Hessian Matrix (Synonymous With Force Constant Matrix)

2016 ◽  
Author(s):  
Vladimir I. Minkin
Keyword(s):  
Force Constant ◽  
Hessian Matrix ◽  
Constant Matrix ◽  
Force Constant Matrix

scholarly journals Ab-initio density-functional lattice-dynamics studies of ice

2003 ◽  
Vol 81 (1-2) ◽  
pp. 115-122 ◽  
Author(s):  
A S Cote ◽  
I Morrison ◽  
X Cui ◽  
S Jenkins ◽  
D K Ross
Keyword(s):  
Ab Initio ◽  
Force Constant ◽  
Density Functional ◽  
Phonon Dispersion ◽  
Scattering Data ◽  
Full Potential ◽  
Dynamical Matrix ◽  
Constant Matrix ◽  
Force Constant Matrix ◽  
Dynamical Properties

We present the results of first-principles computational studies of the dynamical properties of hexagonal ice using both the ab-initio pseudopotential method and the full-potential augmented plane-wave method. Properties obtained using both the generalized gradient approximation (GGA) and the meta-GGA in density-functional theory are compared. The lattice-dynamical properties of the structures are obtained using a finite-difference evaluation of the dynamical matrix and force-constant matrix from atomic forces. Phonon dispersion is evaluated by the direct determination of the force-constant matrix in supercells derived from the primitive molecule unit cells with the assumption that force constants are zero beyond the second molecular nearest neighbors. The k-dependent phonon frequencies are then obtained from the force-constant matrix and dispersion relations, and the Brillouin-zone integrated density of states is evaluated. The importance of phonon dispersion in the various regions of the phonon spectra is then assessed and compared to existing neutron-scattering data. Frozen-phonon calculations are used to compare phonon frequencies evaluated in both the GGA and meta-GGA. PACS Nos.: 61.12Ex, 63.20Dj

Force constants and lattice frequencies

1971 ◽  
Vol 320 (1543) ◽  
pp. 505-526 ◽  
Keyword(s):  
Force Constant ◽  
Force Constants ◽  
Frequency Data ◽  
Transformation Theory ◽  
Continuous Transformation ◽  
Constant Matrix ◽  
Force Constant Matrix ◽  
Usual Procedure ◽  
Wide Range ◽  
Small Force

It is shown that the force constants of a solid cannot be deduced from the lattice frequencies alone. The fact that a set of force constants agrees with the frequencies exactly is no guarantee that the force constants are even approximately correct. Detailed eigenvector measurements would resolve the uncertainty. A solid has many more force constants than frequencies. The usual procedure gets round this by retaining only the largest force constants and then using at least as many frequencies as the number of retained force constants to calculate the latter. This procedure is incorrect, because neglecting the small force constants limits the accuracy to which the frequency data can be used to calculate the retained constants. In fact, the number of data that can be extracted from the frequency measurements is always less than the number of force constants one wants to calculate. Therefore, even if all the lattice frequencies were known exactly they could still be satisfied with a very wide range of very different sets of force constants. A large proportion of these sets cannot be rejected on the basis of physical criteria alone. Using the methods of continuous transformation theory, for diamond all the ways are constructed in which the force constant matrix can be changed continuously without altering the agreement with the frequencies. A numerical example is included.

A Comparative Study of the BJH- and MCYL-Type Potentials Applied to the Gaseous Water Dimer

1991 ◽  
Vol 46 (5) ◽  
pp. 426-432
Author(s):  
Zdenek Slanina
Keyword(s):  
Gas Phase ◽  
Minimum Energy ◽  
Water Dimer ◽  
Energy Structure ◽  
Water Molecules ◽  
Constant Matrix ◽  
Force Constant Matrix ◽  
Molecular Force ◽  
Phase Water ◽  
Derivatives Of

AbstractVarious refined potentials describing the intra- and inter-molecular force fields of water molecules arc used to calculate the properties of the gas-phase water dimer. The intra-molecular parts have been taken from spectroscopic or quantum-chemical sources. The minimum energy structure was found iteratively using the first derivatives of the potential; the force-constant matrix was constructed by numerical difierentation. A quite close agreement between the Bopp-Jancso-Heinzinger and the Matsuoka-Clementi-Yoshimine-Lie potentials is found. The treatment is applied to seven observed water-dimer isotopomeric isomerizations

Calculationof Equilibrium Isotope Effects in a Conformationally Mobile Carbocation

1989 ◽  
Vol 44 (5) ◽  
pp. 480-484 ◽  
Author(s):  
Martin Saunders ◽  
Gary W. Cline ◽  
Max Wolfsberg
Keyword(s):  
Theoretical Calculations ◽  
Isotope Effects ◽  
Stable Solution ◽  
Constant Matrix ◽  
Force Constant Matrix ◽  
Hydride Shift ◽  
Rapid Equilibrium ◽  
Theory And Experiment ◽  
Good Agreement ◽  
C Nmr

The rapid, degenerate 1,2-hydride shift in 2,3-dimethyl-2-butyl cation in stable solution in SbF5/SO2C1F was perturbed by deuterium and 13C leading to splittings observed by 13C NMR spectroscopy over a range of temperatures. Accurate values for equilibrium isotope effects were obtained from these data. Theoretical calculations of the equilibrium isotope effects were performed using the Gaussian-86 program to obtain an optimized geometry and the Cartesian force constant matrix, followed by the program QUIVER which applies the Bigeleisen-Mayer method. When all of the conformers, which are in rapid equilibrium, were considered specifically, quite good agreement between theory and experiment was obtained.

Sign in / Sign up

Export Citation Format

Share Document