Linear atomic density

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. 

Case J = 0. Since there is only one fragment with j fixed by = Bj for the linear-atom and sphere-atom systems, their rotational density of states is times... 

In the case of k, the structures and index values in Table 7 for = 6 reveal that the encoded information relates to the degree of star graph-likeness and linear graph-likeness. Stated in more general terms, encodes information about the spatial density of atoms in a molecule. 

The threshold condition for collective emission (superradiance) or maser action (gain in the medium exceeding the losses) is much lower for Rydberg atoms than it is for the same number density of atoms in low n states. To obtain an order of magnitude estimate for the point at which collective fects occur we estimate the amplitude of the electric dipole field E radiated by an atom at a distance corresponding to a neighbouring atom. That is, if we let L represent a linear dimension in the sample which contains N atoms,... 

Regular (mass) surface density (in g cm ) gives essentially the same accuracy as a real number density (in atoms/cm ). This is why the stopping power expressions in Chap. 8, Vol. 1, contain x as a measure of sample thickness in surface density units (e.g., in g/cm ) rather than a linear distance (in cm) (p. 1611). 

The six electrons go into the lowest three molecular orbitals i, 5, e The molecular orbital with the lowest energy, 1, represents a charge distribution in which the electrons are distributed symmetrically about the ring. The second level, which is degenerate, has associated with it an unsymmetrical charge distribution. 5, for example, represents a charge distribution in which the electron density on atoms a and d is four times greater than the density on any of the other atoms. This does not mean that the resultant electron distribution is unsymmetrical. If we take the normalized linear combinations... 

Dislocations do not move with the same degree of ease on all erystallographie planes of atoms and in all crystallographic directions. Typically, there is a preferred plane, and in that plane there are specific directions along which dislocation motion occurs. This plane is called the slip plane it follows that the direction of movement is called the slip direction. This combination of the slip plane and the slip direction is termed the slip system. The slip system depends on the crystal structure of the metal and is such that the atomic distortion that accompanies the motion of a dislocation is a minimum. For a particular crystal structure, the slip plane is the plane that has the densest atomic packing—that is, has the greatest planar density. The slip direction corresponds to the direction in this plane that is most closely packed with atoms—that is, has the highest linear density. Planar and linear atomic densities were discussed in Section 3.11. 

The embedding function F,(/o) is universal , i.e., it does not depend upon the source which creates the electron density. The atomic densities are usually calculated from the tabulated Hartree-Fock wavefunctions [142] and for transition metals taken as a linear combination of the density created by the valence s- and d-electrons, i.e.,... 

The plastic deformation of crystalline solids proceeds by a process of slip and/or twinning on certain crystal planes and in certain crystal directions. In metals the slip planes (denoted by ) are usually those having the highest atomic density and they are the most widely spaced. The slip directions (denoted by < >) in the plane are those having the highest linear atomic density. A particular combination of slip plane and slip direction is referred to as a slip system. 

A simple measure of the election density distribution over the participating atoms is the Mulliken population [60]. For linear Li—H—Li the alpha spin is... 

Note that the sums of the squares of the coefficients in a given MO must equal 1 (e.g., 0.3717 + 0.6015 + 0.3717 + 0.6015 = 1.0 for Pi) because each of the AOs represents a probability distribution of finding the electron at a given point in space. The total probability of finding an electron in all space for an MO must be unity, exactly as for its constituent AOs. We now can see that the LCAO approximation is only one of many possibilities to describe the electron density (= probability) for MOs. We do not have to express the electron density as a linear combination of the electron densities of AOs centered at the atoms. We could also... 

Highest occupied molecular orbital Intermediate neglect of differential overlap Linear combination of atomic orbitals Local density approximation Local spin density functional theory Lowest unoccupied molecular orbital Many-body perturbation theory Modified INDO version 3 Modified neglect of diatomic overlap Molecular orbital Moller-Plesset... 

Ire boundary element method of Kashin is similar in spirit to the polarisable continuum model, lut the surface of the cavity is taken to be the molecular surface of the solute [Kashin and lamboodiri 1987 Kashin 1990]. This cavity surface is divided into small boimdary elements, he solute is modelled as a set of atoms with point polarisabilities. The electric field induces 1 dipole proportional to its polarisability. The electric field at an atom has contributions from lipoles on other atoms in the molecule, from polarisation charges on the boundary, and where appropriate) from the charges of electrolytes in the solution. The charge density is issumed to be constant within each boundary element but is not reduced to a single )oint as in the PCM model. A set of linear equations can be set up to describe the electrostatic nteractions within the system. The solutions to these equations give the boundary element harge distribution and the induced dipoles, from which thermodynamic quantities can be letermined. 

A basis set is a set of functions used to describe the shape of the orbitals in an atom. Molecular orbitals and entire wave functions are created by taking linear combinations of basis functions and angular functions. Most semiempirical methods use a predehned basis set. When ah initio or density functional theory calculations are done, a basis set must be specihed. Although it is possible to create a basis set from scratch, most calculations are done using existing basis sets. The type of calculation performed and basis set chosen are the two biggest factors in determining the accuracy of results. This chapter discusses these standard basis sets and how to choose an appropriate one. 

LCAO (linear combination of atomic orbitals) refers to construction of a wave function from atomic basis functions LDA (local density approximation) approximation used in some of the more approximate DFT methods... 

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