Periodic solids

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Bloch s theorem states that in a periodic solid each electronic wave function can be expressed as the product of a wave-like component (with wave vector k) and a cell-periodic component/ (r) ... 

Since plane waves are delocalised and of infinite spatial extent, it is natural to perform these calculations in a periodic environment and periodic boundary conditions can be used to enforce this periodicity. Periodic boundary conditions for an isolated molecule are shown schematically in Fig. 8. The molecular problem then becomes formally equivalent to an electronic structure calculation for a periodic solid consisting of one molecule per unit cell. In the limit of large separation between molecules, the molecular electronic structure of the isolated gas phase molecule is obtained accurately. 

Poly(2,5-pyridyl) commonly know as poly(pyridine) has been the subject of considerable research effort as it luminesces in the blue region of the spectrum and may have uses in light emitting diodes (LEDs). Vaschetto and co-workers [103] reported a series of calculations on the molecule and its oligomers. The calculations included both the B3LYP and B3P88 density functions, Hartree-Fock calculations and a periodic solid-state DFT calculation using linear muffin tintype orbitals (LMTO). 

The fourth term on the right-hand side of eq. (11.3) is the electrostatic interaction (Coulomb s law) between pairs of charged atoms i and j, separated by distance r j. Since electrostatic interactions fall off slowly with r (only as r-1) they are referred to as long-range and, for an infinite system such as a periodic solid, special techniques, such as the Ewald method, are required to sum up all the electrostatic interactions (cf. Section 7.1) (see e.g. Leach, Jensen (Further reading)). The... 

The boundary conditions too were known. It would not be as easy as handling an infinite periodic solid, but a number of us set to work. The special demand of chemistry was to quantify very small molecular changes. Successes came slowly, but with the development of computers and a lot of careful, clever work, by the 90s the quantitative problem was essentially solved. The emergent hero of the chemical community was John Pople, whose systematic strategy and timely method developments were decisive. The methods of what is termed ab initio quantum chemistry became available and used everywhere. 

An alternative approach involves integrating out the elastic degrees of freedom located above the top layer in the simulation.76 The elimination of the degrees of freedom can be done within the context of Kubo theory, or more precisely the Zwanzig formalism, which leads to effective (potentially time-dependent) interactions between the atoms in the top layer.77-80 These effective interactions include those mediated by the degrees of freedom that have been integrated out. For periodic solids, a description in reciprocal space decouples different wave vectors q at least as far as the static properties are concerned. This description in turn implies that the computational effort also remains in the order of L2 InL, provided that use is made of the fast Fourier transform for the transformation between real and reciprocal space. The description is exact for purely harmonic solids, so that one can mimic the static contact mechanics between a purely elastic lattice and a substrate with one single layer only.81... 

One area that takes advantage of many of the above formalisms is the application of HF theory to periodic solids. Periodic HF theory has been most extensively developed within the context of the crystal code (Dovesi et al. 2000) where it is available in RHF, UHF, and ROHF forms. Such calculations can be particularly useful for elucidating band structure in solids, assessing defect effects, etc. 

Fig. 28. X-ray diffraction rocking curve of (Ga.Mn)As/GaAs superlaltices with 20 periods (solid line). The nominal thicknesses of the GaAs and (Ga,Mn)As layers and the Mn composition x are 11.4 nm, 12.1 nm, and 0.0S4, respectively. The dashed line shows the simulated rocking curve. The fit to the experimental curve is obtained with GaAs and (Ga,Mn)As thicknesses and x of 11.14 nm, 11.79 nm, and 0.056, respectively (Shen...

Methyl 4,6-O-benzyIidene-a-D-aItropyranoside. Triturate 4.0 g (0.015 mol) of the foregoing anhydro derivative in a mortar with a solution of 5 g of potassium hydroxide dissolved in 140 ml of water. Transfer the suspension to a round-bottomed flask and heat the mixture under reflux until all the solid has dissolved (about 28 hours). During this period solid material tends to creep up the inside of the flask surface shake periodically to re-suspend material. Remove the trace of insoluble matter which remains and neutralise the cooled filtrate with carbon dioxide (use phenophthalein as an indicator). Extract the solution with five 25 ml portions of dichloromethane, wash the combined extracts with a little cold water, dry over anhydrous sodium sulphate and remove the solvent under reduced pressure (rotary evaporator). Crystallise the syrup by scratching a small portion on a watch glass with ether stir the bulk syrup with ether and the seed crystals. Filter off and recrystallise the product from a small quantity of methanol to obtain 3.5 g (83%) of methyl 4,6-0-benzylidene-oc-D-altropyranoside, m.p. 174 °C, [a]D°+115° (c2 in CHCI3). 

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

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. 

The traditional approach to obtain the electronic structure of a periodic solid with ab initio pseudopotentials has been to solve the Kohn-Sham (KS)... 

It has just been stated that a band stracture diagram is a plot of the energies of the various bands in a periodic solid versus the value of the reciprocal-space wave vector k. It is now necessary to discuss the concept of the reciprocal-space lattice and its relation to the real-space lattice. The crystal structure of a solid is ordinarily presented in terms of the real-space lattice comprised of lattice points, which have an associated atom or group of atoms whose positions can be referred to them. Two real-space lattice points are connected by a primitive translation vector, R ... 

The one-electron wave function in an extended solid can be represented with different basis sets. Discussed here are only two types, representing opposite extremes the plane-wave basis set (free-electron and nearly-free-electron models) and the Bloch sum of atomic orbitals basis set (LCAO method). A periodic solid may be considered constmcted by the coalescence of these isolated atoms into extended Bloch-wave functions. On the other hand, within the free-electron framework, in the limit of an infinitesimal periodic potential (V = 0), a Bloch-wave function becomes a simple... 

For solids with more localized electrons, the LCAO approach is perhaps more suitable. Here, the starting point is the isolated atoms (for which it is assumed that the electron-wave functions are already known). In this respect, the approach is the extreme opposite of the free-electron picture. A periodic solid is constructed by bringing together a large number of isolated atoms in a maimer entirely analogous to the way one builds molecules with the LCAO approximation to MO (LCAO-MO) theory. The basic assumption is that overlap between atomic orbitals is small enough that the extra potential experienced by an electron in a solid can be treated as a perturbation to the potential in an atom. The extended- (Bloch) wave function is treated as a superposition of localized orbitals, centered at each atom ... 

As a prelude to our development of the LCAO treatment of solids, it will be beneficial to briefly review the LCAO-MO method. The cyclic tt systems from organic chemistry are familiar, relatively simple, and, more importantly, resemble Bloch functions of periodic solids. Thus, they will be used as the introductory examples. 

---