Inexpensive vibrational anharmonicities from estimated derivatives: Diatomic molecules

Four alternatives are compared for estimating vibrational anharmonicity constants without explicitly calculating the expensive fourth derivatives of the potential curves. In the first, semiempirical approach, fourth derivatives for 53 diatomic molecules are estimated from ab initio second and third derivatives by using the Morse model potential. Vibrational anharmonicities ω_ex_e are then computed from the third and fourth derivatives. The second approach invokes a purely empirical linear correlation between ω_ex_e and the harmonic frequencies ω_e. The third and fourth empirical approaches suppose that the effective harmonic and anharmonic force constants are proportional (with an additive constant in the fourth approach). Experimental values for ω_ex_e are compared with empirical predictions and with semiempirical estimates based upon Hartree–Fock (HF), Møller–Plesset (MP2), and local, nonlocal, and hybrid density-functional theories (DFT), using the small 6-31G* basis set. Ab initio values of ω_e and bond lengths r_e are also compared against experiment. The (U)MP2 results are the worst and include several anomalies. The other semiempirical methods yield results of comparable accuracy for ω_ex_e of hydrides, although the DFT methods are markedly better for ω_e and r_e and for ω_ex_e of nonhydrides. The empirical estimates are nearly as good as the semiempirical ones. We conclude that: (1) both empirical and semiempirical approximations are useful for predicting stretching anharmonicity constants ω_ex_e to precisions of σ≈5 cm⁻¹ for hydrides and σ≈1.5 cm⁻¹ for nonhydrides; and (2) MP2 theory is relatively unreliable for such calculations. In addition, we find the following tests to be useful when evaluating the reliability of vibrational constants calculated at the UMP2 level: (a) the calculated values of ω_e and ω_ex_e should not deviate substantially from the empirical relations; (b) harmonic frequencies and intensities calculated at the MP2 level should be smaller than those calculated at the corresponding HF level; (c) a large distance-dependence of the spin contamination, d〈S²〉/dR≳0.05 Å⁻¹, suggests that calculated constants are too large. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1315–1324, 1998     

Basis set and correlation energy dependence of geometry and harmonic frequencies of difluoroethane,CHF2CH3

Optimum geometries and harmonic frequencies calculated at the Hartree–Fock and the MP2 level are reported for the fluorohydrocarbon CHF₂CH₃; basis sets employed range from STO-3G to 6-311G**. The significantly shortened C? C distance of 1.50 Å is reproduced already with the simplest split-valence basis set; the C? F distance of 1.36 Å on the other hand needs MP2 correction at least at the double-ζ or 6-311G* level. Symmetry coordinates defined in terms of internal coordinates are in qualitative agreement with available experimental evidence. Even the best basis set yields frequencies that differ from experimental (anharmonic) values by up to 200 cm^?1 indicating the well-known necessity of including higher-order force constants if quantitative agreement with experiment is to be achieved.     

Estimating the hessian for gradient-type geometry optimizations

Optimization methods that use gradients require initial estimates of the Hessian or second derivative matrix; the more accurate the estimate, the more rapid the convergence. For geometry optimization, an approximate Hessian or force constant matrix is constructed from a simple valence force field that takes into account the inherent connectivity and flexibility of the molecule. Empirical rules are used to estimate the diagonal force constants for a set of redundant internal coordinates consisting of all stretches, bends, torsions and out-of-plane deformations involving bonded atoms. The force constants are transformed from the redundant internal coordinates to Cartesian coordinates, and then from Cartesian coordinates to the non-redundant internal coordinates used in the specification of the geometry and optimization. This method is especially suitable for cyclic molecules. Problems associated with the choice of internal coordinates for geometry optimization are also discussed.Fellow of the Alfred P. Sloan Foundation, 1981–83     

Dynamic parameters of the hypervalent O3B...O bond

The results of investigations of the common and distinctive features of various chemical bonds, including intermolecular bonds, were considered. The geometries, atom charges, and force constants of the bonds in a dependent system of coordinates were calculated for five boroxide molecular species using the MINDO/3 method. The correlations found between the force constants and the bond energies in the O₃B...O bridge were compared with the analogous relations for the OH...O bridge. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 12, pp. 2237–2241, December, 1999.     

