Interatomic force constants

The real space interatomic force constants can be derived from the total energy E of the system... 

Most of the calculations have been done for Cu since it has the least number of electrons of the metals of interest. The clusters represent the Cu(100) surface and the positions of the metal atoms are fixed by bulk fee geometry. The adsorption site metal atom is usually treated with all its electrons while the rest are treated with one 4s electron and a pseudopotential for the core electrons. Higher z metals can be studied by using pseudopotentials for all the metals in the cluster. The adsorbed molecule is treated with all its electrons and the equilibrium positions are determined by minimizing the SCF energy. The positions of the adsorbate atoms are varied around the equilibrium position and SCF energies at several points are fitted to a potential surface to obtain the interatomic force constants and the vibrational frequency. 

INTERATOMIC FORCE CONSTANTS IN PERIODIC SOLIDS FROM DENSITY FUNCTIONAL PERTURBATION THEORY... 

Interatomic Force Constants (IFCs) are the proportionality coefficients between the displacements of atoms from their equilibrium positions and the forces they induce on other atoms (or themselves). Their knowledge allows to build vibrational eigenfrequencies and eigenvectors of solids. This paper describes IFCs for different solids (SiC>2-quartz, SiC>2-stishovite, BaTiC>3, Si) obtained within the Local-Density Approximation to Density-Functional Theory. An efficient variation-perturbation approach has been used to extract the linear response of wavefunctions and density to atomic displacements. In mixed ionic-covalent solids, like SiC>2 or BaTiC>3, the careful treatment of the long-range IFCs is mandatory for a correct description of the eigenfrequencies. 

In the present paper, a particular technique for the computation of the Interatomic Force Constants of periodic solids in the framework of Density-Functional Theory (DFT) will be described, as well as some of its applications. 

In the description of the technique, the particular aspects that make it different of other schemes aimed at the computation of IFCs in solids or molecules [1-9] will be emphasized. These aspects are connected to the central use of a variational principle in order to find the changes in the wavefunctions due to atomic displacements, on one hand, and to find the change in electronic energy due to the changes in wavefunctions, on the other hand. Some technical details, related with the presence of relatively long-ranged interatomic force constants, caused by... 

Rk is the position of atom k, while a and P label directions in the three-dimensional space. The interatomic force constants appear in the second-order term of this expansion,... 

Interatomic Force Constants in ionic and covalent solids. 

Fig. 1. Longitudinal interatomic force constants in Si, as a function of the distance between the pair of atoms.

Thus, in comparing these four materials, one will monitor the balance between ionic and covalent effects, as well as the anisotropy of each local site. Let us now focus on the potential well that each atom feels, and on the decomposition of the longitudinal interatomic force constants in its dipole-dipole component and its short range part. A more complete account of the results obtained thanks to the application of the above-described formalism can be found in Refs.[15-22], together with comparison with experiments. At the level of vibrational frequencies, the LDA is consistently within 3-5% of the experimental data for all these materials. 

Fig. 2. (lower part of the figure) Oxygen-silicon longitudinal interatomic force constants in quartz, as a function of the distance between the pair of atoms. The full line is a guide for the eyes, and represents a decay proportional to the cube of the distance. The arrow indicates values that are larger than the range allowed by the coordinates. 

Fig. 3. The analysis of oxygen-silicon longitudinal interatomic force constants in stishovite. See the caption of Fig. 2.

X. Gonze and C. Lee, "Dynamical matrices, Bom effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory," Phys. Rev. B 55 (1997), 10355-10368. 

X. Gonze, J.-C. Charlier, D.C. Allan and M.P. Teter, "Interatomic force constants from first-principles the case of a-quartz," Phys. Rev. B 50 (1994), 13035-13038. 

Ph. Ghosez, X. Gonze and J.-P. Michenaud, "Ab Initio phonon dispersion curves and interatomic force constants of barium titanate," accepted for publication in Ferroelectrics. 

Figure 2.5-7 Interatomic force constants for different non-bonded pairs of atoms, = C6H5.

Newton, M. D., M. O Keeffe, and G. V. Gibbs (1980). Ab initio calculation of interatomic force constants in HjSijO, and the bulk modulus of a-quartz and a-cristobalite. Phys. Chem. Mineral. 6, 305-12. 

Collagen [77] is the principal protein constituent of a wide range of mammalian coimective tissue. It is a fibrous protein and the interest is interpreting its mechanical properties in terms of its chemical structure. Properties such as elastic moduli and stress-strain curves depend on the interatomic force constants so vibrational spectroscopy is a necessary... 

X. Gonze, D. C. Allan, and M. P. Teter, Phys. Rev. Lett., 68, 3603 (1992). Interatomic Force-Constants from First Principles - The Case of Alpha-Quartz. 

We have explained in Section 5.2 why the individual interatomic force constants (lx k ) cannot be determined on supercells of reasonable size the force constants fall off with distance rather slowly so that several displaced atoms contribute to forces "detected" on other sites (Fig. 5.2.1). There is, however, a situation where decoupling of the contributions... 

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