Robert Kalescky

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.

Identification of the strongest bonds in chemistry.

Increasing the effective electronegativity of two atoms forming a triple bond can increase the strength of the latter. The strongest bonds found in chemistry involve protonated species of hydrogen cyanide, carbon monoxide, and dinitrogen. CCSD(T)/CBS (complete basis set) and G4 calculations reveal that bond dissociation energies are misleading strength descriptors. The strength of the bond is assessed via the local stretching force constants, which suggest relative bond strength orders (RBSO) between 2.9 and 3.4 for heavy atom bonding (relative to the CO bond strength in methanol (RBSO = 1) and formaldehyde (RBSO = 2)) in [HCNH](+)((1)Σ(+)), [HCO](+)((1)Σ(+)), [HNN](+)((1)Σ(+)), and [HNNH](2+)((1)Σg(+)). The increase in strength is caused by protonation, which increases the electronegativity of the heavy atom and thereby decreases the energy of the bonding AB orbitals (A, B: C, N, O). A similar effect can be achieved by ionization of a nonbonding or antibonding electron in CO or NO. The strongest bond with a RBSO value of 3.38 is found for [HNNH](2+) using scaled CCSD(T)/CBS frequencies determined for CCSD(T)/CBS geometries. Less strong is the NN bond in [FNNH](2+) and [FNNF](2+).

Description of aromaticity with the help of vibrational spectroscopy: anthracene and phenanthrene.

A new approach is presented to determine π-delocalization and the degree of aromaticity utilizing measured vibrational frequencies. For this purpose, a perturbation approach is used to derive vibrational force constants from experimental frequencies and calculated normal mode vectors. The latter are used to determine the local counterparts of the vibrational modes. Next, relative bond strength orders (RBSO) are obtained from the local stretching force constants, which provide reliable descriptors of CC and CH bond strengths. Finally, the RBSO values for CC bonds are used to establish a modified harmonic oscillator model and an aromatic delocalization index AI, which is split into a bond weakening (strengthening) and bond alternation part. In this way, benzene, naphthalene, anthracene, and phenanthrene are described with the help of vibrational spectroscopy as aromatic systems with a slight tendency of peripheral π-delocalization. The 6.8 kcal/mol larger stability of phenanthrene relative to anthracene predominantly (84%) results from its higher resonance energy, which is a direct consequence of the topology of ring annelation. Previous attempts to explain the higher stability of phenanthrene via a maximum electron density path between the bay H atoms are misleading in view of the properties of the electron density distribution in the bay region.

Quantitative Assessment of the Multiplicity of Carbon−Halogen Bonds: Carbenium and Halonium Ions with F, Cl, Br, and I

CX (X = F, Cl, Br, I) and CE bonding (E = O, S, Se, Te) was investigated for a test set of 168 molecules using the local CX and CE stretching force constants k(a) calculated at the M06-2X/cc-pVTZ level of theory. The stretching force constants were used to derive a relative bond strength order (RBSO) parameter n. As alternative bond strength descriptors, bond dissociation energies (BDE) were calculated at the G3 level or at the two-component NESC (normalized elimination of the small component)/CCSD(T) level of theory for molecules with X = Br, I or E = Se, Te. RBSO values reveal that both bond lengths and BDE values are less useful when a quantification of the bond strength is needed. CX double bonds can be realized for Br- or I-substituted carbenium ions where as suitable reference the double bond of the corresponding formaldehyde homologue is used. A triple bond cannot be realized in this way as the diatomic CX(+) ions with a limited π-donor capacity for X are just double-bonded. The stability of halonium ions increases with the atomic number of X, which is reflected by a strengthening of the fractional (electron-deficient) CX bonds. An additional stability increase of up to 25 kcal/mol (X = I) is obtained when the X(+) ion can form a bridged halonium ion with ethene such that a more efficient 2-electron-3-center bonding situation is created.

Direct Measure of Metal–Ligand Bonding Replacing the Tolman Electronic Parameter

The Tolman electronic parameter (TEP) derived from the A1-symmetrical CO stretching frequency of nickel-tricarbonyl complexes L-Ni(CO)3 with varying ligands L is misleading as (i) it is not based on a mode decoupled CO stretching frequency and (ii) a generally applicable and quantitatively correct or at least qualitatively reasonable relationship between the TEP and the metal-ligand bond strength does not exist. This is shown for a set of 181 nickel-tricarbonyl complexes using both experimental and calculated TEP values. Even the use of mode-mode decoupled CO stretching frequencies (L(ocal)TEPs) does not lead to a reliable description of the metal-ligand bond strength. This is obtained by introducing a new electronic parameter that is directly based on the metal-ligand local stretching force constant. For the test set of 181 nickel complexes, a direct metal-ligand electronic parameter (MLEP) in the form of a bond strength order is derived, which reveals that phosphines and related ligands (amines, arsines, stibines, bismuthines) are bonded to Ni both by σ-donation and π-back-donation. The strongest Ni-L bonds are identified for carbenes and cationic ligands. The new MLEP quantitatively assesses electronic and steric factors.