Efficient treatment of out-of-plane bend and improper torsion interactions in MM2, MM3, and MM4 molecular mechanics calculations

Simple and very efficient formulas are presented for four-body out-of-plane bend (used in MM2 and MM3 force fields) and improper torsion (used in the MM4 force field) internal coordinates and their first and second derivatives. The use of a small set of bend and stretch intermediates allows for order of magnitude decreases in calculation time for potential energies and their first and second derivatives, which are required in molecular mechanics calculations. The formulas are eminently suitable for use in molecular simulations of systems with complicated bond networks. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1804–1811, 1997     

A new way of analyzing vibrational spectra. I. Derivation of adiabatic internal modes

A new way of analyzing measured or calculated vibrational spectra in terms of internal vibrational modes associated with the internal parameters used to describe geometry and conformation of a molecule is described. The internal modes are determined by solving the Euler–Lagrange equations for molecular fragments ϕ_n described by internal parameters ζ_n. An internal mode is localized in a molecular fragment by describing the rest of the molecule as a collection of massless points that just define molecular geometry. Alternatively, one can consider the new fragment motions as motions that are obtained after relaxing all parts of the vibrating molecule but the fragment under consideration. Because of this property, the internal modes are called adiabatic internal modes, and the associated force constants k_a, adiabatic force constants. Minimization of the kinetic energy of the vibrating fragment ϕ_n yields the adiabatic mass m_a (corresponding to 1/G_nn of Wilson's G matrix) and, by this, adiabatic frequencies ω_a. Adiabatic modes are perfectly suited to analyze and understand the vibrational spectra of a molecule in terms of internal parameter modes in the same way as one understands molecular geometry in terms of internal coordinates. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67 : 1–9, 1998     

Ab initio calculations of the ultraviolet resonance Raman spectra of uracil

An equation been derived to calculate, ab initio, the frequencies and intensities of a resonant Raman spectrum from the transform theory of resonance Raman scattering. This equation has been used to calculate the intensities of the ultraviolet resonance Raman spectra from the first π-π* excited state of uracil and 1,3-dideuterouracil. The protocol for this calculation is as follows: (1) The force constant matrix elements in Cartesian coordinate space, the vibrational frequencies, and the minimum energy ground and excited state geometries of the molecule are calculated ab initio using the molecular orbital program Gaussian 92, (2) the force constants in Cartesian coordinates are transformed into force constants in the space of a set of 3N – 6 nonredundant symmetrized internal coordinates, (3) the G matrix is constructed from the energy minimized ground state Cartesian coordinates and the GFL = LΛ eigenvalue equation is solved in internal coordinate space, (4) the elements of the L and L^?1 matrices are calculated, (5) the changes in all of the internal coordinates in going from the ground to the excited state are calculated, and (6) these results are used in combination with the transform theory of resonance Raman scattering to calculate the relative intensities of each of the 3N – 6 vibrations as a function of the exciting laser frequency. There are no adjustable parameters in this calculation, which reproduces the experimental frequencies and intensities with remarkable fidelity. This indicates that the Dushinsky rotation of the modes in the excited state of these molecules is not important and that the simplest form of the transform theory is adequate. © 1995 John Wiley & Sons, Inc.     

Vergleichbare Kraftfelder vieratomiger Formylverbindungen

Comparable force fields for HCOO^–, HFCO, HClCO and HDCO have been calculated on the basis of internal coordinates. Linear relations between (i) the carbonyl bond order and the carbonyl stretching force constant, (ii) the sum of the three in-plane bending force constants and the hydrogen out-of-plane force constantf , (iii) a combination of orbital electronegativities andf , have been obtained. The observed in-plane vibrational frequencies have been calculated with an average error of 6.3 cm^–1 or 0.4%.

     

Green's Function Determination of Force Constants-H2O as Example

Treating isotopically substituted molecule as a perturbed system, Green's function for the perturbation are constructed and related to the force field of vibration. By spectral representation, Green's function is diagonalized in the normal coordinates. Then transforming back to the Cartesian coordinates, the Cartesian force constants are generated without solving the secular equation directly. The relations between the internal force constants and the Cartesian force constants ate given and complete internal force field can be obtained. The results for H₂O are discussed.     

