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     

Calculation of molecular vibrations: Selective scaling factors for semiempirical force constants

The complete force constant matrices of a set of 50 aliphatic and aromatic hydrocarbons are calculated at the density functional theory B3LYP/6–31+G(d, p) and semiempirical PM3 levels of theory. After transformation from Cartesian to nonredundant internal coordinates, the errors in the semiempirical force constants are systematically analyzed. The force constants of the C(SINGLE BOND)C stretching coordinates can be easily corrected by a second-order fit. Thus, only two parameters are needed to reduce the mean error from 21.2 to 1.23%. The errors of other internal coordinates, particulary those including torsional modes, exhibit a larger diversity. The performance of the correction scheme in predicting vibrational spectra is shown for several examples including buckminsterfullerene (C₆₀). © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 2050–2059, 1997     

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.     

Polyatomic,anharmonic, vibrational–rotational analysis. Application to accurate ab initio results for formaldehyde

A computer program SURVIB is described for calculating vibrational anharmonicity constants for polyatomic molecules. The program requires as input a grid of calculated energies in the vicinity of a stationary point. This grid is fit, in a least squares sense, to a polynomial function of the internal coordinates. This analytic representation of the energy surface is employed in a normal mode analysis, and the energy is reexpanded as a polynominal function of the normal mode coordinates (expressed as vectors in the mass-weighted atomic Cartesian coordinate space). The resulting coefficients are used in a second-order perturbation theory analysis to obtain the vibrational anharmonicity constants. Also reported is an application of this program to formaldehyde employing ab initio, RHF , MP 2, MP 3, and RHF -CI calculations. The spectroscopic constants obtained for H₂CO are in good agreement with experimentally derived values recently reported by Reisner.     

Calculation of intramolecular force fields from second-derivative tensors

A practical procedure (FUERZA) to obtain internal force constants from Cartesian second derivatives (Hessians) is presented and discussed. It allows a systematic analysis of pair atomic interactions in a molecular system, and it is fully invariant to the choice of internal coordinates of the molecule. Force constants for bonds or for any pair of atoms in general are defined by means of the eigenanalysis of their pair interaction matrix. Force constants for the angles are obtained from their corresponding two-pair interaction matrices of the two bonds or distances forming the angle, and the dihedral force constants are similarly obtained using their corresponding three-pair interaction matrices. © 1996 John Wiley & Sons, Inc.     

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.     

Internal coordinate density of state from molecular dynamics simulation

The vibrational density of states (DoS), calculated from the Fourier transform of the velocity autocorrelation function, provides profound information regarding the structure and dynamic behavior of a system. However, it is often difficult to identify the exact vibrational mode associated with a specific frequency if the DoS is determined based on velocities in Cartesian coordinates. Here, the DoS is determined based on velocities in internal coordinates, calculated from Cartesian atomic velocities using a generalized Wilson's B ‐matrix. The DoS in internal coordinates allows for the correct detection of free dihedral rotations that may be mistaken as hindered rotation in Cartesian DoS. Furthermore, the pronounced enhancement of low frequency modes in Cartesian DoS for macromolecules should be attributed to the coupling of dihedral and angle motions. The internal DoS, thus deconvolutes the internal motions and provides fruitful insights to the dynamic behaviors of a system. © 2015 Wiley Periodicals, Inc.     

The exact isotope rule for symmetrically substituted molecules

The vibrational frequencies of isotopically substituted molecules are related exactly and explicitly to the parent molecules through a mixing parameter, following the method developed by Sheth (C. V. Sheth, J. Phys. B: Atom. molec. Phys. 8, 121 (1975)) using Cartesian displacement coordinates. It is shown that the explicit expressions for vibrational frequencies for symmetrical isotopic substitution of a non-linear X Y²-molecule coincide with that of DeWames and Wolfram (R. E. DeWames and T. Wolfram, J. chem. Phys. 40, 853 (1964)) derived from Green's function. The present procedure has the advantage of working with only symmetry coordinates belonging to genuine vibrations as compared to Green's function technique. Explicit expressions for vibrational frequencies for symmetrically substituted pyramidal X Y³-and tetrahedral X Y⁴-molecules are given for the first time involving properly defined mixing parameters. Using these expressions a theoretical justification of Noether's empirical rule (G. Herzberg, Infrared and Raman Spectra. Van Nostrand, New York (1945); H. D. Noether, J. chem. Phys. 11, 97 (1943)) is given for the first time in the cases considered.     

Normal Coordinate Analysis in Cartesian Space III. The Symmetric Nonplanar XYn Type Molecules

The normal vibrations of tetrahedral hydride and octahedral hexafluoride have been studied in the Cartesian coordinates. Equations relating the force constants with the vibrational frequencies are derived by using symmetry coordinates. Numerical values of the force constants and normal coordinates of these molecules are tabulated.     

Separated internal force constants for the ethylene and ethane molecules and their use for calculation of the force field for the propylene molecule

We consider the question of separation of linear combinations of force constants for ethylene and ethane. Introduction of a perturbation into the matrix of the kinematic coefficients allows us to solve the inverted vibrational problem using the matrix method of successive approximations without eliminating dependent coordinates. Such an approach makes it possible to obtain a sufficient system of equations for determining the separated internal force constants. The separated internal force constants determined for ethylene and ethane are used to calculate the force field for propylene. The calculated separated internal force constants for propylene reproduce its vibrational spectrum and the spectrum or propylene-d₆ with average deviation from experimental frequencies of 8 cm^–1. The numerical influence coefficients for stretching vibrations of the C-H bond are linearly related to the lengths of these bonds.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 22, No. 1, pp. 118–122, Janauary–February, 1986.     

