectra will be discussed. 1. INTRODUCTION the target bond, i.e. it directly relates to the intrinsic bond 6,8 The chemical bond is one of the most important concepts in strength. − chemistry.1 4 Chemists often use the bond dissociation energy In 2000, Cremer and co-workers proposed the local bond − (BDE) as the measure of bond strength.5 7 For example, the stretching force constant as a novel measure of intrinsic bond − strength.7,9 The local bond stretching force constant ka is BDE of the CC single bond in ethane (H3C CH3) is 88 kcal/ n 10−13 mol, while that of the CC double bond in ethylene (H2C derived from the local vibrational modes. The local mode Downloaded via SOUTHERN METHODIST UNIV on April 15, 2019 at 01:30:43 (UTC). 5 fl a fi CH2) is 174 kcal/mol, re ecting that a CC double bond is force constant kn of a speci c bond is the curvature of the π See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. stronger than a CC single bond because of additional potential energy surface (PES) in the direction of this bond bonding. Although the BDE can help chemists to understand with its infinitesimal stretching followed by the relaxation of all chemical bonding in an intuitive way, its deficiencies are other parts of this molecule.14 The local vibrational mode obvious. The BDE is a reaction parameter, which involves the theory has been so far successfully applied to characterize the overall energy changes of a bond dissociation process. It intrinsic bond strength of both covalent bonds7,8,15,16 and − includes the electron density reorganization and the geometry noncovalent interactions including hydrogen,17 20 halogen,21 relaxation of the fragments; therefore, it is not suited as a bond 22 23 24 6,8 pnicogen, chalcogen, and tetrel bonding as well as strength descriptor. Furthermore, the underlying bond 25,26 atom···π interactions. Local stretching force constants were dissociation process into two fragments is not applicable for applied to quantify the intrinsic bond strength of unusual chemical bonds in complex systems, e.g. large water clusters or 15,16,25 metal−organic frameworks (MOFs), in which the dissociation chemical bonding, to explain interesting physicochem- of one bond will drastically change the overall geometry. ical properties, e.g. the fact that warm water freezes faster than 19 fi 8,27−29 Therefore, a better alternative to the BDE as bond strength cold water, and to de ne new electronic parameters 30 descriptor is desired, that can perform an in situ measure of the and rules. bond strength avoiding bond dissociation. Such a bond strength descriptor is expected to reflect all electronic structure Received: December 19, 2018 factors which are responsible for the bonding mechanism of Published: February 18, 2019 © 2019 American Chemical Society 1761 DOI: 10.1021/acs.jctc.8b01279 J. Chem. Theory Comput. 2019, 15, 1761−1776 Journal of Chemical Theory and Computation Article Several theoretical tools have been developed to analyze where fx is the (3N × 3N) dimensional force constant matrix in chemical bonding in periodic systems. Vanderbilt and co- Cartesian coordinates. The (3N × 3N) dimensional diagonal workers proposed a method based on maximally localized mass matrix M contains each atom three times to account for 31,32 × Wannier functions (MLWF), where the band structure is the motion in x, y, and z directions. The (Nvib Nvib) Λ transformed into MLWF-orbitals which are more mathematical dimensional diagonal matrix collects the Nvib vibrational λ μ − and as such do not provide much chemical insight. Crystal eigenvalues μ ( = 1, ..., Nvib with Nvib =3N K) where K orbital Hamiltonian population (COHP)33 and its predecessor equals 5 for linear and 6 for nonlinear molecules. The (3N × 34,35 crystal orbital overlap population (COOP) offer a k-point- Nvib) dimensional matrix L collects the corresponding μ − dependent characterization of bonding being able to vibrational eigenvectors lμ ( = 1, ..., Nvib with Nvib =3N distinguish bonding from antibonding orbitals. The atoms-in- K) as orthonormal column vectors. Harmonic vibrational 36−38 −1 molecules (AIM) approach, which is widely used in both frequencies ωμ in [cm ] and eigenvalues λμ are connected via 2 2 2 47 molecular and periodic systems, partitions the electron density λμ =4π c ωμ where c corresponds to the speed of light. into atomic basins and provides the bond path connecting any Eq 1 can be rewritten in terms of Nvib normal coordinates Q two bonded atoms. From the bond critical points (BCP) and QTx fKLf== L (2) associated bond paths, various bond properties have been × Q derived for the characterization of bonding in molecules as well where the (Nvib Nvib) dimensional force constant matrix f = as in the crystal environment. Alternative topological analysis K is expressed in terms of normal coordinates Q. The T methods based on the electrostatic potential were devel- rectangular matrix L and its transpose L are applied in eq 2. − oped.39 41 Recently, Dunnington and Schmidt succeeded in The local vibrational mode associated with an internal 42,43 fi 47 generalizing the natural bond orbital (NBO) analysis to coordinate qn can be de ned via the Wilson B-matrix, which 44 periodic systems. In this way, results for periodic systems connects the partial derivatives of qn with the Cartesian obtained from plane-wave density functional theory (PW coordinates DFT) can now be interpreted with chemically intuitive ∂q localized bonding orbitals, as it has been widely used for b = n n (3) molecules.45,46 ∂x In this work, we extend Konkoli and Cremer’s local mode The 3N-dimensional row vector b converts the N − n vib theory10 13 from molecules to periodic systems with a vibrational modes collected in L in Cartesian coordinates particular focus on deriving a local mode force constant into the internal coordinates, with the contributions of L to the fl re ecting the intrinsic strength of a bond/weak chemical internal coordinate qn given as interaction in periodic systems, e.g. a hydrogen bond in an ice d = b L (4) crystal. As a result, a deeper understanding of crystal bonding nn will be achieved by a head-to-head comparison of the intrinsic Row vector dn of length of Nvib is then used to obtain the local 10 bond strength of chemical bonds in periodic and molecular mode vector an associated with the internal coordinate qn systems. Kd−1 T The paper is structured in the following way: First, the n an = −1 T original local vibrational mode theory is summarized using dKnn d (5) three different routes of deriving local mode force constants ka n where a is a column vector of length N . It can be for molecules. Based on these routes, the definition of local n vib transformed into Cartesian coordinates via10,48 vibrational modes in periodic systems is worked out, as well as x the derivation of the corresponding local mode properties with an = Lan (6) a focus on the local mode force constant ka. After the n where the superscript x denotes Cartesian coordinate, and Computational Details section, seven examples with different column vector ax has the length of 3N. dimensions in periodicity (one-, two-, and three-dimensional) n The local mode force constant ka associated with internal are discussed in the Results and Discussion section. The n coordinate q is obtained from eq 7. conclusions, along with some general remarks on the n a a T −−11T calculation of local mode force constants kn, are given in the kn ==aKan nnn() dK d (7) last section. a The local mode force constant kn has also been named adiabatic force constant, where the superscript a (adiabatic) 2. METHODOLOGY means “relaxed” and the subscript n stands for the internal 10 2.1. Local Vibrational Modes for Isolated Molecules. coordinate qn leading the local vibration.
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