Peptide free‐energy profile is strongly dependent on the force field: Comparison of C96 and AMBER95

The C96 and AMBER95 force fields were compared with small model peptides Ac‐(Ala)_n‐NMe (Ac = CH₃CO, NMe = NHCH₃, n=2 and 3) in vacuo and in TIP3P water by computing the free‐energy profiles using multicanonical molecular dynamics method. The C96 force field is a modified version of the AMBER95 force field, which was adjusted to reproduce the energy difference between extended β‐ and constrained α‐helical energies for the alanine tetrapeptide, obtained by the high level ab initio MO method. The slight modification resulted in a large difference in the free energy profiles. The C96 force field prefers relatively extended conformers, whereas the AMBER95 force field favors turn conformations. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 748–762, 2000     

Perturbed structures generated using coordinates derived from compliance constants in internal vibrations for QTAIM dual functional analysis: Intrinsic dynamic nature of interactions

Perturbed structures for QTAIM dual functional analysis (QTAIM‐DFA) are proposed to generate using the coordinates corresponding to the compliance force constants in internal vibrations (CIV). In QTAIM‐DFA, total electron energy densities H_b( r _c) are plotted versus H_b( r _c) – V_b( r _c)/2 at bond critical points (BCPs) of interactions in question, where V_b( r _c) are potential energy densities at BCPs. Each plot of an interaction based on the data from both perturbed structures and fully optimized one takes the form (θ_p, κ_p), where θ_p corresponds to the tangent line of the plot and κ_p is the curvature. The θ_p values evaluated with CIV are equal to those obtained by partial optimizations with the interaction distance in question being fixed suitably, within the calculation errors. Very high applicability of CIV is demonstrated to generate the perturbed structures for QTAIM‐DFA. Dynamic nature of interactions based on (θ_p, κ_p) with CIV is called “intrinsic dynamic nature of interactions.”     

The Chemical Bond in C2

Quantum chemical calculations using the complete active space of the valence orbitals have been carried out for H_nCCH_n (n=0–3) and N₂. The quadratic force constants and the stretching potentials of H_nCCH_n have been calculated at the CASSCF/cc‐pVTZ level. The bond dissociation energies of the C?C bonds of C₂ and HC≡CH were computed using explicitly correlated CASPT2‐F12/cc‐pVTZ‐F12 wave functions. The bond dissociation energies and the force constants suggest that C₂ has a weaker C?C bond than acetylene. The analysis of the CASSCF wavefunctions in conjunction with the effective bond orders of the multiple bonds shows that there are four bonding components in C₂, while there are only three in acetylene and in N₂. The bonding components in C₂ consist of two weakly bonding σ bonds and two electron‐sharing π bonds. The bonding situation in C₂ can be described with the σ bonds in Be₂ that are enforced by two π bonds. There is no single Lewis structure that adequately depicts the bonding situation in C₂. The assignment of quadruple bonding in C₂ is misleading, because the bond is weaker than the triple bond in HC≡CH.     

Interaction potential models for bulk Zns,Zns nanoparticle,and Zns nanoparticle‐PMMA from first‐principles

An ab initio derived transferable polarizable force‐field has been developed for Zinc sulphide (ZnS) nanoparticle (NP) and ZnS NP‐PMMA nanocomposite. The structure and elastic constants of bulk ZnS using the new force‐field are within a few percent of experimental observables. The new force‐field show remarkable ability to reproduce structures and nucleation energies of nanoclusters (Zn₁S₁‐Zn₁₂S₁₂) as validated with that of the density functional theory calculations. A qualitative agreement of the radial distribution functions of Zn? O, in a ZnS nanocluster‐PMMA system, obtained using molecular mechanics molecular dynamics (MD) and ab initio MD (AIMD) simulations indicates that the ZnS–PMMA interaction through Zn? O bonding is explained satisfactorily by our force‐field. © 2015 Wiley Periodicals, Inc.     

Ab initio Study of Anharmonic Force Field and Spectroscopic Constants for Germanium Dichloride

Ab initio study of the equilibrium structure, spectroscopy constants, and anharmonic force field for several isotopomers of germanium dichloride (⁷⁰GeCl₂, ⁷²GeCl₂, and ⁷⁶GeCl₂) have been carried out at the MP2 and CCSD(T) levels of theory using cc-pVTZ basis set. The calculated geometries, rotational constants, vibration-rotation interaction constants, harmonic frequencies, anharmonic constants, quartic and sextic centrifugal distortion constants, cubic and quartic force constants are compared with experimental data. For small mass differences of the Ge isotopes, the isotopic effects for germanium dichloride are much weaker. The agreements are satisfactory for these two methods, but the deviations of CCSD(T) results are slightly larger than that of MP2, because of CCSD(T)'s inadequate treatment of electron correlation in hypervalent Cl atom.     