Local vibrational modes of the formic acid dimer – the strength of the double hydrogen bond

The 24 normal and 24 local vibrational modes of the formic acid dimer formed by two trans formic acid monomers to a ring (TT1) are analysed utilising preferentially experimental frequencies, but also CCSD(T)/CBS and ωB97X-D harmonic vibrational frequencies. The local hydrogen bond (HB) stretching frequencies are at 676 cm−1 and by this 482 and 412 cm−1 higher compared to the measured symmetric and asymmetric HB stretching frequencies at 264 and 194 cm−1. The adiabatic connection scheme between local and normal vibrational modes reveals that the lowering is due to the topology of dimer TT1, mass coupling, and avoided crossings involving the H⋅⋅⋅OC bending modes. The HB local mode stretching force constant is related to the strength of the HB whereas the normal mode stretching force constant and frequency lead to an erroneous underestimation of the HB strength. The HB in TT1 is stabilised by electron delocalisation in the O=C–O units fostered by forming a ring via double HBs. This implies that the CO apart from the OH local stretching frequencies reflect the strength of the HB via their red or blue shifts relative to their corresponding values in trans formic acid.

Relating normal vibrational modes to local vibrational modes: benzene and naphthalene.

Local vibrational modes can be directly derived from normal vibrational modes using the method of Konkoli and Cremer (Int J Quant Chem 67:29, 1998). This implies the calculation of the harmonic force constant matrix Fq (expressed in internal coordinates q) from the corresponding Cartesian force constant matrix fx with the help of the transformation matrix U = WB†(BWB†)−1 (B: Wilson’s B-matrix). It is proven that the local vibrational modes are independent of the choice of the matrix W. However, the choice W = M−1 (M: mass matrix) has numerical advantages with regard to the choice W = I (I: identity matrix), where the latter is frequently used in spectroscopy. The local vibrational modes can be related to the normal vibrational modes in the form of an adiabatic connection scheme (ACS) after rewriting the Wilson equation with the help of the compliance matrix. The ACSs of benzene and naphthalene based on experimental vibrational frequencies are discussed as nontrivial examples. It is demonstrated that the local-mode stretching force constants provide a quantitative measure for the C–H and C–C bond strength.

New approach to Tolman's electronic parameter based on local vibrational modes.

Tolmans electronic parameter (TEP) derived from the A1-symmetrical CO stretching frequency of nickel-phosphine-tricarbonyl complexes, R3PNi(CO)3, is brought to a new, improved level by replacing normal with local vibrational frequencies. CO normal vibrational frequencies are always flawed by mode-mode coupling especially with metal-carbon stretching modes, which leads to coupling frequencies as large as 100 cm(-1) and can become even larger when the transition metal and the number of ligands is changed. Local TEP (LTEP) values, being based on local CO stretching force constants rather than normal mode frequencies, no longer suffer from mode coupling and mass effects. For 42 nickel complexes of the type LNi(CO)3, it is shown that LTEP values provide a different ordering of ligand electronic effects as previously suggested by TEP and CEP values. The general applicability of the LTEP concept is demonstrated.

Accurate determination of the binding energy of the formic acid dimer: The importance of geometry relaxation

The formic acid dimer in its C2h-symmetrical cyclic form is stabilized by two equivalent H-bonds. The currently accepted interaction energy is 18.75 kcal/mol whereas the experimental binding energy D0 value is only 14.22 ±0.12 kcal/mol [F. Kollipost, R. W. Larsen, A. V. Domanskaya, M. Nörenberg, and M. A. Suhm, J. Chem. Phys. 136, 151101 (2012)]. Calculation of the binding energies De and D0 at the CCSD(T) (Coupled Cluster with Single and Double excitations and perturbative Triple excitations)/CBS (Complete Basis Set) level of theory, utilizing CCSD(T)/CBS geometries and the frequencies of the dimer and monomer, reveals that there is a 3.2 kcal/mol difference between interaction energy and binding energy De, which results from (i) not relaxing the geometry of the monomers upon dissociation of the dimer and (ii) approximating CCSD(T) correlation effects with MP2. The most accurate CCSD(T)/CBS values obtained in this work are De = 15.55 and D0 = 14.32 kcal/mol where the latter binding energy differs from the experimental value by 0.1 kcal/mol. The necessity of employing augmented VQZ and VPZ calculations and relaxing monomer geometries of H-bonded complexes upon dissociation to obtain reliable binding energies is emphasized.

Rigid Residue Scan Simulations Systematically Reveal Residue Entropic Roles in Protein Allostery

Intra-protein information is transmitted over distances via allosteric processes. This ubiquitous protein process allows for protein function changes due to ligand binding events. Understanding protein allostery is essential to understanding protein functions. In this study, allostery in the second PDZ domain (PDZ2) in the human PTP1E protein is examined as model system to advance a recently developed rigid residue scan method combining with configurational entropy calculation and principal component analysis. The contributions from individual residues to whole-protein dynamics and allostery were systematically assessed via rigid body simulations of both unbound and ligand-bound states of the protein. The entropic contributions of individual residues to whole-protein dynamics were evaluated based on covariance-based correlation analysis of all simulations. The changes of overall protein entropy when individual residues being held rigid support that the rigidity/flexibility equilibrium in protein structure is governed by the La Châtelier’s principle of chemical equilibrium. Key residues of PDZ2 allostery were identified with good agreement with NMR studies of the same protein bound to the same peptide. On the other hand, the change of entropic contribution from each residue upon perturbation revealed intrinsic differences among all the residues. The quasi-harmonic and principal component analyses of simulations without rigid residue perturbation showed a coherent allosteric mode from unbound and bound states, respectively. The projection of simulations with rigid residue perturbation onto coherent allosteric modes demonstrated the intrinsic shifting of ensemble distributions supporting the population-shift theory of protein allostery. Overall, the study presented here provides a robust and systematic approach to estimate the contribution of individual residue internal motion to overall protein dynamics and allostery.