Geometry optimization of metal complexes using natural internal coordinates: Representation of skeletal degrees of freedom

The geometry optimization using natural internal coordinates was applied for transition metal complexes. The original definitions were extended here for the skeletal degrees of freedom which are related to the translational and rotational displacements of the ηⁿ-bonded ligands. We suggest definitions for skeletal coordinates of ηⁿ-bonded small unsaturated rings and chains. The performance of geometry optimizations using the suggested coordinates were tested on various conformers of 14 complexes. Consideration was given to alternative representations of the skeletal internal coordinates, and the performance of optimization is compared. Using the skeletal internal coordinates presented here, most transition metal complexes were optimized between 10 and 20 geometry optimization cycles in spite of the usually poor starting geometry and crude approximation for the Hessian. We also optimized the geometry of some complexes in Cartesian coordinates using the Hessian from a parametrized redundant force field. We found that it took between two and three times as many iterations to reach convergence in Cartesian coordinates than using natural internal coordinates. © 1997 by John Wiley & Sons, Inc.     

Normal Coordinate Analysis in Cartesian Space II. Planar Symmetric XYn Molecules

Force constants of planar symmetric XY_n type molecules in the Cartesian space have been derived in terms of the vibrational frequencies. The force constants and normal coordinates of some planar XY₃ molecules with available isotoptic data are worked out. Further study of infrared absorption spectra of square planar XY₄ complex ions will be needed for the present normal coordinate analysis.     

Scaled Quantum Mechanical Force Fields for the Isoelectronic Molecules CF3X (X = SiH3, PH2, SH,Cl )

The vibrational properties of the CF₃X (X = SiH₃, PH₂, SH, Cl) series of molecules were studied by means of density functional theory (DFT) techniques. After obtaining the optimized geometrical parameters and conformations, the frequencies corresponding to the normal modes of vibration and the associated force constants were calculated. The original force fields in Cartesian coordinates were transformed to local symmetry coordinates and subsequently scaled to reproduce the experimental frequencies. Some trends observed in the experimental data could be explained on the basis of the calculated atomic charges.     

Canonical force field theory in terms of curvilinear internal coordinates: Application to the methane molecule

This Letter generalizes the Kuczera’s theory of the vibrational canonical force field in order to obtain an unambiguous and uniquely defined expression for the anharmonic force field in terms of curvilinear internal coordinates. By using this generalization it has been shown mathematically that the canonical force constants could be transferred between molecules with the same geometrical structure. As illustration, the theory is used to obtain the expressions for the quadratic and cubic canonical force constants of the methane molecule.     

A general quadratic force field for the pyridazine molecule: out-of-plane B1 normal modes

On the basis of Raman and infrared data for pyridazine-d_o′, −3, 6-d₂, and −d₄ we are carrying out a normal coordinate treatment using a general quadratic force field (GQFF) on a non-redundant basis. The BB^* matrix has been set up and diagonalized to obtain the linear redundancy relations. Afterwards, using the Schmidt's process, we constructed the independent coordinates used in our calculations. To date we have completed the refinement of the calculations of the force constants for the B₁ specie.     

Chiral force constants: Recommendations for the presentation of internal coordinate force constants

Some force constants associated with the internal coordinates that sense handedness or chirality can have opposite signs in the enantiomers of chiral molecules. Examples of such force constants include interaction force constants between a torsional and stretching or bending internal coordinates. The sign reversal for these force constants in the enantiomers of chiral molecules or in opposite-handed molecular segments is best recognized by labeling them as chiral force constants. Recognition of chiral force constants suggests that certain guidelines are to be followed in the presentation of internal coordinate force constants. © 1993 by John Wiley & Sons, Inc. © John Wiley & Sons, Inc.     

Vibrational spectra and force constants of Ch3HgI and CD3HgI

The infrared and Raman spectra of CH₃HgI and CD₃HgI were studied in the solid state. All the fundamental wavenumbers are assigned. A general harmonic force field was used as the basis, and the force constants were modified by means of the Jacobian matrix. The force constants fit the observed wavenumbers better than 1 %. The normal coordinates are also given.     

Structural,elastic, electronic and optical properties of the newly synthesized monoclinic Zintl phase BaIn2P2

The present study explores the structural, elastic, electronic and optical properties of the newly synthesized monoclinic Zintl phase BaIn₂P₂ using a pseudopotential plane-wave method in the framework of density functional theory within the generalized gradient approximation. The calculated lattice constants and internal coordinates are in very good agreement with the experimental findings. Independent single-crystal elastic constants as well as numerical estimations of the bulk modulus, the shear modulus, Young's modulus, Poisson's ratio, Pugh's indicator of brittle/ductile behaviour and the Debye temperature for the corresponding polycrystalline phase were obtained. The elastic anisotropy of BaIn₂P₂ was investigated using three different indexes. The calculated electronic band structure and the total and site-projected l-decomposed densities of states reveal that this compound is a direct narrow-band-gap semiconductor. Under the influence of hydrostatic pressure, the direct D–D band gap transforms into an indirect B-D band gap at 4.08 GPa, then into a B–Γ band gap at 10.56 GPa. Optical macroscopic constants, namely, the dielectric function, refractive index, extinction coefficient, reflectivity coefficient, absorption coefficient and energy-loss function, for polarized incident radiation along the [100], [010] and [001] directions were investigated.     

Application of many-body green's functions to the calculation of molecular ionization potentials

We present a method for calculating the one-particle Green's function for molecules. The scheme is essentially that proposed by Schneider and Taylor [1]. The Green's function is obtained through the Dyson equation. Closed expressions result by using the functional derivation technique to truncate an infinite set of coupled equations. A physical interpretation of the approximation is given and a connection with the equations-of-motion method is pointed out. In a numerical application the ionization potentials are obtained for the molecules N₂, H₂O, NH₃ and CH₄.