Theoretical study of dynamical properties on reaction path in molecular internal coordinates

The dynamical properties on reaction path (IRC) in internal coordinates have been obtained, which in. clude ω_K (frequencies orthogonal to IRC), L_K (vibrational modes), B_(KF) (coupling constants between the IRC and vibra tions orthogonal to it), B_(KL) (coupling constants between every two vibrations orthogonal to IRC). A set of theory of teac. tion path in molecular internal coordinates has been also constructed. The dynamical properties, including ω_K, B_(KF) B_(KL) of the reaction H~1O~2H~3 H~4→H~1O~2 H~3H~4 have been calculated, which explicitly explain the interaction, chang ing trend and contribution of each chemical bond (including bond angle) in the reaction.     

Normal Coordinate Analysis in Cartesian Space I. Planar Symmetric Xn Molecules

Normal coordinate analysis of X_n type molecules can be carried out in the Cartesian space as well as in the internal space. Force constants in Cartesian coordinates for aromatic compounds belonging to D_nh group are calculated. The force constants of benzene are evaluated from vibrational frequencies both in the ground state and the ¹B_1u excited state. The calculated frequencies of planar carbon vibration of annulene of any N are tabulated. The normal coordinates derived from the calculation of 10-annulene are roughly the same of naphthalene derived more elaborated by Scherer. The normal modes in 10-annulene are indeed good approximations to the ones in naphthalene. This conclusion is valid for the other aromatic compounds.     

Optimal scaling factors for CM1 and CM3 atomic charges in RM1-based aqueous simulations

Scaling factors for atomic charges derived from the RM1 semiempirical quantum mechanical wavefunction in conjunction with CM1 and CM3 charge models have been optimized by minimizing errors in absolute free energies of hydration, ΔG_hyd, for a set of 40 molecules. Monte Carlo statistical mechanics simulations and free energy perturbation theory were used to annihilate the solutes in gas and in a box of TIP4P water molecules. Lennard–Jones parameters from the optimized potentials for liquid simulations‐all atom (OPLS–AA) force field were utilized for the organic compounds. Optimal charge scaling factors have been determined as 1.11 and 1.14 for the CM1R and CM3R methods, respectively, and the corresponding unsigned average errors in ΔG_hyd relative to experiment were 2.05 and 1.89 kcal/mol. Computed errors in aniline and two derivatives were particularly large for RM1 and their removal from the data set lowered the overall errors to 1.61 and 1.75 kcal/mol for CM1R and CM3R. Comparisons are made to the AM1 method which yielded total errors in ΔG_hyd of 1.50 and 1.64 kcal/mol for CM1A*1.14 and CM3A*1.15, respectively. This work is motivated by the need for a highly efficient yet accurate quantum mechanical (QM) method to study condensed‐phase and enzymatic chemical reactions via mixed QM and molecular mechanical (QM/MM) simulations. As an initial test, the Menshutkin reaction between NH₃ and CH₃Cl in water was computed using a RM1/TIP4P‐Ew/CM3R procedure and the resultant ΔG^?, ΔG_rxn, and geometries were in reasonable accord with other computational methods; however, some potentially serious shortcomings in RM1 are discussed. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Challenges in predicting ΔrxnG in solution: Hydronium,hydroxide, and water autoionization

Standard non‐semiempirical continuum‐dielectric orbital‐based methods horribly overpredict, by 26‐50 kcal mol⁻¹, the Gibbs energy for the water autoionization reaction 2 H₂O_(l) → H₃O⁺_(aq) + OH^–_(aq). Here, we demonstrate these errors, fully investigate the reasons for these errors, and show that the use of 4 explicit solvent within the continuum (the “semicontinuum,” “cluster‐continuum,” or “hybrid” technique) can reduce the error of a standard continuum model from 50 to 2 kcal mol⁻¹. Results from pure cluster, pure continuum (several versions including semiempirical ones), and semicontinuum modeling are each presented and discussed. We recommend use of 3 waters around hydronium and 4 waters around hydroxide with standard continua whenever these ions are involved in reaction. To the possible surprise of some, time‐consuming molecular‐dynamics simulations are not needed to reproduce this problematic energy.     

Calculation of optimum geometries and force fields by the CNDO/force method

CNDO/Force calculations have been performed on a series of molecules, H₂CO, F₂CO, CF₄, CHF₃, CH₂F₂ and CH₃F. The optimum geometries and force fields are reported. It is found that the method can successfully predict the geometries of polyatomic molecules. The bending force constants and interaction force constants are, in general, comparable with experimental values both with respect to sign and magnitude. The stretching force constants have higher values than the experimental force constants. However, the trend in stretching force constants of a series of molecules is comparable with that of the corresponding experimental values.